Theory of excitations of the condensate and non-condensate at finite temperatures

31 Pages·0.3 MB·English

by  Allan Griffin

Theory of excitations of the condensate
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9 and non-condensate at finite temperatures
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A. Griffin
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a
Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada
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v
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7
1
1
0
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t
a Summary. — We give an overview of the current theory of collective modes in
m
trapped atomic gases at finite temperatures, when the dynamics of the condensate
- and non-condensate must both be considered. A simple introduction is given to
d
the quantum field formulation of the dynamics of an interacting Bose-condensed
n
system, based on equations of motion for the condensate wavefunction and single-
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particle Green’s functions for the non-condensate atoms. We discuss the nature
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: of excitations in the mean-field collisionless region, including the Beliaev second-
v
order approximation for the self-energies. We also sketch the derivation of coupled
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X two-fluid hydrodynamic equations using a simple kinetic equation which includes
collisions between condensate and non-condensateatoms.
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PACS 03.75.Fi, 05.30.Jp, 67.40.-w. – .
1. – Introduction
There are several excellent review articles on trapped Bose-condensed atomic gases
written at a relatively introductory level [1, 2], with emphasis on the dynamics of the
condensateatT =0. Inthe presentlectures,Iwillconcentrateonthe interplaybetween
the condensate and non-condensate components. This topic requires a more sophisti-
cated analysis based on the concepts and methods of many body theory. My lectures
will attempt to give a very basic introduction to this kind of approach. The primary
(cid:13)c Societa`ItalianadiFisica 1
2 A.GRIFFIN
audience I have in mind are graduate students and postdocs coming from atomic and
laserphysics,ratherthancondensedmattertheory. However,Ihopetheexpertswillalso
learnsomethingfromtheselectures. Inhislectures,Fetter[3]hasgivenadetailedreview
of excitations in a trapped dilute Bose gas at T =0. He shows that a convenient way of
discussing these excitations is to start from the time-dependent Gross-Pitaevskii (GP)
equation for the macroscopic wavefunction Φ(r,t). Linearizing around the equilibrium
value Φ (r), one finds of the macroscopic wavefunction are given by the well-knownBo-
0
goliubovcoupledequations. ForauniformBose-condensedgas,theexcitationfrequencies
are those first discussed by Bogoliubov in 1947 [4].
AtT =0,one canassumethatallthe atomsareinthe Bosecondensatedescribedby
Φ(r,t). Incontrast,inthese lectures,mymaintopicwillbe toreviewourunderstanding
of what excitations are in a Bose-condensed gas at finite temperatures, when there is
a large number of atoms in the non-condensate (in trapped gases, the non-condensate
is often referred to as the “thermal cloud”). I will make some contact with the ideas
discussed in the lectures by Burnett [5].
The first half of these lectures (Sections 2 - 7) deals with excitations whose very ex-
istence depends onself-consistentmeanfields (ofvarious kinds!), ratherthan onthe col-
lisions between atoms. In the standardlanguage developed in condensed matter physics
in the 1960’s, this means the excitations are in the “collisionless region”.
Thesecondhalfofthelectures(Sections8-12)dealswithexcitationsinthecollision-
dominated hydrodynamic region. I review the two-fluid hydrodynamic equations such
as given by Landau [6], generalized to include a trap. I give an explicit microscopic
derivation of such two-fluid equations in a trapped Bose gas. This extends recent work
ofZaremba,GriffinandNikuni(ZGN)[7]toincludethe casewhenthe condensateisnot
yet in local equilibrium with the non-condensate atoms.
The type of questions I want to address in these lectures include:
1. What is the difference between an elementary excitation and a collective mode?
2. Can we isolate the dynamical role of the condensate on the nature of the excita-
tions?
3. At T = 0, with a pure condensate, excitations in a gas must be oscillations of
the condensate. In contrast, above T , the excitations are not related to a
BEC
condensate. What happens to an excitation as we go from T = 0 to T > T .
BEC
How does the excitation get rid of its “condensate” dressing?
4. Whatisthe essentialphysicsbehindthedifferentmean-fieldtheoriesofexcitations
whichhavebeendiscussedinthe recentliterature: Gross-Pitaevskii,Hartree-Fock-
Popov, Hartree-Fock-Bogoluibov?
5. WhatisthephysicsbehindthedreadedBeliaevsecond-orderapproximation? This
is the first approximation which includes damping of the elementary excitations
even at T =0 [8].
THEORYOFEXCITATIONSATFINITETEMPERATURES 3
6. WhyaretheexcitationsandcollectivemodesinaBose-condensedsystemuniquely
interesting,comparedtoallothermanybodysystems? Thekeyreasonis,ofcourse,
that the condensate couples and hybridizes single-particle excitations with density
fluctuations. AboveT , these two excitationbranchesare uncoupled. Once this
BEC
is understood, one sees why more detailed and systematic experimental studies of
excitations in trapped atomic gases at finite temperature are of great importance.
To give a careful discussion of all these questions would need 10 lectures. In these 3
lectures, I will give a speeded-up version. I will often use a uniform weakly interacting
Bose gas to illustrate the structure of the theory. While I will only sketch the math, I
will still try to give a flavour of what is involved. The approach I will use to discuss
these questionsis basedonthe field-theoreticformulationofaBose-condensedsystemof
particles. AsIreviewelsewhereinthisvolume[9],thispowerfulformalismwasintroduced
byBeliaev[10]in1957andextensivelydevelopedintheGoldenPeriod: 1958-1965. Iwill
introducethisformalisminaveryschematicmanner-butevenexperimentalistswillfind
itusefultoknowsomeofthe“language”usedinthisapproach. Thereareothermethods
to deal with collective modes in Bose gases but they are not as useful at isolating the
dynamical role of the condensate, or dealing with finite temperatures.
While I will always have trapped atomic Bose gases in mind, much of the general
theory [11, 12] is valid for any Bose-condensed fluid (gas or liquid). Thus I will often
make references to superfluid 4He, pointing out similarities with Bose gases.
2. – Elementary excitations and density fluctuations in normal systems
We work with quantum field operators :
ψˆ(r) =destroys an atom at r
(1) ψˆ+(r) =creates an atom at r.
These operators satisfy the usual Bose commutation relations, such as [ψˆ(r),ψˆ+(r′)] =
δ(r r′). Of course, if we have several hyperfine atomic states, then we have different
−
field operators ψˆ (r), where a is the hyperfine state label (the analogue of a spin label).
a
Allobservablescanbewrittenintermsofthesequantumfieldoperators. Twoimportant
examples are the local density operator:
(2) nˆ(r)=ψˆ+(r)ψˆ(r)
and the Hamiltonian
2 1
(3) Hˆ = drψˆ+(r) −∇ +U (r) µ ψˆ(r)+ g drψˆ+(r)ψˆ+(r)ψˆ(r)ψˆ(r).
ext
2m − 2
Z (cid:20) (cid:21) Z
In these lectures, the two-particle interaction will always be approximated by a s-wave
scatteringlength,appropriatetoadiluteBosegasatverylowtemperatures[5,13]. Thus
v(r r′)=gδ(r r′), with g = 4πa (we set ¯h=1 throughout this article).
− − m
4 A.GRIFFIN
Whatwewanttocalculatearevariouskindsofcorrelationfunctionsinvolvingdifferent
local operators:
′ ′ ′ ′
nˆ(r,t)nˆ(r,t) χ (rt,rt)
nn
h i∼
′
= χ (1,1) density response function
nn
≡
ˆj (r,t)ˆj (r′t′) χ (rt,r′t′)
h α β i∼ jαjβ
′
(4) = χ (1,1) current response function.
jαjβ ≡
In these tutorial lectures, I will not bother to distinguish between the various kinds of
correlationfunctions(time-ordered,retarded,etc)oreventhedifferencebetweencorrela-
tion functions and responsefunctions. These arediscussedinall the standardtextbooks
on many body theory [12, 14] but are only important when one is doing systematic cal-
culations. A convenient summary of the formalism is given in a recent review article on
homogeneous weakly interacting Bose gases [15].
The density response function
(5) χ (1,1′)= ψˆ+(1)ψˆ(1)ψˆ+(1′)ψˆ(1′)
nn
h i
involvesfourquantumfieldoperatorsandisanexampleofatwo-particleGreen’sfunction
′ ′
G (1,1). One can measure χ (1,1) by coupling a weak external field to the local
2 nn
density in the system
′
(6) H (t)= drV (r,t)nˆ(r) d1V (1)nˆ(1).
ex ex
≡
Z Z
Linear response theory gives
δn(1) nˆ(r) nˆ(r)
t eq
≡ h i −h i
′ ′ ′
(7) = d1χ (1,1)V (1)+...
nn ex
Z
forthedensityresponse(seeChapters5and9ofRef.[12]andChapter2ofRef.[14]). For
a uniform system with V (r,t)=V ei(q·r−ωt), we have
ex q,ω
′ ′
χ (1,1) =χ (1 1)
nn nn
−
′ ′
=χ (r r,t t).
nn
− −
Fourier transforming (r r′) q and (t t′) ω, the linear response equation in (7)
− → − →
reduces to
(8) δn(q,ω)=χ (q,ω)V .
nn q,ω
THEORYOFEXCITATIONSATFINITETEMPERATURES 5
If χ (q,ω) 1 has a pole atω =E , (for further details, see Section 3) then when
nn ∼ ω−Eq q
ω and q of the external potential satisfy ω = E , δn(q,ω) as given by (8) can be very
q
large even though V is small. We note that
q,ω
1
(9) χ (q,ω) χ (q,t) e−iEqt
nn nn
∼ ω E → ∼
q
−
and thus clearlythe pole at ω =E is the signature ofan oscillating density fluctuation.
q
The basic correlation function in the field-theoretic approachis given by
(10) ψˆ(1)ψˆ+(1′) G (1,1′).
1
h i∼
This single-particle Green’s function G involves two quantum field operators. It de-
1
scribescreatinganatomat1′ =r′,t′,letitpropagatethroughthesystemto1=r,tand
′
then destroying the atom. All other higher-order correlation functions such as G (1,1)
2
can be constructed out of combinations of G (1,1′). For example, the lowest order con-
1
tributions to the density response function in (5) are
χ (1,1′)= ψˆ+(1)ψˆ(1)ψˆ+(1′)ψˆ(1′)
nn
h i
ψˆ+(1)ψˆ(1) ψˆ+(1′)ψˆ(1′) n(1) n(1′)
≃h ih i→h ih i
+ ψˆ+(1)ψˆ(1′) ψˆ+(1′)ψ(1) G (1,1′)G (1′,1)
1 1
h ih i→
(11) + ψˆ+(1)ψˆ+(1′) ψˆ(1)ψˆ(1′) +...
h ih i
As we discuss in Section 4, the terms in the last line of (11) vanish in a normal Bose
system but are finite for T < T . The poles of G correspond to what are called
BEC 1
elementary excitations (or quasiparticles). In a uniform system, we have
1
′
G (1,1) G (q,ω)
1 → 1 ∼ ω Esp
q
−
q2
(12) G1(q,t)∼e−iEqspt; in a free gas, we have Eqsp = 2m ≡ǫq.
One can show [12] that these single-particle excitations determine the thermodynamic
properties of interacting systems. However, it is very difficult to directly measure the
spectrumofG (1,1′)sinceoneneedsanexternalfieldwhichcouplestoψˆ(1),ie,anatom
1
reservoir. Later we will see that what makes a Bose-condensedsystem unique is that we
′
can easily access G (1,1) as a result of the effects of the Bose condensate.
1
Finally we introduce the key idea of a single-particle self-energy through Dyson’s
equation:
(13) G =G +G ΣG ,
1 0 0 1
6 A.GRIFFIN
where G is the interacting single-particle Green’s function, G is the non-interacting
1 0
single-particle Green’s function and all effects of the two-particle interactions are con-
tained in the self-energyfunction Σ. In a uniformsystem, we canFourier transformthis
Dyson equation to give
(14) G (q,ω)=G (q,ω)+G (q,ω)Σ(q,ω)G (q,ω),
0 0 0 1
where G (q,ω)= 1 . This is easily solved to give
0 ω−ǫq
1
(15) G (q,ω)= .
1
ω [ǫ +Σ(q,ω)]
q
−
Thus we see that G may have a single-particle pole at the quasiparticle energy
1
(16) Esp =ǫ +Σ(q,Esp).
q q q
In general, Σ(q,ω) = Σ +iΣ , where Σ describes the damping of the single-particle
R I I
excitations.
Field-theoretic calculations [11, 12] involve a systemmatic (diagrammatic) procedure
to calculate Σ(q,ω) and from this to obtain G (q,ω). We note that the self-energy Σ is
1
by definition highly non-perturbative. As an illustration, let us consider the self-energy
to first order in g. This Hartree-Fock approximation is shown in Fig.1. In our s-wave
approximation, the total self-energy is simply
(17) Σ =gn+gn=2gn,
HF
and therefore (15) gives
1
(18) G(q,ω)= .
ω [ǫ +2gn]
q
−
Hence the normal HF excitation energy has the dispersion relation
q2
(19) EHF = +2gn.
q 2m
3. – Density fluctuation spectrum in the mean-field approximation
ThesimplestapproximationforthedensityresponsefunctionintroducedinSection2
is [11, 12]
(20) χ (1,1′) χ0 (1,1′)=G (1,1′)G (1′,1).
nn ⇒ nn 1 1
THEORYOFEXCITATIONSATFINITETEMPERATURES 7
In a uniform Bose system, the Fourier transform of this gives
N0(ω ) N0(ω )
(21) χ0 (q,ω) dk dω dω 1 − 2 A(k,ω )A(k q,ω ).
nn ∼ 1 2 ω ω ω 1 − 2
Z Z Z (cid:2) 1− 2− (cid:3)
HereN0(ω)=(eβω 1)−1 istheBosedistributionfunctionandA(k,ω) single-particle
− ≡
spectral density 2ImG (k,ω+i0+). If we use A (k,ω) 2πδ(ω EHF) as givenin
∼ 1 HF ∼ − k
(18), we find (21) reduces to
N0(EHF) N0(EHF)
(22) χ0 (q,ω) dk k − k−q .
nn ∼ (EHF EHF) ω
Z k − k−q −
We note that χ0 has a continuum of poles given by ω = (EHF EHF). It is easy
nn k − k−q
to understand the physics which gives rise to this “ideal-gas” spectrum. The Fourier
transform of the local density operator in (2) is given by (for a uniform system)
(23) nˆq = aˆ+kaˆk−q
k
X
Clearly nˆ creates a “particle-hole” density fluctuation with the following features:
q
change in energy:Ek Ek−q =ω
−
(24) change in momentum:k (k q)=q.
− −
The spectrum(22)describes abroadincoherentsuperpositionofparticle-holestates[14]
and is not a true collective mode, such as we discuss next.
The mean field approximation (MFA) for the density response is also called other
names: SCF (self-consistent field), RPA (randon phase approximation), but all involve
thesamephysics. TheMFAwasintroducedbyBohmandPinesintheperiod1951-1953,
workwhichhadapivotaleffectinourunderstandingcollectiveeffectsinallmany-particle
systems. We recall the linear response expression in (7), where χ is the full density
nn
response function for interacting Bose gas. If we introduce the self-consistent Hartree
mean-field:
′ ′ ′
(25) δV (1)= d1v(1 1)δn(1)=gδn(1),
Hartree
−
Z
then we can approximate the linear response equation in (7) by
(26) δn(1)= d1′χ0 (1,1′)[V (1′)+δV (1′)].
nn ex Hartree
Z
Thesystemisassumedtorespondasiftheatomspropagateindependently(asdescribed
by χ0) but are moving in an effective field V (1′). In a uniform system, the Fourier
eff
8 A.GRIFFIN
transform of (26) is
(27) δn(q,ω)=χ0 (q,ω)[V +gδn(q,ω)],
nn qω
which gives the well-known MFA expression for χ [12, 14]:
nn
χ0 (q,ω)
(28) χ (q,ω)= nn .
nn 1 gχ0 (q,ω)
− nn
This result for χ (q,ω) may have new poles given by zeros of the denominator,
nn
(29) 1 gχ0 (q,ω)=0.
− nn
Assuming thatitisdistinguisablefromthe incoherentidealgasdensity fluctuationspec-
trum given by (22), this pole at ω = Ecoll is called a zero sound mode (a plasmon in
q
charged systems). This language was first introduced in 1957 by Landau in Fermi sys-
tems [14] but the concept is generally applicable in any interacting many body system.
Physically, it is clear that zero sound is a “collisionless” density oscillation arising from
dynamic self-consistent mean fields.
We make a few comments about such “zero sound” collective modes:
1. Ecoll andEsp (thepolesofG (q,ω))arebothstatesofaninteractingmany-particle
q q 1
system. However the collective mode disappears if there are no interactions, while
the single-particle excitations still exist in a non-interacting gas.
2. Because of the low density, dynamic mean fields in normal systems are too weak
to allow the existence of a well-defined (weakly-damped) zero sound mode in the
Bose gases of current interest (for T > T ). If gχ0 (q,ω) 1, we can then
BEC nn ≪
approximate χ (q,ω) χ0 (q,ω). The situation is quite different for T <T ,
nn ≃ nn BEC
when a coherent mean-field due to the condensate is present.
3. Onealsoexpectsacollectivepoletoappearinχ (q,ω)inthecollision-dominated
nn
hydrodynamic region. However this sound wave pole is induced by rapid collisions
producing local equilibrium, as we discuss in Sections 8 - 12. Ordinary sound is
notdescribedbythe MFAdiscussedabove,ie,itisnotthe resultofdynamic mean
fields.
4. – Green’s function formulation of excitations in a Bose-condensed system
AlltheformalismwehavebeendiscussingforT >T canbeextendedinanatural
BEC
waytoBose-condensedsystems[10,11,12]makinguseofthefundamentaldecomposition
which separates out the condensate and non-condensate parts of the quantum fields:
(30) ψˆ(r)= ψˆ(r) +ψ˜(r).
h i
THEORYOFEXCITATIONSATFINITETEMPERATURES 9
Heretheaverageisoverarestrictedensemble[16,17]consistentwith ψˆ =0. Themost
h i6
profound way of doing this is to add a symmetry-breaking field
(31) Hˆ (t)= dr[η(r,t)ψˆ+(r)+H.C.]
sb
Z
and work with Hˆ =Hˆ +Hˆ , taking the limit η 0 at the end. This gives the
tot system sb
→
system a “hunting license” [16] to have finite value of ψˆ and one finds
h i
(32) ψˆ =0 for T >T ; ψˆ =0 for T <T .
sb BEC sb BEC
h i h i 6
Akeypointisthatif ψˆ =0,thenadirectconsequenceistheexistenceof“anomalous”
sb
h i 6
or “off-diagonal” propagators[10]
ψ˜(1)ψ˜(1′) =0
sb
h i 6
(33) ψ˜+(1)ψ˜+(1′) =0.
sb
h i 6
These describe new condensate-induced correlations between non-condensate atoms at
′
different space-time points 1 and 1. In a sense, these anomalous correlation functions
are as important as the macroscopic wavefunction Φ(1). We also note that m˜(1)
≡
ψ˜(1)ψ˜(1) is the “pair” function that Burnett [5] discusses in detail, using a different
sb
h i
formalism. ¿Fromnowon,weleavethesymmetry-breakinglabelontheaveragesimplicit.
In addition, if ψˆ = 0, we find that correlation functions involving three non-
h i 6
condensate field operators can be finite
(34) ψ˜(1)ψ˜(2)ψ˜(3) =0.
h i6
In particular, n˜(1)ψ˜(1′) = 0. This describes the condensate-induced coupling of non-
h i 6
condensate density fluctuations n˜ = ψ˜+ψ˜ and the single-particle field fluctuations (see
Section 7).
Clearly one has to work with a 2 2 matrix single-particle propagator G when
1
×
ψˆ =0, namely
h i6
ψˆ(1)ψˆ+(1′) ψˆ(1)ψˆ(1′)
Gαβ = hψˆ+(1)ψˆ+(1′i) hψˆ+(1)ψˆ(1i′) !
h i h i
Φ(1)Φ∗(1′)+G˜ Φ(1)Φ(1′)+G˜
(35) = 11 12 ,
Φ∗(1)Φ∗(1′)+G˜ Φ∗(1)Φ(1′)+G˜
(cid:18) 21 22 (cid:19)
where we haveintroduceda 2 2 matrix Green’s function for the non-condensateatoms
×
G˜ G˜
(36) G˜ 11 12 .
αβ ≡ G˜ G˜
(cid:18) 21 22 (cid:19)
10 A.GRIFFIN
Beliaev [10] effectively showed (in modern matrix notation)
(37) G˜ =G δ +G Σ G˜ ,
αβ 0 αβ 0 αδ δβ
where we use the standard conventionthat repeated indices (δ =1,2)are summed over.
This is the famous Dyson-Beliaev equation, involving a 2 x 2 matrix self-energy Σ .
αδ
Clearly all components G˜ will share the same single-particle excitation spectrum. In
αβ
a uniform system, we have
(38) G˜ (q,ω)=G (q,ω)δ +G (q,ω)Σ (q,ω)G˜ (q,ω),
αβ 0 αβ 0 αδ δβ
which is a set of linear algebraic equations which are easy to solve for G˜ ( G˜ ) and
11 22
∼
G˜ ( G˜ ).
12 21
∼
We will now use this Beliaev formalism to discuss various simple approximations for
the self-energies Σ . The interaction energy in (3), namely
αβ
1
(39) V = g drψˆ+(r)ψˆ+(r)ψˆ(r)ψˆ(r),
int
2
Z
splitsintovariousdistinctcontributionswhenweuse(30)toseparateoutthecondensate
parts (see Fig. 2). At T =0, we can ignore the V and V contributions because so few
3 4
atomsareinnon-condensate. Thisisthe famousBogoliubovapproximation[4],anditis
equivalenttothe linearizedGPtheory. InBeliaevlanguage,the Bogoliubovself-energies
are shown in Fig. 3. For a uniform gas, these give [10, 12]
ω+ǫ +n g
G˜ (p,ω)= p c
11
ω2 ǫ2+2ǫ n g
− p p c
n g
(40) G˜ (p,ω)= (cid:2)− c (cid:3) .
12
ω2 ǫ2+2ǫ n g
− p p c
(cid:2) (cid:3)
TheseBogoliubovsingle-particleGreen’sfunctionscontainthesamephysicsasdiscussed
in Fetter’s lectures [3]. They clearly have poles at the frequencies ω = E , where
p
±
(41) E =(ǫ2+2ǫ n g)1/2.
p p p c
At low p, the single-particle excitation is phonon-like E = c p, with the Bogoliubov
p Bog
phonon velocity
n g
(42) c2 = c .
Bog m
It turns out that this simple T = 0 Bogoliubov theory already exhibits most of the
structure which will always arise in Bose-condensed systems. This is why it plays the
role of the “H-Atom” in discussions of Bose-condensed fluids [11].