Stochastic growth dynamics and composite defects in quenched immiscible binary condensates

We study the sensitivity of coupled condensate formation dynamics on the history of initial stochastic domain formation in the context of instantaneously quenched elongated harmonically trapped immiscible two-component atomic Bose gases. The spontaneous generation of defects in the fastest condensing component, and subsequent coarse-graining dynamics, can lead to a deep oscillating microtrap into which the other component condenses, thereby establishing a long-lived composite defect in the form of a dark-bright solitary wave. We numerically map out diverse key aspects of these competing growth dynamics, focusing on the role of shot-to-shot fluctuations and global parameter changes (initial state choices, quench parameters, and condensate growth rates), with our findings also qualitatively confirmed by realistic finite-duration quenches. We conclude that phase-separated structures observable on experimental time scales are likely to be metastable states whose form is influenced by the stability and dynamics of the spontaneously emerging dark-bright solitary wave.

Figure 1

Memory of stochastic defect dynamics in an immiscible two-component condensate (schematic, compiled from simulation data). On short postquench time scales, the fastest condensing component (here ${}^{87}\mathrm{Rb}$) contains multiple spontaneously generated defects (top). The ensuing stochastic dynamics generically lead to either destruction of defects or survival of a single composite defect in which the second component (here ${}^{133}\mathrm{Cs}$) preferentially condenses, forming a dark-bright solitary wave (middle row). The long-lived metastable states (bottom row) retain a memory of the prior stochastic dynamics.

Figure 2

Typical numerical evolution of a quenched two-component system. (a)–(b): postquench evolution, up to 2 s, of 2D column densities $(n_{\mathrm{col}},n_{0,\mathrm{col}})$ and 1D integrated density profiles $\left(n_{\mathrm{int}},n_{0,\mathrm{int}}\right)$ for c-field and PO (Penrose–Onsager) condensate respectively. (c): corresponding condensate phase. (d)–(e): evolution of (respectively) condensate number, $N_0$, and occupation ratio $r$ between condensate mode and the next-most-highly occupied mode. (f): single-run evolution of the number of spontaneously generated dark solitons, $N_{\mathrm{sol}}$(t) (obtained, to within the estimated uncertainty shown, by examining density minima and phase jumps, similar to the procedure of [36]). In all simulations in this work, both components are confined in harmonic traps with frequencies $({\omega }_z^{\mathrm{Rb}},{\omega }_{\perp }^{\mathrm{Rb}},{\omega }_z^{\mathrm{Cs}},{\omega }_{\perp }^{\mathrm{Cs}})=2\pi \times (3.89,32.2,4.55,40.2)$ Hz, with longitudinal and transverse shifts of $\sim 1 \mu \mathrm{m}$ in their centers [41, 53], and we use scattering lengths $(a_{\mathrm{Rb},\mathrm{Rb}},a_{\mathrm{Cs},\mathrm{Cs}},a_{\mathrm{Rb},\mathrm{Cs}})=(100,280,650)$ Bohr radii. Temperature is quenched from $T_0=80\mathrm{nK}$ to $T=20\mathrm{nK}$. In the first part of this work, we use initial chemical potentials ${\mu }_{\mathrm{Rb}}/k_{\mathrm{B}}=2.13$ nK, ${\mu }_{\mathrm{Cs}}=0.956{\mu }_{\mathrm{Rb}}$ (${\mu }_{\mathrm{Rb}}=1.38\hslash {\omega }_{\perp }^{\mathrm{Rb}},{\mu }_{\mathrm{Cs}}=1.05\hslash {\omega }_{\perp }^{\mathrm{Cs}}$), and rates ${\gamma }_{\mathrm{Rb}}={\gamma }_{\mathrm{Cs}}=0.263{\mathrm{s}}^{-1}$ (equivalent to $\hslash {\gamma }_k/k_{\mathrm{B}}T={10}^{-4}$), with ${\mu }_{\mathrm{Rb}}^{\prime }={\mu }_{\mathrm{Rb}}$ and ${\mu }_{\mathrm{Cs}}^{\prime }=7.34{\mu }_{\mathrm{Cs}}$. The initial condition is an equilibrated state with no single condensate mode $\left(r\sim 1\right)$. In (f) inset, we also plot the average defect number $\langle N_{\mathrm{sol}}\rangle$ immediately following finite-duration quenches with $(T_k,{\mu }_k)$ ramped linearly from initial to final values (as given above) over time $t_{\mathrm{R}}$. Lines qualitatively show potential defect number scalings with $t_{\mathrm{R}}$.

Figure 3

Evolution of four different postquench random noise sequences, with identical prequench states. (a)–(d) show the c-field density evolution [shown here via $n_k^{\left(z\right)}=n_k(x=0,y=0,z)]$, and (e)–(h) the condensate density profiles after 1 s (insets 2 s; ticks on inset axes correspond to those on the main figure axes). After this time the Rb defect has either been stabilized by Cs infilling (e); fully decayed within Rb (f); or decayed to the left (g) or right (h) edge of the condensate. Quench parameters are as in Fig. 2.

Figure 4

Dependence of evolution on quench parameters. (a) shows condensate number evolution (Rb solid, Cs dashed) for different final chemical potentials ${\mu }_k^{\prime }$; standard parameters ${\mu }_{\mathrm{Rb}}^{\prime }={\mu }_{\mathrm{Rb}},{\mu }_{\mathrm{Cs}}^{\prime }=7.34{\mu }_{\mathrm{Cs}}$ (red and blue circles); ${\mu }_{\mathrm{Rb}}^{\prime }=2.63{\mu }_{\mathrm{Rb}},{\mu }_{\mathrm{Cs}}^{\prime }=7.34{\mu }_{\mathrm{Cs}}$ [black squares, example of c-field density evolution shown in (c)]; ${\mu }_{\mathrm{Rb}}^{\prime }={\mu }_{\mathrm{Rb}},{\mu }_{\mathrm{Cs}}^{\prime }=11.9{\mu }_{\mathrm{Cs}}$ [green triangles, example of c-field density evolution shown in (d)]. Error bars in (a) for standard parameters indicate the standard deviation over six numerical trajectories. (b) shows condensate number evolution for different rates ${\gamma }_k$; standard parameters ${\gamma }_{\mathrm{Cs}}={\gamma }_{\mathrm{Rb}}=0.263{\mathrm{s}}^{-1}$ (red and blue circles), and ${\gamma }_{\mathrm{Cs}}=10{\gamma }_{\mathrm{Rb}}=2.63{\mathrm{s}}^{-1}$ (brown stars).

Figure 5

Evolution of (a) condensate atom number, (b) occupation ratio $r$, and (c)–(f) coupled 1D integrated condensate density profiles for initial states with different amount of coherence in the Rb initial state. Shown are cases with ${\mu }_{\mathrm{Cs}}>0$ and (c) ${\mu }_{\mathrm{Rb}}=1.38\hslash {\omega }_{\perp }^{\mathrm{Rb}}$ (corresponding to “reference” Fig. 2 data, plotted only for ease of comparison here), (d) ${\mu }_{\mathrm{Rb}}=0.01\hslash {\omega }_{\perp }^{\mathrm{Rb}}$, and (e)–(f) ${\mu }_{\mathrm{Rb}}=-2.42\hslash {\omega }_{\perp }^{\mathrm{Rb}}$; (f) differs from (e) only in the larger postquench Rb chemical potential $[{\mu }_{\mathrm{Rb}}^{\prime }=3.02\hslash {\omega }_{\perp }^{\mathrm{Rb}}$ in (f), as opposed to $1.38\hslash {\omega }_{\perp }^{\mathrm{Rb}}$ in (e)]. Note that for (d)–(f) ${\mu }_{\mathrm{Cs}}$ is increased to $2.11\hslash {\omega }_{\perp }^{\mathrm{Cs}}.$