Collective dynamics of Fermi-Bose mixtures with an oscillating scattering length

TL;DR

This paper investigates the collective oscillations and parametric resonances in superfluid Fermi-Bose mixtures with oscillating scattering lengths, comparing PDE and ODE models, and applies findings to recent experiments with lithium isotopes.

Contribution

It introduces a detailed analysis of collective modes and parametric resonances in Fermi-Bose mixtures with time-varying interactions, bridging PDE and ODE modeling approaches.

Findings

Identification of parametric resonances under periodic scattering length variation

Comparison of PDE and ODE models for mixture dynamics

Application to experiments with $^{6}$Li and $^{7}$Li atoms

Abstract

Collective oscillations of superfluid mixtures of ultra cold fermionic and bosonic atoms are investigated while varying the fermion-boson scattering length. We study the dynamics with respect to excited center of mass modes and breathing modes in the mixture. Parametric resonances are also analyzed when the scattering length varies periodically in time, by comparing partial differential equation (PDE) models and ordinary differential equation (ODE) models for the dynamics. An application to the recent experiment with fermionic $^{6}$Li and bosonic $^{7}$Li atoms, which approximately have the same masses, is discussed.

Equations

$$i{\psi }_{b,t}$$

$$i{\psi }_{f,t}$$

$$l_{\perp }=\sqrt{\frac{\hslash }{m_b{\omega }_{\perp }}},{\psi }_{b,f}=\sqrt{l_{\perp }}{\Psi }_{b,f},t=\tau {\omega }_{\perp },$$

$$x=\frac{X}{l_{\perp }},g_n=\frac{2m_b{l}_{\perp }}{{\hslash }^2}{G}_n$$

$$g_{fb}(t)=g_{fb}^{(0)}[1+c_{fb}\sin ({\Omega }_{fb}t)].$$

$$\bar{L}={\int }_{-\infty }^{\infty }L(x,t)dx,$$

$$L(x,t)$$

$${\psi }_n=A_n\exp \left(-\frac{(x-{\zeta }_n{)}^2}{2a_n^2}\right)e^{i(b_n(x-{\zeta }_n{)}^2+k_n(x-{\zeta }_n)+{\varphi }_n)},$$

$$\bar{L}(t)$$

$$N_n={\int }_{-\infty }^{\infty }|{\psi }_n{|}^{2}dx=\sqrt{\pi }{A}_n^2{a}_n,n=b,f.$$

$$\frac{\partial \bar{L}}{\partial {\xi }_i}=\frac{d}{dt}\frac{\partial \bar{L}}{\partial {\xi }_{i,t}}.$$

$${\zeta }_{b,tt}$$

$${\zeta }_{f,tt}$$

$${\zeta }_{b,tt}$$

$${\zeta }_{f,tt}$$

$${\Omega }_{b,f}^2=4{\alpha }_{b,f}-K_{b,f},K_{b,f}=\frac{4g_{fb}{N}_{f,b}}{\sqrt{\pi }(a_b^2+a_f^2{)}^{3/2}}.$$

$$X(t)=\frac{{\int }_{-\infty }^{\infty }x\left(|{\psi }_b{|}^2+|{\psi }_f{|}^2\right)dx}{N_b+N_f},$$

$$X(t)=\frac{N_b{\zeta }_b(t)+N_f{\zeta }_f(t)}{N_b+N_f}.$$

$$X(t)=\frac{N_b⟨{\zeta }_b⟩+N_f⟨{\zeta }_f⟩}{N_b+N_f}(1-\cos (2\sqrt{\alpha }t)).$$

$$\Delta {\zeta }_{tt}+4{\alpha }_R\Delta \zeta =0,$$

$${\alpha }_R=\alpha -\frac{g_{fb}N}{\sqrt{\pi }(a_b^2+a_f^2{)}^{3/2}}.$$

$$\tilde{\omega }=2\sqrt{\alpha -\frac{g_{fb}N}{\sqrt{\pi }(a_b^2+a_f^2{)}^{3/2}}}.$$

$$\Delta {\zeta }_{tt}+{\omega }_{\zeta }^2\left[1-h\sin \left({\Omega }_{fb}t\right)\right]\Delta \zeta =0,$$

$${\omega }_{\zeta }=2\sqrt{{\alpha }_R},h=\frac{g_{fb}^{(0)}N{c}_{fb}}{{\alpha }_R\sqrt{\pi }(a_b^2+a_f^2{)}^{3/2}},$$

$${\Omega }_{fb}=2{\omega }_{\zeta },$$

$${\Omega }_{1,2}^2=\frac{{\Omega }_b^2+{\Omega }_f^2}{2}\mp \sqrt{\frac{({\Omega }_b^2-{\Omega }_f^2{)}^2}{4}+K_b{K}_f}.$$

$${\Omega }_{1,2}^2=2({\alpha }_b+{\alpha }_f)-\frac{K_b+K_f}{2}$$

$$\mp \sqrt{4{\left({\alpha }_b-{\alpha }_f\right)}^2+2\left({\alpha }_b-{\alpha }_f\right)\left(K_f-K_b\right)+\frac{{\left(K_b+K_f\right)}^2}{4}}.$$

$${\Omega }_{1,2}^2\simeq 2\left({\alpha }_b+{\alpha }_f\right)-\frac{K_b+K_f}{2}\mp \frac{K_b+K_f}{2}.$$

$${\Omega }_2=2\sqrt{\alpha -\frac{K_b+K_f}{4}}=2\sqrt{\alpha -\frac{g_{fb}N}{\sqrt{\pi }(a_b^2+a_f^2{)}^{3/2}}}=\tilde{\omega }.$$

$${\Omega }_1^2\simeq 4{\alpha }_b-K_b,$$

$${\Omega }_2^2\simeq 4{\alpha }_f-K_f,$$

$${\zeta }_{b,tt}+4{\alpha }_b{\zeta }_b$$

Full text

I Introduction

Fermi-Bose mixtures have attracted much attention lately, starting with the work Schreck2001 , where the first observation of the mixture of a Bose-Einstein condensate in a Fermi sea has been performed. Among the different types of Fermi-Bose mixtures, particular interest is devoted to a mixture where both components are superfluids. One important example is the 6Li-7Li mixture Abeelen ; Kempen . Recently, experimental production of a mixture of fermionic and bosonic atoms, where both components are superfluids, has been reported for 6Li-7Li mixture Ferrier ; Delehaye . Other interesting systems are the 40K-41K mixture with a tunable interaction between species Wang ; Falke ; CWu and the mixture of 133Cs and 6Li with broad interspecies Feshbach resonances Repp ; Tung . Recently, the two-species superfluid 6Li-41K and 6Li-176Yb has been experimentally realized Yao ; Roy . Among the new effects predicted for mixtures of Bose-Fermi superfluids we mention the existence of vortices in a rotating quasi-two-dimensional Fermi-Bose mixtures WenPRA2014 , Faraday waves Abdullaev2013 , an prediction of the existence of super-counter-fluid phase Kuklov , the existence of dark-bright solitons MitraJLTP2018 , and the multiple periodic domain formation TylutkiNJP2016 . Collective oscillations of the Fermi-Bose mixtures for different settings has earlier been analysed in Ferliano ; Banerjee ; Nascimbene ; Wen . Dipole modes in the BCS-BEC crossover regime has been analytically considered in Wen . In the recent work Wu the dipole oscillations of the Fermi-Bose-mixture with a large mass imbalance have been experimentally investigated. It was observed that when the inter-species interaction strength is varied, the dipole oscillations frequency shifts and the 41K and 6Li components display a resonance like behavior in the upward and downward (radial and axial) directions.

The aim of the present paper is to investigate the collective dynamics of the Fermi-Bose mixture when the scattering length is varied periodically in time. We start in section II with a description of the model. To analyze the collective oscillations we then introduce a variational approach. In section III we investigate the dynamics for the center of mass and for breathing modes. There are different possible resonant regimes depending on what kind of scattering length that are varied. In particular the variation of the fermion-boson scattering length $a_{s,fb}$ will lead to parametric resonance in the oscillations of the relative distance between the centers of mass of both superfluids. Also we study the resonance dynamics of breathing modes of both superfluids. The numerical simulations are discussed in section IV. Finally we conclude in section V that the resonances should be clear experimental signals to search for. The details of the variational approach are described in the Appendix.

II The model

The time-dependent model is described by the system of partial differential equations (PDEs) Adhikari ; Abdullaev2013

[TABLE]

where the parameters $g_{b},g_{fb}$ are proportional to atomic scattering lengths $a_{s,b},a_{s,fb}$, respectively, while $\alpha_{b},\alpha_{f}$ determines the harmonic trapping. Furthermore, $g_{b}=2\hbar a_{s,b}\omega_{\perp}$ is called the one-dimensional coefficient of mean-field nonlinearity for bosons, where $a_{s,b}$ is the scattering length and $\omega_{\perp}$ is the perpendicular frequency of the trap. Similarly, $g_{fb}$ is called the interspecies interaction coefficient Adhikari . We are interested in weak Bose-Bose interactions (we consider small positive $g_{b}$) and attractive Fermi-Fermi interactions such that the superfluid Fermi-Bose system is described by the nonlinear Schrödinger-like equation (LABEL:sys1) Heiselberg ; Bulgac ; Manini ; Astr ; Adhikari2 . In the BCS weak attractive coupling limit the fermionic subsystem coefficient is $\kappa=1/4$, while in the molecular unitarity limit it is $\kappa=1/16$ Adhikari . Finally, for the bosonic Tonks-Girardeau limit 1DTonks with the components $\psi_{b}$ ($\psi_{f}$) being a weakly (strongly) repulsive bosonic species we have $\kappa=1$. The system (LABEL:sys1) is written in dimensionless form using the variables

[TABLE]

where $G_{n}$ ($n=b,fb$) is the coefficient of the mean-field nonlinearity. We also implicitly assume that $m_{b}=m_{f}$ in Eq. (LABEL:sys1). As mentioned, such a condition is approximately realized in the 6Li-7Li and 40K-41K mixtures.

Below we will also consider the situation of a sinusoidal time dependence of the scattering length $a_{s,fb}$. This can be achieved using the Feshbach resonance technics Kagan ; Kagan2 ; Inouye ; Collective . According to these schemes, we can manipulate the scattering lengths by varying an external magnetic field in time near the resonant value Ferliano06 . For the Fermi-Bose mean-field nonlinearity in the model (LABEL:sys1) it means that the interaction parameter will be varied in time as

[TABLE]

To analyse collective oscillations of the mixture with the scattering lengths varying in time, we will employ the variational approach (VA). According to this method we first calculate the averaged Lagrangian

[TABLE]

where $L(x,t)$ is the Lagrangian for the system (LABEL:sys1)

[TABLE]

For the Bose and Fermi wave functions we employ the Gaussian ansatz

[TABLE]

for $n=b,f$ and where the parameters $A_{n}$, $a_{n}$, $b_{n}$, $k_{n}$, $\zeta_{n}$, $\phi_{n}$ are all real functions of time $t$. Substituting the ansatz (II) into Eq. (4) and performing the average in Eq. (3), we obtain the following averaged Lagrangian

[TABLE]

The numbers of atoms corresponding to the Gaussian wavefunctions are equal to

[TABLE]

Variation of the Lagrangian in (5) with respect to $\phi_{b}$ and $\phi_{f}$ shows that $N_{b}$ and $N_{f}$ are constants. The dynamical equations for the parameters, or collective coordinates, $\xi_{i}\in$ $\{\zeta_{n}$, $k_{n}$, $b_{n}$, $A_{n}$, $a_{n}\}$ with $n=b,f$ can be derived from the Euler-Lagrange equations for the averaged Lagrangian (5), leading to the system of ordinary differential equations (ODEs)

[TABLE]

Variation with respect to $\phi_{n}$ has already been taken care off leading to a constant $N_{n}$.

III Collective dynamics

III.1 Centers of mass oscillations modes

For the center of mass coordinates ($\zeta_{b},\>\zeta_{f}$), we obtain from (7) the coupled differential equations (see the Appendix for details)

[TABLE]

We note that for small $|\zeta_{b}-\zeta_{f}|\ll 1$ the above equations have the same form as the system (4)-(5) considered in Ferrier :

[TABLE]

where

[TABLE]

III.1.1 Total center of mass

The total center of mass (COM) is given by (assuming the same atomic mass for the two components)

[TABLE]

which, within the Gaussian ansatz (II), takes the form

[TABLE]

Forming the second-order derivative from Eqs. (III.1) while keeping the widths $a_{b}$ and $a_{f}$ constant, we obtain $X_{tt}=-4(\alpha_{b}N_{b}\zeta_{b}(t)+\alpha_{f}N_{f}\zeta_{f}(t))/(N_{b}+N_{f})$. An interesting limiting case is: $\alpha_{b}=\alpha_{f}=\alpha$, for which $X_{tt}=-4\alpha X$. We can choose to center the trap $n=b,f$ at position $d_{n}$ that is to replace $\alpha_{n}x^{2}\psi_{n}$ by $\alpha_{n}(x-d_{n})^{2}\psi_{n}$ in Eq. (LABEL:sys1) for $t\geq 0$. In this case the corresponding wave functions in Eq. (II) are centered around $d_{n}$, i.e. we have the time averages $\langle\zeta_{n}\rangle=d_{n}$. So for $\alpha_{b}=\alpha_{f}=\alpha$ Eq. (12) have the form

[TABLE]

Hence, we note that unlike the COM of the respective components, the total COM is not effected by the coupling parameter $g_{fb}$, see the lower panel of Fig. 1 (b). Note that we use the initial conditions $\zeta_{n}(0)=0$ in Fig. 1, but as the centers of the traps are displaced from zero this do not imply that the averages $\langle\zeta_{n}\rangle$, $n=b,f$, vanish.

III.1.2 Relative center of mass coordinate

We define $\Delta\zeta=\zeta_{b}-\zeta_{f}$ and then, according to (III.1) with $a_{b}$ and $a_{f}$ constant, we have for the case $|\Delta\zeta|\ll 1$, with $\alpha_{b}=\alpha_{f}=\alpha$ and $N=N_{b}+N_{f}$:

[TABLE]

where

[TABLE]

So for $g_{fb}=0$ we have the frequency $\omega_{0}=\omega_{trap}=2\sqrt{\alpha}$ (equals unity in the parameters in use here). For $g_{fb}\neq 0$ we have the frequency

[TABLE]

Due to the definitions (12) and $\Delta\zeta=\zeta_{b}-\zeta_{f}$, we see from Eq. (16), that the Bose-coordinate $\zeta_{b}$ and the Fermi-coordinate $\zeta_{f}$ both have the two common frequences $\omega_{0,b,f}$ and $\tilde{\omega}_{b,f}$, see Figs. 1 and 2.

III.1.3 Parametric resonance for the center of mass mode

Note that the oscillations of the relative distance between centers of the fermionic and bosonic clouds depends on the fermion-boson scattering length, while the motion of the total center of mass of the system do not depend on $g_{fb}$ (see above). Experiments can use this dependence to study the coupling between the dynamics of bosonic and fermionic superfluids. In particular if we consider the periodic modulations in time for the fermion-boson scattering length $g_{fb}$ according to Eq. (2), we obtain for $\Delta\zeta$ the differential equation

[TABLE]

where

[TABLE]

with $\alpha_{R}$ taken from (15). This is the well known Mathieu equation. Such that when

[TABLE]

we have a first parametric resonance in the oscillations of the relative distance between the clouds (upper panel of Fig. 3). The region of instability is the interval $-|h|\omega_{\zeta}/2<\epsilon<|h|\omega_{\zeta}/2$, with $h$ from (18), around the frequency $2\omega_{\zeta}$, that is $\Omega_{fb}=2\omega_{\zeta}+\epsilon$. For the full PDE model (LABEL:sys1) parametric resonances can be found for driving frequences close to the prediction (19), see the lower panel of Fig. 3.

III.1.4 Different trap frequencies

The frequencies of the normal modes to (9) are

[TABLE]

An explicit expression is given by

[TABLE]

[TABLE]

In Fig. 4 we compare the prediction for the frequencies $\Omega_{1,2}$ from Eq. (21) with simulations of the full PDE model (LABEL:sys1). The figure shows that the relative deviation between full numerical solutions and the prediction from Eq. (21) is less than about $5.4\%$. This is a good agreement supporting the validity of our collective coordinate approach in section II.

From Eq. (21) we also have the two important limiting cases from before for the following parameters:

Small $|\alpha_{b}-\alpha_{f}|\ll K_{b,f}$

Then we have from (21) the frequencies

[TABLE]

Hence, we have the two limiting frequences ($\alpha=\alpha_{b}=\alpha_{f}$): $\Omega_{1}=2\sqrt{\alpha}$; and ($N=N_{b}+N_{f}$)

[TABLE]

We note that the above limit agree with the results of Sec. III.1.2, see e.g. Eq. (16) and Fig. 2;

Small $K_{b,f}\ll|\alpha_{b}-\alpha_{f}|$

For which we get from (21) the frequencies

[TABLE]

i.e., the frequences $\Omega_{b,f}$ from Eq. (10), see Fig. 4.

III.1.5 Parametric resonance for the COM with different trap frequencies

In the case of a time-dependent fermion-boson scattering length, see Eq. (2), the system (9) represents two coupled Mathieu equations

[TABLE]

which in the case of $\alpha_{b}=\alpha_{f}$ leads to Eq. (14) in Sec. III.1.2.

As is showed in the work Hansen , parametric resonances in oscillations are possible at the following driving conditions in Eq. (25):

[TABLE]

See Fig. 5 for two numerical examples.

Let us finally estimate possible experimental parameters. The parameters for the system 6Li-7Li at the conditions of the experiment Ferrier are: the fermion-boson scattering length is $a_{fb}=40a_{0}$; the trap frequencies are $\omega_{trap,b}=2\pi\cdot 15.2$ Hz and $\omega_{trap,f}=2\pi\cdot 16.8$ Hz; with the transverse frequency $\omega_{\perp}=550$ Hz, such that the transverse length is $l_{\perp}\simeq 4$ $\mu$m ; the number of fermionic 6Li-atoms is $N_{f}=3.5\cdot 10^{5}$ and the number of bosonic 7Li-atoms is $N_{b}=4\cdot 10^{4}$; the widths of the bosonic and fermionic clouds are $a_{b}\simeq 2.5l_{\perp}$ and $a_{f}\simeq 10l_{\perp}$; the coupling parameter in (9), for the trapping parameters $\alpha_{b}=0.001$ and $\alpha_{f}=0.0011$ , are $K_{b}\simeq 0.002$ and $K_{f}\simeq 10K_{b}$.

Then we can conclude that the frequency of the bosonic center of mass is reduced due to the fermi-bose interaction from $15.3$ Hz, to $14.4$ Hz, while the fermionic is not practically changed. This agree with the experimental observation reported in Ferrier .

The atomic scattering length dependence on the external magnetic field $B$ in the region of the Feshbach resonance is:

[TABLE]

where $B_{0}$ is the Feshbach resonance position, $\Delta$ is the width of the resonance, and $a_{bg}$ is the background atomic scattering length. For the 6Li-7Li mixture with $B_{0}=360$ G Abeelen ; Kempen the atomic scattering length $a_{fb}$ can be varied by applying the external magnetic field varying periodically in the time near the Feshbach resonance at $B_{0}$, $B(t)=B_{av}+B_{1}\sin(\Omega t)$, where $B_{av}$ is the averaged value of the field.

Then for the frequencies of the parametric resonances in dimension variables we find: $2\Omega_{1}=2\pi\cdot 31$ Hz and $2\Omega_{2}=2\pi\cdot 32.1$ Hz.

III.2 Breathing modes

For the width of the bosonic subsystem we have from (7) the equation (see the Appendix for details)

[TABLE]

where

[TABLE]

For the width of the fermionic subsystem we obtain the equation

[TABLE]

The equilibrium values for the widths of bosonic and fermionic clouds, from here on denoted $a_{b0}$ and $a_{f0}$, are given by the solutions to the equations

[TABLE]

We can find the frequencies of breathing modes by a linearization of Eqs. (28) and (29) near the fixed point $(a_{b0},\>a_{f0})$ by the substitutions $a_{b}=a_{b0}+\delta a_{b}$ and $a_{f}=a_{f0}+\delta a_{f}$. From the equations for $\delta_{b}$ and $\delta_{f}$ we then obtain the following coupled system of equations

[TABLE]

where

[TABLE]

and

[TABLE]

The time dependence in $\epsilon_{1,2}$ comes into play when the interspecies coupling $g_{fb}$ is time dependent (e.g.) according to Eq. (2). For Eq. (31) we then have that the frequencies of the normal modes for the coupled bosonic and fermionic systems are

[TABLE]

To compare the breathing mode frequences given by Eq. (34) against full PDE simulations we study some special cases.

III.2.1 Breathing modes for uncoupled components ($g_{fb}=0$)

First we solve the system (30), which is now uncoupled, and obtain the equilibrium value for the widths of the fermionic subsystem $a_{f0}=\left((3\sqrt{3}+2\pi\kappa N_{f}^{2})/(3\sqrt{3}\alpha_{f})\right)^{1/4}$. In particular we see that the $N_{f}\rightarrow\infty$ limit

[TABLE]

compares with the Thomas-Fermi width Thomas-Fermi except for a numerical factor $\left(\pi/(6\sqrt{3})\right)^{1/4}\simeq 0.74$.

Now let us obtain an analytic expression also for $a_{b0}$ by neglecting the term $4/a_{b}^{3}$ in the upper line of (30), which can be motivated for large values of $N_{b}$ and/or $a_{b}$. Then we have the real positive solution

[TABLE]

which compares with the Thomas-Fermi width Thomas-Fermi except for a numerical factor $\left(2/(9\pi)\right)^{1/6}\simeq 0.64$.

Now we take the (simplified) analytic expressions (35) and (36) for the equilibrium widths and insert them into (32), to obtain

[TABLE]

which gives $\omega_{b}^{2}=12\alpha_{b}$ and $\omega_{f}^{2}=16\alpha_{f}$. Now with $\alpha_{n}=\frac{1}{2}m_{n}\omega_{trap,n}^{2}$, and $m_{n}=1/2$, we finally have

[TABLE]

This agrees with the result in the literature for Thomas-Fermi based models MenottiPRA2002 ; FuchsPRA2003 , and hence is an alternative derivation for the breathing mode frequency of each (uncoupled) component.

In Fig. 6 (a), we illustrate this case, that was used to check the accuracy of the numerical procedures.

III.2.2 Breathing modes for constant $g_{fb}\neq 0$ (not coupled to COM modes i.e. $\zeta_{b}=\zeta_{f}=0$)

Also for the breathing modes the two same frequencies occurs in both components when there is a coupling $g_{fb}\neq 0$. Now we have two frequencies $\omega_{1,2}$ that both differs from the respective uncoupled values given by (38): $\omega_{b}=\sqrt{3}\omega_{trap,b}$ and $\omega_{f}=2\omega_{trap,f}$ ($\omega_{trap,n}=2\sqrt{\alpha_{n}}$ both equals unity in the PDE simulations here), see Fig. 6 (b).

We now solve the coupled system (30) for real positive roots (numerically) for each value of $g_{fb}$ to obtain the equilibrium values for the widths of bosonic and fermionic subsystems. Hence, by doing this we can evaluate Eqs. (32), (33) and (34) numerically for different values of the coupling $g_{fb}$, see Fig. 7, and/or for different trap strengths $\alpha_{b}$ and $\alpha_{f}$.

III.2.3 Parametric resonance of breathing modes

For the periodic modulations in time of the scattering length $a_{s,fb}$ i.e. with $g_{fb}$ according to Eq. (2), the system (III.2) is two coupled Mathieu like equations. This system has been investigated in the work Hansen . Applying the results to our case (with $\zeta_{b}=\zeta_{f}=0$) we accordingly expect resonances in the breathing mode oscillations. The resonances can occur for a driving frequency $\Omega_{fb}$ near twice of the normal modes eigenfrequencies (34) (and their subharmonics) $2\omega_{n}$, and also close to the combination frequencies

[TABLE]

For small $\epsilon_{1,2}\ll\omega_{b,f}$ we have the following estimates from (34)

[TABLE]

such that

[TABLE]

As was shown with a multiscale analysis in the Appendix of Abdullaev2013 $|\omega_{1}-\omega_{2}|$ is stable. Furthermore, the driving frequency $\omega_{1}+\omega_{2}$ have a higher gain than $2\omega_{n}$ and is therefore used in the numerical examples of parametric resonance in the breathing modes presented in Figs. 8 and 9. In the Appendix of Abdullaev2013 also the region of instability for $\omega_{1}+\omega_{2}$ was determined. The exponentially increasing widths of the Bose and Fermi clouds demonstrate the instability of this resonance. However, for these large deviations of the widths, we cannot expect good agreement with full numerical simulations as is evident from the Fig. 8. We note that the oscillations in the lowest panel in Fig. 8 are unsymmetric w.r.t. $w_{f}(0)$ since the full spatial shapes (not presented here) of the clouds in the PDE simulations behaves quantitatively different at large amplitudes for fermions and bosons, as we have studied in detail numerically.

IV Numerical simulations

Possible experimental parameters used for the calculations presented in the figures are the following: we have illustrated the BCS regime ($\kappa=1/4$) for $N_{b}=10^{3}$ number of bosons and $N_{f}=2\cdot 10^{2}$ fermions, trapped with the same strength $\alpha_{b}=\alpha_{f}=1/4$ (except for in Figs. 4, 5 and 9); the intracoupling parameter in use was $g_{b}=g_{b}^{(0)}=1$ (see Sec. II for the translation to dimensionless variables).

Furthermore, phase separation for the components is here expected for a coupling in the order of $g_{fb}\sim 1$ ViveritPRA2000 , which we have confirmed numerically. Therefore, the curves in Figs. 2 and 7 based on the ODE model are dashed from this point on, and no rings from full PDE simulations are plotted, since we are not discussing the effect of phase separation here.

ODEs have been solved with Matlab’s built in solver ode45 while PDEs have been solved with XMDS XMDS on dense enough $x-t$-grids. The groundstates used for the initial conditions of the PDE (LABEL:sys1) was calculated using a damped second order equation (DFPM) SandinPRE2016 , using dynamical constraints Gulliksson2018 to keep the number of particles constant. Frequences of the oscillating observables have been obtained using exponential fitting as described in AnderssonIEEE2014 , which is a development of the well known ESPRIT roy1989esprit . In comparison with experimental data, additional noisereduction may be necesarry in obtaining the frequencies andersson2017fixed .

V Conclusion

In conclusion we have investigated the collective oscillations of superfluid mixtures of ultra cold fermionic and bosonic atoms while varying the scattering lengths periodically in time. The case of varying the fermion-boson $a_{s,fb}$ scattering lengths are studied with respect to excited center of mass modes and breathing modes in the mixture. Parametric resonances in the oscillations are predicted and the properties are analyzed by comparing PDE and ODE models for the dynamics. The resulting oscillations with increasing amplitudes provides clear experimental signals to search for. A specific application is to the recent experiment with fermionic 6Li and bosonic 7Li atoms with oscillating fermion-boson scattering length, which can be realized using the Feshbach resonance technic.

VI Acknowledgments

F. Kh. A. acknowledges hospitality of Örebro University Sweden and partial support from a senior visitor fellowship from Conselho Nacional de Desenvolvimento Cientifico e Tecnológico (CNPq-Brasil). The authors also acknowledge valuable comments from an anonymous referee.

VII Appendix on the ODE model

Variation with respect to each of the time dependent parameters $\xi_{b}$, $k_{b}$, $b_{b}$, $\xi_{f}$, $k_{f}$, and $b_{f}$ in the averaged Lagrangian (5) and invoking the Euler Lagrange equations in (7) results in the dynamical systems

[TABLE]

These equations are coupled to the equations derived from variation with respect to $A_{b}$, $a_{b}$, $A_{f}$ and $a_{f}$. The four final equations constitute a rather sizable system. The first equation arises from variation with respect to $a_{b}$ and reads

[TABLE]

The second equation is derived by variation with respect to $a_{f}$ and reads

[TABLE]

The third equation is obtained from variation with respect to the Bose amplitude $A_{b}$

[TABLE]

The final and fourth equation results from variation with respect to the Fermi amplitude $A_{f}$

[TABLE]

For numerical implementation it is convenient to rewrite Eq. (VII) and Eq. (VII) into the form

[TABLE]

similarly for Eqs. (VII) and VII)

[TABLE]

Note that these two final systems of equations are coupled to the system in (VII).