Quantum entanglement due to a modulated dynamical Casimir effect

We study the creation and entanglement of quasiparticle pairs due to a periodic variation of the mode frequencies of a homogeneous quantum system. Depending on the values of the parameters describing the periodic modulation, the number of created pairs either oscillates or, in a narrow resonant frequency interval, grows exponentially in time. For a system initially in a thermal state, we determine in which cases the final state is quantum mechanically entangled, i.e., where the bipartite state is nonseparable. We include some weak dissipation, expected to be found in any experimental setup, and study the corresponding reduction of the quantum entanglement. Our findings are used to interpret the results of two recent experiments.

Figure 1

Example of the frequency modulation of Eq. (18), square rooted so as to give $\omega /{\omega }_0$. The horizontal axis represents $N={\omega }_{p}t/2\pi$. The amplitude $A=0.1$ (which applies to the squared frequency, being replaced by approximately $A/2$ when the square root is taken) and the total number of oscillations $N=14.4$.

Figure 2

Plot of $\left|\beta \right|$ as a function of $N$ for various values of $R$, from top to bottom: 0 (blue curve), $0.5$ (purple curve), 2 (yellow curve), and 5 (green curve). For all plots, the modulation amplitude $A=0.1$. After an initial linear growth for all curves, those with $R<1$ grow exponentially with $N$, while those with $R>1$ rise and fall periodically, the amplitude and period being approximately proportional to $1/R$. We also note the small rapid oscillations occurring on top of the long-time behavior.

Figure 3

Logarithmic plot of $\left|\beta \right|$ as a function of $R$ for various values of $N$: 25 (blue solid curve), 50 (purple dashed curve), and 100 (yellow dotted curve). For all plots, the modulation amplitude $A=0.1$. We clearly see the emergence of a central peak with increasing $N$, extending from $R=-1$ to $R=1$. For $\left|R\right|>1$, the curves oscillate in a complicated way, as for small values of $\left|\beta \right|$ the small rapid oscillations become more important. Because of these, $\left|\beta \right|$ need not exactly vanish at the completion of a cycle. The number of long-time oscillations increases in proportion to $N$ and their maxima trace out an envelope corresponding to the $1/\sqrt{R^2-1}$ behavior of their maxima.

Figure 4

Contour plot of $ln\left|\beta \right|$ as a function of $N$ and $R$. As in Figs. 2 and 3, the modulation amplitude $A=0.1$. The contour values are 5 (the maximum shown, at the boundary between dark blue and white) and 4 and decrease by steps of 1. We clearly see, with increasing $N$, the emergence of an exponentially growing resonant regime for $\left|R\right|<1$ and an oscillating nonresonant regime for $\left|R\right|>1$. Note also that the contours themselves are not exactly smooth, having jagged edges due to the short-time oscillations of $\left|\beta \right|$.

Figure 5

Final occupation number $n_{\mathrm{out}}$ (left) and $\Delta$ (right), for exact resonance $R=0$, as functions of the initial occupation number $n_{\mathrm{in}}$ (and the initial temperature in units of the mean frequency ${\omega }_0$) for various values of $N$: 5 (blue solid curve), 15 (purple dashed curve) and 25 (yellow dotted curve). It is clear that both $n_{\mathrm{out}}$ and $\Delta$ increase with temperature in an approximately linear fashion. However, whereas increasing $N$ raises both the intercept and slope of the $n_{\mathrm{out}}$ curves, it has the opposite effect on the $\Delta$ curves, yielding a nonseparable state over a progressively wider range of initial temperatures.

Figure 6

Loci of the separability threshold $\Delta =0$ in the $(T_{\mathrm{in}},N)$ plane for various values of $R$, from right to left: 0 (blue curve), $0.99$ (purple curve), 2 (yellow curve), and 5 (green curve). As for previous plots, we work with $A=0.1$. Note once again the splitting of the large-$N$ behavior into resonant ($\left|R\right|<1$) and nonresonant ($\left|R\right|>1$) regimes. For $R<1$, the curves are seen to extend indefinitely towards higher values of $n_{\mathrm{in}}$, meaning that it is possible, for any initial temperature, to reach a nonseparable state if $N$ is made large enough. On the other hand, for $R>1$, the curves reach some maximum value of $n_{\mathrm{in}}$ and then turn around, so the state can only be made nonseparable for temperatures below some maximum value and even then the system will oscillate between separable and nonseparable states. Again we notice the lack of smoothness of the curves due to the short-time oscillations of $\left|\beta \right|$.

Figure 7

Mean occupation number (left) and separability parameter $\Delta$ (right), at exact resonance $R=0$ and for an initial thermal bath with $T_{\mathrm{in}}={\omega }_0$, immediately after the end of the frequency modulation. The amplitude is as before, taken to be $A=0.1$. The various curves correspond to different dissipative rates $\Gamma /{\omega }_0$ (from larger to smaller $n$ and from lower to higher $\Delta$): 0 (blue curve), $0.01$ (purple curve), $0.02$ (yellow curve), and $0.03$ (green curve). The rate of increase of $n$ is seen to be reduced by larger dissipation rates, but, in accordance with the prediction of Sec. 4, it approaches exponential growth for $\Gamma /{\omega }_0<A/4=0.025$ and saturates for $\Gamma /{\omega }_0>0.025$. Similarly, $\Delta$ approaches a limiting value, which increases with the dissipation rate, and as predicted in Sec. 4, the final state is nonseparable only when $\Gamma /{\omega }_0<A/8n_{\text{eq}}\approx 0.021$.

Figure 8

Mean occupation number (left) and separability parameter $\Delta$ (right), at $R=2$ and for an initial thermal bath with $T={\omega }_0$, immediately after the end of the frequency modulation. As before, the amplitude $A=0.1$. The various curves correspond to different values of the dissipative rate $\Gamma /{\omega }_0$, from larger to smaller first oscillation: 0 (blue curve), $0.01$ (purple curve), $0.02$ (yellow curve), and $0.03$ (green curve). We note a smoothing out of the oscillations with increasing dissipative rate, and eventually their disappearance, as in overdamped systems. Here $n$ and $\Delta$ are seen to approach limiting values, which, respectively, decrease and slightly increase with increasing $\Gamma /{\omega }_0$.