Non-Markovian dynamics, nonlocality, and entanglement in quantum Brownian motion

The dynamical aspects of quantum Brownian motion at low temperatures are investigated. Exact calculations of quantum entanglement among two Brownian oscillators are given without invoking the Born-Markov or Born approximation widely used for the study of open systems. Nonlocality due to finite oscillator separation is studied. Our approach is applicable for arbitrary time scale, in particular suitable to probe the short-time regime at cold temperatures where many experiments on quantum information processing are performed. We study separability criteria based on the uncertainty relation, negativity, and entanglement of formation. We found a crossover behavior in a disentanglement process between quantum- and thermal-fluctuation-dominated regimes. The fluctuation-dissipation theorem is violated for two Brownian particles interacting with a common environment. The deviation from the relation originates in the interaction mediated by the environment, which drives two particles into a steady oscillatory state, preventing the system from thermalizing. Consequently there is a residual entanglement at low temperatures for arbitrary coupling strength. This entanglement is generated from the environment nonperturbatively even when two modes are not entangled initially. When the distance between two oscillators is varied, competition between entanglement induced from the environment and modified decoherence due to finite separation leads to the characteristic distance where entanglement is minimized.

Figure 1

The temporal evolution of the uncertainty [${\zeta }_{-}$ in Eq. (32)] is plotted. The initial condition is a two-mode coherent state $(r=0)$. Other parameters are $L=10\mathrm{cm}$, $\gamma =0.01\mathrm{GHz}$, and $\Lambda =30\mathrm{GHz}$. We take $\Omega =1.0\mathrm{GHz}$ for all figures. The time scale for vacuum fluctuations, $t_0=0.33\mathrm{ns}$, and thermal fluctuations, $t_{\theta }=0.1\mathrm{ns}$ (see Sec. 4A).

Figure 2

The temporal evolution of the uncertainty function [${\lambda }_{-}$ in Eq. (34)] after the partial transpose is plotted. The initial condition is a two-mode squeezed state with $r=0.1$ and a coherent state $(r=0)$. The result under the Born-Markov approximation gives only the constant value $1∕4$ for the latter case. Other parameters are $L=0$, $\gamma =0.1\mathrm{GHz}$, $\Lambda =50\mathrm{GHz}$, and $T=0$.

Figure 3

The negativity $\mathcal{N}$, the log negativity $E_{\mathcal{N}}$, and the entanglement of formation $E_{\mathcal{F}}$ are shown. The initial state is a two-mode squeezed state with $r=0.1$. Other parameters are $\gamma =0.2\mathrm{GHz}$, $\Lambda =100\mathrm{GHz}$, $L=0$, and $T=0$.

Figure 4

The negativity $\mathcal{N}$, the log negativity $E_{\mathcal{N}}$, and the entanglement of formation $E_{\mathcal{F}}$ are shown for different temperatures. $\gamma =0.2\mathrm{GHz}$ and other parameters are the same as in Fig. 3.

Figure 5

EPR dispersions $E_{\pm }$ are plotted. The vertical axis is in the logarithmic scale. The initial states are a two-mode squeezed state with $r=0.2$. $\gamma =0.2\mathrm{GHz}$. Other parameters are the same as in Fig. 3.

Figure 6

The logarithmic negativity vs distance between two oscillators at $t=0.2\mathrm{ns}$ is plotted. The initial condition is a two-mode squeezed state with $r=0.1$. Other parameters are $\gamma =0.01\mathrm{GHz}$ and $\Lambda =100\mathrm{GHz}$.

Figure 7

(a) $E_{\mathcal{F}}$ for different temperatures (a) and for different separation distances (b) are plotted. $L=0.0$ for (a). $T=0.0$ for (b). The initial condition is a two-mode squeezed state with $r=0.3$. Other parameters are $\gamma =0.01\mathrm{GHz}$ and $\Lambda =20\mathrm{GHz}$. $t_0=0.13\mathrm{ns}$, $t_{\theta }=0.03\mathrm{ns}$, and $t_L=0.4\mathrm{ns}$.

Figure 8

${\lambda }_{-}$ and $E_{\mathcal{F}}$ are shown as a function of time. The initial condition is a two-mode coherent state (there is no initial entanglement). Other parameters are $T=0.0$, $L=0.0$, and $\Lambda =100\mathrm{GHz}$.

Figure 9

${\lambda }_{-}$ and $E_{\mathcal{F}}$ are shown as a function of squeezing in the initial state. Other parameters are $T=0.0$, $L=0.0$, and $\Lambda =300\mathrm{GHz}$.

Figure 10

The threshold temperatures $T_{*}$ for different $\gamma$ are shown as a crossing point of each curve with the minimum uncertainty ${\lambda }_{-}=1∕4$ line [in (a)] or the temperature when entanglement vanishes [in (b)]. The curve for the Born-Markov approximation in (b) agrees with $E_{\mathcal{F}}=0$, indicating no entanglement generation under the Born-Markov approximation. The initial condition is a two-mode coherent state. Other parameters are the same as in Fig. 9.