Quantum critical temperature of a modulated oscillator

We show that the rate of switching between the vibrational states of a modulated nonlinear oscillator is characterized by a quantum critical temperature $T_{c1}\propto {\hslash }^2$. Above $T_{c1}$ there emerges a quantum crossover region where the switching rate displays a steep and characteristic temperature dependence, followed by a qualitatively different temperature dependence for higher $T$. In contrast to the crossover between tunneling and thermal activation in equilibrium systems, here the crossover occurs between different regimes of switching activated by quantum fluctuations. The results go beyond the standard real-time instanton technique of the large-deviation theory.

Figure 1

(a) The scaled Hamiltonian $g(Q,P)$ of the resonantly driven nonlinear oscillator; $Q$ and $P$ are the dimensionless coordinate and momentum in the rotating frame. In the presence of weak dissipation, the local maximum and minimum of $g$ correspond to the small- and large-amplitude stable vibrational states. The scaled field intensity is $\beta =0.02$. (b) The cross section $g(Q,0)$; the solid lines show the quasienergy states localized about the local maximum $g_{{}_{\mathrm{SVS}}}$ for $\lambda =0.015$. Their scaled quasienergies $g_n$ lie between $g_{{}_{\mathrm{SVS}}}$ and the value of $g(Q,P)$ at the saddle point, $g_{{}_{\mathcal{S}}}$.

Figure 2

(a) The steepness of the distribution eikonal $|{\partial }_{g}R|$ calculated numerically as ${\left({\partial }_{g}R\right)}_{g_n}=\nu {\left(g_n\right)}^{-1}\ln ({\rho }_n/{\rho }_{n-1})$. The results refer to the temperatures marked on the right panel by the corresponding symbols; $\beta =0.0448$ and $\lambda =0.0053$ (for such parameters, the number of states localized about the small-amplitude SVS is $M=20$). The solid and dashed lines show $|{\partial }_{g}R|$ for $\overline{n}=0$ and $\overline{n}\to 0$ but $T>T_{c2}$, respectively. (b) The activation exponent $R_A$ calculated numerically from Eq. (2) (solid curve) and the derivative $d{R}_A/\lambda d{T}^{-1}\approx -d\ln W_{\mathrm{st}}/d{T}^{-1}$ (dashed curve). The units on the right axis are kelvins.

Figure 3

(a) The contour of integration $\mathcal{C}$ for calculating the matrix element (B6) in the WKB approximation and the auxiliary integration contours ${\mathcal{C}}^{\prime }$ and ${\mathcal{C}}_{\mathrm{arc}}$ for $g_m<\sqrt{\beta }$; $Q_B\equiv Q_B\left(g_m\right)$ and $Q_{\mathrm{ext}}\equiv Q_{\mathrm{ext}}\left(g_m\right)$ are the branching point and turning point of $P(Q,g_m)$; see Eq. (B3). (b) Mapping of the $Q$ plane (with a branch cut from $Q_{\mathrm{ext}}$ to $\infty$, the black thin line in (a)) on the interior of a rectangle on the $\tau$ plane for $g_m<\sqrt{\beta }$ by the function $Q(\tau ;g)$ that describes the classical Hamiltonian trajectory with given $g$, $g_n<g<g_m$; ${\tau }_p^{\left(1\right)}$, ${\tau }_p^{\left(2\right)}$, and ${\tau }_{*}$ are the real and imaginary periods and the pole of $Q(\tau ;g)$, respectively. The solid ($\tilde{\mathcal{C}}$), dashed (${\tilde{\mathcal{C}}}_{\mathrm{arc}}$), and dash-dotted (${\tilde{\mathcal{C}}}^{\prime }$) lines are the maps of the corresponding contours in (a). The solid arc in the lower left corner is the map of the real axis of $Q$ from $-\infty$ to $Q_B\left(g\right)$; ${\tau }_B$ and ${\tau }_{\mathrm{ext}}$ are the times for reaching $Q_B\left(g_m\right)$ and $Q_{\mathrm{ext}}\left(g_m\right)$. (c) Integration contours for $g_m>\sqrt{\beta }$. (d) Conformal mapping for $g>\sqrt{\beta }$. The curved red line dividing the rectangle into two parts is the map of the real axis of $Q$ from $-\infty$ to $Q_B\left(g\right)$.

Figure 4

Comparison of Eq. (B9) for $a_{mn}$ calculated as a continuous function of $g_n$ (solid lines) with numerical calculations (symbols). The scaled intensity of the modulating field is $\beta =0.0448$. The other parameter values are $\lambda =0.0053$ and $m=4$ (violet triangles), $\lambda =0.09$ and $m=2$ (blue circles), and $\lambda =0.15$ and $m=1$ (green squares). The parameter $m$ has been chosen so that $g_m\approx g_e=1/4$.