Chaotic dynamics of a Bose-Einstein condensate coupled to a qubit

We study numerically the coupling between a qubit and a Bose-Einstein condensate (BEC) moving in a kicked optical lattice using Gross-Pitaevskii equation. In the regime where the BEC size is smaller than the lattice period, the chaotic dynamics of the BEC is effectively controlled by the qubit state. The feedback effects of the nonlinear chaotic BEC dynamics preserve the coherence and purity of the qubit in the regime of strong BEC nonlinearity. This gives an example of an exponentially sensitive control over a macroscopic state by internal qubit states. At weak nonlinearity quantum chaos leads to rapid dynamical decoherence of the qubit. The realization of such coupled systems is within reach of current experimental techniques.

Figure 1

(Color online) Probability density ${\left|{\psi }_1\left(x\right)\right|}^2$ at $t=10$ for $k=3.4$, $\epsilon =0.5$, $T=2$, and $\delta =0.2$ showing a broad distribution at $g=0$ (red/gray curve, magnified by a factor 100) and a BEC soliton at $g=20$ (blue/black curve). The initial state is given by Eq. (2) at $t=0$ for $g_0=20$, $x_0=0.1$, and $p_0=0.05$ and with qubit state $|\varphi ⟩=\left(|0⟩+|1⟩\right)/\sqrt{2}$. Inset shows data on a smaller scale near the soliton center, with full curves for ${\left|{\psi }_1\left(x\right)\right|}^2$ and dashed curves for ${\left|{\psi }_0\left(x\right)\right|}^2$.

Figure 2

(Color online) Circles show the kick function $f\left(x\right)$ obtained from the BEC dynamics of Eq. (1) with parameters $k=1$, $\epsilon =0.25$, $T=4$, $\delta =0.2$, and $g=20$. Initial state is the same as in Fig. 1. The black solid line shows $f\left(x\right)=\text{sin}x$, while the gray area, induced by the qubit, corresponds to the region delimited by the curves $\left(1\pm \epsilon /k\right)\text{sin}x$. Inset: Poincaré section of the standard map at $K=1$ (small dots); BEC positions are shown by large red/gray dots for parameters $k=1$, $\epsilon =0.5$, $T=1$, $\delta =0.05$, and $g=20$ and initial state as in the main figure.

Figure 3

(Color online) Qubit polarization ${\xi }_z$ (top and middle panels) and purity $\xi$ (bottom panel) as a function of time for $k=3.4$, $\epsilon =0.5$, $T=2$, and $\delta =0.2$. Data are shown for (top panel) $g_{11}=g_{10}=g_{01}=g_{00}=g$ with $g=20$ (red triangles) and $g=0$ (blue circles) and (middle panel) $g_{00}=g_{11}=21$ and $g_{01}=g_{10}=19$ (orange diamonds) and $g_{00}=g_{11}=30$ and $g_{01}=g_{10}=10$ (green squares). In the bottom panel, the black solid line shows the solitonic part $W_s$ (see text); other symbols are as in top and middle panels. Initial state is the same as in Fig. 1.

Figure 4

(Color online) Distance $\delta x$ between solitons with the qubit in the states up ${\psi }_1\left(x,t\right)$ and down ${\psi }_0\left(x,t\right)$ as a function of time $t$ for $g_{00}=g_{11}=g=40$, $g_{01}=g_{10}=0$ (red squares), $g_{00}=g_{11}=g=39.5$, $g_{01}=g_{10}=0.5$ (green circles), $g_{00}=g_{11}=g=28$, $g_{01}=g_{10}=12$ (blue diamonds), and $g_{00}=g_{11}=g_{01}=g_{10}=g=20$ (orange triangles). Other parameters are as in main part of Fig. 1.

Figure 5

(Color online) Density plot of the spectral density $S\left(\nu \right)$ of ${\xi }_z\left(t\right)$ expressed in arbitrary units and shown as a function of the frequency $\nu$ and the nonlinear parameter $g$ for $k=3.4$, $\epsilon =0.5$, $T=2$, $\delta =0.2$, and the rescaled Rabi frequency ${\nu }_R=T\delta /\pi$. Initial state is the same as in Fig. 1. Color is proportional to $S\left(\nu \right)$ which varies from zero marked by blue/black to maximal value marked by red/gray.

Figure 6

(Color online) Density plot of the qubit purity $\xi$ after $t=20$ kicks as a function of the chaos parameter $K=kT$ and the nonlinear parameter $g$ for $T=2$, $\delta =0.2$, and $\epsilon =0.5$. Initial state is the same as in Fig. 1.