1 Bose condensation and Phase Rigidity

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1 1 Bose condensation and Phase Rigidity 1.1 Overview and reference literature In this part of the course will explore various ways of describing the phenomenon of Bose-Einstein Condensation (BEC), and also discuss some important properties of superfluids. We start by reviewing the simple statistical physics arguments pertinent to a non-interacting system. We then develop some theoretical machinery that will allow us to generalize to the interacting case and give a proper local characterization of the BE condensed state. Most of these considerations will be rather elementary, and I will only make use of very basic QFT. For a more formal discussion of spontaneous symmetry breaking, I refer to Stone, ch This exposition in these notes are based on the following texts: [i] pp and of reference [1] (perfect and ideal gases). [ii] [iii] pp of reference [1] (BEC of a perfect gas). P. W. Andersson, Coherent Matter Field Phenomena in Superfluids, which is reprinted in reference [2] pp (the nature of the BEC state). [iv] Chapter III, of reference [3] (phenomenology of superfluids), 25 (the weakly interacting bose gas) and 26 (ODLRO and families of ground states). At some places I will also refer to to the notes on second quantization, referred to as SQ and the corresponding equations SQ.2 etc. For the rest of the section on bosons, I will follow Stone ch to discuss the dilute interacting Bose gas and put it in the context of QFT. In a course on quantum liquids one would now go on and discuss and explain the many fascinating and startling phenomena of superfluids, but in this field theory course we can only touch on this vast subject, which is briefly covered in Stone, ch (We will return to 10.4 when we discuss superconductivity.) For those who want to learn more about the physics of superfluidity, the last item on on the reference list above is good place to start, and the last section in these notes provides some comments to this text. 1.2 Non-interacting Bosons In this section we shall consider the simple case of noninteracting bosons, where BEC can be understood as a macroscopic population of the single particle 1

2 groundstate. You are presumably already familiar with part of this material so this section is of review character. In the following I will use the term perfect gas for a system of noninteracting particles, while the term ideal gas is reserved for the special case of high temperature and low density where the ideal gas law applies. Here I give a derivation of the Bose distribution functions as a simple application of the operator formalism we developed in the previous section. The Hamiltonian and number operator for a collection of non-interacting particles is given by, H = i ˆN = i ɛ i a ia i (1) a ia i, where ɛ i is the i th single particle energy. The grand partition function is now given by, Z = e βω β(h µ ˆN) = Tre (2) where µ is the chemical potential and Ω = pv is the grand canonical, or Landau, potential. We now evaluate the trace using the n-states defined by SQ.6 Z = {n i } n α1, n α2,... e β(h µ ˆN) n α1, n α2,... (3) = e β(ɛ i µ)n i = {n i } i i ( ) e β(ɛ i µ)n i = n i i 1 1 e β(ɛ i µ). We can now use standard methods to calculate any thermodynamic quantity. The average number of particles N = ˆN is given by N = µ Ω = 1 ln(1 e β(ɛi µ) ) (4) β µ i = 1 i e β(ɛ i µ) 1 = n B (ɛ i ), i so we can identify the Bose distribution function n B (ɛ i ) = 1 e β(ɛ i µ) 1. (5) Do the corresponding evaluation of the Fermi distribution function as an exercise! 2

3 1.3 BEC and Phase Coherence Bose-Einstein condensation is a unusual example of a macroscopic quantum phenomena in that it occurs even in the absence of interaction, i.e. for a perfect gas. The main point is that below a critical temperature, that depends on the density, not all the particles can be in thermal equilibrium. The particles that do not fit in the thermal distribution will instead condense into the lowest energy state. I briefly explain this in class, and you can also consult the text [ii]. This elementary argument does however only apply to a perfect gas and can for instance not be used to explain the superfluidity of helium II. The reason is that with interactions present one can no longer assign occupation numbers to the different one-particle states, since the state vector is now a coherent superposition of different n-states. Another difficulty with the simple treatment is that the BEC is not connected to any local observable - the macroscopic population of the ground state is a property of the system as a whole. I will now ellaborate this point, following the treatment in [2]. Assume that we have a BEC of N non-interacting particles in a volume V. If a 0 is the creation operator for the ground state of the one-particle system, and Ψ 0 is the condensate ground state, we have, Ψ 0 a 0a 0 Ψ 0 = N (6) Now imagine dividing the volume into m subsystems each with N/m particles. The corresponding ground state operators a 0α where α = 1... m satisfy the commutation relations [a 0α, a 0β ] = δ αβ. If the total condensate could be described as a sum of the condensates in the sub-volumes, it would be natural to define, a 0 = 1 m m α=1 which satisfies [a 0, a 0] = 1. Let us now calculate N = ˆN, a 0α (7) N = N α = 1 α αβ m Ψ 0 a 0αa 0β Ψ 0 (8) = N m + 1 Ψ 0 a m 0αa 0β Ψ 0 (9) α β For this relation to be satisfied the overlaps Ψ 0 a 0αa 0β Ψ 0 must be of O(N) which means that there are strong correlations between the various subsystems. It is thus not possible to describe a BEC as a collection of sub systems each 3

4 of which is Bose condensed. This is rather unsatisfactory since for a large BE condensed system, such as a macroscopic amount of liquid Helium II, we expect properties like superfluidity to be characteristic of parts of the system. After all we can scoop up a spoon of helium from the bucket and it will still be superfluid! We now proceed to develop a formalism that will give a local characterization of a BEC and also be general enough to allow for interactions. Recall from our previous discussion, that a quantum field is the annihilation operator in the position basis. Let us now consider particles in a periodic box of volume V so the field operator can be expanded as, 1 ˆψ( r) = 1 e i p r a p (10) V where p is the momentum and the lowest energy state of free particle is has p = 0 and E = 0. To study condensation, we separate out the lowest energy state, i.e. we write, p ˆψ( r) =  + 1 e i p r a p (11) V p 0 where  = 1 V a 0 and [Â,  ] = 1 V. (12) In the limit of large N, the operators  and  almost commute and can with a very good approximation be considered as classical, i.e. as c-numbers. The action of the operators  and  on the N-particle ground state N 0 is given by,  N 0 =  N 0 = N V N 1 0 = n N 1 0 (13) N + 1 N n N V where n = N/V is the density of the condensate, and the last approximation becomes exact in the N limit. If we also take the thermodynamic limit V while keeping the density constant, we can replace the operator  with the c-number A = ne iφ. 1 Here I will use ˆψ for the boson field operator in order not to confuse it with the phase operator ˆϕ. 4

5 Now comes the main point: Since only the ground state is macroscopically populated, we have Ψ 0 ˆψ( r) Ψ 0 A = ne iϕ + O( 1 N ) (14) which is a local characterization of the condensate since ˆψ is a local operator with an expectation value related to the density of the condensate, not the total number of condensed particles. It is also not necessary that Ψ 0 is a non-interacting state, since the density n in (14) will in general not be equal to the number density in the lowest one-particle state. In writing (14) we glossed over a rather subtle question: What is really the nature of the state Ψ 0? To appreciate this problem, notice that since ˆψ is a creation operator, its expectation value in any state with fixed number of particles is identically zero, as can be seen directly from (13). The rest of this section will provide an introduction to the article [iii] by Andersson, where this problem is discussed. There are (at least) three ways to characterize the condensed state, and thus to make sence of equation (14). We shall now briefly describe the various options Off-Diagonal Long-Range Order The consept of Off-Diagonal Long-Range Order, ODLRO for short, was introduced by Penrose and Onsager to describe the particular order present in a superfluid. You might already be familiar with the density matrix, descrip- density tion of quantum mechanics. For a pure state, the density matrix, ρ is simply matrix related to the wave fuction, ρ( r 1, r 2... r N ; r 1, r 2... r N) = ψ( r 1, r 2... r N)ψ ( r 1, r 2... r N ) (15) from which it follows that an expectation value can be calculated as. O = Tr( ρo). (16) The advantage of working with density matrices rather than wave functions is that it allows for so called mixed states, i.e. states which are mixtures in the sense of classical probability, rather than quantum mechanical superpostitions. 2 For example, a thermal state, such as a classical gas, will be described mixed states 2 For a short introduction to density matrices, see e.g. chapter 3.4 of reference [4] 5

6 by a density matrix of the form, ρ(β) = 1 Z ρ i e βe i (17) i i.e. the density matrices for the various pure microstates of the gas are added with probabilities given by the corresponding Boltzmann weight. The reduced one-particle density matrix is defined as, ρ( r ; r ) = d 3 r 2... d 3 r N, ρ( r, r 2... r N ; r, r 2... r N ) (18) and we see immediately that the particle density, ρ is given by the diagonal elements ρ( r, r) of this matrix. (You have already encountered the density matrix as the ground state expectation value of the density operator, in our discussion of Wigner functions and phase space distributions for the electron gas.) Now comes the basic point. In a normal statistical mixture, such as the thermal state in (17), the relative phases of the wave functions entering the density matrix are completely random and thus we expect ρ( r ; r ) 0 as r r. However, in a BE condensed state of free particles, the N- particle wave function is given by ψ( r 1, r 2... r N) = N i=1 ψ 0 ( r i ), where ψ 0 is the ground state wavefunction. In a box with periodic boundary condtions, ψ 0 = e iα / V where V is the volume and α some arbitrary phase. Thus, ρ( r ; r ) = Nψ0( r)ψ 0 ( r ) = n which is a finite constant. It is thus natural to define Bose-Einstein condensation by demanding that the density matrix factorizes like, factorization of ρ lim ρ( r ; r ) = φ ( r)φ( r ) (19) r r a where the order parameter φ( r) can be constant or vary slowly with r over order distances which are much larger than the microscopic length a, which is typically an interatomic distance. A state with this property is said to have parameter ODLRO since the off-diagonal elements of the density matrix factorizes. For a homogeneous condensate, φ is just the square root of the condensate density, φ = n. Let us connect this to the interpretation of (14). First you should convince yourself that for a pure state, the one-particle density matrix defined by (18) is noting but the expectation value of the density operator, density operator 6

7 ˆρ( r ; r ) = ˆψ ( r) ˆψ( r ) (20) in the relevant state. Thus the factorization property (19) suggests the identification ρ( r ; r ) = Ψ 0 ˆρ( r ; r ) Ψ 0 Ψ 0 ˆψ ( r) Ψ 0 Ψ 0 ˆψ( r ) Ψ 0 (21) which identifies the order parameter φ( r) with the ground state expectation value Ψ 0 ˆψ ( r) Ψ 0 in (14). For a homogenous state it is natural to identify the constant A in (14) with the, order parameter φ in (19). It is now also clear how to generalize to the general space (and time) dependent case, and thus give a general definition of the left hand side of equation (14). An alternative definition of φ( r) which is not based on asymptotic factorization, is to note that since the density matrix is hermitian, it can always be expanded in its diagonal basis as, ρ( r ; r ) = i n i χ i ( r)χ( r ). (22) BEC occurs when one of the eigenstates, χ 0 has a macroscopic eigenvalue, N 0 = O(N). We can then define the order parameter as φ( r) = N 0 χ r. (23) Neglecting all other states, we see that this definition of φ corresponds to the one given in (14) Families of Groundstates The second way of describing a Bose condensed state, which is explained in 26 in text [iv], is based on the notion of a family of ground states, α, N which differ only by the number of particles. A condensed state is defined as such a family for which the limit lim α, N ˆψ α, N + 1 = n (24) N exists. Here α represents all other quantum numbers that are needed to totally specify the groundstate (as e.g. total angular momentum). For further details you might consult to 26 in [iv]. 7

8 1.3.3 Coherent States The third, and more generally used, method is based on states which are not eigenstates of the number operator ˆN. 3 The coherent states discussed in section 4 or SQ, is an important example which we shall comment upon below. To get a deeper understanding of states which are coherent superpositions of different N-states, it is useful to use a density - phase representation for the operator  introduced in the previous section. For this purpose, we write, from which we can derive the relation (do it!),  = e i ˆϕ ˆρ (25)  = ˆρ e i ˆϕ, e i ˆϕ ˆρ = (ˆρ + 1 V )ei ˆϕ (26) which shows that e i ˆϕ acts as a translation operator for the density operator ˆρ. Since ˆϕ in an angular variable, V ˆρ can be thought of as an angular momentum, and thus follows that ˆρ is quantized in units of 1/V, and that ˆϕ and ˆρ satisfy the commutation relation, [ˆρ, ˆϕ] = i V. (27) We see that in the thermodynamic limit, the density and phase commutes and become classical variables, but for microscopic or mesoscopic systems the ρ - ϕ dynamics and the related fluctuations are important. Let us now consider the following superpostition of N-states, ϕ = N c N e iϕn N (28) where again N belongs to a family of groundstates with different N, and the c N :s are fourier coefficients satisfying N c N 2 = 1. It is easy to show that if the c N :s are strongly peaked around some value N 0, i.e. c N e ( N N 0 )2, with small, we have φ = ϕ ˆψ( r) ϕ = n 0 e iϕ (29) 3 Not everybody agree that this is the best way to describe a BEC. A notable critic is Anthony Leggett who maintains that the correct way to describe a BEC is via the factorization (22) of the density matrix, and that the method of coherent states is prone to... to generate pseudoproblesms and is best avoided [5]. 8

9 where n 0 = N 0 /V is the mean density. This equation is very important - it tells us that a BE condensed state is charaterized not only by a constant density - that is true for any gas or liquid in equillibrium - but by a constant phase of the quantum field! Clearly this is again a local characterization of the BEC and it works equally well for interacting as for non-interacting systems. This provides yet another definition of the order parameter, φ, which is of central importance in the discussion of phase transitions. We can now formulate the condition for BE condensation in the following succinct way: A Bose condensed state is a state where the order parameter φ is non-zero. So far we implicitly assumed a very large system with a constant superfluid density and a constant phase. In fact the above description is much more general. First there is no restriction to costant condensates - in general both the density and the phase will vary. For instance, in an atomic trap the condensed atomic cloud can have many different shapes, and typically has the highest density in the middle. Also, the phase of the order parameter does not have to be constant for a a state to be BE condensed, it is sufficient that it varies slowly compared with the typical inter-particle distances. As we shall learn later, there are excitations in the superfluid corresponding to correlated phase - density fluctuations, which are interpreted as sound waves, and so called vortex configurations, where the phase jumps 2π along some line. Finally we should mentioned that a coherent state of the type discussed in SQ sect.4, which satisfies ˆψ( r) φ = φ( r) φ (30) clearly also satisfies (29). (The hat one ˆψ is reinstated to emphasize that it is an operator.) It is however not true that all Bose condensed states which satiesfies (29) is a coherent state in the sense of being an eigenstate of ˆψ The Gross-Pitaevskii equation To describe varying condensates, one uses a space (and for dynamical processes also time) dependent orderparameter φ( r, t). In our discussion of second quantization, we already encountered the resulting Gross-Pitaevskii differential equation. Here we record it again for the special case of a a local two-particle interaction V ( r r ) = 1 2 V 0δ( r r ) which is appropriate to describe cold 9

10 atoms interacting via s-wave scattering processes, i h d ( ) dt φ( r) = h2 2m 2 + U( r) + V 0 φ( r) 2 φ( r) (31) We derived this equation by first taking the expectation value of the Heisenberg equation of motion for the quantum field, and then factorizing products of operators as φ ˆφ ˆφ φ = φ ˆφ φ φ ˆφ φ 2 = φ φ 2 = φ 2 φ An alternative derivation starting from (22), will be given in class. Further study of non-constant BECs are beyond the scope of this course[6]. 1.4 N - ϕ dynamics, Phase Rigidigy and Supercurrents The most remarkable property of the BE-condensed state is superfluidity, i.e. dissipationless flow which e.g. allows Helium II to flow through very narrow capillaries. In the next section we shall discuss the phenomenology of superfluids, and then later connect to the physics of BECs. Anderson s discussion of supercurrents in the article [iii] provides a good introduction and motivation for why we expect quantum coherent flow in a BEC. Here I give a few comments to the relevant parts in [iii]. First notice that (27) can be rewritten as [ ˆN, ˆϕ] = i (32) which shows that the number and phase operators are canonically conjugate. Heisenbergs equation for ˆN thus becomes i h ˆN = [H, ˆN] = i H ˆϕ (33) Taking the expectation value of this equation, we see that the change in the particle number, i.e. the current, is related to the derivative of the energy w.r.t the phase of the order parameter. Since the condensate is characterized by a constant phase, there must be a term in the Hamiltonian that punishes phase variations. The simplest choise is U = V d 3 r u( r) = h2 d 3 r n s ( ϕ) 2 (34) 2m V where n s is the superfluid density. This expression follows from equation (31), if we neglect density fluctuations. This equation embodies the phase rigidity phase rigidity 10

11 referred to in section As explained in the article by Anderson, the supercurrent density is given by the derivative of U w.r.t. the gradient of the phase, j s = 1 h du d( ϕ) = h 2m n s ϕ. (35) Make sure that you understand how to go from the discrete case with a current density between two subsystems 1 and 2, to the continuos case where the current density is obtained from the variation w.r.t. the gradient of the phase. Also notice that although the current density can be formally written as j s = n s v s with the superfluid velocity v s = ϕ, this cannot in general be identified with a real observable velocity for the reasons given in the text. 1.5 Phenomenology of Superfluids We have now learned what is BEC, both in a perfect gas and in a system of interacting bosons, but still not shown conclusively that it is of relevance for real physical systems. Following reference [3] se shall proceed towards this goal in several steps, the first being an account of the basic physics of superfluids. In 22 there is a entirely phenomenological discusson of a quantum bose liquid based upon an assumption about its excitation spectrum. The most important point is that all the low lying excitations are sound waves. In 23 it shown how this assumption implies superfluidity. It is also important to critical note the existence of a critical velocity above which the flow becomes viscous. velocity The next paragraph is more formal and explains how to describe phonons, i.e. quantized sound waves, using a density - phase formulation. For this recall the discussion in section which dealt with the k = 0 part of the density and phase - the present treatment is an extension to general r dependence. The important results of this section are the commutation relation (24.8) and the final expression for the Hamiltonian for phonons, Ĥ = k u hk(ĉ k ĉ k ). (36) The reminder of this section contains some comments to of [iv]. 22: You might or might not be familiar with the theory of sound. The relevant modes for this discussion are the so called acoustic, or gapless sound waves. 11

12 These can be understood in elasticity theory, and the continuum derivation of the important dispersion relation ω = vq, where v is the speed of sound, is given on pp of reference [1]. This description works for both solids and liquids, while the microscopic description based on normal modes of a collection of oscillators in section 3.3 applies only to solid. In this treatment you get, in addition to the linear acoustic branch, also so called optical branches where ω(q = 0) 0. 23: Note that in this paragraph no argument is given for why one should identify the superfluid with a BEC. This connection will be made later, and you will see that for a dense, and strongly interacting, system like HeII, the condensate fraction can be small even at T = 0 where the normal component, ρ n is zero and the supercomponent ρ s makes up the whole liquid, i.e. ρ s = ρ. 24: The condition of potential flow also implies that the flow is irrotational, v = 0, which is a necessary condition, if the fluctuations in the density is to be small (why?). Later we shall relax this condition and study vortices in superfluids. 1.6 A weakly interacting almost degenerate Bose gas The final step in connecting the phenomenology of superfluids to the presence of a BEC is to derive this Hamiltonian (36) from some realistic bose system. The simplest, but still very important, case is that of an almost perfect bose gas with weak repulsive interactions (note that reference [3] use the term ideal rather than perfect ). This is treated in 25, where the very important Bogolyubov transformation is introduced. The important result of this section is the Hamiltonian (25.12) with the sound velocity (25.11). Notice that in the derivation of this result it was crucial to assume that one single particle level was macroscopically populated, i.e. the presence of a BEC. This section is among the more difficult ones, and I shall try to explain the important steps in class. In 26, finally, we connect to the previous discussion of the definition of a BEC in section and A final comment on real systems. As described in 22, the real excitaion spectrum of Helium II has not only a photon, but also a roton branch. This cannot be understood from the perturbative treatment of the weakly interacting Bose gas. Historically this did in fact make some people doubt that the λ-transition in Helium really had to do with the formation of a BEC. To my knowledge, there is even today no fully satisfactory treatment of HeII, but 12

13 there is a very compelling treatment, due to R. P. Feynman, who used variational arguments to obtain the roton branch of the the dispersion relation[7]. The BECs in cold trapped atomic clouds are however very well described by the above perturbative treatment, suitably modified to take into account the shape and size of the cloud[6]. 13

14 References [1] D. L. Goodstein, States of Matter, Dover Publications, [2] P. W. Andersson, Basic Notions of Condensed Matter Physics, Benjamin- Cumming, [3] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2; volume 9 of Landau and Lifshitz, Course of Theoretical Physics, Pergamon Press, [4] J. J. Sakurai, Modern Quantum Mechanics, Rev. ed., Addison Wesley, [5] A. J. Legget, Bose-Einstein condensation in the alkali gases: Some fundamental concepts, Rev. Mod. Phys., 73, 307 (2001). [6] See e.g. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari Rev. Mod. Phys. 71, (1999) [7] See e.g. chapter 11 in R. P. Feynman, Statistical Mechanics W. A. Benjamin,

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