Beliaev damping of the Goldstone mode in atomic Fermi superfluids

# Beliaev damping of the Goldstone mode in atomic Fermi superfluids

G. Bighin, L. Salasnich, P.A. Marchetti and F. Toigo Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova, Via Marzolo 8, 35131 Padova, Italy
Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy
Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia (CNISM), Unità di Padova, Via Marzolo 8, 35131 Padova, Italy
August 20, 2019
###### Abstract

Beliaev damping in a superfluid is the decay of a collective excitation into two lower frequency collective excitations; it represents the only decay mode for a bosonic collective excitation in a superfluid at . The standard treatment for this decay assumes a linear spectrum, which in turn implies that the final state momenta must be collinear to the initial state. We extend this treatment, showing that the inclusion of a gradient term in the Hamiltonian yields a realistic spectrum for the bosonic excitations; we then derive a formula for the decay rate of such excitations, and show that even moderate nonlinearities in the spectrum can yield substantial deviations from the standard result. We apply our result to an attractive Fermi gas in the BCS-BEC crossover: here the low-energy bosonic collective excitations are density oscillations driven by the phase of the pairing order field. These collective excitations, which are gapless modes as a consequence of the Goldstone mechanism, have a spectrum which is well established both theoretically and experimentally, and whose linewidth, we show, is determined at low temperatures by the Beliaev decay mechanism.

###### pacs:
05.30.Fk, 67.85.Lm, 67.85.De

## I Introduction

The discovery of superfluidity below the -point in liquid He kapitza ; allen provided a stunning demonstration of quantum properties of matter at a macroscopic level, paving the way for the experimental realization, in more recent times, of condensates of atomic Bose anderson and Fermi regal gases and of quasiparticles, like in the recently observed exciton-polariton condensate kasprzak in a semiconductor microcavity.

Even if very different in nature and in their physical features, these phenomena are essentially a manifestation of the Bose-Einstein condensation of interacting bosons (respectively single atoms for He, Cooper pairs/bosonic dimers for ultracold Fermi gases, polaritons in an exciton-polariton condensate); thus from a theoretical point of view it would seem compelling to describe them, at least to some extent, within a common framework.

Indeed, excitations in a superfluid can be described using the quantum hydrodynamics approach developed by Landau landau ; a clear advantage of this formalism is the possibility of describing superfluids with non-contact interactions and with a varying number of particles by introducing higher order terms by means of a perturbative expansion around the mean field solution.

Collective excitations in a superfluid are destroyed either by Landau damping, due to their interaction with the thermal cloud, or by Beliaev damping, due to their decay into two, or more, lower energy excitations. There is competition between these two damping modes: whereas Landau damping is relevant at finite temperatures, with a vanishing cross section as the temperature goes to zero, Beliaev damping remains the only allowed decay mode at .

Therefore the Beliaev decay represents a test of Landau’s hydrodynamic theory. First evidences of a phonon decay have been observed in superfluid liquid He maris ; millis ; more recently the Beliaev decay has been observed in a trapped Bose-Einstein condensate (BEC) of rubidium atoms hodby ; katz ; an analogous process has been proposed in order to explain the absence of equilibrium in one dimensional interacting bosons, see ristivojevic and references therein.

In the present paper we focus on Beliaev decay and derive an improved version of the classical result beliaev ; landau ; stringari based on the observation that while the original derivation requires a nonlinear term in the spectrum, nonetheless it treats the kinematics in a low-momentum approximation as if the spectrum was effectively linear. We show that this treatment can be extended and that, in particular, the inclusion of a gradient term in the Hamiltonian yields a Bogoliubov-like spectrum for the bosonic excitations salasnich2 . We calculate the decay rate for the Beliaev damping and show that even for low momenta and small nonlinearities a realistic spectrum can give appreciable differences with respect to the linear approximation of the standard result.

The result is applied to an attractive Fermi gas: as the attractive interaction between atoms is tuned, the gas at goes with continuity from a Bardeen-Cooper-Schrieffer (BCS) weakly-interacting regime, to a strongly interacting gas of bosonic dimers. This scenario can be described nagaosa ; altland ; stoof by introducing the complex Cooper pairing field, which will acquire a non-zero expectation value below the critical temperature. As the phase of the order parameter is macroscopically locked below the critical temperature goldstone ; nambu , spontaneously breaking the symmetry, its fluctuations correspond to the gapless mode predicted by the Goldstone theorem. These collective modes turn out to be fundamental in quantitatively describing the dynamics of an ultracold Fermi gas strinati ; after briefly analyzing the Goldstone mode, we show that its linewidth gets substantially enhanced due to the Beliaev decay process. We also show that our improved description of the decay yields substantial deviations from the standard approximation.

## Ii Beliaev damping: an improved treatment

We briefly introduce Landau’s quantum hydrodynamics landau ; stringari , a semi-phenomenological description of a superfluid which can be, however, rigorously justified and derived from the microscopical theory as discussed in popov . An exact expression for the internal energy of a classical liquid in a sound wave is , where is the local velocity of the fluid, and the local density. Here represents the internal energy of the fluid for unit mass; Landau’s original treatment landau ; beliaev assumes it to be dependent only on the density , and as a consequence the dispersion relation for the sound waves is linear. On the other hand by adding a gradient term as

 e(ρ)→e(ρ,∇ρ)=e(ρ)+λℏ28m2(∇ρ)2ρ2 (1)

higher order terms appear in the dispersion relation as shown in salasnich2 , being the mass of a fluid particle, being a dimensionless coefficient which can be fixed a posteriori. Within the quantum hydrodynamics framework the velocity and density fields of a fluid are promoted to quantum operators, so that the Hamiltonian for a quantum fluid is:

 ^H=∫d3x[12^v⋅^ρ^v+^ρe(^ρ,∇^ρ)] (2)

where the term involving the velocity operator has been opportunely symmetrized to be Hermitean. We rewrite the velocity in terms of a velocity potential and the density by separating the equilibrium value from its fluctuations as . The new operators can be written expanding in plane waves:

 ^ρ′=1√2V∑|k|≠0Ak(^bkeik⋅r+^b†ke−ik⋅r) (3)
 ^ϕ=1√2V∑|k|≠0iℏBk(bkeik⋅r−b†ke−ik⋅r) (4)

with the volume of the system; the () operators annihilate (create) a bosonic excitation over the fundamental state of the liquid , and obey the canonical commutation relationships.

We impose that and should be canonically conjugate variables

 [^ϕ(r),^ρ′(r′)]=−iℏδ(r−r′) (5)

this constraint being fulfilled by . The exact treatment of a quantum liquid in Eq. (2) can be expanded in powers of the field operators: the first to give a contribution is the second order, here the theory can be diagonalized to a theory of non-interacting bosons, i.e. , and the requirement for to be diagonal fully fixes as:

 Ak=√ℏkρu(1+λℏ28m2k2c2)−14 (6)

and the dispersion for the bosons has the usual Bogoliubov structure

 ωk=uℏk√1+λℏ24m2k2u2 (7)

being the sound velocity of the sound waves in the quantum liquid. Clearly the original linear theory can be recovered by setting and removing the gradient terms. The present formalism, as opposed to the Gross-Pitaevskii equation gross ; pitaevskii , allows for the decay of a collective excitation in a superfluid, in particular extending the treatment to the third order one immediately sees that the decay of one excitation into two is allowed: this process is the Beliaev decay beliaev described above. The third order term of the Hamiltonian is:

 ^H(3)=∫d3r[(∇^ϕ)^ρ′2(∇^ϕ)+16(ddρu2ρ)^ρ′3+−λℏ28m2(∇^ρ′)2^ρ′ρ2] (8)

Before going on with the treatment of the Beliaev decay we briefly comment on the scope of application of the present theory; as already mentioned it can be shown popov that the hydrodynamic Hamiltonian in Eq. (2) can be rigorously derived from a description of the Bose gas; this procedure involves integrating out the “fast fields”, effectively defining a momentum scale below which the perturbative expansion should be valid. Following popov one can estimate this quantity for a weakly interacting Bose gas; here is the momentum marking the separation between a linear spectrum and the free-particle quadratic spectrum, and from Eq. (7) one gets

 ℏkc≃2mu√λ (9)

this condition marking, as argued in popov , the upper limit for the validity of the perturbation theory.

In order to study the Beliaev decay we calculate the matrix element:

 H(3)if=⟨i|H(3)|f⟩ (10)

between the following initial and final states:

 |i⟩ = ^b†p|Ω⟩ (11) |f⟩ = ^b†q1^b†q2|Ω⟩ (12)

The matrix element in Eq. (10), when Eqs. (3) and (4) are plugged in, is essentially the expectation value over of a number of terms composed of six creation/annihilation operators; after a lengthy but straightforward calculation, one obtains

 H(3)fi=(2πℏ)3(2V)32⋅δ(3)(p−q1−q2)⋅3√uρpq1|p−q1|(1+χθρ2u2ddρu2ρ) (13)

where is the angle between and , the other angles being fixed by the condition enforced by the function. We also defined:

 χ−1θ=p−q1|p−q1|(1+cos(θ))+cos(θ) (14)

In deriving Eq. (13) we neglected all the terms containing ; it can be checked that they give and contributions to the decay width, whereas the leading contribution will turn out to be . However the nonlinear dispersion relation is relevant when discussing the kinematics: the differential decay rate is calculated using Fermi’s golden rule 111The square of the function imposing momentum conservation is to be interpreted as in landau : :

 dw=2πℏ|H(3)fi|2δ(Ef−Ei)V2(2πℏ)6d3q1d3q2 (15)

and , is the spectrum as derived in Eq. (7). The integration over is performed using the momentum conservation constraint appearing in , the integration over the angular part of removes the function related to energy conservation, fixing at the same time the decay angle , and finally the radial integration remains explicit. The final result for the decay rate is:

 w=932πρℏ4∫p0q2|p−q|20(1+χθ0ρu2ddρu2ρ)2|f′(cosθ0,p,q)|dq (16)

where for shortness sake, is essentially the energy conservation constraint, is its derivative with respect to the first argument and is the only zero of in the interval , and represents the allowed decay angle given the incoming and outgoing momenta.

Equation (16) is the main original result of the present paper, which we will apply to an attractive Fermi gas. We stress that in Eq. (16) is a function of just , and of the incoming momentum ; moreover the exact form of the spectrum, including the coefficient, contributes indirectly to the final result, by modifying and, consequently, . We also note the kinematics constraints can be satisfied and the decay is allowed only if the aforementioned zero of exists, an equivalent condition being that the spectrum should grow faster than linearly.

Let us make the physical meaning of the last remark clearer, expanding the spectrum in Eq. (7) in powers of :

 ωk=uk+αk3+O(k4) (17)

where has the same sign as . The energy conservation constraint in the low momentum limit reads and can be fulfilled only if , i.e. only if the spectrum grows linearly or more than linearly; for no decay is allowed.

We now focus on the strictly linear case . Energy and momentum can be conserved only if , i.e. the momenta of the decaying excitation and those of the decay products are parallel. Even for very small values of the decay kinematics deviates significantly from the aforementioned linear situation as increases.

We stress that, even if the standard treatment of Beliaev decay landau ; beliaev correctly identifies as a necessary condition for the decay to happen, then its use of in the kinematics is a critical assumption; on the other hand the present treatment by including the gradient term in Eq. (2) allows for a realistic, Bogoliubov-like spectrum.

Let us derive the standard result from the more general Eq. (16): having set for a linear spectrum one has that , and also . Moreover noting that , we recover Beliaev’s original approximation beliaev ; landau , which we report here for the sake of completeness:

 w=p53320πρℏ4(1+ρ23u2ddρu2ρ)2 (18)

To conclude the present section we note that in the case of a weakly-interacting Bose gas Eq. (18) further simplifies, because in this case

 ρ2u2ddρu2ρ=0 (19)

Alternatively the weakly-interacting Bose gas regime can also be investigated, as done in katz , starting from the atomic Hamiltonian, introducing the Bogoliubov approximation and isolating the relevant decay vertices. The present hydrodynamic approach is different because it can be derived, as already mentioned, by separating the fast and slow components of the fields, introducing a momentum scale , whereas the Bogoliubov approximation merely separates the zero-momentum contribution. We expect the two approaches to yield the same results for , as we verified. The hydrodynamic approach, however, is better suited for analyzing the collective excitations of an attractive Fermi gas.

## Iii Beliaev damping for an attractive Fermi gas

Let us consider a three-dimensional, uniform dilute gas of interacting Fermi atoms; the atoms are neutral and have two spin species. This system can be described within the path integral formalism nagaosa ; altland ; stoof in which the fermions are represented by the complex Grassman fields , with the spin index . The partition function for the system at temperature , with chemical potential is:

 Z=∫D[ψσ,¯ψσ] exp{−1ℏ S}, (20)

with the following action and (Euclidean) Lagrangian density:

 S=∫ℏβ0dτ∫Vd3r L (21)
 L=∑σ¯ψσ[ℏ∂τ−ℏ22m∇2−μ]ψσ+g¯ψ↑¯ψ↓ψ↓ψ↑ (22)

as usual , is the Boltzmann constant, is the volume of the system and is the strength of the contact interaction between atoms; this quartic interaction can be decoupled, as usual, through a Hubbard-Stratonovich transformation in the Cooper channel, introducing the pairing field . Now the Euclidean Lagrangian density reads:

 L=∑σ¯ψσ[ℏ∂τ−ℏ22m∇2−μ]ψσ+ +¯Δψ↓ψ↑+Δ¯ψ↑¯ψ↓−|Δ|2g (23)

In order to obtain the partition function one also has to extend the functional integration to the newly introduced pairing fields , . We rewrite the fields , as the sum of their saddle point value plus the fluctuations

 Δ(r,τ)=Δ0+η(r,τ) (24)

Up to this point the theory is exact; the mean field theory of a Fermi gas is simply found by neglecting the fluctuations , . The functional integration defining the partition function can then be performed, yielding:

 Zmf=exp(−βΩmf) (25)

with:

 Ωmf=−∑k(Esp(k)−ϵk+μ)−VΔ20g. (26)

with ; is the spectrum of elementary single-particle fermionic excitations. The number and the gap equations for the system can be readily obtained from the mean-field grand potential :

 N=−∂Ωmf∂μ (27) (∂Ωmf∂Δ0)μ=0 (28)

Lastly the gap equation in Eq. (28) needs to be regularized, and this can be done by replacing with the scattering length , according to the following prescription (see e.g. leggett ):

 m4πas=−1g+1V∑k12ϵk (29)

where is the s-wave scattering length associated to the interatomic potential.

Let us now analyze the fluctuations contribution to the present theory; going back to Eq. (24) and reinstating the fluctuations fields , up to the quadratic (Gaussian) order ohashi2 ; randeria2 the partition function reads:

 Z=Zmf ∫D[η,¯η] exp{−Sg(η,¯η)ℏ}, (30)

having defined the Gaussian action:

 Sg(η,¯η)=12∑q(¯η(q),η(−q)) M(q)(η(q)¯η(−q)) (31)

with , and are the Bose Matsubara frequencies. The matrix in Eq. (31) is the inverse propagator for the pair fluctuations of an interacting Fermi gas, its matrix elements are defined by randeria2 ; tempere2 :

 M11(q)=1g+∑k(u2u′2iωn−E−E′−v2v′2iωn+E+E′) (32) M12(q)=∑kuvu′v′(1iωn+E+E′−1iωn−E−E′) (33)

where , , , , , . The remaining matrix elements are defined by the relations: , . By integrating out the , fields we get the Gaussian contribution to the grand potential randeria2 ; tempere2 :

 Ωg=12β∑qln(detM(q)) (34)

A completely equivalent description can be given, after a unitary transformation, in terms of the (linearized) phase and amplitude of the fluctuation field, which can be decomposed as . The Gaussian level action now reads:

 Sg=12∑q(λ∗θ∗)(ME11+M12iMO11−iMO11ME11−M12)(λθ) (35)

in terms of the even/odd components in of the matrix elements gubankova ; engelbrecht , i.e. . This representation makes clear that as soon as the Cooper pairing field acquires a non-zero expectation value, i.e. under , as a consequence of the symmetry breaking one expects to observe the gapless Goldstone mode nagaosa . More specifically it can be verified from Eq. (35) that for , in the low momentum limit the phase and amplitude fluctuations are decoupled engelbrecht : the diagonal entries in Eq. (35) go to zero, and the phase (Goldstone) mode is gapless, while the amplitude (Higgs) mode exhibits a mass gap. From now on up to the end of the present section we will study the system at . Focusing on the former mode, we observe that, indeed, by solving for the equation

 detM(q,iωn→ω)=0 (36)

we obtain the spectrum of the bosonic collective mode, showing a gapless branch. Notably in the BEC regime , and across the whole crossover for low enough momenta, this mode takes (within very good approximation) the familiar Bogoliubov-like form

 Ecol(q)=√ϵq(λϵq+2mc2s) (37)

and the sound speed , along with the parameter , depends on . We use this spectrum in the deep BEC limit, while in the intermediate regime near unitarity we solve numerically Eq. (36) to get the “exact” spectrum within the present Gaussian approximation scheme. When comparing the “exact” spectrum so obtained with the Bogoliubov approximate form, one also has to remember that a natural momentum scale can be defined by studying whether and when the dispersion enters the two-particle continuum reaching the threshold energy:

 Eth(q)=mink(Ek+Ek+q) (38)

above which a Cooper pair breaks down in two fermions. As far as the present work is concerned it is important noting that grows more (less) than linearly if (), moreover the parameter can be calculated easily either from a numerical solution of Eq. (36) or using the techniques in marini2 . It turns out that is a monotonically increasing function of , where is the Fermi momentum. In particular takes negative values in the deep BCS regime and changes its sign for ; referring to the previous section we can then conclude that no Beliaev decay will happen in the deep BCS region for .

We now want to adapt Eq. (16) to the present theory. We start by noticing that if the spectrum has the form in Eq. (37), then the decay angle defined in the previous section has an analytic expression:

 cosθ0(p,q)=m2c2sλpqℏ2+q2p+p2q−√m2c4s+2mc2sλ(ϵp+ϵq)−2λpqℏ22m√(2mc2s+λϵp)(2mc2s+λϵq)+λ2(ϵ2p+ϵ2q)2λpqℏ22m (39)

We note that for the special case , Eq. (39) coincides with the result in katz .

Finally the more complicated expression inside the parenthesis in Eq. (16) can be expressed using the techniques devised in salasnich as:

 ρ2u2ddρu2ρ=−30ϵ(y)−8yϵ′(y)−3y2ϵ′′(y)+y3ϵ′′′(y)30ϵ(y)−18yϵ′(y)+3y2ϵ′′(y) (40)

as a function of , where is the bulk energy per particle per particle of an interacting Fermi gas; when calculating our final results we compared as fitted in salasnich from experimental data with its mean field counterpart, observing no appreciable differences as far as the quantity Eq. (40) is concerned. Consistently with the result found in Eq. (19) for the weakly-interacting Bose gas, the quantity in Eq. (40) tends to zero in the deep BEC limit.

We calculate the Beliaev decay width for the Goldstone collective mode of an attractive Fermi gas; as previously noted there is no decay in the BCS regime up to , as the spectrum as a function of grows less than linearly. For higher values of we can associate an imaginary part to the Goldstone mode spectrum, as

 Imωp=−ℏw2 (41)

using from Eq. (16). In Fig. (1) we report the real spectra , obtained from Eq. (36), for three values of , from unitarity to the BEC regime (, , ), along with their imaginary part due to the Beliaev decay.

A collective excitation in a superfluid Fermi gas cannot have arbitrarily high energy, as it will be damped either by the dissociation mechanism at the threshold energy , decaying into two fermions, or by the Beliaev mechanism, decaying into two lower frequency collective excitations. Either way a natural energy cutoff can be associated to a bosonic excitation.

Referring to the left pane of Fig. (1) we start at unitarity () where the Beliaev decay width is quite narrow: here a collective excitation will mainly decay by hitting the threshold energy and breaking down into two fermions combescot . On the other hand, approaching the BEC regime (, ) the Beliaev decay width gets larger before the collective spectrum touches : here the preferred decay mode for a collective excitation will be decaying into two lower frequency collective excitations. This trend, i.e. the progressively bigger importance of the Beliaev mechanism approaching the BEC regime, can be observed by comparing the three panes in Fig. (1).

In order to define an energy cutoff due to the Beliaev mechanism, we can match the real and the imaginary part of similarly identifying a scale beyond which a collective excitation is no longer well-defined due to the Beliaev decay. This remark is made clear by looking at the pair fluctuation spectral function

 Aηη(k,ω)=−2ImGηη(k,ω+γk) (42)

plotted in Fig. (2). As noted in klimin , it can be interpreted as the contribution to the density from the fluctuations at a given wave number and a given momentum . In the previous equation is assumed to be real and is the imaginary component of the spectrum due to the Beliaev decay, is the Green’s function obtained by inverting the matrix in Eq. (31) and taking the entry. We observe that for low momenta most of the spectral weight is peaked around the dispersion relation, which is marked by a dashed line, assuming the usual Lorentzian structure. However, as the spectrum continues after , for high momenta the line broadening effect due to the Beliaev decay effectively destroys the collective excitation, and the spectral weight is distributed over a large region. The border between these two regimes can also be approximately found by imposing the aforementioned condition , which can be easily read from Fig. (2): when the real part of the dispersion is bigger than the imaginary part, the expression in Eq. (42) has a narrow peak; as the imaginary part of the spectrum gets bigger the Lorentzian structure of the peak is lost and the excitation is no longer well defined.

We can conclude that, as we go from the BCS to the BEC regime, the dissociation mechanism at gets less and less relevant, as the collective mode spectrum gets further away from ; at the same time, the Beliaev decay channel opens at and gets progressively more relevant. Finally in Fig. (3) we compare the decay width, as predicted by the present theory, with the original linear approximation beliaev ; landau : even for relatively small momenta our treatment shows relevant differences with respect to the standard treatment. The differences get larger in the BEC regime, consistently with the fact that the nonlinearity term in the spectrum is bigger; however we stress that even for nearly linear spectra, see the cases and in Fig. (1), the correction due to the present treatment can amount up to 25% for .

In conclusion we briefly comment on the scope of applicability of the present theory to the fermionic case; adapting Eq. (9) one finds

 kc≃2mcs√λ (43)

and we do not expect the theory to be applicable above this momentum threshold; a direct calculation shows that starting at unitarity, up to the moderate BEC regime we considered in Fig. (1) and Fig. (2), assumes respectively the following values: , , ; in the deep BEC limit we get

 kc≃12Δ20|μ| (44)

We notice that for the cases we considered, the momentum scale marking the breakdown of the perturbation theory is higher or equal to the scale at which a collective excitation is no longer well defined due to a high decay rate; we conclude then that the present treatment is consistent.

## Iv Conclusions

We have extended the standard result of Landau’s hydrodynamic theory of a superfluid, which leads to a purely linear spectrum implying a collinear Beliaev decay. By including a gradient term in the Hamiltonian, we have recovered the Bogoliubov-like spectrum bosonic excitations in a superfluid have a Bogoliubov-like spectrum of excitations producing a larger phase-space for the Beliaev. We have shown that even slight variations from linearity of the spectrum can give important modifications to the decay rate of the process we consider.

We have applied our result to an interacting Fermi gas in the BCS-BEC crossover: we have shown that no decay happens at zero temperature in the deep BCS regime, due to kinematics constraints, as the spectrum grows less than linearly. As the strength of the attractive interaction is increased, the collective mode spectrum increases linearly or faster as , thus allowing the decay of one collective excitation into lower energy excitations and this mechanism becomes more and more relevant as the coupling gets stronger. We observe that in the BCS regime in the low-temperature limit a collective excitation can decay only by breaking down into two fermions at ; on the other hand at unitary and in the BEC regime a collective excitation can also decay in two collective excitations by means of the Beliaev decay.

Finally we have identified the regimes to which the theory we have developed applies. The perturbation theory behind the hydrodynamic treatment of a quantum liquid breaks down at a critical momentum . We have estimated this value, verifiying the internal consistence of our treatment, showing by analyzing the decay width and the pair fluctuation spectral function that in the fermionic case is higher or equal to the momentum scale at which a collective excitation is no longer well defined due to the decay process.

###### Acknowledgements.
Work partially supported by MIUR (Ministero Istruzione Università e Ricerca) through PRIN Project “Collective Quantum Phenomena: from Strongly-Correlated Systems to Quantum Simulators”.

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