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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Dissipative quantum systems with a potential barrier. III. Steady state nonequilibrium flux and reaction rate

Joachim Ankerhold, Hermann Grabert

Published Feb 1, 1997• 9 citations
Publisher

Abstract

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 ${\mathrm{T}}_{\mathrm{c}}$ 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 ${\mathrm{T}}_{\mathrm{c}}$. 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 ${\mathrm{T}}_{\mathrm{c}}$.

DOI
10.1103/physreve.55.1355

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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{\mathrm{T}}_{\mathrm{c}}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{\mathrm{T}}_{\mathrm{c}}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{\mathrm{T}}_{\mathrm{c}}Tc​.
We study the real time dynamics of a quantum system with a potential barrier coupled to a heat-bath environment. The time evolution of the density matrix starting from a general initial state is evaluated explicitly in the semiclassical approximation. As the temperature is decreased below a critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​, large quantum fluctuations render the harmonic approximation near the barrier top insufficient and a caustic arises. As a consequence, anharmonicities of the potential become essential, even in the close vicinity of the barrier top, and the semiclassical density matrix has to be evaluated beyond the Gaussian approximation. The range of validity with respect to temperature and time of the semiclassical analytical results is discussed in detail. We illustrate the outcome of the theory with an explicit example.
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 quantum-mechanical version of the Kramers turnover problem is considered. The multidimensional character of the problem is taken into account via transformation to normal modes. This eliminates the coupling to the bath near the barrier top allowing the use of a simple harmonic transmission coefficient for the barrier dynamics. The well dynamics is described by a continuum form of a master equation for the energy in the unstable normal mode. Within first-order perturbation theory, the equations of motion for the stable normal modes have the form of a forced oscillator. The transition probability kernel is found using the known solution for the quantum forced oscillator problem. An expression for the quantum escape rate is derived. It encompasses all previously known limiting results in the thermally activated tunneling regime. The depopulation factor, which accounts for the nonequilibrium energy distribution is evaluated. The quantum transition probability kernel is broader than the classical and is skewed towards lower energies. Interplay between these two effects, together with a positive tunneling contribution, determines the relative magnitude of the quantum rate compared to the classical one. The theory is valid for arbitrary dissipation. Its use is illustrated for the case of a cubic potential with Ohmic (Markovian) dissipation.
The time dependent density matrix of a system with potential barrier is studied using path integrals. The characterization of the initial state, which is assumed to be restricted to one side of the barrier, and the time evolution of the density matrix lead to a threefold path integral which is evaluated in the semiclassical limit. The semiclassical trajectories are found to move in the complex coordinate plane and barrier penetration only arises due to fluctuations. Both the form of the semiclassical paths and the relevant fluctuations change significantly as a function of temperature. The semiclassical analysis leads to a detailed picture of barrier penetration in the real time domain and the changeover from thermal activation to quantum tunneling. Deep tunneling is associated with quasizero modes in the fluctuation spectrum about the semiclassical orbits in the long time limit. The connection between this real time description of tunneling and the standard imaginary time instanton approach is established. Specific results are given for a double well potential and an Eckart barrier.
The dissipationless quantum-thermal escape rate from a metastable well is often written as a Boltzmann average of the Hill-Wheeler flux over a continuum of energies. This continuum treatment diverges exponentially at low temperature, because it pairs the quantum partition function of the well with a classical flux integral. Retaining the discrete nature of the quasibound spectrum removes the divergence exactly, and the resulting rate is obtained in closed form on both sides of the crossover temperature T0T_0T0​. Evaluating a uniform Kemble transmission probability on Bohr-Sommerfeld levels further removes the parabolic-barrier error and recovers the WKB decay rate of the lowest resonance at zero temperature. Benchmarked against the resonances of cubic and quintic potentials, obtained by complex scaling, this uniform semiclassical rate stays within 9%9\%9% of the exact one over eleven orders of magnitude. Below T0T_0T0​, the rate rises above its zero-temperature plateau with the level spacing as activation energy, and the plateau ends at a temperature T1T_1T1​ noticeably below T0T_0T0​. In an isolated well, whose levels are not replenished, the thermal rate is only the initial slope of a decay which ceases to be single exponential at T1T_1T1​, while at T0T_0T0​ the initial escape moves from tunneling inside the well to the barrier top. We discuss implications for Josephson junctions, single-electron pumps, and nuclear burning in dense stellar matter.