Quantum Landau damping in dipolar Bose-Einstein condensates

# Quantum Landau damping in dipolar Bose-Einstein condensates

## Abstract

We consider Landau damping of elementary excitations in Bose-Einstein condensates (BECs) with dipolar interactions. We discuss quantum and quasi-classical regimes of Landau damping. We use a generalized wave-kinetic description of BECs which, apart from the long range dipolar interactions, also takes into account the quantum fluctuations and the finite energy corrections to short-range interactions. Such a description is therefore more general than the usual mean field approximation. The present wave-kinetic approach is well suited for the study of kinetic effects in BECs, such as those associated with Landau damping, atom trapping and turbulent diffusion. The inclusion of quantum fluctuations and energy corrections change the dispersion relation and the damping rates, leading to possible experimental signatures of these effects.

Quantum Landau damping is described with generality, and particular examples of dipole condensates in two and three dimensions are studied. The occurrence of roton-maxon configurations, and their relevance to Landau damping is also considered in detail, as well as the changes introduced by the three different processes, associated with dipolar interactions, quantum fluctuations and finite energy range collisions. The present approach is mainly based on a linear perturbative procedure, but the nonlinear regime of Landau damping, which includes atom trapping and atom diffusion, is also briefly discussed.

## I Introduction

The study of dipolar systems at low temperature has received considerable attention in recent years (1). In particular, the long range interactions between polar atoms and molecules introduces novel aspects of Bose-Einstein condensation, usually dominated by contact atomic collisions. In some cases, a convenient manipulation of external fields can nearly remove the short-range interatomic forces, and dipolar forces become dominant (2). In the usual mean field approximation (3), the Gross-Pitaevskii (GP) equation is completed with the inclusion of a non-local interaction potential (4); (5). But here we use a more general description of the condensates, which is not restricted to mean-field.

Recently, generalized forms of the GP equation have been proposed (6); (7), with cubic and quartic nonlinearities. The usual cubic term describes two-body collisions at zero energy, and the quartic term represents the Lee-Huang-Yang (LHY) correction associated with quantum fluctuations (8); (9). The LHY correction showed to be essential in explaining the appearance of droplets in dipolar condensates (10). We also have an additional cubic term resulting from a first order energy correction to the two-body collisions (11). These new terms can be relevant to the understanding of elementary excitations and they damping rates. Rotons have been observed in recent experiments (12).

Landau damping in BECs has been considered in the past using the mean-field GP equation (13); (14), and its relevance to dipole condensates has recently been addressed (15). Here we extend the previous analysis to consider both the quantum and quasi-classical regimes, and to discuss the eventual occurrence of atom trapping, quasi-linear diffusion and kinetic instabilities. Our approach is also different from previous analysis, because it is not based on the GP equation but makes use of wave-kinetics. This alternative approach is particularly useful for the understanding of kinetic processes, such as those associated with Landau damping.

The WK description of BECs is based on the Wigner function for the quantum medium. This has been explored in the past (16), and recently extended to condensates at finite temperature (17). In this paper, we use a generalized WK equation, which not only includes long range dipolar interactions, but also the effects associated with quantum fluctuations and with a finite energy range of atomic collisions. This can be derived from a generalized GP equation, using a quasi-probability distribution and following the Wigner-Moyal procedure (18). The inclusion of quantum fluctuations and energy corrections is particularly important to the understanding of the dispersion properties of the elementary excitations in condensates, and to determine the appropriate damping coefficient. This study could therefore lead to possible new experimental signatures of these effects.

This paper is organized as follows. In section II, we state the WK equation for dipolar BECs, which is the basic equation of our present model. In section III, we derive the kinetic dispersion relation of elementary excitations in the medium. These excitations are phonon modes of the dipolar quantum fluid. As particular cases, we consider typical configurations, in three (3D) and in quasi two (2D) dimensions. The 3D case contains unstable regions in the range of large wavenumbers, and the quasi-2D case shows the occurrence of a roton-maxon pair (19). In section IV, the kinetic non-dissipative damping of the phonon modes, also known as Landau damping, will be considered. We discuss the cases of a finite temperate BEC, and show that the dipolar interactions modify the Landau damping rate, in both the 3D and quasi-2D configurations. Both quantum and quasi-classical regimes are considered. We also discuss the possible occurrence of kinetic two-stream instabilities and their relation with the fluid instability studied by (20). We show that Landau damping can still exist for condensates with a finite size, even at zero-temperature. Finite dimensions imply the existence of an residual temperature, as a consequence of the uncertainty principle. This residual temperature is usually very small, but could eventually become relevant near a roton minimum, when the phase velocity approaches zero. Our discussion of Landau damping is based on of the linearized kinetic equation, and uses the standard perturbative procedure. But, in order to be complete, we discuss in section V, the limits of validity of the linear Landau regime. This discussion includes the main processes that could occur in the nonlinear regime, namely atom trapping and atom diffusion. The atom trapping is a consequence of finite amplitude oscillations, and relies on the possible existence of trapped quantum states. As for atom diffusion, it could occur in the centre-of-mass velocity space due to the presence of a broad spectrum of excitations. Finally, in section VI, we state some conclusions.

## Ii Wave-kinetic equation

We consider a dipolar condensate, as described by a modified GP equation of the form

 iℏ∂ψ∂t=(HGP+H′)ψ, (1)

where is the condensate order parameter, describing its ground state, and is the usual GP Hamiltonian as determined by

 HGP=−ℏ22m∇2+V0(r)+g|ψ(r,t)|2. (2)

Here is the confining potential, and is the usual coupling constant describing short-range atomic collisions at zero energy, and the scattering length. The Hamiltonian in eq. (1) describes three additional effects and can be written as

 H′=Q|ψ(r,t)|3+12χ[∇2|ψ(r,t)|2]+∫Ud(r−r′)|ψ(r′,t)|2dr′. (3)

The first term in this expression describes the LHY correction due to quantum fluctuations, determined by the coefficient . The second term is due to the finite energy range of atom collisions, and the corresponding coefficient is , with being the effective range obtained from the second-order expansion of the phase shift (6). Finally, the third term describes the long range dipolar interactions and is characterized by a dipole interaction potential , to be specified later.

Equation (1) describes the mean field plus quantum corrections of the condensate wave function . In alternative, we can describe the condensate considering the autocorrelation function. This new quantity is usually called the Wigner function, and can be defined as

 W(q,r,t)=∫ψ∗(r−s/2,t)ψ(r+s/2,t)exp(iq⋅s)ds. (4)

Starting from the above generalized GP equation, and applying the well-known Wigner-Moyal procedure (18), we can derive an evolution equation for , of the form

 iℏ(∂∂t+vq⋅∇)W=∫Vk(t)ΔWexp(ik⋅r)dk(2π)3, (5)

where is the atom velocity, and is defined as

 ΔW=W−−W+,W±≡W(q±k/2,r,t). (6)

The quantity in Eq. (4) is the space Fourier transform of the total potential . We should notice that the integral of the condensate quasi-probability is equal to the local atom density

 n(r,t)≡|ψ(r,t)|2=∫W(q,r,t)dq(2π)3. (7)

This allows us to write the total potential (2) as , where the dipolar term is determined by

 Vd(r,t)=∫dq(2π)3∫dr′Ud(r−r′)W(q,r′,t). (8)

From the convolution theorem, we have

 ∫Ud(r−r′)W(q,r′,t)dr′=∫Ud(k)Wk(q,t)exp(ik⋅r)dk(2π)3, (9)

where and are the space Fourier transforms of the dipolar potential and the quasi-probability . Using this in eq. (8), we can transform it into

 V(r,t)=∫Vk(t)exp(ik⋅r)dk(2π)3, (10)

with

 Vk(t)=V0(k)+[g−k22χ+Ud(k)]Ênk(t)+Q∫n3/2(r,t)exp(−ik⋅r)dr, (11)

and is the spectral component of the BEC density, as given by the space Fourier transform of Eq. (7). The wave-kinetic equation in (5), together with the expression for in Eq. (11), provides the full phase-space descprition of a dipolar BEC in the presence of quantum corrections.

## Iii Dispersion relation

In order to discuss the elementary excitations of the dipolar BEC, we assume that the quasi-probability can be divided in two distinct parts, . Here, is the equilibrium distribution describing the condensate in steady state, and is a small perturbation such that , describing the elementary excitations of the medium. Let us consider the simple case of a uniform and unbounded medium and assume a plane wave perturbation of the form

 ~W(q,r,t)=~Wk(q)exp(ik⋅r−iωt), (12)

where is the mode frequency. The corresponding density perturbation will be . Linearizing Eq. (5) with respect to the perturbed quantities, we can then easily get

 ~Wk=[g+Q√n0−χ2k2+Ud(k)]ΔW0ℏ(ω−k⋅vq)Ê~nk. (13)

Integrating over the atom momentum, we can then obtain a dispersion relation of the form

 1−[g+Q√n0−χ2k2+Ud(k)]∫ΔW0ℏ(ω−k⋅vq)Êdq(2π)3=0. (14)

This is valid for any condensate with long-range dipolar interactions. The latter can also be written as

 1−[g+Q√n0−χ2k2+Ud(k)]∫W0(q)ℏ[1(ω−−k⋅vq)−1(ω+−k⋅vq)]Êdq(2π)3=0, (15)

with . We first study the dispersion relation for a zero-temperature BEC. This allows us to use the simple equilibrium distribution , where is the unperturbed density and defines a constant drift velocity . Eq. (15) is then reduced to

 1−[g+Q√n0−χ2k2+Ud(k)]n0ℏ[1(ω−−k⋅v0)−1(ω+−k⋅v0)]Ê(2π)3=0. (16)

Rearranging therms and using the Bogoliubov speed, , this can also be written as

 (ω−k⋅v0)2=k2c2s[1+Qg√n0−χ2gk2+1gUd(k)]+ℏ2k44m2. (17)

This can easily be extended to the case of two counter-propagating BEC beams (20). It is useful to consider the product of the phase velocity and the group velocity . Assuming a condensate at rest (), we obtain

 vϕvg=c2s[1+Qg√n0+1gUd(k)]+4k2(ℏ24m2−χ2g). (18)

This shows that, the product is nearly equal to the square of the Bogoliubov speed, with corrections coming from the quantum dispersion term and from the three different processes included in the present model (dipolar potential, quantum fluctuations and finite energy range of close collisions). We can now take particular examples of dipolar potential. For typical dipole condensates, we can use the long range interaction potential (21); (22)

 Missing or unrecognized delimiter for \left (19)

where is the magnetic dipole constant, is the angle between the vector difference and the direction of the external polarization field, and is the angle between the orientation of the dipoles and the -axis. Taking , we obtain the Fourier transform

 Ud(k)=Cdd(cos2θk−13), (20)

where is the angle between the wavevector and the -axis. Replacing this in eq. (17), and assuming a condensate at test (), we obtain the dispersion relation

 ω2=k2c2(θk)+ℏ2k44m2. (21)

where is the angle dependent Bogoliubov velocity, defined by

 c2(θk)=c2s[1+Qg√n0−k22gχ+η(cos2θk−13)]. (22)

Here, is the ratio between the dipole and contact potential strength. As we can see, the dipole interactions introduce important qualitative corrections to the characteristic sound velocity, which can become imaginary for large values of the parameter . In particular, a critical wavenumber can be defined, where , as

 Missing or unrecognized delimiter for \left (23)

This is positive for and . In such case, large scale perturbations corresponding to become unstable, with a finite growth rate determined by . This is physically relevant for , where is the typical size of the condensate.

Another interesting example is the quasi-2D condensate. If a BEC is strongly confined along the -axis, which size much small than its transverse dimension , we can still use the same WK equation, only depending on and , but where is replaced by a renormalized coupling parameter, . In this case, the quasi-2D dipole interaction potential can be represented by (15)

 Ud(k)=CddF(klz/√2), (24)

where and the function , with , is defined as

 F(x)=1−32√πxexp(x2)erfc(x). (25)

Here, we have used the complementary error function, defined by

 erfc(x)=2√π∫∞xexp(−t2)dt. (26)

For our discussion, it is useful to consider the asymptotic expansion for , or , as . Using this new dipole potential in eq. (17), we can then write

 ω2=k2c22D[1+Qg√n0−k22gχ+ϵddF(klz/√2)]+ℏ2k44m2, (27)

where we have defined the new quantities , and . For , we can use the approximate expression

 ω2=k2c22D{1+Qg√n0−k22gχ+ϵdd[1−klz√2π(1−k3l3z3⋅23/2)]}+ℏ2k44m2. (28)

It is well known that these dispersion relations can lead to the occurrence of a roton-maxon pair. In some extreme conditions, this can even lead to the formation of a super-solid, where becomes negative for a well defined wavenumber (and not over a large region , as in the above 3D example). A necessary super-solid condition is , at a critical value , such that

 F(k0lz/√2)≃−1ϵdd(1+ℏ2k204m21c22D). (29)

In this expression, we have neglected the terms in and , which that have been recast in Fig. 3 for full illustration. Comparing this with Eq. (27), we can see that the super-solid instability cannot occur for small values of , but will eventually exist in the region of large wavenumbers.

## Iv Landau damping

The wave-kinetic description is particularly well suited to describe Landau damping and the related kinetic instabilities, as shown next. For that purpose, we go back to eq. (15), which can be rewritten in the form

 1−g′(k)ℏk∫G0(u)[1(u−ω+/k)−1(u−ω−/q)]Êdu2π=0, (30)

with . Here, and represent the atom velocity and momentum components parallel to the direction of propagation,according to.

 vq=ukk+v⊥,q=qkk+q⊥. (31)

We have also used the reduced distribution , such that

 G0(q)=∫W0(q,q⊥)dq⊥(2π)2. (32)

The integrals in eq. (30) includes two integrals, can be written in the form

 ∫G0(u)(u−v±)du=P∫G0(u)(u−v±)du+iπG0(v±). (33)

where , and represents the principal part of the integral, in the Cauchy sense. Using this in eq. (30) we can write it in the form , which can be split into its real and imaginary parts, as . For a real value of , this leads to a complex mode frequency . Assuming , we can determine separately the frequency and the damping rate , by writing

 ϵr(ωr,k)=0,γ=−ϵi(ωr,k)(∂ϵr/∂ω)ωr. (34)

Temperature effects are usually be negligible in what concerns the value of . We are then allowed to use in the first of these equations, which then reduces to

 ϵr(ωr,k)=1−g′(k)k2mG0ω2−ℏ2k4/4m2=0. (35)

Writing this reduces to eq. (17), for a BEC at rest. We can then approximately write

 ∂ϵr∂ω=2gωk2c2sg′(k). (36)

Retaining finite temperature effects in the damping rate (34), we can then obtain

 γ=gkc2s4ℏωr[1+Qg√n0−k22gχ+1gUd(k)]2[G0(v+)−G0(v−)]. (37)

This determines the atomic Landau damping of elementary excitations in dipolar condensates. In thermal equilibrium, we always have , and the damping coefficient is negative, . But, in a disturbed BEC, an inversion of population can eventually occur, such that . In this case the excitations are kinetically unstable. It is important to notice that the quantum fluctuations associated with , and the finite energy collisions described by , never change the sign of in the above expression, because the damping rate only depends on the quantity .

It is also useful to consider the semiclassical limit, valid for . In this case, we can develop the quantities around , and eq. (37) becomes

 γ≃k3c4s4n0ωr[1+Qg√n0−k22gχ+1gUd(k)]2(∂G0∂v)v=ω/k. (38)

For a condensate in equilibrium at a finite temperature , the derivative is always negative and the excitations are damped. In order to be more specific, we need an explicit expression for the reduced distribution . We can use a Bose-Einstein distribution.

 G0(v)=2πn0{e[E(v)−μ]β−1}−1 (39)

where , , and the chemical potential provides the zero of the energy scale. The Landau damping rate becomes

 γ≃−k3c4s4ωr[1+Qg√n0−k22gχ+1gUd(k)]2βe[E(ω/k)−μ]β{e[E(ω/k)−μ]β−1}−2. (40)

We can see that is always negative, for all possible values of the phase velocity . However, out of equilibrium situations can eventually occur, where the BEC is kinetically unstable. This is linked with the possible existence of a supra-thermal atomic stream, with density and mean velocity , as described by

 G0(v)=2πn0e[E(v)−μ]β−1+2πnbe[E(v−vb))−μ]βb−1, (41)

where and is the temperature of the supra-thermal stream. In this case, the sign of the mode damping coefficient will eventually change sign, leading to an unstable region of phase velocities . This is the kinetic counterpart of the two-stream instability discussed in (20).

Finally, it is important to notice that, even a zero temperature , the Landau damping may happen. This is due to the uncertainty principle, which implies that for a BEC with typical size the uncertainty of the atom velocity will be . Therefore, a finite size is equivalent to a residual temperature of order of . In that case, the Heisenberg broadens the zero-temperature distribution with the residual temperature

 T→Tr=12kBℏ2mL2 (42)

For a Dy BEC, with au, chemical potential kHz and m, we obtain pK, much less than the critical temperature nK ((25); (26)). This means that Landau damping will mainly be provided by the thermal part of the condensed gas. However, in a situation where the phase velocity of the elementary excitation is strongly reduced in the viscinity of a roton minimum, Landau damping could eventually be provided by the condensed gas itself, due to the existence of a residual temperature . This feature is illustrated in Fig 2. As we can see, in the mean-field case, Landau damping occurs below the roton minimum, while the inclusion of the quantum LHY correction displaces the Landau damping towards the roton minimum. With the inclusion of the finite range of the atomic collisions, the roton minimum remains practically undamped.

## V Trapping and diffusion

Landau damping results from an energy exchange between the phonon excitations and the atomic mean field. The above linear description of Landau damping needs to be completed by a discussion of other possible effects associated with this energy exchange. The first is atomic trapping, which can take place for an oscillation with a finite amplitude. Another is atom diffusion, when a large spectrum of excitations is excited in the medium. In this case, the exchange of energy between the mean field and the phonon field induces diffusion in the atomic velocity space, associated with the cumulative Landau damping over the photon spectrum, with second-order changes in the mean field. These two aspects, trapping and diffusion, are briefly described next. In this section, we will neglect quantum fluctuations and finite energy range effects, and take and , for simplicity. In the discussion that follows, the latter effects only as small corrections.

Atom trapping occurs in the close vicinity of resonance, when the atom velocity is equal to the phonon phase velocity. It can easily be seen that the centre-of-mass energy of the trapped states falls in the range , where is the phonon amplitude and . This means that the trapped states correspond to wavenumbers in the interval , where

 q±≃ωk√1±k2c2sω22m2ℏ2(~nkn0) (43)

The potential well created by the phonon excitations creates a series of trapped energy states, with energies levels , not exceeding , where is an integer. The bounce frequency for the trapped atoms is given by

 ωB=kcs√~nk2n0[1+1gUd(k)]. (44)

This trapping process is very similar to that occurring for free electrons in a quantum plasmas (23); (24), even if the atoms in a BEC are bosons and the electrons in a plasma are fermions. In particular, we can define a similar trapping parameter, , which gives the approximate number of trapped states. For , we are in the quasi-classical limit, and for trapping will be forbidden. Trapping introduces nonlinear corrections to Landau damping, which can lead to modulations of the mode amplitude at the harmonics of the bounce frequency . However, nonlinear Landau damping is outside the scope of the present work.

Let is now consider the case of a broad spectrum of phonons. This is relevant to a turbulent BEC. A quasi-linear theory, based on the above wave-kinetic equation can then be establish, which is formally identical to that derived in (27) for a laser-cooled gas. Each phonon excitation will be damped with the corresponding Landau damping rate, but due to global energy transfer between the mean field and the turbulent field, the equilibrium distribution will change over a long time-scale, as determined by the diffusion equation

 [∂∂t+vq⋅∇−∂∂q⋅¯¯D⋅∂∂q]W0(q,t)=0, (45)

were is a diffusion tensor in the atomic velocity space, given by

 ¯¯D=πn20∫[1+1gUd(k)]2qq|~nk|2δ(ω−k⋅vq)dk(2π)3 (46)

This expression shows that diffusion results from the accumulation of resonant interactions of the centre-of-mass states with the different Fourier components of the phonon spectrum. A detailed study of the diffusion equations is outside the scope of the present work.

## Vi Conclusions

In this paper, we have described main properties of quantum Landau damping in dipolar condensates. The quasi-classical limit was also discussed. Our model was based on a generalized wave-kinetic equation, with a non-local potential, where quantum fluctuations and the finite energy corrections were also included . We have shown that such a kinetic description is particularly adequate to describe Landau damping and kinetic instabilities associated with deviations from thermal equilibrium.

A general expression for the dispersion relation of elementary excitations in the dipolar BEC, and the corresponding Landau damping rate, were established. Typical dipolar configurations in three and quasi-two dimensions were also examined, which included the formation of maxon-roton pairs and the eventual occurrence of supersolids. Landau damping tends to increase in the presence of a maxon-roton pair, because the roton minimum decrease the phonon phase velocity, bringing the resonant phonon-atom interaction closer to the thermal velocity. The opposite situations occurs for near the maxon region.

Possible kinetic instability regimes were discussed, and a two-stream instability was identified. Landau damping at was also considered. We have shown that a residual temperature limit exists, associated with the finite size of BECs. Finally, atom trapping and atom diffusion in velocity space were discussed, and possible extensions of the Landau damping theory were suggested. We believe that the present work illustrates the relevance of the wave-kinetic description of dipolar BECs, and contributes to the understanding of resonant atom-phonon interactions. This will eventually lead to a consistent model of quantum turbulence.

###### Acknowledgements.
JTM and AG would like to thank the financial support of CNPq Brazil. AG also thanks funding of FAPESP Brazil. HT acknowledges FCT - Fundação da Ciência e Tecnologia (Portugal) through the grant number IF/00433/2015.

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