Dynamic quantum tunneling in mesoscopic driven Duffing oscillators

We investigate the dynamic quantum tunneling between two attractors of a mesoscopic driven Duffing oscillator. We find that, in addition to inducing a remarkable quantum shift of the bifurcation point, the mesoscopic nature also results in a perfect linear scaling behavior for the tunneling rate with the driving distance to the shifted bifurcation point.

Figure 1

Energy diagram of the nondriven Duffing oscillator in (a) the laboratory frame and (b) the rotating frame.

Figure 2

(a), (b) Bifurcation diagram of the DDO based on the EOM of $\tilde{x}\left(\tau \right)$, (a) without and (b) with accounting for the quantum shift of the effective detuning [see Eq. (5)]. The dashed vertical lines indicate the respective bifurcation points. (c) Results from the numerical simulation of $\overline{x}\left(t\right)$ $=$$\mathrm{Tr}\left[x\tilde{\rho }\left(t\right)\right]$ based on Eq. (3). We choose the $\overline{x}\left(t\right)$ at $t=160(2\pi /\Omega )$ to represent the steady state amplitude $\overline{x}$. The circles and squares stand for results from different initial conditions (i.e., the SAS and LAS, respectively), from which we observe the hysteresis behavior. Here, again, the dashed vertical line indicates the bifurcation point in (b), say, $0.77F_c$, which agrees well with the numerical simulation. Parameters: $\kappa =0.01$, $T=5$ mK, $\delta =0.065$, and $\aleph =12$.

Figure 3

(a), (b) Wigner function of the coherent state determined from the steady-state solution of the EOM of $\tilde{x}\left(\tau \right)$, (a) without and (b) with accounting for the quantum shift. (c), (d) Wigner function from the numerical simulation based on Eq. (3) at $t=40(2\pi /\Omega )$ and $160(2\pi /\Omega )$, respectively. The dashed vertical line in each subfigure indicates the valid “$x$” center of the Wigner function of the SAS. Parameters: $\kappa =0.01$, $T=5$ mK, $\delta =0.065$, $\aleph =12$, and $F_0=0.7F_c$.

Figure 4

Scaling behavior of the dynamic quantum tunneling rate with the driving distance to the shifted bifurcation point, say $\eta ={\left(0.77F_c\right)}^2-F_0^2$. Here, the circles are the results from the numerical simulation, while the linear fit gives $\alpha =1.015$ for the tunneling action $R\propto {\eta }^{\alpha }$. Inset: an illustrative example of exponential fitting for the occupation probability of the SAS, under driving $F_0=0.76F_c$. Parameters: $\kappa =0.01$, $T=5$ mK, $\delta =0.065$, and $\aleph =12$.