Photons and Quantum Fluctuations 9781003069539, 9780852742402, 9780367403386, 0852742401

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Table of contents :
Cover......Page 1
Half Title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Table of Contents......Page 6
Preface......Page 8
List of Contributors......Page 10
Nonclassical Light......Page 14
Single-Atom Oscillators......Page 23
Photodetection and Photostatistics......Page 44
Nonclassical Photon Interference Effects......Page 64
Photons and Approximate Localisability......Page 79
Quantum Noise Reduction on Twin Laser Beams......Page 94
Nonclassical Effects in Parametric Downconversion......Page 135
Moving Mirrors and Nonclassical Light......Page 164
Propagation of Nonclassical Light......Page 186
Models for Phase-Insensitive Quantum Amplifiers......Page 203
Quasiprobabilities based on Squeezed States......Page 218
Index......Page 230
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P H O T O N S A N D Q U A N T U M F L U C T U A T IO N S

Other books in series Chaos, Noise and Fractals

edited by E R Pike and L A Lugiato Frontiers in Quantum Optics

edited by E R Pike and S Sarkar Basic Methods of Tomography and Inverse Problems

edited by P C Sabatier Nonlinear Phenomena and Chaos

edited by S Sarkar

MALVERN PHYSICS SERIES General Series Editor: Professor E R Pike FRS

PHOTONS AND QUANTUM FLUCTUATIONS Edited by E R Pike Royal Signals and Radar Establishment, Malvern and Department of Physics, King’s College, London and H Walther Sektion Physik, Universitat Miinchen and Max-Planck-Institut fur Quantenoptik, Garching

CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an i nf or ma business

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 First issued in paperback 2019 © 1988 by Taylor & Francis Group, LLC CRC Press is an imprint o f Taylor & Francis Group, an Informa business No claim to original U.S. Government works ISB N -13: 978-0-85274-240-2 (hbk) ISBN-13: 978-0-367-40338-6 (pbk) This book contains inform ation obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity o f all materials or the consequences o f their use. The authors and publishers have attempted to trace the copyright holders o f all m aterial reproduced in this publication and apologize to copyright holders if perm ission to publish in this form has not been obtained. If any copyright m aterial has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as perm itted under U.S. Copyright Law, no part o f this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, m icrofilming, and recording, or in any inform ation storage or retrieval system, without written permission from the publishers. For perm ission to photocopy or use material electronically from this work, please access w w w .copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-7508400. CCC is a not-for-profit organization that provides licenses and registration for a variety o f users. For organizations that have been granted a photocopy license by the CCC, a separate system o f paym ent has been arranged.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.

British Library Cataloguing in Publication Data Photons and quantum fluctuations. 1. Quantum optics I. Pike, E. R. (Edward Roy). II. Walther, H. (Herbert), 1935- III. Series 535'. 15 ISBN 0-85274-240-1 US Library o f Congress Cataloging-in-Publication Data are available Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

CONTENTS Preface

vii

List of Contributors

ix

Nonclassical Light H P Yuen

1

Single-atom Oscillators H Walther

10

Photodetection and Photostatistics M Le Berre

31

Nonclassical Photon Interference Effects C K Hong, Z Y Ou and L Mandel

51

Photons and Approximate Localisability E R Pike and S Sarkar

66

Quantum Noise Reduction on Twin Laser Beams E Giacobino, C Fabre, S Reynaud, A Heidmann and R Horowicz

81

Nonclassical Effects in Parametric Downconversion J G Rarity and P R Tapster

122

Moving Mirrors and Nonclassical Light S Sarkar

151

Propagation of Nonclassical Light I Abram

173

Models for Phase-insensitive Quantum Amplifiers G L Mander, R Loudon and T J Shepherd

190

Quasiprobabilities based on Squeezed States F Haake and M Wilkens

205

Index

217

PREFACE

There has been considerable progress in the field of quantum optics in recent years. New experimental techniques have enlarged our basic understanding of radiation-matter coupling. Single particle events can now be studied in detail and it has become possible to explore non-classical properties of radiation not only in photon correlation measurements of antibunching (silent light!) but also via the dynamics of the radiation-atom coupling. In cavities electromagnetic fields characterized by fixed photon numbers (Fock states) can be generated and it is possible to study the interaction of atoms in such fields. Furthermore, methods of nonlinear optics allow, via parametric downconversion, two photons to be created simultaneously; by use of the photoelec­ tric detection of one of them as a trigger, a good approximation to the ideal localized one-photon state can be achieved in the other. By means of nonlinear optical processes squeezed states can also be generated which allow measure­ ments with precision beyond the limit set by the zero-point or vacuum fluctuations of the optical field. Among the potential applications of such fields is their use in a large laser interferometer for gravitational wave detection. These exciting and recent developments together with other fundamental theoretical contributions in quantum optics are discussed in this book in detail. The different contributions review talks which were given at a special ONR Seminar held on January 21st and 22nd, 1988 at the Istituto d’Arte in Cortina d’Ampezzo, Italy, and at which we were honoured to join their 100th anniversary celebrations. The spectacular natural environment of Cortina was as inspiring and exciting as the topics of the seminar. This Special Seminar was followed by a NATO ARW on ‘Squeezed and Non-Classical Light’, the proceedings of which will be published by Plenum Press. The two volumes may be regarded as complementary. We would like to acknowledge, on behalf of all the participants, the generous support and encouragement of the US Office of Naval Research in London. We would also like to acknowledge additional financial assistance provided by Ministero Pubblica Istruzione, Universita di Roma ‘La Sapienza’, Consiglio Nazionale delle Ricerche (CNR), Gruppo Nazionale di Struttura della Materia (CNR) and local sponsorship from Olivetti Spa, Consorzio per lo Sviluppo e Turismo di Cortina d’Ampezzo, Municipio di Cortina d’Ampezzo, Istituto Statale d’Arte and Hotel Europa. We express our gratitude for the invaluable assistance of Professor Paolo Tombesi of the University of Rome and the inspiration of Professor Danny Walls to suggest such a timely meeting. Special thanks are due to Professor M Spamponi, Mr G Milani, Ing E Cardazzi, Professor G Demenego, Mr E Demenego and, in particular, Professor G Olivieri. Jim Revill of Adam Hilger provided expert assistance with these

Viii

PREFACE

proceedings and, last but by no means least, we thank our willing hard-worked secretaries Beverley James, Angela Di Silvestro and Marcella Mastrofini. Roy Pike Herbert Waither

May 1988

LIST OF CONTRIBUTORS

I ABRAM

CNET 196 Avenue Henri Ravera 92200 Bagneux Cedex France

M LE BERRE

Laboratoire de PPM Universite de Paris Sud Batiment 213 91405 Orsay Cedex France

C FABRE

Laboratoire de Spectroscopie Hertzienne de TENS Universite Pierre et Marie Curie Tour 12, Premier Etage 4 Place de Jussieu 75252 Paris Cedex 05 France

E G IA CO B IN O

L a b o ra to ire de S pectroscopie H ertzien n e de FEN S

Universite Pierre et Marie Curie Tour 12, Premier Etage 4 Place de Jussieu 75252 Paris Cedex 05 France F HAAKE

Fachbereich Physik Universitat Essen - Gesamthochschule Postfach 103764 4300 Essen 1 FRG

A HEIDMANN

Laboratoire de Spectroscopie Hertzienne de FENS Universite Pierre et Marie Curie Tour 12, Premier Etage 4 Place de Jussieu 75252 Paris Cedex 05 France

X

LIST OF CONTRIBUTORS

C K HONG

Department of Physics and Astronomy University of Rochester Rochester New York NY 14627 USA

R HOROWICZ

Laboratoire de Spectroscopie Hertzienne de PENS Universite Pierre et Marie Curie Tour 12, Premier Etage 4 Place de Jussieu 75252 Paris Cedex 05 France

R LOUDON

Department of Physics University of Essex Wivenhoe Park Colchester C 0 4 3SQ UK

L MANDEL

Department of Physics and Astronomy University of Rochester Rochester New York NY 14627 USA

G L MANDER

Department of Physics University of Essex Wivenhoe Park Colchester C 04 3SQ UK

Z YOU

Department of Physics and Astronomy University of Rochester Rochester New York NY 14627 USA

LIST OF CONTRIBUTORS

E R PIKE

Centre for Theoretical Studies Royal Signals and Radar Establishment St Andrews Road Malvern Worcestershire WR14 3PS UK

and

Department of Physics King’s College Strand London WC2R 2LS UK

J G RARITY

Royal Signals and Radar Establishment St Andrews Road Malvern Worcestershire WR14 3PS UK

S REYNAUD

Laboratoire de Spectroscopie Hertzienne de TENS Universite Pierre et Marie Curie Tour 12, Premier Etage 4 Place de Jussieu 75252 Paris Cedex 05 France

SARBEN SARKAR

Centre for Theoretical Studies Royal Signals and Radar Establishment St Andrews Road Malvern Worcestershire WR14 3PS UK

T J SHEPHERD

Royal Signals and Radar Establishment St Andrews Road Malvern Worcestershire WR14 3PS UK

x ii

LIST OF CONTRIBUTORS

P R TAPSTER

Royal Signals and Radar Establishment St Andrews Road Malvern Worcestershire WR14 3PS UK

H WALTHER

Sektion Physik Universitat Miinchen Am Coulombwall 1 8046 Garching FRG

and

Max-Planck-Institut fur Quantenoptik 8046 Garching FRG

M WILKENS

Fachbereich Physik Universitat Essen - Gesamthochschule Postfach 103764 4300 Essen 1 FRG

H P YUEN

Department of Electrical Engineering and Computer Science Northwestern University Evanston Illinois 60620 USA

NONCLASSICAL LIGHT

H P Y U EN

1. IN T R O D U C T IO N

A synopsis on nonclassical light will be presented, em phasizing the roles of dif­ ferent quan tu m amplifiers, th e problem of overcoming the detrim ental effect of loss, and highlighting certain points not explicitly brought out before. It is very far from a com prehensive review. In particu lar, no experim ent will be described. A useful recent review on squeezed light w ith considerable scope has been given by Loudon and K night (1987). No corresponding review on sub-Poissonian light is available, p artly because it is as yet m uch less developed com pared to squeezed light. 2 . NONCLASSICAL LIG H T — SQUEEZING AND ANTIBUNCHING

T he “vacuum ” is filled w ith free electrom agnetic field in its ground state. For a single m ode w ith annihilation operator a and num ber operator N = a^a, the ground state |0 > is an eigenstate of N as well as a. W hen a m ean am plitude a = oq + i a 2 for the field quad ratu res cq, a 2, a = cq -f ia2, is added to the vacuum , we have a “classically” excited m ode in a coherent sta te \a > , a\a > = a \a > (G lauber 1963). A classical state is, by definition, a coherent state or a random superposition of coherent states, i.e., a sta te w ith a tru e probability density P -representation.

Such states

are obtained in all th e conventional light sources. T here are su b stan tial quantum fluctuations even in a pure coherent state. T he q u ad ratu re fluctuation in \a > is ( A a l) = j ,

= (iia + va*)\fiva > , |/i |2 - \v\2 = 1

(5)

T he photocount statistics of |pit/a > can be sub-Poissonian depending on the p aram ­ eters. It is generally spiked at two photons apart; in p articular, (fivo\n) ^ 0 only for even n. T he m inim al q u ad ratu re fluctuation in \pLva > is (A a l ) = 7 (W - M )2

(6)

w ith th e corresponding m axim al (Aa^o+i ) = \(\p\ + \v\)2 so th a t the uncertainty p ro d u ct ( A a l ) ( A a l +«_) > ^

(7)

achieves its m inim um value a t o, which is a function of pi and v (Yuen 1976b). T he im portance of TCS com pared to other squeezed states in low noise application derives from th e fact th a t th e q u ad ratu re signal-to-noise ratio (—\ U / a , “ (A al)

(8)

is m axim ized, und er th e fixed power constraint t r p N < S', by an ap p ro p riate TCS am ong all states (Yuen 1976a). In particu lar, the m axim um value ( § ) a ^ S = 4 5 ( 5 + 1 ) greatly exceeds th e coherent sta te level (jy )J^ = 4 5 for large 5 . Since ( A N 2) = 0 in a num ber state \n > , it m ay appear th a t as

S N R » s (££>

(9)

3

H P YUEN

approaches infinity, near-num ber states are vastly superior to TCS. However, the discrete n a tu re of \n > greatly reduces this apparent advantage in m ost applications. In m any ways, TCS and num ber states are com parable and com plem entary. 4. C O H ER EN C E P R O P E R T IE S T he characterization of the coherence properties of a field depends on the (quan­ tu m ) m easurem ent we are interested in perform ing on the field. In m any applications, w hether a field is coherent or not is irrelevant, particularly in regard to the so-called higher order coherence. (We do generally need a field traveling in a relatively welldefined direction.) Here, we lim it ourselves to the usual first-order coherence de­ scribing interference, which is m athem atically characterized by the factorization of the correlation function G {riti) r2t 2) = t r p E ^ i r i t ^ E i r f a )

(10)

into the p roduct of two c-num ber fields. Physically, this coherence condition expresses the sp atial-tem poral coherence of the field. It is equivalent (T itu laer and G lauber 1966) to having only one (general) space-tim e m ode excited to an arb itra ry state, while th e other modes rem ain in vacuum. Thus, a num ber state field can be perfectly space-tim e coherent, as can a TCS field. It is also well known th a t a m ultim ode excited coherent sta te field is coherent to all orders (G lauber 1963). How does this reconcile w ith the above single m ode condition? It tu rn s out th a t a m ultim ode coherent sta te field can be described, via a m odal tranform ation, as a single m ode coherent state field. This possibility can be traced to th e fact th a t th e vacuum sta te is a coherent state. In a squeezed vacuum where all the modes are in \fxu0 > w ith the sam e //, v , a m ultim ode excitation |fivcti > would be equivalent to a single m ode excitation in a sim ilar fashion, while an ordinary m ultim ode coherent state field would no longer be space-tim e coherent. This is ju st one illustration of the im p o rtan t fact th a t the vacuum sta te is a coherent state. On the other hand, although the vacuum sta te is also a num ber state, a m ultim ode num ber sta te field is not equivalent to a single-mode one, i.e., not space-tim e coherent. This fact can be understood in term s of the unique linear transform ation properties of G aussian states, i.e., states w ith G aussian characteristic functions.

A detailed

developm ent will be given elsewhere. 5. G EN ERA TIO N , PRO PA G A TIO N , AND D E T E C T IO N T he generation of TCS and NNS via nonlinear optical processes are, in some sense, surprisingly similar. This sim ilarity is in addition to th a t between TCS and

4

N O N C L A S S IC A L L I G H T

num ber-phase m inim um uncertain ty states when the nonclassical effects are sm all (Y. Y am am oto, N. Im oto, and S. M achida 1986). It seems to arise from an underlying intrinsic q u an tu m correlation th a t is w orth fu rth er exploring to great depth. Nonclassical light do not p ropagate well. This is because significant loss, be it radiative, absorption, or otherw ise, is often incurred during propagation and as we will see in th e following section, ordinary linear loss re-introduces coherent state fluctuation into th e field. Each ordinary optical detection scheme corresponds to the qu an tu m m easurem ent of a certain observable — in direct detection one m easures iV, in hom odyne detection cty, and in heterodyne detection a (Yuen and Shapiro 1980).

T hus to utilize the

advantage of a large ( 8 ) one hom odynes, or balance-hom odynes (Yuen and C han 1983) for com pletely elim inating th e local oscillator noise. To get a large (9) one counts photons. However, th e effect of a nonunity quantum efficiency is equivalent to linear loss (Yuen and Shapiro 1978a). It seriously degrades the nonclassical effect as follows. 6 . LOSS AND A M PLIFIC A TIO N

T he effect of linear loss on a m ode a can be represented by the transform ation to an o th er m ode b (Yuen 1975, Yuen and Shapiro 1978b, Yuen 1983), b = rjza + (1 — rj)*d

( 11)

w here th e d-m ode would be in vacuum except when a special m edium is involved (Yuen 1975). It follows im m ediately from ( 11) th a t the resulting q u ad ra tu re fluctu­ ation is, for a rb itra ry a-m ode state, (A b l )

=

tj {

A al) +

(1 - »?)/4 > (1 - »?)/4

(12)

T hus, a noise floor is introduced in addition to the usual signal atten u atio n , thereby greatly reducing th e SNR advantage of a TCS a-m ode. Similarly, the d-m ode would in troduce p a rtitio n noise into th e b-m ode when the a-m ode is in a num ber state. From (11) we have, for any a-m ode state, (A7V62) = 772 2 . A straightforw ard com putation from (14) leads to (A N 2) - {N b) = G 2{ A N 2) + G (G - 2){Na) + (G - l )2

(16)

which shows th a t any sub-Poissonian statistics in a would disappear in b for G > 2 also. These features are, of course, confirmed in detailed calculations involving more explicit linear am plifier models (Hong, Friberg, and M andel 1985; M ander, Loudon, and Shepherd, this volume). It is clear th a t PIA would seriously degrade the SNR for TCS or NNS. From (14)-(15), S N R b. = S N R J

1+

G- 1 1 G 4 1/G. Sim ilar to the PSA case, let the PNA o u tp u t b pass through a linear atten u ato r of final o u tp u t c, w ith the resulting (Yuen 1986b) S N R Nc = S N R n J

(26)

From (26), S N R Nc ~ S N R Na for a sufficiently large G.

(27)

If a PIA is used in place of the PNA, we would have

S N R Nc < S N R jv a for NNS inp u t and S N R jy c ~ ^ S N R mq for coherent state input. Again, detection device noise can be suppressed in a sim ilar m anner. E quations (19), (21) and (24) show th a t different amplifiers are suited fo r different detection schemes, regardless of the nature of the source. Of course, different detec­ tion schemes are n atu rally suited for different nonclassical sources. We sum m arize this situ atio n in the following table: SOURCE Coherent state TCS NNS

D E T E C T IO N heterodyne hom odyne direct

A M PL IFIE R PIA PSA PNA

It should be clear th a t PNA is indeed a n a tu ra l amplifier, com pleting the triad for m atching the three sta n d a rd detection schemes.

We need P SA to preserve the

advantage of squeezing, and P N A that of antibunching. In addition, PSA and PNA are useful even for coherent state sources, as they are still 3db superior to PIA in hom odyne or direct detection th a t are som etim es usefully employed for detecting

8

N O N C L A S S IC A L L I G H T

coherent sta te light. It should also be em phasized th a t they are necessary if one w ants to overcome loss or detection noise by pream plification. 10.

A PPLIC A TIO N S Nonclassical light w ith m atching detection leads to significant im provem ent in

S N R , as we have seen. Even in situations where S N R is not the m ost ap propriate perform ance m easure, it is clear th a t the low noise characteristics of TCS and NNS, or sim ply th eir difference from classical states, could be useful. T he m any applications th a t have been suggested for them , to my knowledge, are listed in the following: • C om m unications (Yuen 1975; Yuen and Shapiro 1978b, 1980; Shapiro, Yuen and M achado-M ata 1979; Yuen 1987). • F ib er T apping (Shapiro 1980; Yuen 1987). • Gyros (D orschner, Haus, et. al. 1980). • Interferom etry (Caves 1981; B ondurant and Shapiro 1985; Yuen 1986b; Yurke, M cCall and K lauder 1986). • O ptical C om puting (Y am am oto). • O ptical M em ory (Levenson).

• Precision M easurem ents (Yuen 1976b; Kimble; Slusher). • Spectroscopy (G ardiner 1986; M ilburn 1986; Yurke and W h ittak er 1987).

REFERENCES B o n d u ran t, R. S. and Shaprio, J. H. 1984, Phys. Rev. D, 30, 2548. Caves, C. M., 1981, Phys. Rev. D, 23, 1693. D orschner, T. A. H aus, H. A., et al, 1980, IE E E J. Q uant. E lectron., 16. 1376. G ardiner, C.W . 1986, Phys. Rev. L ett., 56, 1917. G lauber, R. J., 1963, Phys. Rev., 131. 2766. Haus, H.A., and M ullen, J. A., 1963, Phys. Rev., 128, 2407. Hong, C. K., Friberg, S., and M andel, L., 1985, JO SA -B, 2, 494. Loudon, R., and K night, P. L., 1987, J. Mod. O pt., 34, 709.

H P YUEN

M ander, G., Loudon, R., and Shepherd, T ., this volume. M ilburn G. J., 1986, Phys. Rev. A, 24, 4882. Shapiro, J. H., 1980, O ptics L ett., 5, 351. Shapiro, J. H., Yuen, H. P., and M achado-M ata, J. A., 1979, IE E E Trans. Inform. Theory, 25, 179. T itu laer, U. M., and G lauber, R. J., 1966, Phys. Rev., 145. 1041. Yam am oto, Y., Im oto, N., and M achida, S., 1986, Phys. Rev. A, 33, 3243. Yuen, H. P., 1975, Proceedings of the 1975 Conference on Inform ation Sciences and Systems, John Hopkins U niversity Press, B altim ore, pp. 171-177. Yuen, H. P., 1976a, Phys. L ett. A, 56, 101. Yuen, H. P., 1976b, Phys. Rev. A, 13, 2226. Yuen, H. P., 1983, Q uantum O ptics, E xperim ental G ravitation, and M ea­ surem ent Theory, P. M eystre and M. 0 . Scully, Eds., Plenum , New York, pp.

249-268.

Yuen, H. P.,

1986a, Phys.

L ett. A, 113. 405.

Yuen, H. P.,

1986b, Phys.

Rev. L ett., 56,2176.

Yuen, H. P.,

1987, O ptics

L ett., 12, 789.

Yuen, H. P.,

and C han, V. W . S., 1983, O ptics L ett., 8 , 177.

Yuen, H. P.,

and Shapiro, J. H., 1978a, Proc. of the F ourth R ochester

Conf. on Coherence and Q uantum O ptics, L. M andel and E. Wolf, Eds., Plenum , New York, pp. 719-727. Yuen, H. P., and Shapiro, J. H., 1978b, IE E E Trans. Inform. Theory, 24. 657. Yuen, H. P., and Shapiro, J. H., 1980, IE E E Trans. Inform. Theory, 26, 78. Yurke, B., McCall, S. L., and K lauder, J. R., 1986, Phys. Rev. A, 23, 4033. Yurke, B., and W h ittaker, E. A., 1987, O ptics L ett., 12 , 236.

SINGLE-ATOM OSCILLATORS

H W ALTHER

M o d e rn m e th o d s o f l a s e r s p e c t r o s c o p y a l l o w t h e s t u d y o f s i n g l e a to m s o r

io n s in an u n p e r tu r b e d e n v iro n m e n t.

up i n t e r e s t i n g

new e x p e r i m e n t s ,

o f ra d ia tio n -a to m of th is

c o u p lin g .

In th e

ty p e a r e re v ie w e d : t h e s in g le - a to m

s im p le s t

and

m o st

fu n d a m e n ta l

r a d i a t i o n - m a t t e r c o u p lin g i s

a

a c tin g

o f an

w ith a s i n g l e

c a v ity . s h o rtly th e

T h is

m ode

p ro b le m w as o f

p u re ly

so

s u f f ic ie n t to

th a t

th e

le a d to

g re a t deal

th a n th e o th e r c h a r a c t e r i s t i c e x c ite d

s ta te

life tim e ,

of

a to m -fie ld

of

h o w e v e r, (a )

te s t

ra d ia tio n -m a tte r

in te ra c tio n a

s in g le

s in g le

h o w e v e r, m a trix

a re u s u a lly

p h o to n i s

e v o lu tio n

in a

a tte n tio n th e

not

tim e s h o r t e r

tim e s o f t h e s y s te m ,

e x p e rim e n ta lly th e in te ra c tio n .

su ch a s th e

I t w as t h e r e f o r e

fu n d a m e n ta l t h e o r i e s

T h ese

th e o rie s

and b a s ic e f f e c t s .

a to m

d i s a p p e a r a n c e a n d q u a n tu m

(b )

th e o s c illa to r y

and th e

c a v ity

m o d e,

o f a s in g le

e n e rg y exchange and

re v iv a l o f o p tic a l n u ta tio n

i n a s i n g l e a to m b y a r e s o n a n t f i e l d .

p re d ic t,

T hese in c lu d e th e

t h e s p o n ta n e o u s e m is s io n r a t e

a to m i n a r e s o n a n t c a v i t y , b e tw e e n a

fie ld

of

t h e t i m e o f f l i g h t o f t h e a to m t h r o u g h

som e i n t e r e s t i n g

m o d ific a tio n o f

a to m i n t e r ­

in te re s t:

t h e c a v i t y a n d t h e c a v i t y m ode d a m p in g t i m e . n o t p o s s ib le to

s tu d y in g

A t t h a t tim e ,

a c a d e m ic

fie ld

an

fo r

e le c tro m a g n e tic a

e le m e n ts d e s c r ib in g th e r a d ia tio n - a to m to o s m a ll,

io n .

tw o -le v e l

a f t e r t h e m a s e r w as i n v e n t e d .

p r o b le m

m aser and th e s tu d y s to re d

s y s te m

s in g le

re c e iv e d

s tu d y

f o l l o w i n g tw o e x p e r i m e n t s

o f th e re so n a n c e flu o re s c e n c e o f a s in g le The

T h is h a s opened

am ong th e m t h e d e t a i l e d

(c )

th e

in d u c e d

11

H W A L TH E R

The s i t u a t i o n

c o n c e rn in g

th e

b a s ic e f f e c ts has d r a s tic a lly s in c e

e x p e rim e n ta l t e s t i n g changed

th e l a s t

fe w y e a r s

f r e q u e n c y - t u n a b l e l a s e r s now a l l o w p o p u l a t i o n o f h i g h l y

e x c i t e d a to m ic s t a t e s

c h a ra c te riz e d

num ber n o f t h e v a le n c e

e le c tro n .

c a lle d

s in c e

R y d b e rg

s ta te s

d e s c r ib e d by th e ra d ia tio n -a to m s ta te s

h ig h

m a in q u a n tu m

T h ese s t a t e s

by

a re g e n e ra lly

th e ir

fo r

th re e

r a t e s b e tw e e n

F in a lly ,

th e s e tr a n s it i o n s

la rg e to

have

and G a lla s ,

The s t r o n g c o u p lin g o f

e v o lu tio n

p rin c ip le :

le v e l;

th e

o s c illa tin g

see

to

H a ro c h e and

w ith

th e re fo re

in c re a s in g

in te rm s o f th e n

e x c ite d

th e c la s s ic a l

e le c tro n

fre q u e n c y to co rresp o n d s

(1 9 8 5 ).

re so n a n t ra d ia tio n

can b e u n d e rs to o d

tra n s itio n

a to m

s till tim e s .

lo n g l i f e t i m e s w ith

re v ie w s

a to m s

th e h ig h ly

(th e

s c a le as

c a v itie s th a t a re

For

R y d b erg

fre q u e n c y o f

i d e n t i c a l w ith th e

th e s e

fie ld

L e u c h s, W a lth e r an d F ig g e r

b e tw e e n n e i g h b o u r i n g l e v e l s c o rresp o n d en ce

F irs tly ,

in

i n t h e m i l l i m e t r e w ave

re la tiv e ly

s p o n ta n e o u s d e c a y .

(1 9 8 5 )

e ffe c ts

th e r a d ia tio n a re

be

h ig h ly e x c ite d

e n s u r e r a t h e r lo n g i n t e r a c t i o n

R y d b e rg s t a t e s

re s p e c t to R aim o n d

T he

can

n e ig h b o u rin g l e v e l s

w h ic h a l l o w s l o w - o r d e r m ode

s u ffic ie n tly

le v e ls

re a so n s.

a r e v e r y s t r o n g l y c o u p le d t o

in d u c e d t r a n s i t i o n re g io n ,

e n e rg y

f o r o b s e r v i n g t h e q u a n tu m

c o u p lin g

S e c o n d ly ,

a

s im p le R y d b erg f o r m u la .

a to m s a r e v e r y s u i t a b l e

n4) .

in

o f th e s e

th e to

w ith th e re s o n a n c e fre q u e n c y .

becom es

n e ig h b o u rin g

a la rg e

(T h e d i p o l e

d ip o le m om ent

i s v e r y l a r g e s i n c e t h e a t o m i c r a d i u s s c a l e s a s n 2 .) In

o r d e r to u n d e rs ta n d th e

e m is s io n r a t e in

m o d ific a tio n

in an e x te r n a l c a v ity ,

q u a n tu m e l e c t r o d y n a m i c s

d e n s ity

of

tra n s itio n

th is

ra te

is

fre e

c o n tin u u m o f

sp ace, m o d es

but is

t h e s p o n ta n e o u s re m e m b e r t h a t

d e te rm in e d

m o d es o f t h e e l e c t r o m a g n e t i c f i e l d fre q u e n c y C O q .

a t th e

a to m ic

fre q u e n c y .

in a re s o n a n t changed

in to

c a v ity a

I f t h e a to m i s in s te a d ,

s p e c tru m

is

ra d ia te d

e n e rg y

d is s ip a tio n

a t a w e ll-d e fin e d

w ith in

th e

c a v ity ,

th e

of d is c re te

m o d es w i t h o n e o f th e m b e i n g i n r e s o n a n c e w i t h t h e a to m . th e re

by th e

T he v a cu u m d e n s i t y o f m o d es p e r u n i t

v o lu m e d e p e n d s o n t h e s q u a r e o f t h e n o t in

of

we h a v e t o

a

S in c e p h o to n

fre q u e n c y w i l l b e sm e a re d o u t o v e r

12

th e

SIN G L E -A T O M OSC ILLA TO R S

fu ll

s p e c t r a l w id th A ^ c o f

w i d t h a t h a l f maximum A U)c i s

t h e r e s o n a n t m o d e.

re la te d

to

th e

The f u l l

c a v ity

q u a lity

f a c t o r Q = U)c /A tO c# The

s p o n ta n e o u s d e c a y

enhanced in r e l a t i o n g iv e n by th e r a t i o

ra te to

o f t h e a to m

th a t

in

in th e c a v ity

fre e

space ^

o f t h e c o r r e s p o n d i n g m ode

/c

by a

is

fa c to r

d e n s itie s

(Vc i s

t h e v o lu m e o f t h e c a v i t y ) : = f c (w0)/?f(w 0) = 27rQ/Vcu03 = QX03/4tt2Vc .

1 ch f

For

lo w -o rd e r

c a v itie s

in

vc

A q 3; th e

s p o n ta n e o u s

in c r e a s e d by a f a c t o r o f Q th e decay r a te case

m ic r o w a v e

e m is s io n

ra te

d e c r e a s e s w hen t h e c a v i t y

accept

it,

re g io n

one

i s th u s

ro u g h ly

in a re s o n a n t c a v ity ;

t h e a to m c a n n o t e m i t a

a b le to

th e

p h o to n ,

has

c o n v e rs e ly ,

i s m is tu n e d .

In t h i s

s in c e th e c a v ity

and t h e r e f o r e th e e n e rg y h a s t o

is

not

s ta y w ith

t h e a to m . R e c e n tly ,

q u i t e a few

e x p e rim e n ts h a v e b e e n

c o n d u c te d

w ith

R y d b e r g a to m s t o d e m o n s t r a t e t h e e n h a n c e m e n t a n d i n h i b i t i o n s p o n ta n e o u s

decay

s tru c tu re s .

For

in

th e

e x te rn a l m ost

c a v itie s

or

r e c e n te x p e rim e n t

of

c a v ity -lik e

se e Ih e e t a l .

(1 9 8 7 ) . T h e r e a r e a l s o m o re m ode d e n s i t y :

s u b tle e f f e c ts

ra d ia tio n

t h e a n o m a lo u s m a g n e t i c

c o rre c tio n s

d i p o l e m om ent o f t h e e l e c t r o n

m o d ifie d w ith r e s p e c t t o th e c u la te d u n d er

th e

b o u n d a ry

1987) . The c h a n g e i s

due to th e change

fre e

sp a c e v a lu e

c o n d itio n s

if

ac c u ra cy .

a r e d e te rm in e d by v i r t u a l t r a n s i t i o n s

fo llo w in g ,

a to m m a s e r in te ra c tin g

R o u g h ly

s p e a k in g ,

one can

say th a t and n o t by

a s in th e c a s e o f s p o n ta n e o u s d e c a y . a tte n tio n

i n w h ic h t h e w ith

(B a rto n

j u s t o f t h e o r d e r o f m a g n itu d e o f p r e s e n t

e x p e rim e n ta l

In th e

a re a ls o

th e y a r e c a l ­

o f a c a v ity

th e s e e f f e c ts

re a l tra n s itio n s

o f th e

s u c h a s t h e Lamb s h i f t a n d

a

is

f o c u s e d on d i s c u s s i n g t h e o n e -

id e a liz e d

s in g le

m ode

case of

a

o f a t w o - l e v e l a to m ra d ia tio n

fie ld

is

13

H W A L TH E R

re a liz e d ;

th e th e o ry o f t h i s

s y s te m w as t r e a t e d

by J a y n e s and

C um m ings (1 9 6 3 ) m any y e a r s a g o . We c o n c e n t r a t e o n t h e d y n a m ic s o f th e a to m -fie ld of

th e

in te ra c tio n

p r e d ic te d by t h i s

fe a tu re s a re e x p lic itly

n a tu re o f

th e

a consequence

e le c tro m a g n e tic

d is c r e te n a tu re

of

c h a ra c te ris tic s

such

fie ld :

th e

th e o ry .

o f th e

Some

q u a n tu m

s ta tis tic a l

th e

p h o to n f i e l d

le a d s

to

as

c o lla p s e and

re v iv a ls

new in

and

d y n a m ic th e

R abi

n u ta tio n . F i r s t we

r e v i e w t h e m a in r e s u l t s

w ith r e s p e c t t o t h e a to m ic

d y n a m ic s .

a to m i n t h e e x c i t e d s t a t e w h ic h a fie ld

o f n p h o to n s .

in th e e x c ite d s t a t e

o f t h e J a y n e s - C u m m in g s m o d e l We c o n s i d e r a t w o - l e v e l

e n t e r s a r e s o n a n t c a v i t y w ith

The p r o b a b i l i t y

Pe n o f t h e a to m t o b e

i s th e n g iv e n by

Pe , n ( t ) = 1/2 { l+ c o s[2 0 (n + l)1/ 2t]}

w h e re

n

is

th e

s in g le -p h o to n

R abi

f l u c t u a t i n g num ber o f p h o to n s i n i t i a l l y th e

q u a n tu m

R abi

s o lu tio n s

needs

p r o b a b i lit y d i s t r i b u t i o n p (n ) at t

freq u en cy .

W ith

a

p r e s e n t in th e c a v ity ,

to be

a v e ra g e d o v e r th e

o f h a v i n g n p h o t o n s i n t h e mode

= 0: oo Pe(t ) = 1/2 2 p(n) { l+ c o s[2 n (n + l)1/ 2 t ]} 4a~0

At a

lo w

th e rm a l

a to m ic - b e a m

flu x ,

th e c a v ity c o n ta in s e s s e n tia lly

p h o to n s an d t h e i r num ber

fo rm in g t o

B o s e -E in s te in

is

s ta tis tic s .

g i v e n b y P^.j1(n ) = n 11^ / ( ^ t h + l ) n + 1 *

a ra n d o m

q u a n tity con­

In

case

th is

w ith th e a v e ra g e

p (n )

is

num ber

o f t h e r m a l p h o t o n s b e i n g n t h = [ e x p (h V /k T )

- l ] ” 1 . The d i s t r i ­

b u tio n

an

of

R abi fre q u e n c ie s

o s c illa tio n

Pe ^ th *

p h o to n s s t o r e d chan g es. t

= 0,

If th e

P o is s o n ia n : a d e p h a s in g

re s u lts

h i g h e r a to m ic - b e a m

in th e c a v ity a co h e re n t

p ro b a b ility p c (n )

in

is p re p a re d

d is trib u tio n

p (n )

s ta tis tic s

in th e c a v ity is

g iv e n

The P o is s o n s p r e a d

R abi o s c i l l a t i o n s ,

f i r s t e x h ib its a c o lla p s e .

f l u x e s t h e num ber o f

in c r e a s e s and t h e i r

fie ld

= e x p ( - n ) n n/ n !

o f th e

a p p a r e n t ra n d o m

T h is i s

a

in n g iv e s

and th e r e f o r e

d e s c rib e d

by

at

Pe ^c ( t )

in th e re s o n a n t

14

SIN G L E -A T O M OSC ILLA TO R S

c a s e by t h e a p p ro x im a te e n v e lo p e p e n d e n t o f th e a v e ra g e n o t h o ld f o r n o te d l a t e r

e x p ( - «fl.2t 2/ 2 )

p h o to n num ber

(th is

n o n re so n a n t e x c i ta t i o n ) .

T he c o l l a p s e

t i m e s f o r w h ic h Pe ^c ( t )

L a te r

VQ' C ( t )

s ta rts

o s c illa tin g

th e n

e x h ib its

a g a in

is

a l.,

re c o rre la tio n s

i n a v e r y c o m p le x

w as a l s o is

a ran g e

(re v iv a ls )

w ay.

and

As h a s b e e n

o c c u r a t tim e s

w i t h TR = 2 TT(n) ^ 2/S 1 l ( E b e r l y

= kTR (k = 1 , 2 , . . . ) ,

1 9 8 0 , N a ro z h n y e t a l . ,

1 9 8 3 ).

in d e ­

in d e p e n d e n t o f tim e .

sh o w n b y E b e r l y a n d c o - w o r k e r s t h e r e c u r r e n c e s e t a l.,

is

in d e p e n d e n c e d o e s

i n o t h e r w o rk . A f t e r t h e c o l l a p s e t h e r e

o f in te ra c tio n

g iv e n by t

and

1 9 8 1 , Yoo e t a l . ,

B o th c o l l a p s e a n d r e v i v a l s

1 9 8 1 , Yoo e t

in th e c o h e re n t s t a t e

a r e p u r e l y q u a n tu m f e a t u r e s a n d h a v e n o c l a s s i c a l c o u n t e r p a r t . The i n v e r s i o n

a ls o

c o lla p s e s

c h a o tic B o s e -E in s te in f i e l d th e

p h o to n -n u m b e r s p r e a d

s ta te

is

in th e c a se

R a d m o re ,

f a r l a r g e r th a n

o v e rla p and i n t e r f e r e

i r r e g u l a r tim e e v o lu tio n . s e n te d

by an

A

b u t no r e v i v a l s .

c o n s id e re d a s a

c le a r

q u a n tu m

of a

1 9 8 2 ).

H e re

fo r th e c o h e re n t In a d d itio n ,

to p ro d u c e

c l a s s i c a l th e rm a l

e x p o n e n tia l d i s t r i b u t i o n

show s c o l l a p s e ,

It

re v iv e s

a n d t h e c o l l a p s e t i m e i s m uch s h o r t e r .

r e v i v a l s c o m p le te ly

le s s

and

(K n ig h t an d

o f th e

fie ld

th e

a v e ry re p re ­

in te n s ity

a ls o

T h e re fo re th e r e v iv a ls can be fe a tu re ,

b u t th e c o lla p s e

is

c l e a r - c u t a s a q u a n tu m e f f e c t .

is

in te re s tin g

p ro c e s se s th e

t o m e n ti o n t h a t

in th e

R abi fre q u e n c y tu r n s o u t

c a s e o f tw o -p h o to n

t o b e 2 X l(n + 1 )

ra th e r

t h a n 2 f L ( n + l ) , e n a b l i n g t h e sum s o v e r t h e p h o t o n n u m b e rs i n Pe ( t ) th e

to be c a rrie d

in v e rs io n

seq u en ce

o u t in s im p le

re v iv e s p e r f e c tly

(K n ig h t,

in je c te d

in to

m odel i s a

w ith

fo rm .

a

In t h i s

case

c o m p le te ly p e r io d ic

1 9 8 6 ).

The e x p e r im e n ta l s e tu p u s e d f o r J a y n e s - C u m m in g s

c lo s e d

show n

th e e x p e rim e n ta l t e s t on

F ig .

s u p e rc o n d u c tin g c a v ity w as t h e

The

a to m s a r e

in th e u p p e r s t a t e

63 P 3

of

th e m aser

tra n s itio n .

p o p u la te d

by e x c i t a t i o n w ith fre q u e n c y -d o u b le d l i g h t o f a dye

la s e r.

T h is

1.

o f th e

/ 2

T h e a to m s a r e m o n i t o r e d b y u s i n g f i e l d

d e te c tio n can be

p e rfo rm e d s t a t e - s e l e c t i v e l y

R y d b e rg l e v e l , io n iz a tio n .

T h is

by c h o o s in g t h e

15

H W A L TH E R

p ro p e r f ie ld

s tre n g th .

Liqu id H e li u m Tem perature

F ig .

1

S chem e o f t h e s i n g l e - a t o m m a s e r f o r m e a s u r i n g q u a n tu m

c o lla p s e and r e v i v a l In m ost o f

th e

(R em pe, W a l t h e r ,

e x p e rim e n ts

K le in ,

(M e s c h e d e ,

1 9 8 5 ; R em pe, W a l t h e r a n d K l e i n ,

1987)

1987)

W a lth e r and

th e tr a n s itio n

M u lle r, 63 p ^ / 2

61 d 5y 2 w i t h a f r e q u e n c y o f 2 1 .4 5 6 GHz w as i n v e s t i g a t e d . th e

c a v ity

is

tu n e d

in

re so n a n c e

n u m b e r o f a to m s i n t h e u p p e r s t a t e

to

th is

s p o n ta n e o u s e m is s io n . The tu n i n g o f t h e c a v i t y s q u e e z in g th e c a v ity w ith p i e z o e l e c t r i c a to m s i s v e r y lo w s o c a v ity a t

t i m e o f t h e a to m s w i t h t h e c a v i t y s e le c to r.

t h e e n e r g y e x c h a n g e b e tw e e n in v e s tig a te d .

e le m e n ts .

l e s s th a n u n i t y . fie ld

In t h i s

th e

enhanced

i s p e rfo rm e d by

t h a t th e a v e ra g e num ber o f

a tim e i s u s u a l l y

o f a F iz e a u v e l o c i t y

tra n s itio n ,

d e c r e a s e s o w in g t o

~

When

The f l u x o f a to m s i n t h e

The i n t e r a c t i o n

c a n b e v a r i e d b y m ean s w ay ,

t h e d y n a m ic s o f

t h e a to m a n d c a v i t y

fie ld

can.

be

16

SIN G L E -A T O M OSCILLA TO R S

W ith

v e ry

lo w

e s s e n tia lly q u a n tity

a to m ic - b e a m

th e rm a l

a c c o rd in g

v e lo c ity

to

in te ra c tio n

a to m i c - b e a m

is

a

Pe ( t )

th e p r o b a b ility

c o lla p s e s ;

b o th c o lla p s e

A t h ig h e r

in th e c a v ity

th e p r o b a b ility

d is trib u tio n a fte r

re v iv a ls

and

sp re a d

ch anges.

d is trib u tio n

in n r e s u lt s

in

an d t h e r e f o r e t h e e n v e lo p e

th e in

th e

o f t h e a to m

ra n d o m w a y .

c o lla p s e

i n a v e r y c o m p le x w ay .

and

When

in c r e a s e s and t h e i r s t a t i s t i c s

T h is

a g a in

a ra n d o m

so t h a t t h e num ber o f p h o to n s

case o f a c o h e re n t f ie ld

o s c illa tin g

c o n ta in s

i n t e r a c t i o n v a r i e s w ith

an a p p a re n tly

th e th re s h o ld

P o is s o n ia n .

Pe ( t )

c a v ity

s ta tis tic s .

a fte r

d e p h a s in g o f th e R ab i o s c i l l a t i o n s , of

th e

T h e ir num ber i s

f l u x e s t h e a to m s d e p o s i t e n e r g y

in th e c a v ity

For th e

changed,

s ta te

tim e in

th e m aser re a c h e s s to re d

o n ly .

B o s e -E in s te in

o f t h e a to m s i s

b e in g in th e e x c ite d th e

flu x ,

p h o to n s

PeC ^)

s ta rts

As m e n t i o n e d a b o v e ,

th e c o h e re n t

s ta te

a re

p u re

q u a n tu m f e a t u r e s . The a b o v e -m e n tio n e d m e n ta lly . and r e v iv a l

have

been

o f m e a s u r e m e n ts

(R em pe,

p ro b a b ility

d e m o n s tra te d e x p e r i­

c le a rly

show t h e c o l l a p s e

p r e d i c t e d b y t h e J a y n e s -C u m m in g s m o d e l.

show s a s e r i e s m aser

e ffe c ts

The e x p e r im e n ta l r e s u l t s

W a lth e r

Pe ( t )

l a r g e r N and a

th e s in g le -a to m

1 9 8 7 ).

P lo tte d

is

th e

o f f i n d i n g t h e a to m i n t h e u p p e r m a s e r l e v e l

f o r in c r e a s i n g a to m ic f lu x fo r in te ra c tio n

o b ta in e d w ith

and K le in ,

F ig u re 2

tim e s

N.

The

b e tw e e n

50

s tro n g v a r ia tio n a n d 80

r e v i v a l sh o w s u p f o r N

a c tio n tim e s l a r g e r th a n

140

s.

s

o f Pe ( t )

d is a p p e a rs

= 3000 s " 1

fo r

fo r in te r ­

The a v e r a g e p h o to n num ber in

t h e c a v i t y v a r i e s b e tw e e n 2 . 5 a n d 5 , a b o u t 2 p h o t o n s b e i n g d u e to

th e

b la c k -b o d y

fie ld

in

th e c a v ity

c o rre s p o n d in g

to

a

t e m p e r a t u r e o f 2 . 5 K. T h e re i s

a n o th e r a s p e c t o f th e s in g le -a to m

in te re s tin g : th e

c a v ity .

fo llo w in g .

th e n o n - c la s s ic a l T h is

is

s ta tis tic s b rie fly

o f th e p h o to n s d is c u s s e d

T h e r e a r e tw o a p p r o a c h e s t o t h e

t h e o n e -a to m m a s e r. a p p ro a c h to

p r o b le m

m a s e r w h ic h i s v e r y

F ilip o w ic z e t a l .

d e s c rib e th e d e v ic e .

(1 9 8 6 )

in

in th e

q u a n tu m t h e o r y o f u s e a m ic ro s c o p ic

On t h e o t h e r h a n d ,

L u g ia to e t

H W ALTHER

0s

£ CL

0) ■o

cO cn if)

-t—i— i— |— i— i— i 1— |— i— i— i— i— |— I— r- 30

0.7

T = 2 .5 K

N = 2 0 00 s ‘ 1

^ 0 .6 Q?

|0 ,5

50 "O ® "5 c 60 — ^ co

15 o n o ion trap

endcap ring electrode Mg beam

electron gun

electron beam

i-

10mm

Fig.

3

S c he me of the Paul tr ap u s e d for the e x p e r i m e n t s

(for de t a i l s see D i e d r i c h a n d Walther,

1987)

25

H W ALTHER

CN cn

-20

0

TIME Fig.

4

20

40

(nS )

Re su lt s for the i n t e n s i t y c o r r el at io n.

Antibunching

for a si ng le ion for d i f f e r e n t laser inte ns it ie s, fr om b o t t o m to top

(see D i e d r i c h and Walt he r,

in c r e a s i n g

1987 for details)

26

SIN G L E -A T O M OSCILLA TO R S

b e m e a s u re d f o r n e g a t i v e L . The l a s e r i n t e n s i t y a

to

d

and th e r e f o r e

th e

d e c r e a s e s fro m

a v e r a g e tim e i n t e r v a l

i n w h ic h a

s e c o n d p h o to n f o llo w s a f i r s t one i n c r e a s e s . For

a s in g le

io n

th e

in te n s ity

c o rre la tio n

fu n c tio n

is

g iv e n

by g(2) ( f ) = l - e ' W 4 [cosnt+(3Y/4fi)sinflt] ft2 = Q + A2 - (7/4)2 ,

w h e re

reso n an ce, o rd e r to

check

sq u a re s f i t

th e

to

dependence

on t h e

be

in te n s ity

th e

s ig n a ls o f F ig . c o u ld

b e in g

th e n a tu r a l lin e w id th ,

4 a ls o

e v a lu a te d .

A

th e d e tu n in g . o f IX 2 ,

dependence

d a t a p o i n t s w as in te n s ity

th e R abi fre q u e n c y a t

and

c o u ld

be

q u ite

(D ie d ric h and W a lth e r,

The f l u o r e s c e n c e

a s in g le

in te re s tin g

p ro p e rty :

in

th e

a s m a ll tim e

s to re d

e x p e c te d f o r a P o is s o n ia n d i s t r i b u t i o n ,

1 9 8 7 ).

io n h a s

flu c tu a tio n s o f th e i n t e r v a l 2

rT

(1 -

1*i) A.

ct)

dr.

-T

This expression clearly shows that sub-Poissonian statistics necessarily requires a negative function A(r). A trivial example is the monomode N-photon field which has a constant negative A(r) and

= 0.

The sub-Poissonian character was observed experimentally by Short and Mandel (1963 a-b) in the process of resonance fluorescence emitted from one atom. 2°/ The antibunching effect is another purely quantum effect. It is also due to a non classical behaviour of A(r). For classical fields A(r) has a bump at r=0 (Eq. 11), (’bunching effect’). But this property may be inverted, even in the case where A is positive. It is the case for photons spontaneously emitted in atomic beam experiment : the ’intensity’

40

PH O TO D E T EC T IO N A N D PH OTOSTATISTICS

correlation of the light is minimum for equal times tj = t2 (Kimble, Dagenais and Mandel, 1977 ; Cresser et al., 1982). This photon antibunching effect has a simple quantum explanation : just after the emission of a first photon, the atom is in its ground state and then it cannot immediately emit a second photon. A collective antibunching effect is reported in section VI. These two quantum effects (sub Poissonian statistics and antibunching) are distinct as it is clearly explained by Short and Mandel (1983-b, p 673). (b) Moreover one can imagineother sorts of detectors. Mandel (1966),for example, suggested to use the process of

stimulated emission as a basis for detection. The effectof shining thelight on

these spontaneously decaying atoms is that they emit photons rather than absorb them, therefore the operators would occur in anti-normal order in Eq. (17). Different operator orderings are associated with different sorts of experiments. Symmetrical ordering of the operators is also suggested (Glauber, 1970 , p. 72). The problem of ordering field operators and the correspondance between functions of c-numbers and functions of q-numbers has been systematically treated by Agarwal and Wolf (1968 a-b) and Lax (1968). (c) The above derivations suppose ideal photodetectors with broad band sensitivity function S(u>), much broader than the spectral width of the incident light. These expressions need to be modified when the optical field has a substantial bandwidth over which the detector response varies (see Eq. 14). Glauber (1965)indicates that the correlation function entering into the expression for

...tn) has to be convolved by a certain response function of the detector.

This problem was further investigated by Rousseau (1975) and Kimble and Mandel (1984) : a first order perturbation theory shows that the photocounting distribution is still given by Eq.(17), together with

kW

r rt+T

= I I

d t 1

A-

dt"" E

( t 1)

A+ E

*s lar8er than unity inside a cone of half-angle ec * 1.4 k a (N)1/3

(37)

where N is the number of atoms in an absorption length. The spatial correlation function G^(r ,t ; r ’,t) was investigated, it contains

(36)

46

PH O TO D E T EC T IO N A N D PHOTOSTATISTICS

a) N antibunching terms . b) bunching terms > n*Pc) The emission of two photons by atom p is also described by antibunching terms, , npep,

and

like

like a nd

time,

This and

wave

c a n be is

not

vanish

field

mechanical

at

o p t i c s is ze r o

so b e c a u s e

the

a measurement

mechanical

all

waves

one

at o t h e r

description

a

only

for

can

point

in

point.

This

of

inter­

the

in

the

of

a free

the only

x^

in

Fig.

on

the p h o t o e l e c ­

is p r o d u c e d

of

light

the

description, part

form

of

the

at

the

operator

a single-frequency

classical

1 in

rely

annihilation

measurement

the p o s i t i v e - f r e q u e n c y

d i s t i n g u i s h e d by

light

electron

the p h o t o n

consider

as

description

detectors

in w h i c h

If we

the p o i n t a re

same

in d e s c r i b i n g

source

express

unity

that

quantum

of a p h o t o n ,

role

theory. each

of

We

variation

correlation

identically.

the

the

As p r a c t i c a l l y tric

depth

be

0.

effect.

The

expense

be

fact

affected

in

magnitude might

two

angle

| x ^ - x 2 |.

consequency

be

waves

i.e . ,

+

product

at a

up

term,

the

intensity

classical

together

effects

term

the

+ 2 < IA I B > + 2 < I A I R > c o s [2 tt( x 1 - x 2 )/L] . (4)

o f the

the

with

of

obtain

+ < Ig>

A coming

the

we

spacing

corresponding wavelength

because

average

assumption,

in

plays

the

quantum

contribution

t h e n we

f ie l d ,

(Hilbert-space

a

of

c an

E ^ + ^(x ^ )

at

operators

a caret)

E (' + h x i ) = a Ae x p ( i k r A i ) + a g e x p ( i k r B i ) ,

(6 )

55

C K H O NG , Z Y OU AND L M ANDEL

where

and

f i el ds

g e n e r a t e d by

f o l lo ws s ome

that

narrow p

^

1

l

the

the

s ig n a l

l

characteristic

of

state

th e n

|1

(6),

=

t he r e

calculation

the

However

2

o f the

that

For

detector

value A and

degenerate

appropriate

this

expectation

of

the

operator

B correspond parametric

state

signal

positioned

and

is

the

to

down-

two-

idler photons

state

and

the

in Eq.

(7)

is

the

are

field

operators

easily

evaluated

to

(8)

interference

t erm,

just

as

in

show

up

the

classical

intensity.

fourth-order

6x^

within

Ki 6 x i ,

interference

p r o b a b i l i t y p ^ 2 (x ^ , within

the

because

reduces

is no

of

the

,1 >,

together.

probability

that

at

(7)

expectation

modes

process,

P 1 ( x i ) 6 xi

the

the

(1963a,b)]

l

idler

produced

so

v l

suppose

an d

Eq.

on

It t h e n

a photon

[Glauber

If we

always

the

of d e t e c t i n g

s ta t e .

Fock

g i v en b y

acting

B respectively.

given by

l

a factor

given

conversion

operators

= K . 6 x . ,

photon

an d

is

x^ , a n d < > r e p r e s e n t s

for

A and

the p r o b a b i l i t y

l

is

annihilation

sources

range

(x.)fix.

where at

a^ a re p h o t o n

an d

)6x^6x2

effects

detecting

in w h i c h

case

in the

photons

[Glauber

at

x^

joint and

x2

(1963a,b)]

P l 2 (x i > x 2 ) 6 x i 6 x 2 = K 1 K 2 6 x i 6::2

(9)

+ 2 < a J a g a B a A > { 1 + cos [2tt ( x 1 - x 2 ) / L ] } ) . For

the

right

two-photon

vanish

state

a n d we

tEe

first

two

terms

on

the

obtain

P 12 ( x 1 , x 2 ) 6 x 1 6 x 2 = 2 K 1 K 2 6 x 1 6 x 2 { 1 + cos [2 tt( x ^ - x 2 )/L] } . (10) Interference classical 100

%.

This

position

shows

up

again

description,

can

implies

that

completely

another photon

at

but

as in

the rule

a cosine this

case

detection out

modulation, the

of a p h o t o n

the p o s s i b i l i t y

certain positions.

as

visibility at

in

the

n is one

of d e t e c t i n g

56

N O N C LA SSIC A L PH O TO N IN T E R FE R EN C E EFFECTS

It s h o u l d be particle

nature

separate appears of

the

under

noticed

positions

the

extended

which

same

mechanics

o f an

photons

regions

integrate

th the

the

% visibility

c a n n o t be

time.

This

formally

in

Fock

annihilation

is

due

to

the

detected

at

two

particle the

state

operator

nature

change

of

the

state

to

vacuum

state

the

representing

the

of a p h o t o n .

In p r a c t i c e ,

to

at

100

f r o m a s i n g 1e - p h o t o n

action

detection

the

of a p h o t o n ,

in q u a n t u m field

that

detected

A x ’s c e n t e r e d

the

measured

are not

foregoing

joint

at

x^

at and

expression

detection

a point but x^,

so

o v e r Ax.

probability,

over

t h a t we

ought

F r o m E q . (10),

P ^ 2 ^ X 1 ,X2 ^ ’

§ * ven

by x^+Ax/2

P1

2

(' Xl ’ X2 ' 1

x 20 + A x / 2

Xj - A x / 2 s

"A x/ 2

2 K 1 K 2 ( A x ) 2 {1 Due

to

the

r e d u c e d by is

finite the

applicable

to

fa ct o r was

2.4

The

The

the 0.55

Ax,

the

visibility

[s i n (ttAx/L) / (ttAx/L) ] 2 , a n d classical in

the

(11)

(- a * 7 1 ) cos (2 tt( x 1 - x 2 )/L] } . tA x / L

detector width

factor

this

+

P 1 ? ( X1 ’ X? ) d x i d x ?

calculation

experiment

too.

n has

the

same

been factor

In p r a c t i c e

d e s c r i b e d below.

experiment

outline

of

the

interference

experiment

[Ghosh

and

Mandel

Fig. 2 O u t l i n e o f the t w o - p h o t o n i n t e r f e r e n c e e x p e r i m e n t , [ r e ­ p r o d u c e d f r o m G h o s h , R. a n d M a n d e l , L. 1987, P hys. Rev. Lett. 59 1 9 0 3 - 1 9 0 5 ]

57

C K H O N G , Z Y OU AN D L M ANDEL

( 1987)]

is

shown

oscillating crystal, and

the

where

some

A b e a m stop

cause

angle

the

an

T wo

plates,

g lass

a translator, direct

after

and

so

to

the

into

of

plane

amplification

and

a re

inputs

of a t i m e - t o - d i g i t a 1 c o n v e r t e r .

within

a 5 ns

interval

either

to

’s i m u l t a n e o u s ’ s i g n a l

the

accidental

overlap

subtracted

out,

are

treated

of u n c o r r e l a t e d

the

rate

of

R ELA TIVE

as

together

to

x^

pulses.

coincidence

and

x^

The re­

mm. on

edge

on

pulses,

start

and

stop

arriving counts,

photons

When

and

an

mounted

pairs

counting

SEPARATION

at

L-0.34

and

coincidence idler

Ml

bandwidth.

whose

the

UV

passing

mm and at

LilO

signal

the

is m a g n i f i e d

Pulse

and

to

after

t ub e s ,

fed

long

mirrors

spacing

incoming photons

shaping,

two

rad/s

Ax-0.14

two p h o t o m u l t i p l i e r

cm

±3.3°

come

fringe

thickness

the

of

while to

laser

half-frequency

3 x l 0 13

t his the

a 1.5

crystal,

filter with in

on

angles

latter,

m from

to m a k e

each

split at

argon-ion

f al l s

idler photons

formed as

collect

them

the

f r o m an

li n e

emerge

1.1

pattern

a lens

n m UV

deflects

signal

i m a g e d by

Light

UV p h o t o n s

interference

interference

2.

which

6^2° in a p l a n e

through

and

Fig. 351.1

idler photons,

beam. M2

or

in

the

or

due to

the

latter, are

provides

a

(x-x')/L

Fig. 3 E x p e r i m e n t a l r e s u l t s s u p e r i m p o s e d on the p r e d i c t i o n s of q u a n t u m t h e o r y g i v e n b y E q . (11) ( s o l i d c u r v e ) , a nd o f the cl as si ca l t heo ry with m a x i m u m m o d u l a t i o n (dashed curve), [re­ p r o d u c e d f r o m G h o s h , R. a n d M a n d e l , L. 1987, Phys. Rev. Lett. 59^ 1 9 0 3 - 1 9 0 5 ]

58

N O N C LA SSIC A L PH O TO N IN T E R FE R EN C E EFFECTS

measure

of

the

joint p ro bability

P x 2 ^ X l , X 2 ^ :> aP a r t

^rom a scale

factor. Fig.

3 shows

I x^-x^l* by

the

The

E q . (11)

solid with

adjusted dashed

c ur v e

classical same

are

in

found

in

Ax^0.14

Fig.

The

the

values

3.

clearly

favor

MEASUREMENT

3 .1

3 is

L ^ O .34 mm,

a nd

TIME

t wo

size

are

The

the

with

CORRELATION

by

of 0.275 to

t he

factor

data.

prediction

The

the

for

the

agreement predictions

The p r o b a b i l i t i e s

of

P (y 2 ^ 4 .9)-0 . 1 8 a nd

experimental of

scale

the

theoretical

a n d X 2 U ~0.44.

t his

of

given

with

visibility

the

values

of

agreement

maximum

the p r e d i c t i o n

OF

different

a plot

corresponding

P (y 2 ^ 0 . 4 4 ) - 0 . 9 2 , r e s p e c t i v e l y . f ore

three

of y 2 c o r r e s p o n d i n g

values

of x 2 of

for

for b e s t

X 2 -La s s ~4 . 9

values

Fig.

mm and

3 is

measured

to be

obtaining

curve

theory with

L.

the

obtained

arbitrarily

wave

Ax and

between

results

the

results

quantum

BETWEEN

TWO

there­

theory.

PHOTONS

Background

T he

t ime

correlation

between

in

the p a r a m e t r i c

by

B u r n h a m and W e i n b e r g

10 n s . The with

depended

and

lation

time,

Later

t im e

between

pulses

by

and

signal

making

use

What

they

tim e

between

In the

next

section,

out b y H o n g

ference

of

et a l . ( 1 9 8 7 ) ,

effects.

t ime,

T he

of d e t e c t i n g

signal

experiments

between

the two p h o ­ the

but

discuss

experiment

a l . (198 5 ) ]

for

time

u se

not

of

not

correlation

so m u c h

intense

the

light

only

depended

yielded

idler

on

the

gener­ corre­

pulses.

experiment

exhibited

a nd

the

intense

of s e c o n d - h a r m o n i c

two

corre­

o f the p h o t o - d e t e c t ­

measured

by

therefore

also

of a b o u t

the

(1986)

which

but

et

p s . Both

intervals

measured

time

of

the p r o c e s s

will

first

produced

an u p p e r b o u n d

idler beams

two p h o t o n s we

100

resolving

m e a s u r e d was

correlation

probability

the

[Friberg

from detection

only

coworkers and

lation

time

was

photons

a resolving

of orde r

the

idler

process

with

resulting

limited by

Abram

of

and

repeated

time

provided

a t i on .

second

were

resolving

therefore

ors.

pulses,

(1970)

on m e a s u r e m e n t

photoelectric tons,

down-conversion

measurement

improved

signal

carried

a subpico­

nonclassical measuring

photons

inter­

the

produced

joint in

59

C K H O N G , Z Y OU A N D L M ANDEL

B

Fig.

4 The

schematic

o f the

ti m e

correlation

parametric

down-conversion

after

f i e ld s

m i x e d by

splitter,

3.2

are

The

L et

us

pulse

classical

suppose of

a n d Vg

tha t

are

e a c h s o u r c e in shape

g(t)

are

the

combined expressed

as

shown

in

light

Fig.

so

amplitudes,

the

and

to a b a n d w i d t h

that

assumed

field

4 emits at

a very

same

4.

J g 2 (t)dt

to be

falling

on

random

the

g(t)

short

t ime,

t*16 fi-e l d s f r o m s o u r c e s

complex

again

Fig.

duration

corresponding

is n o r m a l i z e d

f i el ds

and

are

constant

envelope

two

be

that

anc*

pulse

d o w n -converted

description

identical

t hat

a beam

the

experiment.

is

A and

B.

the

re a l

s ha l l

assume

= 1.

The ph as es

of the

and

unrelated.

Then

Ago.

detector

We

and

i at

time

t can

as

V .(t)

= v^v.g(t)

V 2 (t)

= l/fivA g(t)

+ i/ffv g ( t + x ) ,

( 12) w h e r e R a nd T are beam splitter, arrival

t i m es

position across

In Sec.

the

the

reflectivity

w i t h R + T = 1, of

the

two p u l s e s

ar gu m e n t has

the

been

crosssection

2 we

resolving to

+ /TVggCt+x),

ignored

times

any

o f the

description

of

of

and

t is

at

ignored, the

transmissivity the

difference

o f the in

the

the b e a m

splitter.

The

because

the p h a s e

difference

mixed b eam

consideration

detectors, the

and

is

constant.

of

the p u l s e

widths

because

they w ere

irrelevant

interference

effects

discussed.

and

But,

60

N O N C LA SSIC A L PH O TO N IN T E R FE R EN C E EFFECTS

in p r a c t i c e , taneous

what

is

m e a s u r e d by

intensity

of

the

resolving width by

time A t ,

6t*l/Aa)

detectors

calculated

which

o f g(t). 1 and

as

is

t he

their

2.

but

its

assumed

Then

2 and

in Sec.

/ < I 1 (t)>dt

field,

a detector

We

+ if< r B>

/ < I 2 (t)> d t = R < l p >

+ T < I b>

integral

to be

much

integrated

from

the

over

instan­ the

greater

than

intensities

c r o s s -c o r r e 1ation

obtain

= T< 1 p >

is n o t

C(t)

the

measured c a n be

E q . (12)

(13)

and c (T ) = / / < I 1 ( t ) I 2 ( t ' ) > d t d t ’ = TR< I^>

+ ri?

(14)

+ r 2 < I A I B > + ■ff2

- 2 2 7 ? h ( T ) , where

h(x)

It is

clear

in

the

=

f r o m Eqs. (13)

average

no p e r i o d i c x,

is

an

If we

[/g (t ) g (t + t ) d t ]2 £

define

term

that

to

n is

interference

x,

t he

Although

last

te r m ,

effects

show

E q . (14) which

up

exhibits

depends

on

term.

visibility

interference

no

intensity.

variation with

interference

E q . (14)

that

integrated

1.

n in

the

the

usual

constant

always

l ess

term,

way

as

t he

ratio

then

it

follows

t h a n or e q u a l

to

50

%,

of

the

from

for

2 T R h ( t ) < IA Ig>

(15)

0

TR< I2 > + TR< Ig>

The p h y s i c a l previous 5.5 The

quantum we

parametric quencies, only

the

photon

reason behind

example

Although

+

in Sec.

state

represented

the

produced by

the

¥> = /duxf) (u)q /

signal

process

same

as

discussed

in

the

+o), o)q /

a nd

idler

modes

of

as h a v i n g w e l l - d e f i n e d

have

frequencies

linear 2

the

actually

two

the

in

the

description

treated

the p h o t o n s

is

2.

down-conversion

su m of

t his

mechanical

earlier

(T 2 +i?2 ) < IA Ig>

a wide

bandwidth

is w e l l

defined.

down-conversion

process

A gj

the fre­

, and

The

two-

c a n be

superposition 2

-a)) |

1

03o /

2

+UJ ’ coo /

2

- co

(16)

61

C K H O NG , Z Y OU AND L M ANDEL

where is

oj0 is

s ome

The

the

frequency

symmetric

f ie l d s

E ^ + ^(t)

on

detectors

by

the

2,

of the p u m p

E^^Ct) are

= /TEf + h t )

in

the

+ i

= i/ffE< + ) (t)

classical

the

t he

at 00 = 0 .

splitter

signal

and

falling

idler

+ ^ (t + t )

time

(17)

+ / T e J + ) (1: + t )

of

of d e t e c t i n g time

the

t,

photons

which

light,

is

is

by bo th

much

detectors

longer

than

any

giv en by

P 1 2 (t ) = K / / < E { ' ) ( t ) E ^ _ ) ( t ' ) E ^ + ) ( t ' ) E { + ) ( t ) > d t d t ’ and

can be

[Hong et P

1

where

(t)

2

readily

al.

C is

fields

description.

resolving

correlation

the b e a m

to

(J>(ojq / 2 + oj ,o)0 / 2 -oj)

is p e a k e d

*

j oi n t p r o b a b i l i t y

within

past

related

*

E ^ + ) (t)

b e a m a nd

function which

expressions,

~

The

and

1 and

following

E^ + h t )

as

weight

calculated

f r o m E q s . (16)

a nd

(18)

(17).

We

find

(1 9 8 7 ) ]

= C [T2 + R 2 another

2rah(x)]

(1 9 )

constant,

/ g ( t ) g ( t + T ) d t / / g 2 (t)dt,

(20)

h(T)

=

g(t)

= / (a>o / 2 + oj ,ojq / 2 - oj) exp ( - iw t) da)// (4) + ]

The last matrix element can be rewritten as

can

be

regarded

as

a

photodetection

This interpretation becomes clearer if we sum over the states ia> with a

weighting function R(a) incorporating the efficiency of detection and work with a single linear polarisation.

dt'

This gives the probability of detecting a photon to be

dt" S ( t " - t 1)

=

S |2 ( t 2 - i x 2 i) «3.

C—2 D3 - n. a). -------C-ib - ruo)_ -------2 2 |*2 | 1 1 1*1 I

(—2 ] 1

exp

1 1 2

( t j - I Xj 1 ))

2

1 - 2 1

C—1 ] 1 IX Kll

(28)

-jmn,

F

N

(ct) 1 ,o 2 , x 1 ,x2)

where

F J m n (a>1

e/ g.1 « 2

**2 )

\ ( 2 ) imn. N^ 1 -“ 2 ) X (oJo ,fa)l ’aJ2 ) £ i ^ n ^ i

[X2] i [X2] j

and the quantum state of the incoming p u m p is given by a coherent state with amplitude

76

PH O TO N S A N D A P P R O X IM A T E L O C A LISA BILITY

(The field could have a transverse profile but this is incorporated through (27).) the nonlinear susceptibility tensor.

On

performing the r integration we

x(2) is

obtain as a

factor in the remaining a) integration (Pike and Sarkar, 1987)

2' '

r

*3 4

n 2 0>2

0

c

[x 2] 3

n l°>l Cx l ] 3]

|~ 2 |

c

J

|* l|

2' rn2“ 2tX2]l ( n l“ l[x l]ll 4

Since

£3

and

c

£j_ are

IX2 1

large

+

c

compared

|* l|

to

J

kQl,

the

dominant

contribution

to

the

co integration is given by

k

o

=

n2 w 2 c

cos0~ + 2

ni^i c

cos#. 1

(29)

where

-----

|x.|

-

cose.

,

i - 1,2

(31) Pill -------I

=

(,- 11N) is i.n #, .

,

.

1

=

1 0 1 , 2

1

The equations (29) and (30) are the phase-matching conditions. valid in the limit £3 ^ , (satisfying oy[+o^

=

£j_ko

°°*

They would be strictly

Given any two values of oq and a)2>

co0), it is possible to find d \

and

an^

^2

satisfying (29) and

say (30).

Mollow (1973) considered this nearly phase-matched situation and approximated (28) by

77

E R PIKE A N D S SA R K A R

G( 0 >2 ) (x x , iC 2 ’ 1

U

0

(JV

0 do3j do)^

^

r

.

(1

2

2 r2

V

,3 d r exp

exp (-

«i

2-1

r3 4

(32)

i“2 (t2_la2 l)

exp (irj(ko -

Vr "k (2 ) j m n / 0>i0)~ 5(0) -03.,- t O xv /J 1 2 o 1 2

*

~

[X2 ] i [ X2 ] j ]

* 1-1

12

I 1- 2 |

1-2 I

(where the two detectors are taken to be in the plane ( * \ ) 2

= (x2)2 = 0)*

The

o) dependence is kept only in the exponent since in the phase-matched situation this is the most rapidly varying term.

Since we

assume

that the frequencies that we

are

concerned with are far away from any crystal resonances, it is satisfactory to consider X(2)jmn to be independent of frequency.

However in the search for photon tails it is

not adequate as done by Mollow to replace the o^c^ in (28) by o>Jg>2 .

In fact in this

example the interesting aspect of the photon tail comes precisely from the After performing the r integration the o3j and

032

integration is

03^0)2

factor.

78

U

O

PH O TO N S A N D A P P R O X IM A T E LO C A LISA B IL ITY

Uu

2'

O do)^ do)^ exp

i

'

r [°

" 4

n2u 2 [x2] 3 c |X2 |

n l Wl CX1 ] 3~| c IXj , j

rI

r n2w2[ x 2 ] 1 ( n i w i t x i ] i i * 42 C |~21 + C 1*11J

exp

exp [ ” ict)2 t 2] exp [ ~ ia,l t l ]

2(33)

5 (a)0 ~a,i “a)2^

with

t. 1

=

t. l

n. l

PM

(i - 1 ,2 )

c

The o)i integration is automatic with the use of the 5-function.

The Fourier theorem

concerning convolutions allows as to the rewrite the o>2 integration as

exp

dr I

b v i]

[ W r]

(3 4 )

12 ( r )

where

I x (r )

do)

I 2 (t)

do) exp

6(o))

6 ( o )q - o ) )

( o )q - o ) ) o )

ex p (-ia n

)

(35)

and

Here

^ ^Uu^-2Vuj - iu r j

(3 6 )

E R PIKE A N D S SA R K A R

U

-

^ 3

=

£ 3

j^n^cos^^ - n ^ c o s f l j J

79

*t2 + n ^ sinfl^ J

+

(3 7 )

and

V

+

2 r

2

|^kQ - njG)QC o s 0 j J

r

*

[n 2 cos02

- n^cos0^J

^ r ,3^'j n^a)Qs i n 0 j |^i2s i n 0 2 + n ^s i n0 j j

(38)

It is easy to show that

2o)

-ir-

to)

sin

Ia (r)

(3 9 )

and

I2(r)

Jw e x p [ * i H e x p [ - 1r ) e x p [ - * i f ]

Clearly I2 is exponentially localised.

(4 0 )

Ij provides an oscillating power-law tail.

In

contrast to the approximately localised photon formed through a generalised imprimitivity which has a photodetection probability falling off as r - ^, the asymptotic behaviour in parametric down-conversion is

TO)

Ot

Sin

—T r2

Moreover this localisation is highly anisotropic.

(4 1 )

In directions perpendicular to those

defined by 0* and 02 there is, in lowest order, no fall-off. approximate

phase

matching allows

only a limited

contribute to the idler and signal photons.

This is because the

directionality

In the directions

of

wavevectors

to

or 02 although a range

of frequencies are allowed, the range is not large enough to obtain the weaklv localised

80

PH O TO N S A N D A PP R O X IM A T E LO C A LISA B IL ITY

states of Jauch, Piron and Amrein.

The main reason for this restricted range is due to

the details of the parametric down-conversion experiment.

We are thus still some the laboratory.

way off

Indeed it

down-conversion, which

from producing

would seem

that any

themost localisedphotons possible in experiments, such

as

parametric

have a preferred direction ofwavevector (such as in the input

pump) are not likely to have available through atom-field interactions enough different types of wavevector to

give a localised photon states approaching

the optimal weakly

localised states.

ACKNOWLEDGEMENTS

We would like to thank C. M. Caves, J. G. Rarity and P. R. Tapster for discussions.

REFERENCES Amrein, W.O. 1969, Helv. Phys. Acta 42 149. Burnham, D.C. and Weinberg, D.L. 1970, Phys. Rev. Lett. 25 84. Friberg, S., Hong, C.K. and Mandel, L. 1985, Phys. Rev. Lett. 54 2011. Glauber, R.J. 1963, Phys. Rev. 131 2766. Hillery, M. and Mlodinow, L.D. 1984, Phys. Rev. A30 1860. Jakeman, E. and Walker, J.G. 1985, Opt. Commun. 55 219. Jauch, J.M. and Piron, C. 1967, Helv. Phys. Acta 40 559. Mackey, G.W. 1978, Unitary Group Representations in Physics, Cumming, Reading. Mollow, B.R. 1973, Phys. Rev. A8 2684. Newton, T.D. and Wigner, E.P. 1949, Rev. Mod. Phys. 21 400. Pike, E.R. 1986, Coherence, Co-operation and Fluctuations, Ed. H. Haake, L.M. Narducci and D.F. Walls, Cambridge University Press, Cambridge. Pike, E.R. and Sarkar, S. 1986, Frontiers in Quantum Optics, Ed. E.R. Pike and S. Sarkar, Hilger, Bristol. Pike, E.R. and Sarkar, S. 1987, Phys. Rev. A35 926. Pike, E.R. and Sarkar, S. 1987, Power Law Tails ofSingle Photon States inParametric Down Conversion, RSRE Malvern Preprint. Pike, E.R. and Sarkar, S. 1988, Photon Localisation in Parametric Down Conversion, RSRE Malvern Preprint Rarity, J.G., Tapster, P.R. and Jakeman, E. 1987, Opt. Commun. 62 201. Wightman, A.S. 1962, Rev. Mod. Phys. 34 845.

QUANTUM NOISE REDUCTION ON TWIN LASER BEAMS

E GIACOBINO, C FABRE, S REYNAUD, A HEIDMANN AND R HOROWICZ L a b o r a t o i r e de S p e c t r o s c o p i e H e r tz . ie n n e de l ' E . N . S . U niversity 4,

1.

Pierre

place Ju ssieu

e t M arie C u r ie ,

75252 P a r i s

T12-E01,

Cedex 05,

France.

INTRODUCTION

Among

the

processes

classical"

properties,

turns

to

out

Takahasi

be

1965,

the

a

M ol l o w

et

1 9 7 3 ) . The p r o c e s s

fields

at

frequencies w ith

conservation placed

a

requires

Degenerate threshold

and

( L o u i s e l l e t a l . 1961,

Mollow 1973,

cc>2

by

S toler

at

a non-linear

pump

field

that

coq = ^ + co2 ) .

frequency

1974,

coq

crystal (energy

The c r y s t a l

cavity operating

P aram etric

c a n be

e i t h e r below or

et

a l. al.

A m plifiers

(OPA)

(below

to x o>2 ) , w h i c h h a v e b e e n s h o wn b o t h

(M ilburn e t a l .

1981,

1984,

experim entally

(Wu e t

lig ht.

the n o n - lin e a r

A ctually

interactio n

i n v o l v e s g e n e r a t i o n o f t wo " s i g n a l "

co^

o p e ra tio n w ith

C ollett

"non

m ost o f t h e work ha ve c o n c e n t r a t e d on Q u a s i

O ptical

theoretically

param etric

one

1967,

l i g h t w ith

t h r e s h o l d d e p e n d i n g on t h e pump p o w e r .

Up t o now,

1984,

al.

in a resonant o p tic a l

above o s c i l l a t i o n

generate

optical

prototypic

Gr aha m

irradiated

which

1986)

L u g i a t o e t a l . 1982,

C ollett to y ie ld

interaction

et

al.

Yurke

1985)

squeezed s t a t e s taking place

in

and of the

82

OPA

Q U A N T U M N O ISE R E D U C T IO N ON TW IN L A SE R BEA M S

leads

to

d e a m p lif ic a tio n of

some q u a d r a t u r e out of

both

components of th e e l e c t r o m a g n e t i c

is

the

purpose of

theo retical

obtained ( 0P0)

on t h e

and

this

be

experim ental

results

N on-Degenerate O p tic a l

s h o wn

understood highly

that

features

In different

its

that

going

into

corresponding

give

proof (§

of

the

3 ) . The

representation properly

of th e f i e l d s

the no ise

in the g en eral experim ent

for

case.

In the

of

properties

tw in f i e l d s

are

a sim ple tre a tm e n t in a good p h y s i c a l

The s e c o n d o n e i s

is

in ten d ed to

model i n a f u l l y quantum based

upon

a classical

b u t i n w h ic h vacuum f l u c t u a t i o n s (§

4).

The l a s t the

fifth

our

r e d u c t i o n on t h e d i f f e r e n c e (§ 5 ) .

em its

This

"classical"

approach

s p e c tr a of the v a rio u s m easurable q u a n t i t i e s

and p r e s e n t

applications

one

crystal

be

w ill presen t th ree

one i s

corpuscular

th ird

accounted

we

p h o to n s which p r o v i d e s (§ 2 ) .

can

it

paper.

first

th e problem

have

) . In p a r t i c u l a r

The c o r r e l a t i o n

in th is

we

Param etric O s c illa to r

the n o n -lin e a r

of the

The

that

characteristics

photons.

studied

"corpuscular"

t w i n beams

of

the t h e o r e t i c a l s e c tio n ,

in sig ht

y ields

tw in

approaches.

of

a

fact

the i n t e n s i t i e s

the b a sic

terms

most

from t h e

correlated

between

manner

fields

on

paper to g iv e a review of

(above t h r e s h o l d o p e r a t i o n w ith

w ill

fluctuations

the c a v ity .

It

are

t h e vacuum

we w i l l d i s c u s s

observation

between th e

of

quantum

in ten sities

the

noise

of the

t wo

s e c t i o n w i l l d e a l w i t h new p o s s i b l e

tw in photons

g e n e r a tio n of am plitude

section,

t o s p e c t r o s c o p y and t o

s q u e e z e d l a s e r beams



6

).

the

et al

E G IA C O B I N O

2.

CORPUSCULAR MODEL

We in

83

first

present

which p h o to n s

a "corpuscular"

are considered

m o d e l(R e y n a u d 1987)

as c l a s s i c a l

particles.

We

c o n s i d e r o n ly th e ca se of n o n - d e g e n e r a te e m is s io n in which th e two

ty p es of

s ig n a l photons

th eirfreq u en cies

can be

(o^ ^ coz ) o r

d i s t i n g u i s h e d by e i t h e r

th eir polarizations.

s hown

both ex p e rim e n ta lly

( Bu r nh a m e t a l . 1 9 7 0 ,

19 8 5)

and

( Mo ll ow

Gr a ha m

theoretically

1984,

conversion photons

Hong

et

al.

et

1985)

the n o n -lin e a r c r y s t a l

a l . 1967,

that

in

em its p a ir s

1 e t 2 e a c h t i m e a pump p h o t o n i s

It

has been

F riberg

et al

Mollow 1973,

p a r a m e t r i c down of sim u lta n e o u s

annihilated.

Such a

t w i n p h o t o n g e n e r a t o r can be u s e d as t h e pumping m echanism f o r an

op tical

cavity

resonant at

the s ig n a l

a>2 . Above some pump p o w e r t h r e s h o l d ,

the

yields

i.e .

" t w i n p h o t o n " b e a m s Bi

having

highly

correlated

the

cavity

d ecorrelation delivered equal

at

in the

storage

inten sities

In

A ctually,

the c a v ity

b e a ms

since

the

go o u t

induces

some

: th e numbers of p h o to n s

other

a tim e

lo n g compared to th e

words,

the d iffe re n c e

must r e v e a l

and

b e am s a r e e x p e c t e d t o b e n e a r l y

counted d u rin g

S i (co) o f

spectrum

tim e,

t w i n b e am s

t wo o u t p u t

tim e.

system o s c i l l a t e s

c r y s t a l do n o t n e c e s s a r i l y

s ame

between th e

o n l y when

cavity

the

and

t wo l a s e r - l i k e

in ten sities.

tw in-photons produced in the of

a n d B2

frequencies

m easuring the n o ise

I between

t h e s i g n a l beam

a re d u c tio n of photon n o is e

inside

the

c a v ity bandw ith.

The prediction we

w ill

photons em ission

purpose of t h i s fo r the noise

assume are is

that

randomly denoted

the

section

to give a q u a n t i t a t i v e

s p e c t r u m Sj (co) . I n t h e p r e s e n t m o d e l e m issio n s of

distrib u ted . A.

is

In

T he

addition

the v ario u s mean

pairs

rate

each c o l l i s i o n

of

of p a ir o f any

84

Q U A N T U M N O ISE R E D U C T IO N ON TW IN L A SE R B EA M S

signal

photon

probability

w ith

R for

transm itted

the

coupling

m irror

being r e f l e c t e d ,

is

d e s c r i b e d by a

a probability

T fo r being

( w i t h R + T = 1) .

In

this

corresponding

framework,

to

we c a n e v a l u a t e

the d e te c tio n

the

spectrum

scheme s k e t c h e d i n

fig.

1

(to) :

S, ( 1 At w ith I(w)

and I ( t )

F igure

i At

=1

I(t)

= I 1 (t)

is

in serted

dt

(2)

(3)

non-degenerate

constituted

in

separated

exp(-iw t)

- I 2 (t)

1 : The

(NDOPO)

(1)

an

by

optical

optical a

tw in-photon

ca vity.

and

their

in ten sities

photodetectors

(PD) .

T he

n o is e spectrum Si

In instantaneous

photodetectors in teg ratio n ch aracteristic

beam

; I (to)

tim e

(TPG)

t w i n s i g n a l beams a r e and I 2

Ia

spectrum

expressions,

signal

tim e

The

generator

analyzer

a re measured by (SA) g i v e s

o f the i n t e n s i t y d i f f e r e n c e I = ^ - I

(to)

these

param etric o s c illa to r

At ;

is

I i (t)

in ten sities the F o u rie r

is

much

t h e symbol

and

I 2 (t)

are

longer < >

than

any

2

.

the

m o n i t o r e d by t h e

tra n sfo rm of I ( t )

the

two

; the other

in e q . (1 ) d e s c r ib e s

a

E G IA C O B I N O

"classical"

mean v a l u e

A

p articular

random v a r i a b l e s and

k2

and 2. if

event

thequantum

where

8

(t

is

8

detection the

-

tim e

sim plicity,

(

t

the

roundtrip

signal

on t h e

0

1

t

)

)

D irac

( no tim e

-

8

the

we s u p p o s e t h a t

|I(co )|2is

=

2

contributions the

reflectio n s

(t

1

and t h e n

1

t

to

:

t o be

+ k

;

t

2

t

)

(and

),

1

the

:

(4)

)

the

first

possible

the coupling m i r r o r ) , x is

and

)

inside are

t

of

this

the

the

cavity

sa me f o r

event to

) - e x p ( - i c o ( t Q+ k 2

(for

the

t wo

the F o u rier

t )

)

| 2

(5)

- k 2 ) c o t ) ]J

w eighted and

S i (co)

(Eq.l)

by t h e undergo

2

tim e

b y su mmi ng t h e s e

p r o b a b i l i t y TR respectively

m u l t i p l y i n g by

k

TR

1

k

k9 2

and kg

t h e n u m b e r A At o f p a i r :

00

S (co) = 2 A

The

(t 0

photons

em issions d uring the i n t e g r a t i o n oo

-

on

the spectrum

(Eq.5)

photons

assumed

1

thus

[u l - c o s ( ( k

We now o b t a i n

th e numbers k

event to I{ t) is

contribution

| X (co) | 2 = | e x p (-i co( t Q+ k i t

that

are

function

of

follow ing

measured as d e t e c t i o n r a t e s

reflection

m o d e s ) . The

transform

are

p articu lar

+ k

events.

c o u p lin g m i r r o r of th e p h o to n s

efficiencies

c o n t r i b u t i o n of t h i s

=

can be d e s c r i b e d by t h e

: th e em issio n tim e of a p a i r ,

the i n t e n s i t i e s

I(t)

85

taken over the v a rio u s p o s s ib le

of r e f l e c t i o n s If

et al

summation of

(TR

1

the

) (TR

2

) [1 *• - c o s ( k 1 - 2k

series

)

co t

(6 )

]

appearing in Eq.

6

leads

86

Q U A N T U M N O ISE R E D U C T IO N ON TW IN L A SE R B EA M S

S, (co) = So

-

1

( 7)

T2 + 4 R s i n 2

(?))

w i t h S o = 2A It com pletely to

(8)

clearly

suppressed at

the q u a l i t a t i v e

photon

noise

m ultiples

of

is

peaks,

Sj (co)

corresponding = = 2A)

2 : P r e d i c t e d n o i s e s p e c t r u m S^ transm ission

(co)

: T = 5 0 % (a ) ,

fo r

various values

of

the m irror

1 0 % ( b ) , 2% ( c ) . S 0

is

t h e u s u a l p h o t o n n o i s e . Th e f r e q u e n c y u n i t Q i s

the c a v i t y

free spectral range.

For a h ig h f i n e s s e

cavity

(T = 1

w ill

(co)

is

equations as

(co)

e q . (21).

(co) pi variance :

fluctuation

The d i f f e r e n t i a l

algebraic

ryj

w hile qt

the

:

( 29)

be

directly

< | p ° ut (co)

| 2

g i v e n by t h e o u t p u t

>

( 30)

Pi 4.3

F luctuation

We which

the

(r

r

1

=

spectra

w ill

first

lo sse s are = r , u* =

'l

2

in the balanced case

give

the

u,

s ame f o r

=

2

u,,

the

1

be

pump f l u c t u a t i o n s

fluctuations am plitude

r ’=

^

I t can im m ediatly that

the r e s u l t s

-

8

fluctuations

:

6

a1

a2

T

the 2

in the

t wo s i g n a l

= r

sim ple case

in

fields

).

seen

on E q s .

w ill

not

(27-a)

and(27-b)

bec o u p l e d i n t o

. Thus f o r t h e d i f f e r e n c e

of

the the

1

r = —

(Pt

- P

2

)

( 31 )

\ | 2

(w ith get

sim ilar notations

the

t

for

the

i n p u t a n d o u t p u t f i e l d s ) , we

follow ing equation:

r +

2

t ’ r = \f 2 r r

1 0

+ \[ 2 ^ r ’ ln

( 32 )

97

et al

e g ia c o b in o

r\i

We t h u s o b t a i n

r

for

th e F o u r i e r component r

(co) = -------------2 r ’ + icor

The o u t g o i n g f i e l d

r '"

(cflrr

r o ut

is

easily

( 33)

and

d e r i v e d from

^

(34)

to

1

,

the

s i m p l y g i v e n by t h e v a r i a n c e

(co) = V°r ui

In

(10)

( 33)

:

equal

difference

Sj

:

of

the

^>ut

r

2T'+ico'r

(co)

( 35)

i n p u t vacuum f l u c t u a t i o n s

squeezing

on t h e r c o m p o n e n t i s

of

4 wT' + co2 t 2 (co) = ------------------4 T i 2 + co2 t 2

the p re s e n t case,

are

to

2 v p r>

(co) +

the v a ria n c e s

norm alized

leads

ut

(co) = ------------------- r 2T'+ico'r Since

be a ms

’ 1 " (co)))

( 34)

2 T - 2 T 1- i o x r

:°u1

Sr

1" (co) + r

:

= \[2 t r - r 1"

H e n ce c o m b i n i n g Eqs

are

(co) o f r

and

(co) i s

the

( 36)

t h e mean i n t e n s i t i e s noise

proportional

spectrum to

(co)

on t h e

on t h e

SQ i s

equal

to

absence resu lt can

the

shot

t h e sum o f t h e of

as in

internal

be s e e n t h a t

(37)

n o i s e o n a beam w i t h a t o t a l in ten sities losses

the p rev io u s in ternal

in ten sity

:

4 u , r ' + co2 t 2 S J (co) = Sq ---------------------4 t ’ 2 + co2 t 2 where

tw in

of th e

(p/=0 , T ' = T ) , section

losses

tw in beams. one

(recalling

decrease

gets

inten sity In the t h e same

th at T=2r). I t

t h e amount o f n o i s e

98

Q U A N T U M N OISE R E D U C T IO N ON TW IN L A SE R B EA M S

r e d u c t i o n a t co= 0

by a f a c t o r

:

V* f* — = -----T*

( 38)

p,+ T

Note

that

this

factor

is

e m i t t e d p h o t o n s w hich do n o t r e a c h for

example

scattered

m irror d if f e r e n t

by

the

^ have

losses

through the o u tp u t m irro r.

in Fig. 4 fo r v ario u s v alues

the

shot noise

the

cavity

losses

p r o p o r t i o n of

or em itted

from t h e o u t p u t o n e ) . T h is

a s s o c ia te d w ith

4 :N o i s e

the

the p h o to d e te c to r crystal

losses

Figure

ju st

t o be

im plies

(they are through a that

k e p t much s m a l l e r

The n o i s e

spectrum i s

extra than s ho wn

o f y*.

spectrum o f

the i n t e n s i t y d i f f e r e n c e . S i s 2 r°' f o r a b e am o f i n t e n s i t y Ii + 2I , coc = -----is T

bandw ith.

(fj, = 0) , c u r v e

Curve (b)

(a)

to

corresponds

equal

to no i n t e r n a l

transm ission

and l o s s e s

(p = r) , w h i l e t h e t o t a l l o s s e s r' a r e t h e s a m e i n a) a n d b ) . T he f l u c t u a t i o n between S etting

the :

tw in

spectrum

fields

can

of

the

phase

be c a l c u l a t e d

in

difference t h e sa me w a y .

E G IA C O B IN O

s = —

et al

99

( qx - q 2 )

(39)

vl2 one f i n a l l y

S

(co)

gets

= 1 +

Figure t wo

: 4r r ' ( 40)

Noise spectrum

signal

field s.

Th e z e r o - f r e q u e n c y l i m i t

Schawlow-Townes l i m i t

This

the

fluctu ation

of a

to

the

when co t e n d s

phase d if f u s io n

process.

t o 0,

which i s

One c a n d e d u c e

s p e c t r u m SQ (co) o f t h e b e a t f r e q u e n c y b e t w e e n

two s i g n a l mo de s

fi = co

correspond

the

a s s o c ia te d w ith phase d i f f u s i o n .

expression diverges

characteristic the

o f the b e a t fr e q u e n c y between

-co2 =

\J"2 a

:

100

Q U A N T U M N O ISE R E D U C T IO N ON TW IN L A SE R B EA M S

= > Ti

situ atio n .

transm ission

to a n a l y t i c a l l y

components of

the

solve

am plitude

(co) .

here

(co) but ra th e r Hi restrict ourselves

spectrum

unequal

possible

rvy

(co) a n d S w ill

of

is

the

f o c u s on t h e i r p h y s i c a l to

the d is c u s s io n

difference much s m a l l e r

, Tz ) w h i c h

lengthy expressions

Sx (co) i n than the

corresponds

for

content.

of the n o ise

th e c a s e where signal

cavity

to our experim ental

104

Q U A N T U M NOISE R E D U C T IO N ON TW IN LA SER BEA M S

Figure

8 :

N oise spectrum o f

non-halanced

case.

coefficients

and

(T

= 0, 55 %

twin

b ea m

, T2

For

= 0,45%,

in ten sity

shot noise

pump

in

displayed

losses

transm ission p^

mean

a m p l i t u d e p a r a m e t e r o' i s a

the

the i n t e n s i t y d i f f e r e n c e curves,

are

slig h tly

= 0,48%, values

equal

= 0 ,5 2% )

d iffer

to 1 .1 .

beam.

Curve b)

30dB e x t r a

noise

at

e x p o n e n t i a l l y to ze ro

by

C u r v e a)

l i m i t e d p ump

beam w i t h

the

the

d ifferen t so

15%.

that

the

T h e pump

corresponds

corresponds

to

to the

zero frequency decaying

a t high fre q u e n c y w ith a h a l f width o f

0 . 1 5 coc . T he m a i n c o n s e q u e n c e o f t h e u n b a l a n c e in ten sity in

fluctuations

an i n c r e a s e d n o i s e

We d ifferen t only low

from u2

between

for

t h e mean

t h e pump

case

i n w h i c h p^

a n d r 2 . T h e n we f i n d

frequency p a rt and

now c o u p l e d i n t o

close of th e loss

to

is

that

th resh o ld

spectrum .

t h e pump

Sx (co) , r e s u l t i n g

of

shot noise

a n d 7^

the

(cr&l)

giving

and i n gives

is

the very S i (co) f o r

15% d i f f e r e n c e

t wo b e a m s ,

lim ited

are not very

th e added n o is e

F i g . ( 8 -a)

coefficients

in ten sities

field

that

a t a n y f r e q u e n c y on t h e w h o l e s p e c t r u m .

the

im portant very

cr = 1 . 1 ,

that

consider

are

is

and a s s u m in g

a t any f r e q u e n c y

:

E G IA C O B IN O

one

notices

frequencies. l ow

m oderate

Fig.

(8-b)

frequency,

situation

:

co # 0 .

Of

course,

the

are

may a p p e a r .

increased

into

of

observes if

the

a

the

noise

to

a

sharp r i s e

low

more in

But t h e

frequency

pump

breaks

down,

general

conclusion

rem ains c e r t a i n l y

at

realistic for

in ten sity

approach fo r

and m u l t i p l i c a t i v e that

b e c a u s e o f u n b a l a n c e o n l y i n a. l i m i t e d

frequencies

l ow

the n o ise

our sim ple l i n e a r i z e d

dynamics

at

a c c o u n t e x t r a pump n o i s e

corresponds

too la r g e ,

fluctuations

noise

increase

takes

which

one t h e n

fluctuations

5.

a

105

et al

(co) i s

range of

l ow

valid.

EXPERIMENT

F igure

9 :

acousto-optic beam -splitter

Experim ental s e t-u p . m odulator

;

F

FR : F a r a d a y r o t a t o r :

filter

;

; AO :

PBS : p o l a r i z i n g

; SA : s p e c t r u m a n a l y z e r .

We u s e a t w o - m o d e p a r a m e t r i c

o scillato r

o p e r a t e d above

106

Q U A N T U M NOISE R E D U C T IO N ON TW IN L A SER BEA M S

th resh o ld al.

to g e n e ra te

1987).

The

aim

high i n t e n s i t y of

fluctuation

s p e c t r u m on

tw in

beams.

We w i l l

shot

noise

the the

experim ent in ten sity

show t h a t

lim it

tw i n beams is

(Heidmann e t

to

study

difference

the

between the

s i g n i f i c a n t r e d u c t i o n below th e

is observed

over a

broad

range

of

frequencies.

Figure pu mp e d an

by a

shows t h e

single

external

50kHz).

(9)

mode Ar*

Fabry-Perot

(FR)

(AO)

are used

to o p t i c a l l y

back

reflected

light

then

focused in to

type

II

signal

of c u r v a tu r e ,

a large

finesse the

17mm.

is

is

the

flat

high

and t r a n s m i t s

signal

The f o u r phase

which i s

reflectin g

transm itting

fo r the

pump f i e l d .

from t h e

conditions

fo r both s ig n a l

inserted

discrete

series

of

for

cavity

f o r t h e pump b e a m .

0. 8% o f

the

and i d l e r f i e l d s

condition,

the c a v ity and low f o r

cavity

values.

(energy

resonance

In

in terv als

nanom eters)

around

the

well-known 1973).

high s e n s i t i v i t y

Thus t h e

electronic output m onitored

OPO l e n g t h

feedback

intensity. on t h e

of

the

This

is

so

that

it

delivers

For

this

purpose,

we ak c o u n t e r p r o p a g a t i n g

(Smith

stabilized

a nearly

the

(a f e w

cause of the

OPO t o v i b r a t i o n s

h a s t o be a c t i v e l y

for a

practice,

occurs only in very sm all le n g th values.

The

in frared

are only f u l f i l l e d

length

an

infrared

oscillatio n

these

in

w i t h a 2cm

the

C onsequently,

and i d l e r )

is

a 7mm l o n g ,

OPO o s c i l l a t i o n c o n d i t i o n s

matching

strong

t h e OPO. The pump l i g h t

p a r t of the green l i g h t .

conservation,

laser

The i n p u t m i r r o r ,

highly

a n d i d l e r be a ms a n d

output m irror and

isolate

4

is

of

and an a c o u s t o - o p t i c m o d u l a t o r

KTP ( KTiOPO ) c r y s t a l ,

length

The OPO i s

(re sid u a l frequency j i t t e r

coming from

phase-m atched

set-up.

a t 528nm s t a b i l i z e d on

t h e p a r a m e t r i c me di um,

c a v ity of

radius

ion l a s e r

cavity

A Faraday r o t a t o r

optical

experim ental

OPO

by

constant

output

is

i n f r a r e d b eam

(I R)

lig h t

E G IA C O BIN O

which

is

transm itted

deflected stable

by t h e

reference

stabilized

on

domains,

back through

level.

the

As a

side

corresponding

experim ental

conditions,

of

the

a slight

we

in tensities

side

lig h t).

The

(PBS)

which

quantum

regim e.

have

m illiw atts

are

efficiencies

coated.

two

photocurrents

The

subtracted

using

a 180*

difference

is

checked

beam splitter

are

are

w ithin

matched

two at

corresponding analyzer.

200mW o f g r e e n

by

less

of

90%.

b e a ms

are

\xm

and

s t o p p e d by

a polarizing

of

: the

by

a

an tireflectio n and

The n o i s e

then on t h e

spectrum a n a ly z e r

the

detection

im perfections

The

surfaces

analysis.

than 1 % ; the 1 %.

A ll

am plified,

com biner.

m onitored

characteristics

carefully

beam

(80mW

t w i n be a ms

= 1.048

A

are

power

co n n ected to a computer f o r d a ta

pump

interv al,

a n d t h e n f o c u s e d on t wo InGaAs p h o t o d i o d e s

t h e t wo i n f r a r e d

the

for

separated

by

between

behaviour

Above t h r e s h o l d

two c r o s s - p o l a r i z e d

encountered

been

the o s c i l l a t i o n

t r a n s m i t t e d pump beam i s

t w i n b e am s

beam splitter

The

some

a r e d e t e r m i n e d by t h e c o l l i n e a r p h a s e m a t c h i n g

The

resulting

f r e q u e n c i e s . In

w avelengths,

The r e m a i n i n g

filter.

m i s m a t c h b e t w e e n t h e OPO

of

f ew

em ission

X2 = 1 . 0 6 7 p,m, conditions.

a

is

b u t one c an a lw a y s c h o o s e f o r

a b istable

of

t h e OPO l e n g t h

can o b s e rv e a b i s t a b l e

g r e e n l i g h t ) , t h e OPO e m i t s

(with

and n o t

such sm all o s c i l l a t i o n

eigen

length,

right

does not y i e l d

of

cavity

a fu n c tio n of c a v ity

which

a

the

one

m irror

and com pared t o a

consequence,

of

to

and

stabilization

the in p u t

a c o u s to -o p tic m odulator,

frequencies

as

107

et al

of

c h a n n e ls have the p o la riz in g

a m p lifie r voltage gains

overall

common mode r e j e c t i o n

c h a n n e l s h a s b e e n m e a s u re d by m o d u l a t i n g t h e a

frequency

coherent

The r e s u l t

of

of

peak this

10

MHz

reduction

measurement i s

and on

m easuring the

25 d B .

the

spectrum

108

Q U A N T U M N O ISE R E D U C T IO N ON TW IN L A SE R BEA M S

A

key p o i n t f o r

is

the c a l i b r a t i o n

we

have used

the

Ei

rotation

and of

between fields

Ea

the

in

the of

Eg

o f s u c h an e x p e r i m e n t

level.

half-w ave p la te

E2 e m i t t e d

axes and

shot noise

beam splitter

29

the

of

a rotating

polarizing

fields

the r e l i a b i l i t y

(labeled by t h e

p late

are

test,

in f r o n t of

in F i g . 9).

The two

OPO u n d e r g o a p o l a r i z a t i o n where 0 i s

the angle

and o f t h e p o l a r i z e r .

respectively

t h e p o l a r i z i n g beam s p l i t t e r

in serted

A/2

half-w ave p la te ,

the

As a f i r s t

transm itted

The t wo

and r e f l e c t e d

by

:

EA = c o s 2 0 E 1 - s i n 2 0

E2

(50-a)

EB = s i n 2 0

E2

(50-b)

E1 + c o s 2 0

When 0 = 0 ° role

and th e

(modulo 4 5 ’ ),

measured s ig n a l

t w i n beam i n t e n s i t i e s . half-w ave p la te 50% beat

term s

observed gives

between

shot

noise

for

the

signal

Ia - I

(modulo 4 5 * ) ,

tw in

the

t wo

fie ld s is

between th e the

like

varies

an u s u a l the the

mo de s do n o t a p p e a r i n

the

the

measured s ig n a l

f o r a b eam o f i n t e n s i t y

that

system

a b o u t 5 THz,

C onsequently,

level

acts

p l a y s no

experim ental co n d itio n s,

the

One c a n show f r o m E q s . (50)

difference

beam splitter

in our

frequency range.

the

the

When 0 = 2 2 . 5 *

Since

fre q u e n c y between

crossed

is

and p o l a r i z i n g

b eam splitter.

the half-w ave p la te

Ii + I2.

t h e n o i s e p o w e r s p e c t r u m S$ (fi) sinusoidally

as a f u n c t i o n of

the angle 9 :

0 (Q)

= S 1 (Q)

F i g . 10 fixed

c o s 2 40 + S 0 s i n 2 40

shows

the

variation

(51)

of

f r e q u e n c y H / 2 tc = 8MHz. One o b s e r v e s

S$ (O)

recorded

at

a

a stro n g m odulation

E G IA C O B IN O

of

the no ise

noise

level

le v e l w ith the

at 0 ’ is

109

expected p e r i o d i c i t y

checked t h a t

frequencies levels

of

this

test,

(0=22.5*)

t h e YAG

higher the

of

22.5*.

to y ie ld

shot noise

l a s e r was

shot noise

level.

We

lim ited

at

YAG w i t h e q u a l me an i n t e n s i t i e s .

can

assert

that

coincides

w ithin

1%

(dashed l i n e

the

: the

t h a n 2MHz. We h a v e t h e n m e a s u r e d t h e n o i s e

OPO a n d

we

45*

we h a v e u s e d a cw YAG l a s e r

independent c h a r a c te r iz a tio n

have

of

a b o u t 30% l o w e r t h a n t h e o n e a t

As a s e c o n d t e s t , an

et al

From

the upper l e v e l of F ig. w ith

the

shot

noise

22

level

in F i g . 10).

^ )

(a)

A /V W

V

150 pA2/H z (b)

—i---------1---------- 1--------- 1----90° 18 0° Figure

1 0 :a ) V a r i a t i o n

in ten sity

in

terms

level

dashed

the

d i f f e r e n c e S$ (H)

(expressed noise

of

gives

as a f u n c t i o n

o f photodiode

equivalent

lin e

m e a s u r e d n o i s e p o w e r on t h e

to

current noise)

is

5 0 s, w ithout

F i g . 11 the

ratio

0=0*,

to

of

the sh o t n o is e l e v e l

the

input

at

The

t h e same f r e q u e n c y

Scan

t i m e f o r a)

and

vid eo filter.

gives the

; b)

th e whole e l e c t r o n i c n o i s e .

o f a YAG l a s e r h a v i n g t h e s a m e i n t e n s i t y . b)

o f 9 f o r Q/2n=8MHz

the noise re d u c tio n "squeezed"

shot noise

noise

spectrum ,

f a c t o r K{Q)

spectrum , 0 = 22.5*

which i s

recorded at (both s p e c tr a

110

Q U A N T U M N OISE R E D U C T IO N ON TW IN L A SER B EA M S

have

been

clearly

corrected

of

30% ± 5% i s

reduction

In because

electronics

n o i s e ) . The c u r v e i s

below 1 o v er a b ro a d fre q u e n c y r a n g e .

reduction noise

from

is

the

the

rejected

low

large

in

in ten sities

b etter

the

A maximum n o i s e

observed a t a frequency of t h a n 15% f r o m 3 t o

frequency

domain,

13MHz.

the

noise

ex tra noise

o n e a c h beam i s

difference

process.

are not

exactly

in

the

equal,

slight

difference

losses

This

is

a cause fo r a d d itio n a l

into

the measured s i g n a l ,

w h i c h we

in

the

attribute

mean to a

t wo i n f r a r e d b e a m s .

fluctuations

as d is c u s s e d

increases

not com pletely

Moreover,

fo r the

8MHz. The

t o be c o u p l e d b a c k

the

theoretical

§ 4.

1

1 0 M hz I

0 F i g u r e 11 It

is

(9 = 0)

20M hz I

: Experim ental n o ise r e d u c tio n

obtained S(Q),

by recording 3 spectra the

shot

e l e c t r o n i c n o i s e E(co), dashed l in e

At

is

R (f l)

noise

(9

= ( S( Q)

a Lorentzian f i t

high fre q u e n c ies ,

of

R(ft)

=

f a c t o r R(Q) : the

spectrum .

"s q u e e z e d " n o i s e

2 2 , 5 * C)

N(Q)

and th e

- E (co)) / (N (co) - E ( c o ) ) .

The

the exp erim en ta l p o in ts .

is

seen to

go t o 1 : t h e

E G IA C O B IN O

noise

of I i

higher

than

Lorentzian higher 50%,

- Ig rises the fit

than

in agreement

w ith

internal

for frequencies

The d a s h e d l i n e spectrum

The e x t r a p o l a t e d

from t h e e f f e c t o f t h e

6.

bandw idth.

the experim ental

5 MH z ).

which i s

111

back to the sh o t n o ise

cavity

of

et al

the

value value

(for

shows a

frequencies

a t co=0 i s

about

one would e x p e c t

losses.

APPLICATIONS OF TWIN BEAMS

The

OPO

provides

photons. noise

This

In

feature

in se v e ra l

enhance the this

high

sensitivity

of

beams

made o f c o r r e l a t e d

a p p e a r s v e ry p r o m is in g to re d u c e quantum

experim ental

paragraph,

generation

in ten sity

configurations,

of quantum n o i s e

we w i l l

lig h t

o utline

beams

below

and t h e r e f o r e

to

l i m i t e d m easurem ents.

t wo

such a p p l i c a t i o n s

the

shot noise

lim it

:

and

measurement of v ery sm a ll a b s o r p t i o n s .

6.1

R eduction of i n t e n s i t y

n o i s e b e l o w s h o t n o i s e on a s i n g l e

l i g h t beam

As Jakeman a l.

s u g g e s t e d by

et al.

1987),

1986,

on

correction.

Two

F i g . (12).

In

m onitored

by a

such

b e a ms

single possible

both

of

may

b ea m,

m odulator,

t h e s e c o n d one

is in

(Fig.

(S a le h e t a l . 1985,

Yuen 1 9 8 6 ,

be via

used

them,

the

intensity first

to c o r r e c t

( 1 2 -b )),

for

"intensity intensity

s c h e m e s a r e sh own i n

scheme

used to modify th e order

Yamamoto e t

electronic

experim ental

p h o to d io d e . In the

an i n f o r m a t i o n

inten sity In

a

authors

Ha us e t a l . 1 9 8 6 ,

correlated

squeezing"

several

o f beam Bz

is

( F i g . ( 1 2 - a ) ),

t r a n s m i s s i o n o f an t h e pump i n t e n s i t y .

t h e sa me i n f o r m a t i o n i s u s e d

112

Q U A N T U M N OISE R E D U C T IO N ON TW IN L A SE R B EA M S

d irectly

to c o rre c t

The m a i n d i f f e r e n c e loop

process

w ay ,

which i s Let

in

us

as

the

first case

now

way.

t w i n beam 1^

between th e

not the

quantitative in-phase

the

two s c h e m e s i s

one w hic h a c t s in

fluctuations

in

same t e c h n i q u e . that

th ere

is

a

a selfconsistent

the second one.

consider From

by t h e

the

E q s . ( 27)

first

technique

and(28),

(in F o u rie r space)

i n a more

we c a n w r i t e

in

the

the balanced case

:

F i g u r e 1 2 : Two p o s s i b l e s c h e m e s o f i n t e n s i t y inten sity

of

inten sity

b y a c t i n g on an i n t e n s i t y m o d u l a t o r

and i n b)

p ° ut

to c o r r e c t

used

in

a)

to c o r r e c t

t h e i n t e n s i t y o f beam

Bi

(IM)

rvj a p Jn is

(feedback)

feedforw ard) .

sources.

proportional

(52-a)

Let

associated

w ith

a n d it

The e l e c t r o n i c

t o p®u 1

intensity.

(52-b)

the p a rt of flu c tu a tio n s

a n d it

t the

: the

t h e p ump

+ a P J"

\

fluctuations noise

is

B2

= itt + a p J ”

P

where

beam

correction

th e re m a i n in g p a r t due t o o t h e r feedback loop produces

which i s be

the

c o m i n g f r o m t h e pump

a signal

u s e d t o r e a c t o n t h e pump b eam am plitude

am plitude

transm ission

m odulator.

The

pump

factor field

E G IA C O B IN O

am plitude

t r a n s m i t t e d by t h e m o d u l a t o r i s

ot* n = t a n + J^ l - t 2

where

is

because Due

of

to

113

et al

the

in ten sity

(53)

losses

fluctuating

G

field

i n t r o d u c e d by t h e

feedback loop,

fluctuations

t = F

:

ccn n

the n o ise

the

then

the

o f beam 1,

entering

the

system

i n t e n s i t y m odulator.

transm ission

t d e p e n d s on t h e

p°u1 , according

to

:

p j ut

oc 0 where and

t

is

G the

the am plitude effective

sake of s i m p l i c i t y ,

transm ission

loop

f a c to r w ithout feedback

gain fa c to r,

t o be r e a l

assumed h e r e ,

and f r e q u e n c y

for

the

independent.

T h en p °1u 0 + t Sot0 - G*1

The inserted

real

in

eq.

fluctuations

part (52)

^ i

as

the

oc1 otn n1

( 54)

fluctuations value

of

p n0

is

now i n s e r t e d

pump

fluctuations,

which

flu ctuations

in eq.

yields

(54)

the

on beam Bz :

must

t h e n be

t h e pump i n t e n s i t y

t h e OPO, w h i c h g i v e s rvj f o r p ®ut :

| 1i -—t 1 + a ( t p 0lnn + vJN

value

in-phase

such

tc

This

the

\\J^1 l -- tt*0

i m p i n g i n g on

s e l f - c o n s i s t e n t value

p out

of

+

the

follow ing

(55) 1+aG

to give

the e f f e c tiv e

follow ing expression

for

114

Q U A N T U M NOISE R E D U C T IO N ON TW IN LA SE R B EA M S

Ga ° 2

Tt

2 When

1+Ga

to

on

p i U1 " P 2 U1 ' the noise

13

electronic b ea m the

having

in

technique.

B 2 the

fs

:

the

open

intensity

this

loop

feedback c o rre c ted

L

o pe n 1o o p

noise

c o r r e c tio n . So/2

( 56)

expression

feedback loop

fluctuations

of

on

As a r e s u l t ,

in ten sity

co2 t 2 + 4 lct’ = So -------CO2 T 2 + 4 T ' 2

spectrum

is

the

difference.

(co)

In tensity

b ea m

Bz

is

:

( 58)

after

the sh o t n o is e o f a c l a s s i c a l

t h e s a m e mean i n t e n s i t y . Th e me an t r a n s m i s s i o n

in ten sity

losses

beam

the high gain l i m i t ,

s p e c t r u m on t h e

fe e d b a c k

1+Ga

I r\y p ‘ n + vjl-t 2 p " ) 0 0

( t

g ain goes to i n f i n i t y ,

in

on

[V"’]

+

:

As e x p e c t e d , tran sfers

1

th e loop

sim ply reduces

Figure

a

=- I t

the

m odulator OPO.

a)

is

80%

feedback

and

of

t h e r e a r e no i n t e r n a l

t e c h n i q u e , b)

feedforw ard

L et us s t r e s s previously I

(E q.(7)),

+ 1^ = 2 1 i

background the

when

. In

the shot n o ise

corresponds

form ula

reached for

standard

possible

that

for

the

is

due

to

the

to

fact

that,

to

correct

feedback loop a c t u a l l y

u s now c o n s i d e r

c o rre c tin g device.

whe n for

In t h i s

case,

the

the

as l a r g e

as

m onitors

the

fluctuatio ns if

beam

Bz

.

The

transm itted

field

d e v i c e h a s an e x p r e s s i o n s i m i l a r

♦J:1 - t 2 Here th e e x t r a n o i s e

of

t h e quantum

2

in-phase

a m p l i t u d e a*

t o Eq.

term a“ i s

inserted

the unescapable

through t h i s

of

c o u n t e r p a r t of :

a n2

(60)

and f r e q u e n c y i n d e p e n d e n t , beam

B2

after

one f i n d s

that

the v a r ia b le

:

~in

+ Bpo

on

(53):

a c t i n g on beam B2 . T h e n

a r e g i v e n by - Git

on

( 59)

fluctuations

_

flu ctuations

transm itter

+ t a 2 - Gp°1 u 1 + J l - E

again G r e a l

"feed forward"

a"2

transm ission

a 21 = t a

the

in ten sity

a r e u s e d t o a c t on a v a r i a b l e

P2 = t i t

one

th e problem of

Bi

transm itter

( 1 3 - a ) , the

l a r g3---------------------e r t h a n 50%.

adds n o is e

beam

the

tw ice

inten sity

are u n c o rre la ted .

Let

Taking

in Fig.

on I 1 - I 2 i s

beam

a variable

of

i n t e n s i t yJ I i . I t i s t h u s n o i s e on o n e b e am , b u t o n l y

o f beam

fluctuations

beam

therefore

fluctuations Bz , t h e

a

Sq , d e f i n e d a s

a beam o f

factor

Bi

level

displayed

intensity

the noise re d u c tio n

This

(58)

co -*■ oo i s

shot noise

to reduce

115

et al

E G IA C O B IN O

I

( t - G) +

2

1-t

a;

(61)

116

Q U A N T U M NOISE R E D U C T IO N ON TW IN L A SE R B EA M S

In o rd e r Ii

- I2,

like

precise the is

to in

value of

noise

transfer the

previous

the e f f e c t i v e

spectrum of the

g i v e n by

on beam Bz

This

final

to

T2 + 4T' u. S0 ----------------------- + — CO2 T 2 + 4T ' 2

in

the

may

notice

precisely the the

1^ ,

the

in Fig.

obtained

t wo c a s e s ,

50% a t

n o i s e on

choose a

intensity

I2

least

for

the that

required

( 62)

c a n be

feedback

an open lo o p reduce

the

the reaso n given p re v io u s ly .

But

follow ing

is

(13-b),

in

we s e e

_ (1-t2)

differences

to

between th e

t wo

:

(i)

to

:

displayed

form ula

r e d u c tio n of

situations

in

form ula,

the

configuration

one

on

i n which c a se

t o2

t

0

fe e d f o r w a r d

in ten sity

need to

feedfordw ard c o rre c te d

= S

2

noise

we

existing

:

(co)

compared

case,

loop g a in G = t ,

2

SJ

the n o ise

The g a i n fixed

first

in

one.

electronic (ii)

attenuator

value necessary

the

the

factor

close

the n o ise

is

and n e e d s o n l y t o be h i g h

T h i s makes t h e s e c o n d s e t - u p v e r y s e n s i t i v e gain f lu c tu a tio n s .

the noise does not

low ers

second case,

to reduce

entering

m atter

squeezing f a c to r to u n ity

is

in in

the the

the

system v ia

feedback case,

second c ase.

therefore

the v a r ia b le

required

in

whereas

it

A transm ission the

feedforw ard

case. (iii) correcting the

system .

one t a k e s

system

feedback

o scillation,

If

(neglected

system which

into

is

may never

account the in

this

very the

tim e d e la y s

first

easily case

in

the

sim ple a p p ro a c h ), lead

in

the

to

spurious

feedforw ard

117

et al

E G IA C O B IN O

6 . 2 M e a s u r e m e n t o f w ea k a b s o r p t i o n b e y o n d t h e

Intensity reduce

the

principle al.

noise

1 98 6 )

:

an

e(co)

t h e beam

varied

floor

b e a ms

in

absorbing

resonant at

.

are

medi um

a r o u n d cog , a n d t h e

in ten sity

difference

m easurem ents.

having

an

to The

(Fabre e t absorption

a g i v e n f r e q u e n c y coa

( by t e m p e r a t u r e o r a n g l e

lim it

suited

extrem ely sim ple

The p h a s e - m a t c h i n g

co^ v a r i e s

ideally

absorption

of s u c h an e x p e r i m e n t i s

co efficient on

correlated

shot noise

is

in serted

c o n d i t i o n o f t h e OPO i s

tuning)

so t h a t

the

frequency

a b s o r p t i o n d i p i s m e a s u r e d on t h e

1^- I 2 as a f u n c t i o n of fre q u e n c y .

As t h e

b a c k g r o u n d n o i s e p o w e r on s u c h a s i g n a l i s r e d u c e d by a f a c t o r P* — , w ith re s p e c t to s h o t n o i s e , t h i s te c h n iq u e a llo w s us to r ' measure a minimum a b s o r p t i o n c o e f f i c i e n t e (co^ ) s m a l l e r b y a factor

— .

This

method

w ill

be

useful

for

example

m

in ten sity

is

T'

measurements im possible

where

increasing

(biological

samples

But t h e s i m p l e the

s ame k i n d ,

w ill

measured

are in

a

in stan ce).

above,

as o t h e r s

a lw a y s be h a m p ered by t h e u n a v o i d a b l e

then

of low

w hich w i l l p r e s u m a b l y p r e v e n t any h i g h

measurement

techniques

for

laser

technique d e sc rib e d

frequency excess n o ise , sensitivity

the

at

needed,

frequency

zero which

range

frequency. tran sfer

where

M odulation

the s ig n a l

the n o ise

is

t o be

minimum

(Gehrtz e t a l . 1985).

However,

phase

or

f r e q u e n c y co a d d s q u a n t u m n o i s e be

sim ply

techniques

understood

by

am plitude (Yurke a t recalling

m odulation

a

a l . 1987a)

: this

that

m odulation

any

c o u p l e s mode s h a v i n g f r e q u e n c y d i f f e r e n c e s

co. Vacuum f l u c t u a t i o n s

at

equal

can

to

o f e m p t y mode s d i s t a n t b y co f r o m f i l l e d

118

Q U A N T U M NOISE R E D U C T IO N ON TW IN LA SE R BEA M S

m o de s a r e

th en coupled backi n t o

in crease

of

tech n iq u es reduce

fluctuations. such as

the n o ise

squeezed s t a t e s

In

in ten sity

therefore

F igure

One

m ultiple

m odulation

absorption

experim ents

losses

M odulation

signal

It w ithout not

m odulator get _

is

the

to

Yurke e t a l . 1 9 8 7 b ).

as w e ll,

h a v i n g a non z e r o

the m odulator mean v a l u e ,

and

amount o f quantum n o i s e .

scheme

nevertheless

allow ing

itself

but

pulsation, :

let

possible

: this

means su c h

possibility

elaborate

to

transfer

the

(PM) a n d a p o l a r i z i n g

( PB S) .

t h e beam

population

more

t o an

a t g i v e n f r e q u e n c y co w i t h o u t a d d i n g q u a n t u m

adding n o ise

different

rise

squeezing in o rder

n o i s e , by using a p o la r iz a tio n m odulator beam -splitter

giving

needs

frequency

adds a s i g n i f i c a n t

:

then

(G ea-B an aclo ch e 1987,

periodic

14

system ,

in phase-m odulation experim ents w ith the h elp

of

intro d u ces

the

is

the

to m odulate

absorption c o e ffic ie n t o scillating

Stark

etc..

We p r e s e n t

in Fig.

and p o l a r i z i n g follow ing values

insert

on

beam

beam -splitter for

B

I i going out of th e d e v ic e

:

signal

done f o r exam p le by m o d u l a t i n g

as an

us

the

the

Bi

PBS.

sh ift

a

(14)

e (co) , by or le v e l another

polarization

I n s u c h a w ay ,

t wo mean i n t e n s i t i e s

we

I * and

E G IA C O B IN O

et al

119

f I- A = c os^c ot I < 1 1

(63)

I = s i n 2 cot I Vi i Of it

course,

couples

mode

such

a sy stem adds quantum n o i s e ,

the o u tp u t to

having

t h e v a cu u m f l u c t u a t i o n s

a polarization

same f r e q u e n c y .

More p r e c i s e l y

orthogonal

on t h e

t h e beam Bi

the o u tp u t f i e l d s

because input

and th e

ocA a n d otA a r e

g i v e n by :

a A = coscot oc 1

(

ocB = - s i n c o t a i

then

+ s i n c o t ft 1 1 + coscot ft

i

(64)

i

: I A = | a A I2 = c o s 2 cot I + s i n 2 c o t t I 1 1 1 Nl i ^IB = | a B | 2 = s i n 2 cot 1^ - s i n 2 c o t JIT

An one

a b s o r b i n g medium

m onitors

When

there

m odulation

the is

in

no

absorption

c a n c e l s when

the

the

(65)

on t h e the

on beam BA a n d

signal

e x tra noise the

w ithout

m odulation

a d d e d by t h e

sum o f i n t e n s i t i e s

IA i s t h e s ame a s 1 + 1I B 2- I m odulation : s i n c e no p h o t o n i s

process, the

signal

no

noise

is

added.

to

the

At

:

c o s 2 cot e (co )

2co e q u a l

.

IA + IB -

n o i s e on

(66)

hav in g a F o u r i e r component choose

i

ft^

inserted

one m e a s u r e s

f r e q u e n c y co , o n e m e a s u r e s

S = T

then

a b so rp tio n dip

IA 1 +1 I B . T h e r e f o r e the n o i s e on I -I lost

is

ft

(I l /2)

e(co a ) a t

f r e q u e n c y 2co.

f r e q u e n c y o f maximum n o i s e

I f we

reduction,

120 we

Q U A N T U M NOISE R E D U C T IO N ON TW IN L A SE R B EA M S

can

take

full

advantage

of

such a r e d u c tio n

to o b ta in

a b s o r p t i o n measurement beyond s h o t n o i s e . ACKNOWLEDGEMENT This

work

has

been

done

with

the

support

of

EEC

Stimulation

Action Grant Number ST2J0278C.

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1967 P h y s .

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160 1 07 6 a n d

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1987 E u r o p h y s i c s L e t t .

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and T e i c h ,

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and W a l l s ,

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1987 J .

P.,

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197 3 " O p t i c a l A recchi,

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H o l l a n d Amsterdam. S t o l e r , D. T akahasi, Wu,

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1986 P h y s .

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34 4 0 2 5 . Yamamoto, Bjork,

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Imoto,

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Soc.

NONCLASSICAL EFFECTS IN PARAMETRIC DOWNCONVERSION

J G R A R IT Y A N D P R T A PST E R

1

IN T R O D U C T IO N

T he m ixing of three electrom agnetic waves w ithin a m edium w ith a second order refractive non-linearity can lead to several different para­ m etric processes. The sem iclassical theory of these m ixing effects is well reviewed by Yariv (1976). The sem iclassical m ethods hide the funda­ m ental quantum nature of the interaction. In param etric downconversion the quantum effects arise because two photons are created from one pum p photon. T hese photons satisfy the necessary conditions of m om en­ tum (phase) and energy conservation w ithin the non-linear m edium . In non-degenerate downconversion separable photons (either by angle or by polarisation) are created. Burnham and W einberg (1970) dem on­ strated that the two photons are created nearly sim ultaneously, using photon counting coincidence techniques, while M ollow (1973) developed a quantum m echanical treatm ent allowing the two photon coincidence detection probability to be calculated. M ollow suggested that the tim e separation of the two photons would be related to the coherence tim e of the illum inating radiation. This has since been disproved by more recent coincidence experim ents with tim e resolutions of a hundred picoseconds (Friberg et al 1985). U sing non-degenerate type I downconversion two identical trains of photons can be selected by apertures satisfying the phase m atching con­ ditions. D etection of photons in one train can be used to m odify the quantum statistics of the other train to produce antibunched an d /o r sub-poissonian light (section 2). We report here on two experim ents of this type. The first em ploys single photon detection techniques and a fast optical shutter to create antibunched and slightly sub-poissonian light. T he second uses analogue detection and feedback to produce sub© British Crown Copyright 1988

J G R A R IT Y A N D P R T A PSTER

123

poissonian light at picowatt power levels with a Fano factor of 0.78 over a lim ited bandwidth. T his, to our knowledge, is the lowest post-detection Fano factor yet reported. Sub-poissonian light is potentially useful in finite dose transm ission m easurem ents and the high degree of tem po­ ral coincidence of the downconverted photon pairs can be exploited in absolute m easurem ent of detector quantum efficiency (section 3). Up until recently, m easurem ent of the degree of tem poral coincidence has been apparatus lim ited. R ecent theory (Fearn and Loudon, 1987;Ou et al, 1987; Prasad et al, 1987) and experim ent (Hong and M andel, 1987; 1988) has shown that recom bining indistinguishable photon pairs in a beam splitter leads to a fourth order interference effect which allows a direct m easure of the photon overlap. Such experim ents allow m ea­ surem ent of sub-picosecond tim e delays at the single photon level. We show here (section 4) that the detailed shape of the effect can be re­ lated to the fourier transform of the photon bandw idth determ ined by the phase m atching uncertainty and aperture size. Experim ental results are presented which confirm this theory. The m inim um photon length is lim ited by the bandwidth of the crystal non-linearity coupled with the detailed physics of the param etric downconversion process (Pike and Sarkar, 1988).

2

A N T I B U N C H I N G A N D S U B - P O IS S O N IA N L I G H T SOURCES

2.1 Phase m atching conditions for non-degenerate param etric downconversion For non-degenerate param etric downconversion (Yariv, 1976) a non­ linear uniaxial crystal is used. The crystal is aligned with its optic axis at an angle (f) with an incident short wavelength laser beam . A bout 10~8 of the beam is converted into pairs of longer w avelength photons which appear in a cone surrounding the laser beam . The photon pairs satisfy the conditions of energy and wave vector conservation in the crys—* tal. Given an incident photon of wave vector ko and angular frequency — a — * cjo, and downconverted photons with wave vectors fci, &2 and angular frequencies these conditions can be w ritten as ko = k\ + &2

(1)

LOq — U)\ -f-

(2)

124

N O N C LA SSIC A L EFFECTS IN PA R A M E T R IC D O W N C O N V E R SIO N

The geom etry illustrated in figure 1 depicts type I phase m atching, where the incom ing beam is polarised in the plane of the crystal optic axis, and the dow nconverted light is polarised perpendicular to this. In this case equation 1 can be rew ritten (jJqtiq = ujitii cos 0 i + oo2n 2 cos 62 (jJitii sin 0i — 002*12 sin 02

(3)

where n\ and n 2 are the ordinary refractive indices of the crystal at frequencies 00\ and o;2, and n 0 is given by n 0 = n e sin + n 0 cos (j)

(4)

where n 0 is the ordinary and n e is the extraordinary refractive index of the crystal at A s a consequence of equations 2-4 the crystal must be negatively uniaxial (rce < n 0) and the cone angle is m axim ised by choosing (j) = 90°. U sing these equations and the refractive index data for deuterated P otassium D ihydrogen P hosph ate (KD*P) (Yariv, 1976), illum inated w ith 413nm light at this angle, the dow nconverted light at 826nm is em itted into a cone of half angle 9.8°. T his is internal to the crystal, and corresponds to 14.3° externally. A broad spectrum of light is em itted at angles around this.

ooq.

Tw o near identical photon stream s can be selected from the cone by placing apertures at opposite ends of the cone base, at angles satisfying

J G R A R IT Y A N D P R T A PSTER

125

the phase m atching conditions (equations 2-4). A more detailed discus­ sion of the coincidence properties of the photon pairs will be given in section 4.2 2.2 A ntibunching experim ents G iven two identical trains of photons, one can use the detection of a photon in one train (trigger channel) to gate detection of photons in the other train. P u ttin g a dead tim e between gate open events can, in principle, produce an antibunched light source. After each photon there will exist a reduced probability of seeing another for a period equal to this dead tim e. If the photodetection rate in the triggering channel is much higher than the inverse dead tim e (ie we overdrive the system ) the gate open events will becom e regularly spaced. T his leads to a reduction in the noise of the gated beam to below that of Poisson light. The experim ent can be more easily understood by referring to the schem atic diagram shown in figure 2. It is a m odified version of the original apparatus used by Walker and Jakem an (1984; 1985) where the shutter was placed before the non-linear crystal. Splitting of incident UV photons takes place w ithin the non-linear crystal and the pairs of angu­ larly resolved photons so produced are observed by (photon counting) trigger and signal detectors. At the trigger detector an electronic dead tim e td is introduced so that detection events in this channel are at least td apart. W hen each of these events are registered an optical shutter in the other, signal channel, is opened for a short tim e t. An optical delay is included before the shutter to com pensate for electronic and shutter response tim es, etc. Thus only partners of events actually registered in the trigger channel should be detected in the signal channel. In princi­ ple this overcom es the problem of random partnerless events arising in the signal channel due to the low efficiency of the trigger channel detec­ tor, which considerably reduced the the non-classical effects observed in the earlier experim ent (Walker and Jakem an, 1985; 1986). There is, in practice, always som e chance of registering further events in the signal channel during the finite open tim e f. Given N shutter openings per sam ple tim e T with f < T , ^ and assum ing initially that the signal channel detector is 100% efficient we can write the number of counts in the signal channel in a typical sam ple

126

N O N C LA SSIC A L EFFECTS IN PA R A M ET R IC D O W N C O N V E R SIO N

tim e as n = N (1 + m)

(5)

where m is drawn from a Poisson distribution of m ean (1 — rji)rt. Here r is photon pair arrival rate w ithout losses and r)i is the trigger detec­ tor efficiency. U sing equation 5 we can express the norm alised second factorial m om ent of the photocount n in the form n (2) _

< n ( n - 1) > < n >2 < N ( N - 1) 1) > < m > ( 2 + < m >) > w fi , < m > 1+ — — \2 < N >2 ( 1 + < m > ) 2 + < N > (1 + < m > ) 2

( 6) angular brackets denoting ensem ble averages w ith sam ple tim e T. This quantity is not affected by the finite quantum efficiency of the signal de­ tector. T he reduction in m easured signal variance over the conventional Poisson variance,the Fano factor F , is however dependent on signal channel quantum efficiency. B y definition F = (Var n ) / < n > = 1 + < n > (n ^ — 1)

(7)

where < n > = rj2 < N > (1 + < m > ) and is given by eq. 7. As photon counting detector efficiencies are low we do not exp ect a large reduction in Fano factor in this experim ent. On the other hand the second factorial m om ent is m inim ised by choosing T < t q when the first term in eq. 6 vanishes. From the theory of deadtim es (M uller, 1974) < JV > = J i r 3-

l + J?irT

(8)

v ’

we can w rite in this case (Jakem an and Jefferson, 1986) (2) = 1 + n r r n I, ________ 1______ ] W T 1 + (1 - m ) 2r t j

(9)

The first term in eq. 9 is a bunching term as T < tq and is reduced by overdriving the trigger channel (rjirrD > 1). However the second term shows an antibunching effect when rji is large and rt the number of photon pairs arriving in the gate tim e is sm all. Hence a short gate tim e and high trigger channel detection efficiency are desirable. It m ust also be noted that any losses or m isalignm ent of the phase m atched apertures will reduce the effective efficiencies rji and ?/2 appearing in eqs. 7 and 9

127

J G R A R ITY A N D P R T A PSTER

TRIGGER

CHANNEL

Figure 2: Block diagram of the photon antibunching apparatus. These effective quantum efficiencies can be estim ated by m easuring the photodetection event coincidences between trigger and signal chan­ nels C C = tfiq2r (10) and dividing by the mean count rate in each channel nj = Vjr

j = 1 ,2

(11)

To avoid detector afterpulsing effects one can estim ate n ^ from the cross-correlation function g ^ ( r ) from two detectors view ing the signal channel via a beam splitter. It has been shown that J 2)(t] = < "i(0) M r ) > 21 < ni >< n 2 >

i

=

P0)

The accuracy of such a m easurem ent will be lim ited by the excess noise in the detector amplifier and correlations from m acroscopic laser fluctu­ ations occurring in the m easurem ent bandwidth. The broadband nature of the downconverted cone m eans that in prin­ ciple detector quantum efficiencies can be m easured over a wide band­ w idth using a variable m onochrom ator and linked goniom eter to preserve the phase m atching angles. 3.2 Reduced quantum noise in transm ission m easurem ents Changes in the intensity due to scattering,absorption,reflection or re­ fraction by a sam ple are fundam ental to m icroscopy and im aging tech­ niques generally and the absorption spectrom eter is a standard piece of laboratory equipm ent finding uses in many scientific disciplines. In the case, often occurring in bio-m edical m easurem ents, where the spec­ imen can be dam aged or optically changed on illum ination there is an upper lim it to the intensity and exposure tim e. Under this dose lim it or pow er/bandw idth constraint (Yuen 1988) the accuracy of the m ea­ surem ent can be improved by reducing the number uncertainty in the incident beam . However the loss in the sam ple itself will lim it the noise reduction achievable. In a conventional transm ission m easurem ent only one channel is used. The m ean flux of photons from the source r (or the m ean detected flux r' = rjr) can be m easured to arbitrary accuracy before introducing the sam ple into the apparatus. Introducing a sam ple with absorption coefficient 0 < a < 1 into the beam for a finite tim e T and counting (or integrating the photocurrent in an analogue apparatus) the number s of transm itted photons with a detector of quantum efficiency r] we can

138

N O N C LA SSICA L EFFECTS IN PA R A M ET R IC D O W N C O N V ER SIO N

e s tim a te d a from a = 1 —s / r T

(31)

var(d) = < a 2 > —< a > 2— —— r'T

(32)

with m easurem ent variance

given that s is a Poisson random variable. A m easurem ent m ade using a correlated pair source such as our para­ m etric downconversion source (w ithout feedback) would involve placing the sam ple in one channel of the apparatus and m easuring direct, n and tran sm itted, s counts in sam ple tim e T . An estim ate of a can be obtained from ~ S f o(33) o\ a- — n------n where for sim plicity we assum e equal detector quantum efficiencies rj. n and s are Poisson random variables with correlated noise

< n s > — < n > < s > = < A n A s > = rj2{ 1 — a)

(34)

hence to order < n > ~ 2 (Jakeman and Rarity, 1986), = ( l - ( 2, - l ) ( l - q ) ) ( l - « l

r'T This estim ator is slightly biased but the bias is always less than the error of m easurem ent. We can define a variance ratio (or Fano factor) relating the conventional (C ) to the correlated pair (P ) m easurem ent as R - ^ 7^ 1 = 1v a r (a c )

(2*7 - 1)(1 -

a)

(36)

There is no reduction in variance possible until rj is greater than 0.5. U sing an estim ator based on partial use of the pair count

(1 - k) < n > + k n - s a = ----:-------- —------------------- :------(1 — k) < n > + k n

.

. [6 i )

and choosing k ~ rj( 1 — a) one can obtain a result which is always sub-poissonian with R = 1 - rj2 ( l - a) (38) In high sen sitivity spectroscopy techniques where the absorption is extrem ely sm all the stab ility of the source at low frequency can be the lim it to sensitivity. Frequency m odulation techniques are thus used to

J G R A R ITY A N D P R T A PSTER

139

move the signal out of the noise dom inated range and differential in­ tensity m easurem ents (across a spectral feature) are made which can, with care, lead to shot noise lim ited m easurem ent of absorption (Wong and Hall, 1985; Gehrtz et al1985). In these situations both the attenu­ ated and unattenuated beam s have to be estim ated from a lim ited tim e m easurem ent and the estim ator for a is sim ilar to eq. 33. However in a conventional apparatus the correlation (eq. 34) is zero and var(a) =

(39)

When compared with the photon pair subtractive estim ator (eq. 35) this leads to an improvem ent for all detector efficiencies with

w

When a is sm all this result becom es R ~ 1 — r;

(41)

which is equivalent to the result of G iacobino and Fabre (1988). Using our apparatus as a source of correlated photons in a differential absorp­ tion m easurem ent of this type would reduce the variance by a factor of 2 below the shot noise level. In the general case of sam ple illum ination using a sub-poissonian source, either the param etric source described above or a direct source such as a high efficiency light em itting diode (Tapster et al, 1987) or sem iconductor laser (M achida et al, 1987), the reduction factor is given by R = 1 — 7/(1 — a ) ( l — F )

(42)

where F is the source predetection Fano factor. Other estim ators can be based on coincidence m easurem ents (Jake­ man and Rarity,1986) as in the detector quantum efficiency m easure­ ment discussed above. There the detector can be thought of as an ideal unit efficiency device behind an absorber with a = 1 — r/. In all absorption m easurem ents the noise reduction is always lim ited by R > a

(43)

hence the technique is lim ited to highly transm issive sam ples and all optical losses (eg container windows, filters etc.) have to be m inim ised.

140

4

NO N C LA SSIC A L EFFECTS IN PA R A M E T R IC D O W N C O N V E R SIO N

P H O T O N L O C A L I S A T IO N A T A B E A M S P L I T T E R

4.1 Introduction Several sim ple quantum m echanical m odels of beam splitters have re­ cently been published (Fearn and Loudon, 1987;Ou et al, 1987; Prasad et al, 1987). Such a beam splitter appears to interact w ith single beam s of classical and non-classical light in the sam e way as m acroscopic m od­ els. One sim ple result is the binom ial partition noise im parted on an N -photon state due sim ply to the random selection of output direction for each input photon (Brendel et al,1988). The beam sp litter is, how­ ever, a four port device and a less intuitive result is obtained when indistinguishable N -states appear sim ultaneously at both input ports of the beam splitter. The sim plest case of this type occurs when one-photon states are sim ultaneously input into each port of a 5 0 /5 0 beam splitter (Fearn and Loudon, 1987). The theory predicts that each output will contain either two photons or none. A naive (particle) theory would pre­ dict a m ixture of tw o’s and on e’s. The required | i , i > photon state can be selected in non-degenerate param etric downconversion experim ents by photon counting coincidence detection (M ollow, 1973). The disap­ pearance of this coincidence after a beam splitter would be expected if the theory is correct. In a more rigorous theory this effect will occur over a range of path length differences due to the finite length of the photons which can be related, by fourier transform , to their bandw idth. This in turn can be related to the geom etry of the param etric downconversion apparatus and the detailed phase m atching conditions at the crystal. We have constructed a w hite light interferom eter (section 4.4) behind a dow nconversion crystal which ensures that photons from the sam e spatial m ode, and w ithin the sam e bandw idth arrive at the two inputs of a beam sp litter. We dem onstrate that the effect only occurs if the difference of the arrival tim es is w ithin the coherence tim e or inverse bandw idth of the dow nconverted photons. As these photons have a bandw idth of a few nanom etres, this tim e is short (less than lOOfs). 4.2 C oincidence properties of param etric photon pairs T he two photon detection probability P 12, that of d etectin g photons at tim es ti, t2 at points X i , X 2 from a source centred at the origin can

141

J G R A R IT Y A N D P R T A PSTER

be obtained from P n oc< A [ - \ t u

>

(44)

where T W (f) is the positive frequency part of the electric field operator (which can also be described as a photon annihilation operator) at tim e t. For a spontaneous param etric downconversion source (or in the general low flux lim it) M ollow showed that P n oc|< i i +)(ti, Y i ) 4 +)(*2, * 2 ) > |2 A/ | \

—*

A/ I

(45) \

— *

is a good approxim ation. The product A \ } ( t i , X i ) A \ {t 2 , X 2) can be thought of as a two photon wavepacket. For brevity in the ensuing discussion we define the shorthand

(46)

For a finite size crystal evaluation of g12 involves a coherent sum m ation over the illum inated region of all the conjugate frequencies contributing to the two photon wavepacket (M ollow, 1973)

9u oc J JduJidoj2oJico2x ^ { wo, u)i , lj2)8( u)q —

uq —

to2) (47)

x J d h e ' 1 ^ i(*i “ X i / C) +

--->Pn )-

Q

or T*Q,

the

is a linear functional on

In the natural co-ordinate basis

These

are

the

familiar momenta.

Conventionally the co-ordinate basis of the 1-forms is written as (dq*,dq^,...,dqN ), and so a general 1-form on T Q is

155

S SA R K A R

This one form can also be regarded as a one form on the manifold T*Q. dynamics of points on T * Q we need a two form on T*Q.

For the

This special form is known

as a symplectic form a)2 and is written as an exterior derivative

IN

-d

'j

IN

^ Pjdq1

-

Li=l

(9)

^ d q 1 A dp. i=l

(This two form is a linear functional of any pair of tangent vectors at a point of T*Q.) A n y physical observable can be regarded as a smooth mapping from T * Q to the set of all real numbers. 0(q,p).

Consequently an observable O

R, the

can be written as a function

Since it is customary to write the basis of tangent vectors of T * Q as

r 3

3

V

3

3

3

’ 9 ? ........... 9? ’ ^

3 i

’ ^2 ’ " ’ ’ ^

we can associate the tangent vector field O'

3_

I. H ri to an observable O.

30

3 i

3q*

Given any two observables

3o„l 3o„ i-i

( 10)

o i 3p J 3q *1

and O

2

we can easily show that

90, 90 . ( 11) 9q‘

q1

This is just the familiar Poisson bracket; it is thus closely related to the existence of a certain symplectic form on the state space T*Q.

The usual process of going from the

classical theory to the quantum theory is by replacing the Poisson bracket with (i/fi) times the commutator.

The above discussion indicates to us space

of

real

classical

solutions

equivalently, A(x,t) in the cavity.

that we should seek a symplectic structure to

(Moore,

1972)

of the

electromagnetic

the

fields,

or,

In order to have the field kinetic energy bounded

we

will require that A(t,x) is in the function space Hl(I), and 3/3t A(t,x) is in the function space l2(I).

Here I is the closed interval (x 1 o

functions which

is square integrable on

I

and

^ x ^ b(t)}.

is the* set of

H* (I) is the set of functions

whose

156

M O V IN G M IR R O R S A N D N O N C LA SSIC A L LIGH T

x derivative is in L^(I).

The elements of the cotangent bundle are

{A(t,x)

This

is rather

approach

familiar

since

to electromagnetism

,

A,

^

and

A(t,x)j

E

are

canonically

in the radiation gauge.

conjugate

Given

two

in the

standard

classical solutions

A(l)(t,x) and Ap)(t,x) we can introduce the symplectic form

b(t) dx | A ^ ( t , x )

A^Ct.x)

-

A^(t,x)]

A ^ 1 ^ (t ,x)|

( 12)

Owing to (3) and (7) this form is independent of t. set of

real

classical solutions

um (t,x) and

It is always possible to choose a

vn(t,x) such

that

we

have

a

form

of

'orthogonality' relations

(14)

ml n

Such a solution forms a basis for the cotangent bundle. basis will be given presently.)

A(t,x)

where an and

0n

(An explicit construction of the

A n y solution A(t,x) can be written as

=

y (a v (t,x) - P u (t,x)) L n n n n n

(15)

are real numbers.

The un and vn m a y also be regarded as observables since they induce maps into R from the cotangent bundle into R, eg

S SA R K A R

u :

{*•

n

m

157

-

iv



( 16 ) v : n

(A> I f )

— + (A|Vn}

W e can then associate operators pn and qn , in the quantum case, with these observables. In particular (Moore, 1972; Sarkar, 1988)

u

> p

n

n (17)

v

n

-- > q

n

and

L n

(we have

mJ

nI

chosen units so that t

=

m

1.)

nm

If R( ) is a function which

is twice

differentiable and invertible and satisfies (Moore, 1972)

R(t-b(t))

=

R(t+b(t)) - 2

(19)

then it is easy to verify that

u (t,x)

=

— --- r (cos(n7rR(t-x) ) - cos (n7rR( t+x) ) ) (2 nx ) 2

(20a)

v (t,x) n

=

— --- 7 (sin(nxR(t+x) ) - s in(nirR( t -x) ) ) (2 nx)

(20b)

satisfy (13) and (14). written as

In terms of the functions in (20) the field operator A(t,x) can be

158

M O V IN G M IR R O R S A N D N O N C LA SSIC A L LIGHT

A(t,x)

=

I

(pnvn (t,x) - qnun (t,x)}

(21)

n

Fr o m (13) and (14) we deduce that

pm

-

(A um)

(22a)

qm

-

-(A v j

(22b)

If we consider n o w a join of two motions bCQ(t) and b^)(t) at t = t', then A

can be

written as

A ( t , x)

I

,X^I

for

1< l'

^23‘>

^ {p n ^ v n ^ ( I ,x ) - q^2 ) i / 2) ( t , x ) j

for

t > t'

(2 4 )

{pn ^ vn1') ^ ,X') " qn ^ Un ^ ^

and

A (t , x)

=

F r o m the orthogonality relations at t = t' we find

p ,

I

m

I n

i< »

-

(u ( 1 ) |u ( 2 ) )

m

I n

If we had another join to a different motion b(3)(t) at t = t" (> have identical relations with 2

3 and 1

2.

linear relation between the (p^,q3) and (p*,ql).

t’) then we would

Clearly we can obtain in this way a W h e n such a relation is converted into

one between creation and annihilation operators, we find that the destruction operator associated with motion

3

involves both destruction and creation operators for motion

1.

159

S SA R K A R

Consequently if we consider a sequence of motions: no motion, motion and no motion, then, even if there were no photons initially, in the final stationary cavity photons would have been generated.

The physical reason for these photons as mentioned earlier are

accelerating charges in the mirrors; this indicates that motion has to be relativistic in order to get large effects.

Our analysis, hopefully, will indicate the magnitude of these

effects together with qualitative features concerning the nature of the fluctuations of the photon states produced.

The important quantities that need to be calculated are

, (i)I v (i+l)v) (v m I n

for i = 1,2.

and.

, (u (i)I v (i+l)N ) m I n

Let us explicitly categorise the motions:

=■ b ( 2 ) ( t )

-

b( 3 ) (t)

0 < t ^ t

=

b ( 2 ) (t ) o

-

b

o

Moreover b(^)(o) = L, b(2)(o) = 0 and l5(3)(t0) = 0. all t.)

t

o

o

(27)

< t

(Clearly we require b(t) ^

0

for

For b(l)(t) and b(3)(t) it is easy to verify that

R (1 ) (S)

-

|

(28)

R (2)( 0

=

- t J -----

(29)

and

b( ’(‘o)

Consequently we have

160

M O V IN G M IR R O R S A N D N O N C LA SSIC A L LIGH T

..(!)/. U (t,X )

1

=

r

(2nx)

COS

nx(t-x) -------- :---------

1

___ COS

-

nx(t+x)) -------- ----------

L

L

J

(30) (!)/«. \ (t,x)

=

I f . n7r (t+x) . n7r(t-x)] ------ j j^sin — ^ - sin — ^ ----I (2 nx)*

and

“ ■*> -

7(27n-7r) T

H

--

o

R ^o l ) (31)

(3).

Vn

The

.

( t ' X)

important

I

"

f

- I

7 7

.

fnx(t+x))

l Sln l - E

(2 n 7r)

o

missing ingredient is an

indicated a method to calculate this when specify b(^)(t) in detail.

.

fnx(t-x)D

J - Sln [—

expression 16 P)(t) 1

o

JJ

for r P)(£). <


o

o

J

>

(69)

0

0 we can arrange this and so have squeezing for some

values of 6 .

A particularly simple form for b(2)(t) is simple harmonic motion

b

( 2)

( t )

=

CQ

+

with c0 + C| = L and c0 , cj > 0.

b

o

For ojt0

b(t ) - L b(o)

o

=

=

1

(1 — cosoJt ) (L + c.cosoJt )

o

but for a full

Consequently, in this case, we can consists of half a cycle of sinusoidal The squeezing parameter (Walls and

gj

N o w for this motion

2 CJ c.

7r this is non-zero,

(70)

C O SG Jt

1

period

(71)

o

(ie u)tQ =

2%)

this is zero.

only have a squeezed photon state if the motion motion.

For complete cycles there is no squeezing.

Reid, 1986) is easily calculated to be

c 1 (L - c 1)

gj(L-2 c

T 2”n 2 47r

(72)

^)

Although taking the vacuum state to be the initial state is somewhat intriguing, the case when

there

are some

photons in the initial state m a y

be

easier to realise.

For

definiteness we consider the state initially to be

n' > I o

=

K O1>+) ° h

nQ is the mo de number while n^ is the number in the mode. photons in the state generated is calculated as before and we find

The me an number of

170

M OVING M IRRORS A N D N O N C LA SSIC A L LIGHT

< n ' | a < 3)V

3 > | n ’>

36n

n'

o

~z (1-5 nno )

\6 7r4 / (n-n )

nn o

o

n .I oJ

d

o

-

(74)

So if iIq is large enough the number of photons created in modes with m o de number (n (* n0) is proportionately large.

In some sense the presence initially of photons is

stimulating the generation of ne w ones (but not at the same m o de number).

I o

n'Xn' I o o

fl + 2n' L o

5

(77)

V n nJ

For m o d e numbers other than nQ the m i ni mu m uncertainty condition is satisfied.

From

(76) we see that the discussion of squeezing for photons in these m o de numbers goes through as before.

For

all mechanical

mirrors

the

treatment

we

have

given

should

be

adequate.

If

non-mechanical mirrors can be designed to have relativistic motion then corrections will be necessary although our hope is that qualitative features will remain unchanged. check this a non-perturbative calculation of R.P) is needed. in (70) we can take

To

For the case of the motion

171

S SA R K A R

ncot

(78)

Substituting in (19) we obtain

iucjt V I

rn

, .. u-n r e t

J

n-u

(nojc. ) - e 1

There is one such relation for each integer u. such an expansion m a y be convergent.

° J

n-u

(-nucjl 1J

= 26

uo

(79)

Preliminary investigations suggest that

However the calculation of the me an number of

photons, etc, will still be difficult.

W e conclude by giving some estimates of the orders of magnitude of the effects that we are predicting.

For the case of a full cycle of periodic motion

, ( 3) + (3)

< o | a ^ ' ' a ^ ' |o>

1

1

=

(bv~ y (o)L)2

^

m=l

1

^ I n T 2

l(m +l)3

(1 -m)3J

(80)

where r = L/c^ and e = c^o/c.

(Here we have changed units and reintroduced c the

speed of light which we took to be 1 before.)

state, n^

~ 9 x 10^, then T

is large and the average number of

photons produced per second is 6 x 10^, a substantial increase. experimentally detectable.

If we choose for

Such an effect m a y be

For the case when the initial state is a vacuum the squeezing

parameter is (6 x 10“^)/n^.

For an initial state inQ> this result continues to hold for

m o d e numbers n different from nQ .

4.

We

CONCLUSIONS

have shown that in general motion of a mirror (from rest to eventual rest) will

create photons, even if the initial state is a vacuum. from

the

vacuum

squeezed. Without en h an ced

are

super-Poissonian,

Unless the mirror motion

In particular the photons generated

non-thermal,

going to extremely fast motions the amount if

Experiments

the

initial

checking

state these

is

a

and

in

certain

circumstances

is effectively relativisitic the effects are small.

p ure

predictions

n um ber would

be

of photon state

with

important

generation is mu ch a in

large

population.

deepening

our

understanding of quantum field theory in non-trivial geometries.

ACKNOWLEDGEMENTS

I a m grateful to J.G. Rarity, J.S. Satchell and P.R. Tapster for discussions.

REFERENCES Arnold, V.I. 1978, Mathematical Methods of Classical Mechanics, Springer-Verlag, N e w York. Caves, C.M. 1980, Phys. Rev. Lett. 45 75. Caves, C.M. 1981, Phys. Rev. D23 1693. Loudon, R. 1981, Phys. Rev. Lett. 47 815-818. Moore, G.T. 1970, J. Math. Phys. U 2679-1691. Sarkar, S. 1988, J. Phys. A21 971-980. Stoler, D. 1970, Phys. Rev. DI 3217. Yuen, H.P. 1976, Phys. Rev. A13 2226. Walls, D.F. and Reid, M.D. 1986, Frontiers in Quantum Optics, Ed. E.R. Pike and S. Sarkar, Hilger, Bristol, 72-105.

PROPAGATION OF NONCLASSICAL LIGHT

I ABRAM

1.

INTRODUCTION C la s s ic a l o p tic s d e sc rib e s q u ite s u c c e s s fu lly th e

g a tio n

of

la se r lig h t,

s p a r e n t m e d iu m . ra d ia tio n ,

b o th in f r e e - s p a c e and i n s i d e a t r a n ­

The r e a s o n i s t h a t i n

th e e le c tr ic

be

tre a te d

c la ssic a lly

T h e s am e h o l d s a l s o

alw ay s

d e sc rib e d

be

as

a

and th u s t h e i r

f o r th e rm a l

sta tistic a l

o f th e e le c tro m a g n e tic f i e l d .

e q u a tio n s

and

tre a t

On t h e o t h e r h a n d ,

lig h t

c la s s ic a l o p tic s

in

sta te ,

M a x w e ll

e le c tro m a g ­

of th e

in d u ced

f i e l d w ith a p o la riz a tio n .

of

q u a n tu m

th is

p ro b lem

th e

e le c tric

con­

o p tic s can ad d re ss ade­

is

not

fie ld

T he f a i l u r e

su rp risin g : in

such as a s in g le -p h o to n s t a t e

of T he

a n o n c la ssic a l or

does n o t c h a r a c t e r i z e th e p h o to n s t a t i s t i c s .

a squeezed T hus,

its

t h e M a x w e ll e q u a t i o n s c a n n o t g i v e a n y i n f o r m a t i o n

on

use

in

th e

e v o lu tio n

of

th e

p h o to n s t a t i s t i c s

th ro u g h l i n e a r o r n o n lin e a r t i o n a l q u an tu m o p t i c s t h a t t h i s th e o ry m odes.

can

n e ith e r c la s s ic a l o p tic s nor th e

fo rm u la tio n

e x p e c ta tio n v a lu e of sta te ,

w hich

p ro p a g a tin g

q u a te ly th e p ro p a g a tio n o f n o n c la s s ic a l l i g h t .

ra d ia tio n

in

p ro p a g a tio n

c la ssic a l

th e in te r a c tio n

p h e n o m e n o lo g ic a lly th ro u g h th e

v e n tio n a l

of

p e rm it th e c a lc u la tio n o f b o th th e s p a t i a l p r o g r e s ­

fie ld ,

m edium

sta te

m ix tu re o f c o h e re n t

The

s io n and th e te m p o ra l e v o lu tio n o f a n e tic

co herent

t h r o u g h t h e m a c r o s c o p i c M a x w e ll

e q u a tio n s. sta te s

a

a n d m a g n e t i c f i e l d s may b e w r i t t e n

term s o f t h e i r e x p e c ta tio n v a lu e s , may

p ro p a­

is

m ed ia. le ss

as l i g h t p ro p ag ates

The i n a d e q u a c y o f c o n v e n ­

o b v io u s,

and l i e s

in th e

fact

i s b a s e d on t h e H a m i l t o n i a n o f t h e r a d i a t i o n

I t th u s a d d re s s e s o n ly th e tim e - e v o lu tio n o f th e

fie ld

174

PR O P A G A TIO N OF N O N C LA SSIC A L LIGHT

p ro p a g a tin g in

(m o stly )

em pty s p a c e an d i n t e r a c t i n g w i t h i n d i ­

v id u a l m a te ria l p o in ts p o la riz a b le e n titie s

th a t cause

t h e r a d i a t i o n m odes. how ever,

cannot

e m itte rs,

(n o n lin e a r)

A t h e o r y b a s e d on a

address

p ro g re ssio n of th e co p ic ,

re p re se n tin g

fie ld

ab sorbers

in te ra c tio n s m odal

among

H a m ilto n ia n ,

d i r e c t l y t h e p ro b lem o f t h e s p a t i a l o p e r a to r s p ro p a g a tin g th ro u g h m acros­

t r a n s l a t i o n a l l y - i n v a r i a n t , p o l a r i z a b l e m ed ia,

c an n o t produce

or

sp a tia l

d iffe re n tia l

e q u a tio n s

sin c e i t

w h ich

co u ld

a c c o u n t f o r t h e c h a n g e i n mode s t r u c t u r e a n d i n t h e e x c i t a t i o n o f e a c h m ode,

a s d i f f e r e n t m ed ia a r e i n t r o d u c e d

a l i g h t beam .

For t h i s

reason,

one o f t h e m ost b a s i c p ro b lem s

of c la s s i c a l p ro p a g a tiv e o p tic s ,

th e

sp a tia l

p ro g re ssio n

l i g h t t h r o u g h a t r a n s p a r e n t r e f r a c t i v e m e d iu m , tio n

and

re fra c tio n

v a c u u m /d ie le c tric

th a t

o ccurs

in te rfa c e ,

as

in te ra c tio n

te rm

th a t,

w hen

lig h t

cro sses T here

fie ld

due to

is

fre e -fie ld

can d e s c rib e th e m o d ific a tio n o f th e s p a t i a l

sio n of th e

a

a t a l l w ith in

o p tic s:

added to th e

of

and th e r e f l e c ­

can n o t be t r e a t e d

t h e c o n v e n t i o n a l f o r m a l i s m o f q u a n tu m to n ia n ,

in th e p a th of

no

H am il­

progres­

re fra c tio n . d e sc rib e d

by

s e c o n d - q u a n t i z a t i o n o p e r a t o r s a c t i n g o n to t h e vacuum s t a t e

N o n c la ssic a l s t a t e s

|0>

of th e ra d ia tio n v a lu e s

of

C le a rly ,

th e

of lig h t

fie ld ,

and

e le c tric

are

th e

u su a lly

c o rre sp o n d in g

e x p e c ta tio n

and m a g n e tic f i e l d s a r e o f t e n

th e d e s c r ip tio n o f p ro p a g a tio n o f n o n c la s s ic a l

re q u ire s

zero. lig h t

a d i r e c t q u a n tu m m e c h a n i c a l t r e a t m e n t o f t h e s p a t i a l

p ro g re ssio n of th e

fie ld

d e v e lo p e d

t h e f o r m a l i s m o f P r o p a g a t i v e Q u an tu m O p t i c s

(A bram , th e

w ith in

1987;

1 9 8 8 ), w h ich c a n a d d r e s s t h e p ro b le m

in d e p e n d e n tly o f i t s

In th e

of

(re fra c tiv e ) in te rfa c e th e

tem p o ral e v o lu tio n .

fo llo w in g S e c tio n s ,

p ro p a g a tiv e

p a rtic u la rly

of

Such a t r e a t m e n t h a s b een

q u an tu m m e c h a n i c a l d e s c r i p t i o n o f t h e s p a t i a l p r o g r e s s i o n

of lig h t

of

A bram ,

o p e ra to rs.

q u a n tu m

o p tic s

th e tre a tm e n t of m edium

(S ectio n 4 ). p h y sic a l

(S e c tio n

(S ectio n 2 ),

p ro p a g a tio n 3),

In o rd e r to

id e a s,

g e o m e try w hich g r e a t l y

we o u t l i n e t h e

we

use

and

fe a tu re s

a n d e x a m i n e m o re

th ro u g h acro ss

fa c ilita te a

m ain

th e

p lan e-w av e

s im p l i f i e s th e m a th e m a tic a l

a

lin e a r

a re fra c tiv e d isc u ssio n p ro p a g a tio n tre a tm e n t.

175

I ABRAM

We t h u s c o n s i d e r e l e c t r o m a g n e t i c p l a n e w a v e s p r o p a g a t i n g a l o n g th e + z -a x is

i n a r e f r a c t i v e m e d iu m , w i t h t h e e l e c t r i c

fie ld

and th e m a t e r i a l p o l a r i z a t i o n P b o th a lo n g th e x - a x i s , m ag n etic f i e l d a x is.

We

H and th e m a g n e tic in d u c tio n B

c o n s i d e r n o n m a g n e tic m a t e r i a l s ,

t h i s g e o m e t r y we c a n t r e a t a l l d ire c tio n Q ua n tu m

b e in g

a lw ay s

m e c h a n ic al

c o rre sp o n d in g

q u a n titie s

im p lic it.

o p e ra to rs

c la ssic a l

are

We

a lo n g

as

by

th e

+Y”

sc a la rs,

In

th e ir

G au ssian u n i t s .

d istin g u ish e d

q u a n titie s

and th e

s o t h a t B = H.

use

E

from

th e c a re t

th e

( A) a n d we

t a k e H = 1.

2.

BASIC CONSIDERATIONS OF PROPAGATIVE QUANTUM OPTICS T he d e s c r i p t i o n o f l i g h t p r o p a g a t i o n r e q u i r e s

tio n

of

le a st)

th e v a r ia tio n of th e e le c tro m a g n etic f ie ld

tw o d i m e n s i o n s :

d in a te .

tim e p lu s

C o n v en tio n al

m ech an ical th e o r ie s )

q u a n tu m

(at le a st) o p tic s

fie ld ,

(lik e

m o tio n

of

m ost

th e

q u a n tu m

H am ilto n ian

th e f ie ld

(i.e .

th e

t h a t can be o b ta in e d

equa­

from t h e

H e is e n b e r g e q u a t i o n ) , th r o u g h w hich t h e t i m e - e v o l u t i o n o f norm al

m odes o f t h e

p l e t e quan tu m re q u ire s

an

fie ld

m e c h a n ic al e x te n sio n

can be c a lc u la te d . d e sc rip tio n

of

sp a tia l

o f q uantum o p t i c s t o

d i f f e r e n t i a l eq u a tio n s.

q u i t e e a s i l y by u s e o f t h e tro m a g n e tic f i e l d 6,

momentum

C le a rly ,

lig h t

c a lc u la tio n s of th e s p a tia l p ro g re ssio n o f th e d ire c t

(at

and th u s w ith in t h i s th e o r y o n ly

te m p o r a l d i f f e r e n t i a l e q u a t i o n s c an be s e t up of

a lo n g

one s p a t i a l c o o r­

g iv e s a c e n tr a l r o le to

o f th e e le c tro m a g n etic tio n s

c o n sid e ra ­

th e

a com­

p ro p a g a tio n

in c lu d e e x p l i c i t fie ld ,

th ro u g h

T h is can be a c h ie v e d

o p e ra to r

of

th e

e le c ­

s i n c e by d e f i n i t i o n (2 . 1)

or eq u iv a le n tly

6 = -i w here

'V

is

[M ]

any w a v e fu n c tio n and 0 any o p e r a t o r .

(2 .2)

176

P R O P A G A T IO N OF N O N C LA SSIC A L LIGHT

In c a l c u l a t i n g th e H a m ilto n ia n th e

fie ld

is

tre a te d

q u a n tu m

and

momentum

m e c h a n ic a lly th ro u g h seco n d -

q u a n tiz a tio n o p e ra to rs.

On t h e

in tro d u c e d

th ro u g h

in d u c e d p o l a r i z a t i o n ,

th ro u g h i t s

o p tic a l s u s c e p tib ility

p h en o m en o lo g ical

its

tre a tm e n t

o th e r

of

hand,

^

th e

=

th e

m ed iu m

P /E .

m a te ria l

T h is

q u a si-

m edium p e r m i t s fie ld

from t h e

d e s c rip tio n of th e fie ld -m a tte r in te ra c tio n .

o p tic a l s u s c e p tib ility

is

or e q u iv a le n tly

s e p a r a t i o n o f t h e p ro b le m o f p r o p a g a tio n o f t h e m ic ro sc o p ic

o p e ra to rs,

can be c a lc u la te d b e fo re h a n d

T he

( f o r exam ­

p l e th ro u g h th e d ia g ra m m a tic te c h n iq u e s d e v e lo p e d in n o n lin e a r o p t i c s b y Yee a n d G u s t a f s o n c h a ra c te ristic p a g a tio n . tre a te d

(1978))

Thus, w ith in t h i s in

a

a

approach,

lig h t

p ro p a g a tio n

is

m anner c o m p le te ly a n a lo g o u s t o th e m ac ro sc o p ic

M axw ell e q u a t i o n s , o p tic s.

and th e n u se d sim p ly a s

c o n s t a n t o f t h e m edium i n t h e t r e a t m e n t o f p r o ­

on w h ic h

T h is s i m i l a r i t y

is

based

c l a s s i c a l p ro p a g a tiv e

su g g e sts t h a t th e r e s u l t s

c a l o p t i c s may s e r v e a s a g u i d e i n

c h o o sin g

th e

of c la s s i­ a p p ro p ria te

q u a n t u m m e c h a n i c a l e n e r g y a n d momentum o p e r a t o r s t h a t d e s c r i b e l i g h t p ro p a g a tio n p ro p e rly . From c l a s s i c a l wave

e n te rs

o p tic s,

we know t h a t w hen a h a r m o n i c l i g h t

a tra n sp a re n t

e v o lu tio n does n o t change: sa m e

frequency.

n e tic a ll

fie ld

in th e

fie ld

of

re m a in s

E q u iv a le n tly ,

p ro p a g a tin g

f i e l d , and i s

consequence m a te ria l

re fra c tiv e

it

energy

m e d iu m ,

its

tim e -h a rm o n ic

at

tim e th e

th e energy o f th e e le c tro m a g ­

th ro u g h

d iffe re n t

m ed ia,

i n d e p e n d e n t o f t h e m e d iu m . c o n se rv a tio n ,

sin c e

a

re m a in s T h is i s

a

tra n sp a re n t

d o e s n o t e x t r a c t any e n e r g y from t h e e l e c t r o m a g n e t i c

p ro p a g a tin g th ro u g h i t .

On t h e o t h e r

hand,

th e

e le c ­

t r o m a g n e t i c e n e r g y - d e n s i t y , g iv e n by

U

is as

=

o TT

(E 2

+

H2

+

4TTPE)

l a r g e r i n s i d e t h e r e f r a c t i v e m edium ( t h a n i n it

e le c tric tio n s,

in c lu d e s th e e f f e c ts

of th e

free

(2 .3 ) space),

in d u ced p o l a r i z a t i o n .

For

a n d m a g n e t i c f i e l d s c o n s i s t e n t w i t h t h e M a x w e ll e q u a ­ th e

e n e rg y -d e n sity

i n d e x o f t h e m ed iu m

is p ro p o rtio n a l to th e r e f r a c tiv e

(n = J 1 + 4irp() , a n d i s

th u s r e la te d to th e

177

I ABRAM

fre e -fie ld

e n e rg y by u. R

n u

w here t h e s u b s c r i p t s R and o

(2 .4 )

o

refer

to

th e

d e n sity

r e f r a c t i v e m edium a n d i n e m p ty s p a c e r e s p e c t i v e l y .

in

th e

T h is means

t h a t t h e quan tu m m e c h a n ic a l d e s c r i p t i o n o f

lig h t

m ust

in d e p e n d e n t o f th e

be

based

on

a

m edium o f p r o p a g a t i o n , e ig e n v a lu e s

are

H a m ilto n ia n t h a t i s

b u t on a n e n e r g y - d e n s i t y o p e r a t o r w hose

p ro p o rtio n a l to th e r e f r a c tiv e

th e H a m ilto n ia n and e n e r g y - d e n s ity o p e r a to r s each

o th e r

p ro p a g a tio n

in d ex .

are

S in ce

re la te d

to

by i n t e g r a t i o n o v e r t h e v o lu m e V o f t h e c a v i t y o f

q u a n tiz a tio n ,

Ot = V u th is

(2 .5 )

im p lie s t h a t e n e rg y -c o n s e rv a tio n upon p ro p a g a tio n can

in tro d u c e d

in

q u a n tu m o p t i c s t h r o u g h t h e a p p r o p r i a t e d e f i n i ­

tio n of th e c a v ity of q u a n tiz a tio n , A t t h e s am e t i m e , th e

p a th

of

be

as w ill be seen l a t e r .

i n t r o d u c t i o n o f a r e f r a c t i v e m edium

a l i g h t beam , m o d i f i e s i t s

in

s p a tia l p ro g re ssio n :

t h e w a v e l e n g t h o f a h a r m o n ic wave i s

re d u c e d by th e r e f r a c t i v e

in d e x

t h e momentum ( w a v e v e c t o r )

o f t h e m e d iu m .

E q u iv a le n tly ,

o f a h a r m o n i c w a v e d e p e n d s o n t h e m edium o f p r o p a g a t i o n a n d i s p ro p o rtio n a l to th e re f r a c tiv e

in d ex .

T h is i n t u r n means t h a t

t h e a p p r o p r i a t e momentum o p e r a t o r t h a t d e s c r i b e s p ro g re ssio n m ed iu m , to

th e

of

and i t s

th e

fie ld

e ig e n v a lu e s

re fra c tiv e

m a te ria l

m e d iu m ,

(th e w av ev ecto rs)

in d ex .

a r e s e v e r a l w ays o f

"m ech an ical"

d e fin in g because

of

momentum.

th e above re q u ire m e n ts i s th e and

corresponds

th e

fie ld

th e

tio n a ry sity

is

m edium

momentum

d iffe re n t

so -c a lle d to

a

w ays

in sid e

a

th a t th e

"e le c tro m a g n e tic "

M in k o w sk i

19 8 5 ) .

a v e c to r d i r e c t e d a lo n g th e

momentum,

p s e u d o - m o m e n tu m

in v a ria n c e of th e f i e l d

(P e ie rls,

are p ro p o rtio n a l

The d e f i n i t i o n t h a t c o n fo rm s t o

e sse n tia lly

d e sc rib e s th e s p a tia l

sp a tia l

In c l a s s i c a l e le c tro d y n a m ic s th e r e

" t o t a l 11 momentum may b e p a r t i t i o n e d b e t w e e n and

th e

q u an tu m m e c h a n i c a l l y d e p e n d s on t h e

in sid e

a

th a t sta ­

T h e M in k o w s k i momentum d e n ­ z -a x is,

w hose

v a lu e

is

178

P R O P A G A T IO N OF N O N C LA SSIC A L LIGHT

g iv e n by

(2 .6) w here D =

is th e e le c tr ic

E +

4-rcP =

(1

+

4 rtjx)

(2 .7 )

E

d isp lacem en t.

We t u r n now t o t h e c a v i t y o f q u a n t i z a t i o n . is

u su a lly

tak en

i n t e g r a l number o f in te re st,

to

be

fin ite

w av elen g th s

of

obeys

T h is

th e

ra d ia tio n

stu d y th e

can

cover

fie ld

a ll

in i n f i n i t e

of

(in fin ite )

space,

in tro d u c e d in

th e

box,

p e rio d ic

boundary

fille d

T hus,

th u s

in

th re e

In o rd e r to

su ffic ie n t

to

space,

th e d im en sio n s

co m b in ed w i t h e q .

c a v ity of q u a n tiz a tio n , its

and

th e

p la n e s

th e

on

p e rio d ic -b o u n d a ry

is

V o n

p e rio d ic -b o u n d a ry

th e n

of

w i t h a r e f r a c t i v e m e d iu m ,

re q u ire

th a t

in d ex ,

c o n d i t i o n s a r e o b ey ed g e t c l o s e r by

T h is e q u a tio n ,

m e d iu m .

is

space.

When a t r a n s p a r e n t r e f r a c ­

t h e v o lu m e

R

te n ts,

th e

p e rio d ic -b o u n d a ry c a v ity change s in c e a l l w a v e le n g th s

t h e sa m e f a c t o r . c a v ity

it

(in fin ite )

becom e s h o r t e r by t h e r e f r a c t i v e w h ich

of

mode s p a c i n g i s b e y o n d

At o p p o site s id e s o f

ex am in e one " r e p r e s e n t a t i v e u n i t " . th e

an

p e r io d ic boundary c o n d itio n s and th u s th e c a v ity

t i v e m e n d iu m i s of

is

waves

c o n s t i t u t e s a " r e p r e s e n t a t i v e u n i t 11 w h o s e r e p e t i t i o n d im e n sio n s

c a v ity

b o x , w hose l e n g t h

b u t l a r g e enough so t h a t i t s

e x p e rim e n ta l r e s o lu tio n . fie ld

a

(2 . 8)

(2 .4 )

in d ic a te s

c o n d itio n s

in d e p e n d e n tly

of

its

energy c o n te n ts a re a ls o

th a t

if

we

be obeyed by th e m a te ria l

con­

in d e p e n d e n t o f th e

That is , (2 .9 )

and th u s t h e H a m ilto n ia n tro m a g n e tic

fie ld

(and t h e t i m e - e v o l u t i o n )

q u a n tiz e d

in

such

a c a v ity

of th e e le c ­ is

t h e sam e,

179

I ABRAM

w h e th e r

th e

c a v ity

r e f r a c t i v e m ed ia. tiv e

o p tic s

e m p ty

or

c o n ta in s

(one

or

m o re )

The e n e r g y c o n s e r v a t i o n f e a t u r e o f p r o p a g a ­

is th u s e q u iv a le n t

p e rio d ic -b o u n d a ry tio n

is

to

th e

p re se rv a tio n

of

th e

c o n d itio n s used fo r th e c a v ity o f q u a n tiz a ­

i n quan tu m o p t i c s . S im ila rly ,

th e

M in k o w s k i

p e rio d ic -b o u n d a ry c a v ity

momentum

co n ta in e d

in

th e

is, V

G = VRg R = T hus, th e

(1 +

E B = n Vo g o

a c c o rd in g to t h i s d e f i n i t i o n ,

( 2 - 10)

t h e momentum

c a v ity a re p ro p o rtio n a l to th e r e f r a c tiv e

c o n te n ts

in d ex ,

t h e y d e p e n d on t h e m a t e r i a l c o n t e n t s o f t h e c a v i t y way a s t h e w a v e v e c t o r .

of

and th u s

i n t h e sam e

T h i s i m p l i e s t h a t t h e q u an tu m m e c h a n i­

c a l o p e r a t o r t h a t c o r r e s p o n d s t o t h e M i n k o w s k i momentum s h o u l d d e sc rib e

s u c c e s s fu lly th e s p a tia l p ro g re ssio n of th e

fie ld

in

a n y m e d iu m .

3.

PROPAGATION IN A REFRACTIVE MEDIUM T he d i s c u s s i o n o f S e c t i o n 2 may e a s i l y b e a p p l i e d t o

q u a n tiz e d fie ld

e le c tro m a g n e tic

o p e ra to rs

(A b ram ,

fie ld .

The e l e c t r i c

q u a n tiz e d in f re e space

may

be

th e

and m a g n e tic w ritte n

as

1 98 7)

£ (z ,t)

=

)

( 3 . la )

(£j+6_j)

(3 -lb )

ej

and

~iSj|~v Ij j w here

is th e

S j = +1

is

( t o w a r d s + z)

th e

j

freq u en cy o f th e j - t h sig n

of

and n e g a tiv e

j , fo r a

h a rm o n ic p la n e wave,

p o sitiv e

fo r

b ac k w a rd -g o in g

and

a fo rw a rd -g o in g (to w ard s

-z)

180

PR O P A G A T IO N OF N O N C LA SSIC A L LIGHT

wave.

The

th e j - t h

o p e ra to rs

mode o f t h e

re la tio n s. c re a tio n

They

d iffe r

im p lic it

d e fin e d in eq s.

and fo llo w

in

in

(3 .1 ),

6 t(6 j).

Bose

th a t

co m m u tatio n

th e

sp a tia l

The

are

not

a sso c ia te d

w ith

a

sin g le

in d ic e s,

wave,

fie ld .

and

and do n o t

They a r e

sim p ly

sin c e th ey co rresp o n d to th e

sp a tia l

and m a g n e tic f i e l d s

The f r e e - s p a c e e l e c t r i c us

e^ a n d fij

a r e com posed o f a l i n e a r c o m b in a tio n o f

v i d u a l co m p o n en ts o f t h e

p e rm it

and

a re u n d er­

o p e ra to rs

and a n n i h i l a t i o n o p e r a to r s o f o p p o s ite

a m a th e m a tic a l co n v e n ie n c e,

(3 .1 )

a p h o to n in

h o w e v e r from t h e t r a d i t i o n a l p h o to n

o p e ra to rs,

re p r e s e n t p h y s ic a l o b se rv a b le s o f th e

e le c tric

(a n n ih ila te )

o s c illa tio n s of th e ele c tro m a g n etic f i e l d

sto o d to be

th u s

fre e -fie ld ,

(a n n ih ila tio n )

te m p o ra l

c re a tio n

c re a te

F o u rie r

ex p a n sio n

in d i­

of

th e

re sp e c tiv e ly . and

to convert a l l

m ag n etic

fie ld

o p e ra to rs

c la s s ic a l o b se rv a b le s o f th e

e le c tro m a g n e tic f i e l d

i n t o t h e i r q u an tu m m e c h a n i c a l e q u i v a l e n t

o p e ra to rs,

f r e e - s p a c e a n d i n s i d e a m a t e r i a l m e d iu m .

T h is

b o th

in

c o n v e r s io n can be done by sim p ly r e p la c in g

and m a g n e tic f i e l d s

th e

e le c tric

in th e c la s s ic a l d e f in itio n o f th e o b serv ­

a b le by th e c o rre s p o n d in g o p e r a to r s o f e q s. In p a r tic u la r ,

th e

energy

d e n sity

(3 .1 ). o p e ra to r

in sid e

a

l i n e a r m edium i s

u = jA

( £ 2 + ft2 + 4n^fe2 ) =

= si

j®-j +

(3 .2 a )

+ 4^ gj®-j)

4 4 w h e r e we h a v e a s s u m e d t h a t t h e e n e r g y - d e n s i t y u n ifo rm ly

in

space,

term s t h a t o s c i l l a t e B ose

c re a tio n

a n d t h u s we e l i m i n a t e d sp a tia lly .

In term s

and a n n i h i l a t i o n o p e r a t o r s ,

(3,2b) is

d istrib u te d

f r o m t h e sum , a l l

of

th e

fre e -fie ld

each in d iv id u a l j -

c o m p o n e n t o f t h e e n e r g y - d e n s i t y o p e r a t o r may b e w r i t t e n

aj = ^

as,

{6j 6j + 6- j 6-j " 2*2(6j " 6- j )(6-j " 6j )} (3-3)

181

I ABRAM

T hus,

in s id e a lin e a r d i e l e c t r i c th e e n e rg y -d e n sity o p e ra to r u

i s n o t d i a g o n a l w hen e x p r e s s e d i n te r n s o f f r e e - f i e l d a n d i n c l u d e s a c o u p l i n g b e t w e e n t h e tw o m e m b e rs o f

o p e ra to rs each

p a ir

o f c o u n t e r - p r o p a g a t i n g m odes.

T h is c o u p lin g c o n s i s t s o f te rm s

t h a t do n o t c o n s e r v e e n e r g y t o

first-o rd e r,

and

are

of

th e

form

2^ ( 6 j A j T h e s e i n t e r a c t i o n t e r m s may b e d e n sity o p e ra to r

becom es

r e la te d to th e of

a

w hen

and

th e

energy-

ex p ressed

in

T he r e f r a c t e d - w a v e o p e r a t o r s

th e ^ j/^ j

f r e e - s p a c e o p e r a t o r s & j / £ j t h r o u g h a Bogo-

liu b o v tr a n s f o r m a tio n . a p p lic a tio n

(3 .4 )

e lim in a te d ,

d ia g o n a l

re fra c ted -w a v e b a s is s e t. are

+ 6 j6 _ j)

T h is tr a n s f o r m a tio n c o rre s p o n d s to th e

u n ita ry o p e ra to r

(sim ila r to th e

"squeeze"

o p e r a t o r u s e d f o r exam ple i n t h e p ro b le m o f p a r a m e t r i c g e n e r a ­ tio n

(S to le r,

0

1970; S t o l e r ,

1 9 7 1 ; Y uen,

1976)

of th e

form

= e x p { J ^ ( 6 _ .6 _ _ .- 6 t6 * ..) } = i

(3 .5 a )

= e x p ( ^ ( 6 . 6__.-£*&*_.) )

(3 .5 b )

i w here

\ w ith

n b e in g th e

tra n sfo rm a tio n tio n )

I n ( 1 + 4 irpj) = |

re fra c tiv e

in d e x

In(n) of

th e

(3 .6 ) m e d iu m .

r e la te s th e re fra c ted -w a v e c re a tio n

o p e r a t o r s fit(fi_.)

T h is

(a n n ih ila ­

to th e c o rre sp o n d in g f r e e - f i e l d

o pera­

t o r s th ro u g h

f i t = ft 1 f i t = 6 1

fit

=

6 fit

ft = c o s h j

fit

- sin h ^

fi_j

ft = c o s h ^ £>j - sin h 'jj fi*j

ft”"1 = c o s h ^

fit

+ s i n h j j fi_^

(3 .7 a )

(3 .7 b )

(3 .7 c )

182

P R O P A G A T IO N OF N O N C LA SSIC A L LIGHT

( 3 . 7d)

The r e l a t i o n s h i p b e tw e e n t h e w ave

o p e ra to rs

th e r e f r a c tiv e

(eqs. in d e x ,

3 .7 )

fre e -fie ld

and

re fra c te d -

may a l s o b e e x p r e s s e d i n t e r m s o f

sin c e (3 .8 a ) (3 .8 b )

In se rtin g to r

e q s . (3 .7 )

in to eq.

(3 .3 ),

th e energy d e n s ity

op era­

i n t h e r e f r a c t e d - w a v e b a s i s s e t may b e o b t a i n e d a f t e r som e

a lg e b ra as

(3 .9 )

The

e n e rg y -d e n sity

re fra c te d -w a v e

o p e ra to r

b a sis s e t,

is

th u s

and a l l

its

d iag o n al

in

th e

e ig e n v a lu e s a re r e la te d

to th e

fre e -fie ld

e n e r g y - d e n s ity e ig e n v a lu e s by th e r e f r a c t i v e

in d ex

n.

is

o b ta in e d

T h is

in c la s s i c a l

M ax w ell e q u a t i o n s in

free-sp ace.

o p e ra to r

e x a c tly

in

o p tic s

fo r th e

Thus,

t h e sa m e (eq. 2 .4 ) ,

fie ld

o p tic s

w h en

as

th a t

so lv in g

th e

i n s i d e a r e f r a c t i v e m ed iu m a n d

d ia g o n a liz a tio n

q u an tu m

re la tio n sh ip

is

of

th e

e n e rg y -d e n sity

e q u iv a le n t to th e s o lu tio n of

t h e M a x w e ll e q u a t i o n s i n c l a s s i c a l o p t i c s . The p r o p o r t i o n a l i t y o f t h e e n e r g y - d e n s i t y e i g e n v a l u e s th e r e f r a c tiv e

in d ex im p lie s t h a t in o r d e r t o have e n e rg y con­

s e r v a t i o n up o n p r o p a g a t i o n i n quan tu m o p t i c s in

order

c a v ity

to

(or e q u iv a le n tly ,

to p re s e rv e th e p e rio d ic -b o u n d a ry c o n d itio n s o f th e

of q u a n tiz a tio n ),

q u a n tiz a tio n c a v ity ,

is

th e

v o lu m e

re la te d to th e

of th e

re fra c te d -w a v e

fre e -fie ld

q u a n tiz a tio n

v o lu m e by

V as d isc u sse d

in S e c tio n 2.

V o n

W ith t h i s

(3 .1 0 ) q u a n t i z a t i o n v o lu m e,

th e

183

I ABRAM

r e fra c te d -w a v e H a m ilto n ia n i s

(3 .1 1 ) and i s th e

id e n tic a l

i n f o r m a n d w i t h t h e sa m e e i g e n f r e q u e n c i e s a s

f r e e - s p a c e H am ilto n ian . T he e q s .

o p e ra to rs

(3 .7 )

p e rm it

r e la tin g th e f r e e - f i e ld us

to

re fra c ted -w a v e b a s is s e t. e le c tric

convert

T hus,

and m a g n e tic f i e l d

a ll

and r e f r a c te d - w a v e o p e ra to rs

su b stitu tin g

in eqs

to

th e

(3 .1 )

th e

o p e ra to rs can be e x p re sse d in th e

re fra c ted -w a v e b a s is s e t as

(3 .1 2 a )

and

(3 .1 2 b ) j T he r e f r a c t e d - w a v e momentum may b e o b t a i n e d

in a

sim ila r

fa sh io n as

(3 .1 3 )

J w h e r e t h e j - t h momentum e i g n e n v a l u e

(3 .1 4 )

K. = D j i s t h e w a v e v e c to r o f t h e j - t h r e f r a c t e d w ave. P ro p a g a tio n o f th e e le c tro m a g n e tic m ed iu m ,

fie ld

in sid e a

i s c o m p le te ly d e s c rib e d th ro u g h th e H a m ilto n ia n

a n d momentum ( 3 . 1 3 )

o p e ra to rs,

lin e a r (3 .1 1 )

w hich p e r m i t t h e c a l c u l a t i o n

of

184

PR O P A G A T IO N OF N O N C LA SSIC A L LIGHT

th e

te m p o ra l

to rs

and a l l

e v o lu tio n and s p a t i a l p r o g r e s s io n o f a l l sta te s

tro m a g n e tic

( c l a s s i c a l and n o n c l a s s i c a l )

fie ld .

In p a r tic u la r ,

e v o lu tio n o f any f i e l d tio n .

F o r ex am p le,

o p e ra to r,

eq.

(3 .1 1 )

opera­

of th e

e le c ­

g iv e s th e tim e -

th ro u g h th e H e ise n b e rg

equa­

fo r th e a n n ih ila tio n o p e ra to r of th e j - t h

mode we h a v e ,

3 k,

A

= i [ # , k j] = In te g ra tin g

t h i s e q u a tio n o f m o tio n ,

(3.15a)

we o b t a i n

atim e -h a rm o n ic

e v o l u t i o n a t f r e q u e n c y UK f o r t h e r e f r a c t e d w a v e s ,

= e 1^ t

k. (t) id e n tic a l to

th a t

S im ila rly ,

th e

th ro u g h eq.

(2 .2 )

sp a tia l Thus,

of

= k .(0 )e

th e

sp a tia l

c o rre sp o n d in g

^

(3 .1 5 b )

fre e -fie ld

p ro g re ssio n of th e

fie ld

is

m odes. o b ta in e d

w h ich c a n be c o n s i d e r e d a s a H e i s e n b e r g - l i k e

e q u a tio n

of

fo r th e j - t h

m o tio n

i n v o l i n g t h e momentum o p e r a t o r .

p l a n e w a v e i n s i d e a l i n e a r m e d iu m , ^ §. = -i

In te g ra tin g

t h i s e q u a t i o n we o b t a i n ,

th e o s c illa to r y

(z)

fa c to r

of

Thus, a its

lin e a r

n

freq u en cy

(w a v e v e c to r)

o p tic s,

(3 .1 6 b ) ren o rm a liz ed

by

w ith r e s p e c t to th e c o rre sp o n d in g f r e e - f i e l d

as in c la s s ic a l o p tic s , d ie le c tric ,

its

w hen a l i g h t be a m t r a v e r s e s

fre q u e n c y rem ain s unchanged, w h ile th e

re fra c tiv e

of

th e

The t i m e - e v o l u t i o n o f t h e r e f r a c t e d w av es i s

th u s

i n d e p e n d e n t o f t h e m edium o f p r o p a g a t i o n : rem a in s

in c l a s s i c a l

a s e x p e c t e d from c l a s s i c a l c o n s i d e r a t i o n s .

w a v e v e c to r i s m o d ifie d by

d ie le c tric .

as

(3 .1 6 a )

= e " i 6 z k j e i 6 z = § j e lK j Z

how ever w ith a s p a t i a l a

= iK jlj

s p a t i a l p r o g r e s s i o n o f a r e f r a c t e d wave,

|

w a v e v e c to r,

[6 ,k j]

tim e -h a rm o n ic

at

th e

in d ex

a t i m e - h a r m o n i c w ave

s am e f r e q u e n c y .

On t h e o t h e r

185

I ABRAM

hand,

its

s p a tia l p ro g re ssio n i s

quency o f o s c i l l a t i o n

a lte re d ,

as i t s

(o r e q u iv a le n tly i t s

s p a tia l

fre­

phase v e lo c ity )

are

changed by r e f r a c t i o n .

4 . REFLECTION IN PROPAGATIVE QUANTUM OPTICS In c l a s s i c a l o p tic s , in te rfa c e

b e tw een

w hen a l i g h t beam i s

em pty

space

b e t w e e n tw o r e f r a c t i v e m e d i a ) is

tra n sm itte d

forw ard

and

it

w aves,

in c id e n t,

a r e f r a c t i v e m edium ( o r

sp lits

in to

th e

r e f l e c t e d b a c k i n t o e m p ty s p a c e .

i n t o tw o

beam s:

m edium w h i l e t h e o t h e r i s

t a n g e n t i a l c o m p o n en ts o f t h e e l e c t r i c

in s u r e s t h a t th e

and m a g n e tic f i e l d s

d isc o n tin u ity

in r e f r a c tiv e

c o rre sp o n d s t o an a b ru p t momentum

o p e ra to rs

change

o p tic s,

in d ex a c r o s s th e

of

th e

upon p r o p a g a tio n ,

energy

"sudden ap p ro x im a tio n " a t th e

i n c i d e n t on a r e f r a c t i v e c o rre sp o n d in g

g e ts

face

of th e e le c tr ic

Thus,

of

re la te s,

th ro u g h

(in c id e n t) to rs. to rs,

on th e

f o r e x am p le i n

re fra c ted -w a v e

eq u a tio n s)

th e e le c tr ic

th e o p e ra to rs

re la te d

and

e ith e r eq.

o p e ra to rs

and u s in g e q s.

(3 .7 )

th a t

th e

acro ss th e fie ld

th e

th e

con­ in te r­

to

th e

o p e ra to rs.

in te rfa c e (3 .5 )

are w h ich

fo rw a rd -g o in g

t h e m edium t o t h e (re fle c te d )

th e

A lte rn a tiv e ly ,

tra n sfo rm a tio n

(3 .7 a ),

in

of

o n to

co rresp o n d s

m ag n etic sid e

u n ita ry

and b a c k w a rd -g o in g

W ritin g eqs

th e

fo rw a rd -g o in g

fre e -fie ld

opera­

i n t e r m s o f t h e i n c i d e n t w ave o p e r a ­

(3 .8 ),

we o b t a i n t h e o p e r a t o r

o f th e F re s n e l fo rm u la s o f c l a s s i c a l o p t i c s , re fle c tio n

A f r e e - f i e l d w av e

assu m in g

and m a g n e tic f i e l d s

( i m p o s e d b y t h e M a x w e ll

c o n tin u ity

and in v o k in g

p ro je c te d

w a v e s o f t h e r e f r a c t i v e m e d iu m .

t h e p r o j e c t i o n may b e c a l c u l a t e d b y tin u ity

or

a n d t h u s may b e t r e a t e d

in te rfa c e :

in te rfa c e

th e

in te rfa c e

d e n sity

by e x am in in g t h e s p a t i a l p r o g r e s s i o n o f t h e f i e l d th e

are

in te rfa c e .

W i t h i n t h e f o r m a lis m o f p r o p a g a t i v e q u an tu m abrupt

one

The c o e x i s t e n c e o f t h e t h r e e

t r a n s m i t t e d and r e f l e c t e d ,

c o n tin u o u s a c ro s s th e

i n c i d e n t on t h e

and r e f r a c t i o n o f l i g h t :

e q u iv a le n t

th a t d e sc rib e th e

186

PR O P A G A T IO N OF N O N C L A SSIC A L LIGH T

j

n+ 1

j

n+ 1

A+ _ 2 / n a + j n+1 j

(4 .1 b )

-j

n-1 a n+1 - j

(4 .1 c )

A _ 2Jn a n - 1 a+ j n+1 j " n+1 -j

( 4 . Id)

T h e s e e q u a t i o n s d e s c r i b e q u an tu m m e c h a n i c a l l y t h e re fle c tio n

and

n a tu ra lly th e c ie n ts

as

re fra c tio n

fa m ilia r

at

th e

in te rfa c e ,

tra n sm issio n

re su ltin g d ire c tly

tio n th a t re la te s

an

and

re fle c tio n

i n c i d e n t on t h e

(4 .1 )

c o e ffi­

and r e f r a c t e d w aves. p la n e

wave

1963)

Io is

of

from t h e B o g o liu b o v t r a n s f o r m a ­

fre e -fie ld

When a q u a s i - c l a s s i c a l w a v e p a c k e t o f t h e j - t h (G la u b e r,

p rocess

and g iv e q u ite

and

> = e

in te rfa c e ,

sp lits

in to

-1 it

-1 |o >

(4 .2 )

i s p r o je c te d a c c o rd in g to eq.

a t r a n s m i t t e d and a r e f l e c t e d

q u a si-

c l a s s i c a l w av ep ack ets:

- i * j ^ S ( f t + §+) = e

+6 + ) e

|0 >

w hose e x p e c t a t i o n v a l u e s o f t h e e l e c t r i c

et

-

K?T EI

"t - S I are th u s

HI

and m a g n e tic f i e l d s

ER -

S IT EI

“r -

-

(4 .4 a ,b )

S IT " l

i d e n t i c a l to th e c o rre sp o n d in g c l a s s i c a l sa tisfy

e le c tric

th e

c o n tin u ity

co n d itio n

(4 .3 )

fo r

(4 .5 a ,b ) e x p re ssio n s

and

th e ta n g e n tia l

and m a g n e tic f i e l d s .

The quan tu m m e c h a n ic a l F r e s n e l f o r m u la s

(4 .1 )

can

a lso

I ABRAM

d e s c rib e th e b e h a v io r o f re fra c tiv e

in te rfa c e .

187

n o n c la ssic a l

Thus,

lig h t

in c id e n t

a sin g le -p h o to n s t a t e

f r e e - f i e l d w ave, w h ich ca n b e d e s c r i b e d a s an th e e le c tr ic

fie ld

on

of th e j - t h

o sc illa tio n

w i l l be tra n sfo rm e d a t a d i e l e c t r i c (4 .1 ),

< v s j )|0 > + S i

to g iv e an o s c i l l a t i o n r e f r a c t e d w ave, Thus, F resn el

(4.6) in te rfa c e ,

a c co rd in g

to

as

M

j-th

of

o f t h e form

(6j + fit) |0>

eqs.

a

w ith in

fo rm u la s

i o>

of a sin g le -p h o to n s t a t e

o r in th e - j th is

(4 .1 )

(re fle c te d )

fo rm alism ,

th e

e i t h e r in

th e

f r e e - f i e l d wave.

q u an tu m

m ech an ical

p e rm it to d e s c rib e b e a m s p littin g

m anner c o m p le te ly a n a lo g o u s t o t h a t o f c l a s s i c a l o p t i c s ,

in a w h ile

a t t h e sa m e t i m e t h e y a c c o u n t f o r t h e q u a n t u m s t a t i s t i c a l p r o ­ p e rtie s

o f b o th th e r e f l e c t e d

and th e r e f r a c t e d w aves.

5. CONCLUSIONS I n q u an tu m o p t i c s a g e n e r a l s t a t e fie ld

c o n sists

fie ld For

of

a

fu n c tio n

of th e

e le c tro m a g n e tic

of th e e le c tr ic

and m a g n e tic

o p e r a t o r s a p p l i e d o n to t h e e l e c t r o m a g n e t i c vacuum s t a t e . sta te s

th a t

m ag n etic f i e l d correponds

to

have a c l a s s i c a l a n a lo g u e ,

o p e ra to rs p re se n t th e

an

th e e le c tr ic

e x p e c ta tio n

co m p lex o s c i l l a t i n g

c la ssic a l

p ro p a g a tio n o f such s t a t e s th ro u g h a p o la r iz a b le th u s

be

d e sc rib e d

v a lu e

and th a t

fie ld .

T he

m e d iu m ,

may

t h r o u g h t h e m a c r o s c o p i c M axw ell e q u a t i o n s

w h ich c a n b e u s e d t o c a l c u l a t e b o th t h e tim e e v o l u t i o n and t h e s p a tia l p ro g re ssio n of th e se e x p e c ta tio n v a lu e s. s i c a l l i g h t on t h e o t h e r h a n d , e le c tric

and

th e s t a t e g a tio n

of

m ag n etic

of th e f ie ld . such s t a t e s ,

For n o n c la s­

th e e x p e c ta tio n v a lu e s

of

th e

f i e l d s do n o t c h a r a c t e r i z e c o m p le te ly T hus, it

in o rd e r to d e s c rib e th e p ro p a­

i s n e c e s s a ry to c a l c u l a t e b o th th e

188

PR O P A G A T IO N OF N O N C L A SSIC A L LIGHT

te m p o ra l e v o lu tio n and th e s p a t i a l p r o g r e s s io n

of

th e

fie ld

o p e ra to rs. P r o p a g a t i v e q u an tu m o p t i c s tio n a l

th eo ry

of

is

an e x te n s io n o f th e conven­

quan tu m o p t i c s t h a t r e l i e s

o p e ra to r o f th e e le c tro m a g n e tic f i e l d , th e

sp a tia l

p ro g re ssio n

of lig h t,

o n t h e momentum

fo r th e c a lc u la tio n

i n a d d i t i o n t o t h e H a m il­

to n ia n t h a t d e s c r ib e s th e te m p o ra l e v o lu tio n . th is

th eo ry t r e a t s

fie ld

p h e n o m e n o lo g ic a lly

T h e m edium

th ro u g h

its

c o n ta in s,

at

te n ts.

The p r e s e r v a t i o n o f

its

m e d ia , as

is

fie ld

th e o ry

M a x w e ll

o p e ra to r is

p e rio d ic -b o u n d a ry

th a t as

lig h t

p ro p a g a te s

eq u a tio n s:

o p t i c s and re d u c e s t o

it

by

an a lo g y

w ith

g iv e s r i s e

fie ld ,

th e

(w ith

re n o rm a liz e d w av ev ecto r)

A t t h e sam e

th e o ry

c la ssic a l

p ropaga­

th e

o p tic s.

T he

th e

tem ­

s p a tia l p rogres­ in

th e

m edium

an d t o a r e f l e c t e d wave a t t h e c o e ffic ie n ts,

a re g iv e n w ith in t h i s

d i r e c t l y by th e tr a n s f o r m a tio n t h a t

d ia g o n a liz e s

fam ­

fo rm a lism

th e

energy

The d i r e c t c o r r e s p o n d a n c e o f t h e r e s u l t s

t h e o r y w ith t h e w ell-k n o w n r e s u l t s o f c l a s s i c a l

o p tic s d em o n stra te s th e v a l i d i t y a p p ly t h i s

equa­

g iv e s

affect

b o t h t o a r e f r a c t e d w ave

c la s s ic a l o p tic s,

M ax w ell

(c la ssic a l)

The t r a n s m i s s i o n and r e f l e c t i o n

d e n sity o p e ra to r.

d iffe re n t

i s t h e d i r e c t q u an tu m

b u t changes i t s

it

from

in con­

in th e a p p ro p ria te lim it.

sio n :

in te rfa c e .

c o n d itio n s

d ia g o n a liz a tio n o f th e energy d e n s ity

e q u iv a le n t to th e s o lu tio n of

e x p e c te d

m a te ria l con­

in c l a s s i c a l o p tic s .

i n d u c e d p o l a r i z a t i o n o f t h e m edium d o e s n o t

us to

is

i n c o r p o r a t e s a q u an tu m m e c h a n i c a l v e r s i o n o f

p o ra l e v o lu tio n of th e

th is

A key

fie ld

th ro u g h

When a p p l i e d t o a l i n e a r m e d iu m , t h i s

ilia r

q u a si-

th e e le c tro m a g n e tic energy i s

T h u s , p r o p a g a t i v e q u an tu m o p t i c s

re su lts

th e

of its

m e c h a n ic al e q u iv a le n t o f th e c o n v e n tio n a l tiv e

in tro d u c e d th a t

irre sp e c tiv e

is th e case a lso

th is

tio n s.

b o u n d a rie s

in s u re s

serv ed in th e

th e

is

i n s u c h a w ay t h a t p e r i o d i c c o n d i t i o n s a r e

obeyed

tim e ,

way,

i n a c a v i t y w hose p h y s i c a l d i m e n s i o n s d e p e n d on t h e

m e d iu m i t

m e d ia ,

th is

in d u ced p o l a r i z a t i o n .

f e a t u r e o f p r o p a g a t i v e q u an tu m o p t i c s

a ll

In

th e s p a t i a l and te m p o ra l c o o r d in a te s o f th e

o n t h e sa m e f o o t i n g .

q u a n tiz e d

of

of th is

approach,

of

(lin e a r)

and p e r m its

fo rm a lism t o th e d e s c r i p t i o n o f t h e e v o lu tio n

189

I ABRAM

o f t h e q u an tu m s t a t i s t i c s

of n o n c la ssic a l s ta te s

o f l i g h t upon

t h e i r p r o p a g a tio n i n l i n e a r o r n o n l i n e a r m ed ia.

REFERENCES A bram ,

I.

1987,

P h y s. Rev. A 35,

A b ra m ,

I.

1988,

P h y s . R e v . A,

G lau b er,

R .J.

P e ie rls,

R.

Ed. F.

1963,

4661

in p re s s

P h y s. Rev.

131,

2766

1985, H ig h lig h ts o f C o n d e n se d -M a tte r P h y s ic s , B a ssa n i,

A m s te r d a m , p p .

F . Fumi a n d M .P . T o s i ,

N o rth -H o lla n d ,

237-255

S to le r,

D.

1970, Phys.

Rev. D 1,

3217

S to le r,

D.

1971, Phys.

Rev. D 4,

1 925

Y uen, H .P .

1976, P hys.

Rev. A 13,

Y e e , T .K a n d G u s t a f s o n ,

T .K .

1978,

2 22 6 Phys.

Rev. A 18,

1597

MODELS FOR PHASE-INSENSITIVE QUANTUM AMPLIFIERS

G L M A N D ER, R LOUDON AND T J SH EPH ERD

1.

INTRODUCTION

S tu d y o f t h e p r o p e r t i e s

of lin e a r o p tic a l a m p lifie rs

i s m o ti­

v a t e d by b o th ac a d e m ic and t e c h n o l o g i c a l i n t e r e s t i n t h e f i e l d . In pure s c ie n c e ,

e x p e rim e n ta l t e s t s

a t e m easu rin g a p p a r a tu s , s t a g e o f w hich i s in to

th e

o f t h e o r i e s m ust in c o r p o r ­

l a s t e s s e n t i a l q u a n tu m -m e c h a n ica l

a h ig h -g a in a m p lifie r to b o o st th e

o n e w h i c h may b e r e g a r d e d a s c l a s s i c a l f o r m e a s u r e m e n t

p u rp o ses.

In a p p lie d s c ie n c e ,

a m p lif ie r th e o ry

t o t h e co m m u n icatio n s i n d u s t r y w h e re , to r

sig n a l

is

f o r e x am p le,

of re le v a n c e se m ico n d u c­

l a s e r a m p l i f i e r s may h a v e u s e a s r e c e i v e r p r e - a m p l i f i e r s

and n o n - r e g e n e r a t i v e r e p e a t e r s

in o p tic a l

The e x i s t e n c e o f s q u e e z e d l i g h t ,

and i t s

fib re

lin k s.

p o te n tia l use to c a rry

o p t i c a l i n f o r m a t i o n w ith an im proved s i g n a l - t o - n o i s e r a t i o re la tiv e

to

co h eren t l i g h t,

degree of g a in t h a t i s sq u e e z in g , is

p o ssib le

a n e e d t o know t h e

in t h e sy stem b e f o r e t h e

and a l s o o t h e r quan tu m p r o p e r t i e s o f l i g h t s u c h a s

a n tib u n c h in g , la tio n s

h as le d to

are

lo st.

A d ete rm in in g f a c to r in th e s e c a lc u ­

t h e am ount o f n e c e s s a r i l y - a d d e d q u an tu m n o i s e i n h e r ­

e n t to t h e sy ste m .

A lth o u g h t h e r e

e x is t p h a se -se n sitiv e

lin e a r

a m p l i f i e r s t h a t add no n o i s e t o one q u a d r a t u r e p h a s e o f t h e fie ld ,

f o r e x am p le, d e g e n e r a te p a r a m e tr ic a m p l i f i e r s ,

in

g e n e r a l t h e s i g n a l - t o - n o i s e r a t i o w i l l be d e g r a d e d by t h e a m p lific a tio n p ro cess,

a n d q u a n t u m s t a t i s t i c a l p r o p e r t i e s may

b e w ashed o u t by t h e ra n d o m iz in g e f f e c t o f t h e n o i s e f i e l d . We c o n c e n t r a t e o u r a t t e n t i o n h e r e on r e v i e w i n g a n d e x t e n d i n g t h e w o r k on t h e p a r t i c u l a r c l a s s o f l i n e a r a m p l i f i e r s know n a s in v e rte d p o p u la tio n a m p lifie rs .

We d i s c u s s f i r s t

th e

191

G L M A N D E R , R L O U D O N A N D T J SH E P H E R D

f u n d a m e n ta l r e q u i r e m e n t s dem anded by q u an tu m m e c h a n ic s from any t h e o r y o f l i n e a r a m p l i f i e r s ,

and th e n d e s c r i b e t h e b a s i c

model f o r t h e a to m ic a m p l i f i e r i n a c lo s e d c a v i t y . t h e e x te n s io n o f t h i s m odel t o a llo w e x p l i c i t lin g of th e in te r n a l c a v ity fin a lly tig a te fie ld 2.

fie ld

We c o n s i d e r

in c o h e re n t coup­

to e x te rn a l f i e l d s ,

an open sy ste m w ith c o h e r e n t f i e l d

c o u p lin g .

and t r e a t We i n v e s ­

i n p a r t i c u l a r t h e c o n d i t i o n s u n d e r w h ich t h e a m p l i f i e d r e t a i n s q u an tu m p r o p e r t i e s

such as sq u e e z in g .

FUNDAMENTAL REQUIREMENTS OF LINEAR AMPLIFIER THEORY

S i n c e no s y s t e m c a n b e t r u l y

iso la te d

from t h e r e s t o f t h e

u n i v e r s e , b e i n g s u r r o u n d e d a t t h e v e r y l e a s t by a vacuum f i e l d , t h e r e w i l l alw ay s be f l u c t u a t i o n s due t o

its

and d i s s i p a t i o n

in te r a c ti o n w ith th e c o n ta in in g

en v iro n m en t.

L ax

(1 9 6 6 )

su g g e ste d t h a t th e r e s e r v o i r c o u ld be

e l i m i n a t e d from t h e c a l c u l a t i o n s , tio n s

p r o v i d e d t h a t t h e m ean e q u a ­

o f m o tio n i n c o r p o r a t e t h e r e s e r v o i r - i n d u c e d

sh ifts

and d i s s i p a t i o n , source is

t o add

c le a rly

N a iv e ly , andi d e a l l y ,

o u tp u t f i e l d

out

one

T he e f f e c t o f t h i s

and f o r c e s an a m p l i f i e r

o f n o i s e t o anys i g n a l i t m ig h t w ish t o w r i t e ,

d e s c r i b e d by t h e a n n i h i l a t i o n

d e r i v e d from an i n p u t f i e l d a

added.

irre m o v a b le ,

a c e r t a i n minim um a m o u n t

cesses.

pro­ f o r an

o p e r a t o r a Qut and

d e s c r i b e d by a i n ;

.= u a . m

(2 .1 )

s o t h a t t h e mean p h o t o n n u m b e r a m p l i f i c a t i o n = IuI 2 < n . > out 1 1 m

and t h e power g a in

(2 .2 )

(lo ss)

c u s s io n ab o v e, and a l s o

is

G = |u |2.

H ow ever, fro m t h e d i s ­

from t h e u n i t a r i t y

re q u ire m e n ts of

q u a n t u m m e c h a n i c s , we m u s t w r i t e , a

out

,= ua. m

w here F i s of th e

+F

(2 .3 )

t h e n o i s e o p e r a t o r , w hich c an be e x p r e s s e d

i n t e r n a l a m p l i f i e r m odes.

in term s

The i n p u t and o u t p u t f i e l d s

th e n obey t h e n e c e s s a r y co m m u tatio n r e l a t i o n s ,

v iz

192

M O D ELS FO R P H A SE -IN SE N SIT IV E Q U A N T U M A M PLIFIERS

[a . , a + .] L o u t' out if

= [a. ,a + ] = 1 L m ' m J

(2 .4 )

th e n o is e o p e ra to r s a t i s f i e s [ F , F +] = 1 - G

(2 .5 )

Now e x p e c t a t i o n v a l u e s o f h e r m i t e a n s q u a r e s o f o p e r a t o r s m u s t be r e a l and n o n - n e g a t i v e ,

so t h a t

_> 0 >_ 0 and

(5)

and

(6)

(2 .6 ) t o g e t h e r g iv e t h e fu n d a m e n ta l th e o re m f o r

p h a se -in se n sitiv e

lin e a r am p lifie rs

i n C a v e s 1 (1 9 8 2 )

form ,

>_ G - 1 T h is i s

(2 .7 )

t h e m a th e m a tic a l e x p r e s s i o n o f t h e f a c t t h a t su c h an

a m p lify in g d e v ic e m ust n e c e s s a r i l y d eg rad e th e s i g n a l : added n o is e o n ly v a n is h e s f o r G = 1, I f th e boson f i e l d s

carry

i.e .

th e

a p a s s i v e com p o n en t.

co h eren t in fo rm a tio n th en th e expec­

t a t i o n v a l u e s < a j_n > ^ < a o u t > a r e n o n “ v an :*-s h ^ n 9 • D e f i n i n g t h e i r f l u c t u a t i o n s as t h e s y m m e tric a l c o r r e l a t i o n f u n c t i o n s , f o r any boson o p e r a to r b < | Ab | 2 > = ^ < b ^ b + bb^> - w hich g i v e s (3)

< | Ab | 2 > >_ %.

(2 .8 )

From t h e b a s i c a m p l i f i e r e q u a t i o n

th e n < IAa | 2 > = G < |A a . I 2 > + < I A F l2 > 1 o u t1 1 m 1 1 1

U sin g t h e e q u i v a l e n t n o i s e f a c t o r o f C aves A

= > %(i - h G

-

“'

(2 .9 ) (1982),

(2 .io)

G

we h a v e , < | Aa J 2 > > G(h + h) 1 o u t1 — T h i s show s e x p l i c i t l y tio n

o p e ra to rs,

fo r sy m m e tric a lly o rd e re d f l u c tu a ­

th e a m p lific a tio n

t o one h a l f p h o to n a t in p u t flu c tu a tio n o p e ra to rs

th a t,

(2 .1 1 )

is

th e

p ro c e ss adds n o is e e q u iv a le n t

a m p l i f i e r i n p u t , w h e r e t h e m inim um

a ls o one h a l f p h o to n .

If th e flu c tu a tio n

a re n o rm ally o rd e re d , th e n th e e q u iv a le n t n o is e a t

193

G L M A N D E R , R L O U D O N A N D T J SH E P H E R D

th e a m p lif ie r in p u t i s

one p h o to n .

Any v i a b l e t h e o r y o f l i n e a r a m p l i f i e r s m u s t s a t i s f y m e n ta l th eo rem .

th e funda­

U s u a l l y one w orks w i t h t h e q u an tu m d y n a m ic s

of th e a m p lify in g d e g re e s o f freedom ,

so t h a t u se o f t h e e f f e c ­

t i v e q uantum n o i s e s o u r c e s can be a v o i d e d .

T h is a ls o has th e

a d v a n t a g e t h a t t h e a m p l i f i e r p a r a m e t e r s may b e r e l a t e d d i r e c t l y t o t h e p h y s i c a l m e c h a n is m o f t h e p r o c e s s . in v e rte d p o p u la tio n a m p lif ie r s ,

F or e x am p le,

th e g a in i s

in term s o f th e a to m ic i n v e r s i o n .

d ire c tly

in

e x p re ssib le

The a m p l i f i e r a d d e d n o i s e

may a l s o b e e x p r e s s e d i n t e r m s o f t h e a t o m i c p r o p e r t i e s . p re c ise ly th is

sh o u ld be i n t r i n s i c a l l y

irre v e rsib le

When t h e a to m d e c a y s ,

subsequent lo ss

co m p risin g th e a m p lif ie r .

t h e r e m a i n i n g a to m s f o r m a p h o t o n s i n k .

I f a p h o to n c o h e r e n t w ith t h e f i e l d b ed by a n o t h e r a to m ,

That a m p lific a tio n

can b e s e e n by c o n s i d e r i n g

a n y p a r t i c u l a r a to m i n t h e c o l l e c t i o n

its

is

n o i s e t h a t b r e a k s t h e t i m e s y m m e tr y i n h e r e n t i n

t h e b a s i c e q u a t i o n s o f q uantum m e c h a n ic s .

random , t h i s

It

is

e m itte d and th e n a b s o r ­

i t may b e r e r a d i a t e d from t h e beam .

process is

c le a rly

sp o n ta n e o u s ly , w ith

S p o n tan eo u s e m is s io n b e in g

not re v e rsib le :

a tte m p t t o ru n t h e a m p l i f i e r backw ards

in d e e d any

(G lau b er,

1986), t h a t i s ,

t o a t t e n u a t e t h e beam , w o u ld o n l y r e s u l t i n t h e f u r t h e r d e g ra d a tio n of th e s ig n a l. 3.

THE CLOSED SYSTEM

T h i s m o d e l w as p r o p o s e d i n

1963 b y G o r d o n , W a l k e r a n d L o u i s e l l

(1 9 6 3 ) , a n d h a s s i n c e b e e n g r e a t l y

used in in v e s tig a tio n of

t h e e f f e c t o f o p t i c a l a m p l i f i c a t i o n on t h e p r o p e r t i e s o f q u a n ­ tum l i g h t f i e l d s . its

We w i l l t h e r e f o r e

l o o k i n some d e t a i l

at

m ain f e a t u r e s .

3 .1

M odel

The s y s t e m i s

com posed o f a s i n g l e

r a d i a t i o n mode i n a l o s s l e s s

c a v i t y c o u p le d t o a l a r g e num ber o f n e a r l y (F ig .

1).

E a c h a to m i s

in d e p e n d e n t atom s

assum ed t o have a f i n i t e

s e t o f e q u a lly spaced energy l e v e l s ,

and i t

is

supposed t h a t

t h e r e a r e s u f f i c i e n t l y many a to m s w i t h t r a n s i t i o n c lo se to th e f i e l d

or in fin ite fre q u e n c ies

f r e q u e n c y t h a t t h e a to m d e n s i t y o f s t a t e s

may b e t a k e n a s c o n t i n u o u s .

O n ly a t o m s w i t h t r a n s i t i o n

194

M O D ELS FO R P H A SE -IN SE N SIT IV E Q U A N T U M AM PLIFIERS

fre q u e n c ies n ear th e f ie ld to

th e f ie ld .

F u rth e r,

co u p led to a h e a t b a th , b u tio n .

fre q u e n c y c o u p le w ith any s t r e n g t h

t h e ato m s a r e t a k e n t o b e w e a k ly so t h a t t h e y h a v e a B o ltzm an n d i s t r i ­

The s t a t i s t i c a l d i s t r i b u t i o n

by v i r t u e o f t h e i r

o f t h e ato m s i s

l a r g e n u m b e r a n d w e ak c o u p l i n g t o

to be c o n s ta n t o v e r th e i n t e r a c t i o n serv e to ensure l i n e a r i t y ,

i.e .

tim e .

ta k e n , th e f ie ld ,

T hese a ssu m p tio n s

n o n - s a tu r a tio n o f th e a m p lif ie r .

Closed Cavity N 2 - level atoms 3 .2 If

N, + N2 = N

A n a ly sis a (t)

is

and a j (t)

th e a n n ih ila tio n o p e ra to r o f th e is

th e lo w e rin g o p e r a to r f o r th e

H a m ilto n ia n o f th e

in te rn a l jth

fie ld ,

a to m t h e n t h e

s y s te m h a s t h e u s u a l form i n t h e r o t a t i n g -

w ave a p p r o x i m a t i o n ,

- +*

N

w here

is

z

N

+ h u I ^oo”i . o"i . + u7 3 3 j= i j =l

H = ftoo o a a

*+*

th e ato m ic i n v e r s i o n o p e r a t o r ,

a to m - f ie ld co u p lin g c o n s ta n t.

- -* +

fiKH. (ai . a + a “i. a ) and

is

(3 .1 ) th e

The r e l e v a n t c o m m u ta tio n r e l a ­

t i o n s g iv e th e H e ise n b e rg e q u a tio n s o f m o tio n f o r th e

fie ld

and a to m ic o p e r a t o r s as da d T" t = -loo o a

i

3

da . 1

-■-- v *

dt

w here i t

=

is

.

*3 :

- l o o . a .

7

k

• 3

.a . 3 3

, .

+

-z j

j

assum ed t h a t th e p o p u l a t i o n i n v e r s i o n

m a in ta in e d in o rd e r to o b ta in in te re ste d

( 3.2)

i K . < a . > a

in

is

lin e a riz e d e q u a tio n s.

th e e v o lu tio n o f th e

cavity

e x te rn a lly We a r e

mode, and t h e

195

G L M A N D E R , R L O U D O N A N D T J SH E P H E R D

H e i s e n b e r g e q u a t i o n may b e s o l v e d i n t h e W i g n e r - W e i s s t t o p f a p p ro x im atio n to g iv e th e r e s u l t a (t)

= u (t)a (0 )

Il v^. V . ( t )) S a. j= l 1 J

(3 .3 )

+ (N2- N 1 ) y Lt ]

(3 .4 )

+ +

w ith u (t ) = e x p [ - i a > Qt

YL = 5 = P ( G - l )

j

3

3

(3 .5 )

3

w h e r e now t h e g a i n G i s d e f i n e d by G = | u (t ) | 2 = e x p [ 2 (N2 ~N1 ) y Lt ] and P i s

th e p o p u la tio n

(3 .6 )

fa c to r

N2

p =T v v

(3-7)

The r e s u l t o b t a i n e d i s a ( t = 0)

tim e -v a ry in g ,

a s t h e i n p u t mode a n d a ( t )

so one c h o o s e s

to id e n tify

a s t h e o u t p u t mode

o f th e

sy ste m , a in = a(0) ^out = a (t)

(3 -8>

The i n p u t - o u t p u t r e l a t i o n sta n d a rd r e s u l t

(2 .3 )

t h a t th e sy stem

w i l l o n ly

isfa c to ry

(3 .3 )

th e n ta k e s th e

form o f t h e

fo r p h a se -in se n sitiv e a m p lifie rs.

c o n d itio n th a t

N o tic e

show g a i n u n d e r t h e p h y s i c a l l y

sa t­

> N ^.

U n i t a r i t y i s o b e y e d o n l y on t h e a v e r a g e ,

J a tx.o m s = 1 c o n s is te n t w ith th e

lin e a riz in g

(3 .9 ) a p p ro x im a tio n .

196

M O D ELS FO R P H A SE -IN SE N SIT IV E Q U A N T U M A M PLIFIERS

T h e p r o p e r t i e s o f t h e o u t p u t f i e l d may b e e v a l u a t e d i n t e r m s of th o se o f th e

in p u t u sin g th e fo llo w in g r e l a t i o n ,

:r -

, r+ s

< [aof u t J] [La o u .t]J =

r F q=0

(J') O. t k .a 7 e ^ l - J - 3 -------------j oo . — co + i r 3 0

Wi th r = %( y j + y 2 ) -

(5.5)

(n2 - n 1 ) y l

as th e in v erse c a v ity

life tim e ,

a n d yt g i v e n b y L

(14).

R ead in g

f r o m l e f t t o r i g h t , we h a v e t h e o u t p u t f i e l d e x p r e s s e d i n t e r m s of re fle c te d

in p u t f i e l d ,

(sp o n ta n e o u s e m issio n )

a m p lifie d in p u t f ie ld s

and th e n o i s e

field.

As w i t h t h e f i r s t m o d e l , u n i t a r i t y

is

o b e y e d on t h e a v e r a g e ,