Stabilizing effect of driving and dissipation on quantum metastable states

We investigate how the combined effects of strong Ohmic dissipation and monochromatic driving affect the stability of a quantum system with a metastable state. We find that, by increasing the coupling with the environment, the escape time makes a transition from a regime in which it is substantially controlled by the driving, displaying resonant peaks and dips, to a regime of frequency-independent escape time with a peak followed by a steep falloff. The escape time from the metastable state has a nonmonotonic behavior as a function of the thermal-bath coupling, the temperature, and the frequency of the driving. The quantum noise-enhanced stability phenomenon is observed in the investigated system.

Figure 1

Potential $V$ [Eq. (3), with $\mathrm{\Delta }U=1.4\hslash {\omega }_0$ and $\epsilon =0.27\sqrt{M\hslash {\omega }_0^3}]$ and the first six energy levels (horizontal lines). In the lower part, the probability densities $|q_i{\left(x\right)|}^2=|\langle x|q_i\rangle {|}^2$ associated with the DVR eigenfunctions are shown, the initial state $|q_3\rangle$ being highlighted by a solid line. Vertical lines indicate the position eigenvalues in the DVR. The metastable region of the potential is to the left of the so-called exit point $c$.

Figure 2

Left well population $P_{\text{left}}={\rho }_{11}+{\rho }_{22}$ as a function of time (in log scale) for $\overline{A}=0.15$, $\overline{T}=0.1$, $\gamma /{\omega }_0=0.2$, and five driving frequency values, namely $\mathrm{\Omega }/{\omega }_0=0.2,0.58,0.75,1.0,1.35$. Here we introduced the dimensionless quantities of the amplitude and temperature, $\overline{A}=A/\sqrt{M\hslash {\omega }_0^3}$ and $\overline{T}=k_{B}T/\hslash {\omega }_0$. Horizontal line: $1-d$, where $d=0.95$ is the threshold value. For $\mathrm{\Omega }/{\omega }_0=0.75$ no escape occurs, as $P_{\text{left}}$ remains above the value $1-d$. This is better shown in the inset. Note the nonmonotonic behavior of the steady-state values of $P_{\text{left}}$ as a function of $\mathrm{\Omega }$.

Figure 3

Escape time vs coupling strength for three driving settings, namely $\mathrm{\Omega }/{\omega }_0=0,0.2,0.7$. Upper panel: dimensionless temperature $\overline{T}=0.1$. At $\mathrm{\Omega }/{\omega }_0=0.7$, no escape occurs for $\gamma /{\omega }_0\lesssim 0.25$. Lower panel: dimensionless temperature $\overline{T}=0.3$. For $\mathrm{\Omega }/{\omega }_0=0.2$ and $\mathrm{\Omega }/{\omega }_0=0.7$, the escape occurs starting from $\gamma /{\omega }_0\lesssim 0.25$ and 0.55, respectively. For both panels, the driving dimensionless amplitude is fixed at the value $\overline{A}=0.15$. Solid lines, in both panels, give the behavior in the absence of driving $\mathrm{\Omega }=0$.

Figure 4

Escape time vs driving frequency for three values of the dimensionless driving amplitude $\overline{A}=A/\sqrt{M\hslash {\omega }_0^3}$, namely $\overline{A}=0.1,0.15,0.2$. Other parameters are $\gamma /{\omega }_0=0.2$ and $\overline{T}=0.1$.

Figure 5

Escape time as a function of the coupling strength and the driving frequency for dimensionless amplitudes $\overline{A}=0.15$. The dimensionless temperature is set to the value $\overline{T}=0.1$.

Figure 6

Escape time as a function of the coupling strength and the driving frequency for dimensionless amplitudes $\overline{A}=0.20$. The dimensionless temperature is set to the value $\overline{T}=0.1$.

Figure 7

Left well population $P_{\text{left}}={\rho }_{11}+{\rho }_{22}$ as a function of the time (in log scale), at driving frequency $\mathrm{\Omega }/{\omega }_0=0.75$, for three values of the scaled coupling parameter, namely $\gamma /{\omega }_0=0.2$ (magenta dashed line), $\gamma /{\omega }_0=0.6$ (green dotted line), and $\gamma /{\omega }_0=1.0$ (red dashed-dotted line). Parameters are $\overline{A}=0.15$ and $\overline{T}=0.1$. Horizontal line: $1-d$, where $d=0.95$ is the threshold value. The critical value of the scaled coupling parameter is ${\gamma }_c/{\omega }_0\simeq 0.75$.