Collective oscillation modes of a superfluid Bose-Fermi mixture

In this work, we present a theoretical study for the transverse and monopole modes of a mixture of Bose and Fermi superfluids in the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC) of molecules in harmonic trapping potentials with cylindrical symmetry of experimental interest. To this end, we start from the coupled superfluid hydrodynamic equations for the dynamics of Bose-Fermi superfluid mixtures and use the scaling theory that has been developed for a coupled system. The collective oscillation modes of Bose-Fermi superfluid mixtures are found to crucially depend on the overlap integrals of the spatial derivations of density profiles of the Bose and Fermi superfluids at equilibrium. By applying a perturbative analysis for the equilibrium density profiles, we present the explicit expressions for the overlap density integrals, as well as the frequencies of the collective modes provided that the effective Bose-Fermi coupling is weak. Subsequently, the valid regimes of the analytical approximations are demonstrated by numerical calculations in realistic experimental conditions. In the presence of a repulsive Bose-Fermi interaction, we find that the frequencies of the collective modes of the Bose and Fermi superfluids are all upshifted, and the frequency shifts can characterize the different groundstate phases of the Bose-Fermi superfluid mixtures in the BCS-BEC crossover for different trap geometry. Our results may provide a further understanding for experiments on the collective oscillation modes of a mixture of Bose and Fermi superfluids.

The Bose-Fermi superfluid mixture [1][2][3][4][5][6] , which is different from other coupled systems obtained previously [41][42][43][44] , can provide a unique setting for studying and understanding the properties of interacting quantum systems belonging to different quantum statistics. The fermion-fermion interactions can be widely tuned with a magnetic field Feshbach resonance.
For the strongly-repulsive interaction, it is a mixture of two BEC, in which one made of atoms and the other of molecules. For the weakly-attractive interaction it is a mixture of a Bose superfluid with a BCS-type superfluid. Such strong interactions are difficult to generate with bosonic atoms. However, three body losses and recombination processes are significantly lower with fermions due to the Pauli principle. The lifetime of a mixture of Bose and Fermi superfluids has been demonstrated experimentally to be in the order of a few seconds 1 . Such stability allows us to observe this mixture oscillating back and forth in harmonic traps over numerous periods without visible damping, and study how the Bose-Fermi interaction affects the dipole modes of Bose and Fermi superfluids. The long-lived center-of-mass oscillations have been observed in the Bose-Fermi superfluid mixtures of 7 Li- 6 Li 1,2 , 41 K-6 Li 6 , and 174 Yb-6 Li 5 , and the Bose-Fermi interaction gives rise to a rich behavior.
In the presence of a repulsive Bose-Fermi interaction, the frequencies of the dipole oscillations of the Bose and Fermi superfluids are both downshifted in the weakly confined direction 1,5,6 , and the frequency shifts increase monotonically from the BCS side to the BEC side 46,47 .
In contrast 6 , the frequency in the tight confinement for the Bose superfluid is upshifted, whereas the frequency for the Fermi superfluid is still downshifted. The frequency shifts show a non-monotonic and resonantlike behavior in both directions around the BCS side 6 , which may be originated from the effects of fermionic pairs breaking 48 .
It is naturally to ask how the Bose-Fermi interaction affects collective modes of Bose and Fermi superfluids in the BCS-BEC crossover, which is of great interest recently. However, a theoretical study for the collective oscillation modes of Bose-Fermi superfluid mixtures in a realistic experimental situation is highly nontrivial. In this work, to study the collective oscillation modes in cylindrically symmetric traps, we start from the coupled superfluid hydrodynamic equations describing the dynamics of the Bose-Fermi superfluid mixtures.
The scaling method for coupled systems is then applied, and the eigenvalue equations for the frequencies of the transverse and monopole oscillation modes are obtained, which are found to be crucially sensitive to the overlap integrals of the spatial derivations of the Bose and Fermi densities at groundstate. To present the explicit expressions for these integrals, we use a perturbative method for the coupled Bose and Fermi density profiles and in the overlap region the Fermi density is approximated by a constant value in the center of the trap. The analytical results for the frequencies of collective oscillation modes are calculated in the USTC experimental parameters 6 , and the valid regimes of the analytical approximations are confirmed by the numerical calculations. In the presence of a repulsive Bose-Fermi interaction, we find that the frequencies for the transverse and monopole modes of the Bose and Fermi superfluids are all upshifted, and the frequency shifts for the Fermi superfluid are smaller than the bosons, due to a larger number of the particles. However, for a fixed repulsive Bose-Fermi interaction, the frequency shifts increase from the BCS side to the BEC regime, and the different speeds of the increases of frequency shifts, especially for the Fermi superfluid, can be used to characterize different ground configurations of the mixtures 49-52 in different trap geometry. This is because that the frequency shifts are not only originated from the Bose-Fermi interaction, but also sensitive to the spatial distributions of the equilibrium Bose and Fermi densities. The Bose superfluid is localized in a small region at the trapping center within the Fermi superfluid, which produces a depletion of the Fermi density in the center due to the repulsive Bose-Fermi interaction. As the interaction energy of the Fermi superfluid decreases from the BCS side to the BEC side, such depletion becomes more pronounced and its boundary are steeper. Thus the overlap integrals of the spatial deviations of these two densities increase, which result in a stronger coupling and larger frequency shifts.
In the BEC regime the depletion of the Fermi density in the center is completely, which is accompanied by fastest increases of the frequency shifts. In the recent experiments on the degenerate Bose-Fermi mixture of 41 K-6 Li 44 , a rapid upshift of the breathing frequency of the bosons is observed, attributed by the emergent interface when the mixture undergoes phase separation 53 by increasing the repulsive interspecies interaction. We hope that our theoretical results can provide a reference for further experiments on collective oscillation modes of Bose-Fermi superfluid mixtures in the BCS-BEC crossover. This paper is organized as follows. The coupled hydrodynamic equations are introduced in Sec. IIA, and within the scaling theory the coupled set of differential equations for the

A. Superfluid hydrodynamic model
We consider a mixture of bosonic atoms and two spin components of fermionic atoms which is prepared in superfluid state at a low enough temperature. The dynamic properties of the Bose-Fermi superfluid mixture can be described by coupled hydrodynamic equations for superfluid. For bosons the hydrodynamic equations are given by 19,54 where Eq.(1a) is the equation of continuity for atomic density n b (r, t) and the total number of bosons is normalized by N b = n b (r, t)dr, and Eq.(1b) for the velocity field v b (r, t) establishes the irrotational and inviscid nature of the superfluid motion. The trapping potential acting on bosons is cylindrically symmetric with the form V b ext (r) = m b [ω 2 b⊥ (x 2 + y 2 ) + ω 2 bz z 2 ]/2, where ω b⊥ and ω bz denotes the trapping frequencies and m b is the mass of a bosonic atom. The boson-boson interaction strength is related to the s-wave scattering The superfluid hydrodynamic equations for fermions in terms of the density n f (r, t) and velocity field v f (r, t) are given by 20,55 , respectively where the trapping potential acting on fermions is V f /2 with m f the mass of a fermionic atom, and the total number of fermions in superfluid state is normalized by N f = n f (r, t)dr. In contrast to a simple expression of the interaction strength in the bosonic part, the two-spin fermionic interaction is characterized by the equation of state µ(n f ). In order to obtain analytical results in various superfluid regimes in a unified way, we take a polytropic approximation 56,57 , i.e.
where the reference chemical potential µ 0 is proportional to the Fermi energy ǫ f = f ⊥ ω f z ) 1/3 defined in a cylindrically symmetric trap and reference atomic number density is given by the density of noninteracting Fermi gas at the trapping center n 0 = (2m f ǫ f ) 3/2 /(3π 2 3 ). In order to be close to experimental observations, σ(η) is based on the explicit expressions of fitting functions from ENS experimental data 58 . The effective polytropic index γ and reference chemical potential µ 0 are determined by σ(η) as a function of the dimensionless interaction η = 1/k f a f , which have been plotted in Ref. 46 . To describe the coupling between these two types of superfluid, we have introduced the bosonfermion interaction g bf = 2π 2 a bf /m bf at the mean-field level with boson-fermion scattering length a bf and reduced mass m bf = m b m f /(m b + m f ). It is worth noting that in the BEC limit where 1/k f a f ≫ 1 and a f is comparable to a bf , the boson-fermion interaction should be replaced by the boson-dimer interaction 8,9 .
For the coupled hydrodynamic equations (1) and (2), they actually work in the Thomas-Fermi (TF) regime, in which they are analytically simpler to handle in the absence of the quantum pressure terms. The TF approximation is valid, provided that the interactomic interaction energy is large enough to make the kinetic energy pressure negligible, i.e. in the large particle limit and collective excitations are of sufficiently long wavelength. By including the proper quantum pressure terms 59,60 , the coupled hydrodynamic equations are equivalent to the coupled order-parameter equations 47,61 . In addition, the superfluid hydrodynamic equations only describe the dynamics of superfluid components, ignoring single particle excitation, the normal components and temperature effects.

B. Scaling theory for a coupled system
To account for collective oscillation modes in a coupled system, we resort to a scaling theory 62,63 . The basic idea behind the scaling method is to take appropriate scaling ansatz and simplify the time dependent problems into solving differential equations for the scaling parameters. It is specially suited for three dimensional hydrodynamic equations, whereby numerical simulations are very expensive. Moreover, it is enable to derive analytical ap- where n 0 b and n 0 f are equilibrium density distributions for the Bose and Fermi superfluids, respectively. The scaling anzatz for the velocity fields can be obtained by inserting the scaling anzatz Eqs.(4) into the equation of continuity Eqs.(1a) and (2a) Substituting the scaling ansatz for the densities (4) and velocities (5) into the Eqs. (1b) and (2b), we arrive at the differential equations for the scaling parameters where we have introduced the time-dependent coordinates and . It is seen that we transfer the time-dependent problems for the coupled hydrodynamic equations into solving the ordinary differential equations for b i (t) and a i (t), with i = x, y, z respectively, and the disturbations of density profiles are expressed by these scaling parameters. In the equilibrium states Eqs.(6) reduce to In order to obtain scaling solutions for a coupled system, i.e. in the presence of the bosonfermion interation g bf , a useful strategy is developed that is assuming the scaling form of the solution a priori and fulfilling it on an average by integrating over the spatial coordinates 35,65 .
Combining the differential equations (6) with the equilibrium states (7) and carrying out the spatial integration, we obtain the following expressions The dimensionless parameters proportional to g bf are given by where we have replaced the variables R b and R f by r for simplicity in Eqs. (10), because the each integration is relevant to either R b or R f .

C. Collective oscillation modes
In order to clearly understand the collective oscillation modes obtained from the scaling method, let us first discuss the results without the boson-fermion interaction g bf . In this case, all dimensionless parameters (10) disappear and Eqs.(9a) and (9b) are decoupled, describing the Bose and Fermi superfluids alone, respectively. In a cylindrical symmetry, there are two ways of linearization to obtain the eigenvalue equations. One deviation for the scaling parameters in the transverse directions i = x, y is given by  By further examining the signs of the eigenvectors, one can find that high-lying mode ω + features the radii in the transverse plane oscillating in phase with each other, while low-lying mode ω − is an oscillation out of phase with each other, which are illustrated in Fig. 1(a).
The high-lying mode is also called transverse (radial) breathing (compression) mode due to the oscillation in phase, while low-lying mode is referred to transverse (radial) quadrupole mode. Other linearization is given in the forms of b i (t) = 1 + δb i (t) or a i (t) = 1 + δa i (t) with i =⊥, z, which is called monopole mode, and their two eigenfrequencies are, respectively, given by 23,25,27 with the trap anisotropy λ b = ω bz /ω b⊥ and λ f = ω f z /ω f ⊥ . As shown in Fig. 1(b), the signs of the eigenvectors of the high-lying monopole mode ω + suggest an in-phase oscillation for all radii in the transverse and axial directions, while low-lying mode ω − corresponds to two radii in transversal direction oscillating in phase with each other, but out of phase with the radius in the axial direction. Differently from the transverse modes, the monopole modes are sensitive to the anisotropy of the trap. In a highly elongated trap (λ b,f ≪ 1), the frequency of the high-lying monopole mode reduces to ω bm+ = 2ω b⊥ (ω f m+ = 2(γ + 1)ω f ⊥ ), which coincides with the transverse mode shown in Eq.(11a), and the low-lying monopole mode is ω bm− = 5/2ω bz (ω f m− = (3γ + 2)/(γ + 1)ω f z ). This is the reason why such high-lying and low-lying monopole modes are also called transverse (radial) breathing mode and axial mode 25,27,40 , respectively. In the oblate limit (λ b,f ≫ 1), the frequency of the high-lying mode reduces to ω b+ = √ 3ω bz (ω f + = √ γ + 2ω f z ), and the low-lying monopole the frequency of the high-lying mode is In the presence of the Bose-Fermi interaction g bf , the collective modes of the Bose and Fermi superfluids are coupled each other and their frequencies are varied. For the transverse mode, the linearation of Eqs.(9) around the equilibrium states result in the eigenvalue function d 2 P/dt 2 = M T P, where the vector notation is P T ≡ (δb x , δb y , δa x , δa y ) and matrix M T is written in a cylindrical coordinate (⊥, z) by parameters B ρρ , F ρρ , and F ρ , which are integrals in terms of the density profiles of the Bose and Fermi superfluids at equilibrium, are found to be responsible for the coupling of the Bose and Fermi components.
For the monopole mode, the eigenvector corresponds to P T = (δb ⊥ , δb z , δa ⊥ , δa z ) and the matrix M M is defined by with to g bf in the cylindrical coordinates (α, β = ρ(⊥), z) take the forms with the mean square radii in the α =⊥, z direction given by R 2 α b ≡ (1/N b ) dr r 2 α n 0 b for the Bose superfluid and R 2 α f ≡ (1/N f ) dr r 2 α n 0 f for the Fermi superfluid.

D. Analytical approximations for the integral terms
In the previous subsection, one can find that the dimensionless parameters are the spatial overlap integrals in terms of the equilibrium density profiles of the Bose and Fermi superfluids, and the frequencies of collective oscillation modes of Bose-Fermi superfluid mixtures crucially depend on these integrals. Eqs. (1b) and (2b) at groundstates give the density profiles of the Bose and Fermi superfluids coupled each other, which are written in cylindrical coordinate as where the bulk chemical potentials µ b and µ f are determined by the total numbers of bosons and fermions, respectively. Without the boson-fermion interaction g bf , the explicit expressions for the density profile of the Bose superfluid are 19,54 and the Fermi density profiles along the BCS-BEC crossover 57 are .
The density profiles n 00 b and n 00 f in the absence of g bf can be regarded as the zero-order approximation for the density profiles (16a) and (16b). By a perturbative expansion, the density profiles n 01 b and n 01 f at the first-order approximation can be naturally obtained by replacing n 0 f by the zero-order results n 00 f in Eq.(16a) and n 0 b by n 00 b in Eq.(16b), correspondingly. Differently from the previous works by the numerical methods 35,65 , we apply analytical approximations for the integrals 46 that holds in the recent experimental situations, then give explicit expressions for the dimensionless parameters.
As a example of the integral I ρρ = ρdρdz ∂n 0 b ∂ρ ρ 2 ∂n 0 f ∂ρ , we derive it as following where ( immersed in Fermi superfluids. Second, the boson-fermion interaction g bf is small. One can find that these two characteristics lead to a weakly-coupled Bose and Fermi density profiles from Eqs.(16a) and (16b). Although µ b (N b ) is much smaller than µ f (N f ), n f is much smaller than n b due to the stronger interaction and the last terms of (16a) and (16b) are relatively smaller compared with other terms. By using a perturbative method, in the first step of the integration (19a), we have substituted the density profiles n 0 b and n 0 f by the first-order approximations n 01 b and n 01 f , respectively. Since the Bose superfluid is weakly interacting and the atomic number is smaller than the fermionic counterpart, the spatial distribution of the Bose superfluid only overlaps with the Fermi superfluid in a small central region.
In the second step (19b), we approximate the Fermi density by the central value and the region of integration is the volume V B of the Bose superfluid. Based on the above analysis for the parameters and these approximations, the explicit expression for the integral I ρρ is presented in (19c). The same analysis can be also applied at the zero-order approximation, i.e. the density profiles in the integrals are replaced by n 00 b and n 00 f , and the integral is given by ). Compared with the first-order approximation (19c), the result of the zero-order approximation is lack of the last term which is relevant to the ratio of boson-fermion interaction g bf to boson interaction g b . In a recent work 46 , we use the scaling theory to study the dipole mode of the Bose-Fermi superfluid mixture, and the result for the frequency shift at the zero-order approximation is equivalent to the mean-  (13) and (14), one can find that if the elements for the couplings of the amplitudes of the bosonic and fermionic excitations in the matrix are small which implies that the effective coupling is weak, the explicit expressions for the frequencies of the collective oscillation modes can be presented 35 . For the transverse mode, the high-lying ω + t± and low-lying ω − t± frequencies are given by, respectively For the monopole mode, the expressions for the high-lying ω + m± and low-lying ω − m± frequen-cies are more complicated and given by, respectively where the ratios of radial amplitude to axial amplitude are given by with ω bm± and ω f m± being the frequency of the monopole modes of the Bose and Fermi superfluid alone in Eqs. (12), and the index referring to high-lying (+) and low-lying (-) modes, respectively. The positive (+) and negative (-) roots of the high-lying ω + t±,m± (low-lying ω − t±,m± ) frequencies correspond to the frequencies of the high-lying (low-lying) transverse and monopole modes of the Bose and Fermi superfluids, respectively.

III. RESULTS AND DISCUSSIONS
In this section, we apply the theoretical results of the previous section to a realistic experimental situation as an example to discuss how the Bose-Fermi interaction affects the and bosons (lower panels) in the BCS-BEC crossover 46 . One can find that for a fixed Bose-Fermi interaction, as the interaction energy of the Fermi superfluid decreases from the BCS side ( Fig. 3(a)) to the BEC regime ( Fig. 3(e)), the depletion of the Fermi superfluid density in the center caused by the bosons are more pronounced, until it is completely in the BEC regime (see Fig 3(d) and 3(e)). Thus as we pass from the BCS side to the BEC regime, the analytical approximation of substituting the Fermi density distribution in the integral region by the central density becomes more unreliable. It should be pointed that for the cases of zero Fermi density in the central region, the spatial overlaps of these two densities still remain.
In addition, for the mean square radii of the Bose and Fermi superfluids shown in Fig. 2(g) and 2(h), compared with the analytical results (dashed lines) actually corresponding to the noninteracting Bose and Fermi superfluids, numerical results (discrete data) show that the repulsive Bose-Fermi interaction has a very slight affect on the Fermi superfluids due to a larger number of particles, while the radii of the Bose superfluids decreases obviously, suggesting that the bosonic cloud is compressed by the outer shell of fermions.
In Fig. 4  BEC crossover are smaller than the Bose counterpart. Without the Bose-Fermi interaction, the frequency of the high-lying mode of the Fermi superfluid shows a non-monotonical variation as a function of the dimensionless parameter 1/(k f a f ) in Fig. 4(a) and low-lying mode with a fixed frequency in Fig. 4 Fig. 6(a) and Fig. 7(a), respectively, one can find the frequency shifts are almost as the same as the transverse modes (see Fig. 4(a) and Fig. 5(a)), implying that in the presence of the Bose-Fermi interaction such symmetry of the system is still preserved.
For the low-lying monopole modes of the Bose-Fermi superfluid mixtures in the cigar-shaped and disk-shaped traps, we plot the analytical results for the frequencies of the Fermi superfluids separately in the insets of Fig. 6(b1) and Fig. 7(b1), and the frequency squares for the Bose counterpart in the insets of Fig. 6(b3) and Fig. 7(b3), respectively. The reason for the failure of the analytical expressions for the low-lying modes in the highly-anisotropic traps is because by examining the explicit expressions (21) and the eigenvalue matrix (14), one can find the elements A ± 12,21 characterizing the coupling of the Bose and Fermi superfluids, are not only determined by the dimensionless parameters as the transverse mode, but also rely on the ratio M ± b,f of the radial excitation amplitude to axial excitation amplitude. We find that for the cases of the low-lying modes in the highly-anisotropic traps, M − b,f leads to the coupling terms A ± 12,21 comparable to the non-coupling terms A ± 11,22 , however, the analytical expressions are only justified for a weak coupling, i.e. the couping part is smaller than the non-coupling one.

IV. CONCLUSION
Since the realization of a mixture of Bose and Fermi superfluids in the BCS-BEC crossover in ultracold atoms the investigation of the properties of collective modes will be of particular interest, which serve as a powerful tool to understand the physics of many-body systems.
In this work, we study the transverse and monopole modes of the Bose-Fermi superfluid mixtures in the BCS-BEC crossover by using the coupled hydrodynamic superfluid equations and the scaling theory for the coupled system. The analytical analysis for the frequencies of collective oscillation modes are presented, and the valid regimes of the approximations are demonstrated by the numerical calculations in currently experimentally feasible setups.
We find that for a repulsive Bose-Fermi interaction, the frequencies of the transverse and monopole modes of the Bose and Fermi superfluids are all upshifted, and the frequency shifts of the Fermi superfluid are smaller than the bosonic counterpart due to a larger number of particles. However, for a fixed repulsive Bose-Fermi interaction, as we pass from the BCS side to the BEC regime where the interaction energy of the Fermi superfluid decreases, the frequency shifts of the collective oscillation modes increase, especially for the Fermi superfluid. The different change speeds of the frequency shifts in different superfluid regimes can characterize the different groundstates of the Bose-Fermi superfluid mixtures.
The theoretical results obtained here from the scaling method for a coupled system may provide a useful reference for future experiments on the collective excitations of Bose-Fermi superfluid mixture in the BCS-BEC crossover, on the other hand, the availability of Bose-Fermi superfluid mixtures with a long-lived lifetime can provide a unique probability for testing the scaling theory of a coupled system, which has been confirmed experimentally for a single superfluid. However, the topic for collective oscillation modes of Bose-Fermi superfluid mixtures in the BCS-BEC crossover is far from simplicity as we study. Under the TF approximation, we ignore the density gradient corrections, which prevent the densities from changing abruptly. In the vicinity of sharp edges, the density will be smoothed out by including the corrections. So the overlap integrals of the spatial derivations of the Bose and Fermi density profiles have been overestimated, as well as the rapid increase of the frequency shifts, but the TF approximation is still expected to provide qualitatively correct results. Furthermore, we ignore all decay processes, resulting from normal components, single particle excitations, and the nonlinear coupling of high amplitude oscillations, which should be further considered theoretically and experimentally.