Fidelity for the quantum evolution of a Bose-Einstein condensate

We investigate fidelity for the quantum evolution of a Bose-Einstein condensate (BEC) and reveal its general property with a simple two-component BEC model. We find that, when the initial state is a coherent state, the fidelity decays with time in the ways of exponential, Gaussian, and power law, depending on the initial location, the perturbation strength, as well as the underlying mean-field classical dynamics. In this case we find a clear correspondence between the fast quantum fidelity decay and the dynamical instability of the mean-field system. With the initial state prepared as a maximally entangled state, we find that the behavior of fidelity has no classical correspondence and observe an interesting behavior of the fidelity: periodic revival, where the period is inversely proportional to the number of bosons and the perturbation strength. An experimental observation of the fidelity decay is suggested.

Figure 1

Periodic snapshot of the orbits for $\mu =T=g_c=1$, $K=2$, where the $x$ axis $\varphi$ is the azimuthal angle. It shows one big island and four small islands. Inside the islands motions are stable, outside the islands motions are mainly unstable or chaotic.

Figure 2

Contour plot of the fraction of chaotic motions for $\mu =T=1$.

Figure 3

(Color online) Fidelity decay in a classically chaotic case, with $K=2$, $g_c=4$, $\mu =1$, $T=1$. Upper panel: An example in the perturbative regime with $S=100$, $ϵ=2\times {10}^{-4}$, obtained from one initial coherent state. The dashed curve is the theoretical prediction in Eq. (11). Lower panel: Two examples in the FGR and Lyapunov regimes, respectively. In the FGR regime, $S=200$, $ϵ=3\times {10}^{-3}$, and an average is performed over 20 initial coherent states taken randomly. The dashed line is the FGR decay. In Lyapunov regime, $S=500$, $ϵ=1\times {10}^{-2}$, with an average over 1000 initial coherent states. The solid line is $e^{-\gamma t}$ with $\gamma =\Lambda$ $(t=6)$ in Eq. (14).

Figure 4

(Color online) Fidelity decay in a classically nearly integrable case, with $K=2$, $g_c=0.2$, $\mu =1$, $T=1$, $N=200$, and $ϵ=0.003$. Upper panel: Fidelity of four randomly chosen initial coherent states, with the smooth solid curve being the Gaussian fit to one of them. Lower panel: Averaged fidelity, with an average performed over 50 initial coherent states chosen randomly.

Figure 5

(Color online) Fidelity decay in the mixed system whose classical phase space structure is shown in Fig. 1. $L=500$ and $ϵ=0.0006$, so that $\sigma =ϵ∕\hslash =0.3$. The almost nondecaying solid line is the fidelity of an initial coherent state lying within the largest regular region. The other two solid curves, one close to the FGR decay, the other first having a Lyapunov decay then an approximately FGR decay, correspond to the fidelity of two initial coherent states lying in the chaotic region of the classical system.

Figure 6

Contour plot of the fidelity $M\left(t\right)$ at $t=200$, for $K=2$, $g_c=\mu =T=1$, $N=1000$, and $ϵ=0.0006$, with the classical phase space possessing a mixed structure shown in Fig. 1.

Figure 7

(Color online) Fidelity for the two Fock states and the GHZ entangled state, with $L=500$, $ϵ=2\times {10}^{-5}$, and $K=2$. The upper panel is with $g_c=0.2$. Inset of upper panel: $100∕T_{\mathrm{ent}}\text{vs}ϵL$ for $ϵL$ from 0.001 to 0.1, with $L=50$ (solid line) and $L=100$ (circles), showing the linear dependence of $T_{\mathrm{ent}}$ on $1∕ϵL$. The lower panel is for $g_c=1$, in which the fidelity of the GHZ state can be well fitted by a Gaussian decay.

Figure 8

Contour plot of the fidelity $M\left(t\right)$ at $t=1500$, with respect to the parameters $g_c$, $K$. The initial quantum states are the GHZ entangled state.