Transition and Damping of Collective Modes in a Trapped Fermi Gas between BCS and Unitary Limits near the Phase Transition

We investigate the transition and damping of low-energy collective modes in a trapped unitary Fermi gas by solving the Boltzmann-Vlasov kinetic equation in a scaled form, which is combined with both the T-matrix fluctuation theory in normal phase and the mean-field theory in order phase. In order to connect the microscopic and kinetic descriptions of many-body Feshbach scattering, we adopt a phenomenological two-fluid physical approach, and derive the coupling constants in the order phase. By solving the Boltzmann-Vlasov steady-state equation in a variational form, we calculate two viscous relaxation rates with the collision probabilities of fermion’s scattering including fermions in the normal fluid and fermion pairs in the superfluid. Additionally, by considering the pairing and depairing of fermions, we get results of the frequency and damping of collective modes versus temperature and s-wave scattering length. Our theoretical results are in a remarkable agreement with the experimental data, particularly for the sharp transition between collisionless and hydrodynamic behaviour and strong damping between BCS and unitary limits near the phase transition. The sharp transition originates from the maximum of viscous relaxation rate caused by fermion-fermion pair collision at the phase transition point when the fermion depair, while the strong damping due to the fast varying of the frequency of collective modes from BCS limit to unitary limit.

Scientific RepoRts | 5:15848 | DOi: 10.1038/srep15848 However, a theoretical description of this unitary regime is still challenging, particularly at nonzero temperature for sharp transition and strong damping of the collective modes.
There are different strong-coupling theories to study the collective excitations of superfluid Fermi gases in the BCS-BEC crossover. One of them is microscopic theory based on a model Hamiltonian either with a one-channel model for a broad (or weak) FR or with a two-channel model for a narrow FR. The link of these two models is well described in 23 . There are numerous efforts to develop the strong-coupling perturbation theories of interacting fermions. For example, the thermodynamic potential (or action) approach [24][25][26][27][28] , the diagrammatic method [29][30][31][32][33][34][35][36][37][38][39] , and the many-body T-matrix fluctuation theories [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] . Leggett's 45 mean-field theory and then Randeria et al. 46 by adding fluctuations get some qualitative correct results at zero temperature. The Quantum Monte Carlo (QMC) simulations 47,48 and the pseudogap approach 43,[49][50][51] have better results in the BCS-BEC crossover. Because the strong coupling atomic Fermi gases are trapped in a finite space at finite temperatures, the inhomogeneous feature of the system, strong pairing fluctuations, and finite temperatures are important keys in considering real cold Fermi gases, which makes the pure microscopic approach difficult to deal with, especially in studying the collective excitations.
 0 8 14 . The transition is accompanied by very strong damping. The corresponding features cannot be explained on the basis of available theoretical models and new physics is in great need in this regime. How to explain this feature is an open question by now. We still lack a full discussion on the transition and damping of collective modes, especially compared with the experimental results [5][6][7][8][9][10][11][12][13][14]63,64 . This is the major motivation for our present study of different collective modes under similar experimental conditions where the system is trapped around the critical temperature T c .
In this paper, we determine the sharp transition and strong damping of the collective modes at − 2 < η < 0 around T c after solving the Boltzmann-Vlasov kinetic equation in a scaled form, in which we have combined both the many-body T-matrix fluctuation theory in the normal phase at T ≥ T c and the mean-field theory in the order phase at T, T c . We first need to get the expressions of the viscous relaxation time, τ nn and τ ns . Here τ nn is related to the scattering between fermions in the normal states, and τ ns is related to the scattering between normal fermions and superfluid fermions. In order to do so, the two-fluid approach is a link to the microscopic and kinetic descriptions of many-body Feshbach scattering. We then calculate two viscous relaxation rates by solving the Boltzmann-Vlasov steady state equation in a variational form. We next calculate the collective-mode frequencies and their damping as a function of the temperature and interaction strength for a trapped gas. We finally compare our results to experiments.

Model and ingredients.
For the broad resonance regime of the interacting fermions, a one-channel microscopic model of the strong Feshbach resonance is the most appropriate. For N fermions of two species σ = ↑ and σ = ↓ with fermionic field ψ σ (r), its Hamilton density is 2 2 is the external potential with λ being the anisotropic parameter, and g = 4πħa sc /m is the interaction coupling constant. In the normal phase at T ≫ T c , the system can be treat as a normal fluid. But near above T c , fermions in the system may exist pre-pairing. While in the order phase at T < T c , parts of fermions become either the Cooper pairs due to the many-body effect at the BCS side or the bosonic molecule due to two-body bound state at the BEC side, and the pair fluctuations become important to the system near the unitary. However, it is hard to find anact solution of the model to describe this picture, also hard to find the expressions for the viscous relaxation rates of the all scattering processes.
Firstly, we start from the Boltzmann-Vlasov kinetic equation and find its scaling solutions. The phenomenological two-fluid physical picture is about N n normal fermions and N s = (N − N n )/2 superfluid fermion pairs (Cooper pairs or molecules). Generally, we should write GP equation to describe the dynamic of the superfluid parts. However, our interest focus on the system's behavior and novel phenomenon near T c , where the effects that form and break pairs play a more important role. So we freeze all superfluid fermion pairs at zero center-mass momentum and omit the dynamics of the superfluid fermions. This approximation is only satisfied at the BCS side. Therefore we focus on two kinds of scattering processes of fermions from BCS limit to unitary limit. One is scattering with the normal fermions in the normal phase by joining the T-matrix approximation at T ≥ T c , the other is with the fermion pairs in the superfluid phase by adopting the mean-field approximation at T < T c . Secondly, in order to consider the pairing and depairing effects of fermions, and to get the pair-pair and fermion-pair coupling constants in the order phase, we need to connect the microscopic and kinetic descriptions of many-body Feshbach scattering. So we solve the one-channel microscopic model in the mean-field approximation. We write the two-liquid phenomenological model and determine the coupling constants by equalizing all the physical quantities obtained from the phenomenological model with those obtained from the microscopic model. Thirdly, we derive the expression of the two viscous relaxation time of fermions, and calculate their relaxation time, τ nn and τ ns , respectively. Finally, we calculate the collective mode frequencies and their damping as a function of the temperature and interaction strength for a trapped gas. We compare our theoretical results to experiments, and conclude that the transition from collisionless to hydrodynamic behavior occurs at the maximum of viscous relaxation rate of normal fermion-superfluid pair collision at phase transition point.

Boltzmann-Vlasov kinetic equation.
We first briefly review the Boltzmann-Vlasov kinetic equation. We consider a two-component gas of Fermi atoms with mass m and different spin σ near its normal phase. We assume that the dynamics is described by a semiclassical distribution function f(r, p, t) for each component. We omit the index of spin because of the system symmetry. f(r, p, t) satisfies the Boltzmann-Vlasov kinetic equation 18,67,68 and C ns [ f, ρ s ] are the collision integrals of normal fermion-normal fermion and normal fermion-superfluid pair, respectively. They are a functional of f(r, k, t) and density of pairs ρ s (r) near below T c . k = p/ħ, and ħ = h/2π with h being the Plank constant. ε is the mean-field interaction energy. Many physical quantities are very sensitive to the equation of state which is given by in a polytropic approximation ε(ρ) = cρ γ with c being the constant with given η. Here γ is an index defined as the logarithmic derivative γ = ∂ ln ε/∂ ln ρ 55 , and ρ is the particle density for each component of Fermi gas. Interparticle interactions enter Eq. (2) in two different ways. One way is to modify the effective potential through the mean-field term ε which affects the streaming part of the Boltzmann kinetic equation. The mean-field term is essential linear in a sc and this theory has no dissipative term. Another way is to consider the two-body interactions in the collision integral − I[ f, ρ s ] which is quadratic in the scattering length and describes the dissipative processes.
The two collision terms in Eq.
(2) at Born approximation level are given by 67,68   with f i ≡ f(r, k i , t) for i = 1, 2, 3, 4. Here g n = g = 4πħa sc /m. g ns is the fermion-pair coupling constant determined below. The binding energy  b of a fermion pair is always negative below T c . In order to study the damping rate, we need to calculate the viscous relaxation rate 1/τ = 1/τ nn + 1/τ ns . In the normal phase at T ≥ T c , τ = τ nn as usual, while in the superfluid phase at T < T c , we have to estimate τ ns . We first write is the equilibrium distribution function without the interactions between two particles and δf is its deviation with the interactions. The Boltzmann-Vlasov kinetic equation (2) is a differential and integral equation, and it is hard to find its solution exactly. In the relaxation time approximation, for any trial function Φ by using a variational method. Here  means an integral over all phase space (both coordinate and momentum space), i.e., ∫ ∫ ( ) . We will model the scatting process and calculate Eq. (5) below.

The scale solutions of Boltzmann-Vlasov kinetic equation.
We investigate the low-lying collective oscillations with both the mean-field and the dissipative contributions by means of the scaling factor method. We take ansatz 17,69 The particle density takes the form of ρ ρ The dependence on time is contained in the parameters b i and θ i . Following refs 17,69, we substitute this ansatz into Eq. (2), integrate in phase space, and calculate the average moment of R i V i . This leads to ( ) By linearizing the Eqs. (7) and (10) around equilibrium (b i = θ i = 1) which gives the generalized Virial theorem of 1 − ζ i − ξ i = η i with i = (⊥ , z) in the axial symmetry external potential, we find the following dispersion law Here subscripts hd and cl denote the hydrodynamic (τ → 0) and collisionless (τ → ∞) regimes, respectively. And When we study the collective modes in the absence of the interacting term of ε but with the dissipative contribution (ξ i = 0 and ζ i = 1 at T ≥ T c , the collision integral represents the two-body interactions), Eq. T-matrix fluctuation theory above the critical temperatures T c . For a trapped system, we use the local density approximation (LDA) and write the chemical potential as µ µ where μ 0 is chemical potential at the center of trap. μ is determined by N = ∫drρ(r). The total potential energy is To find a expression for ρ(r) and discuss the strong-coupling effects in the BCS-BEC crossover above T c , we include pairing fluctuations within the T-matrix approximation [29][30][31][32][33][34][35][37][38][39][40][41][42][43][44] . In ladder approximation, the single-particle thermal Green's function is where ω n = (2n + 1)π/β is the fermion Matsubara frequency for = , ± , ± , n 0 is the free fermion Green's function. The self-energy correction describes effects of pairing fluctuations, where ν n = 2nπ/β is the boson Matsubara frequency and δ is the infinitesimal positive constant. The particle-particle scattering matrix is given by  Mean-field theory below T c . We choose the mean-field theory to describe the system below T c .
In order to calculate the viscous relaxation rate 1/τ ns and ratios ξ i and ζ i for i = (⊥ , z), we express the fermion-pair and pair-pair coupling constants g ns and g s by combining the mean-field microscopic theory with the two-fluid approach that the system consists of N n normal fluid fermions and N s superfluid fermion pairs at T < T c . On the one hand, we calculate the physical quantities within the mean-field theory, such as the particle number density, chemical potentials, energy gap, excitation energies, and the total energy. On the another hand, the two-fluid approach gives the effective energy H eff = K + V + U, where K, V and U are the kinetic energy, potential energy and interacting energy, respectively. We make a connection between H eff and A in the Leggett's mean-field theory to make sure they describe the same system at the level of E, μ, and ρ(r). Thus we can obtain the expressions for g ns , g s , and ρ s (r) in this approach. Meanwhile, we can obtain the ratios ξ i and ζ i for i = (⊥ , z) in this way. Firstly, the one-channel microscopic theory with mean-field approximation gives the action  In the mean-field approximation, the total particle density distribution in the phase space is can be used to determine the chemical potential μ. Since the free energy is F = A/β and the entropy is S = − k B ∂(TA)/∂T, the energy has the form Then in order to calculate the viscous relaxation time τ ns in the normal fermion scatting with superfluid pairs, we want to get the fermion-pair coupling constant g ns in the order phase. As described above for getting the expression for g ns , we combine the one-channel microscopic theory with the two-liquid approach. We suppose that this system can also be described phenomenally as a mixture of normal fluid formed by N n fermions and superfluid formed by N s pairs composed of two fermions in the whole regimes with the total atomic number conservation: N = N n + 2N s . This is just the two-fluid model. For a trapped system, we have being the interaction density. Since we lack the knowledge of the expressions for g s and g ns in the one-channel microscopic model even at the mean-field level, we derive them in the phenomenological way. The mean superfluid pair-superfluid pair(normal fermion-normal fermion) interaction energy per particle is We take the same approximation as in the mean-field theory of the one-channel microscopic model, and assume the N s pairs have no fluctuations and are all frozen at the ground state with q = 0. So only fermions in the normal fluid contribute to K. The kinetic energy is only from N n fermions and the potential energy is from all N fermions: The total energy is Here the interacting energy is  ∫ =Û dr and the binding energy is  b . From the chemical equilibrium condition, we have μ = μ F = μ B /2. Since Since two viscous relaxation time τ nn and τ ns are within the Boltzmann-Vlasov kinetic theory, our approach is combined with the two-fluid physical picture and the one-channel microscopic theory. This theory is within both the T-matrix fluctuation theory in the normal phase and within the mean-field theory in the order phase.
Remarkably, this mean-field theory in the one-channel microscopic model is the simplest theory to study the BCS-BEC crossover system at T < T c . It catches the main characteristics of the system and includes the pairing and depairing effects. Strictly speaking, the mean-field theory only makes sense at the deep BCS side and BEC limit because it omits the pair fluctuations which play an important role near the unitary limit. However, we only use the Boltzmann-Vlasov kinetic theory to study the collective mode and damping rate in this work. The mean-field theory proves a qualitative correct picture at the BCS side where the phase transition locates and includes the pairing and depairing effects which is our main interest. So our theory makes sense for it combines the phenomenological Boltzmann-Vlasov kinetic theory with the mean-field theory at T < T c to study the collective mode and damping rate including the pairing and depairing effects.

Two viscous relaxation time of a trapped strongly interacting Fermi gas.
We now calculate the viscous relaxation time. Take the trial function 68 δ , the denominator on the right side of Eq. (5) turns to Here we have used the fact in the symmetry of the integral functions that Φ ∝ k k 12 x y Under the Galilean invariance of the collision process, it is convenient to use the center-of-mass wave vector k 0 and the relative wave vector coordinates k r and ′ k r instead of the wave vectors of the incoming and outgoing particles k 1 , k 2 and k 3 , k 4 : For atomic gases close to a Feshbach resonance under consideration in this work, a multichannel effective theory 20,73 for atom-atom scattering must be taken into account. Under the broad band approximation close to a resonance, the particle-particle scattering cross section is At T ≥ T c , only τ nn contributes to τ. This term is still important at T < T c . From 67 , the first numerator on the right side of Eq. (5) turns to Using the symmetry of the collision integral under interchange of incoming and outgoing particle momenta, we may write Φ (Φ + Φ − Φ − Φ ) = Φ + Φ − Φ − Φ There are at least two channels contribute to τ ns . One is elastic with unbreakable pairs, the other is un-elastic with breakable pairs after scattering. The first channel only changes the momentum of fermions and fermion pairs, and it is important to τ ns at T ≪ T c . It will complicates the calculations duo to the pair fluctuations. The second channel changes the momentum and energy of fermions as well as the fermion's particle numbers of the normal fluid or superfluid, and it contributes to τ ns near T c . In this work we only consider the second channel and write the second numerator on the right side of Eq. (5) to Here the incoming particles are a fermion pair with zero momentum and energy  b and a fermion with wave vector k 1 and energy ε 1 , while the outgoing particles are three fermions with wave vectors k i and energies ε i for i = 2, 3, 4. Since the kinetic energies do not conserve due to  ≠ 0 b in the scatting process, we can't easily choose a center-of-mass frame to simplify the calculations of delta function. The conversions of the momentum and energies in the scattering processes are expressed as In the multidimensional momentum integral of Eq. (37), according to the symmetry of the integral function, the factor Φ 1 ( 3 for a given function h in Eq. (37). We then choose the first wave vector elements 1 in spherical polar coordinates (k 1z , θ 1 , φ 1 ) with the radial wave vector fixed along the z-axial k 1z in Cartesian coordinates; the second wave vector elements dk 2 = k 2x dk 2x dφ 2 dk 2z in cylindrical coordinates with plane polar coordinates (k 2x , φ 2 ) and z-axial coordinate k 1z ; and the combination . Consequently, the integral function F(k 1 , k 2 , k 3 ) and the integral limits are all given above and we can carry out the calculation of Γ ns in Eq. (39).

Comparison with experiments.
We have first used the T-matrix fluctuation theory to calculate the critical temperature T c as a function of the interaction strength. With the experimental parameters, we then have calculated the gap Δ , coupling constants g s and g ns and density of bosons ρ s (r) below T c , and later the chemical potential μ, density of fermions ρ(r), ratios ζ ⊥ , ξ ⊥ , ζ z , and ξ z , index γ, and two viscous relaxation time τ nn and τ ns , and so on, around T c between the BCS and unitary limits. Finally, we have solved Eq. (11) with different physical parameters (T, η, λ, N). The following is our numerical results and discussions on the properties of the system. Numerical calculations show in Fig. 1 for the temperature dependent (a) frequency Reω and (b) damping Imω for the quadrupole mode (upper plots) and compressions mode (lower plots) at the unitarity limit, respectively. The curves are given for T ≥ T c = 0.3T F . The red solid lines are the numerical results with the full scattering matrix in Eq. (31). As a comparison, we also show the experimental points and theoretical plots from 10 . Our results are essentially consistent with the experimental values. Figure 2 shows (a) frequency Reω and (b) damping Imω in the units of ω ⊥ versus η at T = 0.1T F for the quadrupole mode. The points are experimental values from 1,14 , while the red full lines represent our computing results. From Fig. 2, we can see that η around η − .  0 8 0 , the frequency exhibits a pronounced jump from the hydrodynamic (η > η 0 ) to the collisionless (η < η 0 ) frequency due to the maximum of 1/τ ns at T ≤ T c 74 . This transition is accompanied by a pronounced maximum of the damping rate. This is a striking transition from hydrodynamic to collisionless behavior and it comes from the minimum of τ ns at T c . The present work provides strong evidence that quasistatic hydrodynamic theory does not apply to collective modes of a strongly interacting fermionic superfluid, when the oscillation frequencies approach the pairing gap. The sharp transition occurs at the maximum of viscous relaxation rate of normal fermion-superfluid pair collision at phase transition point. Above this point the pairing gap is breaking. Meanwhile, the strong damping is due to the fast varying of the frequency of collective modes along the BCS-unitarity crossover.
From the BCS limit to the left of the shaded region in Fig. 2, the system is at the normal phase and behaves as a collisionless Fermi gas at T > T c ; while from the unitarity limit to the right of the shaded region, the system is at the order phase and behaves as superfluid fermion pairs at T < T c . Inside the shaded region in Fig. 2 both above approaches for η i = 0 and η i ≠ 0 are not applicable with η i = 1 − ζ i − ξ i for i = (⊥ , z). In this region the system is neither normal phase nor superfluid phase, which is a critical region and the system will have complex behaviors. Figure 3 shows (a) frequency Reω and (b) damping Imω versus η for the radial compressions mode (upper plots) and axial compression mode (lower plots). In the regime of a strongly interacting Fermi gas, an abrupt change in the collective excitation frequency occurs, we show that it is a signature of a transition from a superfluid to a collisionless phase. The measurements on the radial compression mode show three surprises 8 . The abrupt change of the excitation frequency and the large damping rate are not expected in a normal degenerate Fermi gas, where the collective excitation frequency is expected to vary smoothly from the hydrodynamic regime to the collisionless one. Furthermore, for the damping rate of Scientific RepoRts | 5:15848 | DOi: 10.1038/srep15848 the radial compressions mode in the transition regime, a maximum value is ω ω / . ⊥  Im 0 2. The measured damping rate of ω ω / . ⊥  Im 0 5 is clearly inconsistent with our prediction for the normal hydrodynamic regime. Of course, for the experimental parameters of T = 0.1T F , N = 2 × 10 5 , and λ = 70/1500 from 6 , the transition occurs at η 0 = 0.79 with a smooth value of frequency Reω and a maximum damping rate of Imω = 0.2. Anyway, both the sudden change of the collective frequency and a strong damping may due to a transition from the superfluid phase to the normal phase and we need to study the superfluid phase in more details.
In summary, we have determined the transition and damping of collective modes in a trapped Fermi gas near the unitarity limit, based on the Boltzmann-Vlasov kinetic equation, combined with the many-body T-matrix fluctuation approximation in the normal phase and the many-body mean-field approximation in the order phase, and joined the two-fluid approach to connect the microscopic and phenomenal theories. We have calculated the dependence of temperature and scattering length on the frequency and damping of the collective modes, by using both theoretical and available experimental knowledge of the equation of state and two theoretical viscous relaxation time with the collision  probability of fermion-fermion scatting with and without fermion pairs, including a many-body scattering effect. Our results agree well with the experimental observations, not only qualitatively but also quantitatively, particularly for the sharp transition and strong damping of the collective modes in the BCS side near the phase transition when breaking the pairing gap. This theory provides a link between the microscopic and kinetic descriptions of many-body Feshbach scattering. Notes that we have not considered the superfluid dynamics below T c , and omit the pair fluctuations which have the important effects near the Feshbach resonance. We may propose a new theory valid in the whole regime by adding these aspects further.