Dynamical Tunneling in Macroscopic Systems

We investigate macroscopic dynamical quantum tunneling (MDQT) in the driven Duffing oscillator, characteristic for Josephson junction physics and nanomechanics. Under resonant conditions between stable coexisting states of such systems we calculate the tunneling rate. In macroscopic systems coupled to a heat bath, MDQT can be masked by driving-induced activation. We compare both processes, identify conditions under which tunneling can be detected with present day experimental means and suggest a protocol for its observation.

Figure 1

Illustration of the Hamilton function and the potential landscape. (a) $-p_{S,L}^2(x,E)$: the potential changes with $E$; classical turning points are found at $p(x_i,E)=0$. (b),(c) $H_0^{(\delta )}(x,p)$; in (b) white corresponds to high, black to low quasienergy; (b) white lines corresponds to the $L$, black ones to the $S$ branch; continuous lines correspond to real and dashed ones to imaginary valued momentum.

Figure 2

(a) Quantized energies: eigenvalues (EV) of ${\widehat{H}}_0^{(\delta )}$ versus WKB. ${\widehat{H}}_0^{(\delta )}$ was represented in the number state basis considering $2N$ levels. (b) Tunneling-induced energy splittings at level crossings. Frequency sweep at $m\Omega /\hslash =2$, $\gamma =m{\Omega }^2/24$, $\kappa {\omega }_c/{\Omega }^2=0.1$, and $F_0=0.5F_c(\nu )$.

Figure 3

(a) The ratio of tunneling and activation rates from the small well at the avoided level crossings. (b) Corresponding tunneling rates compared to ${\kappa }_{\mathrm{eff}}$ (where ${\Gamma }_{\mathrm{t}}/{\Gamma }_{\mathrm{a}}>1$). Driving frequency sweep at $F_0=0.7F_c(\nu )$; values of $\Omega$ (in GHz): 1(●), 2(□), 3(◆), 4(△), 5(▾), 6(⊳), 7(+), 8(×), at parameters specified in text.