Hydrodynamic collective modes for cold trapped gases

Abstract

We suggest that collective oscillation frequencies of cold trapped gases can be used to test predictions from quantum many-body physics. Our motivation lies in both rigid experimental tests of theoretical calculations and a possible improvement of measurements of particle number, chemical potential or temperature. We calculate the effects of interaction, dimensionality and thermal fluctuations on the collective modes of a dilute Bose gas in the hydrodynamic limit. The underlying equation of state is provided by the non-perturbative Functional Renormalization Group or by Lee–Yang theory. The spectrum of oscillation frequencies could be measured by response techniques. Our findings are generalized to bosonic or fermionic quantum gases with an arbitrary equation of state in the two-fluid hydrodynamic regime. For any given equation of state P(μ, T) and normal fluid density nn(μ, T), the collective oscillation frequencies in a d-dimensional isotropic potential are found to be the eigenvalues of an ordinary differential operator. We suggest a method of numerical solution and discuss the zero-temperature limit. Exact results are provided for harmonic traps and certain special forms of the equation of state. We also present a phenomenological treatment of dissipation effects and discuss the possibility of exciting the different eigenmodes individually.

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arXiv:1103.5342v1 [cond-mat.quant-gas] 28 Mar 2011
Collective modes for precision measurements on cold trapped gases
Igor Boettcher,1Stefan Floerchinger,1,2and Christof Wetterich1
1Institut f¨ ur Theoretische Physik, Universit¨ at Heidelberg,
Philosophenweg 16, D-69120 Heidelberg, Germany
2Physics Department, Theory Unit, CERN, CH-1211 Gen` eve 23, Switzerland
We suggest that collective oscillation frequencies of cold trapped gases can be used as high pre-
cision observables for quantum many-body physics. Our motivation lies both in rigid experimental
tests of theoretical calculations and accuracy improvements of measurements of particle number,
chemical potential or temperature. We calculate the effects of interaction, dimensionality and ther-
mal fluctuations on the collective modes of a dilute Bose gas. The underlying equation of state
is provided by non-perturbative Functional Renormalization Group or by Lee–Yang theory. The
spectrum of oscillation frequencies could be measured by response techniques. Our findings are
generalized to an arbitrary quantum gas in the two-fluid hydrodynamic regime. The collective oscil-
lation frequencies in a d-dimensional isotropic harmonic potential for an equation of state P(µ,T)
and normal fluid density nn(µ,T) are found to be the eigenvalues of an ordinary differential oper-
ator. We suggest a method of numerical solution and discuss the zero-temperature limit. Exact
results are provided in certain cases.
PACS numbers: 03.75.Kk, 67.85.-d, 05.30.Jp
I.INTRODUCTION
Ultracold quantum gases link fundamental theoretical
physics to modern experiments. The feasibility of con-
trolling and tuning the parameters of a trapped gas with
external fields allows us to explore different regimes of
quantum many-body physics. [1–3] Theoretical methods
for interacting systems can therefore be tested under a
large variety of circumstances. Of particular theoretical
interest are systems for which effects beyond mean-field
theory become visible. Powerful methods like the Renor-
malization Group, Quantum Monte Carlo simulations,
2PI approaches, extensions of Density Functional The-
ory, conformal field theory, etc. provide many results
waiting for a rigid and systematic test by experiment.
We are thus lead to an important question: Which ob-
servables can resolve differences in the theoretical treat-
ment and can be measured with sufficient precision in the
laboratory in order to provide stringent tests for methods
beyond mean-field theory? This is a challenge for both ex-
perimentalists and theoreticians. While the former have
to create experimental settings going beyond the present
accuracy, the latter need to make clear statements on
what we expect to see and what it tells us about the sys-
tem and its theoretical understanding. We suggest that
collective oscillation modes may be used for this purpose.
For a trapped gas collective modes describe small de-
viations from static equilibrium. They typically show an
oscillatory behavior with characteristic frequencies. The
number density of atoms in a trap obeys for a particular
oscillation mode
n(? x,t) = n0(? x) + δn(? x)cos(ωt),(1)
where n0(? x) denotes the equilibrium density profile and
we assume small amplitudes, δn ≪ n0. We will show how
ω can be calculated for a d-dimensional harmonic trap in
the ideal two-fluid hydrodynamic regime for an arbitrary
equation of state, for which pressure and normal fluid
density are given as functions of chemical potential and
temperature, P(µ,T) and nn(µ,T), respectively.
Hydrodynamics describes a situation where the micro-
scopic time and length scales of the system, i.e. time
of local equilibration and mean free path of the parti-
cles, are much smaller than the macroscopic scales of the
motion under consideration, which are given here as the
inverse oscillation frequency and the radius of the static
equilibrium cloud. We neglect dissipative processes like
viscosity or heat transfer. Obviously, to provide the re-
lated transport coefficients from an underlying theory is
an even more complicated task than getting access to the
equation of state. The conditions of ideal hydrodynam-
ics might be satisfied for sufficiently high densities and
strong interactions.
Collective oscillation frequencies have been a promis-
ing observable since the first experiment on trapped cold
gases. Therefore, a great amount of literature already
exists on measurement [4–11] and calculation [12–22]
of these modes for weakly interacting Bose gases, the
BEC-BCS crossover and other systems at both zero and
nonzero temperature. The agreement of theory and ex-
periment is very good for some regimes and motivates
us to ask: If the behavior of the collective modes is well
understood, can we use them to measure other quantities
like particle number, temperature, etc. to a high preci-
sion? As we will see, this question is of a subtle nature,
because the frequencies depend on the equation of state
of the trapped gas, which is of course in general unknown
and has to be provided by an underlying microscopic the-
ory.
Our paper is organized as follows. In Sec. II we con-
sider the weakly interacting Bose gas. We are interested
in contact to experiment and focus on easy readability,
giving only the relevant equations without derivation.
The reader will find references to the subsequent sec-

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tions, where derivations of the formulas are presented in
a general framework. We successively improve the equa-
tion of state at zero temperature and also discuss lower-
dimensional systems. We present results at nonzero tem-
perature using the equation of state provided by Lee–
Yang theory and give an outlook on other systems. In
Sec. III we derive the eigenvalue problem for an arbi-
trary equation of state at zero and non-vanishing tem-
perature. This will yield the formulas already used in
Sec. II. A numerical implementation to obtain the col-
lective frequencies, which is used throughout this paper,
is given in Sec. IIID. We then come to our conclusion
in Sec. IV. In App. A our chosen system of units is
explained and App. B summarizes the theory of Lee and
Yang for the dilute Bose gas. In App. C spherical har-
monics in d ≤ 3 dimensions are introduced. The exact
results from the zero temperature Bose gas are derived
in App. D. Details of our phenomenological discussion
of response techniques are given in App. E.
II.OSCILLATIONS OF A BOSE GAS
In this section we demonstrate our ideas for the exam-
ple of a Bose gas. Due to the simplicity of the latter we
expect our findings to be generic for a broad class of ther-
modynamic quantum systems. We keep our presentation
very brief at this point and refer the reader to the next
section where the general theory of collective oscillations
is provided in detail.
At low temperatures and low densities interactions in
a homogeneous Bose gas can be described completely in
terms of contact interactions. In this regime the pre-
cise form of the interaction potential is irrelevant, im-
plying microscopic universality. In three dimensions the
associated coupling constant g3Dhas dimension of length
and is related to the s-wave scattering length a through
g3D = 4π?2a/m for bosons with mass m. For compos-
ite bosons all formulas remain valid with small modifica-
tions. For example, if we aim at describing the weakly
interacting BEC-side of the BEC-BCS crossover for two
component fermions, a has to be replaced by the dimer-
dimer scattering length, which is proportional to the
fermion scattering length [23], and m becomes the dimer
mass, which is two times the fermion mass. This cor-
respondence for the equation of state has been nicely
demonstrated in Ref. [24]. In two dimensions the cou-
pling constant becomes dimensionless. The effects arising
in this interesting situation will be investigated later in
this section.
A.Three-dimensional dilute Bose gas at zero
temperature
For a three-dimensional Bose gas the condition na3≪
1, where n is the density, corresponds to a dilute and
weakly interacting system. In this case, the mean-field
result for the energy density at zero temperature is given
by [2, 3, 25] (? = kB= m = 1, see App. A for units)
ε(n,a) =g3D
2
n2= 2πan2.(2)
Taking a derivative with respect to n, we get the chemical
potential µ = g3Dn which enters the Gibbs-Duhem rela-
tion dP = ndµ for the pressure. The equation of state
written in the grand canonical variables then becomes
P(µ,a) =
µ2
8πa,
(3)
which is of polytropic type P ∝ µα+1with α = 1. At
zero temperature the system is completely superfluid and
will be described by superfluid hydrodynamics.
The oscillation frequencies of this system in a spherical
parabolic trap Vext =
eigenvalue problem
m
2ω2
0r2are found by solving the
Ag(z) =
?ω
ω0
?2
g(z)(4)
for the differential operator
A = −Pµ(z)
Pµµ(z)
?
4z∂2
∂z2+ 2(2l + d)∂
∂z
?
+
?
2z∂
∂z+ l
?
(5)
acting on a function g(z). Here d = 3, see Sec. III.
The operator A depends on the equation of state through
Pµ(µ) and Pµµ(µ), where a superscript denotes differen-
tiation with respect to µ. The former of these quanti-
ties corresponds to the density Pµ= n while their ratio
Pµ/Pµµ= ∂n/∂P = c2is related to the velocity of sound
c. Since only this ratio appears in the operator A, the
prefactor of Eq. (3) is of no importance and the frequen-
cies will not depend on the interaction strength on the
mean field level. Indeed, Stringari [13] found the analytic
expression ωnl= (2n2+2nl +3n+l)1/2ω0, where n and
l are integers. Obviously, no thermodynamic conclusions
can be drawn from this formula and it may at best help to
determine the trapping frequency when interactions and
thermal effects are known to be very small. Stringari’s
formula is a special case of
ωα,n,l=
?2n
α(α + n + l + d/2 − 1) + l
?1/2
ω0,(6)
which holds for d-dimensional spherically symmetric,
harmonic traps and arbitrary polytropic index α.
we will show in App. D, this formula can be obtained by
applying a simple polynomial ansatz g(z) =?n
known in the literature [26, 27].
Eq. (6) reveals that the possible oscillation frequencies
of a trapped system form a discrete set. We will see in
the following that this property is true for an arbitrary
equation of state P(µ,T). This feature does not arise as
a result of imposed boundary conditions but is related
As
k=0ak¯ zk
with ¯ z = z/R2to Eq. (4) for P ∝ µα+1. It is already

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to the domain of definition of the operator A in a differ-
ent way. We will discuss this issue in Sec. IIID where
we solve the eigenvalue problem (4) on a finite grid by
discretization. Here, we only remark that the indices n
and l of Eq. (6) have the same meaning as in the proba-
bly more familiar case of a quantum mechanical particle
in a spherical symmetric potential. Similar to the hy-
drogen atom, n is related to the number of nodes of the
collective mode, where g(z) plays a role analogous to the
wave function. Rotation symmetry implies a conserved
angular momentum l which enters the operator A as a
parameter. For l = 0 the collective motion is isotropic,
while for l = 1 it is dipolar, and so on. Details can be
found in Sec. III where we derive Eq. (5). We expect for-
mula (6) to be applicable only for sufficiently small values
of n and l, because many nodes or a complicated angular
structure may enter in conflict with the assumptions of
hydrodynamics.
Apparently, for each l > 0 there is a mode with n = 0
and frequency√lω0which is independent of the equation
of state. In particular, the lowest dipole mode is exactly
at the trapping frequency. Up to now many measure-
ments focused on the breathing and lowest quadrupole
mode, n = 1,l = 0 and n = 1,l = 2, respectively. We
emphasize that the measurement of two, three or more
frequencies could achieve high precision when combined
with theory: For a given equation of state an arbitrary
number of eigenfrequencies can be obtained by our nu-
merical procedure. Comparison to measurements in well-
understood parameter regions tests the ability of a given
theoretical method to compute the equation of state ac-
curately. On the one hand, it will at least give us an esti-
mate on the theoretical errors. On the other hand, if the
agreement is very good, we can take advantage of a non-
trivial dependence of the frequencies on thermodynamic
quantities like temperature, density, chemical potential
or equation of state parameters like a. For example, if we
predict the lowest lying three monopole modes to have a
specific temperature dependence, we may conclude from
measuring these three modes in what temperature region
we are.
Motivated by this we are looking for non-trivial re-
lations between collective modes and thermodynamic
quantities. We have already seen that a system purely
described by mean field theory does not show such a be-
havior. We expect the mean field picture to be valid for
very small gas parameter na3≪ 1. However, as this
parameter increases, higher order interaction effects be-
come relevant in the equation of state. The leading order
correction to the ground state energy density (2) at zero
temperature has been calculated by Lee, Huang and Yang
and is found to be [28]
εLHY(n,a) = 2πan2
?
1 +
128
15√π(na3)1/2
?
.(7)
The corresponding pressure as a function of chemical po-
tential to the same order in perturbation theory is given
10-4
10-3
10-2
10-1
10-5
10-4
10-3
?n(0)a3
10-2
10-1
LHY
Wu, B=8
FRG
δωB/ωB
FIG. 1: Shift of the mean-field breathing mode ωB =√5ω0for
the zero temperature Bose gas in three dimensions. We show
predictions for different methods: Lee–Huang–Yang-formula
(8), Wu-correction (10) with B = 8 and equation of state from
Functional Renormalization Group [31] (FRG). The solid line
corresponds to Eq. (9), a is the scattering length and n(0)
the density in the center of the cloud.
by
P(µ,a) =
µ2
8πa−
8
15π2µ5/2.(8)
Apparently, the interaction strength can no longer be
eliminated by a rescaling of P(µ) such that the ratio
Pµ/Pµµentering the operator A will always depend on
a. Thus, the frequencies will depend on the coupling
constant. While for a homogeneous system the thermo-
dynamic observables are constant in space, for a confined
gas one usually refers to their values at the center of the
trap. Denoting the density in the center of the external
potential by n(0), a3n(0) provides a small parameter and
one expects a shift of the breathing mode (n = 1,l = 0,
i.e. ωB=√5ω0) [29, 30]
=63√π
128
δωB
ωB
?a3n(0)?1/2
(9)
as compared to the Stringari formula above. We con-
clude that we may use the shift for small values of n(0)a3
in order to determine either n(0) or a very precisely if
the corresponding second quantity is known from another
method.
From this example we can deduce a method to make
high precision measurements using collective modes. If
we assume the shifts of the frequencies ωn,l to be con-
tinuous in n(0)a3or µ(0)a2, where µ(0) is the chemi-
cal potential in the center of the trap, then for small
gas parameters, n(0)a3≪ 1 or µ(0)a2≪ 1, the shift
will be proportional to some power of the gas parame-
ter. In the case of Eq. (9) we found this power to be
1/2. However, this will depend on the system under con-
sideration. Nevertheless, driving the system through a

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certain range of the gas parameter, e.g. by the use of
a Feshbach Resonance, and then collecting the result in
a double-logarithmic plot one will find the results to lie
on a straight line for small gas parameter. This simple
scaling behavior can be used for very precise measure-
ments of n(0)a3after a proper calibration. The accuracy
of this method is directly related to the accuracy in mea-
suring the frequency shifts. In Fig. 1 we show the shifts
beyond mean field due to the LHY-correction which we
obtained by our numerical implementation described in
Sec. IIID. We reproduce the prediction to a high pre-
cision in the regime of small gas parameter, where it is
valid. Although this plot only shows the breathing mode
we emphasize that it is possible to calculate the shifts of
the whole frequency spectrum which are found to show
a similar scaling behavior with
that an experimental accuracy for frequency shifts bet-
ter than 10−2would be useful for exploring the regime
of small gas parameters.
The calculation of higher order corrections to the LHY
equation of state (7) is a difficult task.
next-to-leading order has been derived by Wu [32, 33]
for hard-sphere bosons. The energy density receives two
additional terms
√3
?n(0)a3. Fig. 1 shows
The next-to-
εLHY + 2πan2
?
8
?4π
3
−
?
na3
?
(log(na3) + B).
(10)
Neglecting the logarithmic term, this leads to a correction
which is proportional to µ3a in Eq. (8). In Fig. 1 we set
B = 8. Obviously, the general behavior of the frequency
shifts is not influenced.
For large values of the gas parameter perturbation
theory is no longer applicable and more sophisticated
methods are necessary. The Functional Renormalization
Group (FRG) for the effective average action [34] is a
non-perturbative quantum field theoretical method and
is in particular able to describe the full thermodynamics
of systems which are realized in cold atom experiments.
It does not rely on the expansion in a small parameter
and therefore the most striking effects are expected in
strongly interacting systems. For more information on
the FRG see Refs. [35]. It is still under debate whether
the regime a → ∞ can be reached for a Bose gas at very
low temperatures. One expects the condensate to get
unstable then because of increasing importance of three-
body losses.
In Fig. 1 we show the frequency shift obtained from
the equation of state calculated with FRG at zero tem-
perature [31]. We see that higher values of the gas pa-
rameter n(0)a3can no longer be identified with a unique
δωB/ωB. This is a fully non-perturbative effect. We re-
call that one of the assumptions in the beginning was
that the interaction can be approximated to be point-
like. However, this is only valid if the scattering length a
is much larger than the microscopic distance Λ−1where
one can resolve the details of the interaction potential.
For cold bosonic atoms this length is given by the typ-
ical range of the Van der Waals potential. An effective
ultraviolet cut-off Λ is necessary in any field theoretic
treatment of the system where the contact interaction
appearing in the Lagrangian of the non-relativistic sys-
tem is not renormalizable in three dimensions. Therefore,
the introduction of this scale is no relict of approxima-
tions but has a clear physical meaning. In perturbation
theory one assumes Λ to be infinity and thus the equa-
tion of state of the homogeneous system only involves
one dimensionless parameter, the gas parameter na3. A
proper treatment has to account for the appearance of an
additional length scale Λ−1such that two possible combi-
nations, na3and aΛ, describe the system. As a result the
shift of the breathing mode should rather be plotted as a
two-dimensional surface depending on these two param-
eters. For small gas parameter the additional parameter
aΛ does not play a role – if we take equations of state
P(µ,a) for different values of a and then vary n(0), all
obtained shifts will lie on a line in the double-logarithmic
plot. This is the expected scaling behavior. For higher
densities it makes a difference whether we calculate the
frequencies from P(µ,a,Λ1) or P(µ,a,Λ2) with Λ1?= Λ2,
as reflected by the spread of the points in Fig 1.
B. Lower-dimensional systems
With regard to formula (6) we may wonder whether the
cases d = 2 and d = 1 refer to highly anisotropic traps.
These are of great relevance for experiments where one is
often not dealing with a spherical symmetric potential.
For example, the early experiments in the JILA group
were performed in a disk-shaped confinement [4] while
the MIT group used a cigar-shaped trap [5].
A really lower-dimensional system is obtained in a trap
where the quantum gas is in its ground state in one or
two directions. This will be the case for very tight con-
finement. When calculating the equation of state one can
then neglect quantum and thermal fluctuations in these
directions. The mean field Bose gas in this scenario is
described by Eq. (6) when inserting d = 2 or d = 1
and α = 1. In these cases the system is isotropic in a
lower-dimensional geometry.
A different situation arises if the system is not in its
ground state in the directions of tight confinement and
thus still three-dimensional or in a crossover between
three and lower dimensions. Formula (6) cannot be ap-
plied in this case because the assumption of spherical
symmetry is not justified. The solution of the hydrody-
namic equations for these anisotropic cases is more in-
volved.
We now consider the two-dimensional Bose gas. Its
mean field equation of state is µ = g2Dn and the fre-
quency of the breathing mode is found from Eq. (6) to
be ωB= 2ω0. The equation of state beyond mean field
can be obtained by a Functional Renormalization Group
approach [31]. The coupling constant g2D, which satisfies
µ = g2Dn for small g2D, is dimensionless in two dimen-
sions and shows a logarithmic running with the physi-

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5
10-4
10-3
10-2
10-5
10-4
10-3
10-2
g2D=0.02
g2D=0.04
g2D=0.08
g2D=0.10
g2D=0.30
g2D=0.50
g2D=0.80
g2D=0.90
g2D=1.00
µ(0)Λ−2
δωB/ωB
FIG. 2: Shift of the breathing mode relative to ωB = 2ω0 for
a two-dimensional dilute Bose gas for different values of the
coupling constant g2D. We show the dependence on µ(0)Λ−2,
with chemical potential µ(0) in the center of the trap and
effective UV cut-off Λ. The equation of state is provided by
Functional Renormalization Group.
cal scale on which the experiment is performed. It will
vanish for an infinitely large system. However, realistic
experiments are always performed in traps so that in a
harmonic potential the oscillator length ℓ0=??/mωxy
under considerations. Therefore, when calculating the
equation of state one only has to include quantum fluc-
tuations with momenta bigger than ℓ−1
an infrared cut-off. This is also the reason why Bose–
Einstein condensation is observed experimentally in two
dimensions for small temperatures 0 < T < Tc. For an
infinitely extended system the long-range order would be
destroyed by fluctuations for all non-vanishing tempera-
tures as required by the Mermin-Wagner theorem [36].
In Figs. 2 and 3 we plot the shift of the breathing
mode corresponding to ωB= 2ω0for the two-dimensional
Bose gas at zero temperature with equation of state
P(µ,g2D) from Functional Renormalization Group calcu-
lations [31]. Since the coupling constant is dimensionless
there is no gas parameter for such a system. As we men-
tioned already in the three-dimensional case one always
has a physical ultraviolet cut-off Λ when dealing with a
contact interacting in d ≥ 2 dimensions. Since Λ has
dimension of inverse length, the dimensionless variable
involving the chemical potential (or similar for the den-
sity) is µ(0)Λ−2, where µ(0) is the chemical potential of
the gas in the center of the trap. A good choice for Λ−1is
the oscillator length of tight trapping. We observe from
Fig. 2 that the frequencies for small interactions depend
only weakly on µ(0)Λ−2. For larger coupling g2Dwe find
deviations from this behavior in Fig. 3.
We arrive at two interesting experimental scenarios to
be investigated. On the one hand, by measuring the col-
lective modes one can distinguish whether one is work-
ing with a system which is still three-dimensional (disk,
provides the largest possible length scale of the physics
0, which acts as
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
10-3
10-2
10-1
g2D=1.1
g2D=1.2
g2D=1.3
g2D=1.4
g2D=1.5
g2D=1.6
µ(0)Λ−2
δωB/ωB
FIG. 3: The same setting as in Fig. 2 but for higher values of
the coupling g2D. We find rather large negative shifts, which
have a minimum.
cigar) or a truly lower dimensional system. On the other
hand, if the latter regime is reached experimentally it is
of course tempting to verify the predictions of Fig. 3.
The rather large negative frequency shifts can both test
theoretical predictions like the occurrence of a minimum,
and determine µ(0) or the corresponding density n(0).
C.Three-dimensional dilute Bose gas for
non-vanishing temperature
We now extend our considerations to non-vanishing
temperature and allow for fluctuations of both density
and temperature. If we stay below the critical tempera-
ture of Bose–Einstein condensation the system possesses
both non-vanishing superfluid and normal fluid density
and its hydrodynamic behavior has to be described by
two-fluid hydrodynamics [37–39].
As explained in Sec. III, we again find an eigenvalue
problem which has to be solved in order to get the col-
lective frequencies of the system. It has the general form
?A B
C D
??g(z)
h(z)
?
=
?ω
ω0
?2?g(z)
h(z)
?
,(11)
where the differential operators A, B, C and D are de-
fined in Eqs. (64) - (70). These operators depend on
the equation of state P(µ,T) and the normal fluid den-
sity nn(µ,T). Both quantities have to be provided by
an underlying microscopic theory. The structure of the
eigenvalue problem is similar to the simple zero tempera-
ture case. We will show below that the zero temperature
limit of Eq. (11) is indeed given by Eq. (4) and this
behavior will also be found in the frequencies.
We see that calculating the frequencies at non-
vanishing temperature is as straightforward as it was at
T = 0. However, results on the temperature dependence
of hydrodynamic collective modes are rare in the liter-
ature [16].This is because the equation of state and

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0
200
400
600
800
1000
1200
1400
1600
0 5 10 15 20 25 30 35
T/Tc=0.25
T/Tc=0.5
T/Tc=0.75
r
neq
FIG. 4: Equilibrium density profile as a function of the radius
obtained from Lee–Yang theory using local density approxi-
mation for a spherical symmetric potential. We observe a
peak in the center of the trap. In the outer regions the gas is
in its normal phase.
normal fluid density are in general not known or only
in certain ranges of temperature and chemical poten-
tial. But when applying the local density approximation
µ → µ − Vext(r) one drives through a wide interval of
values for the chemical potential and therefore a com-
plete equation of state in µ has to be provided. Our mo-
tivation is in applying the Functional Renormalization
Group, which is capable of providing the full thermody-
namics of cold quantum gases, especially the functions
P(µ,T) and nn(µ,T). For the three-dimensional weakly
interacting Bose gas the equation of state has been calcu-
lated by Lee and Yang [40]. Their formulas are valid for
small gas parameter na3for temperatures not too close
to the critical temperature. We comment on the criti-
cal behavior later in this section. The Lee–Yang equa-
tion of state is summarized in App. B. We neglect the
next-to-leading order LHY-correction (7) because we are
interested in thermal effects.
Before discussing the collective modes of a Bose gas
at non-vanishing temperature we comment on some as-
pects of the static configuration of the trapped gas which
are necessary for the interpretation of our results. For a
homogeneous system with temperature T we can calcu-
late the critical chemical potential µc(T) of the superfluid
phase transition. Within the local density approximation
in the trapped system there will be a radius rc corre-
sponding to the phase boundary between the superfluid
and the normal regions of the cloud. This radius fulfills
µc= µ(0) − Vext(rc) with the chemical potential µ(0) in
the center of the trap. The characteristic picture of the
density profile consists in a narrow condensate peak in
the center of the trap which is surrounded by a broad
thermal cloud of the normal gas. Of course, there is also
a non-vanishing contribution of the normal component
to the inner regions. We visualize this behavior in Fig 4.
It is now apparent that for the description of a trapped
gas at any nonzero temperature one has to know the
105
106
107
108
0 0.2 0.4 0.6
T/Tc
0.8 1 1.2
N
FIG. 5: Particle number in a spherical trap as a function
of T/Tc for chemical potential in the center of the trap
µ(0) = 10?ω0 and scattering length a/ℓ0 = 0.0005, where
ℓ0 = (?/mω0)1/2is the oscillator length. These are the pa-
rameters that are also used in Figs. 6 and 7. The critical
temperature is defined in Eq. (12).
equation of state for both the superfluid and normal
phase. In particular the presence and dynamics of the
normal gas are of importance. There can be oscillations
of the thermal cloud itself and, furthermore, it provides
a non-trivial background for the oscillations of the con-
densate. As we approach zero temperature the thermal
cloud vanishes. Condensate oscillations correspond to so-
lutions δn of the hydrodynamic equations which are only
nonzero inside a sphere with radius rc.
The critical temperature Tcof the trapped gas is de-
fined as the temperature where the condensate peak ap-
pears in the center of the cloud. This can be reformulated
as µ(0) = µcor equivalently rc= 0. For Lee–Yang theory
the critical temperature is given by
Tc= 2π
?
µ(0)
2gζ(3/2)
?2/3
.(12)
If we set µ(0) = µc and calculate the number of par-
ticles N in the trap, we find the ideal gas prediction
T(0)
parameter. There are several possibilities to change the
parameter T/Tc. In this work we choose to keep the
chemical potential in the center of the trap fixed, relat-
ing the fixed µ(0) to Tcby Eq. (12). For fixed µ(0), the
particle number will change when decreasing the temper-
ature. Another way of going to low temperatures is to
keep the particle number N fixed. Then the chemical
potential in the center of the trap has to be adjusted ap-
propriately. In Fig. 5 we plot the change of N with T/Tc
for fixed µ(0) and the parameters which are used for Fig.
6.
In Fig. 6 the isotropic (l = 0) temperature depen-
dent oscillation frequencies are shown for µ(0)/?ω0= 10
and a/ℓ0= 0.0005, where ℓ0 =
tor length. This corresponds to a critical temperature
c (N) = ?ω0(N/ζ(3))1/3to be satisfied for small gas
??/mω0 is the oscilla-

Page 7

7
0
2
4
6
8
10
12
14
16
0 0.2 0.4 0.6
T/Tc
0.8 1 1.2
ω/ω0
FIG. 6: Oscillations frequencies of collective modes (l = 0) of both the condensate and thermal cloud. We show the temperature
dependence for a fixed value of the chemical potential in the center of the trap with the same choice of parameters as in Fig.
5. The equation of state is provided by Lee–Yang theory.
Tc/?ω0 = 284. This particular choice of parameters is
arbitrary but suffices to show the characteristics of the
frequency dependent modes. We observe a rich spectrum
of frequencies. The oscillations can be classified as con-
densate oscillations and oscillations of the thermal cloud.
The condensate oscillation correspond to the branches
which disappear at Tc. The computability of frequencies
above the critical temperature by our method is a mani-
festation of the fact that the Landau two-fluid model re-
mains formally valid for vanishing superfluid component.
Of course, the frequencies for T ≥ Tccould be computed
equivalently with a one-fluid model for the thermal liquid.
For an ideal gas one expects the oscillation frequencies to
be integer multiples of ω0. We find deviations from this
behavior.
For low temperatures we find the oscillation frequen-
cies in Fig. 6 to scatter. This is a discretization effect
related to our numerical implementation and not a short-
coming of the two-fluid hydrodynamics. Indeed, with a
different discretization, which puts the emphasis on the
region r < rcwhere the condensate is present, the tem-
perature dependence of the frequencies is found to be
smooth. This is demonstrated in Fig. 7 where the con-
densate oscillations are shown for very low temperatures.
This plot supplements Fig. 6. As T/Tcincreases, there
is a systematic error in Fig. 7 because the thermal cloud
is not taken into account properly. We further comment
on this issue in Sec. IIID.
From Fig. 6 it is clear that there are level-crossing
frequencies over wide ranges of T/Tc. These branches can
be used as a thermometer by measuring a certain number
of frequencies and then locating them in the plot. How
this can be done experimentally by response techniques
is explained in Sec. IID.
From our discussion of the equilibrium density profile
we see that for all temperatures below the critical temper-
ature Tcthere is a radius rcwhere the critical equation of
state P(µc,T) has to be known. The vicinity of the point
µc might be small but the behavior of the thermody-
namic functions in this interval could nevertheless influ-
ence the frequencies significantly. Indeed, the superfluid
phase transition is of second order and we expect the spe-
cific heat at constant pressure CP = T(∂S/∂T)P,N and
the isothermal compressibility κT = −V−1(∂V/∂P)T,N
to be singular at µc. Their behavior in the critical re-
gion as a function of temperature is CP ∼ |T − Tc|−α
and κT ∼ |T − Tc|−γwith the critical indices [41] of
the three-dimensional XY-universality class. The spe-
cific heat shows a cusp while the isothermal compress-
ibility diverges. A systematic study of these effects on
the collective modes would be very interesting. Lee–Yang
theory does not allow for such an investigation.
Using the Lee-Yang formulas given in App. B we can
calculate numerically the coefficient functions appearing
in the operators A, B, C, and D in Eq. (11). For very
low temperatures we observe that B and C vanish while
A approaches its zero temperature expression, given by
Eq. (5),
?A B
C D
?
T→0
−→
?A(T = 0)0
0D(T = 0)
?
.(13)
This corresponds to a decoupling of superfluid and ther-
mal degrees of freedom. This is expected to happen in
the zero temperature limit since the few atoms in the
thermal cloud have no influence on the dynamics of the
condensate. The operator entering the eigenvalue prob-
lem is now block diagonal and all zero temperature fre-
quencies are reproduced. They correspond to Stringari’s
mean field formula given above. (Note that we have ne-
glect the LHY-correction.) It is interesting that D does
not vanish. Indeed, the coefficient functions a1, d1 and
d2entering A and D from Eqs. (68) and (70) satisfy
a1
d1
This can be seen by evaluating the integrals for P(µ,T)
and nn(µ,T) numerically or applying an expansion for
T→0
−→1
3,d2
T→0
−→ −1
6. (14)

Page 8

8
0
1
2
3
4
5
6
7
8
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
T/Tc
ω/ω0
FIG. 7: Zero temperature limit of the collective oscillations
for the same choice of parameters as in Figs. 5 and 6. In
this figure, there is a systematic error for high temperatures,
because the thermal cloud is not treated appropriately. The
solid lines over the whole range of temperatures correspond
to formula (6) for superfluid oscillations at T = 0. The fre-
quencies given by Eq. (16) are indicated by solid lines only
for small temperatures, where this formula is expected to be
valid.
T ≪ µ using only the phonon part of the spectrum, see
App. D. It can be shown that a1/d1corresponds to the
ratio of the squared first and second velocities of sound,
c2
2, respectively. Therefore we find the well-known
relation
1and c2
c2
2=c2
1
3
(15)
to be satisfied for very low temperatures.
We show in App. D that the behavior of the coeffi-
cient functions of D as T → 0, Eq. (14), can be used to
derive the eigenvalues of D by slight modifications from
the eigenvalues of A, which are given by Eq. (6) with
α = 1. The corresponding eigenfrequencies are found to
be
ωnl=
?4n(n + l) − l
6
?1/2
ω0, (n ≥ 1).(16)
The experimental verification of this formula will be dif-
ficult, however. Although for low T (and T → 0) these
frequencies are solutions to the two-fluid hydrodynamic
equations, there are only very few atoms left in the nor-
mal phase to oscillate. From Eq. (16) it is apparent that
for n = 1,l = 0 one obtains
trapping frequency. This cannot be achieved by formula
(6) with an effective polytropic index αeff. The exis-
tence of such modes below the trapping frequency might
be a special feature of superfluid hydrodynamics just as
the appearance of a second velocity of sound. Indeed, in
their early experiments Stamper-Kurn et al. [5] used a
cigar-shaped harmonic potential with trapping frequency
ω0z/2π = 18.04(1) Hz in axial direction. They observed
?2/3ω0which is below the
an out-of-phase oscillation between condensate and ther-
mal cloud of a Bose gas at T = 1µK in the hydrodynamic
regime. The critical temperature was around 1.7µK.
The measured frequency of axial motion was found to
be ω0/2π = 17.26(9) Hz, which is below ω0z.
In Fig. 7 we show our numerical results for the frequen-
cies in the zero temperature limit using the full Lee–Yang
equation of state for the same parameters as in Fig. 6.
We find formulas (6) and (16) to be in perfect agreement
with our data.
D.Measuring the spectrum with response
techniques
We have seen that information is contained both in
the precise values of single modes and in the whole spec-
trum of the oscillating system. While the lowest lying
modes can be excited by varying the parameters of the
trapping potential (e.g. the potential depth) and then
measuring the frequency in a free expansion, the whole
spectrum could be obtained by an other method. In prin-
ciple one can fit a superposition of arbitrary many modes
to the freely expanding cloud but the contribution of the
higher modes will only be weak. We suggest to use a
response measurement instead to locate the frequencies
of the trapped cloud. In order to keep the notation short
we restrict this discussion to the zero temperature case.
The eigenfrequencies and corresponding eigenfunctions
obtained so far represent normal coordinates of an os-
cillating system, the trapped cold gas.
of motion ∂2
tδµ + Aδµ = 0 has solutions δµnlm =
¯ gnl(r)rlYlme−iωnltwhich are harmonic oscillations with
frequency ωnl. (More precisely, one should use the oper-
ator E defined below Eq. (E2) instead of A. This is of no
relevance here, because they have the same eigenvalues.)
To describe an experimental situation we need to in-
clude dissipation effects into our description. We there-
fore introduce a phenomenological damping term by re-
placing ∂2
t+ Γ∂t, with damping constant Γ. The
new eigenfrequencies of the system will be denoted by
Ωnl. We use Fourier decomposition and write δµ(t) =
?dΩδµ(Ω)e−iΩt. Since we already know the eigenvalues
(−Ω2
for possible solutions Ωnl. The latter simplify due to the
fact that t2= 1/Γ represents a characteristic time scale
over which damping takes place.
oscillations this scale should be substantially larger than
t1 = 1/ωnl. The other choice would correspond to the
overdamped case. Hence ω2
The equation
t→ ∂2
of A we immediately obtain the quadratic equation
nl− iΓΩnl+ ω2
nl)δµnlm(Ω) = 0(17)
In order to observe
nl≫ Γ2and we have
Γ2
8ω2
nl
Ωnl≃ ωnl
?
1 −
?
−iΓ
2
(18)
for the new eigenfrequencies of the system. Due to damp-
ing effects they acquire an imaginary part and their real

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9
part is slightly shifted. This phenomenological treatment
gives no explanation of the microscopic nature of the
damping terms. It would be interesting to derive the
relation between the damping constants Γ and the five
transport coefficients η,κT,ζ1,ζ2and ζ3of two-fluid hy-
drodynamics [39]. In principle, one expects Γ to depend
on the mode in the form of some continuous function
Γ(Ω). For simplicity, we use here the same phenomeno-
logical Γ for all modes.
The picture of the trapped gas as an oscillating system
will provide us with an intuition how individual (and thus
also higher lying) modes can be addressed by virtue of
response techniques. Consider a classical point particle
with Hamiltonian H =
term included its equation of motion will have the form of
Eq. (17) and is therefore independent of the mass of the
particle. However, when applying a small external force
f(t) the equation of motion for the Fourier components
of x(t) becomes
1
2mp2+m
2ω2x2. With a damping
(−Ω2− iΓΩ + ω2)x(Ω) =1
mf(Ω).(19)
Obviously, the response χ(Ω) = x(Ω)/f(Ω) is propor-
tional to the inverse of the mass m. It is therefore more
difficult to excite a heavy particle than a lighter one.
We introduce an analog quantity mnlmwhich represents
the “mass” for the collective oscillation δµnlm. We call
mnlmthe “response coefficients”. The relation between
the amplitude of the oscillations of the atom cloud and
the oscillation of the external perturbation f(Ω) is gov-
erned by mnlm.
Let us consider a periodic perturbation of the harmonic
trapping potential,
Vext(? x) → Vext(? x) + cos(Ωt)F(r)rlYlm.
The perturbation is assumed to be small in comparison
to Vext. As we show in App. E this small variation acts
as a driving force on the oscillating system (17). There
it is also shown that in order to excite an oscillation with
angular dependence δµnlm∝ Ylm, δVexthas to have the
same angular dependence. In Eq. (E12) we give a general
formula for the response coefficient mnlmof the collective
mode. Given Ω ≃ ωnlwe then have
1
mnlm
(Ω2− ω2
for the imaginary part of the response function defined in
Eq. (E11). The latter is related to dissipation of energy
in the system.
For a demonstration we restrict ourselves to the case of
a three-dimensional mean-field Bose gas, i.e. P(µ) ∝ µ2,
and consider only the isotropic modes with l = 0. In
addition we assume a monomial perturbation F(r) = rq
with q ≥ 0. This includes the cases of a spatially con-
stant background field δVext(? x,t) = cos(Ωt) (q = 0)
and a time-variation of the trapping frequency, δVext=
r2cos(Ωt) (q = 2). Interestingly, we find in App. E that
(20)
Imχ(Ω) =
ΓΩ
nl)2+ Γ2Ω2
(21)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 1 2 3 4 5 6 7 8
q=2
q=4
q=6
q=8
q=10
Ω/ω0
Imχ(Ω)[a.u.]
FIG. 8: Response of the zero temperature mean-field Bose gas
to a perturbation δVext = rqcos(Ωt) of the external potential.
not all values of q can be used to excite the whole spec-
trum. For example, there is no response at all to q = 0,
while for q = 2 only the breathing mode will be excited
and for q = 4 only the lowest two modes oscillate. We
show this behavior in Fig. 8. (In our intuitive picture de-
veloped above the absence of response corresponds to an
infinite mass.) The normalization in Fig. 8 is such that
the lowest mode always has the same amplitude. The rel-
ative amplitudes, both for oscillations with different fre-
quencies, and for oscillations with a given frequency, but
for different excitations, can be used as a further observ-
able for testing the system. The computation of ratios
of amplitudes for different Ω needs knowledge of Γ(Ω).
In principle, this can be measured from the width of the
resonance. Ratios of amplitudes for fixed Ω and differ-
ent radial or angular dependence of the perturbation are
independent of Γ(Ω) and mnlm. In principle, amplitudes
as in Fig. 8 can be measured both by dissipation (Imχ)
or by the amplitude of the cloud oscillations (Reχ).
We conclude that the variation of the external poten-
tial with a certain frequency and radial and angular de-
pendence allows for an excitation of individual modes.
The oscillation can then be detected by imaging the den-
sity profile or, hopefully, by new non-invasive methods.
E.Outlook on other systems
We mentioned in the beginning of this section that
we expect the features of the Bose gas to be generic for
other systems of cold atoms. At nonzero temperature this
requires the applicability of the two-fluid model which is
related to a two-component order parameter. The latter
can be written as φ(? x) = |φ(? x)|eiθ(? x)at each space-time
point. If θ(? x) is varying slowly in space one can define
the superfluid velocity as its gradient, ? vs∝ ∇θ. Thus our
description remains valid for systems which are described
by a complex scalar field which acquires a non-vanishing
expectation value. This includes in particular composite

Page 10

10
bosons like the atom-atom dimers on the BEC-side of the
BEC-BCS crossover. For an imbalanced system it would
then be interesting to include the conserved spin degrees
of freedom of the remaining fermions as an additional
macroscopic variable in the hydrodynamic equations.
At zero temperature any system with hydrodynamic
equations of motion might be considered. An example
can be found in Ref. [42] where the collective oscilla-
tion frequencies of a one-dimensional dipolar quantum
gas have been calculated.
The polytropic equation of state P ∝ µα+1is of par-
ticular interest for scale-invariant systems like ultracold
Fermi gases at the unitary point where the scattering
length a diverges. At zero temperature only the chemical
potential remains as a dimensionful quantity, the pres-
sure has to be proportional to µ(d+2)/2. Indeed, intro-
ducing the Bertsch parameter ξ we can write the chem-
ical potential at the unitary point as µ = ξεF, where
εF ∝ n2/dis the Fermi energy density. Using dP = ndµ
we arrive at P(µ) ∝ µ(d+2)/2as claimed. This time we
know the equation of state exactly, but only for zero tem-
perature. At the unitary point and for T = 0 the oscil-
lation frequencies obey the exact Eq. (6), with α = 3/2
for a three-dimensional spherical trap.
Around the unitary point and in the dilute, three-
dimensional regime,the equation of state can be
parametrized as [43]
ε(n) =3
5εF(n)
?
ξ −
ζ
kF(n)a+ O
?
1
(kFa)2
??
(22)
with an additional parameter ζ implying a shift
δω2
ω2
B
B
=
256
525π
ζ
ξ
1
kF(n(0))a
(23)
of the breathing mode.
III.COLLECTIVE MODES - GENERAL
THEORY
In this section we derive the eigenvalue problem (11)
for collective modes at T ≥ 0 from Landau’s two-fluid
ideal hydrodynamics [37, 39].
drodynamic description lies in the expansion of general
constitutive equations in terms of derivatives of the con-
tributing fields. The range of applicability of this de-
scription has already been discussed in the introduction.
The constitutive relations consist in conservation laws for
particle number, momentum, energy and additional con-
served quantities. The lowest order of this expansion is
static equilibrium and thus thermodynamics. The next
order is called ideal hydrodynamics and describes non-
equilibrium processes without dissipation. The latter is
included in higher orders of the expansion.
The heart of every hy-
A.The eigenvalue problem from two-fluid
hydrodynamics
Observations on quantum fluids suggest that there is
a critical temperature Tcsuch that for 0 ≤ T ≤ Tcthe
macroscopic hydrodynamic motion can be divided into
two kind of flows, a superfluid (subscript s) and a nor-
mal fluid (subscript n) one. Their velocity fields ? vs(? x,t)
and ? vn(? x,t), respectively, enter the momentum density
? g = ns? vs+nn? vnwith coefficients satisfying n = ns+nn,
where n(? x,t) is the density of the gas or fluid under con-
sideration.We might be lead to the conclusion that
the substance is built out of two different kinds of flu-
ids. However, the true nature of superfluidity consists
in quantum physics and our description is only formally
correct. The two flows can be distinguished by the follow-
ing properties. The superfluid velocity field is a gradient
field and hence irrotational. Furthermore, entropy is only
carried by the normal fluid part. The full hydrodynamic
equations under these constraints were set up by Lev. D.
Landau in a famous and still very worth reading paper in
1941 [37]. We concentrate on ideal hydrodynamics where
dissipation terms are neglected and entropy is conserved.
In the homogeneous case, the underlying conservation
laws read
∂n
∂t+ div(? g) = 0,
∂gi
∂t+ ∂kΠik= 0,
∂s
∂t+ div(s? vn) = 0,
?v2
(24)
(25)
(26)
∂? vs
∂t
+ ∇
s
2
+ µ
?
= 0 (27)
with stress tensor Πik= nnvn,ivn,k+nsvs,ivs,k+Pδik, en-
tropy density s, pressure P = −ε+Ts+µn+nn(? vn−? vs)2,
chemical potential µ and internal energy density in the
rest frame of the gas, ε.These equations have to be
supplemented by the equation of state P(µ,T) and the
normal fluid density nn(µ,T). For dissipative hydrody-
namics one would replace Eq. (26) by the conservation
of energy and include an additional term in Eq. (27). We
use dimensionless units described in appendix A, setting
? = kB= m = 1.
In order to understand what changes when applying an
external field, we first derive the local density approxima-
tion from a thermodynamic point of view. For this pur-
pose consider a gas contained in a potential Vext(? x) such
that local equilibrium is reached at each point of space. If
we take two neighboring small but yet macroscopic parts
of the gas their energies E1and E2and particle numbers
N1and N2will adjust in a manner maximizing the en-
tropy S = S1+ S2 under the constraints E1+ E2= E
and N1+ N2= N, respectively. This implies tempera-
ture and the full chemical potential to be constant inside
the trap. However, the full chemical potential is given
by the Gibbs free energy per particle and thus reads

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11
µfull(? x) = µhom(P(? x),T) + Vext(? x), where µhom(P,T)
is the equilibrium chemical potential of the homogeneous
system as a function of pressure P and temperature T.
We conclude that a system where the chemical potential
µ in the homogeneous equilibrium functions is substi-
tuted according to Phom(µ,T) → Phom(µ − Vext,T) etc.
behaves like a system trapped in an external potential
of large spatial extend. This rule is called local density
approximation.
An external potential can be included in Eqs. (25) and
(27) by virtue of force terms,
∂gi
∂t+ ∂kΠik= −n∂iVext,
?v2
(28)
∂? vs
∂t
+ ∇
s
2+ µ
?
= −∇Vext.(29)
The static solution denoted by a subscript 0 gives the
expected expressions µ0(? x) + Vext(? x) = ¯ µ and ∇P0(? x) +
n0(? x)∇Vext(? x) = 0 of static equilibrium.
Gibbs-Duhem relation dP
∇P0 = s0∇T0+ n0∇µ0 at every space point, we con-
clude s0∇T0= 0 in equilibrium. We recognize the right
thermodynamic behavior to emerge naturally. Pressure
and densities of mass, entropy and energy are constant in
time but space dependent, while temperature is constant
all over the trap and the chemical potential follows the
rule of local density approximation. The parameter ¯ µ
can be adjusted in order to get a certain particle number
N. The fluid velocities ? vsand ? vnvanish in equilibrium.
The next step towards our eigenvalue problem consists
in expanding the two-fluid equations in small fluctuations
around their static equilibrium solution such that time-
dependent quantities only appear linearly. We write
Using the
= sdT + ndµ and thus
µ(t,? x) = µ0(? x) + δµ(t,? x),
T(t,? x) = T + δT(t,? x),
? vs,n(t,? x) = δ? vs,n(t,? x).
(30)
(31)
(32)
In the same way we linearize P = P0+ δP,n = n0+
δn,s = s0+ δs. Eqs. (24), (26), (28) and (29) then
become
∂tδn + div(δ? g) = 0,
∂tδ? vs+ ∇δµ = 0,
∂tδ? g + ∇δP + δn∇Vext = 0,
∂tδs + div(s0δ? vn) = 0
(33)
(34)
(35)
(36)
with δ? g = ns,0δ? vs+ nn,0δ? vn. Using the relation ∇P0=
s0∇T0+n0∇µ0we get ∇δP = s0∇δT+δn∇µ0+n0∇δµ =
s0∇δT − δn∇Vext+ n0∇δµ, since ∇T0= 0 and ∇µ0=
−∇Vext. This can be used to cast Eq. (35) into the form
nn,0∂t(δ? vn− δ? vs) + s0∇δT = 0.
In the following we will consider a spherically symmet-
ric harmonic trapping potential Vext(? r) =
generalization of our formalism to other radial poten-
tials Vext(? r) = Vext(r) is straightforward. We assume the
(37)
m
2ω2
0r2. The
fluctuations of the physical quantities to be a harmonic
oscillation in time with frequency ω,
δµ(? x,t) = e−iωtδµ(? x),(38)
and similar expressions for the other time-dependent
functions. The remaining spatial deviations are again de-
noted by δµ etc. Since time has completely dropped out
of our system of partial differential equations, this slight
abuse of notation hopefully does not lead to confusion.
We recognize our obtained set of equations
iωδn = div(ns,0δ? vs+ nn,0δ? vn),
iωδ? vs= ∇δµ,
iω(δ? vn− δ? vs) =
iωδs = div(s0δ? vn)
(39)
(40)
s0
nn,0∇δT,(41)
(42)
to be an eigenvalue problem for a differential operator
acting on δµ, δT, δ? vsand δ? vn− δ? vs.
B.Zero temperature
We derive the zero temperature eigenvalue problem in
terms of an equation of state given in the form P(µ). We
introduce the notation µ0(r) = µ(0) − Vext(r) = µ(0) −
ω2
express all dimensionful quantities with respect to ω0, see
App. A. In the following we set ω0= 1 in all formulas,
which means that we are measuring time in units of the
inverse trapping frequency. Thus, µ0(r) = µ(0) − r2/2.
Furthermore, we define
0r2/2. Since we have set ? = kB = m = 1, we can
Pµ
0(r) =
?∂P
∂µ
?
(µ0(r)), Pµµ
0(r) =
?∂2P
∂µ2
?
(µ0(r)),
(43)
where µ(0) is the chemical potential in the center of the
trap. We can write δn(? x) = Pµµ
ndµ. There are no temperature fluctuations present at
T = 0 and thus the only degrees of freedom are δµ and
δ? vsdescribed by Eqs. (39) and (40). The latter equations
can be decoupled. Writing n0(r) = Pµ
the Stringari wave equation [13]
0(r)δµ(? x) because dP =
0(r) we arrive at
ω2Pµµ
0δµ + div(Pµ
0∇δµ) = 0. (44)
So far, δµ depends on the d-dimensional spatial coordi-
nate ? x. Since we are dealing with a spherically symmetric
trap we can classify the possible solutions to the eigen-
value problem via an ansatz
δµ(? x) = ¯ g(r)rlflm, (45)
where flmare spherical harmonics. (For a definition of
spherical harmonics in d ≤ 3 dimensions we refer to App.
C. Note that we have l = 0,1 for d = 1.) When applying
this ansatz to Eq. (44) we benefit from the facts that

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12
∇µ0· ∇flm= 0 and ∂rP(µ0(r)) = −rPµ
we use the relations
0(r). In addition
? er· ∇δµ(? x) =
?
¯ g′(r) +l
r¯ g(r)
?
rlflm,(46)
∆δµ(? x) =
?
¯ g′′(r) +2l + d − 1
r
¯ g′(r)
?
rlflm,(47)
where ? erdenotes the unit vector pointing in radial direc-
tion. We arrive at
0 = ω2¯ g(r) − r¯ g′(r) − l¯ g(r)
+Pµ
Pµµ
0(r)
0(r)
?
¯ g′′(r) +2l + d − 1
r
¯ g′(r)
?
. (48)
When substituting z = r2the eigenvalue problem is given
by
Ag(z) = ω2g(z)(49)
for g(z) = ¯ g(r) and the differential operator
A = −Pµ(z)
Pµµ(z)
?
4z∂2
∂z2+ 2(2l + d)∂
∂z
?
+
?
2z∂
∂z+ l
?
(50)
.
This equation has to be fulfilled on the interval z ∈ [0,R2]
where R is the radius of the static cloud defined by n0(r =
R) = 0. We do not allow for fluctuations to change the
cloud radius since this would be a second order small
effect. Since we have set ω0to 1, only ω2appears in Eq.
(49) instead of ω2/ω2
0.
C.Non-vanishing temperature
For temperatures T ≥ 0 we have to implement tem-
perature fluctuations and the full thermodynamic func-
tions P(µ,T) and nn(µ,T). Extending the notation in-
troduced in Eq. (44) we write
PµT
0
(r) =
∂2P
∂T∂µ(µ0(r),T).(51)
In the same manner Pµ
defined for T ≥ 0, where
0, Pµµ
0,PT
0, PTT
0
, ˜ n0, and ˜ nµ
0are
˜ n =s2
nn.(52)
The static solutions n0and s0are connected to µ0(r) =
µ(0) − r2/2 and T0= T via n0= Pµ
again with independent variables δµ and δT we get
0,s0= PT
0. Working
δn = Pµµ
δs = PTµ
0δµ + PµT
0 δµ + PTT
0
δT,(53)
0
δT.(54)
From our analysis at zero temperature we know that due
to the spherically symmetry the eigenmodes are most eas-
ily obtained by eliminating the velocity fields. We find
ω2δn + div(n0∇δµ + s0∇δT) = 0,
ω2δs + div(s0∇δµ + ˜ n0∇δT) = 0,
(55)
(56)
or in terms of P(µ,T),
ω2Pµµ
0δµ + ω2PµT
0
δT + div(Pµ
0∇δµ + PT
0∇δT) = 0,
(57)
ω2PTµ
0 δµ + ω2PTT
0
δT + div(PT
0∇δµ + ˜ n0∇δT) = 0
(58)
determining ω. Note that ˜ n0supplements the equation of
state in the latter equation. We have chosen ˜ n0instead
of nn0in order to keep the notation simple.
In their present form, Eqs. (57) and (58) do not ex-
hibit the canonical form of an eigenvalue problem and
are unsuitable for a numerical implementation. We build
the more useful linear combination
div(Pµ
− PµT
ω2detP0δT = PµT
− Pµµ
where we assumed PµT
0
= PTµ
Pµµ
0
−(PµT
case we classify the solutions via an ansatz
−ω2detP0δµ = PTT
0
0∇δµ + PT
0∇δµ + ˜ n0∇δT),
0 div(Pµ
0div(PT
0∇δT)
0 div(PT
(59)
0∇δµ + PT
0∇δµ + ˜ n0∇δT),
and defined detP0 =
)2. Analogously to the zero temperature
0∇δT)
(60)
0
0PTT
0
δµ(? x) = g(r2)rlflm,
δT(? x) = h(r2)rlflm
(61)
(62)
where l = 0,1,2,... and flmare the corresponding spheri-
cal harmonics. The resulting equations simplify on going
over to the new variable z = r2. We arrive at
?A B
where the operators A,B,C,D are defined by
C D
??g(z)
h(z)
?
= ω2
?g(z)
h(z)
?
, (63)
A =a1(z)
?
?
4z∂2
∂z2+ 2(2l + d)∂
4z∂2
∂z
?
?
+
?
2z∂
∂z+ l
?
?
(64)
B =b1(z)
∂z2+ 2(2l + d)∂
∂z
+ b2(z)2z∂
∂z+ l
?
(65)
C =c1(z)
?
?
4z∂2
∂z2+ 2(2l + d)∂
4z∂2
∂z
?
?
(66)
D =d1(z)
∂z2+ 2(2l + d)∂
∂z
+ d2(z)
?
2z∂
∂z+ l
?
(67)
with z-dependent coefficients
a1=PµT
0
PT
0− PTT
detP0
0
Pµ
0
,b1=PµT
0
˜ n0− PTT
detP0
0
PT
0
,
(68)
b2=PTT
0
PµT
0
detP0
− PµT
0
˜ nµ
0
,c1=PµT
0
Pµ
0− Pµµ
detP0
0PT
0
,
(69)
d1=PµT
0
PT
detP0
0− Pµµ
0 ˜ n0
,d2=Pµµ
0 ˜ nµ
0− (PµT
detP0
0
)2
.
(70)

Page 13

13
Again, ω =√l is a solution for constant g(z) and vanish-
ing h(z), which is independent of the equation of state.
An additional complication to the case of vanishing
temperature arises from the fact that the normal part
of the background density nn0is not restricted to a re-
gion z ≤ zmaxas it is the case for the superfluid density
ns0. In contrast it decreases exponentially for large z
(c.f. Fig. 4). In praxis, we expect that the region z > R2
for a sufficiently large value of R2is not affecting the
oscillation frequencies since only an exponentially small
number of particles is in that region. We restrict our
numerical treatment therefore to a finite sphere z ≤ R2.
Given a1,...,d2 over the whole range z ∈ [0,R2] the
problem of determining the collective frequencies is as
straightforward as it was for T = 0. Indeed, as we show
in Sec. IIID a simple and yet efficient method to solve
for the eigenvalues will be given by replacing the opera-
tors by matrices. Thus, if the more complicated coeffi-
cient functions (68) - (70) are given to a sufficient degree
of accuracy, the finite temperature collective modes are
easily obtained.
D.Numerical implementation
Now that we have identified the ordinary differential
operator(s) describing two-fluid hydrodynamic modes,
we present a numerical method to determine the corre-
sponding eigenfrequencies. The input for our approach
consists in the equation of state P(µ,T) and normal fluid
density nn(µ,T) on the interval µ ∈ [µmin,µ(0)]. Since
we restricted ourselves to the case of spherically symmet-
ric trapping the partial differential hydrodynamic equa-
tions in d variables x1,...,xdare projected onto an or-
dinary differential equation in z. This is a disadvantage
of our method when compared to the fact that most ex-
periments are performed in asymmetric traps.
In order to introduce our numerical procedure we con-
sider the case of T = 0 since the operators for T ≥ 0 have
an identical structure but different coefficient functions.
We solve the eigenvalue problem (4), (5) by discretiz-
ing the interval z ∈ [0,zmax] via i = 0,1,...,M and
∆z = zmax/M such that zi = i∆z. The function g(z)
becomes a vector g = (g(zi))i=0,...,Mand A is represented
by a matrix (Aij) having components
Aij= 2ziδi+1,j− δi−1,j
2∆z
+ lδij−Pµ
+ 2(2l + d)δi+1,j− δi−1,j
0(zi)
Pµµ
0(zi)
×
?
4ziδi+1,j+ δi−1,j− 2δij
∆z2
2∆z
?
(71)
for i = 1,...,M − 1 and j = 0,...,M with δij being the
Kronecker delta. The cases i = 0 and i = M require
some care. We define
A0,j=lδ0,j−Pµ
AM,j=2zmaxδM,j− δM−1,j
0(0)
Pµµ
0(0)2(2l + d)δ1,j− δ0,j
∆z
,(72)
∆z
+ lδM,j
−
Pµ
Pµµ
0(zmax)
0(zmax)2(2l + d)δM,j− δM−1,j
∆z
(73)
having replaced the second-order difference scheme by a
first order one and set the second derivative g′′at the
boundary z = zmaxto zero. The eigenvalues of (Aij) can
now be determined using standard procedures.
Our discretization of A is closely related to the un-
derlying physical problem. The functions g(z) on which
A operates describe macroscopic hydrodynamic fluctua-
tions of the chemical potential. Thus they are implic-
itly assumed to be finite and free of a rich local struc-
ture including jumps and kinks. This applies in partic-
ular to the lowest lying modes with only a few radial
nodes. Therefore we can safely assume g(z) to be suf-
ficiently smooth such that the approximations g′(zi) =
(gi+1−gi−1)/(2∆z)+O(∆z2) and g′′(zi) = (gi+1−2gi+
gi−1)/(∆z2)+O(∆z2) hold in the open interval (0,zmax).
However, at the points z0= 0 and zM= zmaxwe run into
problems as z−1and zM+1are not defined. A first guess
might be to assign certain boundary values to g. But
investigating the breathing mode for α = 1 and d = 3 in
Eq. (D6), g(z) = 1−5z/6µ(0) for z ≤ zmax, 0 otherwise,
we recognize none of the sensible boundary conditions
g(zmax) = 0 or g′(zmax) = 0 to be satisfied. This does
not come as a surprise since our description necessarily
breaks down in the outer regions of the cloud where the
static density goes to zero. The crucial point is to pre-
vent the solution from diverging at the boundaries. So we
demand it to be Taylor expandable even there, although
the physical solution being zero for z > zmaxmight have
a discontinuity, and replace the second-order difference
scheme in z−1 and zM+1 by a first-order one, i.e. for
z0= 0
g1− g−1
2∆z∆z
=g1− g0
+ O(∆z). (74)
This can be solved for g0and yields g0≃ (g1+ g−1)/2.
The latter mean-value property is true for any func-
tion which can be linearized around z = 0 provided
a sufficiently small step size.
approximation of g′(zmax) in Eq.
fied. Setting g′′(zmax) = 0 in the same equation implies
gM+1− 2gM+ gM−1= 0 in a discretized version, which
leads us to the same mean-value property.
This simple approach is very efficient. As an example
we show in Tab. I the first ten eigenfrequencies corre-
sponding to a polytropic equation of state P(µ) ∝ µα+1
for different choices of α, d and l compared to the exact
solution given by Eq. (6). The underlying grid of points
is of size M = 2000. We recognize the (more interesting)
lower lying frequencies to converge faster. Of course, the
accuracy can be improved by further increasing M.
In the same way, the
(73) can be justi-

Page 14

14
(i) exact (i) num. (ii) exact (ii) num. (iii) exact (iii) num.
3.46410 3.464101.50214
8.00000 7.999822.35339
12.4900 12.48943.13786
16.6706 16.96933.89609
21.4476 21.44524.64095
25.9230 25.91915.37802
30.3974 30.39166.11010
34.8712 34.86316.83880
39.3446 39.33357.56510
43.8178 43.80318.28963
1.50214
2.35337
3.13777
3.89591
4.64064
5.37752
6.10936
6.83775
7.56367
8.28773
1.00000
2.16025
3.10913
4.00000
4.86484
5.71548
6.55744
7.39369
8.22598
9.05539
1.00000
2.16025
3.10913
4.00000
4.86484
5.71548
6.55744
7.39369
8.22598
9.05539
TABLE I: Exact and numerical frequencies in units of ω0
obtained for an equation of state P = µα+1for (i) α = 0.1,
d = 1, l = 0, (ii) α = 3.9, d = 1, l = 0 and (iii) α = 3, d = 3,
l = 1. All results correspond to a grid of M = 2000 points.
For a particular physical situation it may be favor-
able to use a non-uniform grid, i. e. the points zishall
not be equidistant. For example when we are dealing
with a Bose condensate peaked in the center of the trap
surrounded by a large thermal cloud an equidistant grid
would underestimate the influence of the condensate be-
cause only few points of the matrix correspond to the
central region. Of course, the discretized operator (Aij)
can also be derived for a non-uniform grid. Such a non-
uniform discretization has been applied to calculate the
data points for Fig. 6.
For a Bose gas at nonzero temperature the outer re-
gions of the cloud are in the normal phase and the density
decays only exponentially for large radii, n ∼ Li3/2(eβµ),
where Liν(z) =
?
minimal chemical potential µmin. This is not a physi-
cal quantity but necessary for discretization. Since the
number of points of the grid, M, should not be arbitrary
large for an acceptable computation time, it is favorable
to use an temperature dependent µminwhich takes into
account the shrinking of the thermal cloud as one ap-
proaches T = 0. For Fig. 6 we found it convenient to
use µmin= −4T(T/Tc)2. The size of the thermal cloud
then decreases sufficiently fast as we approach low tem-
peratures. Although the thermal cloud is very small at
low temperatures, it influences the stability of the numer-
ics which leads to scattering in the data. Neglecting the
thermal cloud in the outer regions, i.e. setting µmin= µc,
the data appears smooth. We conclude that this region
requires a more careful numerical implementation.
kzk/kνis the polylogarithm. For
our numerical solution we define an effective radius or
IV.CONCLUSIONS AND OUTLOOK
In this paper we suggest that collective oscillations
may be used as precision observables for trapped ul-
tracold atom gases. The frequencies of these oscilla-
tions reflect the thermodynamic equation of state. Their
measurement can therefore shed light on the validity
of non-perturbative methods for interacting many-body
systems which are used for the computation of the equa-
tion of state. In parallel, the simultaneous measurement
of several frequencies could permit the determination of
thermodynamic variables, in particular the temperature,
with high precision.
In the two-fluid hydrodynamic regime, the excitation
and measurement of collective oscillations results in a
set of discrete numbers which can be used to compare
different theoretical methods. An experimental improve-
ment of the precision of the measurement of collective
modes would set stringent bounds on predictions from
many-body calculations. Indeed, we have shown that
the equation of state allows us to determine the oscilla-
tion frequencies as eigenvalues of a differential operator.
Thus, if broad collections of data on collective oscillation
frequencies would exist, this would necessarily rule out
certain equations of state and theoretical approaches.
The dependence of the collective modes on system pa-
rameters as the scattering length is of particular interest.
We have seen that interaction effects can lead to shifts
of the zero temperature mean field results for the oscil-
lation frequencies of a dilute Bose gas. For the three-
dimensional case these shifts are rather small, however.
Nevertheless, even there non-perturbative effects can be
made visible. The appearance of an effective UV-cutoff
related to the s-wave scattering approximation manifests
itself in a dependence of the equation of state on the gas
parameter and this cutoff. Beyond lowest order perturba-
tion theory we describe the effects of the LHY-correction
for the oscillation spectrum. We also explore the range
of larger scattering length, where perturbation theory is
no longer valid, by means of an equation of state from
Functional Renormalization Group (FRG). For gas pa-
rameters n(0)a3above 10−4the difference between vari-
ous methods for the relative frequency shift reaches the
percent level and may be measurable.
The analysis of the dilute Bose gas in two dimensions
revealed further interesting many-body effects.
again the equation of state from FRG there are sub-
stantial deviations of the collective modes from mean
field theory for large values of the dimensionless coupling
constant. In these cases the difference of the breathing
mode from its mean field value was found to be of order
1% − 10% and showed an extremum for a specific value
of the chemical potential in the center of the trap. Very
interesting would be the investigation of these effects in
a dimensional crossover from three to two dimensions.
A main result of this paper concerns the temperature
dependence of the oscillation frequencies.
to use this as a precision thermometer. As an example
we have computed the temperature dependent oscilla-
tion spectrum based on an equation of state obtained
from Lee–Yang theory. The collective oscillations of the
condensate and the thermal cloud show a rich spectrum
with non-trivial dependence on the temperature, includ-
ing level-crossing and the disappearance of the superfluid
Using
We suggest

Page 15

15
modes at the critical temperature.
The treatment of the oscillation spectrum at non-
vanishing temperature has to account for the coexistence
of two interacting fluids, one for the superfluid conden-
sate and the other for the thermal cloud of uncondensed
atoms. The thermal cloud disappears for T → 0, while
the condensate vanishes as the critical temperature is ap-
proached, T → Tc. Any computation of the spectrum has
to reproduce these limiting cases correctly. In particular,
we show how the zero temperature limit of the two-fluid
hydrodynamic equations has a solution corresponding to
a zero temperature formula for the condensate oscilla-
tions given by Stringari. We also derived a new analytic
result for frequencies at very low temperatures which are
carried by the components of the thermal cloud. Our
method of calculation covers the critical temperature as
well as T > Tc, where the thermal cloud is the only fluid
component. It will be interesting to study the effects
of critical physics related to the second order superfluid
phase transition on the oscillation frequencies.
To measure not only the lowest collective oscillation
frequencies but rather parts of the whole spectrum we
suggest to use response techniques and excite frequencies
individually. Once not only the lowest lying frequencies
are accessible within an appropriate accuracy, but also
higher modes, these will allow for precise verifications of
the consistency of theoretical models.
The formal developments of this paper concern a
method for the computation of oscillation frequencies in
a trap, given the equation of state and normal fluid den-
sity for a homogeneous liquid. Starting from Landau’s
ideal two-fluid hydrodynamics we have derived the gen-
eral expression for a differential operator whose input
is the equation of state given as pressure P(µ,T) and
normal fluid density nn(µ,T). The eigenvalues of this
operator are the collective oscillation frequencies of the
trapped gas. For simplicity we restricted ourselves to
the case of an isotropic, d-dimensional, harmonic trap-
ping. While the extension to an arbitrary spherically
symmetric trapping potential Vext(r) is straightforward,
anisotropy would invalidate our symmetry assumptions.
Although experiments are often performed in not com-
pletely spherically symmetric arrangements, we expect
our findings to be qualitatively correct even for these
cases. We propose a method for numerical implemen-
tation which is based on discretization. This can in prin-
ciple also be done for an anisotropic trapping potential
but with a higher computational effort.
The restriction to the case of ideal hydrodynamics can
be released. To first order, our equations would be mod-
ified due to the five transport coefficients of superfluid
hydrodynamics, including shear viscosity and heat con-
ductivity. Here, immediately many interesting questions
arise: How are these transport coefficients related to
the damping rates of the modes, which appear as the
phenomenological introduced widths in our treatment of
linear response? Can we use collective modes for pre-
cision measurements on the transport coefficients, e.g.
η/s? Is it possible to derive, say, η(µ,T) from a micro-
scopic quantum field theory just like the equation of state
P(µ,T)?
We conclude that collective modes have a high poten-
tial to be used as precision observables, which can quan-
tify different aspects of many-body physics.
worth to improve the techniques used for a precise mea-
surement of oscillation frequencies. In particular, a non-
invasive simultaneous measurement of several frequen-
cies would allow for a precision estimate of the thermo-
dynamic variables temperature and density or chemical
potential.
It seems
Appendix A: System of units
In a relativistic system one usually sets ? = kB= c = 1
in order to express all quantities in terms of energies.
However, for a non-relativistic system the velocity of light
c does not appear in the equations and it is preferable to
set ? = kB = m = 1, where m is a mass. (In our case
this is the mass of the gas atoms.) All quantities can
now be expressed in terms of a length scale of choice.
The spherically symmetric trapping potential
Vext(? x) =m
2ω2
0r2
(A1)
with r2= x2
the oscillator length ℓ0=
quantities are expressed in terms of ℓ0, the latter will no
longer appear in the equations. In other words, we can
set ℓ0 = 1 in our formulas and understand all quanti-
ties to be measured in units of ℓ0. Since ? = m = 1
this is equivalent to expressing everything in powers of
ω0. Thus, the eigenvalues of Ag(z) = ω2g(z) have to be
understood as dimensionless quantities ω/ω0.
We remark that due to m = 1 mass density equals
particle number density, ρ = n, which simplifies our hy-
drodynamic equations.
1+ ··· + x2
dprovides such a scale given by
??/mω0. If all dimensionful
Appendix B: Lee–Yang theory for Bose gas
The equation of state for a weakly interacting Bose gas
in three dimensions has been calculated by Lee and Yang
in Refs. [40]. Starting from the microscopic dispersion
relation ωpthey compute the canonical partition function
and derive the free energy density and other thermody-
namic observables. The system is found to have two dis-
tinct phases, a normal and a superfluid one. Besides this
Lee and Yang also deduce the two-fluid hydrodynamics
for the weakly interacting Bose gas. These equations in-
clude that the mass m∗of the collective excitations of
the normal fluid component does not have to coincide
with the mass m of the atoms. For the present work we
neglect such details and only give a short summary of
the equation of state which enters the Landau two-fluid
hydrodynamics.