Using a generalized version of Langer's "imaginary part of the free energy" method the quantum decay rate Γ = A exp (−B) of a metastable ohmic oscillator — with harmonic frequency ω0 and quantum friction coefficient α — is studied for an almost symmetric barrier and a "free" exit space, involving an "inflection scattering" center. The unstable anomalous fluctuation mode (of the "breathing bounce" configuration in the partition function) is analytically continued in a fully "dynamical" manner. At zero friction and zero temperature the result for Γ agrees with the escape rate found from the pertinent Schrodinger equation by means of an extended WKB "outgoing waves" analysis. At slightly elevated temperatures — removing the earlier restrictions 12B exp (− ħω0/2k B T) → 0 and 2πk B T/ħω0 → 0 arising from the "two-state" and "sudden-flip" approximations — in the weak quantum damping regime (α ≲ 1) the decay rate shows a characteristic transition from quantal to thermal dependence on the bias energy ε. In the strong quantum damping regime (α ≳ 1) this dependence is always thermal while — in that case — at zero bias the oscillator becomes "dissipatively frozen" at zero temperature.
Using a generalized version of Langer's "imaginary part of the free energy" method the quantum decay rate Γ = A exp (−B) of a metastable ohmic oscillator — with harmonic frequency ω0 and quantum friction coefficient α — is studied for an almost symmetric barrier and a "free" exit space, involving an "inflection scattering" center. The unstable anomalous fluctuation mode (of the "breathing bounce" configuration in the partition function) is analytically continued in a fully "dynamical" manner. At zero friction and zero temperature the result for Γ agrees with the escape rate found from the pertinent Schrodinger equation by means of an extended WKB "outgoing waves" analysis. At slightly elevated temperatures — removing the earlier restrictions 12B exp (− ħω0/2k B T) → 0 and 2πk B T/ħω0 → 0 arising from the "two-state" and "sudden-flip" approximations — in the weak quantum damping regime (α ≲ 1) the decay rate shows a characteristic transition from quantal to thermal dependence on the bias energy ε. In the strong quantum damping regime (α ≳ 1) this dependence is always thermal while — in that case — at zero bias the oscillator becomes "dissipatively frozen" at zero temperature.
The stabilizing effect of quantum fluctuations on the escape process and the relaxation dynamics from a quantum metastable state are investigated. Specifically, the quantum dynamics of a multilevel bistable system coupled to a bosonic Ohmic thermal bath in strong dissipation regime is analyzed. The study is performed by a non-perturbative method based on the real-time path integral approach of the Feynman-Vernon influence functional. We consider a strongly asymmetric double well potential with and without a monochromatic external driving, and with an out-of-equilibrium initial condition. In the absence of driving we observe a nonmonotonic behavior of the escape time from the metastable region, as a function both of the system-bath coupling coefficient and the temperature. This indicates a stabilizing effect of the quantum fluctuations. In the presence of driving our findings indicate that, as the coupling coefficient γ increases, the escape time, initially controlled by the external driving, shows resonant peaks and dips, becoming frequency-independent for higher γ values. Moreover, the escape time from the metastable state displays a nonmonotonic behavior as a function of the temperature, the frequency of the driving, and the thermal-bath coupling, which indicates the presence of a quantum noise enhanced stability phenomenon. Finally, we investigate the role of different spectral densities, both in sub-Ohmic and super-Ohmic dissipation regime and for different cutoff frequencies, on the relaxation dynamics from the quantum metastable state. The results obtained indicate that, in the crossover dynamical regime characterized by damped intrawell oscillations and incoherent tunneling, the spectral properties of the thermal bath influence non-trivially the short time behavior and the time scales of the relaxation dynamics from the metastable state.
We study the real time dynamics of a dissipative quantum system in a metastable state which may decay by crossing a potential barrier. Starting from an initial state where the system is in thermal equilibrium on one side of the barrier, the time evolution of the density matrix is evaluated analytically in the semiclassical approximation for coordinates near the barrier top. In a region about a critical temperature Tc large quantum fluctuations render the harmonic approximation of the potential insufficient and anharmonicities become essential. Accounting for non-Gaussian fluctuation modes, we show that the density matrix approaches a quasistationary state with a constant flux across the barrier. This extends our earlier results [Phys. Rev. E 51, 4267 (1995)] on the quantum generalization of the Kramers flux state to the region about Tc. By matching the flux state onto the equilibrium state on one side of the barrier, we determine the decay rate out of the metastable state. The rate constant shows a changeover from thermally activated decay to quantum tunneling for temperatures below Tc.
Based on the dynamical quantization method we derive a quantum phase-space non-Markovian Smoluchowski equation describing the non-inertial Brownian motion of a harmonic oscillator immersed in a generic environment. In the long-time regime we investigate the tunneling phenomenon by evaluating the quantum Kramers escape rate of a Brownian particle over a potential barrier. As far as a quantum thermal reservoir is concerned, it is found that our steady quantum Kramers rate may depend upon no friction constant in the strong friction domain. This preposterous feature may suggest the existence of non-dissipative quantum tunneling at the low-temperature range, including the zero temperature case. Lastly, we predict that our quantum escape rate non-exponentially decays on the edge of the breakdown of the dissipation-fluctuation relation.
Decay of the metastable state is analyzed within the quantum Kramers model in the weak-to-intermediate dissipation regime. The decay kinetics in this regime is determined by energy exchange between the unstable mode and the stable modes of thermal bath. In our previous paper [Phys. Rev. A 42, 4427 (1990)PLRAAN1050-294710.1103/PhysRevA.42.4427], Grabert's perturbative approach to well dynamics in the case of the discrete bath [Phys. Rev. Lett. 61, 1683 (1988)PRLTAO0031-900710.1103/PhysRevLett.61.1683] has been extended to account for the second order terms in the classical equations of motion (EOM) for the stable modes. Account of the secular terms reduces EOM for the stable modes to those of the forced oscillator with the time-dependent frequency (TDF oscillator). Analytic expression for the characteristic function of energy loss of the unstable mode has been derived in terms of the generating function of the transition probabilities for the quantum forced TDF oscillator. In this paper, the approach is further developed and applied to the case of the continuous frequency spectrum of the bath. The spectral density functions of the bath of stable modes are expressed in terms of the dissipative properties (the friction function) of the original bath. They simplify considerably for the one-dimensional systems, when the density of phonon states is constant. Explicit expressions for the fourth order corrections to the linear response theory result for the characteristic function of the energy loss and its cumulants are obtained for the particular case of the cubic potential with Ohmic (Markovian) dissipation. The range of validity of the perturbative approach in this case is determined (γ/ω_{b}<0.26), which includes the turnover region. The dominant correction to the linear response theory result is associated with the "work function" and leads to reduction of the average energy loss and its dispersion. This reduction increases with the increasing dissipation strength (up to ∼10%) within the range of validity of the approach. We have also calculated corrections to the depopulation factor and the escape rate for the quantum and for the classical Kramers models. Results for the classical escape rate are in very good agreement with the numerical simulations for high barriers. The results can serve as an additional proof of the robustness and accuracy of the linear response theory.
Quantum decay rates for barrier potentials driven by external stochastic and periodic forces in the strong damping regime are studied. Based on the quantum Smoluchowski equation derived recently by Ankerhold, Pechukas, and Grabert [Phys. Rev. Lett. 87, 086802 (2001)] explicit analytical and numerical results are presented for the case of the resonant activation phenomenon in a bistable potential and the escape from a metastable well with oscillating barrier, respectively. The significant impact of quantum fluctuations is revealed.
The decay time, τ, of a dissipative object out of the metastable minimum of a (cubic) potential due to quantum noise is determined by a numerical investigation of the quasiclassical Langevin equation. In the zero temperature limit, and for moderate to large damping (γ\gtrsimω o ), we find that σ=ln (ω o τ/2π) can be approximated by a σ≃κ·γΔV/\hbarω o 2 where the constant κ is κ≃3.20. This is in good agreement with an analytic treatment of the Langevin equation, but about a factor three smaller than the result obtained by Caldeira and Leggett by the instanton method.