Phase separation and multistability of a two-component Bose-Einstein condensate in an optical cavity

We examine the multistability associated with miscibility-immiscibility conditions for a two-component Bose-Einstein condensate coupled to the light field in an optical cavity. For a strongly immiscible condition, the system exhibits a variety of density structures, including a separated state, a stripe state, and their coexistence. The multistability arises from these spatial structures of the two-component condensate, which significantly alter the hysteresis curve with respect to the intensity of cavity pumping. We present a variational approach to confirm our numerical results.

Figure 1

Schematic illustration of the two-component BEC-cavity system driven by laser pumping along the cavity axis. The cavity field is pumped at a rate of $\eta$ and decays at a rate of $\kappa$. The cavity mode near the BECs is approximated as $cos\left(kx\right)$. The inset shows the level scheme of the two components. The energy difference between the two components does not affect the results, and is neglected in Eqs. (1) and (2).

Figure 2

Steady-state intracavity photon number ${\left|\alpha \right|}^2$ as a function of laser pump intensity $\eta$, where $\eta$ is increased from $\eta =0$ to 6000 and then decreased to 0 in the imaginary-time evolution. The density profiles of the two components at points a, b, and c are shown in the insets, where the size in the $y$ direction is reduced to 1/2. The parameters are $\tilde{g}=1000, U_1=0.25, U_2=0.125, {\delta }_c=1200, \kappa =400$, and $N=16\tilde{N}=1.2\times {10}^5$. (a) $\tilde{g}{}_{12}=1000$ and (b) $\tilde{g}{}_{12}=1020$. See the Supplemental Material [59] for movies showing the $\eta$ dependence of the atomic density distributions.

Figure 3

Steady-state intracavity photon number ${\left|\alpha \right|}^2$ as a function of laser pump intensity $\eta$ for (a) ${\tilde{g}}_{12}=1030$ and (b) ${\tilde{g}}_{12}=1040$. The density profile at each point is shown in the inset. Other parameters are the same as those in Fig. 2. See the Supplemental Material [59] for movies showing the $\eta$ dependence of the atomic density distributions.

Figure 4

Steady-state intracavity photon number ${\left|\alpha \right|}^2$ as a function of laser pump intensity $\eta$ for (a) ${\tilde{g}}_{12}=1010$, (b) ${\tilde{g}}_{12}=1030$, and (c) ${\tilde{g}}_{12}=1050$. The solid and dashed lines indicate energetically stable and unstable states, respectively. The insets provide magnification. Other parameters are ${\tilde{g}}_{11}={\tilde{g}}_{22}=1000, N=1.2\times {10}^5, U_1=0.25, U_2=0.125, {\delta }_c=1200$, and $\kappa =400$.

Figure 5

(a) Effective energy $E_{\mathrm{eff}}$ as a function of laser pump intensity $\eta$. (b) Volume ratio of region b, $v_b$, as a function of laser pump intensity $\eta$. The intercomponent interaction strength is ${\tilde{g}}_{12}=1030$. Other parameters are the same as those in Fig. 4. Curves of stable states are only shown.

Figure 6

Steady-state intracavity photon number ${\left|\alpha \right|}^2$ as a function of effective detuning ${\delta }_c$ for (a) $\eta =1000$ and (b) $\eta =4000$. The value of intercomponent interaction is ${\tilde{g}}_{12}=1050$. Other parameters are the same as those in Fig. 4.

Figure 7

(a) Steady-state intracavity photon number ${\left|\alpha \right|}^2$ as a function of laser pump intensity $\eta$ for $g_{12}=0.9g$. (b) Density profiles of two components on the $y=0$ plane. The size of each panel is $115.2\mu \mathrm{m}\times 115.2\mu \mathrm{m}$ and the color bar ranges from 0 to $5\times {10}^{-5}$ in units of $N{k}^3$. Each colored point in (a) corresponds to the density profile in (b) for each value of $\eta$. The parameters are $a_{11}=100a_0$, with $a_0$ being the Bohr radius, $U_1=0.025, U_2=0.0125, {\delta }_c=1200$, and $N=6\times {10}^5$.

Figure 8

(a) Steady-state intracavity photon number ${\left|\alpha \right|}^2$ as a function of laser pump intensity $\eta$ for $g_{12}>g$. (b) Density profiles of two components on the $y=0$ plane. The size of each panel is $115.2\mu \mathrm{m}\times 115.2\mu \mathrm{m}$ and the color bar ranges from 0 to $5\times {10}^{-5}$ in units of $N{k}^3$. Each colored point in (a) corresponds to the density profile in (b) for each value of $\eta$. The intercomponent scattering length is $a_{12}=110a_0$, where $a_0$ is the Bohr radius. All other parameters are the same as those in Fig. 7.