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Received November 24, 2012, in final form March 18, 2013
Abstract

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 the obtained quantities 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.

\keywords

dilute Bose gas, Bose-Einstein condensate (BEC), microscopic theory, sound, Landau mechanism, dispersion relations, speed of sound, damping rate \pacs05.30.-d, 05.30.Jp, 67.85.Hj, 67.85.Jk, 03.75.Hh, 03.75.Kk

Abstract

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. \keywordsрозрiджений бозе-газ, бозе-ейнштейнiвська конденсацiя (БЕК), мiкроскопiчна теорiя, звук, механiзм Ландау, дисперсiйнi спiввiдношення, швидкiсть звуку, показник загасання

\issue

201316223004 \doinumber10.5488/CMP.16.23004 Mechanism of collisionless sound damping in dilute Bose gas with condensate]Mechanism of collisionless sound damping
in dilute Bose gas with condensateYu. Slyusarenko, A. Kruchkov]Yu. Slyusarenko\refaddrlabel1,label2thanks: E-mail: slusarenko@kipt.kharkov.ua , A. Kruchkov \refaddrlabel2thanks: E-mail: aleks.kryuchkov@gmail.com \addresses \addrlabel1 Akhiezer Institute for Theoretical Physics, National Science Center Kharkiv Institute of Physics and Technology, 1 Akademichna St., 61108 Kharkiv, Ukraine \addrlabel2 Karazin National University, 4 Svobody Sq., 61077 Kharkiv, Ukraine \authorcopyrightYu. Slyusarenko, A. Kruchkov, 2013

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 spatially homogeneous case.

The direct experimental observation of BEC [3, 4, 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, 9, 10, 11, 12, 13, 14, 15, 16, 17].

It is currently assumed that the Landau damping is the most probable mechanism of sound relaxation in the so-called trapped Bose condensates. This mechanism consists in collisionless absorption of oscillation energy by quanta of elementary excitations [12, 13, 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 homogeneous 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.

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 complicates 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 [19, 20, 10, 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 momentum. 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 between 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 collisionless 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, 28, 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, , whereas the results of [1, 12, 13] give 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

(1)

where is Hamiltonian of the system, consisting of the ideal gas Hamiltonian

(2)

and binary interaction Hamiltonian

(3)

Equation (1) is written in the units in which Planck’s constant is equal to unity. In the formulae (2), (3) m is boson mass, is binary interaction potential, which depends only on the distance between particles, and are field operators. The quasiparticle distribution function , the order parameter and the superfluid velocity 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 , and . Since the general form of equations from [20] is not required in the present paper, we introduce it here in collisionless approximation:

(4)

where instead of description parameter we have introduced a new variable , that is condensate density [22]

(5)

and the quantities and are given by expressions:

(6)

where

(7)

and the following notations were also introduced

(8)

where is an arbitrary function of .

Let us recall that the basis of Bogoliubov equilibrium state theory of Bose gas in the presence of interaction is the assumption that , see [18], where quantity characterizes the smallness of the interaction between the particles, . Furthermore, it is believed that the order of magnitude of does not change after differentiation with respect to , . In the formulae (2)–(2), the notation means the quantity in order of magnitude of .

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)–(2) 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 vanishes by substituting the stationary Bose distribution function

(9)

with the chemical potential of quasiparticles being equal to zero; here is temperature in energy units. The chemical potential having vanished reflects the fact that the number of quasiparticles is not conserved during collisions [19, 22].

3 Sound dispersion equations in diluted gas with BEC

To investigate the propagation of sound in gas with BEC, we linearize coupled equations (2), (2), (7) with respect to spatially homogeneous equilibrium state according to the following formulae

(10)

where is the equilibrium value of the condensate density in the system at the temperature . The equilibrium value of the velocity is considered to be equal to zero in the second formula of (3). Thus, the velocity is supposed to be of the order of magnitude of and . Then, the equilibrium distribution function of quasiparticles pursuant to (9) is given by the expression

(11)

In this formula, like in all subsequent expressions, we omit the index ‘‘0’’ in the designation of the equilibrium value of to avoid encumbering the computations. Note that the quantity defined by (2) represents the chemical potential of atomic Bose gas (see in this regard [19, 20]).

Deviations of the corresponding quantities from their equilibrium values were denoted by , and .

The equations of motion for these variables can be represented as follows:

(12)

where we have introduced the following notation

(13)

Recall that in the present paper, the Planck constant is considered to be equal to unity. Note also that while deriving equations (3), we have discarded the terms that are higher than the first order of magnitude . The circumstance of (5) stating that or i.e. was taken into account.

Proceeding further to the Fourier transform of the quantities , , in equations (3) as provided by formula

(14)

where should be understood as either of the considered quantities, we obtain

(15)

where we denote [see (3), (13)]

(16)

In the formulae (3), (16) in Fourier transforms of the quantities , , we omit the ’tilde’ sign. Further expressing the values and from the first two equations of (3) in terms of ,

(17)

the third equation of (3) can be written in the form

(18)

where is referred to as speed of zero sound in Bose gas

(19)

As is readily seen, equation (18) subject to (16) is an integral equation for distribution function Fourier transform . The solution of this equation can be represented as in [30]:

(20)

where we have introduced the following notation:

(21)

and is an arbitrary function, which is required to impose the following restriction: the distribution function , calculated in accordance with (3), (14) and (20) should be small compared with the equilibrium distribution function .

It implies also from (14) that must satisfy the following relation:

(22)

We denote the whole valid set of such functions by , where is a continuous or discrete symbolic parameter such that the functions may depend on . The reason for introducing this index may consist, for example, in the following: the set of functions should be sufficient to build an arbitrary value for the distribution function at the initial time (in this context see also [31]).

Formula (20) permits to find the value in terms of functions:

(23)

where

(24)

and

(25)

The expression (20) for the given values subject to (23)–(25) can now be represented in the form:

(26)

We note that in case of charged particles gas, the quantity [see (25)]

(27)

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

(28)

Moreover, the wave decrement is determined by an imaginary part of the value . For this reason, weakly damped oscillations in the system

(29)

can exist if only

(30)

besides, as the consequence of (27)–(30) (see in this regard [26, 27]), the frequency can be found from the equation

(31)

and the damping rate is given by expression

(32)

Despite the fact that in the present paper we investigate a neutral system, the existence of longitudinal oscillations is also associated with the existence of zeros of the function . 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 (26) is such that weakly damped waves may also be excited in the system investigated, in accordance with (17) and (14), and the dispersion law is determined by (28)–(30). As we shall show in the following section, such waves would represent sound waves in weakly nonideal Bose gas with condensate. The value obtained according to (26)–(29) 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 and imaginary parts of the quantity . For this purpose in the expression (25) we use formula

(33)

where the symbol means that the further integration is taken in the sense of Cauchy principal value. After some hackneyed transformations one can obtain the following expressions for and :

(34)

Further calculations are not possible without specifying the explicit form of that is the Fourier transform of interaction potential [see (3), (8)]. To simplify calculations it is often accepted to replace the interaction potential by the following effective potential (see [6, 7])

(35)

where is the so-called -wave scattering length (for details see [6]). Thus, we have . Taking it into account, we have and the value [see (7), (21)] can be expressed in the form

(36)

Formulae (35) and (36) enable us to perform integration in (4) over the angle between vectors and , and then can be represented in the form

(37)

After the same angle integration, the second expression of (4) can be written as follows:

(38)

where is Heaviside step function.

We now determine, following works [12, 13], the sound damping rate in the case of low () and high ( but ) temperatures. To study the case of low temperatures it is convenient to use the variable in the integration in (37) and (38). The variables , , in terms of the variable can be expressed using the explicit expression (7) and (36). After carrying out rather cumbersome but necessary calculations, the result can be summarized as follows:

(39)

where functions and are given by:

(40)

and the following notations were introduced [recall still given by (19)]:

(41)

Deriving the last formula of (41) it is necessary to use the expressions (19) and (35). The quantity is referred to as gas parameter, see [6, 7]. That is, in diluted gases the following relation should be satisfied:

(42)

It is easy to verify that the values and can be represented as functions of one dimensionless variable

(43)

where

(44)

and for these functions the symmetry conditions are valid

(45)

Expressions (31) and (32) 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):

(46)

As is easily seen, these oscillations have linear spectrum. In virtue of (4) and (46) the structure of dispersion equation is such that the unknown value , which determines the oscillation frequency as a function of wave vector, does not depend on wave vector itself.

In accordance with (31) and (46), the dependence of oscillation frequency on wave vector in this system should be determined by the solution of equation

(47)

We note that dispersion equation (47) 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 appearing in (46), the damping rate can be presented as follows:

(48)

where

(49)

We emphasize that the above mentioned expressions (4)–(49) are exact, despite the fact that we have modified them to the form suitable to study the low temperature regime.

Figure 1: (Color online) Dependence of dimensionless speed of sound on temperature [dimensionless quantity , see (41)] obtained by a numerical solution of (47) for (solid line). Dots shows the intersection with line (dashed one) in points and .

As is readily seen, equation (47) in the general case can be solved only by numerical methods. Figure 1 shows the dependence [see (4) and (41)] obtained as a result of numerical solution of equation (47) for . It is evident that the function behaves nonmonotonously as changes. At low region (the case of low temperature), an increase of the function is observed. At the point it reaches maximum, and then decreases monotonously as increases, and in the point it is equal to unity again. As mentioned above, the increase of the value is restricted at least to the critical temperature, see (41). At zero temperature () we deal with a classical zero sound in Bose system, because , and hence due to (4). This result is naturally expected. However, we note once again that at the point there also holds 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 equations 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 , that is (or ) for , requires an individual physical interpretation. Some more comments regarding this point will be given below.

In the regions of low () and high ( but ) temperatures a solution of (47) can be expressed in analytical form. Thus, one can obtain analytical expressions for the quantity 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 mentioned above)

(50)

and the rapid decrease of the function as [see (4)], functions and can be expanded in power series of in (4)–(45):

(51)

As will become apparent from the subsequent formulae, such an expansion corresponds to the development of perturbation theory with respect to a small parameter . Taking into account the expansions (51), the dispersion equation (47) can be written as:

(52)

In the region of small , i.e., low temperatures, as we have seen in figure 1, . That is to say, the value should be much less than unity, . Therefore, the equation (4) admits further simplification

(53)

When and [see (41), (46)] the solution of (53) can exist only in the domain. This inequality makes it possible to neglect the value within the logarithm term in the integrand in (53) as the first approximation of perturbation theory. As a result, we obtain

(54)

where the notation is

(55)

Formula (54) is also confirmed by numerical calculations. Figure 2 shows the dependence , , computed according to expression (53)

(56)
Figure 2: Dependence plotted numerically according to (56) for .
Figure 3: (Color online) Analytical dependence of dimensionless speed of sound on temperature (dimensionless quantity ) in case of low temperatures, plotted according to expressions (54), (57) for , and respectively.

As can be seen from figure 2, the equality for holds up to the second decimal. As a consequence, in the expression for that is given above, the dependence on can be neglected. Then, one can derive:

(57)

where the second term was calculated numerically, and, deriving the first term, the following integral value was used:

Thus, the solution (54) shows the following dependence of the sound frequency in dilute gas with BEC on the low temperature range:

(58)

Analytical dependence in case of in accordance with (54) and (59) is shown in figure 3 for three different values of . 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 (57), (58), corrections in [27] have temperature dependence.

The damping rate of sound in BEC calculated according to formulae (48), (49) and using (51), (54)–(58) in the first order approximation with respect to is given by a fairly simple expression