Optomechanics with Quantum Vacuum Fluctuations (Springer Theses) 3031430514, 9783031430510

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Springer Theses Recognizing Outstanding Ph.D. Research

Zhujing Xu

Optomechanics with Quantum Vacuum Fluctuations

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists. Theses may be nominated for publication in this series by heads of department at internationally leading universities or institutes and should fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder (a maximum 30% of the thesis should be a verbatim reproduction from the author’s previous publications). • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to new PhD students and scientists not expert in the relevant field. Indexed by zbMATH.

Zhujing Xu

Optomechanics with Quantum Vacuum Fluctuations Doctoral Thesis accepted by Purdue University, USA

Zhujing Xu Harvard University Cambridge, MA, USA

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-031-43051-0 ISBN 978-3-031-43052-7 (eBook) https://doi.org/10.1007/978-3-031-43052-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

To my family

Supervisor’s Foreword

It is my great pleasure to introduce the doctoral thesis of Dr. Zhujing Xu. Dr. Xu joined my research team at Purdue University in 2016 and completed her Ph.D. in 2022. Her research endeavors spanned several fascinating domains, such as the exploration of the Casimir effect and precision measurements with optomechanical systems. During her Ph.D. study in my research group, Dr. Xu built a sophisticated multi-cantilever atomic force microscope (AFM) system with fiber optic interferometers in a vacuum, starting from just a conceptual sketch. She then used this system to measure the Casimir force and energy transfer with quantum fluctuations. According to quantum mechanics, a vacuum is not empty but full of fluctuations due to zero-point energy. Such quantum vacuum fluctuations can lead to an attractive force between two neutral plates in a vacuum, known as the Casimir effect. The Casimir effect has attracted great attention in both fundamental and practical work because it is a macroscopic evidence of quantum electromagnetic fluctuations. Besides, the Casimir force can dominate the interaction between neutral surfaces at small separations. In this thesis, Dr. Xu presents the first realization of non-reciprocal energy transfer between two cantilevers by quantum vacuum fluctuations. This diode-like transport in a vacuum is a breakthrough in Casimir-based devices. It presents an efficient and robust way of transporting and regulating energy along one preferable direction. Besides, three-body Casimir effects are investigated in this thesis and were used to realize a transistor-like three-terminal device with quantum vacuum fluctuations. These two works pave the way for exploring and developing advanced Casimir-based devices with potential applications in quantum information science. This thesis also includes a study of the non-contact Casimir friction, which will enrich the understanding of quantum vacuum fluctuations. Dr. Xu’s exceptional achievements have merited her several recognitions, including the Karl Lark-Horovitz Award and the Bilsland Dissertation Fellowship at Purdue University. Moreover, she was also granted the prestigious Harvard Quantum Initiative (HQI) Postdoctoral Fellowship from Harvard University. Her work on

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Supervisor’s Foreword

optomechanics was featured in the American Physical Society’s “Highlights of the Year” for 2018 and Optics .& Photonics News’ “Optics in 2022.” West Lafayette July 2023

Prof. Dr. Tongcang Li

Parts of this thesis have been published in the following journal articles

1. “Observation and control of Casimir effects in a sphere-plate-sphere system”, Zhujing Xu, Peng Ju, Xingyu Gao, Zubin Jacob, Tongcang Li, Nature Communications 13, 1 (2022) 2. “Non-reciprocal energy transfer through the Casimir effect”, Zhujing Xu, Xingyu Gao, Jaehoon Bang, Zubin Jacob, and Tongcang Li, Nature Nanotechnology 17, 2 (2022) 3. “Enhancement of rotational vacuum friction by surface photon tunneling”, Zhujing Xu, Zubin Jacob, Tongcang Li, Nanophotonics, 10, 537 (2021) 4. “Ultrasensitive torque detection with an optically levitated nanorotor”, Jonghoon Ahn, Zhujing Xu, Jaehoon Bang, Peng Ju, Xingyu Gao, Tongcang Li, Nature Nanotechnology, 15, 89 (2020) 5. “Detecting Casimir torque with an optically levitated nanorod”, Zhujing Xu, Tongcang Li, Physical Review A 96 (3), 033843 (2017)

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Acknowledgments

It is really an honor and privilege for me to pursue my Ph.D. with extraordinary support. Many people deserve acknowledgment for this work and here I highlight some of the most important roles that enabled this work in the past six years. First and foremost, I would like to acknowledge Prof. Tongcang Li, for being an excellent advisor and a brilliant collaborator. Tongcang is an exceptional researcher with an outstanding scientific vision, talent, and ambition, that I could not ask for more to work with. I joined his group in the fall of 2016. With the support of Tongcang, I have had an unusual experience (also a privilege) in working on different topics, including quantum optomechanics, quantum information science, and precision measurements, both theoretically and experimentally. I have benefited a lot from the discussion with him. Especially, we have worked together on building a Casimir vacuum system from scratch, which is the most valuable experience for me. The scientific progress and results in this dissertation will not be possible without his contributions. I also want to express my gratitude to the committee members, Prof. Zubin Jacob, Prof. Francis Robicheaux, and Prof. Yong Chen. I am fortunate to collaborate with Prof. Zubin Jacob over the years. I thank Prof. Zubin Jacob for his deep insights and sharp intuitions, which are valuable assets to our Casimir diode, Casimir transistor, and Casimir friction work. I want to thank Prof. Francis Robicheaux for his insightful discussions when we worked on the levitated optomechanical system. I also want to thank Prof. Yong Chen for his support on the boron nitride project. Most of my understanding of quantum mechanics comes from the four courses that I took with Prof. Sherwin Love. Those four courses are my favorites at Purdue. I am always impressed by the remarkably clear explanation and teaching from Sherwin. I want to thank Sherwin for his patient guidance and continuous encouragement. The scientific atmosphere in the Li Lab is always a precious asset to me. I have learned a lot from the group members. When I first joined the group, I worked with Dr. Jonghoon Ahn on the optical levitation experiment and the table-top color center experiment. I thank Jonghoon for his rigorous attitude and thoughtful feedback which leads to a great progress in the project. I am also truly grateful for the xi

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Acknowledgments

support from Dr. Jaehoon Bang. I appreciate the technical guidance from him and the emotional support which helps me go through some difficult times. I enjoyed working with Xingyu Gao when we built the Casimir vacuum system together in 2018. His creative thoughts and fruitful discussions enabled some of the essential experimental results demonstrated in this dissertation. I appreciate the scientific support from Peng Ju. His fabrication techniques and discussions are crucial for the Casimir transistor and Casimir friction experiment. I am also grateful to have the opportunity to work with Kunhong Shen, Yuanbin Jin, and Sumukh Vaidya in the Li Lab. I am extremely lucky to have many friends who have encouraged and supported me during the six years of graduate school. Thanks to Daisy Zheng, Zihao Zhan, Guanyu Tao, Jianyu-Chen, Yun Huang, Fan Zhang, Pu Huang, Yu Shi, Qingyue Niu, Amandeep Singh, Alireza Karbakhsh Ravari, Jijun Chen, and Kevin Ro. Last but not the least, I want to thank my family for their endless love and support. It is really a privilege to have been immersed in a scientific environment since I was a kid. My father is not only a role model scientist but also a person that can truly understand the difficulty I have been through. The emotional support from my mother always encourages me to overcome the obstacles. I want to especially thank the support from Fan Wu for his patience and care. West Lafayette, IN, USA July 14, 2022

Zhujing Xu

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Casimir Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Casimir Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Quantum Vacuum Friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Multi-cantilever System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Levitated Optomechanical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Overview of This Dissertation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 5 6 7 8 9

2

Measurement and Calculation of Casimir Force . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Dual-Cantilever-Fiber Interferometer System . . . . . . . . . . . . . . . . . 2.1.2 Vacuum System and Pneumatic Isolation . . . . . . . . . . . . . . . . . . . . . 2.1.3 Piezo Actuator and Mounting System . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Force Measurement: Frequency Modulation Method . . . . . . . . . . . . . . . . . 2.3 Experimental Procedures and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Displacement Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Calibration of Spring Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Calibration of Effective Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Measured Casimir Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Calculation of Casimir Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 15 17 18 21 22 24 24 25 26 28 28 34

3

Experimental Realization of a Casimir Diode: Non-reciprocal Energy Transfer by Casimir Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Energy Transfer Between Two Cantilevers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Dynamical Control Near the Exceptional Point . . . . . . . . . . . . . . . . . . . . . . . 3.3 Effective Hamiltonian of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 43 48 54

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Contents

Experimental Realization of a Casimir Transistor: Switching and Amplifying Energy Transfer in a Three-Body Casimir System . . . 4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Measurement of Casimir Force in the Three-Cantilever System . . . . . 4.3 Casimir Vibrational Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Casimir Switch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Casimir Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 External Gain to the System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Requirement for the Steady Condition. . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Amplify the Casimir Mediated Energy Transfer . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 58 62 66 68 68 70 70 72

Proposal on Detecting Rotational Quantum Vacuum Friction . . . . . . . . . 5.1 Ultrasensitive Torque Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Rotational Vacuum Friction Torque on a Silica Nanosphere Near a Silica Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Enhancement of Rotational Vacuum Friction Torque by Surface Photon Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75

6

Proposal on Detecting Casimir Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Schematic Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Trapping Potential of the Nanorod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Calculation of Casimir Torque and Casimir Force . . . . . . . . . . . . . . . . . . . . 6.4 Torque and Force Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 86 88 91 93

7

Conclusion and Outlook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 A Preliminary Design of a Casimir MEMS Accelerometer. . . . . . . . . . . 7.1.1 Casimir Parametric Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 The Closest Stable Separation Before Pull-in . . . . . . . . . . . . . . . . . 7.1.3 Acceleration Detection Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Switching Casimir Force with Phase-Transition Materials . . . . . . . . . . . 7.3 Repulsive Casimir Force in Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97 97 98 99 100 101 103 105

76 80 82

Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

List of Abbreviations

AFM BST COM LDOS MEMS PFA PLL PSD QVF RMS SEM SNR VO.2 YIG

Atomic Force Microscope Barium Strontium Titanate Center of Mass Local Density of States Microelectromechanical Systems Proximity Force Approximation Phase-Locked Loop Power Spectral Density Quantum Vacuum Friction Root Mean Square Scanning Electron Microscope Signal-to-Noise Ratio Vanadium Dioxide Yttrium Iron Garnet

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Chapter 1

Introduction

Abstract Quantum mechanics predicts an infinite number of random fluctuations in vacuum because of the zero-point energy of the electromagnetic fields. Quantum vacuum fluctuations lead to the Casimir force between two neutral macroscopic objects, the Casimir torque between two anisotropic bodies, and the quantum vacuum friction between two moving bodies. In this thesis, we focus on using optomechanical systems (the multi-cantilever system and the levitated optomechanical system) to study Casimir force, Casimir torque, and quantum vacuum friction.

1.1 Casimir Force In 1948, Hendrik Casimir predicted an attractive force between two ideal metals due to the fluctuated electromagnetic fields in vacuum. The conductive plates only allow the virtual photons with certain wavelengths to exist between two plates (Fig. 1.1), and this results in a net energy difference with and without the plates. By studying the change of electromagnetic zero-point energy, the Casimir force between two perfectly conducting parallel plates per unit area was derived as [1] F =−

.

hcπ ¯ 2 , 240d 4

(1.1)

where d is the distance between two plates. Compared to other forces in classical and quantum physics, a unique feature of Casimir force is that it does not depend on mass, charge, or other coupling constants. Instead, the Casimir force only depends on the physical constants and the distance. The first observation of the Casimir effect was reported in 1958 [2] between neutral metallic plates using the spring balance. However, the measurement error is 100%, and the experiment proved the difficulty of separating two macroscopic plates by submicron distances. After 40 years, the first precise measurement of the Casimir force was reported in 1997 [3] by using a torsional pendulum in vacuum. The measurement agrees with the theory at separations from 0.6 to 6 .μm with a systematic error of 5%. One year later in 1998, a precision measurement from © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 Z. Xu, Optomechanics with Quantum Vacuum Fluctuations, Springer Theses, https://doi.org/10.1007/978-3-031-43052-7_1

1

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1 Introduction

Fig. 1.1 Scheme of Casimir force between two parallel metal plates. Because of the boundary conditions, the number of fluctuated electromagnetic field modes between two plates is fewer than the number of modes outside two modes. The energy difference leads to the attractive Casimir force

Vacuum fluctuations 0.1 to 0.9 .μm was demonstrated by an atomic force microscope (AFM) system [4]. The experiment was presented with a deviation of 1% from the theory at small separations. These two precision measurements showed the possibility of verifying the finite-temperature correction and finite-conductivity correction in the calculation. After that, a rapidly increasing number of precision measurements of Casimir effect were carried out by different systems, including interferometer-based vacuum systems [5], modified AFM systems [6], and microelectromechanical torsional oscillator systems [7–11]. The effect of surface roughness [12, 13], electrostatic patch potentials [14–19], and thermal contributions [20, 21] have also been discussed. Moreover, the Casimir forces between different materials [22–31], between different geometries [4, 5, 32–35], and at low temperatures [36–39] have been extensively studied in both theory and experiment. By further increasing the precision in the measurement, it is possible to detect the short-range gravitational force and set new constraints on the Yukawa potential [40, 41]. Another intriguing study is the dynamical Casimir effect that was first observed in a superconducting circuit [42]. Multiple theoretical studies and schemes have been proposed to detect the dynamical Casimir effect in different systems [43– 46]. The Casimir force dominates the interactions at small separations. The Casimir pressure between two surfaces at a distance of 10 nm is close to one atmosphere pressure. The universal attractive Casimir force can lead to several undesirable effects such as stiction, adhesion, and deflection in micro- and nano-devices [47]. In the past twenty years, a number of studies were focusing on switching the Casimir force from attraction to repulsion. The first observation of repulsive Casimir force was reported in 2009 [48]. The repulsion was realized by a gold–bromobenzene– silica structure. The special relation of three dielectric functions changes the sign of the Casimir force. Similar principles were adapted to realize stable Casimir equilibrium and quantum trapping [49]. An experimental realization of Casimir trapping of gold nanoplates in ethanol near teflon-coated gold surfaces was presented in 2019 [49]. In addition to engineering the Casimir force by the optical properties of materials, non-monotonic Casimir force was demonstrated experimentally between

1.1 Casimir Force

3

two objects with a novel geometry in 2017 [33]. The Casimir force changes from the attractive to the repulsive at a certain separation. Theoretical proposals of detecting repulsive Casimir force in vacuum have been demonstrated between topological insulators [50], birefringent materials [51], metamaterials [52], zeroindex structures [53], and magnetodielectric plate configurations [54, 55]. Despite the fact that the Casimir force in vacuum is usually attractive and difficult to be canceled, it would be useful if one can utilize such inevitable attractive Casimir force to realize some functions for the devices. Several experiments have been carried out to leverage the nonlinearity and ubiquity of Casimir force. The first observation of phonon coupling across vacuum between two mechanical membranes by Casimir force was reported in 2019 [56]. This work demonstrates the first energy transfer of phonons by quantum vacuum fluctuations and reveals a previously undiscovered mechanism of phonon energy transfer. In 2020, Casimir spring and dilution in macroscopic optomechanics were realized by coupling a membrane to a microwave cavity [57]. Self-assembled Casimir microcavities were realized experimentally in 2021 [58]. In this experiment, the joint interaction of the repulsive electrostatic force and the attractive Casimir force between gold nanoflakes in liquid leads to a stable equilibrium and forms Casimir microcavities spontaneously. Although a number of experiments on leveraging the Casimir interaction have been introduced, a Casimir device with non-reciprocity still remains an unexplored frontier. This brings us to the motivation and novelty of the work in this thesis. One of the main goals of this thesis is to build Casimir-based devices that can control the dynamics and energy transfer of mechanical systems at nanoscale and realize complex operations that have never been stated before. This is useful for manipulating phonons in a desirable way by quantum vacuum fluctuations. Experimentally, we build a dual-cantilever system to study Casimir effect and energy transfer between two cantilevers. Different from the conventional Casimir setup, our system has the ability of coupling two cantilevers by Casimir force with strong tunability. The system also allows us to engineer the spectrum to possess an exceptional point in the parameter space. By dynamically controlling the system near the exceptional point, we realize the non-reciprocal energy transfer through quantum vacuum fluctuations. This is the first Casimir diode system that can regulate energy transfer along one direction [59]. Moreover, we build a threecantilever system to study the Casimir effect. The first measurement of Casimir force in a three-object system is demonstrated in this thesis [60]. We propose and experimentally realize a Casimir transistor system that can switch and amplify the energy transfer between three cantilevers by quantum vacuum fluctuations. These two Casimir-based devices represent a crucial advancement for controlling virtual photons at nanoscale and will have potential applications in sensing [61, 62] and information processing [63, 64].

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1 Introduction

1.2 Casimir Torque When two isotropic plates are close to each other, the zero-point energy difference inside and outside the plates leads to the Casimir force. While for the anisotropic materials, the direction of the optical axis also plays an important role in tailoring the Casimir energy. For the anisotropic case, quantum vacuum fluctuations induce the Casimir torque that was first predicted in 1972 [65]. When there is a non-zero angle .θ between two optical axes of the parallel birefringent plates (Fig. 1.2), the virtual photons are polarized by the optical axes, and this leads to an energy difference from the zero-angle case. The birefringent plates would experience a torque that tries to minimize the energy and align two optical axes parallel to each other. This torque is known as the Casimir torque [65]. Casimir force has been successfully measured and studied many times in the past 20 years. However, there are very few studies of Casimir torque. Several theoretical proposals were presented to detect the Casimir torque between birefringent plates [66, 67], anisotropic nanostructures [68, 69], and liquid crystals [70] with different mechanical detection systems. The challenge of detecting the Casimir torque comes from the limited torque detection sensitivity of mechanical systems and the difficulty in placing two plates parallel to each other at a small separation. In 2018, the first observation of Casimir torque was realized between a liquid crystal and a solid birefringent crystal [71]. To ensure the parallelism of two surfaces, the vacuum gap was replaced by an isotropic material [71]. However, the measurement of Casimir torque between two objects with a vacuum gap has never been explored. In this thesis, we propose to detect the Casimir torque by our optically levitated nanorotor system that has achieved an unprecedented torque sensitivity [72]. We propose to optically levitate a nanorod near a birefringent plate. If there is a non-zero angle between the nanorod and the birefringent plate, the nanorod will experience a Casimir torque. The Casimir torque will induce a torsional motion that can be detected optically [73, 74]. In this theoretical proposal, we present the calculation of Casimir torque in the levitation system. Compared to the torque sensitivity, we prove that it is feasible to detect the Casimir torque under realistic conditions in the near future. Fig. 1.2 Scheme of Casimir torque between two birefringent plates. The optical axis (shown in the red line) can polarize the virtual photons. The angular momentum of the virtual photons induces the Casimir torque on the birefringent plate when there is a non-zero angle .θ between two optical axes

Birefringent plates

Vacuum fluctuations

1.3 Quantum Vacuum Friction

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1.3 Quantum Vacuum Friction Quantum friction predicts that two neutral bodies with a relative motion will experience a friction force due to quantum vacuum fluctuations (Fig. 1.3). A physical depiction is that instantaneous charges and electrical dipoles on one surface by vacuum fluctuations will induce image charges and image electrical dipoles on the other surface. When two bodies have a relative motion, the interaction between charges and electrical dipoles leads to the non-contact quantum vacuum friction force [75]. The dissipation essentially comes from the Doppler shift of the electromagnetic waves reflected by two moving surfaces, and the energy eventually dissipates through the electrical resistance of the materials. Quantum vacuum friction was first studied in 1997 [75–81]. There have been a number of debates over the existence of this small quantum vacuum friction force [82–84]. It is worthwhile to test its existence for two reasons. The first is that the measurement of fundamental physics might be eventually limited by the quantum vacuum friction force. The second is that the measurement can help us understand better the mechanism of non-contact friction. Efforts have been put to detect the quantum vacuum friction force by the atomic force microscope system [85–87]. However, the measured results do not agree with the theory because of the extremely weak quantum friction force, the undesirable noise, and the inhomogeneous electric fields on the sample. The challenge of detecting quantum vacuum friction comes from the extreme requirements for measurement sensitivity, and the difficultly in separating the quantum vacuum friction from other non-contact friction such as electrostatic dissipative friction and phononic friction from surface deformation [88, 89]. Novel methods have been proposed theoretically to detect such small friction in different systems [90–96]. There have been no direct observations of quantum vacuum friction so far. In this thesis, we propose to detect the long sought-after rotational quantum vacuum friction torque by our levitated nanorotor system [93]. A fast-rotating neutral nanoparticle in vacuum will experience the rotational vacuum friction torque Fig. 1.3 Scheme of quantum vacuum friction between two moving plates. The left plate is moving upward with a velocity v. Quantum vacuum fluctuations induce instantaneous charges and dipoles on one surface and hence generate image charges and dipoles on the nearby surface. The relative motion will lead to the dissipative quantum friction force

Stationary

Vacuum fluctuations

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1 Introduction

because of the interaction with the fluctuated electromagnetic fields [97]. This rotational vacuum friction can be enhanced by several orders when the particle is placed near a surface because of a large local density of electromagnetic states [90, 98]. The calculation of rotational quantum vacuum friction torque will be introduced in this thesis. We also propose a method to further enhance the rotational vacuum friction torque. We investigate on the perovskite ferroelectric material barium strontium titanate (BST) [99]. BST can have large dielectric responses at GHz frequencies, which can be resonated with the mechanical rotating frequency of the levitated nanoparticle [93, 100]. It can lead to a huge enhancement of rotational vacuum friction torque because of resonant photon tunneling [99]. We will show that the calculation of rotational vacuum friction torque at resonant frequencies is several orders larger than the minimum detectable torque that was demonstrated experimentally [93]. Therefore, it is promising to observe the rotational vacuum friction torque experimentally in the near future.

1.4 Multi-cantilever System Experimentally, we build a dual-cantilever vacuum system to study the Casimir effect and energy transfer. The schematic of the dual cantilever is shown in Fig. 1.4a. We use the commercial AFM cantilevers to construct the system. One cantilever is modified with a polystyrene sphere attached to one free end, while the other one remains the bare cantilever. Both cantilevers are coated with gold films for

(a)

(b)

Vacuum fluctuations

Vacuum fluctuations

Fig. 1.4 Schematics of the multi-cantilever vacuum system. (a) Dual-cantilever system. It consists of two modified atomic force microscope (AFM) cantilevers. We use this system to study the Casimir effect and Casimir mediated non-reciprocal energy transfer that will be discussed in Chap. 3. (b) Three-cantilever system. We use this system to study the three-body Casimir effect that will be discussed in Chap. 4

1.5 Levitated Optomechanical System

7

measuring the force between metallic surfaces and providing good conductivity. The motion of two cantilevers can be monitored by the fiber interferometers independently. We can measure the Casimir force gradient at different separations by tracking the frequency shift. This frequency modulation technique is inherently insensitive to the amplitude vibration noise; hence, it can enhance the sensitivity of measurements. The experimental results agree well with the calculation based on the Lifshitz formula for finite temperature and finite conductivity. Compared to the conventional Casimir force measurement systems, our system has a unique feature of two independent mechanical resonators that can be controlled independently. Thus, we can study the coupling and energy transfer between mechanical resonators by quantum vacuum fluctuations. More advanced and complicated operations can be applied to the system. In this thesis, we utilize the dual-cantilever Casimir system to realize a diode-like non-reciprocal energy transfer by Casimir interaction [59]. All the measurements of Casimir force were reported between two bodies before. The three-object Casimir effects have not been observed yet. Experimentally, we build a novel three-cantilever system as shown in Fig. 1.4b. In this thesis, we report the first observation of Casimir interactions between three isolated macroscopic objects [60]. Moreover, we propose and demonstrate a Casimir transistor system that can switch and amplify energy transfer by quantum vacuum fluctuations.

1.5 Levitated Optomechanical System The optical tweezer was first demonstrated by Ashkin [101], and it is widely used in biology, chemistry, and physics since it can manipulate particles with various sizes. An optically levitated nanoparticle in vacuum can have an ultrahigh mechanical quality factor (Q.> 109 ) since it is well-isolated from the surrounding environment. The system is an excellent candidate for precision measurement [73, 102–108] without the requirement of cryogenic temperature. The center-of-mass motion [104], the torsional motion [109], and the rotational motion [100] of a levitated particle have been observed in vacuum. Optical levitation systems have shown the ability of trapping particles with different geometries, including nano- and microspheres [110, 111], nanodumbells [100], nanorods [74], and microdisks [112]. Besides, the ground-state cooling is reported recently in three different experiments [113–115], which can be used to study the quantum phenomenon for macroscopic objects. As a great tool for precision measurement, force sensing at the .10−21 N level with an optically levitated nanosphere [111] and torque sensing at the .10−28 Nm level by an optically levitated nanorotor [93] have been reported. A schematic of a nanodumbbell levitated by a linearly polarized laser in vacuum is presented in Fig. 1.5. When the laser is linearly polarized, the nanodumbbell will be confined by the restoring torque from the laser and will do torsional vibrations around the polarization direction as shown in Fig. 1.5a. We can use this torsion balance to detect

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1 Introduction

Fig. 1.5 Schematics of the optical levitation system. A nanodumbbell is optically levitated in a trapping laser beam. (a) When the laser is linearly polarized, the optical tweezer provides the restoring torque that confines the orientation of the nanodumbbell. The nanodumbbell will do torsional vibrations around the polarization direction of the laser beam. (b) When the laser is circularly polarized, the nanodumbbell will be driven to rotate

a small torque on the nanodumbbell [100]. When the laser is circularly polarized, the nanodumbbell will be driven to rotate up to a frequency of 5 GHz [93]. The fast rotation can also be used for torque sensing. In this dissertation, we propose to detect the Casimir torque and rotational vacuum friction torque by the ultrasensitive levitated torque sensor [72, 93]. To detect the Casimir torque, an optically levitated nanorod will be placed near a birefringent plate, and the torsional motion of the nanorod induced by the Casimir torque can be optically detected [72]. To detect the rotational quantum vacuum friction, a levitated fast-rotating nanosphere will be placed near a surface, and the rotational quantum vacuum friction torque will also be detected optically [93].

1.6 Overview of This Dissertation In this dissertation, we focus on using the optomechanical systems to study the Casimir effects. The dissertation includes the experimental demonstration of building a Casimir diode [59] and a Casimir transistor system [60] by mechanical resonators. The dissertation also includes two theoretical proposals about detecting the Casimir torque [72] and the rotational vacuum friction torque [93, 99] by the levitated optomechanical system. In Chap. 2, we first introduce the measurement of Casimir force by our unique home-built dual-cantilever vacuum system. The measurement agrees with our calculation. In Chap. 3, we introduce the parametric coupling scheme that allows us to couple two cantilevers with different resonant frequencies. With a dynamical control near the exceptional, the first experiment of non-reciprocal energy transfer

References

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by Casimir interactions is demonstrated. In Chap. 4, we present the first threebody Casimir system that can switch and amplify the energy transfer by Casimir interactions. Two theoretical proposals about measuring rotational quantum vacuum friction torque and Casimir torque are discussed in Chaps. 5 and 6.

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78. L.S. Levitov, Van der Waals’ friction. Europhys. Lett. (EPL) 8(6), 499–504 (1989). https://doi. org/10.1209/0295-5075/8/6/002 79. V. Mkrtchian, Interaction between moving macroscopic bodies: viscosity of the electromagnetic vacuum. Phys. Lett. A 207(5), 299–302 (1995). https://doi.org/10.1016/03759601(95)00670-X 80. B.N.J. Persson, Z. Zhang, Theory of friction: coulomb drag between two closely spaced solids. Phys. Rev. B 57, 7327–7334 (1998). https://doi.org/10.1103/PhysRevB.57.7327 81. A.I. Volokitin, B.N.J. Persson, Theory of friction: the contribution from a fluctuating electromagnetic field. J. Phys. Condens. Matter 11(2), 345–359 (1999). https://doi.org/10.1088/09538984/11/2/003 82. T.G. Philbin, U. Leonhardt, No quantum friction between uniformly moving plates. New J. Phys. 11(3), 033035 (2009). https://doi.org/10.1088/1367-2630/11/3/033035 83. J.B. Pendry, Quantum friction–fact or fiction? New J. Phys. 12(3), 033028 (2010). https://doi. org/10.1088/1367-2630/12/3/033028 84. A.I. Volokitin, B.N.J. Persson, Comment on ’No quantum friction between uniformly moving plates’. New J. Phys. 13(6), 068001 (2011). https://doi.org/10.1088/1367-2630/13/6/068001 85. I. Dorofeyev, H. Fuchs, G. Wenning, B. Gotsmann, Brownian motion of microscopic solids under the action of fluctuating electromagnetic fields. Phys. Rev. Lett. 83, 2402–2405 (1999). https://doi.org/10.1103/PhysRevLett.83.2402 86. B. Gotsmann, H. Fuchs, Dynamic force spectroscopy of conservative and dissipative forces in an Al-Au(111) tip-sample system. Phys. Rev. Lett. 86, 2597–2600 (2001). https://doi.org/10. 1103/PhysRevLett.86.2597 87. B.C. Stipe, H.J. Mamin, T.D. Stowe, T.W. Kenny, D. Rugar, Noncontact friction and force fluctuations between closely spaced bodies. Phys. Rev. Lett. 87, 096801 (2001). https://doi. org/10.1103/PhysRevLett.87.096801 88. B. Gotsmann, Sliding on vacuum. Nat. Mater. 10(2), 87–88 (2011). https://doi.org/10.1038/ nmat2947 89. M. Kisiel, E. Gnecco, U. Gysin, L. Marot, S. Rast, E. Meyer, Suppression of electronic friction on Nb films in the superconducting state. Nat. Mater. 10(2), 119–122 (2011). https://doi.org/ 10.1038/nmat2936 90. R. Zhao, A. Manjavacas, F.J. Garcia de Abajo, J.B. Pendry, Rotational quantum friction. Phys. Rev. Lett. 109, 123604 (2012). https://doi.org/10.1103/PhysRevLett.109.123604 91. F. Intravaia, M. Oelschlager, D. Reiche, D.A.R. Dalvit, K. Busch, Quantum rolling friction. Phys. Rev. Lett. 123, 120401 (2019). https://doi.org/10.1103/PhysRevLett.123.120401 92. M.B. Farias, W.J.M. Kort-Kamp, D.A.R. Dalvit, Quantum friction in two dimensional topological materials. Phys. Rev. B 97, 161407 (2018). https://doi.org/10.1103/PhysRevB.97.161407 93. J. Ahn, Z. Xu, J. Bang, P. Ju, X. Gao, T. Li, Ultrasensitive torque detection with an optically levitated nanorotor. Nat. Nanotechnol. 15(2), 89–93, (2020). https://doi.org/10.1038/s41565019-0605-9 94. A.I. Volokitin, B.N.J. Persson, Quantum friction. Phys. Rev. Lett. 106, 094502 (2011). https:// doi.org/10.1103/PhysRevLett.106.094502 95. M.B. Farias, C.D. Fosco, F.C. Lombardo, F.D. Mazzitelli, Quantum friction between graphene sheets. Phys. Rev. D 95, 065012 (2017). https://doi.org/10.1103/PhysRevD.95.065012 96. J. Marino, A. Recati, I. Carusotto, Casimir forces and quantum friction from Ginzburg radiation in atomic Bose-Einstein condensates. Phys. Rev. Lett. 118, 045301 (2017). https://doi.org/10. 1103/PhysRevLett.118.045301 97. A. Manjavacas, F.J. Garcia de Abajo, Vacuum friction in rotating particles. Phys. Rev. Lett. 105, 113601 (2010). https://doi.org/10.1103/PhysRevLett.105.113601 98. A. Manjavacas, F.J. Rodriguez-Fortuno, F.J. Garcia de Abajo, A.V. Zayats, Lateral Casimir force on a rotating particle near a planar surface. Phys. Rev. Lett. 118, 133605 (2017). https:// doi.org/10.1103/PhysRevLett.118.133605 99. Z. Xu, Z. Jacob, T. Li, Enhancement of rotational vacuum friction by surface photon tunneling. Nanophotonics 10(1), 537–543 (2021). https://doi.org/doi:10.1515/nanoph-2020-0391

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1 Introduction

100. J. Ahn, Z. Xu, J. Bang et al., Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor. Phys. Rev. Lett. 121, 033603 (2018). https://doi.org/10.1103/ PhysRevLett.121.033603 101. A. Ashkin, Acceleration and trapping of particles by radiation pressure. Phys. Rev. Lett. 24, 156–159 (1970). https://doi.org/10.1103/PhysRevLett.24.156 102. Z.-Q. Yin, A.A. Geraci, T. li, Optomechanics of levitated dielectric particles. Int. J. Mod. Phys. B 27(26), 1330018 (2013). https://doi.org/10.1142/S0217979213300181 103. A.A. Geraci, S.B. Papp, J. Kitching, Short-range force detection using optically cooled levitated microspheres. Phys. Rev. Lett. 105, 101101 (2010). https://doi.org/10.1103/PhysRevLett. 105.101101 104. T. Li, S. Kheifets, D. Medellin, M.G. Raizen, Measurement of the instantaneous velocity of a Brownian particle. Science 328(5986), 1673–1675 (2010). https://doi.org/10.1126/science. 1189403 105. O. Romero-Isart, M.L. Juan, R. Quidant, J.I. Cirac, Toward quantum superposition of living organisms. New J. Phys. 12(3), 033015 (2010). https://doi.org/10.1088/1367-2630/12/3/ 033015 106. D.E. Chang, C.A. Regal, S.B. Papp et al., Cavity opto-mechanics using an optically levitated nanosphere. Proc. Natl. Acad. Sci. 107(3), 1005–1010 (2010). https://doi.org/10.1073/pnas. 0912969107 107. H. Shi, M. Bhattacharya, Optomechanics based on angular momentum exchange between light and matter. J. Phys. B At. Mol. Opt. Phys. 49(15), 153001 (2016). https://doi.org/10. 1088/0953-4075/49/15/153001 108. H. Shi, M. Bhattacharya, Coupling a small torsional oscillator to large optical angular momentum. J. Mod. Opt. 60(5), 382–386 (2013). https://doi.org/10.1080/09500340.2013. 778341 109. T.M. Hoang, Y. Ma, J. Ahn et al., Torsional optomechanics of a levitated nonspherical nanoparticle. Phys. Rev. Lett. 117, 123604 (2016). https://doi.org/10.1103/PhysRevLett.117. 123604 110. T. Li, S. Kheifets, M.G. Raizen, Millikelvin cooling of an optically trapped microsphere in vacuum. Nat. Phys. 7(7), 527–530 (2011). https://doi.org/10.1038/nphys1952 111. G. Ranjit, M. Cunningham, K. Casey, A.A. Geraci, Zeptonewton force sensing with nanospheres in an optical lattice. Phys. Rev. A 93, 053801 (2016) https://doi.org/10.1103/ PhysRevA.93.053801 112. G. Winstone, Z. Wang, S. Klomp et al., Optical trapping of high-aspect-ratio NaYF hexagonal prisms for kHz-MHz gravitational wave detectors. arXiv e-prints arXiv:2204.10843, arXiv:2204.10843 (2022). arXiv: 2204.10843 [physics.optics] 113. U. Deli´c, M. Reisenbauer, K. Dare et al., Cooling of a levitated nanoparticle to the motional quantum ground state. Science 367(6480), 892–895 (2020). https://doi.org/10.1126/science. aba3993 114. F. Tebbenjohanns, M.L. Mattana, M. Rossi, M. Frimmer, L. Novotny, Quantum control of a nanoparticle optically levitated in cryogenic free space. Nature 595(7867), 378–382 (2021). https://doi.org/10.1038/s41586-021-03617-w 115. L. Magrini, P. Rosenzweig, C. Bach et al., Real-time optimal quantum control of mechanical motion at room temperature. Nature 595(7867), 373–377 (2021). https://doi.org/10.1038/ s41586-021-03602-3

Chapter 2

Measurement and Calculation of Casimir Force

Abstract Measurement of Casimir force has been progressively improved since the first observation of Casimir force in 1958. In this chapter, we will first introduce our home-built dual-cantilever vacuum system and the detection scheme. We will present the measurement methods and results of Casimir force between two mechanical cantilevers in vacuum. At a separation of x, the Casimir force between an ideally conductive sphere with radius R and an ideally conductive plate is π 3 h¯ c R 0 .F (x) = − C 360 x 3 . However, we need to consider the finite conductivity, finite temperature, and material dispersion in the calculation. Our measurements agree well with the calculation based on Lifshitz theory. We will also discuss the thermal contributions and the effect of gold thin film thickness on the Casimir force. Parts of the contents in this chapter have been published in Xu et al. (Nat Nanotechnol 17(2):148–152, 2022).

2.1 Experimental Setup 2.1.1 Dual-Cantilever-Fiber Interferometer System The system consists of two modified AFM cantilevers and two fiber interferometers (Fig. 2.1a). A 35-.μm-radius polystyrene sphere is adhered to the left cantilever. The right side is a bare cantilever with a size of 500 .×100 .μm. The sizes of the sphere and the cantilever are both far larger than the distance between two surfaces so we can apply the proximity force approximation (PFA), and the cantilever can be approximated as an infinitely large plane. A 70-nm gold layer is coated on the front and back sides of the sphere and the cantilever to create a metallic surface and a better reflectivity. The thicknesses of the gold film on both sides are nearly equal to reduce the material stress and bending of the cantilever. The motion of two cantilevers can be monitored by their fiber interferometers (Fig. 2.6a). A fiber interferometer system has an excellent performance for both DC and AC signals. Several studies have shown that the interferometric detection

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 Z. Xu, Optomechanics with Quantum Vacuum Fluctuations, Springer Theses, https://doi.org/10.1007/978-3-031-43052-7_2

15

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2 Measurement and Calculation of Casimir Force

Fig. 2.1 Schematics of the dual-cantilever Casimir system. (a) The home-built dual-cantilever atomic force microscope (AFM) system consists of two modified AFM cantilevers and two fiber interferometers. A polystyrene sphere is attached to one cantilever to form the sphere-plate geometry. Both surfaces are coated with a 70-nm gold layer for good conductivity. (b) The optical image of this dual-cantilever system

system can have high sensitivity, excellent low-frequency stability, and mechanical robustness [11, 12]. A 50-.μW 1310-nm laser beam is directed to the cantilever and then gets reflected. The reflection is guided into the same fiber and interferes with the reflection light from the fiber-air interface. We can use the interference pattern to calibrate the cantilever motion that will be discussed later. The optical image of the system is shown in Fig. 2.1b. We build a fiber interferometer system as shown in Fig. 2.2. The incident light is illuminated by a 1310-nm single mode laser (Thorlabs S3FC1310 benchtop laser source) with a linewidth smaller than 1 nm. The laser is first connected to a .1 × 2 fiber coupler to two independent paths and they are symmetric. The incident light then passes the isolator and is directed to Port 1 of the .2 × 2 fiber coupler that splits the laser beam equally to Port 2 and Port 3. The light from Port 2 is directed to the left cantilever (Fig. 2.2). Approximately, .4% of the laser power in port 2 is reflected from the fiber-air interface. The other .96% exits the fiber and impinges on the back side of the cantilever. Part of the reflected light goes back to the same fiber and interferes with the reflection at the fiber-air interface, as shown in Figs. 2.2 and 2.6a. The combination of these two reflected light goes through Port 2, and half of its power is guided to Port 4. The path of Port 4 is directed to the Input .+ channel of InGaAs balanced amplified photodetectors (Thorlabs PDB 425 C). On the other side, the light from Port 3 monitors the power of the incident light since the power in Port 3 is half of the power in Port 1. Through an optical attenuator, the light from Port 3 is directed to the Input.− channel of the same balanced detector. The balanced detector subtracts the signal from two channels and gives us the interference signal that is sensitive to the distance between the cantilever and the fiber.

2.1 Experimental Setup

17

Fig. 2.2 The schematic of the fiber interferometers. The 1310-nm laser goes through the fiber coupler to two branches that are prepared for the left and right laser interferometers. The incident light goes though Port 1 and Port 2 of the 2.×2 fiber coupler and then is directed to the left cantilever. The cantilever and the fiber end form a Fabry-Perot cavity. The reflected light will be guided back to Port 2, and half of its power splits Port 4 and finally received by the balanced detector. The half of the incident light from Port 1 goes through Port 3 and an attenuator and directed to the other channel of the balance detector. The subtraction of the signal from port 3 and port 4 gives the interference signal

2.1.2 Vacuum System and Pneumatic Isolation The dual-cantilever-fiber system is placed in a vacuum chamber as shown in Fig. 2.3. An optical monitoring system is placed on top of the vacuum chamber to align the fiber and the cantilever. Two homemade fiber feedthroughs are installed by modifying the blank flanges. Two sets of high-vacuum compatible subminiature-D connectors are used for electrical feedthroughs. A photo of the dual-cantilever-fiber system inside the vacuum chamber is shown in Fig. 2.4. Five XYZ translational stages (Newport 9062-XYZ) are placed on a .10 × 10 × 1 vacuum breadboard for controlling the position of the objective lens, two cantilevers, and two fibers. The objective lens (Thorlabs RMS10X-PF, 10X Olympus Microscope Fluorite Obj) has a numerical aperture (NA) of 0.3 and a working distance of 10 mm. Two piezo chips were mounted to drive the cantilever and adjust the position more precisely, as shown in Fig. 2.6a. To minimize the vibrational impact from the ground, the vacuum chamber is set on two pneumatic vibration isolators as shown in Fig. 2.5. We use a turbo pump to have a pressure around .10−5 Torr and reduce the air friction and hydrodynamic force between two cantilevers. A schematic of the vacuum system is shown in Fig. 2.5. An additional acoustic isolator is installed outside the vacuum system to reduce the effect from the environment.

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2 Measurement and Calculation of Casimir Force

Fig. 2.3 Photo of the experimental setup. The dual-cantilever-fiber interferometer system is placed inside a .18 × 18 × 18 vacuum chamber. The chamber is placed on a double-pneumatic-isolator optical table. An optical monitoring system is placed on top of the vacuum chamber to align the fibers and the cantilevers. The electric feedthroughs are implemented on the left side of the chamber to control the voltage on the piezos, picomotors, and the cantilever surfaces. The fiber feedthroughs are implemented on the right side Fig. 2.4 Photo of the dual-cantilever-fiber system inside the chamber. Five XYZ translational stages are implemented on a    .10 × 10 × 1 vacuum breadboard for controlling the position of the objective lens, two cantilevers, and two fibers

2.1.3 Piezo Actuator and Mounting System The cantilever is glued on a 20 .× 8 . ×1 mm copper plate by the vacuum compatible silver epoxy (inset of Fig. 2.6a). The cantilever-copper-plate sample is fixed with the piezo chips by a 4–40 nut. The nut is glued on top of two piezo chips by vacuum compatible epoxy (Thorlabs Torr Seal) as shown in Fig. 2.6c. The piezo actuator

2.1 Experimental Setup

19

Camera

2” mirror Fiber feedthrough

Connected to pump Obj

Electrical feedthrough XYZ

XYZ

XYZ

XYZ

Gauge

1310 nm laser LED

Pneumatic vibration isolator

Fig. 2.5 Schematic of the vacuum system. On top of the vacuum chamber, there is a home-built optical monitoring system, which can be utilized to align two cantilevers and two fibers. There are two sets of pneumatic isolators used for reducing the seismic vibration

system is designed to be the sandwich structure, where a piezo chip (PA4HKW) is placed between two sapphire plates to reduce the capacitance effect. The photo of the piezo structure is shown in Fig. 2.6c. Each piezo chip has a dimension of 10 .× 10 .× 3 mm and a total travel length of .3.5 .μm under the external voltage of 150 V. The whole sandwich structure (piezo chips) is mounted on an aluminum plate by vacuum compatible epoxy. The aluminum plate is mounted on the XYZ translational stage for coarse displacement control. Picomotor actuators (Newport 8301-V) are implemented in the translational stage as shown in Fig. 2.6b. The picomotor has a travel length of .12.7 mm and is vacuum compatible. In the experiment, we first use picomotor to control the 3D translational stage (Fig. 2.6b). In this way, we can coarsely control the cantilevers and move two cantilevers from far away to a closer separation around 1 .μm. Then we switch to the precise control of the separation by using the piezo chips.

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2 Measurement and Calculation of Casimir Force

4-40 nut Sapphire

Piezo chips

Fig. 2.6 Piezo actuators and mounting system. (a) The schematic of the dual-cantilever system. Each cantilever’s motion is monitored by a fiber interferometer. The light reflection from the fiber end and from the cantilever will interfere with each other. The interference signal relates to the separation between the cantilever and the fiber. If the fiber is static, we can then detect the motion of the cantilever. The piezo chips are used to precisely control the motion of the cantilever. Inset: A photo of the cantilever-copper-plate sample. A cantilever is glued on the copper plate by epoxy. (b) The picomotors are implemented in the translational stage to roughly control the position of the cantilevers and the fibers. (c) A photo of the cantilever mounting system that consists of two piezo chips sandwiched by the sapphire plates to reduce the capacitance effect

2.1 Experimental Setup

21

Fig. 2.7 The scanning electron microscope (SEM) image of the samples. (a) A bare cantilever. (b) A cantilever with a polystyrene microsphere on top

2.1.4 Sample Preparation Casimir force measurement between different geometries has been explored in previous studies [3, 4, 13]. Among them, the sphere–plate geometry is most studied because of the difficulty in maintaining parallelism in the plate–plate geometry or other configurations. Here we employ the sphere–plate configuration with two cantilevers (Fig. 2.7). The bare cantilever on the right side has a dimension of 3 .500 × 100 × 1 .μm. (NanoAndMore Arrow Au). The frequency and the spring constant are estimated to be around 6 kHz and 0.03 N/m. A 70-nm gold film is deposited on both the front and back sides of the cantilever by E-beam evaporator (Fig. 2.7a). For the sphere–cantilever structure, we use another type of cantilever with a dimension of .450 × 50 × 2 .μm.3 (NanoAndMore Arrow TL-CONT) that has an expected frequency of 13 kHz and a spring constant of 0.2 N/m when there is no sphere. A solid polystyrene microsphere is placed on one end of the cantilever and is mounted by the vacuum compatible silver epoxy. The polystyrene sphere (Duke Science 4270 A) has a diameter of .69.1 ± 0.9 .μm and a density of .1.05 g./cm3 . The cantilever–sphere system is also coated with a 70-nm gold layer by E-beam evaporator (Fig. 2.7b). The measurement of Casimir force at small separations is very sensitive to the roughness [14, 15]. Therefore, it is important to improve the surface smoothness by careful cleaning and evaporation process. We have characterized the roughness of our sample by taking an AFM image of the gold-coated cantilever surface as shown in Fig. 2.8. The total topographic range is .6.2 nm, and the rms roughness is .0.8 nm over the area. The roughness is comparable with the roughness reported in a precision measurement experiment [16]. This measured roughness is far smaller than the distance between two surfaces in the experiment (60 to 800 nm) so we can neglect surface roughness in our measurement.

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2 Measurement and Calculation of Casimir Force

(a)

(b)

3.5 nm 2.0

0.0 -1.0

RMS = 0.84 nm

Topography (nm)

1.0

-2.0 -2.7

Fig. 2.8 Surface roughness of the gold-coated sample. (a) An atomic force microscope (AFM) scan of a gold-coated cantilever surface. The root mean square (rms) roughness is .0.8 nm. (b) The topography of the sample surface

2.2 Force Measurement: Frequency Modulation Method In the experiment, we apply the dynamical force measurement scheme. If there is a force gradient, the resonant frequency will shift. We can then measure the force gradient by detecting the frequency shift. In our experiment, the mechanical cantilevers can be treated as damped harmonic oscillators driven by an external force, and the equations of motion will be given by m

.

d 2x mω0 dx + k(x − x0 ) = Fdri (t) + Fint (x), + 2 Q dt dt

(2.1)

where .ω0 is the intrinsic natural frequency, .Q = ω0 /  is the quality factor, k is the spring constant, and .Fdri = F0 cos(ωt) is the external driving force for the dynamical measurement. When the vibration amplitudes are small, the interaction force between two cantilevers .Fint (x) can be Taylor expanded at the equilibrium position to the first order as 

∂Fint .Fint (x) = Fint (x0 ) + (x − x0 ) ∂x

 (2.2)

. x=x0

Therefore, the effective spring constant becomes .keff = k − ( ∂F∂xint )x=x0 , and the    resonant frequency under the interaction is .ω(x0 ) = ω0 1 − k1 ∂F∂xint . Under x=x0

the condition that .( ∂F∂xint )x=x0  k, the resonant frequency shift is approximated as δω = ω − ω0 = −

.

ω0 ∂Fint ( )x=x0 . 2k ∂x

(2.3)

Under a steady condition, the motion of the cantilever can be given by .x(t) = A cos(ωt + φ). By solving Eq. 2.1, we can get the oscillating amplitude A and the phase .φ that

2.2 Force Measurement: Frequency Modulation Method

23

Fig. 2.9 The interface of the Zurich Instrument MFLI device. We use the sweeper function in the Zurich Instrument interface to locate the phase for the resonant condition. Then we can use the phase-locked loop (PLL) function to lock the cantilever and track the resonant frequency of the cantilever

A=

.



F0

,. mω02 (1 − ω2 /ω02 )2 + (ω2 /Q2 ω02 )  ωω0 /Q −1 φ = tan . ω2 − ω02

(2.4)

(2.5)

If the phase .φ is maintained at 90 degrees, the cantilever will be resonantly driven. In this way, we can track the resonant frequency by applying the phase-locked loop (PLL). We can then measure the force gradient at each separation using Eq. 2.3. This frequency modulation force measurement method can improve the force sensitivity since it is insensitive to the amplitude vibration noise and change of the damping rate. To implement the frequency modulation method, a feedback system needs to be installed. In the experiment, we use the Zurich instrument (MFLI2) PLL (phaselocked loop) system to measure the resonant frequency by maintaining the phase .φ at 90 degree. We show an interface of the Zurich instrument in Fig. 2.9. We use the sweeper function to locate the phase for the resonant condition. Then we can use the PLL function to lock the cantilever at a specific phase and track the resonant frequency. We also show an interface of a LabVIEW program in Fig. 2.10. We can use this program to monitor the power spectral density (PSD) of the cantilevers and add PID control that will be useful for the Casimir diode and Casimir transistor experiment.

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2 Measurement and Calculation of Casimir Force

Fig. 2.10 The interface of the LabVIEW program. We can use this LabVIEW program to monitor the PSD of the cantilever and add PID control that will be used in the Casimir diode and Casimir transistor experiment

2.3 Experimental Procedures and Results 2.3.1 Displacement Calibration We use the fiber interferometer system to calibrate the motion of the cantilevers since the interference signal is sensitive to the separation between the cantilever and the fiber. If the fiber is static and the cantilever is moving with a displacement x, the signal recorded from the balanced detector will be   4π x + φ + V1 , (2.6) .V = V0 sin λ where .V0 is the interference pattern amplitude and .V1 is the offset. The interference signal has a period of .λ/2. When the interference signal is at .Vmax or .Vmin (Fig. 2.11), a movement of the cantilever will only induce a very small change of the interference signal, which makes the signal-to-noise ratio (SNR) very bad. In order to get a better sensitivity, the fiber–cantilever spacing is always kept at THE quadrature point (Q-point) where the signal is at the center of the interference pattern (Fig. 2.11). At the Q-point, the interference signal response . V for a small movement . x should be V = V0

.

4π x . λ

(2.7)

An example of the interference signal for displacement calibration is shown in Fig. 2.11. We can extract .Vmax and .Vmin from the signal. At the Q-point, the measurement gives a conversion of 4.4 mV/nm.

2.3 Experimental Procedures and Results

25

Fig. 2.11 Calibration of the displacement. The fiber interference signal when the piezo is applied with a voltage from 0 to 120 V. The red dots correspond to the quadrature point (Q-point), and the green stars correspond to the position when the voltage meets the maximum and minimum

2.3.2 Calibration of Spring Constant Since we measure the force gradient by using the equation . ∂F∂xint = −2k δω ω0 , we need to calibrate the spring constant k. Here we use the thermal Brownian motion to calibrate the spring constant. Without any external force, the equation of motion of a cantilever is given as d 2x dx + 2 x = Fthermal , (2.8) + 0 2 dt dt √ where √ .0 is the damping rate, . = k/M is the resonant frequency, .Fthermal = ξ(t) 2kB T 0 M is the Brownian stochastic force, M is the mass of the cantilever, and .ξ(t) is a normalized white-noise process. After the Fourier transform of .x(t), the single-sided PSD is expected to be in a Lorentzian form as .

S(ω) =

.

4kB T

2 0 . M 2 ( 2 − ω2 )2 + ω2 02

(2.9)

In the experiment, the displacement .x(t) of the cantilever is recorded by the fiber interferometer for a total time length of 10 s and a sampling rate of 29.3 kHz. One example of the measured PSD (blue line) and the fitting (red curve) is shown in Fig. 2.12. From the PSD fitting, we can calibrate the damping coefficient .0 = 2π × 4.5 Hz, the resonant frequency . = 2π × 4564.6 Hz, the spring constant −10 kg. .k = 0.16 N/m, and the mass .M = 1.92 × 10

26

2 Measurement and Calculation of Casimir Force

Fig. 2.12 The power spectral density (PSD) of the thermal Brownian motion of the cantilever. A Lorentzian function fitting of the PSD is shown in the red curve. The temperature is 300 K

2.3.3 Calibration of Effective Separation In addition to the Casimir force, there is electrostatic force due to the patch potentials on two surfaces. Patch potential originates from the polycrystalline structure of the materials, and it leads to an electrostatic potential difference .Vc for a certain interaction area, even under the condition that both cantilevers are electrically grounded. This potential difference .Vc needs to be canceled by applying an additional voltage on the surface. In the experiment, we apply a voltage .V0 on the right cantilever, while the left sphere is electrically grounded. When the separation x is far smaller than the size of the sphere and the cantilever, proximity force approximation (PFA) can be used here, and the electrostatic force .Fe between two surfaces is approximated as [16–18] Fe (x) = −

.

 π 0 R

2 (V0 − Vc )2 + Vrms , x

(2.10)

where .Vrms is the rms voltage fluctuations. The frequency shift due to the electrostatic force and the Casimir force is then given by δω = −

.

 ω dFC ω π 0 R

2 , (V0 − Vc )2 + Vrms − 2 2k x 2k dx

(2.11)

C where . dF dx is Casimir force gradient. Our measurements show that the contribution from .Vrms is far smaller than the contribution from Casimir interaction in the range of separation we considered in the experiment. When we apply a voltage that .V0 = Vc , the electrostatic force from patch potential is minimized and the frequency shift .δω dominantly comes from the Casimir force.

2.3 Experimental Procedures and Results

27

Fig. 2.13 Measurement of frequency shift under the Casimir force and electrostatic force. (a) Measured frequency shift is presented at different external voltages on the cantilever. A parabolic fitting is shown in the red solid curve. The separation is calibrated to be 103 nm, and the patch potential is .0.305 V. (b) The measurement of frequency shift is repeated at three different separations. (c) Measured residual patch potential is presented for different separations. (d) After canceling out the patch potential, the frequency shift dominantly from Casimir force gradient is plotted at different separations

One example of the measured frequency shift is shown in Fig. 2.13a. The voltage is swept from .−0.46 to .−0.15 V. The red solid curve is the parabolic fitting based on Eq. 2.11. The contact potential difference in this case is measured to be .0.305 V, and the separation is calibrated to be 103 nm. By this method, we can calibrate the patch potential .Vc , the separation x, the frequency shift due to Casimir interaction, and C hence the Casimir force gradient . dF dx . We repeat the measurement and parabolic fitting at different separations. Three more examples are shown in Fig. 2.13b. After canceling out the patch potential, we can get the frequency shift dominantly coming from the Casimir interaction at different separations (Fig. 2.13d). The measured residual patch potential is shown in Fig. 2.13c. The fluctuation of patch potential at different separations is less than 2.%. Therefore, the contribution from inhomogeneous patch potentials in this case is negligible.

28

2 Measurement and Calculation of Casimir Force

Fig. 2.14 Measurement of Casimir force and Casimir force gradient. (a) The measured Casimir force gradient divided by the radius of the microsphere at different separations. The measurements are in good agreement with the calculation for real gold films. (b) The Casimir force is obtained by integrating the force gradient over the separation

2.3.4 Measured Casimir Force The measured Casimir force gradient calculated from the frequency shift is shown in Fig. 2.14a. We notice that the measured Casimir force gradient agrees with Lifshitz formula in Eqs. 2.17 and 2.19 for real gold films that will be introduced later. The Lifshitz theory considers the effect of finite temperature and finite conductivity. The effect of voltage fluctuations .Vrms on the surfaces in Eq. 2.10 is negligible for the separations considered in this experiment since the measurement is in good agreement from 70 to 500 nm. By integrating the force gradient over the separation, we can plot the measured Casimir force in Fig. 2.14b.

2.4 Calculation of Casimir Force In this section, we will introduce the Lifshitz theory [9] to calculate the Casimir force between real materials with finite conductivity at finite temperatures. We will also discuss the thermal Casimir effects. To calculate the Casimir force between real materials, we use the dielectric function to characterize the material response in the frequency domain. We first show the calculation for zero-temperature case. The Casimir energy per unit area at a finite separation x is [9] h¯ .E0 (x) = 4π 2





∞ 0

k⊥ dk⊥

0

∞

dξ {ln[1 − rT2 M (iξ, k⊥ )e−2xq ]

+ ln[1 − rT2 E (iξ, k⊥ )e−2xq ]},

(2.12)

2.4 Calculation of Casimir Force

29

 where .ξ is the imaginary frequency and .k⊥ = kx2 + ky2 is the wave vector parallel to the surface. The reflection coefficients for the transverse magnetic and electric modes at a certain imaginary frequency and wave vector are rT M (iξ, k⊥ ) =

.

(iξ )q(iξ, k⊥ ) − k(iξ, k⊥ ) , (iξ )q(iξ, k⊥ ) + k(iξ, k⊥ )

(2.13)

q(iξ, k⊥ ) − k(iξ, k⊥ ) , q(iξ, k⊥ ) + k(iξ, k⊥ )

(2.14)

and rT E (iξ, k⊥ ) =

.

2 + ξ 2 /c2 and .k 2 (iξ, k ) = k 2 + (iξ )ξ 2 /c2 . Therefore, the where .q 2 (iξ, k⊥ ) = k⊥ ⊥ ⊥ Casimir pressure is

h¯ .P0 (x) = − 2π 2





∞

k⊥ dk⊥

0

∞ 0

dξ q{[rT−2M (iξ, k⊥ )e2xq − 1]−1

+ [rT−2E (iξ, k⊥ )e2xq − 1]−1 }.

(2.15)

At zero temperature, the Casimir interaction only comes from quantum vacuum fluctuations. At a non-zero temperature, both thermal and quantum vacuum fluctuations contribute to the Casimir interaction. The Casimir energy per unit area and the Casimir pressure at a non-zero temperature T and at a finite separation x are given by E(x, T ) =

.

∞ kB T  ∞ k⊥ dk⊥ {ln[1 − rT2 M (iξl , k⊥ )e−2xq ] 2π 0 l=0

+ ln[1 − rT2 E (iξl , k⊥ )e−2xq ]},

(2.16)

and P (x, T ) = −

.

∞ kB T  ∞ qk⊥ dk⊥ {[rT−2M (iξl , k⊥ )e2xq − 1]−1 π 0 l=0

+ [rT−2E (iξl , k⊥ )e2xq − 1]−1 }.

(2.17)

For the non-zero-temperature case, the imaginary frequency .ξ is replaced by the Matsubara frequency .ξl = 2π kh¯B T l . The prime on the summation indicates that the term for .l = 0 needs to be multiplied by .1/2. In the experiment, we adapt the proximity force approximation (PFA) here to calculate Casimir force between a sphere and a plate, that is, [19] FC (x, T ) = −2π RE(x, T ).

.

(2.18)

30

2 Measurement and Calculation of Casimir Force

Fig. 2.15 Quantum and thermal contributions to the Casimir force. (a) Calculated Casimir force between a gold sphere and a gold plate at different separations x for the zero-temperature condition and for the room-temperature condition. (b) The ratio between Casimir force at 300 K and at 0 K

Similarly, the Casimir force gradient divided by the radius is given by .

1 dFC (x, T ) = −2π P (x, T ). dx R

(2.19)

Our Casimir system consists of a polystyrene sphere and a silicon bare cantilever. Both surfaces are coated with 70-nm-thick gold films. In the calculation, we treat the system as a gold sphere and a gold plate since the coating layers are much thicker than the skin depth (we will discuss the effect of thickness later) [20]. The dielectric function of the gold film can be characterized by the plasma model such that (ω) = 1 −

.

ωp2 ω2

.

(2.20)

The parameters for gold can be found in the handbook [21]. The plasma frequency ωp is 9 eV/.h. ¯ We calculate the Casimir force (absolute value) at 0 K and at 300 K as shown in Fig. 2.15. For separations smaller than 1 .μm, the contribution from thermal fluctuations at 300 K is less than 6.%, and the effect of quantum fluctuations dominates the Casimir interaction [22]. A typical separation between two cantilevers in our Casimir energy transfer experiment is around 100 nm (which will be mentioned in Chaps. 3 and 4). Therefore, the energy transfer is mainly mediated by quantum fluctuations in vacuum instead of thermal fluctuations. In the real case, the 70-nm gold layers may play a role in the Casimir interaction. Here we discuss the effect of gold film thickness. We will show that the calculation of the Casimir force gradient based on Eqs. 2.17, 2.19, 2.20 for infinitely thick films only introduces an error less than 0.1.% compared to the real case of 70-nm thin films. The real configuration of our samples is shown in Fig. 2.16a. The left surface is composed of a thin gold film and a thick polystyrene plate. The right surface

.

2.4 Calculation of Casimir Force

31

Fig. 2.16 Calculation of Casimir force between a gold-coated polystyrene plate and a gold-coated silicon plate. (a) A gold film with thickness t is coated on a polystyrene plate (left). A gold film with the same thickness t is coated on a silicon plate (right). (b) The calculated Casimir pressure for different thicknesses of the gold films. The red dashed curve corresponds to the Casimir pressure between two bulk gold plates. The separation here is 200 nm. (c) The ratio of the Casimir pressure between the layered structures over the Casimir pressure between two bulk gold plates is presented as a function of the thickness. The ratio becomes 0.9994 at a thickness of 70 nm. (d) The ratio is presented as a function of separation when the thickness is 70 nm

consists of a thin gold film and a thick silicon plate. The Casimir pressure for the layered structure is [23] P (x, T ) = −

.

∞ kB T  ∞ −1 −1 qk⊥ dk⊥ {[r1p (iξl , k⊥ )r2p (iξl , k⊥ )e2xq − 1]−1 π 0 l=0

−1 −1 + [r1s (iξl , k⊥ )r2s (iξl , k⊥ )e2xq

− 1]−1 },

(2.21)

where .r1p , .r1s , .r2p , .r2s are the reflection coefficients of the left and right surfaces for the p and s polarization. Here we introduce the transfer matrix method to get the reflection coefficients for both surfaces.

32

2 Measurement and Calculation of Casimir Force

For a structure that has a thin layer and a thick plate underneath, the transfer matrix for the .p(s) polarization components is [23, 24] p(s)

p(s)

M p(s) = D0→1 P1 (t)D1→2 ,

.

(2.22)

where t is the thickness of the gold layer. The subscript j denotes the j th layer. For the gold film on a polystyrene substrate case, the layer .j = 0 is vacuum, the p(s) layer .j = 1 is gold, and the layer .j = 2 is polystyrene. .Dj →j +1 is the transmission matrix between layer j and .j + 1, that is, Dj →j +1

.

  p(s) p(s) 1 1 + ηj,j +1 1 − ηj,j +1 = , p(s) p(s) 2 1 − ηj,j +1 1 + ηj,j +1

(2.23)

p(s)

where .ηj,j +1 is written as p

ηj,j +1 =

.

j (iξ )Kj +1 , j +1 (iξ )Kj

s ηj,j +1 =

Kj +1 . Kj

(2.24)

 2 + (iξ )ξ 2 /c2 . .P (t) is the propagation matrix for the thin layer Here .Kj = k⊥ 1 with a thickness t, and it is given as   Kt 0 e 1 . .P1 (t) = 0 e−K1 t

(2.25)

With the transfer matrix M, the reflection coefficient is p(s)

p(s)

rp(s) = M21 /M11 ,

.

p(s)

(2.26)

p(s)

where .M11 and .M21 are the components of the transfer matrix M. The dielectric function of silicon is given by [25] Si (iξ ) = ∞ +

.

( 0 − ∞ )ω02 ξ 2 + ω02

,

(2.27)

where . ∞ = 1.035, . 0 = 11.87, .ω0 = 6.6 × 1015 rad s.−1 . The dielectric function of polystyrene is written as [26] poly (iξ ) = 1 +

.

n20 − 1 0 − n20 + , 1 + (ξ/ωI R )2 1 + (ξ/ωU V )2

(2.28)

where . 0 = 2.6, .n0 = 1.564, .ωI R = 5.54 × 1014 rad s.−1 , .ωU V = 1.354 × 1016 rad s.−1 .

2.4 Calculation of Casimir Force

33

Fig. 2.17 Calculation of Casimir force between two thin gold films. (a) Two gold films with a thickness t are separated by a distance x. (b) The calculated Casimir pressure for different thicknesses. The separation here is 200 nm. (c) The ratio of the Casimir pressure of the thin film case over the bulk gold plate case. At a thickness of 70 nm, the ratio is 0.9992. (d) The ratio is presented at different separations. Here the thickness is 70 nm

The calculated Casimir pressure between a gold-coated polystyrene plate and a gold-coated silicon plate using this transfer matrix method is shown in Fig. 2.16. We notice that with a thickness of 70 nm and a separation of 200 nm, the pressure between two gold-coated structures is 0.9994 of the pressure between two infinitely thick gold plates. Therefore, the calculation under the thick-plate approximation (based on Eq. 2.17) only introduces an error of 0.06%, which is negligible for our measurement. We also calculate the Casimir pressure between two gold thin films (no substrate) as shown in Fig. 2.17. We show that the Casimir pressure between two 70-nm gold thin films is 0.9992 of the Casimir pressure between two bulk materials. The difference here is also very small.

34

2 Measurement and Calculation of Casimir Force

References 1. M. Sparnaay, Measurements of attractive forces between flat plates. Physica 24(6), 751–764 (1958). https://doi.org/10.1016/S0031-8914(58)80090-7 2. S.K. Lamoreaux, Demonstration of the Casimir force in the 0.6 to 6 μm range. Phys. Rev. Lett. 78, 5–8 (1997). https://doi.org/10.1103/PhysRevLett.78.5 3. G. Bressi, G. Carugno, R. Onofrio, G. Ruoso, Measurement of the Casimir force between parallel metallic surfaces. Phys. Rev. Lett. 88, 041804 (2002). https://doi.org/10.1103/PhysRevLett. 88.041804 4. U. Mohideen, A. Roy, Precision measurement of the Casimir force from 0.1 to 0.9 μm. Phys. Rev. Lett. 81, 4549–4552 (1998). https://doi.org/10.1103/PhysRevLett.81.4549 5. J.N. Munday, F. Capasso, Precision measurement of the Casimir-Lifshitz force in a fluid. Phys. Rev. A 75, 060102 (2007). https://doi.org/10.1103/PhysRevA.75.060102 6. M. Bordag, U. Mohideen, V. Mostepanenko, New developments in the Casimir effect. Phys. Rep. 353(1), 1–205 (2001). https://doi.org/10.1016/S0370-1573(01)00015-1 7. G.L. Klimchitskaya, U. Mohideen, V.M. Mostepanenko, The Casimir force between real materials: experiment and theory. Rev. Mod. Phys. 81, 1827–1885 (2009). https://doi.org/10. 1103/RevModPhys.81.1827 8. J.N. Munday, F. Capasso, V.A. Parsegian, Measured long-range repulsive Casimir-Lifshitz forces. Nature 457(7226), 170–173 (2009). https://doi.org/10.1038/nature07610 9. E.M. Lifshitz, The theory of molecular attractive forces between solids. Sov. Phys. JETP 2, 73–83 (1956). 10. Z. Xu, X. Gao, J. Bang, Z. Jacob, T. Li, Non-reciprocal energy transfer through the Casimir effect. Nat. Nanotechnol. 17(2), 148–152 (2022). https://doi.org/10.1038/s41565-021-010268 11. D. Rugar, H.J. Mamin, P. Guethner, Improved fiber-optic interferometer for atomic force microscopy. Appl. Phys. Lett. 55(25), 2588–2590 (1989). https://doi.org/10.1063/1.101987 12. U. Sharma, X. Wei, Fiber optic interferometric devices, in Fiber Optic Sensing and Imaging, ed. by J.U. Kang (Springer, New York, 2013), pp. 29–53. https://doi.org/10.1007/978-1-46147482-1_2 13. J.L. Garrett, D.A.T. Somers, J.N. Munday, Measurement of the Casimir force between two spheres. Phys. Rev. Lett. 120, 040401 (2018). https://doi.org/10.1103/PhysRevLett.120. 040401 14. P.J. van Zwol, G. Palasantzas, J.T.M. De Hosson, Influence of random roughness on the Casimir force at small separations. Phys. Rev. B 77, 075412 (2008). https://doi.org/10.1103/PhysRevB. 77.075412 15. P.J. van Zwol, G. Palasantzas, M. van de Schootbrugge, J.T.M. de Hosson, V.S.J. Craig, Roughness of microspheres for force measurements. Langmuir 24(14), 7528–7531 (2008). https://doi.org/10.1021/la800664f 16. J.L. Garrett, D. Somers, J.N. Munday, The effect of patch potentials in Casimir force measurements determined by heterodyne kelvin probe force microscopy. J. Phys. Condens. Matter. 27(21), 214012 (2015). https://doi.org/10.1088/0953-8984/27/21/214012 17. C.C. Speake, C. Trenkel, Forces between conducting surfaces due to spatial variations of surface potential. Phys. Rev. Lett. 90, 160403 (2003). https://doi.org/10.1103/PhysRevLett.90. 160403 18. W.J. Kim, A.O. Sushkov, D.A.R. Dalvit, S.K. Lamoreaux, Surface contact potential patches and Casimir force measurements. Phys. Rev. A 81, 022505 (2010). https://doi.org/10.1103/ PhysRevA.81.022505 19. J. Blocki, J. Randrup, W.J. Swiatecki, C.F. Tsang, Proximity forces. Ann. Phys. 105(2), 427– 462 (1977). 20. M. Lisanti, D. Iannuzzi, F. Capasso, Observation of the skin-depth effect on the Casimir force between metallic surfaces. Proc. Natl. Acad. Sci. 102(34), 11989–11992 (2005). https://doi. org/10.1073/pnas.0505614102

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21. E.D. Palik (ed.), Front Matter (Academic Press, Boston, 1998), p. iii. ISBN: 978-0-12-5444224. https://doi.org/10.1016/B978-0-08-055630-7.50001-8 22. A.O. Sushkov, W.J. Kim, D.A.R. Dalvit, S.K. Lamoreaux, Observation of the thermal Casimir force. Nat. Phys. 7(3), 230–233 (2011). https://doi.org/10.1038/nphys1909 23. L. Ge, X. Shi, Z. Xu, K. Gong, Tunable Casimir equilibria with phase change materials: from quantum trapping to its release. Phys. Rev. B 101, 104107 (2020). https://doi.org/10.1103/ PhysRevB.101.104107 24. T. Zhan, X. Shi, Y. Dai, X. Liu, J. Zi, Transfer matrix method for optics in graphene layers. J. Phys. Condens. Matter. 25(21), 215301 (2013). https://doi.org/10.1088/0953-8984/25/21/ 215301 25. A. Lambrecht, I. Pirozhenko, L. Duraffourg, P. Andreucci, The Casimir effect for silicon and gold slabs. Europhys. Lett. (EPL) 77(4), 44006 (2007). https://doi.org/10.1209/0295-5075/77/ 44006 26. D.B. Hough, L.R. White, The calculation of Hamaker constants from Lifshitz theory with applications to wetting phenomena. Adv. Colloid Interface Sci. 14(1), 3–41 (1980). https://doi. org/10.1016/0001-8686(80)80006-6.61

Chapter 3

Experimental Realization of a Casimir Diode: Non-reciprocal Energy Transfer by Casimir Force

Abstract The Casimir effect is essential for micro- and nanotechnologies because it is inevitably huge at a small separation. The Casimir interaction has been used to realize various advanced function. However, a non-reciprocal Casimir device is still unexplored. In this chapter, we will introduce our recent realization of nonreciprocal energy transfer between two micro-mechanical oscillators by Casimir effect. We will first show how we realize a strong coupling and energy transfer by the parametric coupling scheme. This will be followed by a measurement of the exceptional point in the parameter space by a careful design. We will then present the results of non-reciprocal energy transfer by dynamically driving the system near the exceptional point. In the last section of this chapter, we will show detailed derivations of the effective system Hamiltonian and the exceptional point. Parts of the contents in this chapter have been published in Xu et al. (Nat Nanotechnol 17(2):148–152, 2022).

3.1 Energy Transfer Between Two Cantilevers The direct coupling strength is usually smaller than the frequency difference between two cantilevers in our experiment. To couple two cantilevers, we apply the parametric coupling scheme by modulating the separation at a slow rate .ωmod that is .d(t) = d0 +δd cos(ωmod t) (Fig. 3.1a). When the modulation frequency meets the condition that is .ωmod = |ω2 − ω1 |, two resonators are parametrically coupled (Fig. 3.1b). When the modulation amplitude is far smaller than the separation (.δd  d), the effective coupling strength is approximately proportional to the modulation amplitude .δd . Meanwhile, we can easily control the coupling time, the coupling strength, and the effective detuning of two cantilevers. The parametric coupling scheme has also been realized on other systems for motion transduction and tunable coupling [6, 7].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 Z. Xu, Optomechanics with Quantum Vacuum Fluctuations, Springer Theses, https://doi.org/10.1007/978-3-031-43052-7_3

37

38

3 Experimental Realization of a Casimir Diode: Non-reciprocal Energy Transfer. . .

(a)

(b)

mod

Fig. 3.1 Parametric coupling scheme. (a) Parametric modulation of the separation .d(t) = d0 + δd cos(ωmod t) is applied to couple two cantilevers. (b) When the modulation frequency .ωmod matches the frequency difference between two cantilevers such that .ωmod = |ω2 − ω1 |, a down/upconversion process is realized and two cantilevers are coupled

Now we discuss the dynamics and couplings of this dual-cantilever Casimir system by showing the details of the system equations. Under the parametric coupling scheme, the system equations are m1 x¨1 + m1 γ1 x˙1 + k1 x1 = Fmod (t) + FC (x),

.

m2 x¨2 + m2 γ2 x˙2 + k2 x2 = Fdri (t) − FC (x),

(3.1)

where .m1 and .m2 are the masses of two cantilevers and .k1 and .k2 are the natural spring constants of the two cantilevers. .FC (x) = −2π RE(x, T ) is the Casimir force between two cantilevers calculated by Eqs. 2.16 and 2.18. .Fmod is the slow modulation applied to cantilever 1 to couple two cantilevers, and .Fdri is the resonant driving force applied to cantilever 2. These two forces can be explicitly written as Fmod = k1 δd cos(ωmod t),

.

Fdri = F2 cos(ω2 t + φ),

(3.2)

where .δd is the modulation amplitude of parametric coupling, .ωmod is the modulation frequency, .F2 is the driving force amplitude, and .φ is the phase of the driving. .ω2 is the intrinsic frequency of cantilever 2 under the Casimir interaction. The modulation on cantilever 1 is effectively modulating the separation between two cantilevers, and hence, the equations of motion are 2 m1 x¨1 + m1 γ1 x˙1 + m1 ω10 x1 = FC (d0 + δd cos(ωmod t) + x1 − x2 ),

.

2 m2 x¨2 + m2 γ2 x˙2 + m2 ω20 x2 = −FC (d0 + δd cos(ωmod t) + x1 − x2 )

+ F2 cos(ω2 t + φ),

(3.3)

3.1 Energy Transfer Between Two Cantilevers

39

where .d0 is the equilibrium distance between two cantilevers when no parametric modulation is added to the system. .ω10 and .ω20 are the natural resonant frequencies of two cantilevers. When the vibrational amplitudes of the cantilevers meet the conditions that .δd , |x1 |, |x2 |  d0 , the Casimir force can be expanded as FC (d0 + δd cos(ωmod t) + x1 − x2 ) =FC (d0 ) +

.

+

dFC |d (δd cos(ωmod t) + x1 − x2 ) dx 0

1 d 2 FC |d (δd cos(ωmod t) + x1 − x2 )2 . 2 dx 2 0 (3.4)

In this experiment, we have two cantilevers with a frequency difference over 700 Hz so the direct coupling is neglected. The first and second terms in Eq. 3.4 only shift the intrinsic frequency to a lower value and do not contribute to the energy transfer because they are off-resonant. The energy transfer comes from the term d 2 FC | δ cos(ωmod t)(x1 − x2 ). Therefore, we can rewrite the equations as . dx 2 d0 d m1 x¨1 + m1 γ1 x˙1 + m1 ω12 x1 =  cos(ωmod t)(x1 − x2 ),

.

m2 x¨2 + m2 γ2 x˙2 + m2 ω22 x2 =  cos(ωmod t)(x2 − x1 ) + F2 cos(ω2 t + φ), (3.5) 2

where . = ddxF2C |d0 δd . We generalize the displacements .x1 (t) and .x2 (t) to complex values .z1 (t) and .z2 (t) such that .x1 (t) = Re[z1 (t)] and .x2 (t) = Re[z2 (t)]. Considering a steady state that two cantilevers oscillate at their resonant frequencies with constant oscillating amplitudes .A1 and .A2 , we can write z1 (t) = A1 exp(−iω1 t),

.

z2 (t) = A2 exp(−iω2 t).

(3.6)

By substituting Eq. 3.6 into Eq. 3.5, the equations become .

− iω1 γ1 A1 e−iω1 t =

 cos(ωmod t)(A1 e−iω1 t − A2 e−iω2 t ), m1

−iω2 γ2 A2 e−iω2 t =

F2  cos(ω2 t + φ). cos(ωmod t)(A2 e−iω2 t − A1 e−iω1 t ) + m2 m2 (3.7)

When the modulation frequency .ωmod matches .ω2 − ω1 , the parametric coupling is realized and the equations become .

− iω1 γ1 A1 e−iω1 t =

  A1 e−iω2 t − A2 e−iω1 t , 2m1 2m1

−iω2 γ2 A2 e−iω2 t =

F2 −iω2 t   A1 e−iω2 t + e , A2 e−iω1 t − 2m2 m2 2m2

(3.8)

40

3 Experimental Realization of a Casimir Diode: Non-reciprocal Energy Transfer. . .

where we have neglected the fast-rotating terms. Therefore, we can get the solutions that |A1 | = |

.

1 F2 ω1 ω2 |, 4k1 k2 γ1 γ2 1 + 2 ω1 ω2 2k1 k2 γ1 γ2

|A2 | = |

1 F2 ω2 |. 2k2 γ2 1 + 2 ω1 ω2

(3.9)

2k1 k2 γ1 γ2

In this way, we can get the ratio of two oscillating amplitudes as |

.

d 2 FC ω1 δd ω1 A1 |=| | . |=| 2γ1 k1 A2 dx 2 2γ1 k1

(3.10)

In this experiment, the natural resonant frequency and damping rate of the two cantilevers are .ω10 = 2π × 4826 Hz, .ω20 = 2π × 5582 Hz, .γ10 = 2π × 2.65 Hz, and .γ20 = 2π × 2.68 Hz. By applying the parametric coupling scheme, we have observed the energy transfer when cantilever 2 is resonantly driven (Fig. 3.2). The ratio of the vibrational amplitude of two cantilevers is shown in Fig. 3.2a for two different separations. We notice that the ratio reaches the maximum value when the modulation frequency meets the resonant condition such that .ωmod = ω2 − ω1 . A smaller separation gives a larger transduction ratio since the coupling strength is larger. This agrees with the prediction in Eq. 3.10. The maximum transduction ratio .A1 /A2 at different separations is shown in Fig. 3.2b. This condition is met when we tune the modulation frequency to the resonant case. The theoretical prediction is calculated from Eq. 3.10. Therefore, we have realized Casimir mediated energy

(a)

(b)

Fig. 3.2 Energy transfer between two cantilevers by the parametric coupling scheme. (a) The ratio of the vibrational amplitudes .A1 /A2 at different modulation frequencies .ωmod /2π . (b) The ratio of the vibrational amplitudes at different separations when the modulation frequency meets the resonance

3.1 Energy Transfer Between Two Cantilevers

41

transfer between two microcantilevers. A stronger Casimir coupling strength can be achieved by decreasing the separation x or increasing the modulation amplitude .δd . We also observe strong coupling in this two-body Casimir system. We show the measurement of PSD of the Brownian motion of two cantilevers when modulation frequency is 600 and 750 Hz in Fig. 3.3 as examples. There is no extra driving .Fdri here. The blue and red lines show the measured Brownian motion of cantilever 1 and cantilever 2, respectively. The total measurement time is 4 s. Fig. 3.3a shows the case when the modulation frequency is off-resonant and two cantilevers are still uncoupled. The slow modulation of 600 Hz adds side peaks for cantilevers without significant energy transfer. By contrast, Fig. 3.3b shows the case when the modulation frequency .ωmod is close to .ω2 − ω1 . Under such conditions, phonon conversion is realized and level repulsion is observed. To show the level repulsion in a clearer way, we present the PSD of cantilever 2 in Fig. 3.4a. Near resonance, frequency anti-crossing is observed in Fig. 3.4b. This is an evidence of entering the strong coupling regime with a coupling strength around 30 Hz. We also investigate on the effect of nonlinearity in the mechanical system and discuss its contribution to the parametric coupling. We present the measured thermal PSD of the cantilever and an example of the measured frequency response under a driving as shown in Fig. 3.5. The measurement can be well described by a Lorentzian function, indicating that the mechanical resonator is experiencing harmonic oscillation near the equilibrium position. The fitting gives a resonant frequency of 4548.9 .± 0.3 Hz for the thermal motion and a resonant frequency of 4548.6 .± 0.3 Hz for the motion under driving with an amplitude of 7.6 nm. This small frequency difference might come from the detection noise, fitting uncertainty, and nonlinearity in the mechanical system. We use the frequency shift to estimate the upper bound of the nonlinear coefficients. For a driven nonlinear oscillator, the equation can be described as

Fig. 3.3 Power spectral density (PSD) at two different modulation frequencies. (a) A off-resonant parametric modulation with a frequency 600 Hz is applied to the system. The motion of two cantilevers is not coupled. (b) A resonant parametric modulation with a frequency 750 Hz is applied to the system. A frequency splitting appears in the peak of the PSD indicating that two cantilevers are coupled

42

3 Experimental Realization of a Casimir Diode: Non-reciprocal Energy Transfer. . .

Fig. 3.4 Level repulsion from Casimir coupling. (a) Power spectral density (PSD) of the cantilever 2 as a function of the modulation frequency .fmod and PSD frequency. (b) is the refined scan of the white box shown in (a)

Fig. 3.5 Nonlinearity of the mechanical resonators. The measured PSD and frequency response of a cantilever are shown in blue, and the red curves are the Lorentzian fittings. The measurement can be well described by a Lorentzian function, indicating that the mechanical resonator is experiencing harmonic oscillation near the equilibrium position. (a) PSD of the thermal motion. The fitting gives a resonant frequency of 4548.9.±0.3 Hz. (b) Frequency response under a driving force with a driving amplitude of 7.6 nm. The fitting gives a resonant frequency of 4548.6.±0.3 Hz. This small frequency difference might come from the detection noise, fitting uncertainty, and nonlinearity in the mechanical system

mx¨ + mγ x˙ + kx + βx 2 + αx 3 = Fdri cos(ωt),

.

(3.11)

where .α is the Duffing constant, .β is the quadratic nonlinear constant, m is the mass, .γ is the damping rate, and k is the spring constant. These three parameters can be calibrated in the PSD measurement. When the driving is small, the nonlinear response is negligible and the cantilever can be treated a damped harmonic

3.2 Dynamical Control Near the Exceptional Point

43

oscillator. However, effects from nonlinearity arise when the external driving is strong and leads to a frequency shift in the frequency response [8]. From the frequency shift, we can estimate an approximated Duffing coefficient upper bound of .α = 2 × 107 N/m.3 and an approximated quadratic coefficient upper bound of 2 2 .β = 7.5 × 10 N/m. . This nonlinearity is a few orders smaller than the Casimir nonlinearity reported in [1]. At a separation of 100 nm and a modulation amplitude of 10 nm, the coupling strength for the Casimir force .gC , the Duffing nonlinearity .gD , and the quadratic nonlinearity .gQ will be √ d 2 FC δd /2 m1 m2 ω1 ω2 = 2π × 14.0, dx 2 √ gD = 6αδd2 /2 m1 m2 ω1 ω2 = 2π × 3.9 × 10−4 , √ gQ = 2βδd /2 m1 m2 ω1 ω2 = 2π × 0.5. gC =

.

(3.12)

The coupling strength from nonlinearity is more than one orders smaller compared to the strength from Casimir interaction. In reality, the real nonlinearity should be smaller than this since the measurement of frequency shift has some uncertainty. Therefore, we can neglect the contribution from mechanical nonlinearity.

3.2 Dynamical Control Near the Exceptional Point We have introduced strong coupling and energy transfer between two micromechanical cantilevers. However, the energy transfer discussed so far is symmetric. It would be interesting if one can break the symmetry and realize the nonreciprocity. In this experiment, the first observation of non-reciprocal energy transfer between two micro-mechanical cantilevers is achieved by dynamically driving the system near the exceptional point. Parametric coupling scheme allows us to control the coupling strength and detuning of the system independently over time. In the interaction picture, the simplified system Hamiltonian is (more details of the Hamiltonian derivation will be discussed in Sect. 3.3)  Hint =

.

−i γ21 g 2

g 2

−i γ22 − δ

 ,

(3.13)

2 √ where .γ1,2 are the damping rates of two resonators, and .g = ddxF2C δd /2 m1 m2 ω1 ω2 is the coupling strength. .δ = ωmod − (ω2 − ω1 ) is the detuning of the mechanical system. The exceptional point of this two-resonator open system is located at .g = |γ1 −γ2 | and .δ = 0. At this specific exceptional point, two eigenvalues .λ± of the 2 Hamiltonian are degenerate for both real and imaginary parts (Fig. 3.6a, b). The eigenvalues are plotted as two surfaces in the parameter space. The special nontrivial

44

3 Experimental Realization of a Casimir Diode: Non-reciprocal Energy Transfer. . .

Fig. 3.6 The eigenvalues of the two-cantilever system in the parameter space. A nontrivial topological structure exists near the exceptional point (EP)

topological structure near the exceptional point allows us to break the time reversal symmetry by a well-designed dynamical control. We first engineer the exceptional point of the system and try to measure the location of it experimentally. For the natural condition, the damping rate difference of two resonators is close to zero since two resonators have very similar damping rates. The exceptional point is located at .δd = 0, which corresponds to a zero coupling strength. Thus, it is impossible to break the symmetry and realize energy transfer under such natural condition. To shift the exceptional point to a reasonable location, we need to apply additional loss or gain to the system. For simplicity, we apply additional loss to one cantilever by a PID feedback control and modify the damping rate to be .γ2 = 2π × 13.82 Hz. To locate the exceptional point, we measured the PSD of cantilever 2 (Fig. 3.7). We notice that the exceptional point here is located approximately at .δd = 5.5 nm and .fmod = 727 Hz. We design a control loop near the exceptional point as shown in Fig. 3.8 with a total loop time of 80 ms. During the loop, the modulation amplitude .δd is continuously swept from .6.7 ± 0.6 nm to .13.3 ± 0.6 nm. The modulation frequency .fmod is swept from 680 Hz to 785 Hz. The system evolves in an interesting way because of the special design of the control loop. The rectangular control loop consists of four sessions in the parameter space and each session lasts 20 ms. For the part when .δd is fixed at 6.7 nm and .fmod is swept from 680 Hz to 785 Hz, the minimum energy difference between two eigenvalues is small compared to the operation speed. Therefore, this process is non-adiabatic. During this process, the energy tends to be transferred to the cantilever with a smaller damping rate. The process is adiabatic for other three parts of the rectangular loop since the energy difference is large compared to the operation speed. In this way, time reversal symmetry is broken after driving the system along the control loop shown in Fig. 3.8. The measurements of the non-reciprocity process are shown in Fig. 3.9. Here we present the measurements for the clockwise (CW) and the anti-clockwise (ACW)

3.2 Dynamical Control Near the Exceptional Point

45

Fig. 3.7 Experimental measurement of the exceptional point. PSD intensity of cantilever 2. (a). When .δd is fixed at .5.5 ± 0.6 nm. (b). When .fmod is fixed at 727 Hz Fig. 3.8 A clockwise (CW) control loop in the parameter space (.fmod and .δd )

CW 785 Hz

680 Hz 6.7 nm

13.3 nm

mod ,

loops for the same loop size and same starting point but with different initially excited cantilevers. We use normalized energy .E1 /(E1 + E2 ) and .E2 /(E1 + E2 ) to quantify how much energy is transferred during the process. Here .E1 and .E2 are the energy of two cantilevers by measuring the vibrational amplitudes of the cantilevers. One resonator is first driven resonantly from .t = 0 to .t = 80 ms. We then apply the dynamical control loop to the system as shown in the gray shaded area. The measurements in Fig. 3.9a show that the energy can be transferred from cantilever 1 to cantilever 2 under a clockwise loop. However, the energy transfer is suppressed from cantilever 2 to cantilever 1 (Fig. 3.9b). At the end of the clockwise loop, cantilever 2 dominates the energy no matter what the initial state is. In contrast, cantilever 1 dominates the energy after the anti-clockwise loop as shown in Fig. 3.9c, d. Non-reciprocity with high contrast is realized for the first time in a Casimir system. This system can be used to control the energy transfer directionality in a desired way. We also simulate the transfer process by using the following equations: m1 x¨1 + m1 γ1 x˙1 + m1 ω12 x1 = FC (d0 + δd (t) cos(ωmod (t)t) + x1 (t) − x2 (t)),

.

m2 x¨2 + m2 γ2 x˙2 + m2 ω22 x2 = −FC (d0 + δd (t) cos(ωmod (t)t) + x1 (t) − x2 (t)) + F2 H (t0 − t) cos(ω2 t + φ),

(3.14)

46

(a)

3 Experimental Realization of a Casimir Diode: Non-reciprocal Energy Transfer. . .

Clockwise (CW) 1 to 2

(b)

T12

(c)

Anti-clockwise (ACW) 1 to 2

T12

Clockwise (CW) 2 to 1

T21

(d)

Anti-clockwise (ACW) 2 to 1

T21

Fig. 3.9 Measurement of the normalized energy in the transfer process. One cantilever is first driven resonantly from 0 ms to 80 ms (white region). The dynamical control starts at 80 ms (shaded in gray) for clockwise (CW) and anti-clockwise (ACW) loop and lasts for 80 ms. A clockwise loop allows the energy transferred from 1 to 2 (T12) as shown in (a) and avoids the transmission from 2 to 1 (T21) as shown in (b). A diode-like energy transfer is realized in the system. An anti-clockwise loop allows the energy transferred from 2 to 1 as shown in (d) and avoids the reverse direction as shown in (c)

where .δd (t) and .ωmod (t) are the time-dependent parameters that are controlled dynamically in the transfer process. .ω1 and .ω2 are the resonant frequencies of the two cantilevers under the Casimir interaction. .F2 is the driving force applied on cantilever 2. .H (t0 − t) is the Heaviside step function, and it represents that the driving is applied from 0 to .t0 (in our experiment .t0 = 80 ms). In this way, we can perform the simulation by using the Runge–Kutta methods based on Eq. 3.14. We show the simulation for all four conditions in Fig. 3.10. The simulation shows the same key features with the experimental results. We record the transfer efficiency .η = E2 /(E1 + E2 ) at t = 160 ms for different control loops (Fig. 3.11). Here cantilever 1 is always excited before the control loop

3.2 Dynamical Control Near the Exceptional Point

47

Clockwise (CW) 1 to 2

Clockwise (CW) 2 to 1

(a)

(b)

Anti-clockwise (ACW) 1 to 2

Anti-clockwise (ACW) 2 to 1

(c)

(d)

Fig. 3.10 The simulated Casimir mediated energy transfer process. (a), (b) Clockwise (CW) control loop. (c), (d) Anti-clockwise (ACW) control loops. The separation here is 76 nm.

max starts. As shown in the inset of Fig. 3.11, the maximum modulation frequency .fmod in the loop is sweeping, while the minimum modulation frequency at 680 Hz is fixed. The modulation amplitude .δd is maintained the same as the loop in Fig. 3.9. When the control loop includes the resonant modulation frequency .fmod = f21 , energy is transferred to cantilever 2 when we first excite cantilever 1. When the loop does not include .f21 , energy cannot be transferred. In summary, the first realization of Casimir mediated non-reciprocal energy transfer is reported. Strong coupling and energy transfer by Casimir effects are achieved by the parametric coupling scheme. With a well-designed control loop near the exceptional point, non-reciprocity with high contrast is observed for the first time in a Casimir system. Our work develops a novel method to regulate quantum vacuum fluctuations and build functional Casimir devices. Compared to the conventional optomechanical coupling that requires a great number of real

48

3 Experimental Realization of a Casimir Diode: Non-reciprocal Energy Transfer. . .

Fig. 3.11 Efficiency of the Casimir mediated non-reciprocal energy transfer. The minimum .fmod in the loop is fixed at 680 Hz, and the maximum .fmod is sweeping. The control on .δd is the same as before. When the control loop (inset) includes the resonant condition that .fmod = f21 (as shown in the green dashed line), energy is transferred from cantilever 1 to cantilever 2. Little energy is transferred when the maximum .fmod is lower than .f21

photons in a high-quality cavity [9–12], the Casimir coupling mediated by virtual photons will not suffer from cavity loss. Our Casimir diode can potentially apply to quantum information processing.

3.3 Effective Hamiltonian of the System In this section, we will introduce the detailed derivation of the effective system Hamiltonian in Eq. 3.13. The eigenvalues of the Hamiltonian will also be discussed. We start with a condition when no driving force .Fdri is applied to the system. The equations of motion under the parametric modulation and Casimir interaction are m1 x¨1 + m1 γ1 x˙1 + m1 ω12 x1 =  cos(ωmod t)(x1 − x2 ),

.

m2 x¨2 + m2 γ2 x˙2 + m2 ω22 x2 =  cos(ωmod t)(x2 − x1 ),

(3.15)

2

where we have . = ddxF2C δd . Similarly, we generalize the displacements .x1 (t) and .x2 (t) to complex values .z1 (t) and .z2 (t) such that .x1 (t) = Re[z1 (t)] and .x2 (t) = Re[z2 (t)]. We separate the fast-rotating term and the slow-varying term for .z1 (t) and .z2 (t) such that z1 (t) = A1 (t)e−iω1 t ,

.

z2 (t) = A2 (t)e−iω2 t ,

(3.16)

3.3 Effective Hamiltonian of the System

49

where .A1 (t) and .A2 (t) are slow-varying amplitudes. Therefore, the first and second derivatives of .z1 (t) and .z2 (t) are z˙1 = −iω1 A1 (t)e−iω1 t + A˙1 (t)e−iω1 t ,

.

z˙2 = −iω2 A2 (t)e−iω2 t + A˙2 (t)e−iω2 t , z¨1 = −ω12 A1 (t)e−iω1 t − 2iω1 A˙1 (t)e−iω1 t + A¨1 (t)e−iω1 t , z¨2 = −ω22 A2 (t)e−iω2 t − 2iω2 A˙2 (t)e−iω2 t + A¨2 (t)e−iω2 t .

(3.17)

By substituting the derivatives in Eq. 3.15 by the explicit expression in Eq. 3.17, we can rewrite the equation as z¨1 + γ1 z˙1 + ω12 z1 = −iω1 γ1 A1 (t)e−iω1 t + (γ1 − 2iω1 )A˙1 (t)e−iω1 t + A¨1 (t)e−iω1 t ,

.

z¨2 + γ2 z˙2 + ω22 z2 = −iω2 γ2 A2 (t)e−iω2 t + (γ2 − 2iω2 )A˙2 (t)e−iω2 t + A¨2 (t)e−iω2 t . (3.18) We assume that .A1 (t) and .A2 (t) are the slow-varying amplitudes, and hence, we can neglect the second derivative terms related to .A¨1 (t) and .A¨2 (t). Besides, we have the small damping condition that .γ1  ω1 and .γ2  ω2 . Therefore, the equations of motion can be rewritten as .

− iω1 γ1 A1 (t)e−iω1 t − 2iω1 A˙1 (t)e−iω1 t =

 eiωmod t + e−iωmod t (A1 e−iω1 t − A2 e−iω2 t ), 2 m1

− iω2 γ2 A2 (t)e−iω2 t − 2iω2 A˙2 (t)e−iω2 t =

 eiωmod t + e−iωmod t (A2 e−iω2 t − A1 e−iω1 t ). m2 2

(3.19)

Under the rotating wave approximation, we can neglect the fast-rotating terms, and hence, the equations are .

− iω1 γ1 A1 (t)e−iω1 t − 2iω1 A˙1 (t)e−iω1 t =

 (A1 (t)e−i(ω1 +ωmod )t − A2 (t)e−i(ω2 −ωmod )t ), 2m1

− iω2 γ2 A2 (t)e−iω2 t − 2iω2 A˙2 (t)e−iω2 t =

 (A2 (t)e−i(ω2 −ωmod )t − A1 (t)e−i(ω1 +ωmod )t ). 2m2

(3.20)

Now we apply the transformation such that .A1 (t) = A1 (t) and .A2 (t) = A2 (t)eiδt , where .δ = ω1 + ωmod − ω2 . Then the equations of motions are

50

3 Experimental Realization of a Casimir Diode: Non-reciprocal Energy Transfer. . .

.

− iω1 γ1 A1 (t) − 2iω1 A˙1 (t) =

 (A (t)e−iωmod t − A2 (t)), 2m1 1

 (−iω2 γ2 − 2ω2 δ)A2 (t) − 2iω2 A˙2 (t) = (A (t)eiωmod t − A1 (t)). (3.21) 2m2 2 We can neglect the fast-rotating term, and hence,  γ1 A (t), i A˙1 (t) = −i A1 (t) + 2 4m1 ω1 2

.

γ2  i A˙2 (t) = (−i − δ)A2 (t) + A (t). 2 4m2 ω2 1

(3.22)

The vibrational motions  can be quantized asphonons. By introducing the normalized amplitudes .c1 = m1h¯ω1 A1 and .c2 = m2h¯ω2 A2 , the equations of motion for phonon modes as    −i γ21 c˙1 .i = √  c˙2 4 m m ω 1

2 1 ω2

√  4 m1 m2 ω1 ω2 −i γ22 − δ

  c1 . c2

(3.23)

Therefore, the effective Hamiltonian in the interaction picture is  Hint =

.

−i γ21 √  4 m1 m2 ω1 ω2

√  4 m1 m2 ω1 ω2 −i γ22 − δ

 .

(3.24)

The eigenvalues of this Hamiltonian are 

γ 1 + γ2 1 (γ1 − γ2 )2 δ ± + δ 2 + g 2 − i(γ1 − γ2 )δ, − λ± = − − i 2 4 2 4

.

(3.25)

2

 = d∂xF2C δd 2√m m1 ω ω is the coupling strength and .δ = where .g = 2√m m 1 2 ω1 ω2 1 2 1 2 ω1 + ωmod − ω2 is the detuning. The exceptional point is located at .δ = 0 and |γ1 −γ2 | .g = . The eigenstates of this Hamiltonian are 2

 v± =

.

iγ2 +2δ g

− g1 ( i(γ12+γ2 ) + δ ∓



 2 2) 2 + g 2 − i(γ − γ )δ) − (γ1 −γ + δ 1 2 4 . 1 (3.26)

To understand the system better, we first consider the weak coupling strength condition that .g  |δ|, and we neglect damping .γ1 and .γ2 for now. Under such limit, the eigenvalues and eigenstates can be approximated to

3.3 Effective Hamiltonian of the System

λ+ =

.

v+ =

.

51

g2 g2 , λ− = −δ − . 4δ 4δ   1 g 2δ

, v− =

(3.27)

 −g  2δ

1

.

(3.28)

The above results are calculated in the interaction picture. To understand our system, we need to go back to the original frame with two states .|0 and .|1 that correspond to the motions of two cantilevers. If we assume that at .t = 0 two frames coincide at .|1, which means .|ψ(0)int = |ψ(0) = |1. Then the state evolves as g 2 −i(ω2 +δ+g 2 /4δ)t e |1 4δ 2 g g 2 2 + e−i(ω1 +g /4δ)t |0 − e−i(ω1 −δ−g /4δ)t |0. 2δ 2δ

|ψ(t)int = e−i(ω2 −g

.

2 /4δ)t

|1 +

(3.29)

We notice that there are four terms in the expression that correspond to four branches as shown in Fig. 3.4a. From top to down, four branches have frequency .ω2 − g 2 /4δ, 2 2 2 .ω2 +δ+g /4δ, .ω1 +g /4δ, and .ω1 −δ−g /4δ when .δ < 0. The first term is the main motion of cantilever 2. It has the maximum amplitude compared to other three, and it also agrees with the experimental measurement. There is another signal for the motion of cantilever 2, and it comes from the parametric coupling with cantilever 1. When the detuning is far from resonance, the amplitude of this term is very weak. The bottom two are the motions of cantilever 1 but coupled to cantilever 2. The amplitudes are .g/2δ. It is also interesting to see the eigenvalues of each state near the exceptional point. We analyze the frequency and damping for each state by changing the coupling strength g and the detuning .δ (equivalently, .ωmod in the experiment). We first assume that detuning .δ = 0 and look at the trend when coupling strength g varies. Under such limit, the eigenvalues and eigenstates are  1 (γ1 − γ2 )2 λ1 + λ2 + g2, ± − .λ± = −i 2 4 4  v± =

.

i(γ2 −γ1 ) 2g

±

1 g

(3.30)



 2 2) 2 + g − (γ1 −γ 4 . 1

(3.31)

If the initial state is .|1, which means .|ψ(0)int = |ψ(0) = |1, the state evolves as

52

3 Experimental Realization of a Casimir Diode: Non-reciprocal Energy Transfer. . .

|ψ(t)int = 

.

1 2) − (γ1 +γ +g 4 2

1 g { e−i(ω1 + 2 2 2 1 g − e−i(ω1 − 2 2

 γ +γ (γ −γ )2 − 1 4 2 +g 2 −i 1 4 2 )t

|0

 (γ −γ )2 γ +γ − 1 4 2 +g 2 −i 1 4 2 )t

|0

  1 γ2 − γ1 −i(ω2 + 1 − (γ1 −γ2 )2 +g 2 −i γ1 +γ2 )t (γ1 − γ2 )2 4 2 4 )e + g2 − i +( |1 − 4 2 4   1 (γ1 − γ2 )2 γ2 − γ1 −i(ω2 − 1 − (γ1 −γ2 )2 +g 2 −i γ1 +γ2 )t 2 4 2 4 )e +g +i −( − |1}. 4 2 4 (3.32) When .g > state as

|γ1 −γ2 | , .c 2

|ψ(t)int =

.

=



2) + g 2 is a real number, and we can rewrite the − (γ1 −γ 4 2

γ1 +γ2 c g −i(ω1 + c )t − γ1 +γ2 t g 4 2 e e |0 − e−i(ω1 − 2 )t e− 4 t |0 2c 2c 1 i(γ2 − γ1 ) −i(ω2 + c )t − γ1 +γ2 t 4 2 e +( − |1 )e 2 4c 1 i(γ2 − γ1 ) −i(ω2 − c )t − γ1 +γ2 t 4 2 e )e −( + |1. 2 4c

(3.33)

It is obvious that when the coupling strength is larger than the damping difference 2| between two cantilevers such that .g > |γ1 −γ , the four frequencies split and 2 2 correspond to .ω1 ± c/2 and .ω2 ± c/2. They have identical damping . γ1 +γ 4 . Under 2| , four frequencies become .ω1 ± g and .ω2 ± g. We an extreme limit that .g  |γ1 −γ 2 2| . Here we also analyze the case for small coupling strength such that .g < |γ1 −γ 4   2 2 2) 2) + g 2 = ia, where .a = (γ1 −γ − g 2 is a real number. The state have . − (γ1 −γ 4 4 evolves as |ψ(t)int =

.

g −iω1 t −( γ1 +γ2 + a )t g −iω1 t −( γ1 +γ2 − a )t 4 2 |0 − 4 2 |0 e e e e 2ia 2ia 1 γ2 − γ1 −iω2 t −( γ1 +γ2 − a )t 4 2 |1 )e e +( − 2 4a 1 γ2 − γ1 −iω2 t −( γ1 +γ2 + a )t 2 |1. 4 )e (3.34) e −( + 4a 2

We can see that two frequencies are degenerate as .ω1 and .ω2 , while the damping 2 opens up for each state as . γ1 +γ ± a2 under such conditions. The calculation above 4 also agrees with the experimental results in Fig. 3.7b. As the coupling strength g (essentially modulation amplitude .δd ) goes up, the frequencies of two states are first degenerate and then open up and become split.

3.3 Effective Hamiltonian of the System

53

We are also interested in the dependence of the PSD frequency on .ωmod near the exceptional point. To simplify the derivation, we assume a special case when |γ2 −γ1 | . The eigenvalues and eigenstates of the system become .g = 2  γ 1 + γ2 1 2 δ ± δ + iδ(γ2 − γ1 ), λ± = − − i 2 4 2

.

 v± =

.

2δ+i(γ2 −γ1 ) 2g

+

1 g

 δ 2 + iδ(γ2 − γ1 ) . 1

(3.35)



(3.36)

If the initial state is at .|1, the state evolves as 1 2δ + i(γ2 − γ1 ) |ψ(t)int = ( −  )v1 e−iλ1 t 2 4 δ 2 + iδ(γ2 − γ1 )

.

2δ + i(γ2 − γ1 ) 1 +( +  )v2 e−iλ2 t 2 4 δ 2 + iδ(γ2 − γ1 ) √ γ1 +γ2 δ 1 (γ2 − γ1 )2 2 =  e−i(ω1 − 2 + 2 δ +iδ(γ2 −γ1 )−i 4 )t |0 8g δ 2 + iδ(γ2 − γ1 ) √ γ1 +γ2 δ 1 (γ2 − γ1 )2 2 −  e−i(ω1 − 2 − 2 δ +iδ(γ2 −γ1 )−i 4 )t |0 8g δ 2 + iδ(γ2 − γ1 ) 1 δ +( −  2 δ 2 + iδ(γ2 − γ1 )  δ 2 + iδ(γ2 − γ1 ) −i(ω2 − δ + 1 √δ 2 +iδ(γ2 −γ1 )−i γ1 +γ2 )t 2 2 4 − )e |1 4δ δ 1 +( +  2 2 δ + iδ(γ2 − γ1 )  δ 2 + iδ(γ2 − γ1 ) −i(ω2 − δ − 1 √δ 2 +iδ(γ2 −γ1 )−i γ1 +γ2 )t 2 2 4 )e + |1. 4δ (3.37) Here we consider two limits that .|δ| the state can be approximated as |ψ(t)int =

.

|γ1 −γ2 | 2

and .|δ|

|γ1 −γ2 | 2| . When .|δ| |γ1 −γ , 2 2

(γ2 − γ1 )2 −i(ω1 + (γ1 −γ2 )2 )t − γ1 t 16δ e e 2 |0 8gδ −

(γ2 − γ1 )2 −i(ω1 −δ− (γ1 −γ2 )2 )t − γ2 t 16δ e e 2 |0 8gδ

54

3 Experimental Realization of a Casimir Diode: Non-reciprocal Energy Transfer. . .

(γ1 − γ2 )2 −i(ω2 + (γ1 −γ2 )2 )t − γ1 t 16δ e 2 |1 e 16δ 2

−

+ (1 + Four frequencies are .ω1 + (γ1 −γ2 )2 16δ .

(γ1 − γ2 )2 −i(ω2 −δ− (γ1 −γ2 )2 )t − γ2 t 16δ )e e 2 |1. 16δ 2 (γ1 −γ2 )2 16δ ,

ω1 − δ −

.

(γ1 −γ2 )2 16δ ,

ω2 +

.

(3.38)

(γ1 −γ2 )2 16δ ,

and .ω2 −

δ− Two of them are close to the natural frequency .ω1,2 , and the other two increase linearly proportional to .δ when .δ is large enough. 2| When .|δ| |γ1 −γ , the state is approximated as 2 |ψ(t)int =

.



(γ2 − γ1 )2

8g δ 2 + iδ(γ2 − γ1 ) e −

−(

e

−i(ω1 − 2δ +

γ1 +γ2 1 √ √ 4 + 2 2 (γ2 −γ1 )|δ|)t



(γ2 − γ1

1 √ √ (γ2 −γ1 )|δ|)t 2 2

|0

)2

8g δ 2 + iδ(γ2 − γ1 ) −i(ω1 − 2δ −

1 √ √ (γ2 −γ1 )|δ|)t 2 2

e

−i(ω2 − 2δ +

1 √ √ (γ2 −γ1 )|δ|)t 2 2

e

−i(ω2 − 2δ −

1 √ √ (γ2 −γ1 )|δ|)t 2 2

e

−(

γ1 +γ2 1 √ √ 4 − 2 2 (γ2 −γ1 )|δ|)t

−(

γ1 +γ2 1 √ √ 4 + 2 2 (γ2 −γ1 )|δ|)t

−(

γ1 +γ2 1 √ √ 4 − 2 2 (γ2 −γ1 )|δ|)t

|0  δ δ 2 + iδ(γ2 − γ1 ) 1 ) +( −  − 4δ 2 4 δ 2 + iδ(γ2 − γ1 ) e

|1  δ δ 2 + iδ(γ2 − γ1 ) 1 ) +( +  + 2 4 δ 2 + iδ(γ2 − γ1 ) 4δ e

e

|1. (3.39)

1 √ (γ2 − γ1 )|δ| and Near the exceptional point, four frequencies are .ω1 − 2δ ± √ 2 2 √ √ (γ −γ )|δ| γ +γ 1 δ 2 1 1 2 √ (γ2 − γ1 )|δ|. The dampings are . 4 ± , and they depend .ω2 − ± √ 2 2 2 2 2 on .δ near the exceptional point.

References 1. H.B. Chan, V.A. Aksyuk, R.N. Kleiman, D.J. Bishop, F. Capasso, Nonlinear micromechanical Casimir oscillator. Phys. Rev. Lett. 87, 211801 (2001). https://doi.org/10.1103/PhysRevLett. 87.211801.

References

55

2. R. Zhao, L. Li, S. Yang et al., Stable Casimir equilibria and quantum trapping. Science 364(6444), 984–987 (2019). https://doi.org/10.1126/science.aax0916 3. K.Y. Fong, H.-K. Li, R. Zhao, S. Yang, Y. Wang, X. Zhang, Phonon heat transfer across a vacuum through quantum fluctuations. Nature 576(7786), 243–247 (2019). https://doi.org/10. 1038/s41586-019-1800-4 4. J.M. Pate, M. Goryachev, R.Y. Chiao, J.E. Sharping, M.E. Tobar, Casimir spring and dilution in macroscopic cavity optomechanics. Nat. Phys. 16, 1117–1122 (2020). https://doi.org/10.1038/ s41567-020-0975-9 5. Z. Xu, X. Gao, J. Bang, Z. Jacob, T. Li, Non-reciprocal energy transfer through the Casimir effect. Nat. Nanotechnol. 17(2), 148–152 (2022). https://doi.org/10.1038/s41565-021-010268 6. P. Huang, P. Wang, J. Zhou et al., Demonstration of motion transduction based on parametrically coupled mechanical resonators. Phys. Rev. Lett. 110, 227202 (2013). https://doi.org/10. 1103/PhysRevLett.110.227202 7. J.P. Mathew, R.N. Patel, A. Borah, R. Vijay, M.M. Deshmukh, Dynamical strong coupling and parametric amplification of mechanical modes of graphene drums. Nat. Nanotechnol. 11(9), 747–751 (2016). https://doi.org/10.1038/nnano.2016.94 8. P. Huang, J. Zhou, L. Zhang et al., Generating giant and tunable nonlinearity in a macroscopic mechanical resonator from a single chemical bond. Nat. Commun. 7(1), 11517 (2016). https:// doi.org/10.1038/ncomms11517 9. H. Xu, D. Mason, L. Jiang, J.G.E. Harris, Topological energy transfer in an optomechanical system with exceptional points. Nature 537, 80 EP (2016). [Online]. Available: https://doi.org/ 10.1038/nature18604 10. C. Yang, X. Wei, J. Sheng, H. Wu, Phonon heat transport in cavity-mediated optomechanical nanoresonators. Nat. Commun. 11(1), 1–6 (2020). https://doi.org/10.1038/s41467-020-184264 11. M. Aspelmeyer, T.J. Kippenberg, F. Marquardt, Cavity optomechanics. Rev. Mod. Phys. 86, 1391–1452 (2014). https://doi.org/10.1103/RevModPhys.86.1391 12. S. Barzanjeh, A. Xuereb, S. Groblacher, M. Paternostro, C.A. Regal, E.M. Weig, Optomechanics for quantum technologies. Nat. Phys. 1–10 (2021). https://doi.org/10.1038/s41567-02101402-0

Chapter 4

Experimental Realization of a Casimir Transistor: Switching and Amplifying Energy Transfer in a Three-Body Casimir System

Abstract The Casimir interaction can be used to couple two mechanical resonators and improve the quality factor of a mechanical resonator. While the Casimir effect between two objects has been extensively studied, the Casimir force between three objects is still unexplored. It is interesting to study the three-body interactions by quantum vacuum fluctuations. In this chapter, we will demonstrate the first observation of Casimir effects between three separate objects. A transistor-like energy transfer is realized in this three-channel Casimir system, which can switch and amplify the Casimir mediated energy transfer. We will first introduce the threecantilever system. This is followed by the the measurement of Casimir force in this three-body system. Later on, we will present the observation of level repulsion in the three-body system due to Casimir coupling. The coupling and level repulsion can be well explained by an effective Hamiltonian that will also be discussed. In the last part of this chapter, we will show how we realize the transistor-like energy transfer in this three-object Casimir system. Parts of the contents in this chapter have been published in Xu et al. (Nat Commun 13(1):6148, 2022).

4.1 Experimental Setup The three-object system consists of three closely spaced micro-mechanical cantilevers (Fig. 4.1a). The motion of three cantilevers can be monitored by three fiber interferometers. Each two neighboring cantilevers experience a separationdependent Casimir force between them because of quantum vacuum fluctuations. The optical image is shown in Fig. 4.1b. The size of the left and right cantilevers is .450 × 50 × 2 .μm.3 (NanoAndMore Arrow TL-CONT). The size of the center cantilever is .500 × 100 × 1 .μm.3 (NanoAndMore Arrow Au). Two polystyrene spheres with a diameter of 70 .μm (Duke Science 4270A sphere) are adhered to the left cantilever and the right cantilever to form the sphere–plate–sphere geometry. A 100-nm gold film is coated on cantilever surfaces for good conductivity and reflectivity.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 Z. Xu, Optomechanics with Quantum Vacuum Fluctuations, Springer Theses, https://doi.org/10.1007/978-3-031-43052-7_4

57

58

4 Experimental Realization of a Casimir Transistor: Switching and Amplifying. . .

(a)

(b) Fiber 2

Cantilever 1

Fiber 1

1

3

Input

2

Cantilever 2

Cantilever 3 Fiber 3 Output

Fig. 4.1 Casimir force in the three-object system. (a) Three modified AFM cantilevers are closely spaced. The resonant frequencies are .ω1 , .ω2 , and .ω3 . The vibration amplitudes are .A1 , .A2 , and .A3 . (b) The optical image of the three-terminal Casimir system

4.2 Measurement of Casimir Force in the Three-Cantilever System We assume that three objects are ideal metals and the center cantilever is sufficiently thick. Under the additivity approximation, the Casimir force on the center cantilever is [9] 0 .F2,C

π 3 hc ¯ = 360



R1 d13

−

R2 d23

 ,

(4.1)

where .d1 and .d2 are the separations between each two cantilevers (inset of Fig. 4.2a). R1 and .R2 are the radii of the microsphere. The Casimir force between real gold films can be calculated by the Lifshitz formula [3, 17], similar to the Casimir force calculation in Chap. 2. Similar to the measurement process in Chap. 3, we still apply the dynamic force measurement method in the three-body system. We first fix the positions of cantilever 1 (left) and cantilever 3 (right) with a relation that .d1 + d2 = 760 nm (Fig. 4.2a). At the same time, we change the position of the cantilever 2 (center). The measured force gradient on cantilever 2 reaches the minimum value when .d1 = d2 if .R1 = R2 (Fig. 4.2a). At this particular position, the net Casimir force on cantilever 2 (center) is zero (Fig. 4.2b). Under force additivity approximation, the measurement agrees well with the calculation based on Lifshitz’s formula. The force additivity approximation is further validated for this three-cantilever system in Fig. 4.2c, d. Although Casimir force has been extensively studied experimentally, all measurements were conducted between two bodies. Before this work, most experts in this field thought it would be too challenging to measure the Casimir effect between three objects experimentally, severely limiting the investigation of the Casimir effect and its applications. Our work demonstrates the first observation of the Casimir .

4.2 Measurement of Casimir Force in the Three-Cantilever System

(a)

(b)

(c)

(d)

59

Fig. 4.2 Casimir force in the three-cantilever system. (a) Measured Casimir force gradient on cantilever 2 (center) for different separations. The positions of the left and right cantilevers are fixed with a relation that .d1 + d2 = 760 nm. The center cantilever is moving from right to left to change .d2 . (b) Measured Casimir force on cantilever 2 (center) for different .d2 . (c) Measured Casimir force gradient on cantilever 2 (center) for different .d1 when .d2 is fixed at 310 nm. The blue circles are the measured Casimir force gradient only from cantilever 1 (left). The red diamonds are the total measured Casimir force gradient. The gray dashed line is the gradient only from cantilever 3 (right). Here we adapt the additivity approximation. (d) Measured Casimir force gradient on cantilever 2 (center) for different .d2 when .d1 is fixed at 276 nm

force between three bodies, and this is an experimental breakthrough in the field. In the future, we can study the Casimir force between more complicated geometries and configurations. We can also investigate on the non-additivity of the Casimir force by replacing the center cantilever with a thinner film [18, 19]. To get a better understanding of the three-body Casimir system, we performed more advanced calculation for the three-body Casimir force without the additivity approximation. Here we discuss the effect of the center plate thickness on the nonadditivity component. As shown in Fig. 4.3a, the Casimir system consists of three gold plates. To study the Casimir force on the left gold plate (L), we treat the system as a gold plate (L) and a multi-layer structure, and they are separated by a distance .d1 in vacuum. The multi-layer structure consists of three layers that are a thin gold membrane with thickness .L1 , a vacuum with thickness .d2 , and an infinitely thick gold plate. The Casimir pressure on the gold plate (L) is given by [20]

60

4 Experimental Realization of a Casimir Transistor: Switching and Amplifying. . .

Fig. 4.3 Calculation of Casimir force for a multi-layer system. (a) We treat our Casimir system as a left gold plate (L) and a multi-layer system and calculate the pressure on the gold plate (L). The multi-layer structure consists of three layers that are a middle gold membrane (M), a vacuum, and a right gold plate (R). The interface between the gold plate (L) and the multi-layer structure is vacuum. The left gold plate (L) and the right gold plate (R) are assumed to be infinitely thick. (b) The calculated Casimir pressure on the gold plate (L) is shown as a function of .d2 for different membrane thicknesses .L1 . Here .d1 is set to be 100 nm. When the thickness of the gold membrane (M) is 10 nm, the Casimir pressure on the gold plate (L) changes by .7.6 .% when the separation .d2 increases from 20 nm to 500 nm. If the thickness of the gold membrane (M) is 100 nm, the Casimir pressure on the gold plate (L) only changes by .9 × 10−4 .% when .d2 changes from 20 nm to 500 nm. (c) The calculated Casimir pressure on the gold plate (L) is shown as a function of the thickness .L1 , while the separation .d1 and .d2 are fixed. When the thickness .L1 is larger than 50 nm, the direct Casimir force interaction between the gold plate (L) and the gold plate (R) is negligible.

P (x, T ) = −

.

∞  kB T   ∞ −1 −1 qk⊥ dk⊥ {[r1p (iξl , k⊥ )r2p (iξl , k⊥ )e2xq − 1]−1 π 0 l=0

−1 −1 +[r1s (iξl , k⊥ )r2s (iξl , k⊥ )e2xq − 1]−1 }. (4.2)

Here .r1p and .r1s are the reflection coefficients of the gold plate (L) for the p and s polarizations, and they are written as r1p (iξl , k⊥ ) =

.

(iξl )q(iξl , k⊥ ) − k(iξl , k⊥ ) , (iξl )q(iξl , k⊥ ) + k(iξl , k⊥ )

(4.3)

q(iξl , k⊥ ) − k(iξl , k⊥ ) . q(iξl , k⊥ ) + k(iξl , k⊥ )

(4.4)

and r1s (iξl , k⊥ ) =

.

r2p and .r2s are the reflection coefficients of the multi-layer structure (gray area in Fig. 4.3a). Similar to the calculation in Chap. 2, we apply the transfer matrix method to obtain the reflection coefficients in the multi-layer structure. The transfer matrix of the multi-layer structure in Fig. 4.3a is written as [20, 21]

.

4.2 Measurement of Casimir Force in the Three-Cantilever System p(s)

p(s)

61 p(s)

M p(s) = D0→1 P1 (L1 )D1→2 P2 (d2 )D2→3 ,

.

(4.5) p(s)

where each layer is labeled as .0, 1, 2, 3 as shown in Fig. 4.3a. .Dj →j +1 is the transmission matrix between layer j and .j + 1 and is given as   p(s) p(s) 1 1 + ηj,j +1 1 − ηj,j +1 = , p(s) p(s) 2 1 − ηj,j +1 1 + ηj,j +1

(4.6)

j (iξ )Kj +1 , j +1 (iξ )Kj

(4.7)

p(s) .D j →j +1

p(s)

where .ηj,j +1 is written as p

ηj,j +1 =

.

s ηj,j +1 =

Kj +1 . Kj

 2 +  (iξ )ξ 2 /c2 . .P (L ) and .P (d ) are the propagation matrices Here .Kj = k⊥ j 1 1 2 2 for layer 1 and layer 2, and they are given as P1 (L1 ) =

KL 0 e 1 1 , 0 e−K1 L1

(4.8)

P2 (d2 ) =

Kd 0 e 2 2 . 0 e−K2 d2

(4.9)

.

and .

We can then get the reflection coefficients of the multi-layer structure as p(s)

p(s)

r2p(s) = M21 /M11 ,

.

p(s)

p(s)

(4.10)

where .M11 and .M21 are the components of the transfer matrix .M p(s) based on Eq. 4.5. The calculated Casimir pressure on the gold plate (L) is shown in Fig. 4.3b. Here we consider a case that .d1 = 100 nm. When the thickness of the middle gold membrane (M) is 10 nm, the Casimir pressure on the gold plate (L) changes by 7.6% when the separation .d2 increases from 20 to 500 nm. If the thickness of the gold membrane (M) is 100 nm, the Casimir pressure on the gold plate (L) only changes by .9 × 10−4 % when .d2 changes from 20 to 500 nm. In our experiment, the thickness of the middle gold membrane (center gold-coated silicon cantilever) is about .1 .μm, and hence, the non-additivity Casimir effect is negligible in our system. To observe the non-additivity Casimir effect, the thickness of the center gold needs to be reduced to below 30 nm (Fig. 4.3c). Our system can be modified to study this non-additivity Casimir effect in the near future.

62

4 Experimental Realization of a Casimir Transistor: Switching and Amplifying. . .

4.3 Casimir Vibrational Coupling We now introduce how we couple three cantilevers by Casimir interaction. The natural resonant frequencies and damping rates of three cantilevers are .ω1 = 2π × 5661 Hz, .ω2 = 2π × 6172 Hz, .ω3 = 2π × 4892 Hz, .γ1 = 2π × 3.22 Hz, .γ2 = 2π × 6.06 Hz, and .γ3 = 2π × 3.58 Hz when they are far apart. The PSDs of three cantilevers are shown in Fig. 4.4a. Similar to the Casimir diode experiment, we apply parametric coupling scheme to the three-object Casimir system. The separation between each two nearby surfaces is modulated at an offresonant frequency .ωmod1,2 [3, 22]. Experimentally, the position of the cantilever 2 is modulated as .δd1 cos(ωmod1 t) + δd2 cos(ωmod2 t), relative to its equilibrium position. The motion of three cantilevers can be coupled when .ωmod1 = |ω1 − ω2 | and .ωmod2 = |ω3 − ω2 | (Fig. 4.4b). We now discuss the coupling and dynamics with more details to get a better understanding of the three-cantilever coupled system. When a slow parametric modulation is applied to cantilever 2, the separations are explicitly written as d1 (t) = d10 − δd1 cos(ωmod1 t) − δd2 cos(ωmod2 t) + x1 (t) − x2 (t),

.

d2 (t) = d20 + δd1 cos(ωmod1 t) + δd2 cos(ωmod2 t) + x2 (t) − x3 (t).

(4.11)

Here .d10,20 is the equilibrium separation, .δd1,d2 is the modulation amplitude, and ωmod1,2 is the modulation frequency. .x1 (t), .x2 (t), and .x3 (t) are the vibration amplitudes of three cantilevers. The motions of this three-cantilever system are given by

.

m1 x¨1 + m1 γ1 x˙1 + m1 ω12 x1 = FC (d1 (t)),

.

m2 x¨2 + m2 γ2 x˙2 + m2 ω22 x2 = −FC (d1 (t)) + FC (d2 (t)), m3 x¨3 + m3 γ3 x˙3 + m3 ω32 x3 = −FC (d2 (t)).

(4.12)

Fig. 4.4 Parametric coupling scheme in the three-cantilever system. (a) The PSD of the Brownian motion of three cantilevers. (b) Parametric modulation with two frequencies .δd1 cos(ωmod1 t) + δd2 cos(ωmod2 t) is applied to cantilever 2

4.3 Casimir Vibrational Coupling

63

When the separations are far larger than the modulation amplitudes and the oscillation amplitude of three cantilevers (.d10,20  δd1 ,.δd2 ,.|x1 |,.|x2 |,.|x3 |), the Casimir force term .FC (d1,2 ) can be expanded to the second order with respect to the term .−δd1 cos(ωmod1 t) − δd2 cos(ωmod2 t) + x1 (t) − x2 (t) and .δd1 cos(ωmod1 t) + δd2 cos(ωmod2 t) + x2 (t) − x3 (t). In our experiment, we utilize parametric modulation to couple each two cantilevers and the direct coupling is neglected. Similar to the two-body condition, the coupling and energy transfer only come from the second derivative of the Casimir force .FC (d1,2 ). Under the limit of small modulation amplitudes and small oscillation amplitudes of three cantilevers, the equations can be written as .

x¨1 + γ1 x˙1 + ω12 x1 =

1 cos(ωmod1 t)(x1 − x2 ). m1

x¨2 + γ2 x˙2 + ω22 x2 =

1 cos(ωmod1 t)(x2 − x1 ) m2 +

x¨3 + γ3 x˙3 + ω32 x3 =

2 cos(ωmod2 t)(x2 − x3 ). m2

2 cos(ωmod2 t)(x3 − x2 ). m3

(4.13)

2

Here we have . 1,2 = ddxF2C |d10,20 δd1,2 . We now generalize the displacements .x1,2,3 (t) to complex values .z1,2,3 (t) such that .x1,2,3 (t) = Re[z1,2,3 (t)]. We separate the fast-rotating term and the slowvarying term for .z1,2,3 (t) to solve Eq. 4.13 such that z1,2,3 (t) = B1,2,3 (t)e−iω1,2,3 t ,

.

(4.14)

where .B1,2,3 (t) is the slow-varying oscillating component for three cantilevers, and hence, the second derivative terms .B¨ 1,2,3 (t) are negligible. Under the small damping limit such that .γ1,2,3  ω1,2,3 , the equations of motion become .

=

− iω1 γ1 B1 (t)e−iω1 t − 2iω1 B˙ 1 (t)e−iω1 t 1 (B1 (t)e−i(ω1 +ωmod1 )t − B2 (t)e−i(ω2 −ωmod1 )t ), 2m1 − iω2 γ2 B2 (t)e−iω2 t − 2iω2 B˙ 2 (t)e−iω2 t

=

1 (B2 (t)e−i(ω2 −ωmod1 )t − B1 (t)e−i(ω1 +ωmod1 )t ) 2m2 +

2 (B2 (t)e−i(ω2 −ωmod2 )t − B3 (t)e−i(ω3 +ωmod2 )t ), 2m2

− iω3 γ3 B3 (t)e−iω3 t − 2iω3 B˙ 3 (t)e−iω3 t =

2 (B3 (t)e−i(ω3 +ωmod2 )t − B2 (t)e−i(ω2 −ωmod2 )t ), 2m3

(4.15)

64

4 Experimental Realization of a Casimir Transistor: Switching and Amplifying. . .

where the rotating wave approximation is taken and the fast-rotating terms are neglected. We apply the transformation that .B1 (t) = B1 (t), .B2 (t) = B2 (t)eiδ2 t , and  iδ t .B (t) = B3 (t)e 3 , where .δ2 = ω1 +ωmod1 −ω2 and .δ3 = ω1 +ωmod1 −ωmod2 −ω3 . 3 Then, the equations of motion are rewritten as ⎞⎛ ⎞ ⎛ γ ⎞ 1 0 −i 21 B˙1 (t) B  (t) 4m1 ω1 ⎟ ⎝ 1 ⎠ ⎜ ˙  ⎟ ⎜ 1 γ2 2 .i ⎝B (t)⎠ = ⎝ 4m2 ω2 −i 2 − δ2 4m2 ω2 ⎠ B2 (t) . 2 γ3 2  ˙ B3 (t) B3 (t) 0 −i 4m3 ω3 2 − δ3 ⎛

(4.16)

Under the steady condition, .B˙ 1 , .B˙ 2 , and .B˙ 3 all equal to zero. The vibration amplitudes .A1,2,3 are the absolute values of the slow-varying components .B1,2,3 (t) so we can have .A1,2,3 (t) = |B1,2,3 (t)|. When cantilever 1 is driven resonantly and the parametric modulation is on resonance such that .δ2 = δ3 = 0, the ratio of .A3 /A1 is .

A3 B3 1 2 =| |=| |. A1 B1 4m2 m3 ω2 ω3 γ2 γ3 + 22

(4.17)

In the strong coupling regime that .|g23 | = | 2√m m2 ω ω |  |γ2,3 |, the transduction 2 3 2 3 amplitude is approximated to be .

A3 1 = | |. A1 2

(4.18)

Under the weak coupling limit that .|g23 | = | 2√m m2 ω ω |  |γ2,3 |, the transduction 2 3 2 3 amplitude is approximated to be .

A3 1 2 =| |. A1 4m2 m3 ω2 ω3 γ2 γ3

(4.19)

The vibration amplitudes can be quantized as phonons.  By introducing normalized   m3 ω3  m1 ω1  m2 ω2  amplitudes .c1 = h¯ B1 , .c2 = h¯ B2 , and .c3 = h¯ B3 , the equations of motion for the phonon modes will be ⎞⎛ ⎞ ⎛ ⎞ ⎛ γ1 g12 −i 2 c1 c˙1 0 2 γ g g 2 23 12 ⎠ ⎝ ⎠ ⎝ ⎝ .i −i 2 − δ2 c2 ⎠ , c˙2 = 2 2 γ3 g23 0 −i 2 − δ3 c3 c˙3 2

(4.20)

2 √ 1 = ddxF2C |d10 δd1 2√m m1 ω ω , and .g23 = 2√m m2 ω ω 2 m1 m2 ω1 ω2 2 3 2 3 1 2 1 2 d 2 FC 1 √ . Therefore, the effective system Hamiltonian is | δ dx 2 d20 d2 2 m2 m3 ω2 ω3

where .g12 =

=

4.3 Casimir Vibrational Coupling

65

Fig. 4.5 The eigenvalues of the three-cantilever system based on the Hamiltonian in Eq. 4.21. The calculated eigenvalues are shown as a function of .δ3 when .δ2 = 0 and .|g12 | = |g23 | = 2π × 20 Hz

⎞ ⎛ γ1 g12 0 −i 2 2 g12 g23 ⎠. .H = ⎝ −i γ22 − δ2 2 2 γ3 g23 0 −i 2 − δ3 2

(4.21)

The eigenvalues near resonance are shown in Fig. 4.5. We notice that there is a clear two-fold anti-crossing when the detunings meet the condition that .δ3 = δ2 = 0. Here the couplings are set to be .|g12 | = |g23 | = 2π × 20 Hz. Our experimental measurement of level repulsion from the Casimir coupling in the three-cantilever system is shown in Fig. 4.6a, b. The modulation frequency .ωmod1 here is fixed at .ωmod1 = |ω1 − ω2 |. Three branches in the PSD correspond to the hybrid modes of three cantilevers. A clear level repulsion behavior with the dark line is observed in Fig. 4.6a, b. Thus, we have experimentally observed the strong coupling of the three-object system by Casimir interactions. Simulations of the PSD based on Eq. 4.21 are shown in Fig. 4.6c, d, and they are in good agreement with experimental results. The separations in the simulation are .d1 = 88 nm and .d2 = 90 nm. We notice that in a special case that .γ1 = γ2 = γ3 = 0, .g12 and √ = g23 = g, √ .δ2,3 = 0, the eigenvalues of the Hamiltonian are .λ1 = 0, λ2 = g/ 2, λ3 = −g/ 2. The corresponding eigenvectors are ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 1 1 1 ⎝ ⎠ 1⎝ √ ⎠ 1 ⎝√ ⎠ .|e1 = √ 0 , |e2 = − 2 . 2 , |e3 = 2 2 2 −1 1 1

(4.22)

Interestingly, .|e1 corresponds to the case when cantilever 1 and cantilever 3 oscillate with the same amplitude, but cantilever 2 does not oscillate at all. Thus only two modes will show up in the motion of cantilever 2 under these special conditions, as expected in Fig. 4.6b, d where one mode disappears in the PSD.

66

4 Experimental Realization of a Casimir Transistor: Switching and Amplifying. . .

Fig. 4.6 Level repulsion from Casimir coupling in the three-body system. (a), (b) Measured PSD of cantilever 3 and cantilever 2. Here .ωmod1 is fixed at .|ω1 −ω2 | ∼ 440 Hz. Modulation amplitudes are fixed at .δd1 = 10.4 nm and .δd2 = 14.1 nm. .ωmod2 is swept near the resonance. (c), (d) Simulated PSD when the separations are .d1 = 88 nm and .d2 = 90 nm

4.4 Casimir Switch Parametric coupling in the three-object Casimir is controllable with high flexibility and allows us to take more complicated operations. Inspired by the field effect transistor (Fig. 4.7b), we realize a three-channel Casimir system that can switch and amplify energy transfer. The switch function can be easily realized by turning on and off the modulation on cantilever 2. The schematic is shown in Fig. 4.7c. In Fig. 4.7d, e, we drive cantilever 1 resonantly and record the motion of cantilever 1 and cantilever 3. If the modulation is turned on and the modulation frequencies .ωmod1 and .ωmod2 are on resonance, energy can be partially transferred from cantilever 1 to cantilever 3 (Fig. 4.7e). If the modulation is turned off (.δd1 = δd2 = 0), energy cannot be transferred since they are off-resonant (Fig. 4.7d). We also study the frequency dependence of the energy transfer. The measured transduction ratio .A3 /A1 is presented at various modulation frequencies .ωmod2 in Fig. 4.8a. Here the other modulation frequency .ωmod1 is fixed at the resonant frequency, which matches the frequency difference between cantilever 1 and cantilever 2. Cantilever 1 is resonantly driven. The transduction ratio .A3 /A1 can

4.4 Casimir Switch

67

Fig. 4.7 Switching Casimir mediated energy transfer. (a) A symbolic switch. (b) A symbolic field effect transistor. (c) The schematic of the Casimir switching system. The Casimir mediated energy transfer between cantilever 1 and 3 can be switched on and off by controlling the parametric modulation on cantilever 2. (d) Measured time-dependent vibrational motion of cantilever 1 and cantilever 3 when the modulation is off. (e) Measured time-dependent vibrational motion of cantilever 1 and cantilever 3 when the resonant modulation is on. Here ωmod1 = 2π × 465 Hz, ωmod2 = 2π × 1230 Hz, δd1 = 6.0 nm, and δd2 = 8.5 nm. The separations are d1 = 100 nm and d2 = 105 nm.

reach up to 0.44 when both modulations are on resonance, and close to zero when the modulation is off-resonant. Thus we have realized the Casimir switching function of energy transfer with high contrast experimentally. In Fig. 4.8b, the measured transduction ratio .A3 /A1 is shown at various modulation amplitudes .δd1 when .δd2 = 1.42δd1 . The transduction ratio .A3 /A1 increases as .δd1 raises under a resonant coupling condition (blue circles) when the modulation frequencies meet the condition that .ωmod1 = |ω1 − ω2 | and .ωmod2 = |ω3 − ω2 |. Our experimental results are in good agreement with the prediction in Eq. 4.17 and the simulation results from the system equations of motion. We also measured the off-resonance case that .ω1 still equals .|ω1 − ω2 |, but .ω2 is off by about 80 Hz. The transduction ratio .A3 /A1 is nearly zero for the off-resonant case (red diamonds).

68

4 Experimental Realization of a Casimir Transistor: Switching and Amplifying. . .

Fig. 4.8 Properties of the Casimir switch. (a) The transduction ratio A3 /A1 when ωmod1 is fixed at the resonant frequency |ω1 − ω2 | and ωmod2 is swept near resonance. Here cantilever 1 is driven resonantly with frequency ω1 , and we record the vibrational amplitude A1 and A3 . (b) The transduction ratio when the modulation frequencies are fixed at ωmod2 = 2π × 1231 Hz (resonant, blue circles) and ωmod2 = 2π × 1150 Hz (off resonant, red diamonds). ωmod1 is fixed at |ω1 − ω2 | for both resonant and off-resonant cases Fig. 4.9 The scheme of the Casimir amplification. In addition to the resonant parametric modulation, an additional gain G is added to cantilever 2 (center) by PID feedback control

Modulaon on (

)+

(

)

Gain on (G) Driven Casimir coupling

4.5 Casimir Amplifier To realize a high efficiency energy transfer in the three-body system, we apply an additional gain to the system to amplify the energy transfer. The schematic of the Casimir amplification is shown in Fig. 4.9.

4.5.1 External Gain to the System Now we introduce how we add an external gain to amplify energy transfer. In the experiment, we use a PID feedback control (Fig. 4.10a) to modify the effective damping rate of cantilever 2 that is .γ2 = γ20 − G, where .γ20 is the natural damping of cantilever 2 and G is the gain coefficient. When .G < γ20 , .γ2 is still positive. We use the ringdown scheme to calibrate the damping rate under such conditions as shown in Fig. 4.10b. Cantilever 2 is first driven resonantly to a large amplitude.

4.5 Casimir Amplifier

(a) Piezo chips

Feedback control

69

(b)

(c)

(e)

(f)

Single mode fiber

(d)

Fig. 4.10 Additional gain is introduced to cantilever 2 by a PID feedback control loop. (a) Schematic of the three-cantilever and feedback control system. The motion of cantilever 2 is recorded from the fiber interferometer and is sent to the computer. The computer does the derivative of the signal and drive the piezo chips accordingly. (b) and (c) We add a gain to the system such that .γ2 = γ20 − G > 0. (b) We use ringdown scheme for damping rate .γ2 calibration. We first drive cantilever 2 resonantly to a large amplitude. We turn off the driving voltage and turn on the feedback control (gain) at Time .= 0 s. (c) We extract the upper envelope of (b) and fit the amplitude by a function .x2 = A × exp(− γ22 t) and get the value .γ2 . (d) and (e) We add a gain to the system such that .γ2 = γ20 − G < 0. We turn on the feedback control (gain) at Time .= 0 s and fit the envelope with the same function .x2 = A × exp(− γ22 t) to extract the negative damping rate .γ2 . (f) The calibrated damping rate .γ2 is shown at each derivative gain value in the PID control

At Time .= 0, we turn off the driving voltage and record the displacement .x2 . By extracting the envelope of the oscillating displacement and fitting it with a function γ2 .x2 = A × exp(− 2 t), we can calibrate the real damping rate .γ2 under a gain as shown in Fig. 4.10c. When .G > γ20 , the damping rate of cantilever 2 is negative. When we turn on the gain at Time .= 0, the motion of cantilever 2 will be amplified over time as shown in Fig. 4.10d. We fit the envelope with the same function that is γ2 .x2 = A×exp(− 2 t) to extract the damping rate .γ2 here. In the experiment, we add a gain by a derivative function in the PID control. The calibrated .γ2 at each derivative control parameter .Kc ∗ Td /Ts is shown in Fig. 4.10f, where .Kc is the proportional gain, .Td is the derivative time, and .Ts is the PID loop time. In our system, the PID loop rate is 50,000 Hz. We can control .γ2 from .2π × 6 Hz to .−.2π × 6 Hz as shown in Fig. 4.10f. A larger gain will lead to instability of the three-cantilever system.

70

4 Experimental Realization of a Casimir Transistor: Switching and Amplifying. . .

4.5.2 Requirement for the Steady Condition Now we discuss the requirement of the steady condition for the three-body system under an external gain. Equation 4.21 gives the effective Hamiltonian of the system. Here we consider a special case that .g12 = g23 , .γ1 = γ3 , and .δ2,3 = 0. The eigenvalues of Eq. 4.21 are λ1 = −i

.

γ1 , 2



γ1 + γ2 + λ2 = −i 4 γ1 + γ2 − λ3 = −i 4

2 − (γ − γ )2 8g12 1 2



4

2 − (γ − γ )2 8g12 1 2

4

Under the strong coupling condition that .|g12 | > 2 − γ1 +γ 4 .

,

|γ1 √ −γ2 | , 2 2

.

(4.23)

we have .I m(λ2 ) =

The steady state requires that γ1 + γ2 ≥ 0.

(4.24)

.

Under the weak coupling condition that .|g12 | < 

2 + − γ1 +γ 4

2 (γ1 −γ2 )2 −8g12 . 4

we have .I m(λ2 ) =

The steady state requires that

γ1 + γ2 −

.

|γ1 √ −γ2 | , 2 2



2 ≥ 0. (γ1 − γ2 )2 − 8g12

(4.25)

4.5.3 Amplify the Casimir Mediated Energy Transfer When a gain is applied to cantilever 2, the energy possessed by cantilever 1 is first transferred to cantilever 2 and get amplified, and transferred to cantilever 3. We record the time-dependent vibrational amplitude of cantilever 1 and cantilever 3 when an extra gain of .G = 2π × 8.73 Hz is applied to cantilever 2. As shown in Fig. 4.11, we observe the amplification of energy transfer by Casimir force. Other parameters (modulation amplitudes, separations, modulation frequencies) are the same as those in Fig. 4.7e. More properties of our Casimir amplification system are presented in Fig. 4.12. The energy transfer with an extra gain shows a similar modulation frequency and modulation amplitude dependence as the no-gain case (Fig. 4.12a, c). It also shows a striking enhancement of the transduction ratio by more than eight times. A larger gain will also give a higher transduction ratio (Fig. 4.12d) until the system becomes

4.5 Casimir Amplifier

71

Fig. 4.11 Amplifying Casimir mediated energy transfer. An extra gain of .G = 2π × 8.73 Hz is added to cantilever 2. Cantilever 1 is resonantly driven. The energy is transferred to cantilever 3 with amplification. Modulation amplitudes, separations, and modulation frequencies are the same as in Fig. 4.7e

Fig. 4.12 Properties of the Casimir amplification. (a) The transduction ratio .A3 /A1 when .ωmod1 is on resonance and an external gain is added to cantilever 1. No-gain case is also presented as a comparison. (b) Under a resonant coupling condition, the vibrational amplitude .A3 is recorded over different .A1 for three different gain coefficients. (c) The transduction ratio .A3 /A1 is measured at different modulation amplitudes .δd1 and .δd2 . The relation of two modulation amplitudes are maintained at .δd2 = 1.42δd1 . (d) The transduction ratio .A3 /A1 is presented as a function of gain G

72

4 Experimental Realization of a Casimir Transistor: Switching and Amplifying. . .

unstable. In this way, we realize the amplification of energy transfer in a three-object Casimir system. The amplification function can potentially apply to zeptometer metrology [23] and ultrasensitive magnetic gradiometry [24]. In conclusion, we have introduced the first observation of Casimir force between three objects in this chapter. We demonstrate the strong coupling of three micromechanical cantilevers through virtual photons experimentally. We also realize the novel Casimir-based switching and amplification, which will have potential applications in sensing [23, 24] and information processing [25, 26].

References 1. O. Di Stefano, A. Settineri, V. Macri et al., Interaction of mechanical oscillators mediated by the exchange of virtual photon pairs. Phys. Rev. Lett. 122, 030402 (2019). https://doi.org/10. 1103/PhysRevLett.122.030402 2. K.Y. Fong, H.-K. Li, R. Zhao, S. Yang, Y. Wang, X. Zhang, Phonon heat transfer across a vacuum through quantum fluctuations. Nature 576(7786), 243–247 (2019). https://doi.org/10. 1038/s41586-019-1800-4 3. Z. Xu, X. Gao, J. Bang, Z. Jacob, T. Li, Non-reciprocal energy transfer through the Casimir effect. Nat. Nanotechnol. 17(2), 148–152 (2022). https://doi.org/10.1038/s41565-021-010268 4. J.M. Pate, M. Goryachev, R.Y. Chiao, J.E. Sharping, M.E. Tobar, Casimir spring and dilution in macroscopic cavity optomechanics. Nat. Phys. 16, 1117–1122 (2020). https://doi.org/10.1038/ s41567-020-0975-9 5. M. Sparnaay, Measurements of attractive forces between flat plates. Physica 24(6), 751–764 (1958). https://doi.org/10.1016/S0031-8914(58)80090-7 6. G. Bressi, G. Carugno, R. Onofrio, G. Ruoso, Measurement of the Casimir force between parallel metallic surfaces. Phys. Rev. Lett. 88, 041804 (2002). https://doi.org/10.1103/PhysRevLett. 88.041804 7. S.K. Lamoreaux, Demonstration of the Casimir force in the 0.6 to 6 μm range. Phys. Rev. Lett. 78, 5 (1997). https://doi.org/10.1103/PhysRevLett.78.5 8. U. Mohideen, A. Roy, Precision measurement of the Casimir force from 0.1 to 0.9 μm. Phys. Rev. Lett. 81, 4549–4552 (1998). https://doi.org/10.1103/PhysRevLett.81.4549 9. H.B. Chan, V.A. Aksyuk, R.N. Kleiman, D.J. Bishop, F. Capasso, Quantum mechanical actuation of microelectromechanical systems by the Casimir force. Science 291(5510), 1941– 1944 (2001). https://doi.org/10.1126/science.1057984 10. J.N. Munday, F. Capasso, V.A. Parsegian, Measured long-range repulsive Casimir-Lifshitz forces. Nature 457(7226), 170–173 (2009). https://doi.org/10.1038/nature07610 11. J.L. Garrett, D.A.T. Somers, J.N. Munday, Measurement of the Casimir force between two spheres. Phys. Rev. Lett. 120, 040401 (2018). https://doi.org/10.1103/PhysRevLett.120. 040401 12. L. Tang, M. Wang, C.Y. Ng et al., Measurement of non-monotonic Casimir forces between silicon nanostructures. Nat. Photonics 11(2), 97–101 (2017). https://doi.org/10.1038/nphoton. 2016.254 13. H.B.G. Casimir, On the attraction between two perfectly conducting plates. Proceedings 51, 793–795 (1948) 14. L.M. Woods, D.A.R. Dalvit, A. Tkatchenko, P. Rodriguez-Lopez, A.W. Rodriguez, R. Podgornik, Materials perspective on Casimir and van der Waals interactions. Rev. Mod. Phys. 88, 045003 (2016). https://doi.org/10.1103/RevModPhys.88.045003

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15. T. Gong, M.R. Corrado, A.R. Mahbub, C. Shelden, J.N. Munday, Recent progress in engineering the Casimir effect – applications to nanophotonics, nanomechanics, and chemistry. Nanophotonics 10(1), 523–536 (2021). https://doi.org/10.1515/nanoph-2020-0425 16. Z. Xu, P. Ju, X. Gao, K. Shen, Z. Jacob, T. Li, Observation and control of Casimir effects in a sphere-plate-sphere system. Nat. Commun. 13(1), 6148 (2022). ISSN: 2041-1723. https://doi. org/10.1038/s41467-022-33915-4 17. E.M. Lifshitz, The theory of molecular attractive forces between solids. Sov. Phys. JETP 2, 73–83 (1956). 18. R. Messina, M. Antezza, Three-body radiative heat transfer and Casimir-Lifshitz force out of thermal equilibrium for arbitrary bodies. Phys. Rev. A 89, 052104 (2014). https://doi.org/10. 1103/PhysRevA.89.052104 19. K.A. Milton, E.K. Abalo, P. Parashar et al., Three-body effects in Casimir-Polder repulsion. Phys. Rev. A 91, 042510 (2015). https://doi.org/10.1103/PhysRevA.91.042510 20. L. Ge, X. Shi, Z. Xu, K. Gong, Tunable Casimir equilibria with phase change materials: from quantum trapping to its release. Phys. Rev. B 101, 104107 (2020). https://doi.org/10.1103/ PhysRevB.101.104107 21. T. Zhan, X. Shi, Y. Dai, X. Liu, J. Zi, Transfer matrix method for optics in graphene layers. J. Phys. Condens. Matter. 25(21), 215301 (2013). https://doi.org/10.1088/0953-8984/25/21/ 215301 22. P. Huang, P. Wang, J. Zhou et al., Demonstration of motion transduction based on parametrically coupled mechanical resonators. Phys. Rev. Lett. 110, 227202 (2013). https://doi.org/10. 1103/PhysRevLett.110.227202 23. J. Javor, M. Imboden, A. Stange, Z. Yao, D.K. Campbell, D.J. Bishop, Zeptometer metrology using the Casimir effect. J. Low Temp. Phys. 1–13 (2022). https://doi.org/10.1007/s10909021-02650-3 24. J. Javor, Z. Yao, M. Imboden, D.K. Campbell, D.J. Bishop, Analysis of a Casimir-driven parametric amplifier with resilience to Casimir pull-in for MEMS single point magnetic gradiometry. Microsyst. Nanoeng. 7(1), 73 (2021). https://doi.org/10.1038/s41378-021-00289-4 25. G. Benenti, A. D’Arrigo, S. Siccardi, G. Strini, Dynamical Casimir effect in quantuminformation processing. Phys. Rev. A 90, 052313 (2014). https://doi.org/10.1103/PhysRevA. 90.052313 26. X.-F. Liu, Y. Li, H. Jing, Casimir switch: Steering optical transparency with vacuum forces. Sci. Rep. 6(1), 27102 (2016). https://doi.org/10.1038/srep27102

Chapter 5

Proposal on Detecting Rotational Quantum Vacuum Friction

Abstract Quantum friction predicts that two neutral bodies with a relative motion will experience a friction force due to quantum vacuum fluctuations. However, quantum vacuum friction has never been observed experimentally due to its small amplitude. In this chapter, we propose to detect the rotational vacuum friction torque by a levitated nanorotor near a surface. We will first introduce our ultrasensitive torque sensor with an unprecedented torque sensitivity. After that, we will present the calculation of rotational quantum vacuum friction on a rotating silica sphere near a silica surface. Compared to the torque sensitivity, our system will be able to detect the rotational vacuum friction torque. Besides, we also investigate on the rotational vacuum friction torque for a barium strontium titanate (BST) system. The rotational vacuum friction torque can be enhanced by several orders at a rotating frequency around GHz because of the resonant photon tunneling. Parts of the contents in this chapter have been published in Ahn et al. (Nat Nanotechnol 15(2):89–93, 2020) and Xu et al. (Nanophotonics 10(1):537–543, 2021).

5.1 Ultrasensitive Torque Sensor Experimentally, we can levitate a silica nanoparticle optically in vacuum by a circularly polarized 1550-nm laser [1] as shown in Fig. 5.1a. We can control the laser polarization with a quarter waveplate (.λ/4 waveplate). A 1020-nm laser is applied to add an external torque on the levitated nanoparticle. The trapping laser will pass a collimation lens and will be directed to balanced photodetectors to monitor the center-of-mass motion, the torsional motion, and the rotational motion of the levitated silica nanoparticle. If there is an external torque on the nanorotor, the rotating frequency will shift. We can measure the external torque by recording the rotating frequency shift. Experimentally, we are able to achieve a torque sensitivity of .4.3×10−27 Nm Hz.−1/2 (Fig. 5.1b) at a pressure of .1.3×10−5 torr [1]. Previously, the state-of-the-art torque sensitivity is realized in a nanofabricated torque sensor at low temperature [3]. The torque sensitivity realized by our room-temperature levitation system is several orders improved compared to the previous one.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 Z. Xu, Optomechanics with Quantum Vacuum Fluctuations, Springer Theses, https://doi.org/10.1007/978-3-031-43052-7_5

75

76

5 Proposal on Detecting Rotational Quantum Vacuum Friction

(a)

1020 nm

DM

OBJ NP

Lens

/2

DET

/4 /4

PBS

(b)

Torque Sensitivity (Nm/ Hz)

1550 nm

8x10-27 4x10-27 0

0.1

1

Frequency (Hz)

10

Fig. 5.1 Optical levitation setup and measured torque sensitivity. (a) A silica nanoparticle is optically levitated in vacuum with a tightly focused and circularly polarized 1550-nm laser beam. A 1020-nm is also directed to the nanoparticle to add an external torque. Silica nanospheres and nanodumbbells can be stably levitated √ by this system (inset of (a)) The scale bar here is 200 nm. (b) Measured torque sensitivity . ST obtained from the rotational PSD when there is no external torque. The yellow dashed line indicates a torque sensitivity of .4.3 × 10−27 Nm Hz.−1/2 by averaging the PSD in the frequency range

5.2 Rotational Vacuum Friction Torque on a Silica Nanosphere Near a Silica Surface We can use the ultrasensitive torque sensor to detect the rotational vacuum friction torque [4–7]. A fast-rotating neutral nanoparticle in vacuum will experience vacuum friction because of the interaction with the fluctuated electromagnetic fields [5]. The vacuum friction torque is extremely weak in free space but can be enhanced by several orders when the particle is placed near a surface because of a large LODS [6, 7]. Although several theoretical studies have predicted the existence rotational vacuum friction, the experimental observation of rotational vacuum friction has not been realized yet. In this section, we will introduce calculations of rotational vacuum friction and show that the friction torque on a silica nanosphere with a rotation frequency of 1 GHz near a silica surface will be detectable under realistic conditions. Assume that there is a silica sphere of radius R rotating at a frequency . along z-axis near a planar surface as shown in Fig. 5.2. The separation between the center

5.2 Rotational Vacuum Friction Torque on a Silica Nanosphere Near a Silica. . . Fig. 5.2 The schematic of a sphere rotating near a surface

77

z

1

Ω

d

x

y

0

of the sphere and the surface is d. The temperatures of the sphere and the surface are T1 and .T0 , respectively. Adapting the theory presented in [5, 6], the vacuum friction torque on the sphere along the z-axis is

.

Mvacuum = −

.



2h¯ π

∞ −∞

¯ [n1 (ω − ) − n0 (ω)]Im{α(ω − )}ImG(ω)dω,

(5.1)

where .nj (ω) = [exp(h¯ ω/kB Tj ) − 1]−1 is the Bose–Einstein distribution function at a temperature .Tj . .j = 0, 1 stands for the plate and the sphere. .α(ω) is the sphere ¯ polarizability, and .G(ω) is the green function to connect the dipole moment on the sphere and the induced electromagnetic field on the surface. ¯ The Green function used in the calculation is .G(ω) = [Gxx (ω) + Gyy (ω)]/2, where .Gij is the electromagnetic Green tensor. When R and d are much smaller than the wavelength of the electromagnetic fluctuation, the Green tensor components for this near planar surface geometry are obtained as Gxx (ω) = Gzz (ω) = Gyy (ω)/2 =

.

1 4π 0 (2d)3



 sub (ω) − 1 , sub (ω) + 1

(5.2)

where .sub (ω) is the dielectric function of the substrate material. The polarizability of the sphere is given as  α = 4π 0 R

.

3

 sp (ω) − 1 . sp (ω) + 2

(5.3)

sp (ω) is the dielectric function of the sphere. The dielectric function here is written as a summation over multiple classical harmonic oscillators such that

.

 (ω) = ∞ 1 +

.



2 − ω2 ωLO TO

ωT2 O − ω2 − j ωγ

 ,

(5.4)

where .ωLO is the longitudinal polar-optic phonon frequency, .ωT O is the transverse polar-optic phonon frequency, .γ is the damping coefficient, and .∞ is the permittivity in the high-frequency limit. The optical phonon parameters for SiO.2 , Si.3 N.4 , and SiC can be obtained from [7, 8], and they are shown in Table 5.1. Only the

78

5 Proposal on Detecting Rotational Quantum Vacuum Friction

Table 5.1 Optical phonon frequencies of three common phonon polaritonic materials Material SiO.2

.∞

Si.3 N.4

3.90

SiC

6.70

2.09

[cm.−1 ] 1186 1112 1106 814 525 948 962 1084 1710 21162 969

.ωLO

[cm.−1 ] 1167 1064 1058 799 434 826 925 1060 1189 17873 793

.ωT O

[cm.−1 ] 4.43 15.53 0.42 12.94 54.14 6 10 6 3253 0 4.76

.γ

Reference [8]

[8]

[7]

Fig. 5.3 The dielectric properties of different materials. (a). The imaginary part of the polarizability of SiO.2 nanosphere with a radius of 75 nm. (b). The imaginary part of the Green function ¯ of the connection between the SiO.2 sphere and three different surface materials. The red .G(ω) solid curve is for the SiO.2 surface, while the black dashed line and the blue dotted line are for the Si.3 N.4 SiC, respectively

imaginary parts of the polarizability and the green function contribute to the vacuum friction. Here we assume that the sphere is made of silica, which is the case for our experiment, while the substrate can be made of various materials. Three common phonon polaritonic materials are used in the calculation. The imaginary part of the polarizability of SiO.2 nanosphere with a radius of 75 nm is shown in Fig. 5.3a. ¯ Similarly, the imaginary part of the Green function .G(ω) connecting between the sphere and the SiO.2 , Si.3 N.4 , and SiC surfaces is shown in Fig. 5.3b. Note that the resonance peak frequencies are around .1014 rad/s for three materials. The torque on a silica nanosphere near a silica surface at different rotating frequencies is shown in Fig. 5.4. The friction torque .Mvacuum is proportional to the rotating frequency. The torque on a silica sphere near three different surfaces is shown in Fig. 5.5a. We notice that a silica surface will generate the largest rotational vacuum friction torque on a silica nanosphere because their phonon polariton modes

5.2 Rotational Vacuum Friction Torque on a Silica Nanosphere Near a Silica. . .

79

Fig. 5.4 Calculated rotational vacuum friction torque is shown as a function of rotating frequency. The temperatures of the sphere and the plate are assumed to be 1000 K

(a)

(b) -27

200 nm SiO2

-28

10

Torque (Nm)

Torque (Nm)

10

Si3 N4

10-29 10-30

SiC

10-31 200

300

400

500

600

700

10-27

300 nm 10−9 Torr

10-28

10-29

10−10 Torr 200

Separation (nm)

400

600

800

1000

Temperature (K)

Fig. 5.5 Calculated rotational vacuum friction torque on a fast-rotating silica nanosphere. (a). Calculated rotational vacuum friction torque is presented as a function of the separation. The silica nanosphere has a radius of 75 nm and a rotating frequency of 1 GHz. The temperature for both the sphere and the surface is 1000 K. (b). Comparison of the air damping torque and the rotational vacuum friction torque for different temperatures

match. The rotational vacuum friction torque can reach up to .10−27 Nm when the separation is smaller than 300 nm. This calculated friction torque is comparable to the minimum detectable torque demonstrated by our levitation system. Smaller torques can be detected at a lower pressure and for a longer measurement time. We also discuss the effect of air damping on the rotational vacuum friction torque detection here. Here the air damping torque on a single sphere is [9]

Mair

.

πpD 4  = 11.976

 2mgas , π kB T

(5.5)

where D is the diameter of the sphere, . is the angular rotating frequency, p is the air pressure, .mgas = 4.6 × 10−26 kg is the air molecule mass, and T is the temperature of the surrounding air molecules. The vacuum friction torque and the air damping torque are plotted in Fig. 5.5b. As the temperature increases, the rotational vacuum friction torque goes up and the air drag torque goes down. Here the air pressure is assumed to be constant. At a pressure of .10−9 torr, the rotational vacuum friction torque is larger than the air damping when .T > 300K and

80

5 Proposal on Detecting Rotational Quantum Vacuum Friction

d = 200 nm, or when .T > 590K and .d = 300 nm (Fig. 5.5b). These temperatures and pressures are realistic to reach experimentally [10, 11]. Near surface optical levitation has been realized in vacuum with a separation less than 400 nm [12, 13]. It is feasible to observe the rotational vacuum friction torque by our levitated fastrotating nanosphere under realistic conditions.

.

5.3 Enhancement of Rotational Vacuum Friction Torque by Surface Photon Tunneling In this section, we propose to detect the rotational vacuum friction torque on a fast-rotating barium strontium titanate (BST) near a BST surface. BST is a perovskite ferroelectric material that can have a large dielectric resonance at GHz frequency [14] that can be resonated with the mechanical mode of GHz rotation. This will lead to an enhancement of rotational friction by resonant photon tunneling. There are two resonances corresponding to the normal and anomalous Doppler effect in the rotating frequency domain [15–17]. The resonant tunneling is realized when the rotating frequency of the nanoparticle matches the sum or the difference of the surface plasmon frequency and surface phonon polariton frequency. At resonant conditions, the mechanical energy can be converted into the electromagnetic energy. Our levitated sphere has been rotated beyond 5 GHz in vacuum [1, 18] and hence can match the polariton frequency of BST. In this way, the rotational vacuum friction torque can be enhanced. The calculation of the vacuum friction torque on a BST sphere is still based on Eq. 5.1. The dielectric functions can be described by Lorentz oscillator model, that is,   2 − ω2  ωL T .(ω) = ∞ (5.6) 1+ , ωT2 − ω2 − i ω where .∞ = 2.896, .ωL = 1.3 × 1010 s.−1 , .ωT = 5.7 × 109 s.−1 , and . = 2.8 × 108 s.−1 [14]. The rotational vacuum friction torque will be enhanced significantly when the rotating frequency . meets the condition . = |ω1p ± ω2p |. Here .ω1p and .ω2p are surface polariton frequencies that meet the conditions .Re((ω1p )) = −2 and 10 s.−1 .Re((ω2p )) = −1. Two polariton frequencies for BST are .ω1p = 1.06 × 10 10 −1 and .ω2p = 1.15 × 10 s. . For the BST system, the resonant rotating frequency will be . = |ω1 − ω2 | = 2π × 150 MHz and . = ω1 + ω2 = 2π × 3.52 GHz. The calculated imaginary Green function for the BST system is shown in Fig. 5.6a. As a comparison, the imaginary part of the Green function for a silica surface is presented in Fig. 5.6b. The surface polariton frequency of a silica surface [8] is several orders larger than the surface polariton frequency of BST. To meet the resonant photon tunneling condition for the silica system, we need to reach a rotating frequency of 13 Hz that is far beyond the record of the experiments. .10

5.3 Enhancement of Rotational Vacuum Friction Torque by Surface Photon. . .

(a)

102

(b) BST

10-2

-4

107

102 SiO2

100

100

10

81

108

109

10

-2

10

-4

10

-6

1010

1010

Freuqency (Hz)

1011

1012

1013

1014

1015

Frequency (Hz)

Fig. 5.6 Green function for a BST system and a silica system. (a). The imaginary part of the Green function (Eq. 5.2) that can connect a BST surface and a nanosphere. (b). The imaginary part of the Green function that can connect a SiO.2 surface and a nanosphere

(b) 10

-22

10

-24

(c) BST sphere and BST plate SiO2 sphere and SiO2 plate

10

10-26

-28

10-28

10-30 10

SiO2 sphere and SiO2 plate

10-24

10-26 10

-22

Torque (Nm)

Torque (Nm)

(a)

7

10

8

10

9

Rotating frequency (Hz)

10

10

10-30 10 10

10

11

10

12

10

13

10

14

Rotating frequency (Hz)

10

15

Fig. 5.7 Calculated rotational vacuum friction on a BST/silica sphere near a BST/silica plate. (a). The rotational vacuum friction torque is presented for different separations. The rotating frequency is 150 MHz. (b). The rotational vacuum friction torque is presented at different rotating frequencies. The separation here is 300 nm. The resonant rotating frequencies are 150 MHz and 3.52 GHz. (c). The rotational vacuum friction torque on a SiO.2 sphere near a SiO.2 surface is presented at a higher frequency range. The resonance occurs at rotating frequency beyond .1013 Hz. The separation is 300 nm

The calculated rotational vacuum friction torque for the BST and silica system at different rotating frequencies is shown in Fig. 5.7b. The separation here is 300 nm. The sphere radius is 75 nm, and the temperature for both sphere and plate is 300 K. For the BST system, two resonances occur at rotating frequencies 150 MHz and .3.52 GHz. The resonance enhances the torque by several orders, compared to the non-resonant part. In particular, the torque can reach .8.06 × 10−23 Nm at 3.52 GHz. For the silica system, the vacuum friction torque is much lower than the one for BST. This is explained by a much lower surface polariton frequency for BST. At 150 MHz and .3.52 GHz, we notice an enhancement of more than five orders by comparing torque for BST and for silica (Fig. 5.7b). We also present the calculation of rotational vacuum friction torque for the silica system at a higher frequency range in Fig. 5.7c. Due to resonant photon tunneling,

82

5 Proposal on Detecting Rotational Quantum Vacuum Friction

we also notice resonant peaks for the silica case. When the resonance occurs, the torque is also significantly enhanced compared to the non-resonant case. At around 100 THz, the vacuum friction torque can reach up to .10−22 Nm. However, this rotating frequency is far beyond the current experimental limit. Now we fix the rotating frequency at 150 MHz. The friction torque for a BST system and a silica system at this frequency is shown in Fig. 5.7a. The enhancement is calculated to be more than five orders over all separations. As mentioned above, our ultrasensitive torque detection experiment has demonstrated a minimum detectable torque of .5 × 10−28 Nm with a levitated nanorotor at .10−5 torr [1]. At a distance of 300 nm and at a rotating frequency of 150 MHz, the rotational vacuum friction torque on a BST sphere is more than four orders larger than the minimum detectable torque. Similar to the previous case, we also need to consider the air damping torque. We can easily estimate the air damping torque by Eq. 5.5. At 300 K, the air damping torque on a 75-nm-radius sphere is −24 Nm when the rotating frequency is 150 MHz and the pressure is .10−4 .4.5 × 10 torr. This air drag friction torque is one order smaller than the rotational vacuum friction torque. .10−4 torr is a moderate pressure that can be easily realized for levitated nanosphere system. Compared to the silica system, a BST system can significantly relax the strict requirement of pressure. In conclusion, we propose to detect the rotational vacuum friction torque by our ultrasensitive torque sensor. We have introduced our experimental setup and the calculation of vacuum friction torque on a rotating silica sphere near a silica surface. We also show the enhancement of rotational vacuum friction torque for a BST system. In the future, it will be interesting to investigate on the rotational friction torque between more complex metamaterials [19] and topological materials [20].

References 1. J. Ahn, Z. Xu, J. Bang, P. Ju, X. Gao, T. Li, Ultrasensitive torque detection with an optically levitated nanorotor. Nat. Nanotechnol. 15(2), 89–93 (2020). https://doi.org/10.1038/s41565019-0605-9 2. Z. Xu, Z. Jacob, T. Li, Enhancement of rotational vacuum friction by surface photon tunneling. Nanophotonics 10(1), 537–543 (2021). https://doi.org/10.1515/nanoph-2020-0391 3. P.H. Kim, B.D. Hauer, C. Doolin, F. Souris, J.P. Davis, Approaching the standard quantum limit of mechanical torque sensing. Nat. Commun. 7(1), 13165 (2016). https://doi.org/10.1038/ ncomms13165 4. M. Kardar, R. Golestanian, The “friction” of vacuum, and other fluctuation-induced forces. Rev. Mod. Phys. 71, 1233–1245 (1999). https://doi.org/10.1103/RevModPhys.71.1233 5. A. Manjavacas, F.J. Garcia de Abajo, Vacuum friction in rotating particles. Phys. Rev. Lett. 105, 113601 (2010). https://doi.org/10.1103/PhysRevLett.105.113601 6. R. Zhao, A. Manjavacas, F.J. Garcia de Abajo, J.B. Pendry, Rotational quantum friction. Phys. Rev. Lett. 109, 123604 (2012). https://doi.org/10.1103/PhysRevLett.109.123604 7. A. Manjavacas, F.J. Rodriguez-Fortuno, F.J. Garcia de Abajo, A.V. Zayats, Lateral Casimir force on a rotating particle near a planar surface. Phys. Rev. Lett. 118, 133605 (2017). https:// doi.org/10.1103/PhysRevLett.118.133605

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8. J. Kischkat, S. Peters, B. Gruska et al., Mid-infrared optical properties of thin films of aluminum oxide, titanium dioxide, silicon dioxide, aluminum nitride, and silicon nitride. Appl. Opt. 51(28), 6789–6798 (2012). https://doi.org/10.1364/AO.51.006789 9. J. Corson, G.W. Mulholland, M.R. Zachariah, Calculating the rotational friction coefficient of fractal aerosol particles in the transition regime using extended Kirkwood–Riseman theory. Phys. Rev. E 96, 013110 (2017). https://doi.org/10.1103/PhysRevE.96.013110 10. B.R. Slezak, C.W. Lewandowski, J.-F. Hsu, B. D’Urso, Cooling the motion of a silica microsphere in a magneto-gravitational trap in ultra-high vacuum. New J. Phys. 20(6), 063028 (2018). https://doi.org/10.1088/1367-2630/aacac1 11. F. Tebbenjohanns, M. Frimmer, A. Militaru, V. Jain, L. Novotny, Cold damping of an optically levitated nanoparticle to microkelvin temperatures. Phys. Rev. Lett. 122, 223601 (2019). https://doi.org/10.1103/PhysRevLett.122.223601 12. R. Diehl, E. Hebestreit, R. Reimann, F. Tebbenjohanns, M. Frimmer, L. Novotny, Optical levitation and feedback cooling of a nanoparticle at subwavelength distances from a membrane. Phys. Rev. A 98, 013851 (2018). https://doi.org/10.1103/PhysRevA.98.013851 13. L. Magrini, R.A. Norte, R. Riedinger et al., Near-field coupling of a levitated nanoparticle to a photonic crystal cavity. Optica 5(12), 1597–1602 (2018). https://doi.org/10.1364/OPTICA.5. 001597 14. A.O. Turky, M. Mohamed Rashad, A.E.-H. Taha Kandil, M. Bechelany, Tuning the optical, electrical and magnetic properties of Ba0.5Sr0.5TixM1-xo3 (BST) nanopowders. Phys. Chem. Chem. Phys. 17, 12553–12560 (2015). https://doi.org/10.1039/C5CP00319A 15. Y. Guo, Z. Jacob, Singular evanescent wave resonances in moving media. Opt. Express 22(21), 26193–26202 (2014). https://doi.org/10.1364/OE.22.026193 16. Y. Guo, Z. Jacob, Giant non-equilibrium vacuum friction: role of singular evanescent wave resonances in moving media. J. Opt. 16(11), 114023 (2014). https://doi.org/10.1088/20408978/16/11/114023 17. A. Volokitin, Resonant photon emission during relative sliding of two dielectric plates. Mod. Phys. Lett. A 35(03), 2040011 (2020). https://doi.org/10.1142/S0217732320400118 18. J. Ahn, Z. Xu, J. Bang et al., Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor. Phys. Rev. Lett. 121, 033603 (2018). https://doi.org/10.1103/ PhysRevLett.121.033603 19. X. Ni, Z. J. Wong, M. Mrejen, Y. Wang, X. Zhang, An ultrathin invisibility skin cloak for visible light. Science 349(6254), 1310–1314 (2015). https://doi.org/10.1126/science.aac9411 20. Q.-D. Jiang, F. Wilczek, Quantum atmospherics for materials diagnosis. Phys. Rev. B 99, 201104 (2019). https://doi.org/10.1103/PhysRevB.99.201104

Chapter 6

Proposal on Detecting Casimir Torque

Abstract Many efforts were made to improve the precision in the Casimir force measurement and to harness the Casimir effect in various devices. Similar to the Casimir force that comes from the linear momentum of virtual photons, virtual photons also carry angular momentum, and this induces the Casimir torque between anisotropic materials. In spite of significant interests and theoretical proposals toward detecting Casimir torque, there is only one reported experimental observation of Casimir torque in 2018. The measurement was conducted between a liquid crystal and a solid birefringent crystal. To ensure the parallelism of two surfaces, the vacuum gap is replaced by an isotropic material. To our knowledge, Casimir torque between two objects with a vacuum gap has never been detected yet. In this chapter, we propose a scheme to detect the Casimir torque by our ultrasensitive optical levitation system. We will present the calculation of the Casimir torque between a levitated nanorod and a birefringent surface at a sub-wavelength distance. Compared to the torque sensitivity of the levitation system, we will demonstrate that it is promising to detect the Casimir torque across a vacuum gap in the near future, and our levitation setup is expected to provide a better accuracy for the Casimir torque measurement. Parts of the contents in this chapter have been published in Xu and Li (Phys Rev A 96:033843, 2017).

6.1 Schematic Illustration The optically levitated nanoparticle system can have an extremely high-quality factor since it is well-isolated from the thermal environment. Therefore, it is a good platform for precision measurement. Here we propose to measure the Casimir torque on a levitated nanorod when a birefringent surface is placed at a close distance (Fig. 6.1a). The nanorod will be levitated by a linearly polarized 1064-nm laser beam, as shown in Fig. 6.1c. The nanorod will experience a large optical torque from the trapping laser beam and attempt to stay along the polarization direction. If there is a non-zero angle between the long axis of the nanorod and the optical axis of the birefringent plate, the nanorod will experience the Casimir torque (Fig. 6.1a,b).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 Z. Xu, Optomechanics with Quantum Vacuum Fluctuations, Springer Theses, https://doi.org/10.1007/978-3-031-43052-7_6

85

86

6 Proposal on Detecting Casimir Torque

Fig. 6.1 Schematics of the Casimir torque detection system. (a) A silica nanorod is levitated in a linearly polarized optical tweezer. A birefringent plate is placed at a close distance. The measurement will be operated in vacuum. (b) A non-zero Casimir torque will exist on the nanorod when there is a relative angle .θ. (c) The levitated nanorod torsional motion detection scheme

This Casimir torque can be measured by optically detecting the nanorod’s torsional vibration and the nanorod’s orientation relative to the laser polarization (Fig. 6.1c).

6.2 Trapping Potential of the Nanorod A silica nanorod is trapped by a linearly polarized 1064-nm laser beam in vacuum. The length is .l = 200 nm, and the radius is .a = 20 nm. Under the paraxial approximation, the electric field of the laser beam is given by Ex (x, y, z) = E0

.

    x2 + y2 −(x 2 + y 2 ) ω0 exp ikz + ik − iζ (z) ,. exp 2R(z) ω(z) [ω(z)]2 (6.1) Ey (x, y, z) = Ez (x, y, z) = 0,

(6.2)

where z is the axial distance from the focus of the laser beam, and x and y are the coordinates in the plane perpendicular to the propagation direction of the beam. .E0 is the amplitude of the electric field at the origin, and .R(z), .ω(z) and .ζ (z) are the parameters for a Gaussian beam. If the aspect ratio of the nanorod is large enough (.l/a  1), the polarizability will be .α = V 0 (r − 1) and .α⊥ = 2V 0 (r − 1)/(r + 1) [23]. The absorption of silica at the laser wavelength 1064 nm is negligible, so we can assume the permittivity .r to be a real value. Rayleigh approximation can be applied when the wavelength of the laser is far larger

6.2 Trapping Potential of the Nanorod

87

than the size of the nanorod. In the presence of the laser beam, the induced dipole on the nanorod will be .p = αx Ex xˆ N + αy Ey yˆ N + αz Ez zˆ N . The nanorod will attempt to align with the laser beam polarization. If the vibrational amplitude is small, the vibrations along different directions can be uncoupled, and the optical potential will be harmonic around the laser focus. Here we investigate on the optical potential for the center-of-mass z motion and the rotation around z-axis. The potential is written as U (z, φ) = −

.

1 [α − (α − α⊥ ) sin2 φ]Ilaser (z) . 2c0

(6.3)

Here .0 is the vacuum permittivity, c is the speed of light, .φ is the angle between the nanorod and the polarization of the laser beam, and .Ilaser (z) is the laser intensity at nanorod position. .Ilaser = P k02 NA2 /(2π ) is the laser intensity at the focus. P is the laser power and .k0 is the magnitude of the wave vector of the laser beam. For the calculation, in this chapter, we assume the numerical aperture to be NA.= 0.85. Here, we also need to take into account the reflection of the birefringent plate, which will combine with the incident wave to create a standing wave and increase the trapping potential at the position of .λ/4 (Fig. 6.1). We assume that the center of the laser beam is .λ/4 = 266 nm away from the birefringent surface. The refractive index for vacuum is .n0 = 1. The refractive indices for the birefringent crystal .BaTiO3 along two optical axes are .no = 2.269 and .ne = 2.305 at 1064 nm [24]. Therefore, the reflectances are .Ro = 0.16 and .Re = 0.15 if the trapping laser propagation direction is normal to the surface. If the NA is 0.85, the angular aperture will be .58◦ so parts of the light are not normal to the surface. The surface of a birefringent crystal has a wide range of incident angles so the reflectance will depend on the position and orientation. Besides, 50% of the incident laser beam will be p-polarized and the other 50% will be s-polarized. The average reflectance at the maximum incident angle .58◦ is 0.19, which is not far away from .Ro = 0.16 and .Re = 0.15. For simplicity, we use approximated .Ro = 0.16 and .Re = 0.15 in the simulation. The polarization of the laser beam is set at .π/4 relative to the optical axis of the birefringent plate. When the nanorod is aligned with the polarization of the laser beam (.φ ≈ 0), the potential near .z = 0 (or .d = d0 ) is ω02 1 U (z) = U (d − d0 ) ≈ − α E02 [ 4 [ω(d − d0 )]2    ω02 ω02 Ro + Re ], Ro + Re cos(2kd) + + 2 ω(d − d0 )ω(d + d0 ) 2 [ω(d + d0 )] (6.4) .

where .d0 = 266 nm, .P = 100 mW, and the waist radius .w0 is 400 nm. The calculation of the trapping potential based on Eq. 6.4 is shown in Fig. 6.2a. The calculated trapping potential is around .2.2 × 104 K, which is deep enough to prevent losing the levitated nanorod from thermal motions at 300 K.

88

6 Proposal on Detecting Casimir Torque

0

× 104 0

-20

Casimir

-30 100

300

F(N)

U/kB (K)

(a) -1

G/k (K) B

10-14 -10

(b)

Barium titanate Calcite

10-16

500

d(nm)

-2

100

300

500

d(nm)

10-18 100

300

500

d(nm)

Fig. 6.2 Calculated trapping potential and Casimir free energy. (a) Calculated trapping potential is shown as a function of the distance d. Inset: Calculated Casimir free energy is shown as a function of the distance d. The angle between the nanorod and the birefringent plate is .θ = π/4. The substrate is made of barium titanate. (b) Calculated Casimir force for two birefringent materials is shown as a function of distance d when the relative orientation is at .θ = π/4

6.3 Calculation of Casimir Torque and Casimir Force To calculate the Casimir torque, we follow the methods in Refs. [15, 25]. The Casimir free energy .g(d, θ ) per unit length of the cylinder, between a single cylinder and a half-space plate, is [15] 

2π ∞ ∞ kB T a 2 −2dρ3 N ,  QdQ dφ e .g(d, θ ) = 4π D 0 0

(6.5)

n=0

where    − ⊥ {Q2 sin2 (φ + θ ) × [f˜(φ)1,⊥ (Q2 sin2 φ(ρ1,⊥ + ρ3 ) .N = 2 + ρ1,⊥ ρ3 (ρ3 − ρ1,⊥ )) + (1,⊥ − 3 )(ρ3 (ρ1,⊥ + 2ρ3 ) − Q2 )] − 2f˜(φ)1,⊥ ρ1,⊥ ρ32 [2Q2 sin φ cos θ sin(φ + θ ) + ρ32 sin2 θ ] + f˜(φ)1,⊥ ρ32 [Q2 sin2 φ(ρ1,⊥ − ρ3 ) + ρ1,⊥ ρ3 (ρ1,⊥ + ρ3 )] + ρ32 (3 − 1,⊥ )(Q2 + ρ1,⊥ ρ3 )} + 2f˜(φ)⊥ 1,⊥ [Q2 sin2 φ(Q2 ρ1,⊥ − ρ33 ) + ρ1,⊥ ρ32 (Q2 cos(2φ) + ρ1,⊥ ρ3 )] − ⊥ (1,⊥ − 3 ) × [(Q2 + ρ32 )(Q2 + ρ1,⊥ ρ3 ) + (Q2 − ρ32 )(Q2 − ρ1,⊥ ρ3 )] (6.6) and D = ρ3 (ρ1,⊥ + ρ3 ){1,⊥ f˜(φ)[Q2 sin2 φ − ρ1,⊥ ρ3 ] + 1,⊥ ρ3 + 3 ρ1,⊥ } . (6.7)

.

6.3 Calculation of Casimir Torque and Casimir Force

89

In the equations above,

ρ1,⊥ =

.

Q2 +

1,⊥ ωn2 , c2

(6.8)

Q2 +

3 ωn2 , c2

(6.9)

ρ3 =

.

f˜(φ) =

 2 −ρ Q2 ((1, /1,⊥ ) − 1) cos2 φ + ρ1,⊥ 1,⊥

.

2 Q2 sin2 φ − ρ1,⊥

.

(6.10)

Here we have .⊥ = (2,⊥ − 3 )/(2,⊥ + 3 ), . = (2, − 3 )/3 . T is the temperature of environment, .3 is the dielectric function of the isotropic medium between the cylinder and the half-space, .1,⊥ and .1, are the dielectric functions of the birefringent material, and .2,⊥ and .2, are the dielectric functions of the cylinder material. We use the imaginary Matsubara frequencies .ωn = 2π kh¯B T n for the calculation in Eq. 6.5, similar to the calculation of Casimir force in Sect. 2.4. All the frequency response of the dielectric functions can be replaced by the discrete imaginary Matsubara frequencies, i.e., as .3 ≡ 3(n) = 3 (iωn ), .1,⊥ (iωn ), .1, (iωn ), .2,⊥ (iωn ), and .2, (iωn ). For most inorganic materials, the dielectric function can be described by two undamped oscillators such that [26–28] (iξ ) = 1 +

.

CI R CU V + , 2 1 + (ξ/ωI R ) 1 + (ξ/ωU V )2

(6.11)

where .ωI R and .ωU V are the characteristic absorption angular frequencies in the infrared and ultraviolet ranges, respectively, and .CI R and .CU V are the absorption strengths. Separate dielectric functions are used to describe the dielectric properties for the ordinary and extraordinary axis of the birefringent materials. The parameters for the dielectric functions are summarized in Table 6.1. The calculated Casimir free energy .G(d, θ ) = g(d, θ ) × l based on Eqs. 6.5– 6.11 is shown in the inset of Fig. 6.2a. Here l is the length of the cylinder [29].

Table 6.1 Parameters for the dielectric function of calcite, barium titanate, and silica [12, 26, 27] .CU V

.ωI R (rad/s)

.ωU V (rad/s)

.1.683

14 .2.691 × 10

.6.300

.1.182

14 .2.691 × 10

3595 .145.0 .0.829

.4.128

× 1014 14 .0.850 × 10 14 .0.867 × 10

× 1016 16 .2.134 × 10 16 .0.841 × 10 16 .0.896 × 10 16 .2.034 × 10

.CI R

Calcite. Calcite.⊥ Barium titanate. Barium titanate.⊥ Silica

.5.300

.4.064 .1.098

.0.850

.1.660

90

6 Proposal on Detecting Casimir Torque

Comparing the Casimir free energy to the optical trapping potential, we notice that the absolute value of Casimir free energy is far smaller than the absolute value of the optical potential at separations .d > 100 nm. Under such condition, the nanorod can be trapped stably near the center of the trapping laser beam. When the separation is smaller than 100 nm, the diameter of the nanorod is comparable to the separation and the method we use in the calculation will fail. In this chapter, we only consider the condition when the separation is larger than 100 nm. After getting the Casimir free energy, we can calculate the Casimir force by F =−

.

∂G(d, θ ) , ∂d

(6.12)

and the Casimir torque induced by the birefringent plates by [10] M=−

.

∂G(d, θ ) . ∂θ

(6.13)

The calculation of the Casimir force based on Eq. 6.12 is shown as a function of separation in Fig. 6.2b. Here the angle .θ is set to be .π/4 and the temperature is 300 K. The calculated torque at different separations based on Eq. 6.13 is shown in Fig. 6.3a. We notice that the Casimir force has the same power law dependence of the separation for different birefringent materials. However, there is no single power law dependence that can adequately characterize the Casimir torque for all materials. The amplitude of the Casimir torque depends on the dielectric function difference between two axes of the birefringent plates, and this dielectric difference is inequable for different materials. We also show the Casimir torque at different relative orientations in Fig. 6.3b. The separation is set to be 266 nm. The Casimir torque has a periodicity of .π , and

×10 Barium titanate Calcite -24

M(Nm) 10

Barium titanate Calcite

4

M(Nm)

10

-25

0

-26

-4 (b)

(a)

100

300

d(nm)

500

0

π θ

2π

Fig. 6.3 Calculation of Casimir torque. (a) Calculated Casimir torque on a silica nanorod near a birefringent crystal is shown as a function of the separation when the relative orientation is .θ = π/4. (b) Calculated Casimir torque is shown as a function of the angle between the nanorod and the birefringent plate

6.4 Torque and Force Sensitivity

91

the torque can be described by .M = M0 sin(2θ ). The Casimir torque reaches the maximum value when the angle is at .θ = π/4 and .θ = 3π/4. Different birefringent plates give different Casimir torque values but with the same periodicity. Materials with a larger birefringence will lead to a larger torque, and hence, the Casimir torque on a silica nanorod near a barium titanate plate is larger than that near a calcite plate.

6.4 Torque and Force Sensitivity From the calculation in Fig. 6.3, the maximum Casimir torque on a silica nanorod is 3.2 × 10−25 Nm for barium titanate and .4.6 × 10−26 Nm for calcite at a separation of 266 nm. To prove that it is realistic to observe the Casimir torque, we will show the calculated torque sensitivity of our levitation system in this section. When the pressure is high, the measurement noise is dominated by the thermal Brownian motions from the environment. The effect of photon recoil from the trapping laser beam is negligible. However, photon recoil becomes dominant in high vacuum and sets an ultimate bound for torque detection sensitivity. We will analyze the torque sensitivity from thermal and optical, respectively. When the vibrational amplitudes of the nanorod are small, the trapping potential is harmonic. Then the equation of motion for the torsional motion is

.

θ¨ + γ θ˙ + 2r θ = M(t)/I .

.

(6.14)

Here .θ is the angle between the nanorod and the laser polarization, .r is the resonant frequency of the torsional motion, .M(t) is the time-dependent fluctuating torque, and I is the moment of inertia of the nanorod. .γ is the damping rate that can be separated into the thermal and radiation parts .γ = γth + γrad . The minimum detectable torque from the thermal Brownian noise is [30]  Mth =

.

4kB T I γth . t

(6.15) 2 3

a l Here .t is the measurement time, and T is the environment temperature. .I = ρπ12 is the moment of inertia of the nanorod. .ρ is the density of the silica nanorod. The damping coefficient from thermal noise is .γth = fr /I , where .fr = kB T /Dr is the rotational friction drag coefficient. .Dr is the rotational diffusion coefficient of the nanorod in the free molecular regime and is given by [31]

 1 1 Dr = kB T Kn /{π μl 3 ( + 3 ) 6 8β π −4 1  π −2 1 1 ) }, +f ( + + 2+ 48 8β 8 8β 3 8β .

(6.16)

92

6 Proposal on Detecting Casimir Torque

Fig. 6.4 Torque and force sensitivity. (a) Calculated torque sensitivity of a levitated silica nanorod. The noise is dominated by thermal noise at high pressure. As the pressure goes down to .10−7 torr, the noise is dominated by photon recoil. (b) Calculated force sensitivity of a levitated silica nanorod

Nm Hz -1/2

10 -25

10

-27

10

-29

10

-18

10

-20

(a)

Torque sensitivity

Thermal noise Total noise

N Hz-1/2

(b)

Force sensitivity Thermal noise Total noise

10 -22 -9 10

10

-7

10

-5

10

-3

10

-1

Pressure(Torr)

 π kB T where .β = l/a, .Kn = λ/a, .λ = μ p 2mgas , .mgas is the molecular mass, .μ is the gas viscosity, and .f = 0.9 is the momentum accommodation. Therefore, the minimum √ detectable torque from thermal motions .Mth depends on . p. In addition to the thermal limited noise, the inevitable photon recoil from the trapping laser also contributes to the torque detection noise limit. The minimum detectable torque limited by photon recoil is  Mrad =

.

4I d KR , t dt

(6.17)

d KR is the shot noise heating rate of a nanorod in a linearly polarized laser where . dt beam and can be written as [32–34]

8π Jp d . KR = dt 3



k02 4π 0

2 (α⊥ − α )2

h¯ 2 . 2I

(6.18)

Here .Jp is the photon flux that equals to .Jp = Ilaser /h¯ ω0 . .ω0 is the frequency of incident laser beam, and .k0 is the incident wave vector. The total torque sensitivity limit is written as  2 + M2 Mth (6.19) .Mmin = rad . The calculation of torque sensitivity of a nanorod based on Eqs. 6.15–6.19 is shown in Fig. 6.4a. The calculation shows that the torque sensitivity will be dominated by thermal noise under high pressure. When pressure goes below .10−7 torr, the torque sensitivity is dominantly limited by photon recoil from √ the 100 mW trapping laser. The ultimate sensitivity bound is around .10−28 Nm/ Hz. Therefore, the calculated

References

93

Casimir torque near a barium titanate surface at a separation of 266 nm will be three orders larger than the minimum detectable torque in our levitation system. It is promising to detect the Casimir torque in the near future. Similar to the torque sensitivity, the force sensitivity is limited by both thermal noise and photon recoil noise. The calculation of force √ sensitivity is shown in Fig. 6.4b. The ultimate force sensitivity is around .10−21 N/ Hz. The Casimir force is calculated to be around .10−16 N at a separation of 266 nm (Fig. 6.2b). Therefore, the Casimir force is also detectable by our proposed method. Besides using the torsion balance, we can also utilize the rotation motion to measure the external torque. Experimentally, we have realized the world fastest nanorotor by a silica nanodumbbell in a circularly polarized optical tweezer. The nanodumbbell can be driven to rotate beyond 5 GHz (300 billion rpm). By utilizing the fast rotation, we have achieved a unprecedented torque sensitivity of √ −27 Nm/. Hz at room temperature [35]. The rotating nanorotor also has .4.2 × 10 potential in measuring the Casimir torque.

References 1. M. Sparnaay, Measurements of attractive forces between flat plates. Physica 24(6), 751–764 (1958). https://doi.org/10.1016/S0031-8914(58)80090-7 2. S.K. Lamoreaux, Demonstration of the Casimir force in the 0.6 to 6 μm range. Phys. Rev. Lett. 78, 5–8 (1997). https://doi.org/10.1103/PhysRevLett.78.5 3. U. Mohideen, A. Roy, Precision measurement of the Casimir force from 0.1 to 0.9 μm. Phys. Rev. Lett. 81, 4549–4552 (1998). https://doi.org/10.1103/PhysRevLett.81.4549 4. G. Bressi, G. Carugno, R. Onofrio, G. Ruoso, Measurement of the Casimir force between parallel metallic surfaces. Phys. Rev. Lett. 88, 041804 (2002). https://doi.org/10.1103/PhysRevLett. 88.041804 5. J.N. Munday, F. Capasso, Precision measurement of the Casimir-Lifshitz force in a fluid. Phys. Rev. A 75, 060102 (2007). https://doi.org/10.1103/PhysRevA.75.060102 6. J.N. Munday, F. Capasso, V.A. Parsegian, Measured long-range repulsive Casimir-Lifshitz forces. Nature 457(7226), 170–173 (2009). https://doi.org/10.1038/nature07610 7. G.L. Klimchitskaya, U. Mohideen, V.M. Mostepanenko, The Casimir force between real materials: experiment and theory. Rev. Mod. Phys. 81, 1827–1885 (2009). https://doi.org/10. 1103/RevModPhys.81.1827 8. M. Bordag, U. Mohideen, V. Mostepanenko, New developments in the Casimir effect. Phys. Rep. 353(1), 1–205 (2001). https://doi.org/10.1016/S0370-1573(01)00015-1 9. V.A. Parsegian, G.H. Weiss, Dielectric anisotropy and the Van der Waals interaction between bulk media. J. Adhes. 3(4), 259–267 (1972). https://doi.org/10.1080/00218467208072197 10. Y.S. Barash, Moment of Van der Waals forces between anisotropic bodies. Radiophys. Quantum Electron. 21(11), 1138–1143 (1978). https://doi.org/10.1007/BF02121382 11. S.J. van Enk, Casimir torque between dielectrics. Phys. Rev. A 52, 2569–2575 (1995). https:// doi.org/10.1103/PhysRevA.52.2569 12. J.N. Munday, D. Iannuzzi, Y. Barash, F. Capasso, Torque on birefringent plates induced by quantum fluctuations. Phys. Rev. A 71, 042102 (2005). https://doi.org/10.1103/PhysRevA.71. 042102 13. J.N. Munday, D. Iannuzzi, F. Capasso, Quantum electrodynamical torques in the presence of Brownian motion. New J. Phys. 8(10), 244–244 (2006). https://doi.org/10.1088/1367-2630/8/ 10/244

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14. X. Chen, J.C.H. Spence, On the measurement of the Casimir torque. Phys. Status Solidi (b) 248(9), 2064–2071 (2011). https://doi.org/10.1002/pssb.201147150 15. A. Siber, R.F. Rajter, R.H. French, W.Y. Ching, V.A. Parsegian, R. Podgornik, Optically anisotropic infinite cylinder above an optically anisotropic half space: dispersion interaction of a single-walled carbon nanotube with a substrate. J. Vac. Sci. Technol. B Nanotechnol. Microelectr.: Mater. Process. Measurement Phenomena: JVST B 28(3), C4AC4A17–C4AC4A24 (2010). 092003JVB[PII]. https://doi.org/10.1116/1.3416904 16. A. Šiber, R.F. Rajter, R.H. French, W.Y. Ching, V.A. Parsegian, R. Podgornik, Dispersion interactions between optically anisotropic cylinders at all separations: retardation effects for insulating and semiconducting single-wall carbon nanotubes. Phys. Rev. B 80, 165414 (2009). https://doi.org/10.1103/PhysRevB.80.165414 17. R.F. Rajter, R. Podgornik, V.A. Parsegian, R.H. French, W.Y. Ching, Van der Waals–London dispersion interactions for optically anisotropic cylinders: metallic and semiconducting singlewall carbon nanotubes. Phys. Rev. B 76, 045417 (2007). https://doi.org/10.1103/PhysRevB.76. 045417 18. D.A.T. Somers, J.L. Garrett, K.J. Palm, J.N. Munday, Measurement of the Casimir torque. Nature 564(7736), 386–389 (2018). https://doi.org/10.1038/s41586-018-0777-8 19. D.A.T. Somers, J.N. Munday, Rotation of a liquid crystal by the Casimir torque. Phys. Rev. A 91, 032520 (2015). https://doi.org/10.1103/PhysRevA.91.032520 20. K. Yasui, K. Kato, Oriented attachment of cubic or spherical BaTiO3 nanocrystals by Van der Waals torque. J. Phys. Chem. C 119(43), 24597–24605 (2015). https://doi.org/10.1021/acs. jpcc.5b06798 21. X. Zhang, Y. He, M.L. Sushko et al., Direction-specific Van der Waals attraction between rutile TiO2 nanocrystals. Science 356(6336), 434–437 (2017). https://doi.org/10.1126/science. aah6902 22. Z. Xu, T. Li, Detecting Casimir torque with an optically levitated nanorod. Phys. Rev. A 96, 033843 (2017). https://doi.org/10.1103/PhysRevA.96.033843 23. B.A. Stickler, S. Nimmrichter, L. Martinetz, S. Kuhn, M. Arndt, K. Hornberger, Rotranslational cavity cooling of dielectric rods and disks. Phys. Rev. A 94, 033818 (2016). https://doi.org/10. 1103/PhysRevA.94.033818 24. D.E. Zelmon, D.L. Small, P. Schunemann, Refractive index measurements of barium titanate from .4 to 5.0 microns and implications for periodically poled frequency conversion devices. MRS Proc. 484, 537 (1997). https://doi.org/10.1557/PROC-484-537 25. V.A. Parsegian, Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists (Cambridge University Press, 2005). https://doi.org/10.1017/CBO9780511614606 26. L. Bergström, Hamaker constants of inorganic materials. Adv. Colloid Interf. Sci. 70, 125–169 (1997). https://doi.org/10.1016/S0001-8686(97)00003-1 27. D.B. Hough, L.R. White, The calculation of Hamaker constants from Lifshitz theory with applications to wetting phenomena. Adv. Colloid Interf. Sci. 14(1), 3–41 (1980). https://doi. org/10.1016/0001-8686(80)80006-6 28. J. Mahanty, B.W. Ninham, Dispersion forces between oscillators: a semi-classical treatment. J. Phys. A: Gener. Phys. 5(10), 1447–1452 (1972). https://doi.org/10.1088/0305-4470/5/10/009 29. V.M. Mostepanenko, N.N. Trunov, The Casimir effect and its applications. Sov. Phys. Usp. 31(11), 965–987 (1988). https://doi.org/10.1070/pu1988v031n11abeh005641 30. L. Haiberger, M. Weingran, S. Schiller, Highly sensitive silicon crystal torque sensor operating at the thermal noise limit. Rev. Sci. Instrum. 78(2), 025101 (2007). https://doi.org/10.1063/1. 2437133 31. M. Li, G.W. Mulholland, M.R. Zachariah, Rotational diffusion coefficient (or rotational mobility) of a nanorod in the free-molecular regime. Aerosol Sci. Technol. 48(2), 139–141 (2014). https://doi.org/10.1080/02786826.2013.864752 32. C. Zhong, F. Robicheaux, Shot-noise-dominant regime for ellipsoidal nanoparticles in a linearly polarized beam. Phys. Rev. A 95, 053421 (2017). https://doi.org/10.1103/PhysRevA. 95.053421

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33. C. Zhong, F. Robicheaux, Decoherence of rotational degrees of freedom. Phys. Rev. A 94, 052109 (2016). https://doi.org/10.1103/PhysRevA.94.052109 34. B.A. Stickler, B. Papendell, K. Hornberger, Spatio-orientational decoherence of nanoparticles. Phys. Rev. A 94, 033828 (2016). https://doi.org/10.1103/PhysRevA.94.033828 35. J. Ahn, Z. Xu, J. Bang, P. Ju, X. Gao, T. Li, Ultrasensitive torque detection with an optically levitated nanorotor. Nat. Nanotechnol. 15(2), 89–93 (2020). https://doi.org/10.1038/s41565019-0605-9

Chapter 7

Conclusion and Outlook

Abstract In the present thesis, we have introduced our home-built multi-cantilever vacuum system. We use the multi-cantilever systems to measure the Casimir effect and study the energy transfer by quantum vacuum fluctuations. In this chapter, we will introduce several topics that can possibly be studied by our Casimir vacuum system in the near future. We will first introduce our preliminary design of a Casimir microelectromechanical systems (MEMS) accelerometer by utilizing the strong nonlinearity of the Casimir force and parametric amplification scheme. The simulated signal under an external acceleration will be presented to demonstrate the sensitivity of the Casimir MEMS accelerometer. Next, we will investigate on switching the Casimir force between a vanadium oxide (VO.2 ) surface and a gold surface. VO.2 has a metal–insulator transition temperature at 340 K that is accessible in the experiment. The transition from insulator to metal can generate an abrupt Casimir force increment at the transition temperature. The calculation of the Casimir effect between VO.2 and gold will be presented. In the last part of this chapter, we will discuss a few possible schemes to engineer the repulsive Casimir force in vacuum that has never been observed yet.

7.1 A Preliminary Design of a Casimir MEMS Accelerometer Microelectromechanical systems (MEMS) devices can be used in various scenarios. Especially, the systems can be used for temperature, pressure, acceleration, and humidity sensing [1, 2]. MEMS accelerometers are used in nearly every smartphone and many other devices to monitor their orientation and motion. The separation of mechanical resonators in commercial MEMS accelerometers is usually larger than one micrometer to avoid adhesion. By including the Casimir effect in the design, we can achieve a separation around 100 nm. We can reduce the gap between mechanical resonators by one order without adhesion and hence improve its sensitivity by two orders. Besides, the Casimir force between two ideally conductive parallel surfaces is proportional to .1/d 4 , which shows stronger nonlinearity compared to other © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 Z. Xu, Optomechanics with Quantum Vacuum Fluctuations, Springer Theses, https://doi.org/10.1007/978-3-031-43052-7_7

97

98

7 Conclusion and Outlook

interactions such as electrostatic force. Therefore, the Casimir force is very sensitive to the separation and also the displacements of two surfaces. Several theoretical studies have shown the potential in detecting zeptometer displacements [3–5] and sensing extremely small magnetic field gradient by Casimir force [3]. In this section, we propose to build a Casimir accelerometer by utilizing parametric amplification method [6]. The strong nonlinearity of the Casimir force together with the parametric amplification scheme will be beneficial to improve the signal-to-noise ratio (SNR). Our preliminary calculation shows the sensitivity of a MEMS accelerometer can be improved by Casimir parametric amplification.

7.1.1 Casimir Parametric Amplifier Parametric amplifier has been proved to improve the sensitivity for precision measurements in a mechanical system [7]. One way to realize the parametric amplification is modulating the spring constant at .2ω0 frequency, where .ω0 is the natural frequency of the oscillator. Meanwhile, the phase between the driving and the modulation needs to be tuned at .π/2 for obtaining a gain [7]. In our case, we can modulate the separation between two cantilevers at a frequency of .2ω0 , and hence, the Casimir interaction is modulated at .2ω0 . In this way, we can effectively modulate the spring constant with the time-dependent Casimir force gradient. Here we still investigate on a sphere and a cantilever geometry as shown in Fig. 7.1, similar to the Casimir diode system. The equations of the motions are written as m1 ω10 x˙1 + k10 x1 = FCasimir (d, t) + Fdri (t) + Fth1 (t), Q1 m2 ω20 x˙2 + k20 x2 = FCasimir (d, t) + Fmod (t) + Fth2 (t), m2 x¨2 + Q2 m1 x¨1 +

.

(7.1)

where the parameters for two cantilevers are .ω10 = 2π × 5000 Hz, .ω20 = 2π × 4000 Hz, .k10 = k20 = 0.1 N/m, .Q1 = 1250, .Q2 = 1000, and .R = 35 .μm. The Casimir force between a gold sphere and a gold plate is approximated as Fig. 7.1 Schematic of Casimir parametric amplifier. A resonant driving force .Fdri is applied to cantilever 1, and a parametric modulation force .Fmod is applied to cantilever 2 to modulate the separation between two cantilevers at a specific frequency

Cantilever 2

1, 1, 1

2, 2, 2

Cantilever 1

7.1 A Preliminary Design of a Casimir MEMS Accelerometer (a)

(b)

=0

= 10

99 (c)

7.8 nm

3.8 nm

= 15 14.7 nm

Fig. 7.2 The simulated displacements of two cantilevers when an external acceleration .a = 300 .μg at the resonant frequency of cantilever 1 .ω1 is applied to cantilever 1. The separation is .d = 100 nm. (a) No parametric amplification is applied to the system. The oscillating amplitude of cantilever 1 under the acceleration is 3.8 nm. (b) and (c) Parametric amplification scheme is applied with modulation amplitudes of 10 and 15 nm, respectively. The signal due to the acceleration is amplified

FCasimir = 4.82 × 10−23 × (d − x1 (t) − x2 (t))−2.359 × R. The modulation force on cantilever 2 is .Fmod = δdmod × k2 × sin(2ω1 t). The driving force on cantilever 1 is .Fdri = a × m1 × cos(ω1 t + φ1 ). Here the resonant frequency .ω1 under the ω10 Casimir interaction is approximated as .ω1 = ω10 − dFCasimir (d) × 2k . .δdmod is the dx 10 modulation amplitude. a is an external acceleration with the resonant frequency .ω1 and needs to be measured. To reach the condition for parametric amplification, the phase here is .φ1 = π/2. In Fig. 7.2, we show the simulation of the displacements of two cantilevers for different parametric modulation amplitudes. The separation here is set to be .d = 100 nm, and the external acceleration is .a = 300 .μg at the resonant frequency of cantilever 1 .ω1 . We notice that the oscillating amplitude is .x1 = 3.8 nm under such acceleration and without parametric amplification (.δdmod = 0). If the parametric amplification is applied with a modulation amplitude .δdmod = 15 nm, the oscillating amplitude of cantilever 1 becomes .x1 = 14.7 nm. Therefore, our parametric amplification through Casimir interaction can amplify the signal by a few times, which will benefit the acceleration detection. .

7.1.2 The Closest Stable Separation Before Pull-in The Casimir force is expected to increase faster than the linear restoring force from the harmonic oscillators at small separations [6, 8]. To prevent the pull-in effect, we discuss the minimum separation that we can achieve in this sphere and plate geometry. The closest separation occurs when the effective spring constant and the total force become zero [6]. We can then get the condition by solving the equations that are dFCasimir | − k0 = 0, dx = |FCasimir | − k0 δd = 0.

ksys = |

.

Fsys

(7.2)

100

7 Conclusion and Outlook (b)

(a)

8

0

(c)

10

(d)

8 0

Fig. 7.3 The peak value of PSD over 20 independent simulations with and without parametric amplification. (a) and (b) Parametric amplification with a modulation amplitude 10 nm is applied to the system. We can distinguish the acceleration of 8 and 0 .μg. (c) and (d) No parametric amplification is applied to the system. We cannot distinguish the effect of 8 and 0 .μg

We can then get the following equations: k0 = 1.137 × 10−22 × (d0 − δd)−3.359 × R,

.

k0 δd = 4.82 × 10−23 × (d0 − δd)−2.359 × R.

(7.3)

This gives us a pull-in relation that .δd = d0 /3.359, and hence, the closest separation is .d0C = 4.173 × 10−7 × (R/k0 )1/3.359 . The closest separation is 39.05 nm when the spring constant is .k0 = 0.1 N/m and the sphere radius is .R = 35 .μm. Compared to a typical separation of 100 nm, we can engineer the system with a lot of freedom without pull-in and adhesion.

7.1.3 Acceleration Detection Sensitivity In this part, we investigate on the sensitivity improvement by applying Casimir parametric amplification. We simulate the displacement of cantilever 1 when an external acceleration is applied to cantilever 1, similar to the results in Fig. 7.2. Then we analyze the power spectrum density of the motion. The total simulation time is 20 s, and we take a move mean for every 20 data points in the PSD to reduce the noise. The separation .d0 is 100 nm. By extracting the peak value of PSD, we can plot the peak of PSD over 20 independent simulations in Fig. 7.3a,c. Figure 7.3a corresponds

1

−2

101

= = =

Peak amplitude (m∗

Fig. 7.4 The peak amplitude extracted from the PSD is shown for different modulation amplitudes .δdmod and different accelerations

)

7.2 Switching Casimir Force with Phase-Transition Materials

to the case when a modulation amplitude of 10 nm is applied to cantilever 1 for the parametric amplification scheme. Figure 7.3c corresponds to the no parametric amplification case. Under the parametric amplification, we can distinguish a small acceleration of 8 .μg (Fig. 7.3a), which is difficult to be detected by the normal method in Fig. 7.3c. The parametric amplification scheme can improve the SNR by effectively increasing the qualify factor of the sensor. A further enhancement can be realized by optimizing the separation and modulation amplitude. To visualize the effect of parametric amplification, we plot the histograms of the simulation in Fig. 7.3b,d. We also show the simulated peak amplitude of PSD for different accelerations in Fig. 7.4. Proper parametric amplification scheme can help read out the acceleration signal with amplification (two times at .δdmod = 10 nm and four times at .δdmod = 15 nm). This is beneficial for detecting small accelerations, especially for the value below 10 .μg. We notice that parametric amplification can improve the SNR for detection that is limited by the device measurement noise. However, the parametric amplification cannot improve the noise from the thermal limit [9]. The acceleration sensitivity from thermal noise can be calculated as  4ω0 kB T b (7.4) .amin = = 6.4 × 10−5 m/s 2 = 6.4 μg. mQ This also agrees with the simulation in Fig. 7.4. The signal is eventually limited by the thermal noise.

7.2 Switching Casimir Force with Phase-Transition Materials Several interesting Casimir measurements have been carried out between different materials [10–19] and geometries [20–25]. A particularly intriguing direction is to

102

7 Conclusion and Outlook

Fig. 7.5 Temperature-dependent measurement of the optical conductivity of VO.2 from Ref. [40]

generate a “switchable” force between the materials whose optical properties can be controlled by a simple stimulus [11, 17–19, 26]. For example, a theoretical investigation was conducted on Casimir force between high-.Tc superconductors [19]. The optimally doped ceramics YBCO with a critical transition temperature of 93 K is considered in this work. The calculated Casimir force shows an abrupt increment at the transition temperature for separations larger than 1 .μm. This discontinuity can be used for switching Casimir force by controlling the temperature. However, the discontinuity vanishes for separations below 1 .μm so the applications of Casimir force between superconductors are limited. In this section, we propose to use vanadium oxide (VO.2 ) for Casimir force switching. The metal–insulator transition in vanadium oxides V.x O.y has generated significant attention for fundamental research or potential applications [27–39]. Vanadium dioxide VO.2 is particularly important since the transition temperature around 340 K is easily accessible in the experiment. Compared to the Casimir switching by high-.Tc superconductors, the switching condition for a VO.2 system around 340 K is much easier to access. The Casimir force switching in a VO.2 system also shows a larger contrast for the short-range regime compared to the superconductor case. We will present the preliminary calculation results of the Casimir force between a VO.2 surface and a gold surface. The temperature-dependent infrared measurement of optical conductivity for a VO.2 sample on an Al.2 O.3 substrate is shown in Fig. 7.5 (data from Ref. [40]). For simplicity, the two-oscillator Lorentzian model is used for all temperature cases (both metallic and insulating cases), and the optical conductivity .σ1 is written as [40] σ1 (ω) = Re(σˆ ) =

2 

i ω2 σdc

i=1

ω2 + (ωi2 − ω2 )τi2

.

where .σˆ =

2

i ω σdc i=1 ω+j (ω2 −ω2 )τ 2 i i

is the optical property.

,

(7.5)

7.3 Repulsive Casimir Force in Vacuum

103

Table 7.1 The fitting parameters for the optical conductivity of VO.2 Temp (K) 319 328 336 338 340 341 342 343 344 345 346 352 357

.τ1 (s

−1 )

1

1.86.×10−15

1.79.×10−15 1.50.×10−15 1.66.×10−15 1.79.×10−15 2.02.×10−15 1.92.×10−15 1.96.×10−15 1.86.×10−15 2.76.×10−15 2.76.×10−15 4.17.×10−15 4.35.×10−15

.σdc

(cm1 )

585 527.9 461.8 429.5 419.8 416.1 505.4 500 525 700 312 1788 1952

.ω1 (rad/s)

.τ2 (s

−1 )

.σdc

8.20.×1014

0 0 0 0 0 0 0 3.71.×10−15 3.71.×10−15 3.71.×10−15 3.71.×10−15 0 0

0 0 0 0 0 0 0 220 315 600 117 0 0

8.10.×1014 7.83.×1014 7.03.×1014 5.71.×1014 4.93.×1014 4.74.×1014 4.21.×1014 3.58.×1014 2.76.×1014 2.53.×1014 1.23.×1011 2.52.×1013

2

(cm1 )

.ω2

(rad/s)

0 0 0 0 0 0 0 0 0 0 0 0 0

We can easily fit the optical conductivity .σ1 in Fig. 7.5 with Eq. 7.5. The fitting parameters can be found in Table 7.1. The dielectric function at an imaginary frequency is (iξ ) = 1 +

.

σˆ (iξ ) σdc / 0 =1+ . 0 ξ ξ + (ξ 2 + ω02 )τ

(7.6)

The calculated Casimir force gradient between a gold sphere and a VO.2 surface on Al.2 O.3 is shown in Fig. 7.6a. The Casimir force gradient difference between at 357 K (blue solid line) and at 309 K (red dashed line) is about 20 to 30 .% over the short-range separation. This force gradient difference can be measured in our Casimir system. As a comparison, the force gradient between a gold sphere and a gold plate at 357 K (green solid line) and at 309 K (black dashed line) is also shown in Fig. 7.6a. The difference between these two curves is negligible compared to the difference between the blue and the red. Therefore, the thermal Casimir effect is far smaller than the effect of metal–insulator transition. The calculated Casimir force gradient is also presented as a function of temperature in Fig. 7.6b. An abrupt increment about 20.% to 30.% occurs at the transition temperature around 345 K. Therefore, it is promising to realize switching the Casimir effect in a VO.2 system.

7.3 Repulsive Casimir Force in Vacuum The Casimir effect is usually attractive between two neutral and parallel metal plates. However, the attractive Casimir force can result in undesirable adhesion problems that are detrimental to the devices. It would be interesting if one can break the symmetry and engineer a repulsive Casimir force.

104

(a)

7 Conclusion and Outlook

(b)

Fig. 7.6 Calculated Casimir force gradient between a VO.2 surface and a gold-coated sphere. (a) The calculated force gradient at the short-range separations. The metal–insulator transition of VO.2 will introduce a Casimir force gradient difference of 20.% to 30.% at two different temperatures. The calculated Casimir force gradient between two gold surfaces is also included for two different temperatures to show that the thermal Casimir effect is negligible for the short-range separations. (b) Calculated Casimir force gradient is plotted at different temperatures

One intuitive way to realize the repulsion is to engineer the dielectric function with a relation that . 1 (iξ ) > 3 (iξ ) > 2 (iξ ), where the subscripts 1 and 2 indicate two interacting plates and the subscript 3 denotes the intervening medium between two plates. The first observation of repulsive Casimir interaction was demonstrated in a gold–bromobenzene–silica system [41]. Recently, quantum trapping by repulsive Casimir force is demonstrated by levitating a gold nanoplate in ethanol near a teflon-coated gold surface [42]. However, the repulsive Casimir force has only be achieved in liquid. Here we discuss two possible schemes for realizing repulsive Casimir force in vacuum. The first scheme is to construct the repulsive force in magnetodielectric plate configurations [43, 44]. The challenge of realizing Casimir repulsion in vacuum comes from the requirement of high dielectric constant for host materials. However, if there is a system with both non-zero dielectric response and non-zero magnetic response in the broadband wavelength, we can possibly change the sign of the Casimir force with the joint effect of the permittivities . l and permeabilities .μl (.l = 1, 2 denotes two interacting materials). A theoretical proposal shows that a yttrium iron garnet (YIG) plate and a gold plate in vacuum can generate a repulsive Casimir force at some specific separations, while the force is attractive for most of the separations [43]. In the future experiment, we can easily implement a YIG sphere and a gold plate to measure the force and test the theoretical prediction [43]. The second scheme is to generate the repulsive Casimir force between topological insulators [45]. The Casimir force can be switched from the attractive to the repulsive by tuning the distance and the sign of the topological magnetoelectric polarizability [45].

References

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Although there are many theoretical studies showing the potential in detecting repulsive Casimir force in vacuum, there is no experimental realization stated so far. It will be interesting and promising to realize such repulsion in our dual-cantilever vacuum system in the near future.

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Curriculum Vitae

Zhujing Xu [email protected] (765)418-0766 Education Ph.D., Department of Physics and Astronomy, Purdue University, August 2016– August 2022 Bachelor of Science, University of Science and Technology of China, August 2012– June 2016 Work Experience Postdoctoral Fellow, September 2022–present John A. Paulson School of Engineering and Applied Sciences, Harvard University • Project: Spin–phonon coupling in Diamonds – Integrate SiV centers with high-quality factor nanomechanical resonators at low temperature. – Drive the spin of SiV centers coherently with surface acoustic wave.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 Z. Xu, Optomechanics with Quantum Vacuum Fluctuations, Springer Theses, https://doi.org/10.1007/978-3-031-43052-7

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Curriculum Vitae

– Toward building scalable quantum computers and chip-scale quantum networks. • Project: High-quality factor surface acoustic wave resonators – Design and fabricate surface acoustic wave resonators on different heterostructures. – Optimize the design to have high Q, high electromechanical coupling, and small mode volume. – Potentials in quantum phononics and integrated hybrid systems. Research Assistant, June 2018–August 2022 Department of Physics and Astronomy, Purdue University • Project: Quantum optomechanics with virtual photons (Casimir effect) – Built a vacuum dual-cantilever atomic force microscope and achieved precision measurement of Casimir force at separations ranging from 50 to 1000 nm. – Realized non-reciprocal energy transfer through Casimir interaction between two cantilevers. – Realized Casimir mediated transistor-like energy transfer between three cantilevers. – Proposed schemes to detect Casimir torque and vacuum friction with a levitated nanoparticle. – Published in Nature Nanotechnology (2022), Nature Communications (2022), Physical Review A (2017), Nanophotonics (2021). • Project: Precision measurement by levitated optomechanics – Created the world’s fastest nanorotor that rotates beyond 300 billion rpm. – Realized the world’s most sensitive torque detector with an optically levitated nanorotor. – Published in Nature Nanotechnology (2020), Physical Review Letters (2018) (APS “Highlights of the Year” of 2018) • Project: Quantum information and network by quantum emitters – Proposed to realize the scalable network with closely spaced diamond color centers. – Observed stable single-photon emissions from boron nitride nanotubes. – Observed plasmonic-enhanced shallow spin defects in hexagonal boron nitride. – Published in Optics Letters (2018), Optical Materials Express (2019), Nano Letters (2021) Teaching Assistant, August 2016–June 2018 Department of Physics and Astronomy, Purdue University • Modern physics lab, Modern mechanics lab, Physics for life sciences lab

Curriculum Vitae

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Awards Harvard Quantum Initiative (HQI) Postdoctoral Fellowship, Harvard University, March 2022 Bilsland Dissertation Fellowship, Purdue University, May 2021 Karl Lark-Horovitz Award (highest honor, top 2 percentile), Purdue University, May 2020 Poster Award in the International Symposium of Quantum Science and Technology, Purdue University, April 2019 APS Physics top 10 “Highlights of the Year” of 2018, APS Physics, Dec 2018 Graduate School Summer Research Grant, Purdue University, June 2018 Rolf Scharenberg Graduate Fellowship, Purdue University, June 2017 Outstanding Student Award (top 10 percentile), USTC, 2013, 2014, 2015 (three times) Publications 1. “Observation and control of Casimir effects in a sphere-plate-sphere system”, Zhujing Xu, Peng Ju, Xingyu Gao, Zubin Jacob, Tongcang Li, Nature Communications 13, 1 (2022) 2. “Nuclear spin polarization and control in a Van der Waals material”, Xingyu Gao, Sumukh Vaidya, Kejun Li, Peng Ju, Boyang Jiang, Zhujing Xu, Andres E. Llacsahuanga Allcca, Kunhong Shen, Takashi Taniguchi, Kenji Watanabe, Sunil A. Bhave, Yong P. Chen, Yuan Ping, Tongcang Li, Nature Materials 21, 1024 (2022) 3. “Non-reciprocal energy transfer through the Casimir effect”, Zhujing Xu, Xingyu Gao, Jaehoon Bang, Zubin Jacob, and Tongcang Li, Nature Nanotechnology 17, 2 (2022) 4. “On-chip optical levitation with a metalens in vacuum”, Kunhong Shen, Yao Duan, Peng Ju, Zhujing Xu, Xi Chen, Lidan Zhang, Jonghoon Ahn, Xingjie Ni, Tongcang Li, Optica, 8, 1359 (2021) 5. “High-contrast plasmonic-enhanced shallow spin defects in hexagonal boron nitride for quantum sensing”, Xingyu Gao, Boyang Jiang, Andres E Llacsahuanga Allcca, Kunhong Shen, Mohammad A Sadi, Abhishek B Solanki, Peng Ju, Zhujing Xu, Pramey Upadhyaya, Yong P Chen, Sunil A Bhave, Tongcang Li, Nano Letters, 21, 78 (2021) 6. “Enhancement of rotational vacuum friction by surface photon tunneling”, Zhujing Xu, Zubin Jacob, Tongcang Li, Nanophotonics, 10, 537 (2021) 7. “Five-dimensional cooling and nonlinear dynamics of an optically levitated nanodumbbell”, Jaehoon Bang, Troy Seberson, Peng Ju, Jonghoon Ahn, Zhujing Xu, Xingyu Gao, Francis Robicheaux, Tongcang Li, Physical Review Research 2 (4), 043054 (2020) 8. “Ultrasensitive torque detection with an optically levitated nanorotor”, Jonghoon Ahn, Zhujing Xu, Jaehoon Bang, Peng Ju, Xingyu Gao, Tongcang Li, Nature Nanotechnology, 15, 89 (2020)

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Curriculum Vitae

9. “Quantum information processing with closely-spaced diamond color centers in strain and magnetic fields”, Zhujing Xu, Zhang-qi Yin, Qinkai Han, Tongcang Li, Optical Materials Express, 9, 4654 (2019) 10. “High-Temperature Polaritons in Ceramic Nanotube Antennas”, Ryan StarkoBowes, Xueji Wang, Zhujing Xu, Sandipan Pramanik, Na Lu, Tongcang Li, Zubin Jacob Nano Letters, 19, 8565 (2019) 11. “Optically Levitated Nanodumbbell Torsion Balance and GHz Nanomechanical Rotor”, Jonghoon Ahn, Zhujing Xu, Jaehoon Bang, Yu Hao Deng, Thai M. Hoang, Qinkai Han, Ren Min Ma, Tongcang Li, Phys. Rev. Lett., 121, 033603 (2018) 12. “Stable emission and fast optical modulation of quantum emitters in boron nitride nanotubes”, Jonghoon Ahn*, Zhujing Xu*, Jaehoon Bang, Andres E. Llacsahuanga Allcca, Yong P. Chen, Tongcang Li, Optics Letters 43, 3778 (2018) 13. “Detecting Casimir torque with an optically levitated nanorod”, Zhujing Xu, Tongcang Li, Physical Review A 96 (3), 033843 (2017) Selected Presentations 1. “Optomechanics with quantum vacuum fluctuations”, Zhujing Xu Gordon Research Conference: Mechanical Systems in the Quantum Regime, June 2022 (invited talk) 2. “Quantum vacuum mediated energy transfer”, Zhujing Xu, Gordon Research Seminar: Mechanical Systems in the Quantum Regime, June 2022 (invited talk) 3. “Non-reciprocal energy transfer through the quantum vacuum fluctuations”, Zhujing Xu, APS March Meeting, March 2022 (oral presentation) 4. “Optomechanics with quantum vacuum fluctuations and spin qubits”, Zhujing Xu, Quantum Fest, Harvard Quantum Initiative, 2021 (invited talk) 5. “Quantum information processing with high-density diamond nitrogen-vacancy centers in strain and magnetic fields”, Zhujing Xu, Zhang-Qi Yin, Qinkai Han, Tongcang Li, APS March Meeting, March 2021 (oral presentation) 6. “Detecting rotational vacuum friction by optically levitated nanorotor”, Zhujing Xu, Tongcang Li, OSA Frontiers in Optics and Laser Science Conference, September 2020 (oral presentation) 7. “Towards measuring quantum electrodynamic torque with a levitated nanorod”, Zhujing Xu, Jaehoon Bang, Jonghoon Ahn, Thai M Hoang, Tongcang Li, APS Division of Atomic, Molecular and Optical Physics Meeting, June 2017 (oral presentation)