Many-body phenomena in condensed matter and atomic physics 9780738201016, 9780486435039, 9780486819013

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Leonid Levitov

Many-body phenomena in condensed matter and atomic physics

MIT 2003

Physics 8.514

Massachusetts Institute of Technology Physics Department

Fall 03

Many-body phenomena in condensed matter and atomic physics Lectures: Tu, Thr, 9:30-11, in Rm. 26-168, by Prof Leonid Levitov http : //www.mit.edu/people/levitov/8514 Prerequisite: Statistical Mechanics and Quantum Mechanics, introductory level courses The aim of the course is two-fold. First, we shall discuss topics of interest for both condensed matter and atomic physics, focussing on the effects of quantum statistics, interactions, and correlations in many-particle systems. Our second goal will be to provide a gentle introduction to the methods of quantized fields and their applications in many-body physics. We shall try to emphasize the physical and visualizable aspects of the subject. While the course is intended for students with a wide range of interests, many examples will be drawn from condensed matter physics. Course topics (tentative): • Bose condensates (quasiparticles, collective modes, superfluidity, vortices); • Fermi gases and liquids, collective excitations; • Cooper pairing (BCS theory, off-diagonal long-range order, superconductivity); • Atom interacting with an optical field; • Lamb shift, Casimir effect; • Dicke superradiance; • Quantum transport and wave scattering in disordered media, localization; • Tunneling and instantons; • Macroscopic quantum systems, coupling to a thermal bath; • Spin-boson model, tunneling and localization; • Kondo effect; • Spin dynamics and transport in gases and solids; • Cold atoms in optical lattices; • Quantum theory of photodetection and electric noise In this course we shall develop theoretical methods suitable for the description of the manybody phenomena, such as Hamiltonian second-quantized operator formalism, Greens functions, path integral, functional integral, and the quantum kinetic equation. The concepts to be introduced include, but are not limited to, the random phase approximation, the mean field theory (aka saddlepoint, or semiclassical approximation), the tunneling dynamics in imaginary time, instantons, Berry phase, coherent state path integral, renormalization group. Recommended texts: Michael Stone, The Physics of Quantum Fields, (2000, Springer)

References on bosonization (the oscillator picture of a Fermi liquid): K. Sawada, “Correlation Energy of an Electron Gas at High Density”, Phys. Rev. v.106, p. 372 (1957) https://doi.org/10.1103/PhysRev.106.372 Sawada et al., “Correlation Energy of an Electron Gas at High Density: Plasma Oscillations”, Phys. Rev. v.108, p.507 (1957) https://doi.org/10.1103/PhysRev.108.507 G. Wentzel, “Diamagnetism of a Dense Electron Gas”, Phys. Rev. v.108, p.1593 (1957) https://doi.org/10.1103/PhysRev.108.1593 References on Bogoliubov-deGennes equation and Andreev scattering: I read about it in the books by deGennes, Tinkham, and Abrikosov: de Gennes P. G. Superconductivity of Metals and Alloys. New York: W. A. Benjamin, 1966. ISBN 978-0-7382-0101-6. Tinkham M. Introduction to Superconductivity (Second ed.). New York: Dover, 2004. ISBN 978-0-486-43503-9. Abrikosov A.A. Fundamentals of the Theory of Metals. Dover, 1997. ISBN 978-0486819013. Also, the original paper by Andreev is reasonably well-written: Andreev A.F., “Thermal conductivity of the intermediate state of superconductors”, Sov. Phys. JETP. 19: 1228 (1964) Selected articles and lectures for 8.514 reading: R.J. Glauber, “The Quantum Theory of Optical Coherence”, Phys Rev, 130, 2529 (1963) https://doi.org/10.1103/PhysRev.130.2529 M.M. Nieto, “The Discovery of Squeezed States - In 1927”, quant-ph/9708012 https://arxiv.org/abs/quant-ph/9708012 M.M. Nieto, “Displaced and Squeezed Number States”, quant-ph/9612050 https://arxiv.org/abs/quant-ph/9612050 P. Coleman, “Many Body Physics: Unfinished Revolution”, cond-mat/0307004 https://arxiv.org/abs/cond-mat/0307004

8.514: Many-body phenomena in condensed matter and atomic physics

1

Last modified: September 24, 2003

Lecture 1. Coherent States.

We start the course with the discussion of coherent states. These states are of interest because they provide • a method to describe on equal terms both particles and photons; • a connection to classical physics (mechanics and electrodynamics); • tools for the construction of path integral, to be discussed later The coherent states also provide a natural entry point into the method of second quantization that will be introduced in the next lecture.

1.1

Harmonic oscillator; the creation and annihilation operators

Particle in a parabolic potential: mω 2 2 p2 + q , p = −i¯h∂q , [q, p] = i¯h 2m 2 The ground state width can be found by minimizing energy: H=

hHi =

h ¯2 mω 2 λ2 + → min 2mλ2 2

(1)

(2)

which gives λ = (¯h/mω)1/2 . It will be convenient to use nondimensionalized variables q = λ˜ q , p = (¯h/λ)˜ p, so that the classical phase volume is rescaled by h ¯ . Thus we obtain H=

 h ¯ω  2 p˜ + q˜2 , 2

p˜ = −i∂q˜ ,

[˜ q , p˜] = i

(3)

We shall study the Hamiltonian (3) below having in mind the quantum-mechanical particle problem. However, later we shall find that the quantized electromagnetic field is also described by a set of harmonic oscillators of the form (3). The canonical creation and annihilation operators are defined as 1 a = √ (˜ q + i˜ p) , 2

1 a+ = √ (˜ q − i˜ p) 2

(4)

They can be used to express q, p and H as follows:

  h ¯  + λ  q = √ a + a+ , p = i √ a −a 2 2λ    h ¯ω  + 1 H= ¯ ω a+ a + a a + aa+ = h 2 2

1

(5) (6)

The operators a and a+ obey the commutation relation [a, a+ ] = 1

(7)

Proof: aa+ − a+ a =

1 ((˜ q + i˜ p)(˜ q − i˜ p) − (˜ q − i˜ p)(˜ q + i˜ p)) = i (˜ pq˜ − q˜p˜) = 1 2

(8)

As a simple application of the operators a and a + , let us reconstruct the main facts of the harmonic oscillator quantum mechanics. 1. The ground state |ψ0 i, also called vacuum state, provides the lowest possible energy expectation value 1 1 hψ0 |H|ψ0 i = h ¯ ω hψ0 |a a|ψ0 i + =h ¯ ω haψ0 |aψ0 i + 2 2 



+





(9)

which gives the condition aψ 0 = 0, i.e., (q + ip)ψ0 = 0. Let us find the ground (vacuum) state in the q-representation. Using the units with the length λ = 1, i.e., q˜, p˜ instead of q, p, we write qψ0 (q) + ψ0′ (q) = 0 ,

ψ0′ /ψ0 = −q ,

ln ψ0 = −q 2 /2

(10)

This leads to a Gaussian wavefunction 



ψ0 (q) = π −1/4 exp −q 2 /2 ,

E0 = h ¯ ω/2

(11)

2. The higher energy states can be obtained from the ground state. Starting with the commutation relations, a+ H = (H − h ¯ ω) a+ ,

aH = (H + h ¯ ω) a

(12)

one can show that the states ψn (q) = (a+ )n ψ0 (q) are the eigenstates. Indeed, consider ψ1 = a+ ψ0 and apply the first relation (12): (H − h ¯ ω) ψ1 = a+ Hψ0 = E0 a+ ψ0 = E0 ψ1

(13)

which gives E1 = E0 + h ¯ ω = 3¯hω/2 and   2 1 ψ1 (q) = √ (q − ∂q ) ψ0 = √ q exp −q 2 /2 2¯h 2¯h

(14)

Subsequently, from ψ1 one obtains the eigenstate ψ2 (q) ∝ (2q 2 − 1) exp (−q 2 /2) with the energy E2 = E1 + h ¯ ω = 5¯hω/2, and so on. The recursion relation ψn = a+ ψn−1 , En = En−1 + h ¯ ω, gives 

ψn = An (a+ )n ψ0 ,

En = h ¯ω n + 2

1 2



(15)

where we inserted the normalization factors A n . The factors An can be determined from 1 = A2n h(a+ )n ψ0 |(a+ )n ψ0 i = A2n hψ0 |a...aa+ ...a+ |ψ0 i   H 1 = A2n hψ0 |an−1 + (a+ )n−1 |ψ0 i h ¯ω 2 = A2n nhψ0 |an−1 (a+ )n−1 |ψ0 i = ... = An n!hψ0 |ψ0 i = An n!

(16) (17) (18)

which gives An = (n!)−1/2 . The normalized oscillator eigenstates 1 |ni = √ (a+ )n |0i , n!

|0i = ψ0

(19)

form an orthonormal complete set of functions, providing a basis in the oscillator Hilbert space. The ground state |0i is also known as the vacuum state. 3. The operators a and a+ written as matrices in the basis of states (19) have nonzero matrix elements only between the states |ni and |n ± 1i: √ √ (20) hn|a+ |mi = nδn,m+1 , hm|a|ni = nδn,m+1 while all other matrix elements are zero. 4. It is convenient to define the so-called number operator n ˆ = a + a which counts the number of energy quanta in the QM particle problem, or the number of photons for quantized E&M field. In the energy basis |ni, the number operator is diagonal: +

n ˆ |ni = a a|ni = n|ni ,

1.2

1 H=h ¯ω n ˆ+ 2 



(21)

Definition of coherent states.

The coherent states are defined as eigenstates of the operator a: a|vi = v|vi

(22)

where v is a complex parameter. Expanded in the energy basis (19), |vi = the coherent state can be reconstructed from a|vi =

∞ X

n=0

P∞

n=0 cn |ni,

∞ X √ vcn |ni cn n|n − 1i =

(23)

n=0

√ Comparing the coefficients, obtain a recursion relation c n = (v/ n)cn−1 , leading to vn √ c0 cn = n!

(24)

The coefficient c0 is determined from normalization 1=

∞ X

n=0

|cn |2 =

∞ X |v|2n

n!

n=0

3

2

|c0 |2 = e|v| |c0 |2

(25)

Finally, |vi = e−|v|

2 /2

∞ X vn

√ |ni = e−|v| n! n=0

2 /2

+

eva |0i

(26)

As an example, consider the distribution of the number of quanta n ˆ = a + a in a coherent state. Since n ˆ |ni = n|ni, the distribution is given by pn = |cn |2 = e−|v|

2

|v|2n n!

(27)

This is a Poisson distribution with the mean n ¯ = |v| 2.

1.3

The quasiclassical interpretation of coherent states

As we shall see below, the coherent states represent the points of the classical phase space (q, p). This can be conjectured most easily from their time dependence. Applying the Schr¨odinger equation i∂t ψ = Hψ to the number states, we have 1

|ni(t) = e−i(n+ 2 )ωt |ni

(28)

for the number states. Combined with (26), this gives −|v|2 /2

|vi(t) = e

∞ X vn

1

√ e−i(n+ 2 )ωt |ni = e−iωt/2 |v(t)i n! n=0

(29)

with v(t) = e−iωt v

(30)

This defines a circular trajectory in the complex v plane, suggesting the correspondence with classical coordinate and momentum, q = c v′ ,

p = c v ′′ ,

v = v ′ + iv ′′

(31)

where c is a scaling factor. The relation of coherent states with the points in a classical phase space will be clarified below. Let us find the form of a coherent state in the q-representation, ψv (q) = hq|vi. As before, we use the units in which the length λ = 1, and write 1 1 vψv (q) = hq|a|vi = √ hq| (q + ∂q ) |vi = √ (q + ∂q ) ψv (q) 2 2 √ Solving the equation qψ + ψ ′ = 2vψ, obtain √ ln ψ = −q 2 /2 + v˜q + const. , v˜ = 2v

(32)

(33)

and, finally, 1 ψv (q) = A exp − (q − v˜)2 , 2 



′′ )2 /2

|A| = π −1/4 e−(˜v 4

= π −1/4 e−(v

′′ )2

(34)

√ with v˜′′ = 2v ′′ . The probability |ψv (q)|2 has a form of a gaussian centered at q = Re(˜ v), which agrees with the above interpretation of v as a point in the phase space (with the √ scaling factor taking value c = 2). A more detailed picture of the phase-space density is provided by the Wigner distribution function Z ∞ 1 1 1 W (q, p) = hq + x|ˆ ρ|q − xieixp/¯h dx (35) 2π¯h −∞ 2 2 where ρˆ is the density matrix. For a pure state ψ(q), the density matrix in position space ′ ¯ is just ρˆq,q′ = ψ(q)ψ(q ), and the matrix element in (35) is ¯ − 1 x)ψ(q + 1 x) h...i = ψ(q 2 2

(36)

The interpretation of the Wigner function as a phase-space density is supported by the following observations. One can check that the function (35) is real and normalized to unity. Also, the coordinate and momentum distributions, obtained by integrating over the conjugate variable, are reproduced correctly. The distribution in q is Z

W (q, p)dp = hq|ˆ ρ|qi

(37)

which is equal to |ψ(q)| 2 for a pure state, while the distribution in p is Z

W (q, p)dq = ... =

1 |ψ(p)|2 2π¯h

(38)

where ψ(p) = ψ(q)e−iqp dq. For a coherent state |vi, the Wigner function is given by R

W (q, p) =

|A|2 2π¯h

Z

∞

−∞

1

1

¯

2

1

1

2

e− 2 (q+ 2 x−v˜) e− 2 (q− 2 x−˜v) eixp dx

(39)

|A|2 ∞ −(q−˜v′ )2 ixp− 1 ( 1 x+i˜v′′ )2 1 −(q−v′ )2 −(p−˜v′′ )2 2 2 = e e dx = e (40) 2π¯h −∞ 2π¯h √ √ with v˜′ = 2v, v˜′′ = 2v. The gaussian distribution, centered at q = v˜′ , p = v˜′′ , evolves in time as if carried by the classical harmonic oscillator phase flow. Since v(t) = e −iωt v, the center of the gaussian packet is circling around the phase space origin: Z

W (q, p, t) =

1 −(q−|˜v| cos ωt)2 −(p−|˜v| sin ωt)2 e 2π¯h

(41)

For any |vi, the width of the Wigner distribution is the same as for the vacuum state |0i. Thus one can conclude that a coherent state can be thought of as a displaced vacuum state. This interpretation will be substantiated in Problem 2, PS#1.

5

1.4

Coherent states vector algebra

Here we discuss the the vector space propeties of coeherent states. Normally, the states appearing in quantum mechanics are orthogonal, or can be made orthogonal in some natural way, which provides an orthonormal basis in Hilbert space. The situation with coherent states is quite different. Let us start with evaluating the overlap: 1

2

1

hu|vi = e− 2 |u| e− 2 |v|

2

∞ X (¯ uv)n

n=0

n!

1

= e− 2 |u|

2 − 1 |v|2 +¯ uv 2

(42)

which shows that the coherent states are not orthogonal. On the other hand, Eq.(42) gives overlap decreasing exponentially as a function of the distance between u and v in the complex plane: 2 |hu|vi|2 = e−|u−v| (43) For generic classical states, |u|, |v| ≫ 1, the overlap is very small, which is consistent with the intuition that different classical states are orthogonal in the quantum mechanical sense. Recalling the interpretation of the complex v plane as a phase space, q˜ = v ′ , p˜ = v ′′ , we see that the overlap falls to zero at the length scale of the order of the √ wavepacket width √ ¯ , δp ∼ h ¯ /λ ∝ h ¯. set by Planck’s constant, i.e. by the uncertainty relation, δq ∼ λ ∝ h Another property of coherent states is completeness in the vector algebra sense. (A set of vectors is called complete if linear combinations of these vectors span the entire vector space.) The property is seen most readily from the formula know as unity decomposition:

d2 v ˆ =1 (44) π Proof can be obtained by evaluating the matrix elements of the operator on the left hand side of Eq. (44) between the number states Z

|vihv|

d2v E ¯m v n d2 v 2 v n = e−|v| √ = π m!n! π Z ∞ 2n −r 2 r = δm,n e dr 2 = δm,n n! 0 D

Z

m

Z

|vihv|

Z

∞ 0

Z

π

−π

e−r

2

r m+n ei(n−m)θ rdrdθ √ (45) π m!n! (46)

(we used polar coordinates v = reiθ ). Using the formula (44), one can express any operator in terms of coherent states: ZZ

ˆ = ˆ1M ˆ ˆ1 = M

|uihv| M(u, v)

d2vd2u π2

(47)

ˆ with the matrix elements M(u, v) = hu| M|vi. This formula can be useful in calculations, as well as in formal manipulations (we shall use it later to derive Feynman path integral). As another application of Eq. (44), let show that the coherent states form an overcomplete set, i.e. they are not linearly independent. Indeed, by writing |vi = ˆ1|vi =

Z

d2 u |uihu|vi = π

Z

− 21 |u|2 − 12 |v|2 +¯ uv d

2

u = π

|ui e

6

Z

v u − 12 |u−v|2 d u ¯v−¯

|ui e

e

2

u (48) π

we express the state |vi as a superposition of the states |ui with |u − v| ≤ 1. The overcompleteness (48) should not come as a surprise. The coherent states, parameterized by complex numbers, form a continuum, and thus there are way too many of them to form an a set of independent vectors. In contrast, the number states, which provide a basis of the oscillator Hilbert space, are a countable set. To summarize, the coherent states are non-orthogonal and form an over-complete set. There have been many attempts to reduce the number of these states to a ‘neccessary minimum,’ by identifying a good subset that could serve as a basis. Even though some of the proposals are very interesting (e.g. Perelomov lattices 1 ) it is probably more natural to use the entire space of coherent states, coping with the overcompleteness and not favoring some of the states to the others.

1.5

Coordinate and momentum uncertainty

We already mentioned, while discussing the Wigner function, that the coherent states form wavepackets in the phase space of width corresponding to the absolute minimum required by the uncertainty relation. Let us estimate coordinate uncertainty of a state |ui:  λ2 λ2 λ2  2 hu|(a + a+ )2 |ui = hu|a2 + a+ + 2a+ a + 1|ui = (u + u¯)2 + 1 2 2 2 2 2  2 λ λ hu|a + a+ |ui = (u + u¯)2 (hu|ˆ q|ui)2 = 2 2 λ2 h ¯ hu|δ qˆ2 |ui = hu|ˆ q 2|ui − (hu|ˆ q|ui)2 = = (49) 2 2mω

hu|ˆ q 2 |ui =

The uncertainty does not depend on u, which is consistent with the observations made using Wigner function. Similarly, for momentum uncertainty,  (i¯h)2 (i¯h)2 h ¯2  + 2 +2 2 + 2 hu|(a −a) |ui = hu|a + a − 2a a − 1|ui = 1− (u− u ¯ ) 2λ2 2λ2 2λ2 2  2 2 (i¯h) (i¯h) + (hu|ˆ p|ui)2 = hu|a − a|ui = (u − u¯)2 2 2 2λ 2λ 2 h ¯ h ¯ mω hu|δ pˆ2|ui = hu|ˆ p2 |ui − (hu|ˆ p|ui)2 = 2 = (50) 2λ 2

hu|ˆ p2|ui =

which is also independent of u. The uncertainty product hδ pˆ2 i1/2 hδ qˆ2 i1/2 equals 12 h ¯ , which is the lower bound required by the uncertainty relation. Below we shall see that coherent states can be naturally generalized to a broader class of states that minimize uncertainty product. 1

One can consider lattices in the complex plane, vm,n = mu1 + nu2 , m, n ∈ Z. Perelomov shown that the lattice {vm,n } generates an undercomplete set of coherent states {|vm,n i} if the area of the lattice unit cell is greater than 2π¯ h, and an overcomplete set if the area is less than 2π¯h. The borderline lattices, having the unit cell area equal to 2π¯ h, are overcomplete just by one vector. After any single vector is removed from such a lattice, it becomes a complete set.

7

8.514: Many-body phenomena in condensed matter and atomic physics

Problem Set # 1 Due: 9/16/03

Coherent states 1. Operator identities. Here we prove two useful theorems from operator algebra that will be used in the problems of this homework and later in the course. a) Let Ab and Bb be two operators that do not necessarily commute. Prove the so-called operator expansion theorem: x2 b b b b b b b b b f (x) = exp(xA)B exp(−xA) = B + x[A, B] + [A, [A, B]] + ... 2!

(1)

with x a parameter. (Hint: consider the derivative f ′ (x) and compare its Taylor series in x with that for f (x).) b B] b = c is a c-number, the series terminates after the In the special case when the commutator [ A, second term, giving b B b exp(−xA) b =B b + cx exp(xA) (2) b h = −id/dq. Apply this result to the coordinate and momentum operators, Bb = qb, Ab = p/¯ b B] b commutes with both A and B (e.g., b) Let A and B be two operators whose commutator [A, b B] b = c a c-number). Prove the Campbell-Baker-Hausdorf theorem: [A, h

i











b = exp xA b exp xB b exp −x2 [A, b B]/2 b exp x(Ab + B) 









(3)

b For that, consider C(x) = exp xAb exp xBb , differentiate both sides with respect to x and, using 



b b + x[A, b B] b C(x). b the operator expansion theorem (1), show that dC(x)/dx = Ab + B Integrate with respect to x like an ordinary differential equation.

2. Displacement operators. a) Consider the displacement operators, defined as 

c D(v) = exp v ab+ − v¯ab



(4)

c+ (v)D(v) c c−1 (v) = D(−v). c with v a complex parameter. Prove unitarity: D = 1, D For a real-valued v show that in the q-representation the displacement operator (4) acts as an argument shift: √ c D(v)ψ(q) = exp (˜ v ∂q ) ψ(q) = ψ(q + v˜) , v˜ = 2λv (5) q

c with the length λ = h ¯ /mω. (Hint: relate D(v) to the Taylor series formula.) c b) Show that the coherent states can be obtained by “displacing” the vacuum state, |vi = D(v)|0i. (Use the operator expansion theorem (1)). c c) Show that the unitary transformation D(v) displaces ab by v, and ab+ by v¯, c+ (v)a c bD(v) D = ab + v ,

c+ (v)a c b+ D(v) D = ab+ + v¯

(6)

For any function of operators ab and ab+ with a power series expansion, show that c+ (v)f (a c b, a+ )D(v) D = f (ab + v, a+ + v¯)

(7)

d) Prove the product formula

′

′

c ′ )D(v) c c + v′) D(v = ev v¯−¯v v D(v

(8)

c c ′ ) commute only when arg(v) = arg(v ′ ). Note that the displacement operators D(v) and D(v

3. Harmonic oscillator excited by an external force. a) Consider a particle moving in a parabolic potential in the presence of a time-dependent force, H = 12 h ¯ ω (pb2 + qb2 ) − F (t)qb. Show that the evolution in time of an arbitrary coherent state can be obtained using the displacement operators (4) studied in Problem 2. Assume that the evolved coherent state remains a coherent state at all times, so that c |αi(t) = D(v(t))|αi = |α + v(t)i

(9)

Obtain a differential equation for the function v(t) and show that its real and imaginary parts correspond to the classical Hamilton equations dq/dt = p, dp/dt = F (t). c c−1 (v(t)) gives a free oscillator Hamiltonian Show that the unitary transformation H ′ = D(v(t))H D with F (t) = 0. It describes the transformation of the quantum problem to the classical co-moving reference frame. How does the function v(t) should evolve in time in order for is such that at all times it remains a coherent state b) The harmonic oscillator of part a), initially in the ground state, was subject to a constant force during the time interval 0 < t < τ . Find the state at t > τ . Determine the distribution of energies. c) For the state found in part b) at t > τ , find the phase-space density, i.e., the Wigner function W (q, p), as a function of time.

8.514: Many-body phenomena in condensed matter and atomic physics

1

Last modified: September 24, 2003

Lecture 2. Squeezed States

In this lecture we shall continue the discussion of coherent states, focusing on their properties as a basis in Hilbert space. Also, we introduce the so-called squeezed states that minimize the uncertainty product δqδp. These states can be viewed as a generalization of the coherent states and, like the latter, are closely related to the properties of the operators a and a+ . The coherent states discussed above are members of a wider class of states having the property that the product of the dispersions of qˆ and pˆ is a minimum. Such states are called ‘squeezed states’. The possibility of reducing the uncertainty in a physical variable, e.g., coordinate or momentum of a mechanical oscillator, or the duration of an optical pulse, provided by the squeezed states, is often useful in applications.

1.1

Uncertainty product

¯. Recall the proof of the uncertainty relation δqδp ≥ 21 h For any two hermitian operators A, B, and any state ψ, consider the quantity ˆ ˆ ˆ B]iα ˆ ˆ 2i F (α) = h(αAˆ + iB)ψ|(α Aˆ + iB)ψi = hAˆ2 iα2 + hi[A, + hB

(1)

where h...i stands for the expectation value hψ|...|ψi. Since F (α) is nothing but the norm 2 ˆ ||(αAˆ + iB)ψ|| , it is non-negative. The quadratic polynomial F (α) thus does not have real roots, which gives ˆ 2 i ≥ 1 h[A, ˆ B]i ˆ 2 hAˆ2 ihB (2) 4 ˆ = pˆ = −i¯h∂q , obtain Applying this to Aˆ = qˆ, B F (α) = h(αˆ q + iˆ p)ψ|(αˆ q + iˆ p)ψi = hˆ q 2 iα2 + hi[ˆ q , pˆ]iα + hˆ p2 i ≥ 0

(3)

¯ 2 . For a state with q¯ = hˆ qi = 0 and p¯ = hˆ pi = 0, the coordinate leading to hˆ p2 ihˆ q 2 i ≥ 14 h and momentum uncertainty is hδq 2 i = h(ˆ q − q¯)2 i = hˆ q2i ,

hδp2 i = h(ˆ p − p¯)2 i = hˆ p2 i

(4)

which gives

1 2 hδq 2 ihδp2 i ≥ h ¯ (5) 4 The uncertainty relation for a more general state ψ(q) with nonzero q¯, p¯ can be obtained from the above. Let us shift the coordinate and momentum as ˜ − q¯) ψ(q) = ei¯pq ψ(q 1

(6)

Since h(ˆ q − q¯)2 iψ = hq 2 iψ˜ and h(ˆ p − p¯)2 iψ = hp2 iψ˜, one can write 1 2 hδq 2 iψ hδp2 iψ = hq 2 iψ˜ hp2 iψ˜ ≥ h ¯ 4

(7)

the uncertainty relation for an arbitrary state. The states that minimize the uncertainty product and have q¯ = p¯ = 0 satisfy (αˆ q + iˆ p)ψ = αqψ + h ¯ ψ′ = 0

(8)

where α is a complex parameter. Integrating this equation, with h ¯ absorbed in α, obtain ψ(q) = π −1/4 (α′ )1/2 exp(−αq 2 /2) ,

α = α′ + iα′′

(9)

(normalizability requires α ′ > 0). More general states with minimal uncertainty can be obtained by a displacement of p and q, as in Eq.(6). The general states that minimize hδq 2 ihδp2i are called squeezed states. As we shall see, a squeezed state may be constructed to have and arbitrarily small width hδq 2 i1/2 .

1.2

Squeezed states and the operators a and a+ .

Suppose we prepared a squeezed state (9) of a harmonic oscillator H = h ¯ ω(a + a + 12 ) with q¯ = p¯ = 0, and ask how will it evolve in time. The simplest way to obtain the time evolution is to rewrite the minimal uncertainty condition αqψ + h ¯ ψ ′ = 0 through the canonical creation and annihilation operators, since the latter evolve in time in a very simple way: a(t) = eiHt a e−iHt = e−iωt a, a+ (t) = eiHt a+ e−iHt = eiωt a+ . Using the relations q = √λ2 (a + a+ ), p = i √¯h2λ (a+ − a) (Lecture 1), we obtain (αˆ q + iˆ p)ψ =

 1  (αλ2 − 1)a + (αλ2 + 1)a+ ψ = 0 2λ

(10)

The time dependent state ψ(t) = e−iHt ψ satisfies 







e−iHt (αλ2 − 1)a + (αλ2 + 1)a+ eiHt ψ(t) = (αλ2 − 1)a(−t) + (αλ2 + 1)a+ (−t) ψ(t) = 0 



(αλ2 − 1)eiωt a + (αλ2 + 1)e−iωt a+ ψ(t) = 0

(11)

This equation has the form of Eq.(10) with a time-dependent α given by λ2 α − 1 2iωt λ2 α(t) − 1 = e , λ2 α(t) + 1 λ2 α + 1

α(t) =

λ2 α cos ωt − i sin ωt λ2 (cos ωt − iλ2 α sin ωt)

(12)

The wavepacket remains gaussian at all times, while its width oscillates. For the ground state of the oscillator, α(t)λ 2 = 1 at all times. If the initial state is squeezed, αλ 2 ≫ 1, the wavepacket width reaches maximum at the times when cos ωt is close to zero, and collapses to the minimal value when sin ωt is near zero. 2

The Wigner function of the squeezed state (9) evolving in time according to (12) is a gaussian distribution in the phase space, centered at the origin and rotating with the frequency ω without changing shape (see Problem 3, PS#2). In contrast with the isotropic phase-space distribution of the ground state (as well as any other coherent state), the squeezed states produce gaussians elongated in one direction and squashed in a perpendicular direction. The major axes rotate according to the classical oscillator phase flow. There is another, more formal, definition of squeezed states based on the so-called ‘squeeze operators’. These are unitary operators which, when applied to the oscillator vacuum state, produce a squeezed state. The simplest example of a squeeze operator is  



U(θ) = exp θ a ˆa ˆ−a ˆ+ a ˆ+ /2



(13)

This operator has the following properties (see Problem 1, PS#2) U + (θ)ˆaU(θ) = cosh θˆa − sinh θˆa+ ,

U + (θ)a+ U(θ) = cosh θˆa+ − sinh θˆa

(14)

from which it follows that  λ  ˆ + aˆ+ U(θ) = e−θ qˆ U + (θ)ˆ q U(θ) = U + (θ) √ a 2  i¯h  + U + (θ)ˆ pU(θ) = U + (θ) √ a ˆ −a ˆ U(θ) = eθ pˆ 2λ

(15) (16)

Using these results, one can show that the state U(θ)|0i is a minimum uncertainty state (9) with the parameter αλ2 = e2θ . Hence the width of the wavepacket hδq 2 i1/2 is eθ times smaller than that of oscillator vacuum. Other squeezed states can be obtained in a similar way using unitary operators U(z) = exp (z (ˆaa ˆ−a ˆ+ aˆ+ ) /2) with complex parameter z. Because of the form of the squeeze operator, the squeezed states are also sometimes called ‘two-photon coherent states’.

1.3

Squeezed states from time evolution

How can one obtain a squeezed state, starting from, e.g., the vacuum state? It turns out that the squeezed states arise quite naturally from oscillator dynamics, provided that the parameters of the oscillator Hamiltonian, the frequency ω and the mass m, are functions of time. Let us consider the Schr¨odinger evolution i¯h∂t ψ = H(t)ψ ,

H(t) =

p2 m(t)ω 2 (t) 2 + q 2m(t) 2

(17)

Integrating formally the evolution equation in the time interval [0, t], have ˆ ψ(t) = S(t)ψ t=0 = lim

N →∞

with tj = j∆t, ∆t = t/N. 3

N Y

j=1

i

e− h¯ H(tj )∆t ψt=0

(18)

We now show that the evolving squeezed state satisfies (P (t)ˆ q − Q(t)ˆ p) ψ(t) = 0

(19)

where P (t) and Q(t) are functions of time to be found. Indeed, let us define the quantity ˆ = S(t) ˆ (P (0)ˆ C(t) q − Q(0)ˆ p) Sˆ−1 (t)

(20)

ˆ ˆ (P (0)ˆ ˆ C(t)ψ(t) = S(t) q − Q(0)ˆ p) Sˆ−1 (t)S(t)ψ =0

(21)

Then, at all times,

ˆ obeys equation of motion of the form The quantity C(t)   i H(t)∆t ˆ ˆ ˆ ˆ = lim 1 e− h¯i H(t)∆t C(t)e h ¯ h ¯ ∂t C(t) − C(t) = i[C(t), H(t)] ∆t→0 ∆t

(22)

ˆ H] is We also note that, if Cˆ is a polynomial in q, p of a degree n, then the commutator [ C, ˆ H] determines the rate of change also a polynomial of exactly the same degree. Since [ C, ˆ ˆ of C with time, and the C starts as a linear function in q and p at t = 0, it is natural to ˆ = P (t)ˆ suppose that Cˆ remains linear in in q and p at all times, C(t) q − Q(t)ˆ p. Using the evolution equation, we find ˆ = i[C(t), ˆ h ¯ ∂t C(t) H(t)] = i (P qˆ − Qˆ p) , h

!   mω 2 2 i P pˆ2 2 = −¯h + qˆ pˆ + mω Qˆ q (23) 2m 2 m

ˆ = P˙ (t)ˆ ˙ p, obtain Substituting ∂t C(t) q − Q(t)ˆ Q˙ = P/m ,

P˙ = −mω 2 Q

(24)

which coincides with the classical Hamiltonian equations for the oscillator (17). The initial conditions, e.g., corresponding to the state (9), are generally speaking complex valued. For instance, for the oscillator vacuum state, have Pt=0 = 1 ,

Qt=0 = −iλ2

(25)

Thus the evolving state has the form (9) at all times with α(t) = −iP (t)/Q(t). Let us consider a simple example, when the oscillator frequency jumps from ω 0 to ω. After that, the former ground state, which is not a ground state any more, starts to evolve in time. To analyze the dynamics of Q and P , it is convenient to introduce the variables z± = Q ± iP/mω which obey z˙+ = −iωz+ ,

z˙− = iωz−

(26)

Solving for the time evolution z + (t), obtain −iωt

z+ (t) = e

−iωt

z+ (t = 0) = e 4



i i − mω mω0



(27)

Similarly, for z − (t), obtain iωt

z˜− (t) = e

iωt

z˜− (t = 0) = −e



i i + mω mω0



(28)

which gives Q(t) =

1 sin ωt cos ωt , (z+ + z− ) = −i 2 mω mω0

P (t) = i



sin ωt cos ωt −i mω0 mω



(29)

The resulting time dependence α(t) = −iP (t)/Q(t) agrees with Eq.(12). As this example demonstrates, any time dependence of the Hamiltonian can be used to generate squeezed states. The underlying physical reason for this general behavior is that the certainty product, proportional to the phase space area of the Wigner density peak, remains constant in time due to the phase volume conservation in Hamiltonian dynamics. However, in practice it is often desirable to get a highly squeezed state by employing only small variations of the oscillator parameters. The above example suggests that this may be difficult, since for the value ω close to ω 0 the wavepacket width varies very little as a function of time. However, large squeezing by small perturbation can be achieved by using the phenomenon of parametric resonance. It is well known (to any child on a swing) that, when the frequency of the oscillator is modulated, ω 2(t) = ω02 + λ cos Ωt

(30)

and the external frequency Ω is close to 2ω 0 , the classical oscillator becomes unstable, with the oscillations of small amplitude growing larger as a function of time. A highly squeezed state is formed near the moments corresponding to the extremal points of classical motion, when P (t) is large and Q(t) small (being complex, Q(t) never completely vanishes, but can become really small at large times). For further discussion of the details of parametric resonance we refer to Problem 2, PS#2 (see also Landau & Lifshits, “Classical Mechanics”).

5

8.514: Many-body phenomena in condensed matter and atomic physics

Problem Set # 2 Due: 9/23/03

Squeezed states 1. Squeeze operators. Consider a unitary operator U(θ) = exp (θ (abab − ab+ ab+ ) /2). a) Prove that U + (θ)abU(θ) = cosh θab − sinh θ ab+ ,

U + (θ)a+ U(θ) = cosh θab+ − sinh θ ab

(1)

(Hint: use the operator expansion theorem, Problem 1 a), PS#1). From that derive the transformation rule for the coordinate and momentum operators, U + (θ)qbU(θ) = e−θ qb ,

b U + (θ)pU(θ) = eθ pb

(2)

b) To show that the operator U(θ), applied to the vacuum state |0i, generates a squeezed state, calculate the coordinate and momentum uncertainty, hδq 2i, hδp2 i, and show that the uncertainty product equals 21 h ¯ , independent of θ. c) To characterize the time evolution of the state ψ 0 = U(θ)|0i, formally given by ψ(t) = e −iHt/¯h ψ0 , find the variance matrix   b + iψ(t) hqb2 iψ(t) h 12 {qb, p} (3) b + iψ(t) h 12 {qb, p} hpb2 iψ(t) b + = qbpb + pbq.) b time dependence. (Here the expectation values h...i ψ(t) = hψ(t)|...|ψ(t)i, and {qb, p}

2. Time-dependent states of a harmonic oscillator. Consider a harmonic oscillator with a time-dependent frequency, H(t) =

p2 mω 2 (t) 2 + q 2m 2

(4)

a) Suppose that ω(t) is a given function of time. Look for a solution of the Schr¨odinger evolution equation i¯h∂t ψ = H(t)ψ of a gaussian form, ψ(q, t) = A(t) exp(−α(t)q 2 /2)

(5)

From the consistency requirement for such an ansatz, obtain a nonlinear differential equation that relates the time-dependent α(t) with ω(t). b) Show that a squeezed state time evolution can be obtained from the condition b ψ(t) = 0 (P (t)qb − Q(t)p)

(6)

where P (t) and Q(t) are complex solutions of the classical Hamilton equations Q˙ = P/m, P˙ = −mω 2 Q. The equations for P and Q are linear, while the equation for α(t) found in part a) is nonlinear. To establish a connection between the two methods, find a substitution that turns the equation for α(t) into a linear equation. c) Consider a harmonic oscillator initially in the ground state. The parabolic potential is abruptly removed at t = 0, and then restored at t = τ . Find the state at 0 < t < τ and at t > τ .

d) A popular practical method of producing squeezed states involves parametric resonance which takes place when the parameters of the oscillator are externally varied at a frequency close to twice the unperturbed normal frequency, ω 2 (t) = ω02 + λ cos Ωt ,

Ω = 2ω0

(7)

Taking the oscillator initially in the ground state and assuming small λ, obtain the time dependence of P (t), Q(t). A note on weakly perturbed oscillator: At small λ, it is convenient to look for a solution of the ¨ + ω 2(t)Q = 0 in the form Q(t) = A(t) cos ω0 t + B(t) sin ω0 t. For unperturbed harmonic equation Q oscillator, at λ = 0, the solution is given by constant A, B. Accordingly, for a weakly perturbed oscillator, the leading time-dependence A(t), B(t) should be slow. Based on this intuition, derive the differential equations for A(t), B(t) by discarding the rapidly oscillating terms (argue that their effect is negligible). To analyze wavepacket evolution, from the solution P (t), Q(t) find the parameter α(t). Qualitatively, sketch the width of the wavepacket as a function of time. 3. The phase-space density of a squeezed state. a) Show that the Wigner function W (q, p) of a squeezed state is a gaussian distribution in the phase space.   P b) For a general gaussian distribution P (x) ∝ exp − 12 nij=1 Dij xi xj of an n-component variable xi , show that   Dij = M −1 , Mij = hxi xj i (8) ij

In other words, the matrix D is fully characterized by the variance matrix M. Consider the Wigner function W (q, p) of a squeezed state. Using the result of Problem 1, part c), find the time dependence M(t) and D(t). c) For the states obtained in Problem 2, parts c) and d), reconstruct and qualitatively describe the time evolution of the phase-space distribution W (q, p). You may find it useful to use numerics for visualization.

8.514: Many-body phenomena in condensed matter and atomic physics

1

Last modified: September 24, 2003

Lecture 3. Second Quantization, Bosons

In this lecture we discuss second quantization, a formalism that is commonly used to analyze many-body problems. The key ideas of this method were developed, starting from the initial work of Dirac, most notably, by Fock and Jordan. In this approach, one thinks of multi-particle states of bosons or fermions as single particle states each filled with a certain number of identical particles. The language of second quantization often allows to reduce the many-body problem to a single particle problem defined in terms of ’quasiparticles,’ i.e. particles ’dressed’ by interactions.

1.1

The Fock space

The many-body problem is defined for N particles (here, bosons) described by the sum of single-particle Hamiltonians and the two-body interaction Hamiltonian: H=

N X

a=1

H(1) (xa ) +

H(1) (x) = −

X

a6=b

H(2) (xa , xb )

h ¯2 2 ∇ + U(x) , 2m x

(1)

H(2) (x, x′ ) = U (2) (x − x′ )

(2)

where xa are particle coordinates. In some rare cases (e.g. for nuclear particles) one also has to include the three-particle, and higher order multiparticle interactions, such as P (3) a,b,c H (xa , xb , xc ), etc. The system is described by the many-body wavefunction Φ(x 1 , x2 , ..., xN ), symmetric with respect to the permutations of coordinates x a . The symmetry requirement follows from pareticles indestinguishability and Bose statistics (i.e., the wavefunction invariance under permutations of the particles). The wavefunction Φ(x 1 , x2 , ..., xN ) obeys the Schr¨odinger equation i¯h∂t Φ = HΦ. Since the number of particles in typical situations of interest is extremely large, solving this equation directly presents a very hard problem. There are several approaches, however, that allow to gain insight. The method of second quatization, historically the first many-body technique developed in the 30’s, will be discussed here. We shall start with defining the Fock space, sometimes also called the ’big’ manyparticle space, V=

N Mn O N

symm

V (1)

o

(3)

— the sum of the N-th symmetric powers of the single particle Hilbert space V (1) . It describes the states of a system containing any number of particles N = 0, 1, 2, 3, ....

1

One can choose the basis in the ’big’ space V in the form of symmetrized products of single particle wavefunctions ϕ p (x) drawn from an orthonormal complete set of states in V (1) , !1/2 X m1 !m2 !... Φm1 ,m2 ,... (x1 , x2 , ..., xN ) = ϕp1 (x1 )ϕp2 (x2 )...ϕpN (xN ) (4) N! P with the sum taken over all permutations of the states ϕ p (x). The numbers mp indicate how many times the function ϕ p (x) appears in the product. The number of permutations P in the sum P is equal to the number of ways to combine N elements into groups containg m1 , m2 , etc., elements each (m1 + m2 + ... = N). This combinatorial factor, equal to N!/m1 !m2 !..., defines the normalization factor in Eq.(4). One can check that the states (4) are orthogonal and form a complete set in V. As an example, consider free Bose particles in a box L × L × L, in which case the single particle states can be chosen as eigenstates of the single particle problem Eϕ(x) = h2 ¯ ∇2 ϕ. Assuming periodic boundary conditions, we have eigenstates of a plane wave − 2m form 1 2π ϕn (r) = √ exp (ikn r) , kn = n , n = (n1 , n2 , n3 ) (5) L V with integer n1,2,3 and V = L3 . The energies of these states are En = is then spanned by the functions 1,

ϕn (r),

1 √ (ϕn (r)ϕm (r′ ) + ϕm (r)ϕn (r′ )) , 2

h2 2 ¯ k . 2m n

ϕn (r)ϕn (r′ ),

The space V ...

(6)

corresponding to the no-particle state, one particle, two particles, etc. The energies of these states are 0, En , En + Em , 2En , ... (7) Note that the structure of the two-particle functions depends on whether the participating single-particle states are different or the same. To make progress, one can introduce the so-called number representation. Allow any total particle number N and focus on the dependence of the state on the occupation numbers mi . This dependence is captured most vividly by the representation in which an auxiliary oscillator, along with the creation and annihilation operators, is associated with each single particle state. The occupation numbers are interpreted in this representation as number of quanta in eachy oscillator. The corresponing Fock states, in the number representation, have the form |m1 , m2 , ...i =

∞ Y

i=1

√

1  + mi a |0i mi ! i

(8)

where |0i is the no-particle state, and mi = N. This representation accounts correctly for the symmetry properties of the states (4) due to Bose statistics. P

2

1.2

Second-quantized operators

In the number representation, the many-body Hamiltonian (1) is represented by a polynomial in the operators a i , a+ i : X

H= (1)

ij

(2) ij

with the quantities H ij , Hkm (1) Hij

(1)

Hij a+ i aj +

(1)

1 X (2) ij + + H km ai aj ak am 2 ijkm

(9)

being the one- and two-particle matrix elements,

= hϕi (x)|H (x)|ϕj (x)i =

Z

ϕ¯i (x)H(1) (x)ϕj (x)dx

′ (2) ′ ′ H (2) ij km = hϕi (x)ϕj (x )|H (x, x )|ϕk (x)ϕm (x )i =

ZZ

(10)

ϕ¯i (x)ϕ¯j (x′ )H(2) (x, x′ )ϕk (x)ϕm (x′ )dxdx′

One can prove the equivalence of this representation to the original many-body problem (1) formulated in terms of many-particle wavefunction Φ(x 1 , x2 , ..., xN ) by directly evaluating the matric elements of the Hamiltonian between all pairs of many body states, and showing that in both representations the results agree. The combinatorics involved in this proof is combersom, albeit completely straightforward. Instead of reviewing it here, we refer to the book by J. R. Schrieffer, “The Theory of Superconductivity” that contains an Appendix describing the analysis in some detail. Another proof can be devised using functional integral, and we shall talk about it later on. The expressions (9),(10) are true for any orthogonal set of functions ϕ i (x). In the case when these functions are chosen to be the eigenstates of the single-particle problem, the matrix elements of H (1) vanish between different states, hϕi (x)|H(1) (x)|ϕj (x)i = Ei δij

(11)

and the one-particle part of the Hamiltonian simplifies to H(1) =

X

Ei a+ i ai

(12)

i

Since n ˆ i = a+ i ai is nothing but the number operator, the eigenvalues of the one-particle Hamiltonian corresponding to the number states (8) are En1 n2 ... = hn1 , n2 , ...|H(1) |n1 , n2 , ...i =

X

Ei ni

(13)

i

In the above example of free bosons in a box, the states are labeled by discrete momenta, and the expression (13) becomes En1 n2 ... =

X

Ek nk

(14)

k

If the bosons are interacting via a two-body potential U (2) = U(r − r′ ), from Eqs.(9),(10) we obtain the two-particle Hamiltonian of the form H(2) =

1 X + hk1 k2 |U (2) |k3 k4 ia+ k1 ak2 ak3 ak4 2 k1 k2 k3 k4 3

(15)

where the matrix element in (15), evaluated on the plane wave states (5), has the form hk1 k2 |U

(2)

|k3 k4 i =

ZZ

1 −ik1 r−ik2 r′ +ik3 r+ik4 r′ e U(r′ − r)d3 rd3 r ′ 2 V

(16)

This expression can be simplified and evaluated by choosing a = r ′ − r as an integration variable instead of r ′ , after which the integral in (16) factors as Z



ei(k4 −k2 )a U(a)d3 a ×

Z



˜ 2 − k4 ) δk1 +k2 =k3 +k4 e−ik1 r−ik2 r+ik3 r+ik4 r d3 r = U(k

(17)

R ˜ where U(k) = e−ikr U(r)d3 r is the Fourier transform of the interaction potential, and

δk1 +k2 =k3 +k4 =



V, k1 + k2 = k3 + k4 0, k1 + k2 6= k3 + k4

(18)

Finally, the two-body Hamiltonian takes the form H(2) =

1 2V

X

k1 +k2 =k3 +k4

+ U˜ (k2 − k4 )a+ k1 ak2 ak3 ak4

(19)

where the sum is taken over all integers parameterizing the plane wave states (5) subject to the constraint k1 + k2 = k3 + k4 . This constraint, as it is clear from the calculation, arises due to translational invariance of the system. Physically, it expresses the conservation of momentum in two particle scattering. The second-quantized interaction Hamiltonian is written in terms of the operators a k , + ak which remove or add particles. One may thus be worried by apparent particle nonconservation. After looking at it closer, however, and taking into account the commutation relations of a, a+ with the number operator n ˆ = a+ a, n ˆ a = a(ˆ n − 1) ,

n ˆ a+ = a+ (ˆ n + 1)

(20)

ˆ = Pk a+ ak commutes with the Hamilone can show that the total number of particles N k tonian and is thus conserved.

1.3

Field operator

A very useful representation of the second-quantized many-body hamiltonian is provided by the field operator, first introduced by Jordan, ϕ(x) ˆ =

X

ϕˆ+ (x) =

a ˆk ϕk (x) ,

k

X

a ˆ+ ¯k (x) kϕ

(21)

k

where x labels configuration space, e.g., x = r in D = 3 The states ϕ k (x) in Eq.(21) can be the eigenstates of a single-particle problem, such as, e.g., the plane waves of the above section, or any other convenient orthonormal basis set.

4

The operators (21) obey commutation relations [ϕ(x), ˆ ϕ(x ˆ ′ )] = [ϕˆ+ (x), ϕˆ+ (x′ )] = 0 ,

[ϕ(x), ˆ ϕˆ+ (x′ )] = δ(x − x′ )

(22)

which can be proven by using the commutation relations of a ˆ k and a ˆ+ k along with the orthogonality of the states ϕk (x). For example, [ϕ(x), ˆ ϕˆ+ (x′ )] =

X

ϕk (x)ϕ¯k′ (x′ )[ˆak , a ˆ+ k′ ] =

k,k′

=

X k

X

ϕk (x)ϕ¯k′ (x′ )δk,k′

(23)

k,k′

ϕk (x)ϕ¯k (x′ ) = δ(x − x′ )

(24)

We note that, although some particular basis set ϕ k (x) was employed to contruct the field operators, their properties, such as the commutation relations (22), are invariant with respect to the choice of basis. Using the field operators, the second-quantized problem (9),(10) can be expressed as a polynomial H=

Z

h ¯2 2 1 ϕˆ (x) − ∇x + U(x) ϕ(x)dx ˆ + 2m 2 !

+

ZZ

ϕˆ+ (x)ϕˆ+ (x′ )U(x − x′ )ϕ(x) ˆ ϕ(x ˆ ′ )dxdx′

(25) in which the quadratic and the quartic parts describe noninteracting particles and their interaction, respectively. It is sometimes helpful to think of ϕ(x) ˆ as a “quantized wavefunction.” In the field operator representation, the many body problem starts looking very much like a single particle problem. Of course, this simplicity is only apparent, since we still have a quartic term in the Hamiltonian, expressing the interactions and leading to “nonlinear” dynamics. Not just the Hamiltonian, but many other quantities also take a simple form in terms P of the field operator. For example, particle density n(x) = a δ(x − xa ) becomes n ˆ (x) =

X ij

due to

R

hϕi (x′ )|δ(x − x′ )|ϕj (x′ )ia+ ˆ+ (x)ϕ(x) ˆ i aj = ϕ

(26)

ϕ¯i (x′ )δ(x − x′ )ϕj (x′ )dx′ = ϕ¯i (x)ϕj (x). Similarly, the particle current operator is    ¯  + ˆj(x) = h ϕ (x)∇x ϕ(x) − ∇x ϕ+ (x) ϕ(x) 2mi

(27)

The density (26) and current (26) obey continuity relation written as an operator equation ∂t n ˆ + ∇ˆj = 0.

5

8.514: Many-body phenomena in condensed matter and atomic physics

1

Last modified: September 24, 2003

Lectures 4, 5. Bose condensation. breaking and quasiparticles.

Symmetry-

In an ideal Bose gas, at sufficiently low temperature, the lowest energy state becomes occupied by a macroscopic number of particles (Bose-Einstein condensate). The density of a gas of free bosons, given by a sum of occupancies of different momentum states, has the form Z   1 n= n0 δ(p) + (2π¯h)−3 n(reg) d3 p , n(reg) = β(p2 /2m−µ) (1) p p e −1 with m the particle mass and µ the chemical potential. At high temperature T > TBEC , with TBEC

h ¯ 2 2/3 =α n , m

α=

2π ζ 2/3 (3/2)

= 3.3142...

(2)

at density n, there is no condensate: n0 = 0, µ < 0. On the other hand, at low temperature T < TBEC , there is a macroscopic number of particles in the ground state, while the chemical potential is zero. In this case, the condensate density is 

n0 = n − nc = n 1 − (T /TBEC )3/2



(3)

The question we shall discuss below is how this behavior is modified by the presence of interatomic interaction. We shall focus on the problem of weakly nonideal Bose gas. This problem, due to the existence of a simple analytical method, serves well to illustrate the new features of Bose condensation of interacting particles: spontaneous symmetry breaking, the off-diagonal long-range order, and collective excitations.

1.1

Spontaneous symmetry breaking

Weakly interacting Bose gas with a short-range interaction, H=

Z

λ h ¯2 2 + −ϕˆ (x) ∇x ϕ(x) ˆ + ϕˆ+ (x)ϕˆ+ (x)ϕ(x) ˆ ϕ(x) ˆ dx 2m 2 !

(4)

where x = r in D = 3. The coupling constant is the two-particle scattering amplitude in the Born approxiR mation, λ = U˜k=0 = U(x − x′ )dx. A more accurate formula: λ = 4π¯h2 a/m, where a is the s-wave scattering length, to be discussed below.

1

The ground state at T = 0 is characterized by large occupation number of the k = 0 state. In number representation, the BEC state of N particles is |BECi = |N k=0 , 0, 0, ...i, i.e. (√ N |BECN −1 i, k = 0 (5) ak=0 |BECN i = 0, k 6= 0

This formula, √ at large N, suggests to replace the number state by a coherent state, a |BECi = N |BECi, which is equivalent to replacing the operator aˆ 0 by a c-number √0 N. This can be achieved if the BEC state is understood as a coherent state, which requires considering the problem (4) in the ’big’ space with all particle numbers allowed. Such an approach gives results equivalent to that of the problem with fixed particle number N in the limit N → ∞, since for a coherent state hδN 2 i1/2 = N 1/2 ≪ N. Uponqsuch a P replacement, the field operator ϕˆ = V −1/2 k ak eikr turns into a classical field ϕ = N/V , where V is system volume. To that end, we are led to consider the coherent states √ m ∞ √   V ϕ X 1 2 √ |ϕi = exp V ϕˆa+ ¯a0 |0i = e− 2 V |ϕ| |mi (6) 0 − ϕˆ m! m=0 which have the desired property ϕ|ϕi ˆ = ϕ|ϕi. These states do not correspond to any specific number of particles, in fact they are characterized by a distribution of particle ˆ = numbers. Accordingly, the states |ϕi are not invariant under the number operator N P + ˆ k ak ak , while the hamiltonian (4) commutes with N. One has to understand why the BEC state apparently does not respect the particle number conservation. We start by noting that ˆ

ˆ

eiαN |ϕi = |eiα ϕi ,

ˆ

eiαN He−iαN = H

(7)

ˆ

i.e. the operator eiαN , applied to |ϕi, produces a state of the same energy, with a phase ′ 2 of ϕ shifted by α. Since the overlap of coherent states obeys |hϕ ′|ϕi|2 = e−V |ϕ −ϕ| , any two different states |ϕi, |ϕ′ i are orthogonal in the limit V → ∞. Thus the states with different phase factors, |ϕi, |eiα ϕi, are macroscopically distinct. This observation demonstrates that the BEC states form a degenerate manifold parameterized by a phase variable 0 < α < 2π. To clarify the origin of this degeneracy, let us find the states |ϕi that provide minimum to the energy (4). Taking minimum at fixed particle density can be achieved by adding ˆ , H → H − µN ˆ . Taking the expectation value, obtain to H a term proportional to N ˆ |ϕi = λ |ϕ|4 − µ|ϕ|2 U(ϕ) = hϕ|H − µN 2

(8)

— the so-called Mexican hat potential. The energy minima are found on the circle |ϕ| 2 = µ/λ, i.e. the phase of ϕ is arbitrary, while the modulus |ϕ| is fixed, thereby giving a 2

relation between the density and chemical potential, µ = λn. 1 From the symmetry point of view, the systuation is quite interesting. The microscopic hamiltonian (4) has global U(1) symmetry, since it is invariant under adding a constant phase factor to the wavefunction of the system, ϕˆ → eiα ϕ. ˆ The ground states, however, do not possess this symmetry: adding a phase factor to the state |ϕi produces a different ground state. This phenomenon, called spontaneous symmetry breaking, is absent in the noninteracting Bose gas. In the interacting system, the U(1) symmetry breaking has a very fundamental consequence: it leads to superfluidity. There is yet another way to understand the phenomenon of U(1) symmetry breaking, due to Penrose and Onsager, that does not require to consider the states with fluctuating particle number. One can instead start with the density matrix of the Bose gas (4) ground state |Φλ i in the coordinate representation, R(x, x′ ) = hΦλ |ϕˆ+ (x′ )ϕ(x)|Φ ˆ λi

(9)

Using the translational invariance, one expects that the quantity (9) will depend only on the distance x − x′ between the two points. By going to Fourier representation, one can transform (9) to the form ′

R(x, x ) =

Z

−ik(x−x′ )

e

d3 k , nk (2π)3

nk = hΦλ |ˆa+ ˆk |Φλ i ka

(10)

In a Bose condensate, the particle distribution n k has singularity at k = 0, nk = n0 (2π)3 δ(k) + f (k)

(11)

where f (k) is a smooth function. Accordingly, the density matrix (10) has two terms, R(x, x ) = n0 + f˜(x − x′ ) , ′

f˜(x − x′ )

Z

′

e−ik(x−x ) fk

d3 k (2π)3

(12)

the constant n0 , independent of point separtion, and the second part, f˜(x − x′ ), that vanishes at large |x − x′ |. A density matrix that does not vanish at large point separtion represents an anomaly (recall that in any ordinary liquid all correlations vanish at several interatomic distances). The finite limit n 0 = lim|x−x′|→∞ hΦλ |ϕˆ+ (x′ )ϕ(x)|Φ ˆ ˆ λ i suggests that the quantities ϕ(x), + ′ iα √ + −iα √ ϕˆ (x ) in some sense have finite expectation values: hϕ(x)i ˆ =e n0 , hϕˆ (x)i = e n0 , with fixed modulus, but an undetermined phase. The name Off-diagonal Long-range Order, or ODLRO, associated with this phenomenon, expresses the fact that in the density matrix the ordering is revealed by the behavio of the off-diagonal component R(x, x′ )|x−x′|→∞ . 1

In a finite, but large system, with fixed particle number, the true ground state (TGS) of a quantummechanical hamiltonian is nondegenerate. This TGS is isotropic in ϕ, due to boundary effects that split the circular manifold. The statement about the absence of degeneracy of TGS in a finite system is formally coirrect, but misleading, since this TGS is not ’pure’. Typically, at any moment of time the state is characterized by a global phase, changing slowly as a function of time. (In D = 3 the mixing of ϕ’s with different phases results from vortices passing across the system, from one boundary to another.)

3

1.2

Quasiparticles

To study the excitations above the ground state, we substitute aˆ 0 = the hamiltonian (4), and keep quadratic terms,

√

N , a c-number, in

  X  (0) 1 X + + ˆ = 1 λn2 V + H − µN ǫk − µ + 2λn a+ ak a−k + ak a−k (13) k ak + λn 2 2 k6=0 k6=0  X  (0) 1 + + + + = λn2 V + ǫk (a+ a + a a ) + λn(a + a ) (a + a ) k k k k −k −k −k −k 2 (k,−k)

(14)

where the sum is taken over pairs (k, −k) Here we used the value µ = λn obtained above. At this stage, it is convenient to introduce the quantities qˆk = √12 (ak + a+ ˆk = −k ), p + + + i √ (a − ak ). These operators are non-hermitian, qˆk = qˆ−k , pˆk = pˆ−k , but obey the 2 −k standard p, q algebra, [ˆ qk , pˆk′ ] = δkk′ , which allows to treat them as coordinate and momentum. In terms of the operators pˆk , qˆk the hamiltonian is represented as a sum of + independent harmonic oscillators. Indeed, since a + ˆ+ ˆk + qˆk+ qˆk , we can k ak + a−k a−k = p kp rewrite the hamiltonian as follows:    X  (0) + 1 (0) ǫk pˆk pˆk + ǫk + 2λn qk+ qˆk (15) H = λn2 V + 2 (k,−k) This hamiltonian, quadratic in qˆk , pˆk , can be brought to the normal form by a rescaling (squeezing) transformation (0)

qˆk = eθk qˆk′ ,

pˆk = e−θk pˆ′k ,

e4θk =

ǫk (0)

ǫk + 2λn

(16)

which acts on the operators ak , a+ k as ak = cosh θk bk − sinh θk b+ −k ,

+ a+ −k = cosh θk b−k − sinh θk bk

(17)

(see Lecture 2). The transformation (17), called Bogoliubov transformation, can be shown to preserve the canonical commutation relations, [b k , b+ k ] = 1. The hamiltonian is now reduced to   X X 1 1 2 + H = λn2 V + Ek b+ b + b b = λn V + Ek b+ (18) k −k k −k k bk 2 2 k6=0 (k,−k) describing a gas of Bogoliubov quasiparticles, the noninteracting bosons created and annihilated by the operators b+ k , bk , having energy Ek =

r 

2

(0)

ǫk + λn

− (λn)2

(19)

The new ground state is annihilated by all the b k . Since for the ground state of the ideal Bose gas ak |Φ0 i = 0, and the transformation (17) can be represented as bk = Uak U −1 , + −1 b+ , with k = Uak U   U = exp 

X

k6=0





+  θk a+ k a−k − ak a−k /2

4

(20)

(see Lecture 2), one can write the new ground state as |Φλ i = U|Φ0 i. The dispersion relation (19), for small k, is linear, Ek = h ¯ c|k| ,

c=

q

λn/m

(21)

which is characteristic for sound waves in a fluid. For higher values of k, the dispersion takes the form of a usual free-particle expression E k = h ¯ 2 k2 /2m + λn. Remarkably, both the collective modes, sound waves, and the single-particle excitations appear on the same dispersion curve, gradually blending into one another at the energy ca. Ek ≃ λn. One can gain some insight into the difference of the modes at large and small k by considering the field equations 2

¯ ˆ =−h i¯h∂t ϕˆ = [ϕ, ˆ (H − µN)] ∇2 ϕˆ + λϕˆ+ ϕˆ2 − µϕˆ (22) 2m ¯2 2 + ˆ ϕˆ+ ] = − h −i¯h∂t ϕˆ+ = [(H − µN), ∇ ϕˆ + λ(ϕˆ+ )2 ϕˆ − µϕˆ+ (23) 2m It is instructive to treat these equations as a q classical dynamics problem, linearizing near stationary solution, ϕ = ϕ 0 + η, where ϕ0 = µ/λ. The linearized equation has solution of the form η(r, t) = aeikr−iωt + ¯be−ikr+iωt (24) r

(0)

2

with h ¯ ω = ± ǫk + λn − (λn)2 the same as Eq.(19). In other words, one can consider condensate with fluctuating amplitude and phase, and show that these fluctuations propagate in just the same way as the collective modes (19). In such an approach, the difference between small and large k follows from the relation between the amplitudes a and b obtained from the dynamical equation. At small k, the sum a + b is much smaller than the difference a − b. This means that the oscillations are predominantly in the phase of the field ϕ, not in the modulus, just as one expects from Goldstone theorem (and the above Mexican hat picture). At large k, however, the normal modes have a + b or a − b nearly equal in magnitude, which means that the oscillation follows a small circle in the complex ϕ plane, i.e. the phase and the modulus of ϕ participate in the collective oscillations roughly equally. We can use the above results to estimate the effect of condensate depletion due to interactions. The total density of all particles in the system can be written as n = hΦλ |a+ 0 a0 + = n0 +



X

k6=0

a+ k ak |Φλ i = n0 +

X

k6=0

sinh2 θk hΦλ |bk b+ k |Φλ i 

(0) ǫk

 + λn 1 X r − 1     2 2 k6=0 (0) ǫk + λn − (λn)2

(25)

(26)

Estimating the sum as O(λ 3/2 ), we find that the condensate depletion is a small effect. In contrast, in superfuid 4 He only few percent of the helium atoms are in the single-particle ground state. 5

8.514: Many-body phenomena in condensed matter and atomic physics

Problem Set # 3 Due: 9/30/03

Bose condensation 1. Quasiparticles. Consider a Bose gas at T = 0 with one quasiparticle with momentum p 6= 0 added on the top. Quasiparticle state can be obtained by applying the quasiparticle creation operator to the nonideal Bose gas ground state: |1p i = bb+ (1) p |0i b+ bp . where bb+ p = cosh θp a p − sinh θp a How many particles are contained in one quasiparticle? To find out, take the number operator + c=P a bk of the original particles and evaluate the difference N k bk a c i − h0|N|0i c ¯p = h1p |N|1 N p

(2)

(Be careful: abp |0i 6= 0, instead bbp |0i = 0.) Express the answer in terms of the Bogoliubov angle θ p . Compare the situation at high and low quasiparticle energy and interpret the result. 2. Landau criterion for superfluidity. A superflow state of a Bose condensate having velocity v is characterized by macroscopic occupancy of a state with nonzero momentum p = mv. The many body state can be constructed by generalizing the scheme used to describe stationary condensate: |Φv i = exp

√

¯ ab+ ) , V (φ(x)abp − φ(x) p 

φ(x) = φ exp



i px h ¯



(3)

c a) Starting from this state, consider the expectation value hΦ v |H − µN|Φ v i and, by minimizing energy in φ, obtain the chemical potential µ of the superflow state. How does µ depend on the superflow velocity v? b) Consider elementary excitations (quasiparticles) in the superflow state. The Bose gas hamiltonian expanded up to second order in ak , a+ k , has the form

H = E0 +

X

k6=0

  1 X 2 + + (0) ¯2 ak a2p−k ǫk − µ + 2λ|φ|2 a+ a + λ φ a a + φ k k k 2p−k 2 k6=0

(4)

To diagonalize this hamiltonian, group together the states with momenta k and 2p − k, b bk = cosh θk abk − sinh θk ab+ 2p−k ,

b b+ b 2p−k b+ k = cosh θk a k − sinh θk a

(5)

Find the parameters θk that diagonalize the hamiltonian, and obtain the quasiparticle dispersion relation E(k). (Hint: Don’t let yourself be dragged into long calculation, — the result can be more or less read off the solution for stationary BEC with slight adjustments.) Find the critical superflow velocity v c above which the energy of quasiparticles E(k) can become negative. Landau argued that the superfluid can sustain nondissipative flows with velocities v < v c , and in this way he could explain the phenomenon of critical velocity observed in superfluid 4 He. At E(k) > 0 the quasiparticles cannot be created spontaneously, while at v > v c the flow is accompanied

by massive quasiparticle creation, and is thus dissipative. Find the critical velocity v c for nonideal Bose gas. c) Can you interprete the result of part b) for quasiparticle dispersion in superflow from the point of view of a Galilean transformation? Note that the microscopic hamiltonian is invariant with respect to changing the reference frame from stationary to moving, x ′ = x + vt, t′ = t. Show that for an excitation with frequency ω and wavevector k this yields ω ′ = ω − kv, k′ = k. How is the quasiparticle energy changed under a Galilean transformation? 3. Condensate depletion. a) In a nonideal Bose gas at T = 0 only a fraction of all the particles is found in the condensate. The reduction of condensate density due to interactions is called “condensate depletion.” (An extreme example is prodided by 4 He, where the majority of the particles — more than 90% — are not in the condensate. To estimate this effect in a weakly nonideal Bose gas, find the expectation value of the total density X b =n b 0 + V −1 bk , n bk = a b+ bk n n (6) ka k6=0

b0 i, while the second term over the ground state. The first term gives the condensate density n0 = ha+ 0a gives the density of the out-of-condensate particles. Find the depletion factor (n − n 0 )/n dependence on the coupling constant λ. b ′ )|0i. Show that it is b) Consider the correlator of the field operators R(x, x′ ) = h0|φb+ (x)φ(x P ′ ik(x−x′ ) related to the particle number distribution as R(x − x ) = k h0|nk |0ie . Describe the behavior of R(x − x′ ) as a function of point separation x − x′ . Find the limits at |x − x ′ | → ∞ and at x = x′ . Estimate the length scale ξ, called BEC healing length, at which the crossover from R(0) to R(∞) takes place.

4. Thermodynamics of a Bose gas. Thermodynamic quantities of Bose-condensed gas can be found by treating the system as a gas of noninteracting Bogoliubov quasiparticles obeying Bose statistics. The thermodynamic potential of the system is Ω ≡ −T ln Z = T

Z



−βE(k)

ln 1 − e

 d3 k

(2π)3

,

E(k) =

q

ǫ(0) (k) (ǫ(0) (k) + 2λn)

(7)

with ǫ(0) (k) = h ¯ 2 k2 /2m. a) Show that simple analytical results for the thermodynamic potential Ω can be obtained at very low temperatures, T ≪ Tλ ≡ λn and at moderately high temperatures, T λ ≪ T ≤ TBEC . (Hint: Given the temperature, low or high, simplify the form of E(k) by ignoring ǫ (0) (k) compaired to λn, or vice versa.) b) Find the entropy, the specific heat, and the normal component density n(T ) in the above two temperature intervals. Compare with the ideal Bose gas.

8.514: Many-body phenomena in condensed matter and atomic physics

1

Last modified: September 29, 2003

Lecture 6. Vortices, superfluidity. Trapped gases. BEC at finite temperature.

To treat hydrodynamics and BEC in spatially varying background, need a more general approach that does not assume condensation into a particular plane wave state. One can formulate theory of BEC so that the wavefunction of the condensate is an arbitrary function in space-time that is determined self-consistently from a classical nonlinear field equation, known as the Gross-Pitaevskii equation.

1.1

Gross-Pitaevskii equation

Let us start again from the Hamiltonian of weakly interacting Bose gas with short-range interaction, H=

Z

h ¯2 2 λ ϕˆ (x) − ∇x + U(x) ϕ(x) ˆ + ϕˆ+ (x)ϕˆ+ (x)ϕ(x) ˆ ϕ(x) ˆ dx 2m 2 !

+

!

(1)

To formulate the condensate dynamics, start with the Heisenberg evolution, i¯h∂ t ϕˆ =   h2 ¯ [ϕ, ˆ H] = − 2m ∇2 + U(x) ϕˆ + λϕˆ+ ϕˆ2 , and replace ϕ, ˆ ϕˆ+ by classical variables ϕ, ϕ, ¯ which gives a classical field dynamics problem, h ¯2 2 i¯h∂t ϕ = − ∇ + U(x) ϕ + λϕϕ ¯ 2 2m

(2)

h ¯2 2 −i¯h∂t ϕ¯ = − ∇ + U(x) ϕ¯ + λϕ¯2 ϕ 2m

(3)

!

!

called the Gross-Pitaevskii equations. It is instructive to write eqs. separately for the modulus and phase ϕ = |ϕ|e iθ . n = |ϕ|2 ,

∂t n + ∇j = 0 ,

j=

and

h ¯ h ¯ (ϕ∇ϕ ¯ − ϕ∇ϕ) ¯ = |ϕ|2∇θ 2mi m

(4)

h ¯2 h ¯ ∂t θ = − U(x) + λn + |∇ϕ|2 (5) 2m Comparing the above expressions for the density and current, obtain the superflow velocity !

h ¯ ∇θ m The flow is irrotational, ∇ × v = 0 (this is true away from singularities in θ), and v=



2



m∂t v = −∇ µ ˜ + mv /2 ,

√ h ¯2 √ ∇2 n µ ˜ = U(x) + λn + 2m n

(compare to the Euler equation). 1

(6)

(7)

1.2

Superfluidity. Vortices.

Let us consider the circulation of velocity in a superflow. It follows from the relation between the velocity and the phase, Eq. (6), that the circulation around any contour C obeys I h ¯ Γ= v · dr = 2πl (8) m C with some integer l. We see that • The circulation is quantized in multiples of h/m; • The flows in multi-connected geometries, such as a ring-shape tube, are discrete; • Quantum leaps are required to change a flow. The fact that a superflow, due to the dicreteness of circulation, cannot be dissipated gradually, but only in dicrete steps, is the origin of superfluidity. The only way to eliminate a superflow is to produce excitations with discrete vorticity and then remove them (along with the vorticity) from the system. Also mention the Landau criterion for superfluidity: The quasiparticle energy ǫ ′ (k) = ǫ(k) − v · k, Doppler-shifted due to the flow, should be positive, to prevent massive production of quasiparticles. This criterion defines a critical velocity vc = min ǫ(k)/|k| k

(9)

above which the flow without excitations is unstable, thus showing that superfluidity can be sustained only at the flow velocity below certain critical value. 1 The Landau criterion points at a neccessary condition for superfluidity. However, since it does not take into account vortices which can be generated in the flow even at velocities below v c , one cannot use it to predict the actual value of critical velocity. The observed critical velocities are system-dependent (non-universal) and are typically several orders of magntude below v c estimated from quasiparticle dispersion. Now let us consider velocity field in a vortex with singularity on the z axis.The flow lines are concentric circles parallel to the (x, y) plane. Constant circulation requires that the velocity falls inversely with the distance ρ from the z axis: v(r) =

h ¯ l θˆ m 2πρ

(10)

where θˆ is the asimuthal unit vector of the cylindrical coordinate system. Density variation is important only close to the vortex core, at distances where the kinetic energy per particle exceeds the interaction energy, 12 mv 2 > λn, which gives ρ≤ξ= √ 1

h ¯ λm

For weakly interacting Bose gas, the critical velocity is equal to Bogoliubov sound velocity.

2

(11)

The so-called healing length ξ determines the size of the vortes core where density is depleted below its bulk value. This qualitative picture can be confirmed by an analysis based on the Gross-Pitaevskii equation. One can look for a solution of the equation that describes a vortes. √ For a single-quantized vortex with unit circulation, l = 1, we expect ϕ(ρ → ∞) = neiθ , and ϕ(ρ ≪ ξ) ∝ ρeiθ . Thus one can take a trial function of the form √ ρ ϕ(r) = n q eiθ (12) ρ2 + r02 √ and minimize the energy H(ϕ, ϕ), ¯ which gives r 0 = 2ξ, in agreement with the estimate above. Let us consider The energy of the vortex, that can be estimated as the kinetic energy of the flow, is positive. Thus vortices do not appear unless the system is driven, or stirred. Let us consider a cylindrical jar rotating with angular velocity Ω, and find the critical rotation velocity at which vortices start to appear. For a vortex located on the symmetry axis of the jar of radius b and height L, the kinetic energy of the flow is Ev =

Z

L 0

Z

0

b

h ¯2 1 2 n(ρ) mv (ρ)2πρdρdz = πn 2 m

Z

0

L

Z

ξ

b

dρ h ¯2 b dz = πn ln L ρ m ξ !

(13)

(According to Eq.(12) the density can be approximated by a constant for ρ > ξ, while the depletion of density in the vortex core, at ρ ≤ ξ, cuts the log divergence at small ρ.) the contribution to the energy due to the core, which can be estimated using the trial 2 function (12), turns out to be approximately πn ¯hm ln 1.464L, which is smaller than our kinetic energy estimate (13). The energy of the vortex in a jar rotating with velocity Ω is E v (Ω) = Ev − ΩM, where M is the angular momentum of the vortex, Mv =

Z

mn r × v d3 r =

Z

0

L

Z

b

n(ρ)mρv(ρ)2πρdρdz = πb2 h ¯ nL

(14)

0

The vortex becomes energetically favorable at Ω>

Ω(1) c

b h ¯ = Ev /Mv = ln 2 mb ξ

!

(15)

Note the inverse square dependence of Ω(1) c on the radius b, which means that it is easier to produce vortices in a larger jar. If the rotation velocity is larger than Ω (1) c and keeps increasing, one can reach the next critical value Ω (2) at which the second vortex appears, and then, at some higher value c (3) Ωc , the third vortes enters the jar, and so on. At high rotation speed, when there are many vortices, one can estimate their number h ¯ N from the requirement that the total circulation due to the vortices, m N, matches the H 2 circulation of a uniformly rotating fluid, v · dr = Ωπb , which gives a linear dependence m (16) N(Ω) ≃ πb2 Ω h ¯ 3

Of course, since N is integer, in reality the number of vortices increases discretely, in steps, on average following the proportionality relation (16).

1.3

Trapped gases.

Bose condensation of confined gasses differs somewhat from BEC in a uniform system that we discussed so far. Most importantly, the BEC transition is accompanied by an abrupt change of density distribution. This is due to the fact that the lowest energy quantum state in which atoms condense is peaked at the trap center and has spatial extent much less than the size of thermal cloud at temperatures slightly above T BEC . In the experiments on BEC in trapped gasses, atoms are confined by a magnetic trap, which can be described by a harmonic potential U(r) = 12qmω 2 r2 . The ground state is a gaussian wavepacket ψ0 (r) ∝ exp(−r 2 /2lω2 ) of width lω = h ¯ /mω. In an ideal Bose gas, in the absence of interactions, in the BEC state at T = 0 all the atoms populate the state ψ0 . One notes that the density of this state, at the peak, n ≃ N/lω3 , can be extremely high when the number of atoms is large. In the presence of interactions, on can easily reach the limit when the interaction energy per particle is much larger than the level spacing in the trap, λn ≫ h ¯ ω. For that, the number of atoms should exceed Nc = h ¯ ωlω3 /λ. However, for realistic parameters the value N c can be 103 − 104 , which is much less than the typical atom numbers N in the experiments. To understand the BEC state at a larger number of atoms, one can start with the Gross-Pitaevskii energy functional and look for a non-uniform state ϕ(x) that minimizes the energy, E(ϕ) =

Z

h ¯2 1 |∇ϕ(x)|2 + ((U(x) − µ)|ϕ(x)|2 + λ|ϕ(x)|4 dx 2m 2 !

(17)

with the particle number N = |ϕ|2 dx being fixed by a chemical potential µ. Let us argue that one can discard the gradient term in the energy functional, since the expected condensate size is much larger than the oscillator ground state width l ω , for which the kinetic and potential energy terms are approximately equal. Indeed for a condensate h2 ¯ of size R, the kinetic, potential, and interaction energies can be estimated as 2m N/R2 , 2 mω NR2 , and λN 2 /R3 , respectively (since the gas density n ≃ N/R 3 ). Comparing the 2 1/5 potential and interaction energy, obtain the condensate size R ≃ (λN/mω 2 ) . At large 2 h2 −2 ¯ N ≫ Nc , the value R is much larger than l ω that satisfies 2m lω = mω l2 . Hence the 2 ω 2 2 h ¯ kinetic energy is small, 2m R−2 ≪ mω R2 , which justifies ignoring it in the estimate of R. 2 Without the kinetic energy term, the functional be written in terms of the density  R 1 2 2 n = |ϕ| only, E(n) = ((U(x) − µ)n + 2 λn dx. After taking the minimum, one has R

µ = U(x) + λn

(18)

One can arrive at this result by making a local density approximation, i.e. treating each small part of the BEC cloud as a uniform system. For the latter, as we already know, 4

the relation between the chemical potential and density is given by Eq.(18). In addition, the chemical potential in equilibrium must be constant throughout the system. This condition fixes the density distribution n(x) so that the term λn in Eq.(18) compensates the potential U(x) variation in space, which gives n(x) =



(µ − U(x)) /λ , U < µ 0, U >µ

(19)

We note that the argument used to find the density distribution is similar to that of the Thomas-Fermi theory of many-electron atoms, based on a local density approximation for electrons moving in an effective potential that is determined selfconsistently from an elecrostatic problem. The local density approximation in trapped BEC, along with Eqs.(18),(19), is often referred to as Thomas-Fermi approximation. For a central-symmetric harmonic trap potential, by relating BEC radius parameter R 2 R2 , from Eq.(19) we obtain density distribution with the chemical potential via µ = mω 2 of the form  mω 2  2 n(r) = R − r2 , r < R (20) 2λ The particle number can be related with R (and thus with µ) as follows: N=

Z

0

R

   4π 1 1 mω 2 5 mω 2  2 2 2 R − r 4πr dr2π − R = mω 2 R5 2λ 3 5 λ 15λ

(21)

which gives a relation between the BEC radius and the number of particles, R=

15λ N 4πmω 2

!1/5

(22)

√ For large N ≫ Nc , the radius R is much larger than the BEC healing length ξ = h ¯ / λnm estimated for typical density n = N/ 4π R3 , which determines the scale of spatial nonlo3 cality in BEC correlations. This means that the Thomas-Fermi approximation is indeed a local density approximation. The above discussion summarizes the situation at T = 0. Let us briefly discuss how the BEC transition affects density distribution. At temperatures above the transition, the gas in the trap forms a cloud of width R T that can be estimated from 21 mω 2 RT2 ≃ T . At T < TBEC , condensate appears forming a much more narrow peak of radius (22) that coexists with the broad thermal distribution. As temperature goes down and becomes very small, the thermal component in the density distribution disappears, and one obtains the zero-temperature state (20).

1.4

Finite T effects: quasiparticle lifetime.

Decay of quasiparticles due to elastic scattering Hint =

X λ a+ a+ ak ak 2 k1 +k2 =k3 +k4 k4 k3 2 1

5

(23)

Golden Rule for transition rate: 2π X Wi→f = |hf |Hint |ii|2 δ(Ef − Ei ) h ¯ f

(24)

In a normal Bose gas, at T > TBEC , the rate of scattering out of the state |ii is dfi 2π X =− |Mij,mn |2 fi fj (1 + fm )(1 + fn )δ(ǫi + ǫj − ǫm − ǫn ) dt h ¯ ij,mn

out :

(25)

while the rate of scattering in |ii is

dfi 2π X = |Mmn,ij |2 (1 + fi )(1 + fj )fm fn δ(ǫi + ǫj − ǫm − ǫn ) dt h ¯ ij,mn

in :

(26)

with Mij,mn = Mmn,ij = λ, fi = ha+ i ai i, etc. The resulting rate is the sum of the in-rate and out-rate dfi /dt = dfi /dt|in + dfi /dt|out , 2π X dfi =− |Mij,mn |2 (fi fj (1 + fm )(1 + fn ) − (1 + fi )(1 + fj )fm fn ) δ(ǫi + ǫj − ǫm − ǫn ) dt h ¯ ij,mn (27) Features: • The rate

dfi dt

vanishes in equilibrium, since 1 + f j = eβǫj fj , etc. (0)

• For near-equilibrium distribution, dfdti = − τ1 (fi − fi ) with scattering rate q (recall: λ = 4π¯h2 a/m, vT = 2T /m)

1 τ

• Despite scattering, quasiparticles are well defined: ǫ(k) ≫

1 τ

= πa2 nvT the classical

In the BEC state, scattering is stimulated by the presence of the condensate: √ + an , a+ N n → a0 , a0 =

(28)

which gives the rate of scattering out of the state |ii as out :

dfi 2π X =− |Mij,m|2 fi fj (1 + fm )δ(ǫi + ǫj − ǫm ) dt h ¯ ij,m

(29)

and the rate of scattering in |ii,

dfi 2π X = |Mm,ij |2 (1 + fi )(1 + fj )fm δ(ǫi + ǫj − ǫm ) dt h ¯ ij,m √ with Mij,mn = Mmn,ij ∝ λ N. The resulting rate, dfi /dt = dfi /dt|in + dfi /dt|out, is in :

dfi 2π X |Mij,m |2 (fi fj (1 + fm ) − (1 + fi )(1 + fj )fm ) δ(ǫi + ǫj − ǫm ) =− dt h ¯ ij,m

(30)

(31)

Due to the presence of Bose condensate, scattering rate is enhanced at T < TBEC . 6

1.5

Finite T effects: two-fluid hydrodynamics, I & II sound.

Momentum can be carried both by the condensate and excitations: j = ρvs + jex = ρvs +

Z

pfp

d3 p (2π¯h)3

Normal fluid is described by quasiparticle distribution fp =

1 exp (β (ǫp − p · (vn − vs ))) − 1

with the quasiparticle energy Doppler-shifted due to relative motion of the normal gas and superfluid. The momentum due to the normal component is jex =

Z

pfp

d3 p = ρn (|vn − vs |) (vn − vs ) (2π¯h)3

At small velocities, have ρn =

Z

p2 d3 p (−∂fp /∂ǫp ) = 3 (2π¯h)3

(

(2π 2 /45¯h3 c5 )T 4 ρ(T /TBEC )3/2

T ≪ Tλ Tλ ≪ T < TBEC

where Tλ = λn is the Bogoliubov sound – free particle crossover energy, and c = is the sound velocity. With ρs = ρ − ρn , the total momentum density can be expressed as

q

λn/m

j = ρs vs + ρn vn To describe dynamics, one needs separate equations for ρ s , vs and ρn , vn . We will not discuss the two-fluid hydrodynamics in full generality. Instead, we describe a particular phenomenon, the II sound, a collective mode that appears in the two-fluid regime. In this mode, the relative fraction of the normal and superfluid component oscillates and can propagate in a sound-like fashion. We consider a uniform system, in the absence of external potential. The total mass and mass current density obey the continuity relation ∂t ρ + ∇ · j = 0

(32)

∂t j = −∇p

(33)

and the momentum conservation law

After eliminating j from (32) and (33), one obtains ∂t2 ρ − ∇2 p = 0 7

(34)

For superfluid velocity, one can write m∂t vs = −∇µ

(35)

This relation, derived above from the Gross-Pitaevskii equation, is in fact very general, and is true for any superfluid. It follows from the relation between the phase of superfluid order parameter and the chemical potential, h ¯ ∂ t θ = −µ, discussed above. In this form it was first introduced by Josephson in the theory of superconductivity. Use the thermodynamic Gibbs relation Ndµ = V dp − SdT with ρ = Nm/V , σ = S/Nm, to relate the gradient ∇µ with the pressure and temperature gradients: ∇µ =

m ∇p − σm∇T ρ

(36)

From Eqs.(36),(35),(33), combined with j = ρ s vs + ρn vn , obtain ∂t (vn − vs ) = −σ

ρ ∇T ρn

(37)

In a sound wave, gas compression is adiabatic, with entropy conserved: ∂t (ρσ) + ∇ (ρσvn ) = 0

(38)

(The entropy is carried by the normal component only!) After linearizing and combining with the mass conservation Eq.(32), have ∂t σ = σ

ρs ∇ (vs − vn ) ρ

(39)

ρs 2 2 σ ∇ T ρn

(40)

Combined with Eq.(37), this yields ∂t2 σ =

Collective modes are obtained by considering small oscillations of density, pressure, temperature and entropy, of the form exp (iqr − iωt). It is convenient to choose density and temperature as independent variables. Linearizing Eqs. (34), (40) in δρ, δT , obtain 

∂p ω 2 δρ − q 2  ∂ρ 

∂σ ω2  ∂ρ

! T

! T

∂p δρ + ∂T

!

!



∂pσ δρ + ∂T

ρ



ρ

δT  = 0

δT  − q 2

ρs 2 σ δT = 0 ρn

(41) (42)

In terms of sound velocity u = ω/q, the equations have a solution if (u2 − c21 )(u2 − c22 ) − u2 c23 = 0 8

(43)

with c21

=

∂p ∂ρ

!

,

c22

T

ρs σ 2 T = , ρn C˜

c23

=

∂p ∂T

!2 ρ

T ρC˜

(44)

The constants c1 and c2 are the isothermal sound velocity and the velocity of temperature waves at constant density, while C˜ = T (∂σ/∂T )ρ is the specific heat at constant volume, per unit mass. The expression for constant c 3 of the form (44) was obtained from Eqs. (41),(42) by using Maxwell relation ∂p ∂T

!

= ρ

∂S ∂V

!

2

T

= −ρ

∂σ ∂ρ

!

(45) T

The I and II sound velocities, obtained from Eq.(43), are u2I,II

 1 2 1q 2 2 2 2 = c1 + c2 + c3 ± (c1 + c22 + c23 ) − 4c21 c22 2 2

(46)

So far, the treatment was completely general, applicable to any Bose system, irrespective the interaction strength and form. Using the result, one can look at several regimes. In a weakly nonideal gas, at low temperatures, T ≪ Tλ = λn, the I sound velocity coincides with that of Bogoliubov quasiparticles at low energies, q (47) uI = λn/m, √ while the II sound velocity is 3 times lower, √

uII = uI / 3 =

q

λn/3m,

(48)

The velocity uII decreases as a function of temperature. In 4 He the II sound represents mostly a temperature wave, and to excite/detect it people had to use oscillatory thermal sources and heat sensors.

9

8.514: Many-body phenomena in condensed matter and atomic physics

Problem Set # 4 Due: 10/07/03

Bose condensation 1. Vortices. H h ¯ h ¯ ∇θ, v · dr = 2π m l with a) Starting from the superflow equations away from singularities, v = m integer l, show that the velocity field v(r) can be found from a ‘magnetostatics problem’ ∇·v = 0,

∇ × v = 2π

h ¯ j(r) m

(1)

Here j(r) is an auxiliary line current that flows along vortex cores, and for each vortex takes an integer value equal to the quantized circulation l. Argue that the velocity field of several vortices can be obtained from superposition principle. Consider two vortices of unit circulation, aligned parallel to each other and separated by a distance d. Find the velocity v(r), the phase θ(r), and the interaction energy of the two vortices. (Consider two situations, with vortices of the same sign and of opposite signs.) b) Consider a vortex near the wall of a container with superfluid. The vortex is aligned parallel to the wall at a distance d from it. The flow around the vortex is distorted due to the presence of the wall. Show that this distortion can be characterized by introducing an image vortex on the other side of the wall, along with extending the flow to the entire space (i.e. removing the wall). What is the sign of the image vortex? Find the superflow velocity v(r) and the interaction energy of the vortex and the wall. Is this interaction attractive or repulsive? c) For a generic vortex configuration, starting from Eq. (1), derive ’Biot-Savart formula’ v(r) =

Z Xh ¯ lα da × (r − r′ ) α

2m

(2)

|r − r′ |3

for superflow velocity. Here the integral is taken over vortex core lines, the sum is taken over all vortices of circulation l α each. Consider a circular vortex ring of radius R. Each portion of the ring is sitting in a flow induced by other parts of the ring. As a result, the ring propels itself as a whole, and moves without changing the shape and radius. Find the velocity of self-propelled ring motion. Describe the dependence of ring velocity on the radius. Show that smaller rings move faster than larger rings. d) Consider a vortex in an infinite system. Suppose that the vortex core line is slightly displaced relative to its initial completely straight configuration. Analyze how this displacement evolves in time and propagates along the core line. For a small displacement amplitude, linearize the problem and find the dispersion relation for long-wavelength vortex oscillations. 2. Collective modes of trapped BEC. a) Consider Bose gas in a harmonic trap, H=

λ h ¯2 2 b b φ(r) b + φb+ (r)φb+ (r)φ(r) φ (r) − ∇r + U(r) φ(r) d3 r 2m 2

Z "

b+

!

#

(3)

with the trap potential U(r) = 12 mω02 r2 = 12 mω02 (x2 + y 2 + z 2 ). Collective modes of this system, in general, depend on the temperature of the gas and on the interaction strength. However, for one

special mode, sometimes called Kohn mode, that corresponds to the center of mass motion of the gas, the behavior is universal. Show that the center of mass operator b = R

Z

φb+ (r)rφ(x)d3 r

(4)

¨ = −ω 2 R, and thus the dynamics of R is characterized by frequency ω 0 irrespective of the obeys R 0 quantum state of the gas. b) Collective modes of Bose condensate can be studied by using Gross-Pitaevskii equation, as the long-wavelength dynamics of the density and phase variables, (i) :

∂t n = −∇ (nv) ,

v=

h ¯ ∇θ ; m

(ii) :

∂t θ = −µ/¯h ,

1 µ = U(r) + λn + mv2 2

(5)

Linearize these equations for small fluctuations in a steady state characterized by density n(r), taking the limit of long wavelength, and obtain the wave equation for collective modes, λ θ¨ = ∇ (n∇θ) m

(6)

Check that for a system at uniform density the collective modes have the same sound-like dispersion as Bogoliubov quasiparticles at low energy. c) Consider collective modes in a trapped BEC sample of radius R at T = 0, with density mω 2 distribution n(r) = λ1 (µ − U(r)) = 2λ0 (R2 − r 2 ) ,. In this case, since n varies in space, the wave equation (6) cannot be solved by Fourier transform. (No plane waves in a finite system!) Instead one has to look for normal modes of the problem (6) which we rewrite as −ω 2 θ =

  ω02  2 R − r 2 ∇2 θ − 2r · ∇θ 2

(7)

Show that there is a class of special solutions with special dependence θ(r) = r l+1 Yl,m (α, β), where Yl,m are spherical harmonics of the spherical angles α, β. (Note that the functions of this form satisfy Laplace equation ∇2 f = 0.) Find the frequency of oscillations as a function of the spherical harmonic number l. Which of these modes correspond to the center of mass motion discussed in part a)? d) Find the spectrum of all collective modes of the problem (7). (Hint: Use the variable f (r) = rθ(r). For a particular spherical harmonic, write f (r) = f (r)Y l,m(α, β), then look for a solution in the form of power series X f (r) = aj r j (8) j=l+1

and use Eq.(7) to obtain a recursion relation for the coefficients a j .)

8.514: Many-body phenomena in condensed matter and atomic physics

Problem Set # 5 Due: 10/14/03

Interacting Fermions 1. Short-range interaction. a) Consider Schr¨odinger equation in D = 1 for two spinless fermions moving in an external potential U(x) = 21 mω 2 x2 and interacting via a short-range potential λδ(x − x ′ ). Find the energy spectrum and the eigenstates for this two-body problem. b) Generalize the result of part a) to any number of fermions. c) Generalize the result of part a) to an arbitrary external potential U(x) and arbitrary space dimension. d) Prove that the energies and states of spinless fermions with short-range interaction are the same as without interaction by starting from the second-quantized hamiltonian H=

Z

h ¯2 λ b b b ψ(x) dx ψ (x) − + U(x) ψ(x) + ψb+ (x)ψb+ (x)ψ(x) 2m 2 !

!

b+

(1)

b and using the algebra of the field operators ψ(x) and ψb+ (x).

2. Cooper pairs. To understand the origin of pairing in the presence of attractive interaction in a Fermi system, Cooper proposed to replace the many-body problem by a toy model of two particles at the Fermi level. To take into account that the states below the Fermi level are filled, and thus are ‘dynamically inaccessible,’ the Hilbert space of states for each of the two particles in this model consists of all single-particle states with energies above the Fermi level. The Hilbert space for the two particles consists of the states + |p, α, p′ , βi = a+ p,α ap′ ,β |0i ,

|p|, |p′ | > pF ,

α, β =↑, ↓

(2)

The energy of the state (2), in the absence of interactions, is E p + Ep′ , where Ep = p2 /2m − p2F /2m. a) Consider short-range interaction Hint

1 Z = λ 2

X

α,β=↑,↓

ψbα+ (x)ψbβ+ (x)ψbβ (x)ψbα (x)dx

(3)

Find the matrix elements of the interaction (3) between the states (2) b) Show that the total momentum P = p + p ′ is conserved, and thus the two-particle Hilbert space decouples into subspaces with fixed value of P. c) Let us fix P = 0, i.e. consider the subspace of two-particle states of the form + |p, α, −p, βi = a+ p,α a−p,β |0i ,

|p| > pF ,

α, β =↑, ↓

(4)

Show that the interaction (3) projected on (or, acting within) this subspace is an operator of rank one. In other words, up to a constant, it maps all vectors onto one specific vector. d) Suppose the hamiltonian has the form H = A + B, where A is a diagonal matrix with the spectrum Ei , and B is an operator of rank one with vector |ui = (u1 , u2 , u3 , ...)

corresponding to its only nonzero eigenvalue λ. Show that the operator (ǫ − A) −1 B is of rank one for ǫ 6= Ei . Use this fact to obtain an algebraic equation for the spectrum of H. e) Apply the result of part d) to the problem of Cooper pair with an attractive interaction λ < 0 and zero total momentum P. Show that the spectrum of the problem consists of a continuum of states with positive energies, and of one discrete state with negative energy. (To cut a divergence in the eigenvalue equation, you may assume that the interaction strength is gradually decreasing at large energies, λ → λ exp(−E p /E∗ ), with E∗ ∼ EF .

8.514: Many-body phenomena in condensed matter and atomic physics

Problem Set # 6 Due: 10/21/03

Bardeen-Cooper-Schrieffer theory 1. Quasiparticles. Consider quasiparticles of a BCS superconductor, H=

X

ǫp a+ p,σ ap,σ +

p,σ

X



¯ p,↑ a−p,↓ + h.c. = ∆a

p

X

Ep b+ p,σ bp,σ

(1)

p,σ

q

where Ep = ǫ2p + |∆|2 and bp,↑ = up ap,↑ + vp a+ −p,↓ , etc., are Bogoliubov quasiparticle operators. a) Find out how many particles are contained in one quasiparticle. For that, consider a state with one quasiparticle added to the BCS ground state, |p, σi = b+ p,σ |0BCS i

(2)

evaluate the expectation value hNi = hp, σ|

a+ q,α aq,α |p, σi

X q,α

(3)

and express the result in terms of the Bogoliubov angle θ p . Can hNi be nagative? Explain. b) Consider momentum and spin of a quasiparticle in the state (2). What are they? Do they depend on the Bogoliubov angle? 2. Gap equation. For a BCS superconductor derive the gap equation ∆ = λν

Z

E∗

−E∗

up v¯p tanh



1 βEp dǫ 2 

(4)

with E∗ the interaction cutoff parameter (E∗ ∼ EF for nonretarded contact interaction). Study the gap ∆ as a function of temperature. Show that ∆ decreases monotonically with T and vanishes at a certain temperature T = T c . Find the value Tc . 3. Gap suppression in a superflow. Critical current. Superflow in a superconductor is described by the order parameter with spatially varying phase, ∆(r) = ∆e2iqr , which is related to the superflow velocity by v s = q/m. BCS quasiparticles in the presence of superflow are described by the Hamiltonian H=

X p,σ

ǫp a+ p,σ ap,σ +

¯ p+q,↑ a−p+q,↓ + h.c. ∆a

X p



(5)

which can be dioganalized by a Bogoliubov transformation in which the states p + q and −p + q are paired up. a) Find the quasiparticle spectrum. Assuming |q| ≪ p F , show that the result can be interpreted in terms of Doppler shift E p′ = Ep + vs p, where Ep is the spectrum in the absence of the flow.

b) Show that the energy gap between the BCS ground state and the first excited state is reduced in the presence of the flow. Find the critical velocity at which the gap vanishes. c) Consider BCS pairing in the frame co-moving with the flow. By using Galilean invariance, or otherwise, argue that the gap equation and thus the order parameter ∆ are not affected by the flow. Combined with the result of part b), this shows that the energy gap and pairing amplitude ∆ aint necessarily have to be equal. They happen to be equal in a clean superconductor in the absence of external pair-breaking fields or flows, but are not equal in general.

8.514: Many-body phenomena in condensed matter and atomic physics Problem Sets # 7,8 Due: 11/04/03 (released late due to instructor illness) Quasiparticle transport in a superconductor 1. Electron tunneling. Consider two metals that can be in a normal or in a superconducting state coupled through a tunnel junction, X Ht = Tk,q c+ (1) k,σ cq,σ + h.c. σ,k,q

where cq,σ and ck,σ are Fermi operators of an electron in the metal and in the superconductor, respectively. a) Consider tunneling current in the presence of voltage V applied across the barrier. Using the Golden Rule dW = 2π |hf |Ht|ii|2 , evaluate the rate of transitions from material h ¯ 1 to the material 2 and show that I1→2 = A

Z

∞

−∞

|T |2 N1 (E)f (E)N2 (E + eV ) [1 − f (E + eV )] dE

(2)

with N1,2 (E) the density of states dN /dǫ in both materials, f (E) the Fermi distribution. P Here A is a proportionality constant and |T | 2 = ǫk ,ǫq |Tk,q |2 . b) Following the route that has led to Eq.(2), find a similar expression for the current I2→1 from material 2 into the material 1. For the total tunneling current I = I 1→2 − I2→1 obtain Z ∞ I=A |T |2 N1 (E)N2 (E + eV ) [f (E) − f (E + eV )] dE (3) −∞

c) Verify that for a pair of normal metals, with constant density of states each, the tunneling current (3) obeys Ohm’s law, I = GV . d) Consider tunneling between a normal metal and a superconductor. Analyze the expression for the current and plot I vs. V at low temperature and at T = 0. Show that the so-called tunneling density of states W (V ) = dI/dV at zero temperature is proportional to the BCS quasiparticle density of states 

W (V ) ∝ N(E) = ν0 ∆/ E 2 − ∆2

1/2

E=eV

(4)

2. Andreev reflection. Charge transport through a clean normal metal-superconductor interface can be described by Bogoliubov-deGennes equation with position dependent pairing amplitude ∆(r), E



u v



=



H ∆(r) ∗ ∆ (r) −H∗



u v



(5)

where H is a single-particle Hamiltonian of noninteracting fermions. a) Consider a one-dimensional problem with a step-like pairing function ∆(x < 0) = 0, h2 2 ¯ ∆(x > 0) = ∆, and H = − 2m ∂x − EF . Consider scattering state of an electron incident on the NS interface with the energy below the BCS gap, |E − EF | < ∆. Show that in

the superconductor the solution is an evanescent wave, and in the metal it describes a reflected hole. b) Now consider incident electron with the energy slightly above the gap, |E − E F | > ∆. Describe the result of scattering of such an electron. c) Generalize the result of part a) to a 3D system. Consider planar normal metalsuperconductor interface with an electron incident at an angle θ to normal. Find the direction of the outgoing hole. Fermi liquid theory 3. Thermodynamic functions, specific heat. a) Thermodynamic potential of the ideal Fermi gas can be evaluated as Ω = −T

Z





ln 1 + e−βǫp d3 p/(2π¯h)3

(6)

Starting from this expression, show that specific heat is a linear function of temperature at T ≪ EF . Find the proportionality constant in the relation C = γT . b) Consider the thermodynamic potential using the particle-hole oscillator representation. We are going to check if it gives the same result as the canonical Fermi representation. In the absence of interactions, H=

X 1

k,p∈Rk

2

∗ πk,p πk,p

1 2 ∗ + ωp,k φk,p φk,p 2



(7)

with ωp,k = (p + k)2 /2m − p2 /2m. Apply the formula for the thermodynamic potential of an ensemble of free bosons, Ω=T

X α





ln 1 − e−β¯hωα = T

X

k,p∈Rk



ln 1 − e−β¯hωp,k



with ωp,k = v · k + k2 /2m, v = p/m, and the crescent domain Rk defined in lecture as an overlap of a displaced Fermi sphere complement |p + k| > p 0 with the undisplaced Fermi sphere |p| < p0 . Compare with the result of part a). To simplify analysis, consider only low temperatures, and find the specific heat for T ≪ EF . c) Consider thermodynamic functions of the system of interacting fermions using the oscillator representation, Hint =

X

1/2

1/2

Vk ωp,k ωp′ ,k φ∗k,pφk,p′

(8)

k,p,p′ ∈Rk

Compare with the noninteracting case. Like in part b), do the calculations assuming T ≪ EF .

4. Screening Consider screening of an external potential in a 3D electron gas with Coulomb repulsion R Vk = (e2 /|r|)eikr d3 r = 4πe2 /k2 between electrons. Show that for a slowly varying potential, the screened potential is described by Vk =

k2 V ext , k2 + rs−2 k

rs−2 = 4πνe2

(9)

with ν the density of states at the Fermi level. The quantity r s is the screening radius, as can be seen from the form of the screened Coulomb potential 1r e−r/rs .

8.514: Many-body phenomena in condensed matter and atomic physics Problem Set # 9 Due: 11/20/03 (released late due to instructor illness) Path Integral Reading: R. P. Feynman and A. P. Hibbs, Quantum Mechanics and Path Integrals; M. Stone, The Physics of Quantum Fields 1. Harmonic oscillator. a) Starting from a path integral representation, obtain the Greens function of a harmonic oscillator: 1/2

mω hx , t|x, 0i = 2πi sin ωt 

′

i imω h 2 2 exp (x + x′ ) cos ωt − 2xx′ 2 sin ωt 



(1)

To find the prefactor, evaluate the infinite product of gaussian integrals using the formula !−1

Y z z2 = 1− 2 2 sin z n>0 π n

(2)

Note the periodicity in t and the singularities at ωt = πn, n ∈ Z. Discuss their meaning and check that in the limit ω → 0 the Greens function of the Schr¨odinger equation for a free particle is recovered. b) Consider the density matrix ρ(x, x ′ ) = hx′ |e−βH |xi of a harmonic oscillator at finite temperature (β = 1/kB T ). It is convenient to evaluate ρ(x, x′ ) using the imaginary time R path integral representation, ρ(x, x′ ) = d[x(τ )]e−S , with the integral take over the paths x(τ )0