Dynamics of a degenerate Cs-Yb mixture with attractive interspecies interactions

We probe the collective dynamics of a quantum degenerate Bose-Bose mixture of ${}^{133}\mathrm{Cs}$ and ${}^{174}\mathrm{Yb}$ with attractive interspecies interactions. Specifically, we excite vertical center-of-mass oscillations of the Cs condensate. We observe significant damping for the Cs dipole mode, due to the rapid transfer of energy to the larger Yb component and the ensuing acoustic dissipation. Numerical simulations based on coupled Gross-Pitaevskii equations provide excellent agreement with experiment and additionally reveal the possibility of late-time revivals (beating), which are found to be highly sensitive to the Cs and Yb atom number combinations. By further tuning the interaction strength of Cs using a broad Feshbach resonance, we explore the stability of the degenerate mixture. We observe collapse of the Cs condensate mediated by the attractive Cs-Yb interaction when $a_{\mathrm{Cs}}<50a_0$, well above the single-species collapse threshold, in good agreement with simulations.

Figure 1

Cs and Yb coupled dynamics. (a) Layout of the bichromatic optical dipole trap. (b) Numerically generated coupled displaced 3D density profiles for Cs (red) and Yb (blue) upon introducing a $+0.95\text{−}\mu \mathrm{m}$ vertical shift of the Cs trap minimum (at time $t=0$) with $(N_{\text{Yb}},N_{\text{Cs}})=(40,6.5)\times {10}^3$. Solid lines are sliced density profiles along the $x, y$, and $z$ axes. (c) Vertical CoM position of the Cs BEC after 40 ms ToF vs hold time in the trap measured experimentally (green circles) and simulated numerically (red line). The solid green line is a fit to the experiment: $A{e}^{-\beta t}sin(\omega t+{\varphi }_0)$ with $A=52\left(8\right)\mu \mathrm{m}, \omega =2\pi \times 260\left(2\right)$ Hz, and $\beta =0.10\left(2\right){\mathrm{ms}}^{-1}$. The numerical plot is based on an average over 204 particle number sets, with the red band indicating the standard deviation. Atom numbers are randomly selected such that their mean is $(N_{\text{Yb}},N_{\text{Cs}})=(40,6.5)\times {10}^3$, with a 20% standard deviation. (d) Corresponding CoM position of the Cs BEC performed in the absence of Yb, yielding $\omega =2\pi \times 251\left(1\right)$ Hz. The dashed green line in (c) reproduces the fit to the measurement shown in (d).

Figure 2

Cs and Yb excitation and energy exchange calculated numerically in a coupled-GPE treatment with $(N_{\text{Yb}},N_{\text{Cs}})=(40,6.5)\times {10}^3$. (a) Extended numerical evolution of the vertical CoM motions. (b) Changes in the total energy of Cs and Yb ($\mathrm{\Delta }{E}_{\text{Cs/Yb}}^{\text{tot}}$), the Cs-Yb interaction energy ($\mathrm{\Delta }{U}_{\text{CsYb}}$), and the main corresponding single-particle energy along the $z$ and (for Yb only) the $y$ axis. Note the more gradual initial increase along $y$, consistent with a secondary excitation mechanism, and the initial out-of-phase oscillations between $z$ and $y$ directions for Yb. The gray region shown in both (a) and (b) indicates the first 22 ms of evolution, where the primary damping and subsequent energy transfer occur.

Figure 3

Analysis of the Cs-Yb coupled dynamics during the first 22 ms [panels at left, (i)] and zoomed in to the first 5 ms [panels at right, (ii)]. All parameters are calculated from the numerical simulations used for Fig. 2. (a) Vertical CoM oscillation of Cs (red) and Yb (blue). The dashed blue line shows the CoM of the central “bulge” region of Yb (defined by $\left|x\right|<10\mu \mathrm{m}, \left|y\right|<6\mu \mathrm{m}$, and $\left|z\right|<3\mu \mathrm{m}$). (b) and (c) Cs and Yb velocity fields along the $z$ axis [defined by $v_{\text{Cs(Yb)}}=\hslash \nabla \text{arg}\left[{\psi }_{\text{Cs(Yb)}}(0,0,z)\right]/m_{\text{Cs(Yb)}}]$. (d) Scaled relative local velocity $(v_{\text{Cs}}-v_{\text{Yb}})/(c_{\text{Cs}}+c_{\text{Yb}})$ along the $z$ axis, where $c_{\text{Cs(Yb)}}=\sqrt{g_{\text{Cs(Yb)}}{n}_{\text{Cs(Yb)}}/m_{\text{Cs(Yb)}}}$ is the local speed of sound evaluated using the local atom density. The velocity plots include a gray mask to filter out regions corresponding to Cs (b) and Yb [(c) and (d)] atomic densities lower than $2\mu {\mathrm{m}}^{-3}$. The pairs of (approximately horizontal) black lines in (c) and (d) denote the vertical extent of the Cs velocity profile shown in (b). The dashed and solid vertical lines mark the Cs CoM turning points at $T_{\mathrm{Cs}}/2\approx 1.9\mathrm{ms}$ and $T_{\mathrm{Cs}}\approx 3.7\mathrm{ms}$, respectively.

Figure 4

The Fourier spectra of $z_{\text{Cs(Yb)}}^{\text{CoM}}$ for $(N_{\text{Yb}},N_{\text{Cs}})=(40,6.5)\times {10}^3$ (based on an evolution of 413.8 ms giving a spectral resolution of ${\omega }_{\mathrm{Cs},z}/650)$. The red and blue solid lines are the spectra of Cs and Yb, respectively. The pink and blue dotted lines are the corresponding spectra for the case where the other component is static (see text for details). In (b) we zoom in to the region around $\omega /{\omega }_{\text{Cs},z}=1$ to show the splitting of the oscillation frequency in Cs.

Figure 5

Dual-species collapse. (a) Cs collapse mediated by ${}^{174}\mathrm{Yb}$ (top row) and Cs single-species collapse (bottom row). Images show the Cs BEC optical depth (OD) profile for varying $a_{\mathrm{Cs}}$. Pink contours show $1/e$ and $1/e^2$ of peak OD. (b) Cs condensate number as a function of Cs scattering length for Cs collapse mediated by ${}^{174}\mathrm{Yb}$ (green circles) and Cs single-species collapse (red circles). The dashed line at $a_{\mathrm{Cs}}=-2.5a_0$ denotes the theoretically predicted single-species collapse threshold for $N_{\mathrm{Cs}}=4\times {10}^3$. The dashed line at $a_{\mathrm{Cs}}=54.3a_0$ corresponds to the cancellation of mean-field interactions $\delta g=0$. The results of GPE simulations with three-body loss are shown with green ($\mathrm{Cs}+\mathrm{Yb}$) and red squares (Cs only); lines are guides to the eye.

Figure 6

(a)–(d) ToF Cs CoM positions along the $z$ direction in the presence of the Yb component predicted numerically (red line) and measured experimentally (green circles). The experimental data set is compared with simulations corresponding to different atom numbers, consistent with atom number variations of up to $1.25\sigma$ from the “primary” values $(N_{\text{Yb}},N_{\text{Cs}})=(40,6.5)\times {10}^3$. In addition to numerical predictions in the actual experimental geometry with $\alpha ={20}^{\circ }$ (solid red lines) (as also presented in the main text), the corresponding dynamics for $\alpha =0^{\circ }$ are also shown (dashed pink lines). The green line shown is the corresponding best fit to the experimental data set.

Figure 7

(a)–(f) Density snapshots of Cs (red) and Yb (blue) at different times during the first $\approx 21$ ms of coupled evolution. Shown are (i) 3D profiles corresponding to density isosurfaces for the $5\%$ peak density of Yb at $t=0$ with corresponding slices shown along each axis and 2D sliced density snapshots across the (ii) $x\text{−}y$ ($z=0$) plane, (iii) $x\text{−}z$ ($y=0$) plane, and (iv) $y\text{−}z$ ($x=0$) plane.