Atom chips and one-dimensional Bose gases

1.2 MB·English

by  Isabelle Bouchoule

Atom chips and one-dimensional Bose gases
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2 I. Bouchoule1, N.J. van Druten2, C. I. Westbrook1

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J 1: Laboratoire Charles-Fabry, CNRS UMR 8501, Institut d’Optique, Palaiseau, France

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2: Van der Waals-Zeeman Instituut, Universiteit van Amsterdam, The Netherlands
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] January 26, 2009
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o Contents
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1 Introduction 2
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y 2 Regimes of1D gases 2
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2.1 Stronglyversusweaklyinteractingregimes . . . . . . . . . . . . . . . . . . . . 4
p
[ 2.2 Nearly idealgasregime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 2.3 Quasi-condensateregime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
v 2.3.1 Densityfluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3
2.3.2 Phase fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
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3 2.4 Exactthermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3
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1 3 1D gasesinthe real world 16
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3.1 Transversetrappingandnearly 1D Bosegases . . . . . . . . . . . . . . . . . . . 16
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0 3.2 Applying1D thermodynamicstoa3D trappedgas . . . . . . . . . . . . . . . . . 17
:
v 3.3 Longitudinaltrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
i
X 3.3.1 Local densityapproximation . . . . . . . . . . . . . . . . . . . . . . . . 18
r 3.3.2 Validityofthelocal densityapproximation . . . . . . . . . . . . . . . . 19
a
3.4 3Dphysicsversus1D physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Experiments 21
4.1 FailureoftheHartree-Fock model . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Yang-Yanganalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3 Measurementsofdensityfluctuations . . . . . . . . . . . . . . . . . . . . . . . 23
4.3.1 A local densityanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3.2 Ideal gas regime: observationofbunching . . . . . . . . . . . . . . . . 25
4.3.3 Quasi-condensateregime: saturationofatomnumberfluctuations . . . . 28
5 Conclusion 29
1
6 Acknowledgments 29
1 Introduction
As this volumeindicates, the technology of atom chips is currently enjoying great success for a
large variety of experiments on degenerate quantum gases. Because of their geometry and their
ability to create highly confining potentials, they are particularly well adapted to realizing one
dimensional (1D) situations [1–9]. This characteristic has contributed to a revival of interest in
thestudyof1DBosegaseswithrepulsiveinteractions,asystemwhichprovidesavividexample
ofanexactlysolvablequantummany-bodysystem[10–12]. Thequantummany-bodyeigenstates
[10,11] and thermodynamics [12] can be calculated without resorting to approximations. In
addition, the 1D Bose gas shows a remarkably rich variety of physical regimes (see Fig. 1)
that are very different both from those found in 2D and in 3D. One dramatic example of the
difference is the tendency for a 1D Bose gas to become more strongly interacting as its density
decreases [10]. Finally, and in a more practical vein, a good understanding of its behavior is
relevantfor guided-waveatomlasers [13]and trapped-atom interferometry [14]. Because ofthe
effects of interactions,theanalogyto themanipulationof lightinsinglemodefibers needs to be
examinedcarefully.
An atom chip is not the only means of producing a 1D Bose gas. Optical trapping has been
used to generate similarly elongated trap geometries. In particular, a 2D optical lattice can be
usedtogeneratea2Darrayof1Dtraps[15–19]. Becauseofthemassivelyparallelnatureofthis
system, it is possible to work with only a few atoms per tube, and still get a sizeable signal per
experimentalcycle. Thus, the stronglyinteracting regimealluded to abovecan be reached. This
regimehasyet tobereached withan atomchip. But aswewillshowhere, akeyfeatureofatom
chips is that they produce individual samples in which one does not intrinsically average over
many realizations. Fluctuation phenomena are therefore readily accessible, an aspect which we
willtreat laterinthischapter.
In the following we first give an introduction to the various regimes of the homogeneous
1D Bose gas, with particular emphasis on the behavior of the density profiles and the density
fluctuations in the context of approximate models. Then we will discuss the exact solution and
how it differs from the approximations. Next, we discuss some of the important issues involved
in realizing 1D gases in a 3D trap. Finally, we describe a series of experiments performed in
Orsay and Amsterdamusingatomchips toexploreand illustratefeatures ofthe1D Bosegas.
2 Regimes of one-dimensional gases
First,wereviewsometheoreticalresultsconcerningtheone-dimensionalBosegaswithrepulsive
interactions. Mostoftheseresultsare derivedin Refs. [10,12,20–24]. Here we willconcentrate
onintuitivearguments,andthereaderisreferredtotheabovereferencesformorecarefuldemon-
2
strations. Thesystemisdescribed bytheHamiltonian
~2 ∂2 g
H = dzψ+ ψ + dzψ+ψ+ψψ, (1)
−2m ∂2 2
Z z Z
whereψisthefieldoperatorinsecondquantization,andgisthecouplingconstantcharacterizing
the interactions between particles. From this coupling constant, one can deduce an intrinsic
lengthscalerelated totheinteractions,
~2
l = , (2)
g
mg
as wellas an energy scale:
mg2 ~2
E = = . (3)
g 2~2 2ml2
g
In thermal equilibrium,the gas is described by the temperatureT and the linear atomicden-
sityn. Rescalingthesetwoquantitiesbytheintrinsicscalesintroducedabove,andsettingBoltz-
mann’s constant equal to unity (i.e., measuring temperature in units of energy) we find that the
propertiesofthegasare functionsofthedimensionlessquantities
T
t = , (4)
E
g
and
mg 1
γ = = , (5)
~2n nl
g
thelatterbeing thefamousLieb-Linigerparameter [10].
It is useful to also introduce two other relevant scales, namely the thermal de Broglie wave-
length,
2π 4π
λ = ~ = l , (6)
dB g
mT t
r r
and thequantumdegeneracy temperature
~2n2 E
T = = g. (7)
d 2m γ2
In the above (t,γ) parametrization, quantum degeneracy (T T , or equivalently nλ 1) is
d dB
≈ ≈
reached around
1
t . (8)
≈ γ2
ThethermalequilibriumforthehamiltonianofEq.(1)hasbeenextensivelystudiedtheoreti-
cally[12,22]. Withoutgoingintogreatdetailhowever,wecanpresentsomeimportantfeaturesof
thissystem. Severalregimesmaybeidentifiedintheparameterspace(γ,t),assketchedinFig.1.
We begin by noting that the region γ 1, t 1 (dark grey area) defines a strongly interacting
≫ ≪
3
Figure1: Physicalregimesofa1DBosegaswithrepulsivecontactinteractionsintheparameter
space (γ,t), adapted from [22]. The dashed diagonal line separates the degenerate and nonde-
generate gases. The strongly interacting regime is shown in dark grey. The weakly interacting
regimeisdividedintothenearlyidealgasregime(alsocalleddecoherentregime)showninwhite
and the quasicondensate regime shown in light grey. Note that the nearly ideal gas can be de-
generate. The quasicondensate regime is divided into the thermal and quantum regimes. The
linesrepresentsmooth(andoftenwide)crossoversratherthanphasetransitions. Thecrossovers
are given in Eqs. (10), (11), (8), (25) and (41). The dashed area shows the parameter space
investigatedin theexperiments presentedin thischapter.
regimethat occurs at low density and low temperature, often referred to as the Tonks-Girardeau
gas[20,25,26].
In the weakly interacting regime, γ < 1, several sub-regimes are identified. These are the
regimes which to date have been accessible in atom chip experiments, and we shall elaborate
further on their nature in the discussion below. The two main regimes are the nearly ideal gas
regime (white area) and the quasi-condensate regime (light grey area). Each one permits an
approximate description that we present later in this section and which allows the identification
of sub-regimes. For the moment we simply wish to emphasize that no phase transition occurs
in the 1D Bose gas and that all the boundaries represent smooth (and often broad) crossovers in
behavior.
2.1 Strongly versus weakly interacting regimes
We first comment on the distinction between strong and weak interactions. Following the ap-
proach of Ref. [20], we study the scattering wave function of two atoms interacting via the
potentialgδ(z z ),wherez andz arethepositionofthetwoatoms. Forthis,weconsiderthe
1 2 1 2
−
4
wavefunctionψ inthecenter-of-massframe,withreducedmassm/2andsubjecttothepotential
gδ(z). Theeffect ofthepotentialisdescribed bythecontinuitycondition
∂ ∂ mg
ψ(0 ) ψ(0 ) = ψ(0) (9)
∂z + − ∂z − 2~2
where 0 (0 ) denotes the limit when z goes to zero through positive (negative) values. Let us
+
consider the−scattering solutionfor an energy E = ~2k2/m. Since we considerbosons, we look
for even wave functions of the form cos(k z + φ). The continuity conditions give φ and thus
| |
thevalueψ(0). Wefind then thattheenergy E givenby Eq.(3)is therelevantenergy scaleand
g
thatfor E E , ψ(0)is closeto zero, while, forE E , ψ(0)is closetoone, as illustratedin
g g
≪ ≫
Fig. 2.
The above results hold for a gas of particles since the continuity relation (9) holds for the
many-body wavefunction when two atoms are close to the same place. Thus, as long as the
typical energy of the particles is much lower than E , the many-body wavefunction vanishes
g
when twoparticlesare at thesameposition: thegasisthen inthestronglyinteracting,orTonks-
Girardeau regime. The vanishingof the wave function when two particles are at the same place
mimics the Pauli exclusion principle and the gas acquires some similarities with a gas of non
interacting fermions. More precisely, in this strong interaction regime, the available wave func-
tions of the many body problem are, up to a symmetrization factor, the wave functions of an
ideal Fermi gas [26]. Since the wave function vanishes when two atoms are at the same place,
the energy of the system is purely kinetic energy and the eigen energies are those of the Fermi
system. Thus the 1D strongly interacting Bose gas and the ideal 1D Fermi gas share the same
energy spectrum. This implies in particular that all thermodynamic quantities are identical for
bothsystems.
To identify the parameter space of the strongly interacting regime, we suppose the gas to be
stronglyinteracting and then requirethat thetypical energy ofthe atoms be smallerthan E . To
g
estimate the typical energy per atom, we use the Bose-Fermi mapping presented above. If the
gasisdegenerate,thetemperatureissmallerthanthedegeneracytemperatureT ,Eq.(7),andT
d d
corresponds to the "Fermi" energy of the atoms. The typical atom energy is therefore T and it
d
isoforderE if
g
γ 1. (10)
≃
The strongly interacting regime thus requires γ 1. If the gas is non degenerate, the typical
≫
energy oftheequivalentFermi gasisT and interactionsbecomestrongwhen T = E or
g
t 1. (11)
≃
Wethen find thatthegasisstronglyinteractingfort 1.
≪
The condition (10) is often derived using the following alternative argument, valid at zero
temperature. At zero temperature, thereare twoextremes forthepossiblesolutionsforthewave
function ψ(z ,z ,...). As seen in Fig. 2, either the wave function vanishes when two atoms
1 2
are at the same place, or the wave function is almost uniform, corresponding to the strongly
and weakly interacting configurations respectively. In the weakly interacting configuration, the
kineticenergyisnegligibleandtheinteractionenergyperparticle,oftheorderofgn,determines
5
Figure 2: Strong interaction versus weak interaction regime. We show the wave function in the
center-of-massframeoftwoatomsfor(a)stronginteractions,scatteringenergyE muchsmaller
than E = mg2/2~2 and (b) weak interactions, E much larger than E . We also plot the wave
g g
functionψ(z ,z ,z ,...)forgivenpositionsofz ,z ,...in(c)thestronglyinteractingregimeand
1 2 3 2 3
(d)theweaklyinteractingregime.
the total energy. In the strongly interacting configuration, on the other hand, the interaction
energy vanishes while the typical kinetic energy per particle is ~2n2/m. Comparing these two
energies, wefind thatthestronglyinteractingconfiguration isfavorableonlyforγ > 1.
2.2 Nearly ideal gas regime
At sufficiently high temperatures, interactions between atoms have little effect and the gas is
well described by an ideal Bose gas. In Ref. [22], thisregimewas referred to as the"decoherent
regime";Wewillcallitthe(nearly)idealBosegasregime. A1DidealBosegasatthermalequi-
libriumiswelldescribedusingthegrandcanonicalensemble,introducingthechemicalpotential
µ. AllpropertiesofthegasarecalculatedusingtheBoltzmannlawwhichstatesthat,foragiven
one-particle state of momentum ~k, the probability to find N atoms in this state is proportional
toe (~2k2/(2m) µ)N/T;notethatµ < 0inthisdescription. Inthefollowing,weuseaquantization
− −
box of size L (tending to infinity in the thermodynamiclimit)and periodicboundary conditions
so that the available states are the momentum states with momentum k = 2πj/L where j is an
integer.
Let us first consider the linear gas density. From the Boltzmann law, we find that the mean
populationistheBosedistribution
1
n = . (12)
h ki e(~2k2/(2m) µ)/T 1
−
−
The atom number, and thus the linear density, is obtained by summing the population over the
statesand onefinds
1
n = g (eµ/T), (13)
1/2
λ
dB
whereg (x) is oneoftheBosefunctions
1/2
∞ xl
g (x) = , (14)
n
ln
l=1
X
6
also known as the polylogarithmicfunctions [27,28]. Unlike in 3D systems, where the excited-
state density is given by ρ = g (eµ/T)/λ3 in this approach [27], no saturation of the excited
e 3/2 dB
states occurs (the function g diverges as πT/µ as µ 0 from below, whereas g (1) =
1/2 3/2
− →
2.612 is finite): in the thermodynamic limit no Bose-Einstein condensation is expected and the
p
gasis welldescribed byathermal gasat anydensity.
Two asymptotic regimes may be identified: the non degenerate regime for which µ T
− ≫
and ~2n2/m T and the degenerate regime for which µ T and ~2n2/m T. In the non
≪ − ≪ ≫
degenerateregime,thelineardensityiswell approximatedbytheMaxwell-Boltzmannformula
1
n = eµ/T, (15)
λ
dB
In this regime nλ is much smaller than unity. In the degenerate regime, the states of energy
dB
muchsmallerthan T arehighlyoccupiedand thelineardensityisgivenby
T m
n = (16)
~ 2µ
r−
Thisdensityismuchlarger than1/λ , i.e. nλ 1.
dB dB
≫
As we willdiscussin theexperimentalsection, fluctuationsare also very importantforchar-
acterizing the gas. It is thus instructive to consider the correlation functions. The normalized
one body correlation function is g(1)(z) = ψ+(0)ψ(z) /n, where ψ is the field operator in
h i
the second quantization picture. Using the expansion of the field operator in the plane wave
basis ψ(z) = a e ikz/√L where a is the annihilation operator for the mode k, we find
k k − k
g(1)(z) = n e ikz/(Ln). Here n = a+a is the atom number operator for the mode k.
khPki − k k k
Simple analytical expressions are found in the nondegenerate and highly degenerate limits. In
P
thenondegeneratelimit( µ T or, equivalentlyn 1/λ ), wefind
dB
− ≫ ≪
2
z
g(1)(z) e−4πλ2dB. (17)
≃
As the gas becomes more degenerate, the correlation length increases and, in the degenerate
regime( µ T or, equivalentlyn 1/λ ), wefind
dB
− ≪ ≫
2πz
g(1)(z) e−mn~T2z = e−nλ2dB. (18)
≃
In thisregimethecorrelationlength, aboutnλ2 , is much largerthan thedeBrogliewavelength
dB
(and themeaninterparticledistance1/n)sinceλ 1/n.
dB
≫
Nextweconsiderthenormalizeddensity-densityortwo bodycorrelationfunction
g(2)(z) = ψ+(z)ψ+(0)ψ(0)ψ(z) /n2. (19)
h i
This function is proportional to the probability of finding an atom at position z and at position
z = 0. Itis givenby
n2g(2)(z) = a+a+a a eik1ze ik4z/L2. (20)
h k1 k2 k3 k4i −
k1Xk2k3k4
7
Using Bose commutation relations and the fact that, since atoms do not interact, different mo-
mentumstatepopulationsareuncorrelated, thesumsimplifiesto:
n2g(2)(z) = n n (1+ei(k1 k2)z)/L2 + a+a+a a /L2. (21)
h k1ih k2i − h k k k ki
kX16=k2 Xk
In the last term, the commutation relations give: a+a+a a = n2 n , and in thermal
h k k k ki h ki − h ki
equilibriumonehas:
n2 = n +2 n 2. (22)
h ki h ki h ki
Therefore wefind:
g(2)(z) = 1+ g(1)(z) 2, (23)
| |
a result which one can also obtain directly from Wick’s theorem [29]. Equation (23) means that
the probability of finding atoms within less than a correlation length in a thermal Bose gas is
twice that of finding two atoms far apart. This phenomenon is often referred to as "bunching"
andhasbeenobservedincoldatomsinseveralexperiments[30–32]. Bunchingiscloselyrelated
to density fluctuations. As one can see from Eq. (22), in a thermal gas, fluctuations in the occu-
pation of a single quantum state, δn 2 = n2 n 2, show a "shot noise" term, n and an
k h ki − h ki h ki
"excessnoise"term, n 2. Thedensityfluctuationexperimentdescribedlaterinthischapterhas
k
h i
demonstratedthisbehavior.
Validity of the ideal gas treatment. The two body correlation function has been used to
characterize the crossover between the ideal gas and quasi-condensate regimes [22]. When in-
teractions become important, they impose an energy cost on density fluctuations and the latter
tend to smoothout. This amountsto a reduction in the valueof g(2)(0). In the quasi-condensate
regime which we discuss in the next section, the bunching effect is absent and g(2)(0) is close
to unity. Theideal Bose gas description fails when the typical interactionenergy per particlegn
is not negligible compared to µ. Using Eq. (16) one finds that the ideal Bose gas description
−
fails when the temperature is no longer much smaller than the crossovertemperature, which we
defineas
T T √γ. (24)
co d
≃
Usingthereduced dimensionlesstemperaturet = T/E ,thiscan bewritten as
g
1
t . (25)
co
≃ γ3/2
Thislineseparates thenearly ideal gasregimefrom thequasi-condensateregimein Fig.1. Note
that, in terms of chemical potential, the domain of validity of the ideal gas model is µ µ
co
− ≫
wherewedefine thecrossoverchemical potentialas
T
µ = . (26)
co t1/3
In making this estimate, we have assumed that the gas is degenerate at the crossover. From
Eq. (24), one can see that if one is in the weakly interacting regime (γ 1) this assumption
≪
8
is indeed true. The experiments described below confirm that one can observe the effects of
degeneracy beforetheonsetofthereductionofdensityfluctuations.
Aprecursorofthereductionofdensityfluctationsisshownbyaperturbativecalculationvalid
inthenearly ideal gasregimewhichgives,to lowestorderin g [22],
g(2)(0) 2 4(T /T)2. (27)
co
≃ −
Toaccuratelytreatthecrossoverregimehowever,itisnecessarytomakeuseoftheexactsolution
tothe1D Bosegasmodel. Theexactsolutionin thecrossoverregimeis discussedin Sec. 2.4.
Thecorrelationlengthsofthegasareimportantparametersofthegasthatwillbeusedinthe
following to estimate the validity criteria of the local density approximation. In the degenerate
regime, the correlation length is l nλ2 (see Eq. (18)). Using Eq. (24), we find that, close to
c ≃ dB
thecrossover,thecorrelationlengthofthegasiscloseto thehealinglength
~
ξ = . (28)
√mgn
2.3 Quasi-condensate regime
On the other side of the crossover, i.e. for T T , the bunching effect is entirely suppressed
co
≪
and the g(2) function is close to unity for any z. This regime is the quasi-condensate regime1.
In this section, we present a description of the gas, valid in the quasi-condensate regime. This
descriptionpermitsasimpleestimateofthedensityfluctuations. Wethusverifyaposteriorithat
thequasi-condensateregimeisobtainedforT T . Wealsogiveasimplecalculationofphase
co
≪
fluctuationsinthequasi-condensateregime.
In the quasi-condensate regime density fluctuations are strongly reduced compared to their
valueinanidealBosegaswherethebunchingeffectisresponsiblefordensityfluctuationsofthe
orderofn2. In otherwords:
δn2 n2 (29)
≪
In thisregime,asuitabledescriptionisrealized bywritingthefield operatoras ψ = eiθ√n+δn
where the real number n is the mean density and the operator δn and the phase operator θ are
conjugate: [δn(z),θ(z )] = iδ(z z ). Notethatthedefinitionofalocalphaseoperatorissubtle
′ ′
−
and the condition Eq. (29) is not well defined since, because of shot noise, δn2 is expected to
divergein a small volume. A rigorous and simpleapproach consists in discretizing the space so
that in each cell alarge numberof atomsis present whilethediscretisationstep is much smaller
thanthecorrelation lengthofdensityand phasefluctuations[33].
Following this prescription, one first minimizes the grand canonical Hamiltonian H µN
−
withrespect ton toobtaintheequationofstate
µ = gn. (30)
1It is also called coherentregime since the g(2) functionis close to unity, as in a coherentstate. On the other
hand,thefirstordercorrelationfunctionstilldecaysandsothegasisnotstrictlycoherentinthissense. Withinthis
terminology,theidealBosegasregimeiscalledthedecoherentregime[22].
9
To second order in δn, this is the correct expression of the chemical potential. This equality
ensures that the Hamiltonian has no linear terms in δn and θ. Linearizing the Heisenberg
∇
equationsofmotioninδnand θ, weobtain[33]
∇
~∂θ/∂t = 1 ( ~2 ∆+2gn)δn
−2√n −2m √n (31)
( ~∂δn/∂t = 2√n( ~2 ∆)θ√n
−2m
Theseequationsaretheso-calledhydrodynamicequations. TheyarederivedfromaHamiltonian
quadratic in δn and θ, that can be diagonalized using the Bogoliubovprocedure [33]. It is not
∇
the purpose of this chapter to detail this calculation and to give exact results within this theory.
We will simply give arguments that enable an estimate of the density fluctuations and of their
correlation length. This estimate will then be used to check that δn2 n2, as assumed in
≪
Eq.(29). Wewillshowthatthisconditionisthesameas theconditionT T whereT given
co co
≪
in Eq. (24). After that, we will give similar arguments to estimate the phase fluctuations. Since
inthefollowingwewillstudythegaspropertiesversusthechemicalpotential,itisinstructiveto
rewrite the condition T T in terms of chemical potential. Using Eq. (30), we find that the
co
≪
quasi-condensateregimeisvalidas longas µ µ whereµ isgivenby Eq.(26).
co co
≫
2.3.1 Density fluctuations
To estimate the density fluctuations introduced by the excitations, it is convenient to divide the
excitationsin twogroups: theexcitationsoflow wavevectorforwhich thephase representation
is most appropriate and the excitations of high wave vector for which a particle point of view is
mostconvenient.
In the following, we use the expansions on sinusoïdal modes θ = √2(θ cos(kz) +
k>0 ck
θ sin(kz)) and δn = √2(δn cos(kz)+δn sin(kz)). Here δn and θ are conjugate
sk k>0 ck sk Pjk jk
variables ([δnjk,θj′k′] = (i/L)δjj′δkk′) where j stands for c or s. For modes of small wave
P
vectork,theexcitationsarephonons,ordensitywaves,forwhichtherelativedensitymodulation
amplitudeδn /nismuchsmallerthanthephasemodulationamplitudeθ . Inthiscase,thelocal
jk jk
velocity of the gas is given by ~ θ/m and the kinetic energy term is simply Ln~2k2θ2 /(2m).
∇ jk
TheHamiltonianforthismodethen reduces to
H = L gδn2 /2+n~2k2θ2 /(2m) . (32)
jk jk jk
This hamiltonian could also have bee(cid:0)n derived from the equation(cid:1)s of motion given in Eq. (31),
provided that the quantumpressure term ~2/(2m)∆δn/n is neglected: indeed, for a givenwave
vector k, the laplacians in Eq. (31) give a factor k2 and Eqs. (31) are simply the equations
of motion derived from the Hamiltonian Eq. (32). For temperatures much larger than ng, the
thermalpopulationofthesephononmodesislarge and classicalstatisticsapply. Thus,themean
energy perquadraticdegreeoffreedom isT/2and weobtain
δn2 = T/(Lg). (33)
h jki
and
θ2 = mT/(Lnk2~2). (34)
h jki
10