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The Journal of Chemical Physics

Extended diffusion in a double well potential: Transition from classical to quantum regime

Paul Blaise, Yuri P. Kalmykov, Adina Ceausu–Velcescu

Published Sep 6, 2012• 4 citations
Publisher

Abstract

The transition between the classical and quantum regimes in the diffusion of a particle in a 2-4 double-well potential is treated via the strong collision model in the high-temperature limit. Both the classical and semiclassical position correlation functions, their spectra, and correlation times are evaluated using the memory function formalism. It is shown that even in the high temperature limit, marked classical-quantum transition effects appear in the observables when collisions are rare.

DOI
10.1063/1.4748145

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The transition between the classical and quantum regimes in the diffusion of a particle in a 2-4 double-well potential is treated via the strong collision model in the high-temperature limit. Both the classical and semiclassical position correlation functions, their spectra, and correlation times are evaluated using the memory function formalism. It is shown that even in the high temperature limit, marked classical-quantum transition effects appear in the observables when collisions are rare.
The author studies the barrier-crossing rate Gamma for a particle in a symmetric double well coupled to a sluggish bath; the aim is to justify the quantum simulation procedure used in related work on the diffusion of hydrogen in metals. He uses path-integral methods to discuss the transition between the low-temperature tunnelling and the high-temperature barrier-jumping regimes, and suggests a unifying formula valid for both, which relates Gamma to the probability of finding the centroid of the quantum path at the barrier top. Numerical results for Gamma are presented for the case of the quartic double well.
Abstract The so-called quantum-to-classical transition is studied in the surface diffusion dynamics. This continuous and smooth transition is carried out from the Liouville-von Neumann equation by scaling Planck’s constant. As an application, the Brownian motion of an adsorbate on a flat surface is analyzed within the Caldeira–Leggett master equation formalism for the reduced density matrix and the two extreme time behaviors, the ballistic and diffusive motions. In the ballistic time regime, the intermediate scattering function displays a Gaussian function which is governed by the thermal velocity in the classical regime, whereas in the quantum regime, by the initial spreading velocity of the wave packet. Adsorbates behave as a classical or quantum ideal gas. Finally, in the diffusive regime, and starting from the Chudley–Elliott model, the well-known H-function is calculated for three different surface temperatures in the diffusion of H and D on a Pt(111) surface. The main goal of this analysis is whether one can discriminate the irreversibility coming from tunneling or thermal activation diffusion.
The stationary Wigner functions (WFs) have been calculated for particles evolving in a quartic double-well potential V(x) = ax2/2+bx4/4(a < 0 and b > 0), at temperature T. In the high temperature limit, the results totally agree with those obtained using Wigner's perturbative method of deriving quantum corrections to the classical distribution function. Comparison with the perturbative approach allows one to establish the range of applicability of the latter. For illustration, the second moments of the position and momentum have been calculated for the double-well potential. Furthermore, the time-evolution of the WFs for a state initially located at one of the wells has been also investigated to show the tunneling effect.
The transition-path-time distribution is formalized for quantum systems and applied to a number of examples. Using a symmetrized thermal density, transition times are studied for the free particle, a $\ensuremath{\delta}$-function potential, a square-barrier potential, and symmetric-double-well dynamics at very low temperature. These studies exemplify extreme nonlocality for motion in $\ensuremath{\delta}$-function potentials, vanishing tunneling times for the square-barrier potential, and varying transit times in the symmetric-double-well potential. In all cases, there are regions where the longer the distance traversed, the shorter the mean transit time is. For the thermal density correlation functions studied here, the Hartman effect exemplifies itself through the independence of the transit time on the barrier height. However, due to the thermal distribution, the transit time does depend on the barrier width, initially decreasing with increasing width but then increasing again.
For suitable parameters, the classical Duffing oscillator has a known bistability in its stationary states, with low- and high-amplitude branches. As expected from the analogy with a particle in a double-well potential, transitions between these states become possible either at finite temperature, or in the quantum regime due to tunneling. In this analogy, besides local stability, one can also discuss global stability by comparing the two potential minima. For the Duffing oscillator, the stationary states emerge dynamically so that a priori, a potential-minimum criterion for them does not exist. However, global stability is still relevant, and definable as the state containing the majority population for long times, low temperature, and close to the classical limit. Further, the crossover point is the parameter value at which global stability abruptly changes from one state to the other. For the double-well model, the crossover point is defined by potential-minimum degeneracy. Given that this analogy is so effective in other respects, it is thus striking that for the Duffing oscillator, the crossover point turns out to be non-unique. Rather, none of the three aforementioned limits commute with each other, and the limiting behaviour depends on the order in which they are taken. More generally, as both $\hbar\To0$ and $T\To0$, the ratio ℏω0/kBT\hbarω_0/k_\mathrm{B}Tℏω0​/kB​T continues to be a key parameter and can have any nonnegative value. This points to an apparent conceptual difference between equilibrium and nonequilibrium tunneling. We present numerical evidence by studying the pertinent quantum master equation in the photon-number basis. Independent verification and some further understanding are obtained using a semi-analytical approach in the coherent-state representation.
The discrete time path integral Monte Carlo with a one-particle density matrix approximation is applied to study the quantum phase transition in the coupled double-well chain. To improve the convergence properties, the exact action for a single particle in a double-well potential is used to construct the many-particle action. The algorithm is applied to the interacting quantum double-well chain for which the zero-temperature phase diagram is determined. The quantum phase transition is studied via finite-size scaling, and the critical exponents are shown to be compatible with the classical two-dimensional Ising universality class--not only in the order-disorder limit (deep potential wells), but also in the displacive regime (shallow potential wells).