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

Collective oscillations of superfluid mixtures of ultracold 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 models and ordinary differential equation models for the dynamics. An application to the recent experiment with fermionic ${}^6\mathrm{Li}$ and bosonic ${}^7\mathrm{Li}$ atoms, which approximately have the same masses, is discussed.

Figure 1

Examples of dynamics of center of mass modes (COM) of the densities from the full PDE model Eq. (1). Colors in use are: blue (b), red (f), and green (tot). (a) At $t=0$ we have moved the trap for both uncoupled ($g_{fb}=0$) components by $d_b=d_f=0.05$. We find only one frequency (equals to unity in the present parameters, ${\omega }_0=2\sqrt{{\alpha }_b}=2\sqrt{{\alpha }_f}=1$) in all of the three signals $b, f$, and tot, plotted on top of each other. (b) Coupled system ($g_{fb}=1/2$) where only the bosonic trap have been translated ($d_b=0.05$ and $d_f=0$) at $t=0$. Using $N_b={10}^3$ and $N_f=2\times {10}^2$, Eq. (13) gives the mean position $\langle X\rangle \simeq 0.042$ for the total COM oscillations, shown by the dashed horizontal black line in the lower panel, while the black circles shows the results for the total COM oscillations from the analytic ODE based model Eq. (14). As indicated by the beating patterns in (b) we find an additional frequency ${\tilde{\omega }}_n$ in both the components $b$ and $f$, but that cancels for the total density $tot$ (lower panel) from the numerical PDE solution, in agreement with Eq. (14). See Fig. 2 for quantitative results. Physical parameters and the method for determining frequencies are discussed in Sec. 4.

Figure 2

Center of mass frequency for different coupling $g_{fb}$. The frequency ${\tilde{\omega }}_n$ normalized with ${\omega }_{0,n}=2\sqrt{{\alpha }_n}$ (equal unity with the parameters in use) is the same for the Bose (b) and Fermi (f) components, but not present in the total COM; see Eq. (14). Rings are from simulations of the full PDE model Eq. (1), the solid (/dashed) curve shows the ODE based approximation Eq. (17) with the widths $a_f=a_{f0}$ and $a_b=a_{b0}$ taken from Eqs. (36) and (37), respectively. The black squares at $g_{fb}=-1/2$ corresponds to the black squares in Fig. 4. Physical parameters and the method for determining frequencies are discussed in Sec. 4.

Figure 3

Parametric resonance for the relative center of mass modes $\mathrm{\Delta }\zeta \left(t\right)$. Upper panel: Solution to the ODE model, from Eqs. (18), (19), and (20), i.e., ${\mathrm{\Omega }}_{fb}=2.2388$, with parameters $g_{fb}^{\left(0\right)}=-1/2, c_{fb}=0.1$, and the initial value $\mathrm{\Delta }\zeta \left(0\right)=0.005$. The widths $a_f=a_{f0}$ and $a_b=a_{b0}$ are taken from Eqs. (36) and (37), respectively. Lower panel: Solution to the PDE model Eq. (1) with sinusoidal interspecies coupling Eq. (2), with parameters $g_{fb}^{\left(0\right)}=-1/2, c_{fb}=0.05, {\mathrm{\Omega }}_{fb}=2.1000$, and an initial translation of the trap for the Bose-component such that $d_b=0.005$ (see text). Physical parameters are discussed in Sec. 4.

Figure 4

Center of mass frequencies ${\mathrm{\Omega }}_{1,2}$ for different traps ${\alpha }_b\ne {\alpha }_f$. Blue (${\mathrm{\Omega }}_1$, lower rings) and red (${\mathrm{\Omega }}_2$, upper rings) are from simulations of the full PDE model Eq. (1), the solid curves shows the ODE based approximation Eq. (22) with the widths $a_f=a_{f0}$ and $a_b=a_{b0}$ taken from Eqs. (36) and (37), respectively. The dashed lines show the $K_{b,f}\ll |{\alpha }_b-{\alpha }_f|$ limits of Eq. (25). Since $g_{fb}=-1/2$ here, the black squares at ${\alpha }_b={\alpha }_f$ corresponds to the data points in the black squares of Fig. 2. The top black square is also the $|{\alpha }_b-{\alpha }_f|\ll K_{b,f}$ limiting result for ${\mathrm{\Omega }}_2$ of Eq. (24), while ${\mathrm{\Omega }}_1=1$. Physical parameters and the method for determining frequencies are discussed in Sec. 4.

Figure 5

Parametric resonance for the relative center of mass modes $\mathrm{\Delta }\zeta \left(t\right)$ for different traps ${\alpha }_b\ne {\alpha }_f$ within the ODE model. Upper panel: Solution to the ODE model, Eq. (26), with the driving ${\mathrm{\Omega }}_{fb}=2{\mathrm{\Omega }}_2=2.2638$, with parameters $g_{fb}^{\left(0\right)}=-1/2, c_{fb}=0.1, {\alpha }_b=1/4, {\alpha }_f=1.08{\alpha }_b$, and the initial value $\mathrm{\Delta }\zeta \left(0\right)=0.005$. The widths $a_f=a_{f0}$ and $a_b=a_{b0}$ are taken from Eqs. (36) and (37), respectively. Lower panel: Solution to the ODE model, Eq. (26), with the driving ${\mathrm{\Omega }}_{fb}=2{\mathrm{\Omega }}_2=2.2152$, with the same parameters as in the upper panel except that ${\alpha }_f=0.9{\alpha }_b$. No qualitative differences are seen for the ${\alpha }_b\ne {\alpha }_f$ case compared to Fig. 3, that shows both ODE and PDE results for the case ${\alpha }_b={\alpha }_f$. Physical parameters are discussed in Sec. 4.

Figure 6

Examples of breathing mode dynamics. We plot the RMS widths of the densities (anticorrelated to the peak density) of the two components, from the full PDE model Eq. (1). Panel (a) is for the uncoupled system. Panel (b) is for the coupled system ($g_{fb}=1/2$). The straight lines in (a) show the RMS width of the Thomas-Fermi profiles [36]. We determined the single frequencies for the RMS signal in each component in (a) to ${\omega }_b=1.732$ and ${\omega }_f=2.000$, which is in agreement with Eq. (39). As indicated by the beating patterns in (b), we then have two frequencies for each RMS signal, the same in both component, which are in agreement with Eq. (35). See Fig. 7 for quantitative results. Physical parameters and the method for determining frequencies are discussed in Sec. 4.

Figure 7

Breathing mode frequencies for different couplings $g_{fb}$. The solid (/dashed) curves shows the predictions from the ODE model, from Eqs. (33), (34), and (35). Rings shows results from the full PDE simulations. For the PDE calculations the initial state was chosen as the groundstate for ${\omega }_{\text{trap},n}=0.995$, i.e., by about $1\%$ lower values of ${\alpha }_n$ compared to the evolution in real time. Physical parameters and the method for determining frequencies are discussed in Sec. 4.

Figure 8

Parametric resonance of breathing modes, i.e., the change in the normalized RMS widths. Two upper panels ($n=b$, blue; $n=f$, red): Solution to the ODE model, from Eqs. (32) and (40), with parameters $g_{fb}^{\left(0\right)}=1/2, c_{fb}=0.2$, i.e., ${\mathrm{\Omega }}_{fb}={\omega }_1+{\omega }_2=3.6222$, and the initial values $\delta a_{b,f}\left(0\right)=0.005$. The widths $a_f=a_{f0}$ and $a_b=a_{b0}$ are obtained numerically from Eq. (31). Two lower panels: Solution to the PDE model Eq. (1) with sinusoidal interspecies coupling Eq. (2), with parameters $g_{fb}^{\left(0\right)}=1/2, c_{fb}=0.2, {\mathrm{\Omega }}_{fb}=3.6358$. An initial increase in the trapping frequencies for both components was used (see caption of Fig. 7), such that the RMS widths initially have a change in the order $\mathrm{\Delta }{w}_{\text{TF},n}\simeq 0.005w_{\text{TF},n}$ [36]. Physical parameters are discussed in Sec. 4.

Figure 9

Parametric resonance of breathing modes for different traps ${\alpha }_b\ne {\alpha }_f$ within the ODE model. Two upper panels ($n=b$, blue; $n=f$, red): Solution to the ODE model for ${\alpha }_b=1/4$ and ${\alpha }_f=1.08{\alpha }_b$, from Eqs. (32) and (40), with parameters $g_{fb}^{\left(0\right)}=1/2, c_{fb}=0.2$, i.e., ${\mathrm{\Omega }}_{fb}={\omega }_1+{\omega }_2=3.6980$, and the initial values $\delta a_{b,f}\left(0\right)=0.005$. The widths $a_f=a_{f0}$ and $a_b=a_{b0}$ are obtained numerically from Eq. (31). Two lower panels ($n=b$, blue; $n=f$, red): Solution to the ODE model according to the same procedure as for the upper panels but with a weaker trap for the fermions ${\alpha }_f=0.9{\alpha }_b$, i.e., ${\mathrm{\Omega }}_{fb}={\omega }_1+{\omega }_2=3.5232$. Physical parameters are discussed in Sec. 4.