Mechanism of collisionless sound damping in dilute Bose gas with condensate
We develop a microscopic theory of sound damping due to Landau mechanism in dilute gas with Bose condensate. It is based on the coupled evolution equations of the parameters describing the system. These equations have been derived in earlier works within a microscopic approach which employs the Pele...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Mechanism of collisionless sound damping in dilute Bose gas with condensate / Yu. Slyusarenko, A. Kruchkov // Condensed Matter Physics. — 2013. — Т. 16, № 2. — С. 23004:1-17. — Бібліогр.: 37 назв. — англ. |
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irk-123456789-1208112017-06-14T03:02:38Z Mechanism of collisionless sound damping in dilute Bose gas with condensate Slyusarenko, Yu. Kruchkov, A. We develop a microscopic theory of sound damping due to Landau mechanism in dilute gas with Bose condensate. It is based on the coupled evolution equations of the parameters describing the system. These equations have been derived in earlier works within a microscopic approach which employs the Peletminskii-Yatsenko reduced description method for quantum many-particle systems and Bogoliubov model for a weakly nonideal Bose gas with a separated condensate. The dispersion equations for sound oscillations were obtained by linearization of the mentioned evolution equations in the collisionless approximation. They were analyzed both analytically and numerically. The expressions for sound speed and decrement rate were obtained in high and low temperature limiting cases. We have shown that at low temperature the dependence of obtained quantities on temperature varies significantly from those one obtained by other authors in the semi-phenomenological approaches. Possible effects connected with non-analytic temperature dependence of dispersion characteristics of the system were also indicated. Побудовано мiкроскопiчну теорiю загасання звуку за механiзмом Ландау у розрiджених газах iз бозе-конденсатом. В основу теорiї було закладено пов’язанi рiвняння еволюцiї для парамерiв опису системи. Цi рiвняння було виведено у бiльш раннiх роботах у мiкроскопiчному пiдходi, що базується на використаннi методу скороченого опису квантових систем багатьох частинок (метод Пелетминського) та моделi Боголюбова для слабко неiдеального бозе-газу з видiленим конденсатом. Отримано рiвняння дисперсiї звукових коливань у системi, що вивчається, шляхом лiнеаризацiї зазначених рiвнянь еволюцiї у беззiткненнєвому наближеннi. Проведено аналiз рiвнянь дисперсiї, як чисельно, так i аналiтично. Одержано аналiтичнi вирази для швидкостi розповсюдження й коефiцiєнта поглинання звуку в розрiджених газах iз бозе-конденсатом у граничних випадках великих та малих температур. Нами продемонстровано, що в областi малих температур температурна залежнiсть знайдених величин суттєво вiдрiзняється вiд тих, що отриманi ранiше iншими авторами в напiвфеноменологiчних пiдходах. Вказано на можливi ефекти, пов’язанi з неаналiтичними залежностями дисперсiйних характеристик системи вiд температури. 2013 Article Mechanism of collisionless sound damping in dilute Bose gas with condensate / Yu. Slyusarenko, A. Kruchkov // Condensed Matter Physics. — 2013. — Т. 16, № 2. — С. 23004:1-17. — Бібліогр.: 37 назв. — англ. 1607-324X PACS: 05.30.-d, 05.30.Jp, 67.85.Hj, 67.85.Jk, 03.75.Hh, 03.75.Kk DOI:10.5488/CMP.16.23004 arXiv:1208.1653 http://dspace.nbuv.gov.ua/handle/123456789/120811 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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| description |
We develop a microscopic theory of sound damping due to Landau mechanism in dilute gas with Bose condensate. It is based on the coupled evolution equations of the parameters describing the system. These equations have been derived in earlier works within a microscopic approach which employs the Peletminskii-Yatsenko reduced description method for quantum many-particle systems and Bogoliubov model for a weakly nonideal Bose gas with a separated condensate. The dispersion equations for sound oscillations were obtained by linearization of the mentioned evolution equations in the collisionless approximation. They were analyzed both analytically and numerically. The expressions for sound speed and decrement rate were obtained in high and low temperature limiting cases. We have shown that at low temperature the dependence of obtained quantities on temperature varies significantly from those one obtained by other authors in the semi-phenomenological approaches. Possible effects connected with non-analytic temperature dependence of dispersion characteristics of the system were also indicated. |
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| author |
Slyusarenko, Yu. Kruchkov, A. |
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Slyusarenko, Yu. Kruchkov, A. Mechanism of collisionless sound damping in dilute Bose gas with condensate Condensed Matter Physics |
| author_facet |
Slyusarenko, Yu. Kruchkov, A. |
| author_sort |
Slyusarenko, Yu. |
| title |
Mechanism of collisionless sound damping in dilute Bose gas with condensate |
| title_short |
Mechanism of collisionless sound damping in dilute Bose gas with condensate |
| title_full |
Mechanism of collisionless sound damping in dilute Bose gas with condensate |
| title_fullStr |
Mechanism of collisionless sound damping in dilute Bose gas with condensate |
| title_full_unstemmed |
Mechanism of collisionless sound damping in dilute Bose gas with condensate |
| title_sort |
mechanism of collisionless sound damping in dilute bose gas with condensate |
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Інститут фізики конденсованих систем НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/120811 |
| citation_txt |
Mechanism of collisionless sound damping in dilute Bose gas with condensate / Yu. Slyusarenko, A. Kruchkov // Condensed Matter Physics. — 2013. — Т. 16, № 2. — С. 23004:1-17. — Бібліогр.: 37 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT slyusarenkoyu mechanismofcollisionlesssounddampingindilutebosegaswithcondensate AT kruchkova mechanismofcollisionlesssounddampingindilutebosegaswithcondensate |
| first_indexed |
2025-07-08T18:37:00Z |
| last_indexed |
2025-07-08T18:37:00Z |
| _version_ |
1837104963914825728 |
| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 2, 23004: 1–17
DOI: 10.5488/CMP.16.23004
http://www.icmp.lviv.ua/journal
Mechanism of collisionless sound damping
in dilute Bose gas with condensate
Yu. Slyusarenko1,2∗, A. Kruchkov 2†
1 Akhiezer Institute for Theoretical Physics, National Science Center Kharkiv Institute of Physics and Technology,
1 Akademichna St., 61108 Kharkiv, Ukraine
2 Karazin National University, 4 Svobody Sq., 61077 Kharkiv, Ukraine
Received November 24, 2012, in final form March 18, 2013
We develop a microscopic theory of sound damping due to Landau mechanism in dilute gas with Bose conden-
sate. It is based on the coupled evolution equations of the parameters describing the system. These equations
have been derived in earlier works within a microscopic approach which employs the Peletminskii-Yatsenko
reduced description method for quantum many-particle systems and Bogoliubov model for a weakly nonideal
Bose gas with a separated condensate. The dispersion equations for sound oscillations were obtained by lin-
earization of the mentioned evolution equations in the collisionless approximation. They were analyzed both
analytically and numerically. The expressions for sound speed and decrement rate were obtained in high and
low temperature limiting cases. We have shown that at low temperature the dependence of the obtained quan-
tities on temperature significantly differs from those obtained by other authors in the semi-phenomenological
approaches. Possible effects connected with non-analytic temperature dependence of dispersion characteristics
of the system were also indicated.
Key words: dilute Bose gas, Bose-Einstein condensate (BEC), microscopic theory, sound, Landau mechanism,
dispersion relations, speed of sound, damping rate
PACS: 05.30.-d, 05.30.Jp, 67.85.Hj, 67.85.Jk, 03.75.Hh, 03.75.Kk
1. Introduction
The study of mechanisms of sound damping in a Bose-Einstein condensate (BEC) has a long history.
Calculation of the sound damping rate in systems with BEC is a rather complicated theoretical problem.
First expressions for damping rate in such systems have been apparently obtained in [1, 2] for the spa-
tially homogeneous case.
The direct experimental observation of BEC [3–5] has stimulated a great number of works devoted to
various aspects of this phenomenon (see, for example, [6, 7] and references therein). A number of papers,
both theoretical and experimental, deal with the problem of propagation and damping of excitations in
Bose gases with the presence of condensate [8–17].
It is currently assumed that the Landau damping is the most probable mechanism of sound relax-
ation in the so-called trapped Bose condensates. This mechanism consists in collisionless absorption of
oscillation energy by quanta of elementary excitations [12–14]. In this regard, recall that the existence of
specific collective excitations in a gas with BEC has been known since the pioneering work of Bogoliubov
[18]. In this paper, a special perturbation theory was proposed for a weakly non-ideal and spatially ho-
mogeneous Bose gas with condensate in which the repulsive interaction acts between atoms. This theory
predicts the elementary excitation spectrum for such system at zero temperature. At small wave vectors,
it coincides with the spectrum of sound oscillations in a condensed Bose gas.
∗E-mail: slusarenko@kipt.kharkov.ua
†E-mail: aleks.kryuchkov@gmail.com
© Yu. Slyusarenko, A. Kruchkov, 2013 23004-1
http://dx.doi.org/10.5488/CMP.16.23004
http://www.icmp.lviv.ua/journal
Yu. Slyusarenko, A. Kruchkov
The first work, which proposed a method to calculate the sound damping rate in trapped BEC due to
Landau mechanism, is apparently the paper [12] (see also [14]). The approach developed by the authors
uses the perturbation theory and is based on the calculation of the difference in probabilities between
emission and absorption of quanta of oscillations by elementary excitations in a system. It should be
noted that trapped condensates represent a spatially inhomogeneous system. This fact essentially compli-
cates analytical calculations. The method of [12, 13] was shown to be suitable for numerical calculations
of sound damping in a trapped condensate (see the same paper). For a spatially homogeneous BEC, this
method has provided analytical formulae for damping according to Landau mechanism [12, 13]. In this
case, the authors reproduced the results obtained in [1, 2].
It should be stressed that the formulae of [12, 13] (and, hence, the results of [1, 2]) can be obtained by
another method involving the kinetic equation for distribution function of elementary excitations (see,
e.g., [6]). In other words, the semi-phenomenological approach to the calculation of the damping rate,
shown in [6], is equivalent to the method of [12, 13]. It was inter alia indicated in [12, 13]. However, it
is clear that in the most general case, the mere use of kinetic equation for excitations is insufficient. The
system should be described by the coupled evolution equations which take into account the mutual effect
of condensate density, phase (or superfluid velocity) and distribution function of elementary excitations.
A consistent derivation of a system of coupled equations can be achieved using a microscopic approach,
proceeding from the first principles. This problem was solved in [10, 19, 20, 22] within a microscopic
approach based on the reduced description of quantum many-particle systems [23, 24] and Bogoliubov
model for a weakly non-ideal Bose gas with condensate [18]. The synthesis of the approaches elaborated
in [18] and [24] made it possible to obtain in [19, 20] the kinetic equation for distribution function of
elementary excitations coupled with the evolution equations for condensate density and superfluid mo-
mentum. Note that the validity of such a system of equations is confirmed by its controlled derivation
within the framework of special perturbation theory with weak interparticle interaction. Furthermore,
the following fact speaks in favour of the mentioned coupled equations: they have been employed in
[21, 22] to derive hydrodynamic equations of a superfluid in which the smallness of the difference be-
tween superfluid and normal velocities is not taken into account. As this difference tends to zero, the
obtained equations are reduced to the well-known Khalatnikov hydrodynamic equations (see e.g. [25]).
Also note that evolution equations from [18, 19] were in fact reproduced in [26] in another microscopic
approach. This circumstance was mentioned in [26] with an appropriate reference link.
Notwithstanding the above considerations, the equations of [20, 26] have not yet been used to study
the propagation and damping of sound in a dilute gas with BEC. However, the same considerations allow
us to hope that the correct solution of evolution equations found in [20] should lead us to the correct
expression for sound damping rate in a gas with BEC. The present paper is devoted to the study of a colli-
sionless mechanism of sound damping in dilute gases with BEC on the basis of general dynamic equations
of such systems obtained in [20] from the first principles. As will be seen later, the results obtained in the
present work significantly differ in some cases from those of [1, 2] and, consequently, of [12, 13] (see also
[27–29]). For example, in the present study it is shown that in the collisionless approximation the sound
damping rate at low temperature is quadratic on temperature, γ ∼ T 2, whereas the results of [1, 12, 13]
give γ∼ T 4 dependence.
2. Kinetics of spatially inhomogeneous Bose gas with the presence of
condensate
Constructing spatially inhomogeneous Bose gas kinetics, authors of [20] started from the Liouville
equation for the statistical operator ρ(t)
i
∂ρ (t)
∂t
= [Ĥ ,ρ (t)], (2.1)
where Ĥ = Ĥ0 + V̂ is Hamiltonian of the system, consisting of the ideal gas Hamiltonian Ĥ0
Ĥ0 =
1
2m
∫
d3r
∂ψ̂+ (r)
∂rk
∂ψ̂(r)
∂rk
(2.2)
23004-2
Mechanism of collisionless sound damping in dilute Bose gas with condensate
and binary interaction Hamiltonian V̂
V̂ =
1
2
∫
d3r
∫
d3Rψ̂+ (r+R)ψ̂+ (r)V (|R|)ψ̂ (r)ψ̂ (r+R) . (2.3)
Equation (2.1) is written in the units in which Planck’s constant ħ is equal to unity. In the formulae
(2.2), (2.3) m is boson mass, V (|R|) is binary interaction potential, which depends only on the distance
between particles, and ψ̂+ (r) , ψ̂ (r) are field operators. The quasiparticle distribution function fp (r, t),
the order parameterψ(r, t) = |Spρ(t)ψ̂ (r) | and the superfluid velocity vk (r, t)= m−1 ∂
∂rk
Imln Spρ(t)ψ̂(r)
were selected as parameters to describe weakly non-ideal Bose gas with condensate in kinetic stage of
system evolution in [19].
Using the method of reduced description [24] combined with the special perturbation theory [18]
made it possible to obtain in [20] the following system of equations for the parameters fp (r, t),ψ(r, t) and
vk (r, t). Since the general form of equations from [20] is not required in the present paper, we introduce
it here in collisionless approximation:
∂ fp
∂t
+
∂ fp
∂rk
∂
∂pk
εp (n,v)−
∂ fp
∂pk
∂
∂rk
εp (n,v)+o (λ) = 0,
∂n
∂t
+ ∂
∂rk
(vk n) = 0,
∂vk
∂t
+ ∂
∂rk
{
v2
2
+h
(
f ,n
)
}
= 0, (2.4)
where instead of description parameter ψ(r, t)we have introduced a new variable n(r, t), that is conden-
sate density [22]
ψ2(r, t) ≡ n(r, t), (2.5)
and the quantities εp (n,v) and h
(
f ,n
)
are given by expressions:
εp (n,v) ≡ωp (n)+p ·v, h
(
f ,n
)
= 1
m
〈
νp
εp
ωp
〉
+ ν0
2m
〈
αp
ωp
〉
+h0 (n)+o
(
λ4
)
,
h0 (n) =
1
V
∑
p,0
1
2m
(
νp
εp −ωp
ωp
+ν0
αp −ωp
ωp
)
+ ν0n
m
, (2.6)
where
ωp (n) ≡
{
εp
[
εp +2βp (n)
]}1/2
, αp (n) ≡ εp +βp (n) , εp ≡ p2
2m
, βp (n) ≡ νpn , (2.7)
and the following notations were also introduced
〈
Ap
〉
≡ 1
(2π)3
∫
d3p fp Ap , ν0 ≡ νp=0 , νp ≡
∫
d3R V (|R|)exp
(
−ip ·R
)
, (2.8)
where Ap is an arbitrary function of p.
Let us recall that the basis of Bogoliubov equilibrium state theory of Bose gas in the presence of
interaction is the assumption that ψ∼ λ−1, see [18], where quantity λ characterizes the smallness of the
interaction between the particles, V (|R|) ∼λ2. Furthermore, it is believed that the order of magnitude of
ψ(r, t) does not change after differentiation ψ(r, t)with respect to r, ∂ψ(r, t)/∂rk ∼ λ−1. In the formulae
(2.4)–(2.6), the notation o (λn )means the quantity in order of magnitude of λn .
We emphasize once again that in our paper we study the mechanism of collisionless sound damping
in a gas with a BEC (Landau mechanism). For this reason, in equations (2.4)–(2.6) we have omitted the
terms associated with the presence of interparticle collision term. The explicit form of the collision term
for quasiparticles can be found in [19, 20]. Here, we note only the fact that the quasiparticle collision term
Lp
(
f ,ψ
)
vanishes by substituting the stationary Bose distribution function f 0
p
Lp
(
f 0,ψ
)
= 0, f 0
p =
[
exp
(ωp −p ·v
T
)
−1
]−1
, (2.9)
with the chemical potential of quasiparticles being equal to zero; here T is temperature in energy units.
The chemical potential having vanished reflects the fact that the number of quasiparticles is not con-
served during collisions [19, 22].
23004-3
Yu. Slyusarenko, A. Kruchkov
3. Sound dispersion equations in diluted gas with BEC
To investigate the propagation of sound in gas with BEC, we linearize coupled equations (2.4), (2.6),
(2.7) with respect to spatially homogeneous equilibrium state according to the following formulae
n(r, t) = n0 + ñ(r, t), v(r, t)= ṽ(r, t), fp(r, t)= f 0
p + f̃p(r, t),
n0 ≫|ñ(r, t)|, f 0
p ≫| f̃p(r, t)|, (3.1)
where n0 = n0 (T ) is the equilibrium value of the condensate density in the system at the temperature
T . The equilibrium value of the velocity v0 is considered to be equal to zero in the second formula of
(3.1). Thus, the velocity |ṽ(r, t)| is supposed to be of the order of magnitude of ñ(r, t) and f̃p(r, t). Then,
the equilibrium distribution function of quasiparticles pursuant to (2.9) is given by the expression
f 0
p =
[
exp
(ωp
T
)
−1
]−1
. (3.2)
In this formula, like in all subsequent expressions, we omit the index “0” in the designation of
the equilibrium value of ω0
p ≡ ωp (n0) to avoid encumbering the computations. Note that the quantity
h
(
f 0,n0
)
defined by (2.6) represents the chemical potential of atomic Bose gas µ = h
(
f 0,n0
)
(see in this
regard [19, 20]).
Deviations of the corresponding quantities from their equilibrium values were denoted by ñ(r, t),
ṽ(r, t) and f̃p(r, t).
The equations of motion for these variables can be represented as follows:
∂
∂t
ñ(r, t)+n0
∂
∂rk
ṽk (r, t)= 0,
∂
∂t
ṽk (r, t)+
ν0
m
∂
∂rk
ñ(r, t)+
∫
d3pKp
∂
∂rk
f̃p(r, t)= 0,
∂
∂t
f̃p(r, t)+
∂ωp
∂pk
∂
∂rk
f̃p(r, t)−
∂ f 0
p
∂pk
∂
∂rk
[
p · ṽ(r, t)+
νpεp
ωp
ñ(r, t)
]
= 0, (3.3)
where we have introduced the following notation
Kp = 1
2m (2π)3
εp
(
ν0 +2νp
)
+νpν0n0
ωp
. (3.4)
Recall that in the present paper, the Planck constant ħ is considered to be equal to unity. Note also
that while deriving equations (3.3), we have discarded the terms that are higher than the first order of
magnitude λ. The circumstance of (2.5) stating that n0 ∼λ−2 or i.e. νpn0 ∼λ0 was taken into account.
Proceeding further to the Fourier transform of the quantities ñ(r, t), ṽ(r, t), f̃p(r, t) in equations (3.3)
as provided by formula
ζ̃(r, t) =
∞
∫
−∞
dω
∫
d3q eiqr−iωtζ(q,ω), (3.5)
where ζ̃(r, t) should be understood as either of the considered quantities, we obtain
ωn(q,ω) = n0q ·v
(
q,ω
)
,
ωvk (q,ω) = ν0
m
qk n
(
q,ω
)
+qk A
(
q,ω
)
,
(
ω−qk
∂ωp
∂pk
)
fp
(
q,ω
)
=−qk
∂ f 0
p
∂pk
[
p ·v
(
q,ω
)
+
νpεp
ωp
n
(
q,ω
)
]
, (3.6)
where we denote [see (3.3), (3.4)]
A
(
q,ω
)
≡
∫
d3p fp
(
q,ω
)
Kp = 1
2m (2π)3
∫
d3p fp
(
q,ω
) εp
(
ν0 +2νp
)
+νpν0n0
ωp
. (3.7)
23004-4
Mechanism of collisionless sound damping in dilute Bose gas with condensate
In the formulae (3.6), (3.7) in Fourier transforms of the quantities ñ(r, t), ṽ(r, t), f̃p(r, t) we omit the
’tilde’ sign. Further expressing the values n(q,ω) and v
(
q,ω
)
from the first two equations of (3.6) in terms
of A
(
q,ω
)
,
n(q,ω)=
n0q2
ω2 −u2
0 q2
A
(
q,ω
)
, vk (q,ω) =
ωqk
ω2 −u2
0 q2
A
(
q,ω
)
, (3.8)
the third equation of (3.6) can be written in the form
(
ω−qk
∂ωp
∂pk
)
fp
(
q,ω
)
=−qk
∂ωp
∂pk
∂ f 0
p
∂ωp
p ·qωωp +νpεp n0q2
ωp
(
ω2 −u2
0 q2
) A
(
q,ω
)
, (3.9)
where u0 is referred to as speed of zero sound in Bose gas
u2
0 =
ν0n0
m
. (3.10)
As is readily seen, equation (3.9) subject to (3.7) is an integral equation for distribution function
Fourier transform fp
(
q,ω
)
. The solution of this equation can be represented as in [30]:
fp
(
q,ω
)
= Bp
(
q
)
δ
(
ω−q ·up
)
−q ·up
∂ f 0
p
∂ωp
p ·qωωp +νpεp n0q2
ωp
(
ω2 −u2
0 q2
)(
ω−q ·up + i0
) A
(
q,ω
)
, (3.11)
where we have introduced the following notation:
up ≡
∂ωp
∂p
, (3.12)
and Bp
(
q
)
is an arbitrary function, which is required to impose the following restriction: the distribution
function f̃p(r, t), calculated in accordance with (3.1), (3.5) and (3.11) should be small compared with the
equilibrium distribution function f 0
p .
It implies also from (3.5) that Bp
(
q
)
must satisfy the following relation:
B∗
p
(
q
)
= Bp
(
−q
)
. (3.13)
We denote the whole valid set of such functions by Bσp
(
q
)
, where σ is a continuous or discrete sym-
bolic parameter such that the functions Bp
(
q
)
≡ Bσp
(
q
)
may depend on σ. The reason for introducing
this index may consist, for example, in the following: the set of functions Bσp
(
q
)
should be sufficient to
build an arbitrary value for the distribution function f̃p(r, t) at the initial time t = 0 (in this context see
also [31]).
Formula (3.11) permits to find the value A
(
q,ω
)
in terms of Bσp
(
q
)
functions:
Aσ
(
q,ω
)
= Bσ
(
q,ω
)
ε−1
(
q,ω
)
, (3.14)
where
Bσ
(
q,ω
)
= 1
2m (2π)3
∫
d3p
ν0n0νp +εp
(
ν0 +2νp
)
ωp
Bσp
(
q
)
δ
(
ω−q ·up
)
(3.15)
and
ε
(
q,ω
)
≡ 1+ 1
2m (2π)3
∫
d3p
∂ f 0
p
∂ωp
(
q ·pωωp +νpεp n0q2
) q ·up
[
ν0n0νp +εp
(
ν0 +2νp
)]
ω2
p
(
ω2 −u2
0 q2
)(
ω−q ·up + i0
) . (3.16)
The expression (3.11) for the given values fp
(
q,ω
)
subject to (3.14)–(3.16) can now be represented in
the form:
fσp
(
q,ω
)
= Bσp
(
q
)
δ
(
ω−q ·up
)
+ε−1
(
q,ω
)
q ·up
∂ f 0
p
∂ωp
p ·qωωp +νpεp n0q2
ωp
(
ω2 −u2
0 q2
)(
ω−q ·up + i0
) Bσ
(
q,ω
)
. (3.17)
23004-5
Yu. Slyusarenko, A. Kruchkov
We note that in case of charged particles gas, the quantity ε
(
q,ω
)
[see (3.16)]
ε
(
q,ω
)
= ε1
(
q,ω
)
+ iε2
(
q,ω
)
(3.18)
represents complex dielectric permittivity of the system (see e.g., [24]). It is known that the presence of
an imaginary term in dielectric permittivity indicates the energy dissipation of electromagnetic waves
with dispersion relation that should be obtained from the equation
ε
(
q,ω0
(
q
)
− iγq
)
= 0. (3.19)
Moreover, the wave decrement γq is determined by an imaginary part ε2
(
q,ω
)
of the value ε
(
q,ω
)
.
For this reason, weakly damped oscillations in the system
∣
∣ω0
(
q
)∣
∣≫ γq (3.20)
can exist if only
∣
∣ε1
(
q,ω
)∣
∣≫
∣
∣ε2
(
q,ω
)∣
∣ , (3.21)
besides, as the consequence of (3.18)–(3.21) (see in this regard [26, 27]), the frequencyω0
(
q
)
can be found
from the equation
ε1
(
q,ω0
(
q
))
= 0 (3.22)
and the damping rate γq is given by expression
γq =
[
∂ε1
(
q,ω
)
∂ω
]−1
ω=ω0
ε2
(
q,ω0
(
q
))
. (3.23)
Despite the fact that in the present paperwe investigate a neutral system, the existence of longitudinal
oscillations is also associated with the existence of zeros of the function ε
(
q,ω
)
. In this case, there is a
complete analogy with the mentioned case of longitudinal oscillations in systems of charged particles.
That is, the structure of solution (3.17) is such that weakly damped waves may also be excited in the
system investigated, in accordance with (3.8) and (3.5), and the dispersion law is determined by (3.19)–
(3.21). As we shall show in the following section, such waves would represent sound waves in weakly
nonideal Bose gas with condensate. The value γq obtained according to (3.17)–(3.20) will determine the
damping of sound in the system investigated.
4. Sound damping rate in dilute Bose gas with condensate
To solve this problem it is necessary to determine real ε1
(
q,ω
)
and imaginary ε2
(
q,ω
)
parts of the
quantity ε
(
q,ω
)
. For this purpose in the expression (3.16) we use formula
1
x + i0
= P
1
x
− iπδ(x) , (4.1)
where the symbol P means that the further integration is taken in the sense of Cauchy principal value. Af-
ter some hackneyed transformations one can obtain the following expressions for ε1
(
q,ω
)
and ε2
(
q,ω
)
:
ε1
(
q,ω
)
= 1+ 1
2m (2π)3
P
∫
d3p
∂ f 0
p
∂ωp
q ·up
(
q ·pωωp +νpεp n0q2
)[
ν0n0νp +εp
(
ν0 +2νp
)]
ω2
p
(
ω2 −u2
0 q2
)(
ω−q ·up
) ,
ε2
(
q,ω
)
=− πω
2m (2π)3
∫
d3p
∂ f 0
p
∂ωp
(
q ·pωωp +νpεp n0q2
)
[
ν0n0νp +εp
(
ν0 +2νp
)]
ω2
p
(
ω2 −u2
0 q2
) δ
(
ω−q ·up
)
. (4.2)
Further calculations are not possible without specifying the explicit form of νp that is the Fourier
transform of interaction potential V (|R|) [see (2.3), (2.8)]. To simplify calculations it is often accepted to
replace the interaction potential V (|R|) by the following effective potential (see [6, 7])
V (|R|) = ν0δ(R) , ν0 =
4πasc
m
, (4.3)
23004-6
Mechanism of collisionless sound damping in dilute Bose gas with condensate
where asc is the so-called s-wave scattering length (for details see [6]). Thus, we have νp ≡ ν0. Taking it
into account, we have ωp ≡ωp and the value up [see (2.7), (3.12)] can be expressed in the form
up = p
p
up , up ≡
∂ωp
∂p
=
εp +ν0n0
ωp
p
m
. (4.4)
Formulae (4.3) and (4.4) enable us to perform integration in (4.2) over the angle between vectors q
and p, and then ε1
(
q,ω
)
can be represented in the form
ε1
(
q,ω
)
= 1− ν0
4π2
(
ω2 −u2
0 q2
)
∞
∫
0
dpp2
ω2
p
∂ f 0
p
∂ωp
(
p
mup
ω2ωp +εp u2
0 q2
)
×
(
n0ν0 +3εp
)
(
1− ω
2qup
ln
∣
∣
∣
∣
ω+qup
ω−qup
∣
∣
∣
∣
)
. (4.5)
After the same angle integration, the second expression of (4.2) can be written as follows:
ε2
(
q,ω
)
=− ν0ω
8πm
(
ω2 −u2
0 q2
)
∞
∫
0
dpp2
ω2
p up
∂ f 0
p
∂ωp
(
p
up
ω2ωp +εpν0n0q2
)
(
n0ν0 +3εp
)
θ
(
qup −|ω|
)
, (4.6)
where θ (x) is Heaviside step function.
We now determine, following works [12, 13], the sound damping rate in the case of low (T ≪ ν0n0)
and high (T ≫ ν0n0 but T < Tc) temperatures. To study the case of low temperatures it is convenient
to use the variable z ≡ωp /T in the integration in (4.5) and (4.6). The variables p , εp , up in terms of the
variable z can be expressed using the ωp explicit expression (2.7) and (4.4). After carrying out rather
cumbersome but necessary calculations, the result can be summarized as follows:
ε1
(
q,ω
)
= 1−
F
a
(
ω2 −u2
0 q2
)
∞
∫
0
dz
∂ f 0 (z)
∂z
g
(
q,ω, z
)
[
1−
ω
2qu0v (z)
ln
∣
∣
∣
∣
ω+qu0v (z)
ω−qu0v (z)
∣
∣
∣
∣
]
,
ε2
(
q,ω
)
= − πF
2a
(
ω2 −u2
0 q2
)
∞
∫
0
dz
∂ f 0 (z)
∂z
g
(
q,ω, z
) ω
qu0v (z)
θ
(
qu0v (z)−|ω|
)
,
f 0 (z) = 1
ez −1
, (4.7)
where functions g
(
q,ω, z
)
and v (z) are given by:
g
(
q,ω, z
)
≡
p
2
√p
1+a2z2 −1
(
3
p
1+a2z2 −2
)
z
(
1+a2z2
)
[
q2u2
0
(
1+a2z2 −
√
1+a2z2
)
+ω2a2z2
]
,
v (z) ≡
p
2
p
1+a2z2
√p
1+a2z2 −1
az
, (4.8)
and the following notations were introduced [recall u0 still given by (3.10)]:
a ≡ T
ν0n0
, F ≡ 1
4π2
ν0m2u0 =
2
p
π
√
n0a3
sc . (4.9)
Deriving the last formula of (4.9) it is necessary to use the expressions (3.10) and (4.3). The quantity
n0a3
sc is referred to as gas parameter, see [6, 7]. That is, in diluted gases the following relation should be
satisfied:
n0a3
sc ≪ 1. (4.10)
23004-7
Yu. Slyusarenko, A. Kruchkov
It is easy to verify that the values ε1
(
q,ω
)
and ε2
(
q,ω
)
can be represented as functions of one dimen-
sionless variable s ≡ω
(
q
)/
u0q
ε1 (s) = 1− F
a
(
s2 −1
)
∞
∫
0
dz
∂ f 0 (z)
∂z
g (s, z)
[
1− s
2v (z, a)
ln
∣
∣
∣
∣
s + v (z, a)
s − v (z, a)
∣
∣
∣
∣
]
,
ε2 (s) =−
πF s
2a
(
s2 −1
)
∞
∫
0
dz
∂ f 0 (z)
∂z
g (s, z)
v (z, a)
θ
(
v (z, a)−|s|
)
, (4.11)
where
g (s, z) ≡
p
2
√p
1+a2z2 −1
(
3
p
1+a2z2 −2
)
z
(
1+a2z2
)
(
1+a2z2 −
√
1+a2z2 + s2a2z2
)
, (4.12)
and for these functions the symmetry conditions are valid
ε1 (s) = ε1 (−s) , ε2 (s) = ε2 (−s) . (4.13)
Expressions (3.22) and (3.23) that determine the dispersion and decrement (or increment) of small
oscillations in the system investigated, can be written in the following form by taking into account (4.11):
ε1 (s0) = 0, s0 ≡
ω0
(
q
)
u0q
, γq = u0q
[
ε2(s)
(
∂ε1(s)
∂s
)−1]
s=s0
. (4.14)
As is easily seen, these oscillations have linear spectrum. In virtue of (4.11) and (4.14) the structure of
dispersion equation is such that the unknown value s, which determines the oscillation frequency as a
function of wave vector, does not depend on wave vector itself.
In accordance with (3.22) and (4.14), the dependence of oscillation frequency ω0 on wave vector q in
this system should be determined by the solution of equation
s2
0 −1= F
a
∞
∫
0
dz
∂ f 0 (z)
∂z
g (s0, z)
[
1− s0
2v (z)
ln
∣
∣
∣
∣
s0 + v (z)
s0 − v (z)
∣
∣
∣
∣
]
. (4.15)
We note that dispersion equation (4.15) is similar to the dispersion equation of zero sound in a normal
Fermi liquid, (compare e.g., with the corresponding formula in [30]). Further, taking into account this
equation, to calculate the derivative ∂ε1 (s)/∂s appearing in (4.14), the damping rate γq can be presented
as follows:
γq
u0q
=−π
2
F s0
a b (s0)
∞
∫
0
dz
∂ f 0 (z)
∂z
g (s0, z)
v (z)
θ
(
v (z)−|s0|
)
, (4.16)
where
b (s0) ≡
[
(
s2 −1
) ∂ε1 (s)
∂s
]
s=s0
=
s2
0 +1
s0
+ F
a
∞
∫
0
dz
∂ f 0 (z)
∂z
[
g (s0, z)
s0
− ∂g (s0, z)
∂s0
]
+F s0
2a
∂
∂s0
∞
∫
0
dz
∂ f 0 (z)
∂z
g (s0, z)
v (z)
ln
∣
∣
∣
∣
s0 + v (z)
s0 − v (z)
∣
∣
∣
∣
. (4.17)
We emphasize that the above mentioned expressions (4.7)–(4.17) are exact, despite the fact that we
have modified them to the form suitable to study the low temperature regime.
As is readily seen, equation (4.15) in the general case can be solved only by numerical methods. Fig-
ure 1 shows the dependence s (a) [see (4.11) and (4.9)] obtained as a result of numerical solution of equa-
tion (4.15) for F = 10−3. It is evident that the function s (a) behaves nonmonotonously as a changes. At
low a region (the case of low temperature), an increase of the function is observed. At the point a ≈ 0.43
it reaches maximum, and then s (a) decreases monotonously as a increases, and in the point a ≈ 0.847
23004-8
Mechanism of collisionless sound damping in dilute Bose gas with condensate
Figure 1. (Color online) Dependence of dimensionless speed of sound s (a) ≡ ω0
(
q, a
)
/u0q on tempera-
ture [dimensionless quantity a, see (4.9)] obtained by a numerical solution of (4.15) for F = 10−3 (solid
line). Dots shows the intersection with line s = 1 (dashed one) in points a = 0 and a ≈ 0.847.
s (a) it is equal to unity again. As mentioned above, the increase of the value a is restricted at least to the
critical temperature, see (4.9). At zero temperature (a = 0) we deal with a classical zero sound in Bose
system, because s = 1, and hence ω0
(
q
)
= u0q due to (4.11). This result is naturally expected. However,
we note once again that at the point a ≈ 0.847 there also holds s = 1 and the frequency of sound in dilute
gas with Bose condensate is again equal to the frequency of zero sound in such a system. The fact that we
discovered had not been mentioned in the literature. This may be due to the use of the evolution equa-
tions of the system in the present work, that were obtained within the microscopic approach, as well as
due to the numerical solution of the dispersion equations. Whatever the case is, the very existence of the
second point of a sound dispersion curve with s = 1, that is a ≈ 0.847 (or T ≈ 0.847 ν0n0) for F = 10−3,
requires an individual physical interpretation. Some more comments regarding this point will be given
below.
In the regions of low (T ≪ ν0n0) and high (T ≫ ν0n0 but T < Tc) temperatures a solution of (4.15)
can be expressed in analytical form. Thus, one can obtain analytical expressions for the quantity γq in
two limiting cases. To do it, as we shall see, one need to involve a numerical analysis as an auxiliary
technique.
Consider first the case of low temperatures. By virtue of inequality (low temperature regime, as men-
tioned above)
a = T
ν0n0
≪ 1 (4.18)
and the rapid decrease of the function f 0 (z) as z →∞ [see (4.7)], functions g
(
q,ω, z
)
and u (z) can be
expanded in power series of az in (4.11)–(4.13):
g (s, z) ≈ a3z2
(
s2 + 1
2
)
, v (z) ≈ 1+ 3
8
a2z2. (4.19)
As will become apparent from the subsequent formulae, such an expansion corresponds to the devel-
opment of perturbation theory with respect to a small parameter a. Taking into account the expansions
(4.19), the dispersion equation (4.15) can be written as:
s2
0 −1 = F a2
(
s2
0 +
1
2
)
∞
∫
0
dz z2 ∂ f 0 (z)
∂z
[
1− s0
2v (z)
ln
∣
∣
∣
∣
∣
s0 +1+ 3
8
a2z2
s0 −1− 3
8 a2z2
∣
∣
∣
∣
∣
]
,
s0 ≡
ω0
(
q
)
u0q
. (4.20)
23004-9
Yu. Slyusarenko, A. Kruchkov
In the region of small a, i.e., low temperatures, as we have seen in figure 1, s ≈ 1. That is to say, the
value δs ≡ s −1 should be much less than unity, |δs| ≪ 1. Therefore, the equation (4.20) admits further
simplification
δs ≈ 3
4
F a2
∞
∫
0
dz z2 ∂ f 0 (z)
∂z
(
1− 1
2
ln 2+ 1
2
ln
∣
∣
∣
∣
δs − 3
8
a2z2
∣
∣
∣
∣
)
. (4.21)
When F ≪ 1 and a ≪ 1 [see (4.9), (4.14)] the solution of (4.21) can exist only in the δs ≪ a2 ≪ 1 do-
main. This inequality makes it possible to neglect the value δs within the logarithm term in the integrand
in (4.21) as the first approximation of perturbation theory. As a result, we obtain
δs = F a2d , δs ≡ s −1, (4.22)
where the notation d is
d =
3
4
∞
∫
0
dz
z2ez
(ez −1)2
(
1
2
ln 2−1−
1
2
ln
3
8
a2z2
)
. (4.23)
Formula (4.22) is also confirmed by numerical calculations. Figure 2 shows the dependence d (δs),
δs ≡ s −1, computed according to expression (4.21)
d (δs) ≡
3
4
∞
∫
0
dz
z2ez
(ez −1)2
(
1
2
ln 2−1− 1
2
ln
∣
∣
∣
∣
δs − 3
8
a2z2
∣
∣
∣
∣
)
. (4.24)
Figure 2. Dependence d (δs) plotted numerically according to (4.24) for a = 0.1.
As can be seen from figure 2, the equality d (δs) ≈ d ≈ 4.4 for a = 0.1 holds up to the second decimal.
As a consequence, in the expression for d that is given above, the dependence on δs can be neglected.
Then, one can derive:
d ≈−π2
4
ln a −1.27, (4.25)
where the second term was calculated numerically, and, deriving the first term, the following integral
value was used: ∞
∫
0
dz
z2ez
(ez −1)2
= π2
3
.
Thus, the solution (4.22) shows the following dependence of the sound frequency in dilute gas with BEC
on the low temperature range:
ω0
(
q
)
≈±u0q
(
1+F a2d
)
, a =
T
ν0n0
≪ 1. (4.26)
23004-10
Mechanism of collisionless sound damping in dilute Bose gas with condensate
Figure 3. (Color online) Analytical dependence of dimensionless speed of sound s ≡ ω0
(
q, a
)
/u0q on
temperature (dimensionless quantity a) in case of low temperatures, plotted according to expressions
(4.22), (4.25) for F1 = 3 ·10−4 , F2 = 10−3 and F3 = 1.5 ·10−3 respectively.
Analytical dependence s (a) in case of a ≪ 1 in accordance with (4.22) and (4.27) is shown in figure 3
for three different values of F . Note that sound speed temperature corrections in a gas with BEC were
also obtained in [27]. In contrast to our results that are shown by formulae (4.25), (4.26), corrections in
[27] have δs ∼ a4 ln a temperature dependence.
The damping rate of sound in BEC calculated according to formulae (4.16), (4.17) and using (4.19),
(4.22)–(4.26) in the first order approximation with respect to a is given by a fairly simple expression
γq
ω0
(
q
) ≈
γq
u0q
≈ π3
8
F a2 = π5/2
2
√
n0a3
sc
(
T
ν0n0
)2
. (4.27)
If we assume that at low temperatures [see (4.18)] all the Bose gas is in condensate [12, 13], n0 (T ) ≈
n0 (0) = n0, the quadratic dependence of decrement γq on temperature follows from (4.26). This be-
haviour of sound decrement at low temperatures significantly differs from that given in [1, 12, 13]. In
these papers the value γq is known to depend on temperature in low-temperature regime as T 4. This
circumstance, as well as the above mentioned difference in δs temperature dependence, is clearly the
consequence of our using the evolution equations of the system studied, derived in microscopic approach
from the first principles. The quadratic dependence of γq on temperature shows that sound damping in
dilute Bose gas with condensate can occur faster than it was expected before.
Figure 4 shows the dimensionless damping rate γq /ω0
(
q
)
at low temperatures (low values of a) plot-
ted for F = 10−3. The solid line reflects the behaviour of dimensionless decrement that follows from the
analytical expression (4.27). The dotted line shows the dependence γq
/
ω0
(
q
)
on the temperature ob-
tained as a result of numerical calculations based on formulae (4.15)–(4.17). It is evident that analytical
expression (4.27) with good accuracy coincides with the damping rate temperature dependence in a di-
lute gas with BEC in a rather wide range of low temperatures.
Here, we should make the following remark. The experimental data (see [3, 4] and [32–36]) show that
under the present experimental conditions, the criterion a =
(
T
/
ν0n0
)
≪ 1 is not generally realized. For
example, approximate estimates of the parameter a obtained in accordance with typical experimental
conditions of the mentioned studies, show the following circumstance. This parameter takes on the value
a ≈ 2 for 85Rb [35], a ≈ 23 for 23Na [4], a ≈ 58 for 1H [32]. To perform these estimates, the scattering
length values were taken from [6]. The exceptions are the data of [33] and [36]: for 7Li we have a ≈ 0.5
[33] and for 133Cs accordingly a ≈ 0.8 [36]. Therefore, to observe the effect of sound damping in dilute
gas with BEC at low temperatures, the most promising are experimental conditions in [33, 36].
Experimental conditions in other studies are rather close to the limiting case contrary to (4.18), that
23004-11
Yu. Slyusarenko, A. Kruchkov
Figure 4. (Color online) Dependence of dimensionless damping rate γq /ω0 on temperature [dimen-
sionless quantity a, see (4.18)]. Solid line represents the analytical approach given by (4.27) and dot-
dashed line represents numerical calculation in accordance with (4.15)–(4.17). Both lines were plotted
for F = 10−3. Dashed vertical line indicates the region where analytical formula (4.18) works with good
accuracy.
is the case of high temperatures. For this reason, we now consider, as in [12, 13], another limiting case
a ≡
T
ν0n0
≫ 1, T < Tc . (4.28)
As before, we assume here that εp ∼ ν0n0. Then, from (4.28) it follows that ωp (n0) ≪ T . In this case,
the following limiting expression for the distribution function f 0
p
(
ωp
)
can be used (see, in this regard
(3.2) and [12, 13])
f 0
p
(
ωp
)
≈ T
ωp
, εp ∼ ν0n0 , ωp (n0) ≪ T . (4.29)
To simplify the further calculations in (4.5) and (4.6), it is convenient to change the integration vari-
able p to variable z, that is introduced in accordance with the formula
p2
2mν0n0
= z .
As a result of the change of integration variable, we obtain the following expressions:
ε1 (s) = 1− F a
(
s2 −1
)
∞
∫
0
dz g (s, z)
[
1− s
2v (z)
ln
∣
∣
∣
∣
s + v (z)
s − v (z)
∣
∣
∣
∣
]
, (4.30)
where now we have introduced the notations
g (s, z) ≡
p
2(3z +1)
[
z
(
s2 +1
)
+2s2 +1
]
p
z (z +1) (z +2)2
, v (z) ≡
p
2
z +1
p
z +2
, (4.31)
and the quantities F and a are still defined by (4.9) and (4.10).
From the condition ε1 (s0) = 0 [see (2.19), (3.23), (4.26), (4.30)], written when s0 (a) ≈ ± [1+δs (a)],
|δs|≪ 1, in the form of
δs = F a
∞
∫
0
dz
(3z +1)(2z +3)
p
2z (z +1) (z +2)2
[
1− 1
2v (z)
ln
∣
∣
∣
∣
1+ v (z)+δs
1− v (z)+δs
∣
∣
∣
∣
]
,
δs (a) ≡ s (a)−1, (4.32)
23004-12
Mechanism of collisionless sound damping in dilute Bose gas with condensate
it follows that equation (4.26) has a solution if only F a ≪ 1 [but a ≫ 1 see (4.28)]. There is no analytical
solution to this equation. However, the numerical analysis reveals that the result of integration with
respect to z in the right-hand side of formula (4.32) almost does not depend on δs if |δs| ≪ 1. In this
regard, the solution of (4.32) in case of |δs|≪ 1 can be written as follows:
δs =−F ad2 , F a ≪ 1, (4.33)
where the constant d2 is given by
d2 ≡
∞
∫
0
dz
(3z +1) (2z +3)
p
2z (z +1)(z +2)2
[
1−
p
z +2
2
p
2(z +1)
ln
∣
∣
∣
∣
∣
p
z +2+
p
2(z +1)
p
z +2−
p
2(z +1)
∣
∣
∣
∣
∣
]
≈ 3.97. (4.34)
Figure 5. (Color online) Analytical dependence of the dimensionless sound speed s ≡ ω0
(
q, a
)
/u0q on
temperature (dimensionless quantity a) in case of high temperatures, plotted according to expressions
(4.33), (4.34) for F1 = 3 ·10−4 , F2 = 10−3 and F3 = 1.5 ·10−3 respectively.
Analytical dependence s (a) in case of a ≫ 1 in accordance with (4.33) and (4.34) is shown in figure 5
for three different values of F .
The value b (s) required in accordance with (4.16), (4.17) to calculate the damping rate of sound at
high temperatures [see (4.28)] is given by:
b (s) ≡
[
(
s2 −1
) ∂ε1 (s)
∂s
]
ω=ω0
=
s2
0 (a)+1
s0 (a)
− F a
s0 (a)
∞
∫
0
dz g
(
s0 (a) , z
) v2 (z)
s2
0 (a)− v2 (z, a)
−
F a
s0 (a)
∞
∫
0
dz
∂g
(
s0 (a) , z
)
∂s0 (a)
[
1−
s0 (a)
2v (z, a)
ln
∣
∣
∣
∣
s0 (a)+ v (z, a)
s0 (a)− v (z, a)
∣
∣
∣
∣
]
, (4.35)
where the functions g (s, z)and v (z) are still defined by (4.31), and s0 (a) is given by (4.32) while tak-
ing (4.33) into account. One can verify that the first term only yields the main contribution to b (s). The
damping rate γq of sound in BEC, calculated according to formulae (4.16) using (4.35), in the main ap-
proximation with respect to F a [see (4.33)] indicates a linear dependence on temperature:
γq
ω0
(
q
) ≈
γq
u0q
≈ π (π+6)
8
F a, (4.36)
and that is a well known result (see for example [2, 12, 13]). Deriving (4.36), the following integral was
used ∞
∫
0
dz
(3z +1) (2z +3)
p
z (z +1)2 (z +2)3/2
=
π+6
2
.
23004-13
Yu. Slyusarenko, A. Kruchkov
Formula (4.36) using (3.10), (4.3), (4.9) can be written as
γq = π+6
8
ascT q . (4.37)
It should be mentioned that this expression differs prima facie from that given in [2, 12, 13], where
γq =
3π
8
ascT q . (4.38)
However, it is easy to verify that the numerical factors in (4.37) and (4.38) coincide with two signif-
icant digits. Note that in [12, 13] it was also indicated that the formula (4.38) obtained therein slightly
differs in a numerical factor from a similar formula of the previous studies. In our case, the method used
to obtain formula (4.37) differs greatly from the approaches used in [12, 13] to derive (4.38). This is what
really caused the difference (although small) in formulae (4.37) and (4.38).
Figure 6. (Color online) Dependence of dimensionless damping rate γq /ω0 on temperature [dimension-
less quantity a, see (4.28)] in the case of high temperatures. The solid line shows the dependence given by
(4.36) and the dashed one reveals the result of numerical computing based on the formulae (4.15)–(4.17).
Both lines were plotted for F = 10−3.
Figure 6 shows the dependence of dimensionless damping rate γq /ω0
(
q
)
at high temperature region
[a ≡ (T /ν0n0) ≫ 1 and T < Tc] plotted for F = 10−3. It is apparent that the graph plotted according to
formula (4.36) or (4.37) (solid line) nearly coincides with the graph obtained by numerical computations
based on formulae (4.15)–(4.17) (dashed line). As it follows from figure 4, the condition of sound existence
in a dilute gas with BEC,
(
γq /ω0
)
≪ 1, is well satisfied even in the region where a ∼ 30. Naturally, one
should be confident that in a particular physical system, the condition T < Tc is also satisfied in this range
of a.
5. Conclusion
We have reported the main results related to the construction of microscopic theory of sound damp-
ing due to Landau mechanism in a dilute gas with Bose condensate. The analytical expressions of prop-
agation velocity and damping rate of sound in a dilute gas with Bose condensate in the limiting cases of
high and low temperatures were obtained. It was shown that at high temperatures these expressions co-
incide with those obtained previously by other authors in various phenomenological approaches. At low
temperatures, the behaviour of collisionless sound decrement obtained in the present paper, significantly
differs from the same decrement obtained by other authors. In our opinion, the distinction is caused by
our use of evolution equations that were obtained in the microscopic approach from the first principles.
We now make a significant remark. At first glance, the presence of general equations (4.14)–(4.17)
makes it possible to find the parameters of propagation and damping of sound in the present system in
23004-14
Mechanism of collisionless sound damping in dilute Bose gas with condensate
any temperature range (i.e., for all values a), at least numerically. Meanwhile, figure 3 and figure 4 do not
display the data for the ‘intermediate’ temperature range, that is from a = 0.7 to a = 1.4. This is caused
by the following circumstance. We have already mentioned that in this range of values of a there are no
analytic methods for solving dispersion equations (4.14)–(4.17). But as it turned out, in this temperature
range, the numerical methods, at least those we have used, also become uncontrollable. Hence, one can-
not trust its results. This is supposed to be due to the fact that in this interval of a, the point a ≈ 0.847 is
located. Recall that it is the point where the value of s0 (a) = ω0
(
q, a
)/
u0q [see (4.14)] is equal to unity.
As it was mentioned earlier, the same value s gains also at the point a = 0 (zero sound). It is that very
neighbourhood of a ≈ 0.847 that gives huge oscillations of sound damping rate in the received outcome
of numerical computations with insignificant changes of the value a. However, one cannot be sure that
these oscillations of the damping rate reflect a real physical picture. The reason for doubting the out-
come lies in the mentioned uncertainty in this temperature range of the numerical methods used in the
present paper. Numerical calculations are poorly controlled due to the slow convergence of integrals in
(4.15)–(4.17) as the manifestation of non-analytic dependence on temperature of the dispersion charac-
teristics of the system in the neighbourhood of the point a ≈ 0.847. Apparently, a more detailed study of
the behaviour of a decrement (or maybe even an increment!) of sound in the neighbourhood of this point
requires the use of more sophisticated numerical methods. The authors intend to address this issue soon.
In this regard, the results in [37] are also of interest, where propagation and absorption of the trans-
verse breathingmode of an elongated BEC were investigated experimentally in 87Rb vapour. The authors
of the mentioned study, attempting to measure the damping rate of these modes at the temperature ap-
proximately equal to 40÷ 60 nK, had found that the behaviour of the perturbation amplitude in this
temperature region differs significantly from the behaviour of the amplitude of a damped sinusoidal sig-
nal. The presence of such a phenomenon was suggested to be explained due to nonlinear effects in the
propagation of the modes studied in the system. We also would like to draw attention to the fact that
temperature range 40÷60 nK corresponds to the values of quantity a in terms of the present paper that
are in the 0.7÷1.0 range. The value of a is naturally calculated according to formula (4.9) and using the
values of the physical characteristics of the system [37]. In other words, in the mentioned case [37], it
deals with the nearest neighbourhood of the point a ≈ 0.847 where the nonanalytic dependence of the
dispersion characteristics on temperature is revealed.
In conclusion, we note a good correlation of our theory in the case of high temperature region with
the experimental data [37]. As can be seen from figure 7, the results of numerical calculations based on
formulae (4.15)–(4.17) for F = 3 ·10−4 [see (4.9)] are in satisfactory concordance with the experimental
data [37]. Experimental data reveal the same good fit with the dependence given by formula (4.36).
The authors are naturally aware of the fact that a direct comparison of the results of the present
Figure 7. (Color online) The dependence of dimensionless damping rate γq /ω0 on temperature (dimen-
sionless quantity a). The solid line displays the result of numerical computation for F = 3 ·10−4 based on
expressions (4.15)–(4.17). Diamonds reproduce the experimental points taken from [33].
23004-15
Yu. Slyusarenko, A. Kruchkov
paper with the experimental data [37] can hardly be considered entirely correct. At least, the reason
is that the authors of [37] have to deal with the trapped BEC. However, such a comparison once again
demonstrates the validity of the statement [12, 13] that the linear dependence of the sound damping rate
on temperature should occur in the case of trapped BEC.
Noting the qualitative similarity in particular cases of our results with the experimental ones, we
especially emphasize once again that we do not claim to microscopically describe the damping of sound
in trapped BEC. This specific case requires a cardinal modification of the whole theory, and it is the issue
the authors are currently working on.
References
1. Hohenberg P.C., Martin P.C., Ann. Phys. (N.Y.), 1965, 34, 291; doi:10.1016/0003-4916(65)90280-0.
2. Szepfalusy P., Kondor I., Ann. Phys. (N.Y.), 1974, 82, 1; doi:10.1016/0003-4916(74)90330-3.
3. Anderson M.H., Ensher J.R., Matthews M.R., Wieman C.E., Cornell E.A., Science, 1995, 269, 198;
doi:10.1126/science.269.5221.198.
4. Davis K.B., Mewes M.-O., Andrews M.R., van Druten N.J., Durfee D.S., Kurn D.M., Ketterle W., Phys. Rev. Lett.,
1995, 75, 3969; doi:10.1103/PhysRevLett.75.3969.
5. Bradley C.C., Sackett C.A., Tollett J.J., Hulet R.G., Phys. Rev. Lett., 1995, 75, 1687; doi:10.1103/PhysRevLett.75.1687.
6. Pethick C.J., Smith H., Bose-Einstein Condensation in Dilute Gases, Cambridge University Press, Cambridge, 2002.
7. Pitaevskii L., Stringari S., Bose-Einstein Condensation, Oxford Science Publications, Oxford, 2003.
8. Jin D.S., Ensher J.R., Matthews M.R., Wieman C.E., Cornell E.A., Phys. Rev. Lett., 1996, 77, 420;
doi:10.1103/PhysRevLett.77.420.
9. Mewes M.-O., Andrews M.R., van Drutten N.J., Kurn D.M., Durfee D.S., Townsend C.G., Ketterle W., Phys. Rev.
Lett., 1996, 77, 988; doi:10.1103/PhysRevLett.77.988.
10. Stringari S., Phys. Rev. Lett., 1996, 76, 1405; doi:10.1103/PhysRevLett.76.1405.
11. Edwards M., Ruprecht P.A., Burnett K., Dodd R.J., Clark C.W., Phys. Rev. Lett., 1996, 77, 1671;
doi:10.1103/PhysRevLett.77.1671.
12. Pitaevskii L.P., Stringari S., Phys. Lett. A, 1997, 235, 398; doi:0.1016/S0375-9601(97)00666-X.
13. Pitaevskii L.P., Physics-Uspekhi , 1998, 41, 569; doi:10.1070/PU1998v041n06ABEH000407 [Usp. Fiz. Nauk, 1998,
168, 641 (in Russian); doi:10.3367/UFNr.0168.199806e.0641].
14. Liu W.V., Schieve W.C., Preprint arXiv:cond-mat/9702122, 1997.
15. Fedichev P.O., Shlyapnikov G.V., Walraven J.T.M., Phys. Rev. Lett., 1998, 80, 2269;
doi:10.1103/PhysRevLett.80.2269.
16. Giorgini S., Phys. Rev. A, 1998, 57, 2949; doi:10.1103/PhysRevA.57.2949.
17. Iigaya K., Konabe S., Danshita I., Nikuni T., Phys. Rev. A, 2006, 74, 053611; doi:10.1103/PhysRevA.74.053611.
18. Bogolyubov N.N., J. Phys. USSR, 1947, 11, 23.
19. Peletminskii S.V., Tsukanov V.D., Yatsenko A.A., Ukr. Fiz. Zh., 1973, 18, 1160 (in Russian).
20. Peletminskii S.V., Sokolovskii A.I., Shchelokov V.S., Theor. Math. Phys., 1977, 30, 35; doi:10.1007/BF01029358
[Teor. Mat. Fiz., 1977, 30, 57 (in Russian)]
21. Peletminskii S.V., Sokolovskii A.I., Schelokov V.S., Theor. Math. Phys., 1978, 34, 51; doi:10.1007/BF01036471 [Teor.
Mat. Fiz., 1978, 34, 81 (in Russian)]
22. Lavrinenko N.M., Peletminskii S.V., Slyusarenko Yu.V., Sov. Low Temp. Phys., 1983, 9, 412 [Fiz. Nizk. Temp., 1983,
9, 795 (in Russian)].
23. Bogolyubov N., Problems of a Dynamical Theory in Statistical Physics, Providence College, Rhode Island, 1959.
24. Akhiezer A.I., Peletminskii S.V., Methods of Statistical Physics, Pergamon Press, Oxford, 1981.
25. Khalatnikov I.M., Theory of Superfluidity, Benjamin, New York, 1965.
26. Kirkpatrick T.R., Dorfman J.R., J. Low Temp. Phys., 1985, 58, 301; doi:10.1007/BF00681309.
27. Giorgini S., Phys. Rev. A, 2000, 61, 063615; doi:10.1103/PhysRevA.61.063615.
28. Jackson B., Zaremba E., New J. Phys., 2003, 5, 88; doi:10.1088/1367-2630/5/1/388.
29. Escobedo M., Manuel C., Phys. Rev. A, 2010, 82, 023614; doi:10.1103/PhysRevA.82.023614.
30. Slyusarenko Yu.V., Low Temp. Phys., 1998, 24, 219; doi:10.1063/1.593588 [Fiz. Nizk. Temp., 1998, 24, 291 (in Rus-
sian)].
31. Slyusarenko Yu.V., Low Temp. Phys., 1998, 24, 393; doi:10.1063/1.593607 [Fiz. Nizk. Temp., 1998, 24, 522 (in Rus-
sian)].
32. Fried D.G., Killian T.C., Willmann L., Landhuis D., Moss S.C., Kleppner D., Greytak T.J., Phys. Rev. Lett., 1998, 81,
3811; doi:10.1103/PhysRevLett.81.3811.
33. Bradley C.C., Sackett C.A., Hulett R.G., Phys. Rev. Lett., 1997, 78, 985; doi:10.1103/PhysRevLett.78.985.
23004-16
http://dx.doi.org/10.1016/0003-4916(65)90280-0
http://dx.doi.org/10.1016/0003-4916(74)90330-3
http://dx.doi.org/10.1126/science.269.5221.198
http://dx.doi.org/10.1103/PhysRevLett.75.3969
http://dx.doi.org/10.1103/PhysRevLett.75.1687
http://dx.doi.org/10.1103/PhysRevLett.77.420
http://dx.doi.org/10.1103/PhysRevLett.77.988
http://dx.doi.org/10.1103/PhysRevLett.76.1405
http://dx.doi.org/10.1103/PhysRevLett.77.1671
http://dx.doi.org/0.1016/S0375-9601(97)00666-X
http://dx.doi.org/10.1070/PU1998v041n06ABEH000407
http://dx.doi.org/10.3367/UFNr.0168.199806e.0641
http://arxiv.org/abs/cond-mat/9702122
http://dx.doi.org/10.1103/PhysRevLett.80.2269
http://dx.doi.org/10.1103/PhysRevA.57.2949
http://dx.doi.org/10.1103/PhysRevA.74.053611
http://dx.doi.org/10.1007/BF01029358
http://dx.doi.org/10.1007/BF01036471
http://dx.doi.org/10.1007/BF00681309
http://dx.doi.org/10.1103/PhysRevA.61.063615
http://dx.doi.org/10.1088/1367-2630/5/1/388
http://dx.doi.org/10.1103/PhysRevA.82.023614
http://dx.doi.org/10.1063/1.593588
http://dx.doi.org/10.1063/1.593607
http://dx.doi.org/10.1103/PhysRevLett.81.3811
http://dx.doi.org/10.1103/PhysRevLett.78.985
Mechanism of collisionless sound damping in dilute Bose gas with condensate
34. Modugno G., Ferrari G., Roati G., Brecha R.J., Simoni A., Inguscio M., Science, 2001, 294, 1320;
doi:10.1126/science.1066687.
35. Cornish S.L., Claussen N.R., Roberts J.L., Cornell E.A., Wieman C.E., Phys. Rev. Lett., 2000, 85, 1795;
doi:10.1103/PhysRevLett.85.1795.
36. Weber T., Herbig J., Mark M., Nagerl H.-Ch., Grimm R., Science, 2003, 299, 232; doi:10.1126/science.1079699.
37. Chevy F., Bretin V., Rosenbuch P., Madison K.W., Dalibard J., Phys. Rev. Lett., 2002, 88, 250402;
doi:10.1103/PhysRevLett.88.250402.
Про беззiткненнєвий механiзм загасання звуку в
розрiджених бозе-газах iз конденсатом
Ю.В. Слюсаренко1,2, О.Ю. Крючков2
1 Iнститут теоретичної фiзики iм. О.I. Ахiєзера, ННЦ ХФТI, вул. Академiчна, 1, 61108 Харкiв, Україна
2 Харкiвський нацiональний унiверситет iм. В.Н. Каразiна, пл. Свободи, 4, 61077 Харкiв, Україна
Побудовано мiкроскопiчну теорiю загасання звуку за механiзмом Ландау у розрiджених газах iз бозе-
конденсатом. В основу теорiї було закладено пов’язанi рiвняння еволюцiї для парамерiв опису системи.
Цi рiвняння було виведено у бiльш раннiх роботах у мiкроскопiчному пiдходi, що базується на викорис-
таннi методу скороченого опису квантових систем багатьох частинок (метод Пелетминського) та моделi
Боголюбова для слабко неiдеального бозе-газу з видiленим конденсатом. Отримано рiвняння дисперсiї
звукових коливань у системi, що вивчається, шляхом лiнеаризацiї зазначених рiвнянь еволюцiї у без-
зiткненнєвому наближеннi. Проведено аналiз рiвнянь дисперсiї, як чисельно, так i аналiтично. Одержано
аналiтичнi вирази для швидкостi розповсюдження й коефiцiєнта поглинання звуку в розрiджених газах
iз бозе-конденсатом у граничних випадках великих та малих температур. Нами продемонстровано, що
в областi малих температур температурна залежнiсть знайдених величин суттєво вiдрiзняється вiд тих,
що отриманi ранiше iншими авторами в напiвфеноменологiчних пiдходах. Вказано на можливi ефекти,
пов’язанi з неаналiтичними залежностями дисперсiйних характеристик системи вiд температури.
Ключовi слова: розрiджений бозе-газ, бозе-ейнштейнiвська конденсацiя (БЕК), мiкроскопiчна теорiя,
звук, механiзм Ландау, дисперсiйнi спiввiдношення, швидкiсть звуку, показник загасання
23004-17
http://dx.doi.org/10.1126/science.1066687
http://dx.doi.org/10.1103/PhysRevLett.85.1795
http://dx.doi.org/10.1126/science.1079699
http://dx.doi.org/10.1103/PhysRevLett.88.250402
Introduction
Kinetics of spatially inhomogeneous Bose gas with the presence of condensate
Sound dispersion equations in diluted gas with BEC
Sound damping rate in dilute Bose gas with condensate
Conclusion
|