Transition of entanglement dynamics in an oscillator system with weak time-dependent coupling

We investigate the entanglement dynamics of two harmonic oscillators with a weak time-dependent coupling. We find strong connections between the dynamical behaviors of the entanglement and the stability of the normal modes. An abrupt transition of the entanglement dynamics from periodic and bounded to monotonically increasing occurs, when one of the normal modes becomes dynamically unstable as the modulation amplitude crosses the critical value. In particular, when the modulation frequency is roughly twice the oscillator frequency, an explicit critical value for the modulation strength is derived to specify the transition. Moreover, it is found that one can significantly change the dynamical behavior of the entanglement by adding an extremely weak time modulation with specific frequency to a constant coupling.

Figure 1

Stability of the Mathieu's equation. In the white areas where the Floquet exponent $\nu$ is complex, the general solution is unstable. In the shaded areas $\nu$ is a noninteger real number and the general solution is bounded. The boundary lines correspond to eigenvalues of the Mathieu's equation ${\alpha }_n\left(\epsilon \right)$ $\left(n=0,1,2,...\right)$ and ${\beta }_n\left(\epsilon \right)$ $\left(n=1,2,3,...\right)$. Note that the shaded zone is symmetrical about the $\delta$ axis.

Figure 2

Entanglement $E_N$ as a function of time for an initial vacuum state. The coupling parameters are $c_0=0.1\omega ,{\omega }_D=1.94\omega$: (a) $c_1=0.03\omega$ and (b) $c_1=0.1\omega$.

Figure 3

Entanglement $E_N$ as a function of time for an initial two-mode squeezed thermal state with squeezing parameter $r=0.6$ and mean occupation number $\overline{n}=0.5$ for both oscillators. The coupling parameters are the same as Fig. 2.

Figure 4

Time evolution of position (blue dashed line) and momentum (red solid line) variances for (a) the $+$ mode and (b) the $-$ mode with an initial vacuum state. The coupling parameters are $c_0=0.1\omega ,{\omega }_D=2.05\omega$, and $c_1=0.11\omega$.

Figure 5

Temporal behaviors of position (blue dashed line) and momentum (red solid line) variances for the $+$ mode at time intervals (a) $\omega t\in [23,29]$, (b) $\omega t\in [49,55]$, and (c) $\omega t\in [74,80]$. The vertical dotted-dashed lines in (a)–(c) denote the selected times at which the Wigner functions for $+$ mode are plotted in Figs. 6–6, respectively. The parameters are the same as Fig. 4.

Figure 6

The Wigner function in the phase space for the $+$ mode at specific chosen times (a) $\omega t=0$, (b) $\omega t=26.6$, (c) $\omega t=52.5$, and (d) $\omega t=77.7$, as marked by the vertical dotted-dashed lines in Fig. 5. The parameters are the same as Fig. 4.

Figure 7

The Wigner function in the phase space for the $-$ mode at specific chosen times (a) $\omega t=0$, (b) $\omega t=23.0$, (c) $\omega t=45.3$, and (d) $\omega t=70.6$. The parameters are the same as Fig. 4.

Figure 8

Stability of Mathieu's equation around $\delta =1$ and $\epsilon =0$. ${\alpha }_1$ and ${\beta }_1$ are eigenvalues Mathieu's equation. $A_{+}\text{−}{D}_{+}$ (blue circles) and $A_{-}\text{−}{D}_{-}$ (red squares) denote, respectively, the positions of $({\epsilon }_{+},{\delta }_{+})$ and $({\epsilon }_{-},{\delta }_{-})$ for normal modes. $A_{\pm }\text{−}{D}_{\pm }$ are calculated from Eq. (22) with the same coupling parameters as lines $A\text{−}D$ in Fig. 9, respectively.

Figure 9

Entanglement $E_N$ as a function of time with an initial vacuum state for increasing $c_1$: (a) $c_1=0.07\omega$ (line $A)$, (b) $c_1=0.09\omega$ (line $B)$, (c) $c_1=0.11\omega$ (line $C)$, and (d) $c_1=0.13\omega$ (line $D)$. The other coupling parameters are $c_0=0.1\omega$ and $\Delta =0.05\omega$.

Figure 10

Entanglement $E_N$ as a function of time with an initial vacuum state for a constant coupling $c=c_0$ and a dynamical coupling $c\left(t\right)=c_0+c_{1}cos\left[(2\omega +\Delta )t\right]$, where $c_0=0.1\omega ,c_1=0.1c_0$, and $\Delta =-0.1\omega$.