Ehrenfest approach to open double-well dynamics

We consider an Ehrenfest approximation for a particle in a double-well potential in the presence of an external environment schematized as a finite resource heat bath. This allows us to explore how the limitations in the applicability of Ehrenfest dynamics to nonlinear systems are modified in an open system setting. Within this framework, we have identified an environment-induced spontaneous symmetry breaking mechanism, and we argue that the Ehrenfest approximation becomes increasingly valid in the limit of strong coupling to the external reservoir, either in the form of an increasing number of oscillators or increasing temperature. The analysis also suggests a rather intuitive picture for the general phenomenon of quantum tunneling and its interplay with classical thermal activation processes, which may be of relevance in physical chemistry, ultracold atom physics, and fast-switching dynamics such as in superconducting digital electronics.

Figure 1

Transition from bistable to monostable behavior induced by the external environment. The absolute value of the minimum of the potential is plotted vs the number of involved harmonic oscillators with uniform distribution of their angular frequency in a range centered around ${\omega }_0=5$ for different values of their bandwidth $\mathrm{\Delta }\omega =0,2,6,10$ (from top to bottom, black, red, blue, and green curves, respectively). The case of a monochromatic bath yields the only deterministic curve and gives a value of $N_{\mathrm{crit}}=2\mu /\left(m{\omega }_0^2\right)=800$ for $\mu =1$ and $m={10}^{-4}$, both in arbitrary units. The opposite case of maximal bandwidth yields instead a value which is smaller by a factor of $1+\mathrm{\Delta }{\omega }^2/\left(12{\omega }_0^2\right)=4/3$, i.e., $N_{\mathrm{crit}}=600$, in good agreement with the observed behavior of the curve at the end point. The inset explicitly shows the changing effective potential for an increasing number of oscillators, $N_{\mathrm{osc}}=1,300,600,900$ (from bottom to top, black, red, blue, and green curves, respectively), with the dashed curve indicating the threshold case of $\langle N_{\mathrm{crit}}\rangle =600$. The bandwidth in this case was taken to be the maximum value, $\mathrm{\Delta }\omega =10$.

Figure 2

The test particle's phase space $(Q,P)$ and fluctuational phase space $(\rho ,\mathrm{\Pi })$ for the case of $\mu =1,\lambda =1, M=1$. Initial conditions in $(Q,P,\rho ,\mathrm{\Pi })$ space are $(1/\sqrt{2},0,0.03,0)$, corresponding to the test particle sitting in the minimum of the potential well with zero initial momentum (top plots), and $(1,0,0.03,0)$, for which the test particle starts away from the minimum of the double well (bottom plots). The time evolution lasts for five periods, where the cycle time is defined by the effective period of the intrawell oscillations, $\tau =2\pi /\sqrt{2\mu }$. Even starting at the minimum of the potential well with zero kinetic energy in real phase space, the kinetic energy associated with fluctuations feeds back on the phase space, giving rise to quantum tunneling.

Figure 3

Same as Fig. 2, but with the test particle in contact with a bath with a temperature $T={10}^{-6}$. The heat bath is made of 2000 oscillators with frequencies uniformly distributed in the range $[0,10]$ and mass $m={10}^{-9}$.

Figure 4

Same as Fig. 3, but with the test particle in contact with a bath with a temperature $T={10}^{-3}$, corresponding to thermally activated hopping processes.

Figure 5

Trajectories in phase space for the classical case ($\hslash =0,\rho =0$) for a particle (left) in a double well without external heat bath and in contact with a heat bath at (middle) $T={10}^{-6}$ and (right) $T={10}^{-3}$, with initial conditions very close to the minimum of the potential, such that there is nearly no evolution in the first case, some random motion without hopping in the second case, and effective thermal activation in the third case.

Figure 6

Interwell period vs the initial position of the test particle (black solid line, left vertical scale) and corresponding potential energy (red dashed line, right vertical scale). A dramatic increase in period corresponds to the initial position at the minimum of the potential, with the interpretation of a lower transition probability. Here the temperature of the bath is assumed to be zero and $\mu =\lambda =1$, and the initial conditions for $P,\rho ,\mathrm{\Pi }$ are the same as in Fig. 2.

Figure 7

Fast Fourier transform of the time-dependent position of the particle in the double-well potential for different temperatures of the reservoir; from top to bottom on the right side of the plot: $T={10}^{-2}$ (green curve), $T={10}^{-4}$ (blue), $T={10}^{-6}$ (red), and $T={10}^{-8}$ (black), in arbitrary units. The regime of thermal activation is marked by the presence of a broad spectrum of frequencies, while at low temperatures the motion has a small set of dominant frequencies, peaked around curves progressively broadened as the temperature is increased. The motion is sampled for $2^{13}=8192$ time steps, with initial conditions and the same system parameters as in the bottom plots of Fig. 2, and 2000 oscillators in the heat bath.