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arXiv:2609.21625

Discrete semiclassical theory of quantum-thermal escape

Luca Salasnich, Cesare Vianello

quant-phcond-mat.othercond-mat.stat-mechSubmitted Sep 18, 2026• Announced Sep 21, 2026• 0 citations• Revised Oct 21, 2026 (v2)
v1 · Sep 18, 2026v2 · Oct 21, 2026
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Abstract

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 $T_0$. 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 $9\%$ of the exact one over eleven orders of magnitude. Below $T_0$, the rate rises above its zero-temperature plateau with the level spacing as activation energy, and the plateau ends at a temperature $T_1$ noticeably below $T_0$. 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 $T_1$, while at $T_0$ 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.

Comments
14 pages, 4 figures

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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.
Thermal escape out of a metastable well is considered in the weak friction regime, where the bottleneck for decay is energy diffusion, and at lower temperatures, where quantum tunneling becomes relevant. Within a systematic semiclassical formalism an extension of the classical diffusion equation is derived starting from a quantum mechanical master equation. In contrast to previous approaches finite barrier transmission also affects transition probabilities. The decay rate is obtained from the stationary nonequilibrium solution and captures the intimate interplay between thermal and quantum fluctuations above the crossover to the deep quantum regime.
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​.
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 problem of metastability due to quantum and thermal fluctuations in a weakly damped system is treated within Feynman's influence-functional theory. It is shown that quite generally the nonequilibrium density matrix may be written as a single path integral. For a cubic metastable potential it has the form of a Feynman propagator with a complex Lagrangian containing second-order derivatives in time and terms noninvariant under time reversal. Dissipative tunneling and thermal activation emerge together as different solutions of the associated Lagrange equations. The low-temperature correction to the exponent of the tunneling rate behaves as Tn+1{T}^{n+1}Tn+1 for a spectrum of bath excitations vanishing as ${\ensuremath{\omega}}^{n}$. It is shown, moreover, how this result may be understood in terms of environment-assisted tunneling. Thus the low-temperature behavior of the tunneling rate provides a method to measure the dissipation mechanism in the quantum regime.
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.
When a Josephson junction in the zero-voltage state is current-biased below its critical current, the rate that it escapes to the finite-voltage state depends on the quantum state of the junction. By employing a slow current sweep, it is possible to observe experimentally the emptying of the thermally populated first excited level as a well-defined feature in the escape rate. This feature provides a simple method of determining the junction's energy relaxation time T1{T}_{1}T1​, a key parameter for evaluating its utility for quantum computation. We discuss the temperature regime where this effect is readily observable and describe how the emptying depends directly on the relaxation time. Our model of the junction dynamics agrees well with the measured escape rate of a $10\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}\ifmmode\times\else\texttimes\fi{}10\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}$ $\mathrm{Nb}\text{\ensuremath{-}}\mathrm{AlOx}\text{\ensuremath{-}}\mathrm{Nb}$ device in the 25 to 300mK300\phantom{\rule{0.3em}{0ex}}\mathrm{mK}300mK temperature range, yielding ${T}_{1}\ensuremath{\approx}4\phantom{\rule{0.3em}{0ex}}\mathrm{ns}$.