Chaos-driven dynamics in spin-orbit-coupled atomic gases

The dynamics, appearing after a quantum quench, of a trapped, spin-orbit coupled, dilute atomic gas is studied. The characteristics of the evolution is greatly influenced by the symmetries of the system, and we especially compare evolution for an isotropic Rashba coupling and for an anisotropic spin-orbit coupling. As we make the spin-orbit coupling anisotropic, we break the rotational symmetry and the underlying classical model becomes chaotic; the quantum dynamics is affected accordingly. Within experimentally relevant time scales and parameters, the system thermalizes in a quantum sense. The corresponding equilibration time is found to agree with the Ehrenfest time, i.e., we numerically verify a $\sim \ln \left({\hslash }^{-1}\right)$ scaling. Upon thermalization, we find that the equilibrated distributions show examples of quantum scars distinguished by accumulation of atomic density for certain energies. At shorter time scales, we discuss nonadiabatic effects deriving from the spin-orbit-coupled induced Dirac point. In the vicinity of the Dirac point, spin fluctuations are large and, even at short times, a semiclassical analysis fails.

Figure 1

Adiabatic potentials of the isotropic (a) and anisotropic (b) SO-coupled models. In both figures, the $E=0$ plane is the one including the DP at $p_x=p_y=0$. A necessary, but not sufficient, condition for the validity of the BOA is that $E<0$. In (a), the lower adiabatic potential $V_{-}(p_x,p_y)$ has the characteristic sombrero shape. By considering an anisotropic SO coupling, the rotational symmetry is broken and $V_{-}(p_x,p_y)$ possesses two global minima at $(p_x,p_y)=(0,\pm v_y)$.

Figure 2

Two examples of classical trajectories $(x\left(t\right),P_x\left(t\right))$ for regular (a) and chaotic (b) dynamics. In (a), typical for regular motion the trajectories evolve upon a tori. Contrary, in (b) the trajectory is much more irregular, which is characteristic for the chaotic evolution. The regular motion is calculated for the SO-coupling strengths $v_x=v_y=30$, and the chaotic motion with $v_x=20$ and $v_y=30$. In both cases, the energy is $E=-192$.

Figure 3

Poincaré sections of the Rashba SO-coupled adiabatic model (5) for the intersections $y=0$ (a) and $p_y=0$ (b). The initial energy is $E=-192$, the SO-coupling strengths $v_x=v_y=30$, and the number of simulated semiclassical trajectories 18.

Figure 4

Poincaré sections of the anisotropic SO-coupled adiabatic model (5) for $y=0$ [(a) and (c)] and for $p_y=0$ [(b) and (d)]. The initial energies are $E=-88$ [(a) and (b)] and $E=-192$ [(c) and (d)], and the SO-coupling strengths $v_x=20$ and $v_y=30$ for both cases. The corresponding maximum Lyaponov exponets have been derive, using the method out lined in Ref. [44], to $\lambda \approx 0.12$ and $0.090$, respectively. The number of semiclassical trajectories is the same as for Fig. 3, namely, 18.

Figure 5

Bloch vector components $R_x$ (dashed lines) and $R_y$ (solid lines). For the upper plot (a), the trap has been displaced in the $y$ direction, $x_s=0$ and $y_s=28$, while in the lower plot (b), the displace direction is the perpendicular, $x_s=28$ and $y_s=0$. In both figures, $v_x=10$ and $v_y=15$, and the average energy $\overline{E}\approx 280$.

Figure 6

Distributions $|\Psi (x,y,t_f){|}^2$ [(a) and (c)] and $|\Phi (p_x,p_y,t_f){|}^2$ [(b) and (d)] at $t_f=400$ for the Rashba SO-coupled model. At time $t=0$, the trap is suddenly displaced from $x_0=y_0=16$ to $x_0=y_0=0$. The initial ground state is then quenched into a localized excited state. The upper two plots (a) and (b) display the results from full quantum simulations of the adiabatic model (5), while the lower plots (c) and (d) show the corresponding semiclassical TWA distributions. The average semiclassical energy $\overline{E}\approx -192$ with a standard deviation $\delta \overline{E}\approx 22$. The dimensionless SO-coupling strengths $v_x=v_y=30$.

Figure 7

Same as Fig. 6, but for the anisotropic SO-coupled model with $v_x=20$ and $v_y=30$. The largely populated regions are so-called quantum scars and derive from properties of the underlying classical model, i.e., they are not outcomes of some coherent quantum mechanism.

Figure 8

Same as Fig. 7, but for an initial energy $E>0$. The dimensionless SO couplings $v_x=14$ and $v_y=21$, while the shifts $x_s=y_s=16$ giving an average energy $\overline{E}=\langle {\widehat{H}}_{\text{SO}}\rangle \approx 36.5$.

Figure 9

Examples of the phase-space area ${\Delta }_x\left(t\right)$ for different $h$ values ($h=1,2,3,...,10$). The upper plot (a) gives ${\Delta }_x\left(t\right)$ without shifting the time, while for the lower one (b), time has been shifted by $\delta =\ln \left(h\right)/\lambda$. The arrow indicates increasing $h$ values. It is clear how the spread in ${\Delta }_x\left(t\right)$ between different $h$ values is suppressed when we shift the time. The trap shifts $x_s=y_s=19$, resulting in an energy $\overline{E}\approx -88$. The maximum Lyaponov exponent $\lambda =0.18$.

Figure 10

Sections of $\left|\psi \right(x,y=0\left)\right|$ for different values on the dimensionless Planck's constant $h$: $h=1$ (black solid line), $h=2$ (blue dotted line), and $h=3$ (red dashed line). The final time $t_f=80$, $x_s=y_s=16$, and $v_x=14$ and $v_y=20$. As a comparison between classical and quantum results, we also include the TWA results as a green solid line, calculated for $h=1$. The green line has been shifted downward with 0.02 for clarity.

Figure 11

Same as Fig. 7, but for the shifts $x_s=20$ and $y_s=0$. For the given dimensionless parameters, the initial state is such that its dynamics should be regular according to the corresponding Poincaré section (Fig. 4). The energy $\overline{E}\approx -250$.