Spontaneous symmetry breaking in a spin-orbit-coupled $f=2$ spinor condensate

We study the ground-state density profile of a spin-orbit-coupled $f=2$ spinor condensate in a quasi-one-dimensional trap. The Hamiltonian of the system is invariant under time reversal but not under parity. We identify different parity- and time-reversal symmetry-breaking states. The time-reversal symmetry breaking is possible for degenerate states. A phase separation among densities of different components is possible in the domain of time-reversal symmetry breaking. Different types of parity- and time-reversal symmetry-breaking states are predicted analytically and studied numerically. We employ numerical and approximate analytic solutions of a mean-field model in this investigation to illustrate our findings.

Figure 1

The $c_2-c_1$ phase plot illustrating ferromagnetic, antiferromagnetic, and cyclic phases. The $c_2=20c_1$ line separating the ferromagnetic and antiferromagnetic phases as obtained from the present analytic consideration is shown. Separated phase is possible above this line and miscible phase is possible below this line.

Figure 2

Component densities of a ${}^{23}\mathrm{Na}$ spinor BEC of $10000$ atoms for $\Omega =0$ and $c_0=242.97,c_1=12.06>0,$ and $c_2=-13.03<20c_1$. The parameters used are (a) $\gamma =\mathcal{M}=0$, (b) $\gamma =\mathcal{M}=0$, (c) $\gamma ={\gamma }^{\prime },\mathcal{M}=0$, and (d) $\gamma ={\gamma }^{\prime },\mathcal{M}=0.5$; here ${\gamma }^{\prime }$ can have any arbitrary real value including zero, and hence denotes any arbitrary strength of SO coupling. Time-reversal symmetry is broken in (d). In this and following similar figures all quantities are dimensionless and only the nonzero density components are shown.

Figure 3

Same as in Fig. 2 for $c_0=242.97,c_1=12.06>0$, and $c_2=65.76<20c_1$. The parameters used are (a) $\gamma =\mathcal{M}=0$, (b) $\gamma =0,\mathcal{M}=0$, (c) $\gamma =1,\mathcal{M}=0$, and (d) $\gamma =1,\mathcal{M}=0.5$. Time-reversal symmetry is broken in all cases.

Figure 4

Same as in Fig. 2 for $c_0=201.36,c_1=-1.81,c_2=24.15>20c_1$. The parameters used are (a) $\gamma =\mathcal{M}=0$, (b) $\gamma =0,\mathcal{M}=0.5$, (c) $\gamma =0.25,\mathcal{M}=0$, (d) $\gamma =0.25,\mathcal{M}=0.5$, (e) $\gamma =1,\mathcal{M}=0$, and (f) $\gamma =1,\mathcal{M}=0.5$. Time-reversal symmetry is broken in (b)–(f).

Figure 5

Same as in Fig. 2 for $c_0=242.97>0,c_1=12.06>0$, and $c_2=459.68>20c_1$. The parameters used are (a) $\gamma =0.5,\mathcal{M}=0$, (b) $\gamma =0.5,\mathcal{M}=0.5$, (c) $\gamma =2,\mathcal{M}=0$, (d) $\gamma =2,\mathcal{M}=0.5$, (e) $\gamma =4,\mathcal{M}=0$, and (f) $\gamma =4,\mathcal{M}=0.5$. Time-reversal symmetry is broken in all cases.

Figure 6

Same as in Fig. 2 for $c_0=201.36,c_1=-1.81,c_2=24.15>20c_1$. The parameters used are (a) $\gamma =0.5,\mathcal{M}=0$, (b) $\gamma =1,\mathcal{M}=0$, (c) $\gamma =0.25,\mathcal{M}=0.5$, and (d) $\gamma =0.5,\mathcal{M}=0.5$. Time-reversal symmetry is broken in all cases.

Figure 7

Real and imaginary parts of the $j=\pm 2$ wave-function components for the parameters of Fig. 2. Plots (a) and (b) were obtained with two real Gaussian input states for $j=\pm 2$ components. Plots (c) and (d) were obtained with an imaginary Gaussian input for $j=2$ and a real Gaussian input for $j=-2$ components.

Figure 8

Dynamics of a phase-separated spinor condensate in the presence of both SO coupling and Rabi term ($\gamma =1.0,\Omega =0.5$, and a zero initial magnetization). The density profiles are shown at (a) $t=0$, (b) $t=70.0$, and (c) $t=275$, the time after which the condensate has recovered its initial shape after one complete oscillation. (d) The component population $N_j=\int {\rho }_j\left(x\right)dx$ vs time during a complete cycle of the periodic oscillation.