Temperature-induced resonances and Landau damping of collective modes in Bose-Einstein condensed gases in spherical traps

Interaction between collective monopole oscillations of a trapped Bose-Einstein condensate and thermal excitations is investigated by means of perturbation theory. We assume spherical symmetry to calculate the matrix elements by solving the linearized Gross-Pitaevskii equations. We use them to study the resonances of the condensate induced by temperature when an external perturbation of the trapping frequency is applied and to calculate the Landau damping of the oscillations.

Interaction between collective monopole oscillations of a trapped Bose-Einstein condensate and thermal excitations is investigated by means of perturbation theory. We assume spherical symmetry to calculate the matrix elements by solving the linearized Gross-Pitaevskii equations. We use them to study the resonances of the condensate induced by temperature when an external perturbation of the trapping frequency is applied and to calculate the Landau damping of the oscillations.

I. INTRODUCTION
Since the discovery of Bose-Einstein condensation in magnetically trapped Bose gases, the study of the lowenergy collective excitations has attracted a big interest both from experimental and theoretical point of view. Mean field theory has proven to be a good framework to study static, dynamic and thermodynamic properties of these trapped gases. In particular, it provides predictions of the frequencies of collective excitations that very well agree with the observed ones. Recently [1,2] the energy shifts and damping rates of these low-lying collective excitations have been measured as a function of temperature. However, these phenomena have not yet been completely understood theoretically.
In this paper we study the influence of thermal excitations on collective oscillations of the condensate in the collisionless regime. Previous papers on this subject has been devoted mainly to calculation of the Landau damping by means of perturbation theory. In Refs. [3][4][5][6] only the uniform system has been considered, whereas in Refs. [7,8] Landau damping in trapped Bose gases has also been studied but using the semiclassical approximation for thermal excitations and the hydrodynamic approximation for collective oscillations. An important point of Refs. [7,8] is that the authors discuss the possible chaotic behavior of the excitations in an anisotropic trap. The frequency shift has also been studied for a trapped condensate in the collisionless regime in Ref. [8].
In the present work we study the interaction between collective and thermal excitations using the Gross-Pitaevskii equation and perturbation theory. We consider spherically symmetric traps, since in this case the spectrum of excitations is easily calculated, avoiding the use of further approximations. Even though the case of anisotropic traps can be significantly different in the final results, a detailed investigation of spherical traps is instructive. We explore, in particular, the properties of monopole oscillations by studying the temperatureinduced resonances that occur in the condensate when an external perturbation of the trapping frequency is applied and, also, the Landau damping associated with the interaction with thermally excited states.
This paper is organized as follows. In section II we introduce the general equations that describe the elementary excitations of the condensate within the Bogoliubov theory [9]. In Sec. III we recall the perturbation theory for a trapped Bose-condensed gas in order to study the interaction between elementary excitations. In Sec. IV we introduce the linear response function formalism and calculate the response function of the condensate when a small perturbation of the trapping frequency is applied. We derive analytic equations for the response function at zero temperature and treat perturbatively the contribution of the elementary excitations, which is related to Landau damping. In Sec. V we discuss the main results.

II. ELEMENTARY EXCITATIONS OF AN ISOTROPIC TRAP
We consider a weakly interacting Bose-condensed gas confined in an external potential V ext at T = 0. The elementary excitations of a degenerate Bose gas are associated with the fluctuations of the condensate. At low temperature they are described by the time dependent Gross-Pitaevskii (GP) equation for the order parameter [10,11]: where dr|Ψ| 2 = N 0 is the number of atoms in the condensate. At zero temperature it coincides with the total number of atoms N , except for a very small difference δN ≪ N due to the quantum depletion of the condensate. The coupling constant g is proportional to the s-wave scattering length a through g = 4πh 2 a/m. In the present work we will discuss the case of positive scattering length, as for 87 Rb atoms. The trap is included through V ext , which is chosen here in the form of an isotropic harmonic potential: V ext (r) = (1/2)mω 2 ho r 2 . The harmonic trap provides a typical length scale for the system, a ho = (h/mω ho ) 1/2 . So far experimental traps have axial symmetry, with different radial and axial frequencies, but experiments with spherical traps are also feasible [12]. The choice here of a spherical trap has two different reasons. First, it greatly reduces the numerical effort and will allow us to study the interaction of oscillations with elementary excitations without any further approximations. Second, the energy spectrum of the excitations in such a trap is well resolved yielding to the appearance of well-separated resonances. In anisotropic traps, conversely, the spectrum of excitations is much denser.
The normal modes of the condensate can be found by linearizing equation (1) , i.e., looking for solutions of the form where µ is the chemical potential and functions u and v are the "particle" and "hole" components characterizing the Bogoliubov transformations. After inserting in Eq. (1) and retaining terms up to first order in u and v, one finds three equations. The first one is the nonlinear equation for the order parameter of the ground state, where H 0 = −(h 2 /2m)∇ 2 + V ext (r); while u(r) and v(r) obey the following coupled equations [11]: Numerical solutions of these equations have been found by different authors [13][14][15][16][17][18]. In the present work, we use them to calculate the response function of the condensate under an external perturbation and the Landau damping of collective modes. When the adimensional parameter N a/a ho is large, the time-dependent GP equation reduces to the hydrodynamic equations [19]: where ρ(r, t) =| Ψ(r, t) | 2 is the particle density and the velocity field is v(r, t) = (Ψ * ∇Ψ−Ψ∇Ψ * )h/(2miρ). The static solution of equations (6)- (7) gives the Thomas-Fermi ground state density, which in the spherical symmetric trap reads in the region where µ > V ext (r), and ρ = 0 elsewhere. The chemical potential µ is fixed by the normalization of the density to the number of particles N 0 in the condensate. The density profile (8) has the form of an inverted parabola, which vanishes at the classical turning point R defined by the condition µ = V ext (R). For a spherical trap, this implies It has been shown [19] that the hydrodynamic equations (6) and (7) correctly reproduce the low-lying normal modes of the trapped gas in the linear regime when N a/a ho is large (see however Ref. [20]).

III. PERTURBATION THEORY
Let us briefly recall the perturbation theory for the interaction between collective modes of a condensate and thermal excitations as it was developed in Ref. [4]. Suppose that a certain mode of the condensate has been excited and, therefore, it oscillates with the corresponding frequency Ω osc . We assume that this oscillation is classical, i.e. the number of quanta of oscillation (n osc ) is very large. Then, the energy of the system associated with the occurrence of this classical oscillation can be calculated as E =hΩ osc n osc with n osc ≫ 1. Due to interaction effects, the thermal bath can either absorb or emit quanta of this mode producing a damping of the collective oscillation. The energy loss can be written aṡ where W (a) and W (e) are the probabilities of absorption and emission of one quantumhΩ osc , respectively. The interaction between excitations is small, so one can use perturbation theory to calculate the probabilities for the transition between a i-th excitation and a k-th one, available by thermal activation Let E i and E k be the corresponding energies and assume E k > E i . Since energy is conserved during the transition process, one has E k = E i +hΩ osc . The interaction term in second quantization is given by In the framework of Bogoliubov theory, the field opera-torΨ can be written as the sum of the condensate wave function Ψ 0 , which is the order parameter at equilibrium, and its fluctuations δΨ, whereΨ = Ψ 0 + δΨ [see Eq. (2)].
The fluctuations can be expressed in terms of the annihilation (α) and creation (α † ) operators of the elementary excitations of the system: where the functions u and v are properly normalized solutions of equations (4)- (5). In the sum (13) one can select a low energy collective mode, for which we use the notation u osc , v osc , α osc , α † osc , and investigate its interaction with higher energy single-particle excitations, for which we use the indices i, k as in (11). These latter excitations are assumed to be thermally excited. Inserting expression (13) into Eq. (12) one rewrites the interaction term V int in terms of the annihilation and creation operators. Since we want to study the decay process in which a quantum of oscillationhΩ osc is annihilated (created) and the i-th excitation is transformed into the k-th one (or viceversa), we will keep only terms linear in α osc (α † osc ) and in the product α † k α i (α k α † i ). And the energy conservation during the transition process will be ensured by the delta function δ(E k − E i −hΩ osc ). This mechanism is known as Landau damping [21].
Assuming that at equilibrium the states i, k are thermally occupied with the usual Bose factor where Let us define the dissipation rate γ through the following relation between the energy of the system E and its dissipationĖ:Ė Using expression (14) γ can be calculated as where the transition frequencies ω ik = (E k − E i )/h are positive. The "damping strength" has the dimensions of a frequency. In this work we calculate the quantities γ ik by using the numerical solutions u and v of Eqs. (2)(3)(4)(5) into the integrals (15). The results will be discussed in section V.

IV. RESPONSE FUNCTION
The results of the previous section can be used also to study the effect that an external perturbation of the trap has on the collective excitations of the condensate. Let us assume the trapping frequency in the form [ω ho + δω ho (t)], where δω ho ∼ exp(−iωt) is a time-dependent modulation. Assuming that the perturbation is small, one can use the response function formalism to describe the fluctuations of the system. Let us briefly recall the basic formalism [22].
The behaviour of a system under an external perturbation can be described by studying the fluctuations that may generate the external interaction to a certain physical quantity of the system. An external perturbation acting on the system is described by a new term in the Hamiltonian of the typê wherex is the quantum operator of the physical quantity that may fluctuate, and f (t) is the "perturbing force". The mean value x is zero in the equilibrium state, in absence of perturbation, and is not zero when it is present.
For a periodic perturbation f (t) ∼ exp(−iωt), the relation between x and f (ω) is where α(ω) is the response function also called generalised susceptibility.
In general α(ω) is a complex function. It can be seen that the imaginary part of the susceptibility determines the absorption of energy Q of the external force f by the system through the following relation: and that the real and imaginary parts of α(ω) satisfy the Kramers-Krönig relation where P means the principal value of the integral. The time-dependent external drive δω ho induces oscillations of the condensate density δρ with frequency ω; ρ(r, t) = ρ(r, 0) + δρ. Expanding the energy due to the confining potential, E ho = V ext ρ dr, with respect to δω ho and δρ one obtains the "mixed" term, corresponding to the Hamiltonian (19): Comparing it with Eq. (19) one can identify the perturbing force and the corresponding coordinate as Note that the first order term mω ho δω ho r 2 ρ(r, 0)dr can be omitted because it gives an additive shift in the Hamiltonian which does not contribute to the equations of motion of the system.
Once we have identified f and x, we can calculate the response function of the condensate α(ω). According to the definition one has Let us present the response function in the form α(ω) = α 0 (ω) + α 1 (ω), where α 0 (ω) corresponds to the response function of the condensate at T = 0, i.e., calculated without elementary excitations, and α 1 (ω) is the contribution of the excitations. At low temperatures it can be assumed that α 1 (ω) ≪ α 0 (ω) and then α 1 can be treated as a perturbation. We proceed as follows. First, we use the hydrodynamic approximation to obtain the response function at T = 0. Then, within a perturbation theory, we introduce the contribution of the elementary excitations at finite T to obtain α 1 (ω).
For a spherically symmetric breathing mode [23], one can easily prove that the hydrodynamic equations of motion (6) and (7) admit analytic solutions of the form [24] These equations are restricted to the region where ρ ≥ 0. Notice that they include the ground state solution (8) in the Thomas-Fermi limit. This is recovered by putting α r = 0, a r = mω 2 ho /(2g), and a 0 = µ/g. Inserting Eqs. (26) into the hydrodynamic equations, one obtains two coupled differential equations for the time dependent coefficients a r (t) and α r (t), while at any time a 0 = −(15N/8π) 2/5 a 3/5 r is fixed by the normalization of the density to the total number of atoms. The form (26) for the density and velocity distributions is equivalent to a scaling transformation of the order parameter. That is, at each time, the parabolic shape of the density is preserved, while the classical radius R, where the density (26) vanishes, scales in time as [25] where the unperturbed radius R(0) is given by Eq. (9). The relation between the scaling parameter b(t) and the coefficient a r (t) is a r = mω 2 ho /(2gb 5 ). Inserting it into Eq. (26) we obtain And the hydrodynamic equations then yield a r =ḃ/b andb The second and third terms of (29) give the effect of the external trap and of the interatomic forces, respectively. From (27) and (29) it follows that at equilibrium b = 1 andḃ = 0. For a small driving strength δω ho , one can assume that the radius of the cloud is perturbed around its equilibrium value, so where It means that δb is the fractional amplitude of oscillations of the radius and, therefore, it is a measurable quantity.
In the small amplitude limit, one can linearize Eq. (29) with respect to δω ho and δb yielding the following equation The solution is where Ω M = √ 5 ω ho corresponds to the frequency of the normal mode of monopole in the hydrodynamic limit [19].
Keeping only the lowest order in the small perturbation δb, Eq. (28) yields ρ(r, t) = ρ(r, 0) + 1 and using the Thomas-Fermi radius at equilibrium (9), it follows that the density fluctuation is given by We can calculate now x using (24) and (35), finding where C = 16πµR(0) 5 /(35g). Then, from Eq. (33), one gets At T = 0 there are no thermally excited states and, hence, α(ω) = α 0 (ω). By comparing the definition (25) with (37) one has This is the response function at zero temperature without including any dissipation. Therefore α 0 (ω) is real, i.e., the induced oscillations at T = 0 are undamped. The energy of oscillation can be calculated as twice the mean kinetic energy associated to the mode, E = dr ρ(r, 0)v 2 . For a monopole mode in an isotropic trap, the calculation [23,24] gives E = 15 7 N µ|δb| 2 , where |δb| is the amplitude of the oscillation of the cloud (31). Using Eqs. (36) and (25) at T = 0, it follows that Now we want to calculate the contribution of the thermally excited states to the response function. We study the low temperature regime, where α 1 ≪ α 0 and the energy of oscillation (39) can be estimated using α 0 instead of α(ω) = α 0 (ω) + α 1 (ω). The effect of α 1 will be introduced within a perturbation theory.
We have already seen that the thermal excitations can either absorb or emit quanta of oscillationhω and thus they will dissipate energy. The contribution of the elementary excitations to the susceptibility will be a complex function, α 1 (ω) = Re[α 1 ] + iIm[α 1 ], whose imaginary part is related to the absorption of energy Q of the external perturbation. However, in a stationary solution which is the case under consideration, the absorption Q must be compensated by the energy dissipation (16) due to the interaction with the elementary excitations. Therefore, Let us rewrite the definition of the damping rate (16) by using (17) and (18) with a generic oscillation frequency ω:Ė Inserting Eq. (39) and defining β(ω) = α 0 (ω)/C = 2/[m(Ω 2 M − ω 2 )] one obtains the energy dissipatioṅ Let us recall that the energy dissipation according to Eqs. (21) and (40) can be calculated also from the imaginary part of the response function α(ω) = α 0 (ω)+ α 1 (ω). Since α 0 (ω) is real, Eq. (21) becomes Comparing Eqs. (42) and (43) one can calculate the imaginary part of α 1 (ω) as And using the Kramers-Krönig relation (22) one finds the real part Now we have all the ingredients to calculate the response function of a spherically symmetric trapped condensate when the monopole mode is excited and a small perturbation of the trapping frequency δω ho ∼ exp(−iωt) is applied. It can be calculated within first order perturbation as α(ω) = α 0 (ω) + Re[α 1 (ω)] + iIm[α 1 (ω)], by using Eqs. (38), (45) and (44), respectively. It is worth stressing that the real part of the susceptibility diverges at Ω M (resonance of the condensate at T = 0) but also at ω ik which are the frequencies of the thermal excited modes that due to the interaction are coupled with the monopole.
Actually, the resonances of the condensate can be found by measuring the fractional amplitude of oscillations of the cloud radius δb at different perturbing frequencies. This measurable quantity can be easily related to the response function α(ω) from Eqs. (36) and (25) Note that the perturbation theory we have used is valid when | α 1 |≪| α 0 |. This condition becomes very restrictive at ω near ω M . However it is not difficult to improve the approximation in this region by taking benefit of the analogy between the response function and the Green function G.
It is well known that the Green function obeys the Dyson equation [26] which relates the perturbed quantity (G) and the unperturbed one (G 0 ) through the inversed functions (G −1 and G −1 0 ) in such a way that a perturbation theory for G −1 has a more wide applicability that for G. Analogously, we will find a relation between the inverse response functions, perturbed (α −1 ) and unperturbed (α −1 0 ). One has and formally with the same accuracy Now the applicability of (48) is restricted only by the condition that the second term is small with compare to m 2C Ω 2 M .
It is worth noting that according to the equation (48) the poles of α(ω) related to the resonances are shifted with compare to frequencies ω ik and are given by the equation: α 1 (ω ′ R )/α 0 (ω ′ R ) = 1. However these shifts are very small.

V. RESULTS
In order to present numerical results we choose a gas of 87 Rb atoms (scattering length a = 5.82 · 10 −7 cm). For the spherical trap we fix the frequency ω ho = 2π187 Hz, which is the geometric average of the axial and radial frequencies of Ref. [1], and corresponds to the oscillator length a ho = 0.791 · 10 −4 cm. We solve the linearized Gross-Pitaevskii equations Eqs. (2)(3)(4)(5) at zero temperature, to obtain the ground state wave function Ψ 0 and the spectrum of excited states E i as well as the corresponding functions u i (r), v i (r). In spherically symmetric traps the eigenfunctions are labeled by i = (n, l, m), where n is the number of nodes in the radial solution, l is the orbital angular momentum and m its projection. The eigenfunctions are u nlm (r) = U nl (r)Y lm (θ, ψ), the energies E nl are (2l + 1) degenerate and the occupation of the thermally excited states is fixed by the Bose factor.
For a fixed number of trapped atoms, N , the number of atoms in the condensate, N 0 , depends on temperature T . At zero temperature all the atoms are in the condensate, except a negligible quantum depletion [18]. At finite temperature the condensate atoms coexists with the thermal bath. In the thermodynamic limit [27] the T -dependence of the condensate fraction is N 0 (T ) = N [1 − (T /T 0 c ) 3 ]. We consider the collective excitations in the collisionless regime. This regime is achieved at low enough temperature. The excitation spectrum at low temperature can be safely calculated by neglecting the coupling between the condensate and thermal atoms [28]. It means that the excitation energies at a given T can be obtained within Bogoliubov theory at T = 0 normalizing the number of condensate atoms to N 0 (T ).
We investigate the monopole mode (l = m = 0 and n = 1). The functions u osc and v osc do not present angular dependence, and from Eq. (15) it is straightforward to see that the matrix element A ik couples only those energy levels (i, k) with the same quantum numbers l and m. That is, the selection rules corresponding to the monopole-like transition are ∆l = 0 and ∆m = 0. It is obvious, also, that different pairs of levels with the same quantum numbers n and l but different m give the same contribution. Therefore, only the integration of the radial part has to be done numerically.
Fixed N 0 and at a given temperature, we calculate the damping strengths (18) for the transitions ω ik coupled with the monopole. In Figure 1 we show the values of γ ik (in units of the frequency of the monopole Ω M ) for N 0 = 50000 87 Rb atoms at k B T = µ. The arrow points to the frequency of the breathing mode Ω M = 2.231 ω ho , and the chemical potential is µ = 15.69hω ho [these values are numerical results of the linearized Gross-Pitaevskii equations (2)- (5) for N 0 = 50000 rubidium atoms]. The position of the bars correspond to the allowed transition frequencies ω ik (in units of ω ho ) whereas their height defines the numerical value of γ ik [29].
One can see that there are two different types of allowed transitions ω ik . The damping strength associated to most of them is very small. Conversely, there are a few transitions which give relatively large values of γ ik . The latter correspond to transitions between the lowest levels (n k = 1, n i = 0) for different values of l (l = 2, 3, 4, 5). The main reason for these "strong transitions" is that the temperature occupation factor for these low-lying levels is large. Moreover the calculation shows that the matrix elements are also enhanced compared to other transitions. This is due to the fact that the radial wave functions involved in the integration have either one (n k = 1) or no node (n i = 0), differently from the oscillating character of the radial wave functions associated to higher levels [20].
The contribution of the other transitions is like a small "background" which is difficult to resolve in the scale of the Figure. A close-up view of the damping strengths of the transition frequencies around the monopole is displayed in the inset of Fig. 1 in order to show the dense background. It is worth stressing that such a distinction between "background" and "strong" transitions depends on the number of condensed atoms in the system and, of course, on temperature. When the number of atoms in the condensate increases, the number of excited states available by thermal excitations also increases, leading to a denser and less resoluble background.
In Figure 2 we present the same as in Fig. 1 but for N 0 = 5000 atoms of rubidium at k B T = µ, where here µ = 6.25hω ho . In this case, one can see that the difference between the "strong" and "weak" transitions is not so impressive as in a bigger condensate since all damping strengths can be appreciated in the same scale.
We can conclude that at large N 0 we have actually two different phenomena. The strong transitions create temperature induced resonances which can be observed in direct experiments. The background transitions give rise to Landau damping of the collective oscillations (See subsection B).

A. Temperature-induced resonances
Using the transition frequencies ω ik and the corresponding damping strength γ ik , we have calculated the response function α(ω). At zero temperature, the response function α 0 (ω) given by Eq. (38), gives a resonance at the monopole frequency Ω M = √ 5ω ho evaluated in the hydrodynamic regime. Due to interaction, thermal excited modes are coupled with the monopole. It means that when one excites the breathing mode of the condensate, the elementary excitations can give rise to other resonances at ω ik , which are the frequencies where Re[α 1 (ω)] diverges [see Eq. (45)]. We will now discuss the conditions for the observation of these effects in actual experiments. In particular, we calculate the contribution of these resonances to the response function and estimate the associated strengths.
Let us study the resonances at k B T = µ for N 0 = 150000 atoms of 87 Rb. The behavior of the damping coefficients γ ik is analogous to the one for 50000 condensate atoms (see Fig. 1) but in this case the difference between " strong" resonances and small background is even bigger: the dense background is not more resoluble in the scale of the strong resonances. There are five resonances that stand out the others, and that we label as ω R and γ R the corresponding damping strength (see table 1 for numerical values).
For perturbing frequencies close to the monopole ω ∼ Ω M , the monopole susceptibility , Eq. (38), can be approximated to Analogously, α 1 (ω) near each resonance ω ∼ ω R can be presented in the form α 1 (ω) ≃ A 1 /(ω R − ω). The ratio A 1 /A 0 is a measure of the relative intensity between temperature-induced and monopole resonance. Table 1 displays the numerical values of the relative intensity for each temperature-induced resonance ω R with respect to the monopole one, for N 0 = 150000 atoms in the condensate at k B T = µ. The relative strenght of the response function (A 1 /A 0 ) at ω R depends not only on the damping coefficient γ R but also on (Ω 2 M − ω 2 R ) −1 . It means that one mode ω R will be easier to excite, i.e., the strength of the response will be bigger, when it is close to the frequency of the monopole. Note also that the resonance strength increases with temperature through γ R .
From table 1 one can see that the biggest resonance occurs at ω R = 2.2576 ω ho which is resoluble from the monopole frequency Ω M = 2.234 ω ho and has a large enough relative strength to be observed. It means that, tuning the perturbation frequency ω to this value, a fluctuation of the fractional amplitude of oscillations can be observed.
In Figure 3 we have plotted the frequency dependence of the real part of response function α(ω) calculated according to equation (48) for N 0 = 150000. The response function is given in arbitrary units, and frequency is in units of ω ho . The dashed line shows the monopole resonance at Ω M , whereas the other divergences of α(ω) correspond to the temperature-induced resonances at ω R . From this figure one can see that the thermal induced resonances are quite distinct one from the other and from the monopole one. Therefore, temperature-induced resonances could be observed in experiments with good enough frequency resolution and good accuracy in the measurement of the radius fluctuations.
We would like to stress that the phenomenon we have discussed is related to quite delicate features of interaction between elementary excitations, and therefore, its observation would give rich information about properties of Bose-Einstein condensed gases at finite temperature.

B. Landau damping of collective modes
From Fig. 1 one can see that the weak background transitions ω ik have, generally speaking, very small frequency separation. To estimate this distance quantitatively let us renominate the resonances by an index i in the order of increasing value of ω. Then, one can define the average distance between resonances ∆ω according to: In a small interval around the collective oscillation 0.82 Ω M < ω ik < 1.
18 Ω M , we sum up all the transition frequencies allowed by the monopole selection rules and find the following values for the average distance between two consecutive transition frequencies: ∆ω/ω ho ≃ 0.0006, 0.001 and 0.006 for N 0 = 150000, 50000 and 5000, respectively. It is hopeless, of course, to try to resolve these resonances. Actually, there are reasons to believe that these resonances are smoothed and overlapped. First of all, because a real trap cannot be exactly isotropic. It means that levels with different m have slightly different energies, only levels with m = ± | m | are exactly degenerated. Therefore, each energy level with a given l will be splitted on l + 1 closer sublevels making more dense the energy spectrum. Furthermore, all excitations at finite temperature have associated a finite life time. Excitations with E ∼ µ which are the ones that mainly contribute in the "background" transitions, have the shortest life time. This can be accounted for phenomenologically by assuming that these levels have a finite lorentzian width ∆. That is, instead of delta functions in the equation for the damping rate (17), we will consider a Lorentzian distribution centered at ω ik with a fixed width ∆: In this case the damping rate becomes a smooth function of Ω osc and its value when Ω osc = Ω M defines the Landau damping of the monopole oscillations. At conditions the damping rate will have only a weak dependence on the exact value of ∆. In Figure 4 we plot the the dimensionless damping rate γ/Ω M as a function of the lorentzian width ∆ (in units of ω ho ) for N 0 = 50000 at different temperatures. The summation in (17) has been done over all resonances excluding of course the "strong resonances" presented in Figures 1, 2 and 3. One can see that the ∆-dependence is weak indeed in the interval ∆/ω ho = 0.05 ÷ 0.2 and γ can be reliable extrapolated from this interval to the value ∆ = 0. We take as Landau damping this extrapolated value of γ. One can estimate the accuracy of this extrapolation procedure to be of the order of 10 % according to the change of γ over this interval. In Figure 5 we plot the damping rate versus k B T /µ for N 0 = 150000 and 50000 atoms in the condensate. As expected, Landau damping increases with temperature since the number of excitations available at thermal equilibrium is larger when T increases. One can distinguish two different regimes in Fig. 5 one at very low T (k B T ≪ µ) and the other at higher T . The behaviour of the damping rate becomes linear at relatively small temperature (k B T ∼ µ) in comparison to the homogeneous system [4] where this regime occurs at k B T ≫ µ. Moreover, the damping rate increases for larger number of condensed atoms because the density of states available to the system also increases. It is interesting to note that the order of magnitude of the damping rate is the same as the one previously estimated for an uniform gas [3][4][5][6] and for anisotropic traps [7,8].

VI. SUMMARY
We have considered the monopole oscillation of a Bosecondensed dilute atomic gas in an isotropic trap. First of all, we have calculated the normal modes of the condensate by solving the time-dependent Gross-Pitaevskii equation within Bogoliubov theory [18] and then we have used the formalism developed in Ref. [4] to calculate the matrix elements associated with the transitions between excited states allowed by the monopole selection rules. Within a first order perturbation theory we have studied the Landau damping of collective modes due to the coupling with thermal excited levels. We have developed the response function formalism to study the fluctuations of the system due to an external perturbation. The contribution of the elementary excitations has been introduced also perturbatively as in the calculation of the damping strength, and we have derived analytic equations for the response function at zero temperature and at low temperature regime. We have seen that when the condensate oscillates with the monopole mode and a small perturbation to the trap frequency is applied, one can excite new resonances at the transition frequencies. These thermalinduced resonances are coupled with the monopole due to interaction effects. One cannot exclude a priori the possibility to observe such resonances also in anisotropic traps. This problem deserves further investigation. Observation of these resonances would give important and unique information about the interaction between elementary excitations in Bose-Einstein condensed gases.
We thank F. Dalfovo, P. Fedichev and S. Stringari for helpful discussions. M.G. thanks the Istituto Nazionale per la Fisica della Materia (Italy) for financial support.