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Journal of the Physical Society of Japan

Quantization of Kramers' Rate Theory

Shunya Ishioka

Published Feb 1, 1980• 11 citations
Publisher

Abstract

With use of the Wigner representation, the Kramers equation for a Brownian particle in a potential field of external force is extended to the quantum mechanical case. As a simple application of the equation, the rate of jump over a potential barrier is estimated to the lowest order approximation in \(\hbar\). The rate increases due to the quantum mechanical effect by the factor \(1{+}(\beta\hbar\omega)^{2}\omega^{2}/6(\eta^{2}{+}\omega^{2}\)), where ω, η and β are the angular frequency of the particle in the potential well, viscosity and the inverse of temperature, respectively.

DOI
10.1143/jpsj.48.367

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With use of the Wigner representation, the Kramers equation for a Brownian particle in a potential field of external force is extended to the quantum mechanical case. As a simple application of the equation, the rate of jump over a potential barrier is estimated to the lowest order approximation in (\hbar). The rate increases due to the quantum mechanical effect by the factor (1{+}(\beta\hbar\omega)^{2}\omega^{2}/6(\eta^{2}{+}\omega^{2})), where ω, η and β are the angular frequency of the particle in the potential well, viscosity and the inverse of temperature, respectively.
The stochastic quantization problem of Brownian particle motion that follows Kramers equation is investigated in one-dimension using invariant related unitary transformation method. The explicit Hamiltonians for Langevin equation are constructed separately for the systems that have several different types of the external forces. In the case which does not have external forces, the spectrum of Schrödinger solution is continuous since the particles are unbound in a finite region. However, when a external force is given by the (time-dependent) harmonic potential, the particles are bound inside the potential, leading the spectrum of quantum wave functions to be discrete. The bound–unbound transitions are important since it is connected to metal–insulator transitions that can be achievable for certain compound semiconductors by increasing the doping concentrations at low temperatures. Additionally, when the Hamiltonian involves a higher order term of x as well as harmonic potential term, we have executed the ordinary perturbation expansion in order to obtain the approximate quantum solutions.
We have presented a simple approach to quantum theory of Brownian motion and barrier crossing dynamics. Based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution of quantum-mechanical mean values of their co-ordinates and momenta we have derived a c number generalized quantum Langevin equation. The approach allows us to implement the method of classical non-Markovian Brownian motion to realize an exact generalized non-Markovian quantum Kramers' equation. The equation is valid for arbitrary temperature and friction. We have solved this equation in the spatial diffusion-limited regime to derive quantum Kramers' rate of barrier crossing and analyze its variation as a function of the temperature and friction. While almost all the earlier theories rest on quasiprobability distribution functions (e.g., Wigner function) and path integral methods, the present work is based on true probability distribution functions and is independent of path integral techniques. The theory is a natural extension of the classical theory to quantum domain and provides a unified description of thermally activated processes and tunneling.
Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.
We make a brief review of the Kramers escape rate theory for the probabilistic motion of a particle in a potential well U(x), and under the influence of classical fluctuation forces. The Kramers theory is extended in order to take into account the action of the thermal and zero-point random electromagnetic fields on a charged particle. The result is physically relevant because we get a non null escape rate over the potential barrier at low temperatures (T -> 0). It is found that, even if the mean energy is much smaller than the barrier height, the classical particle can escape from the potential well due to the action of the zero-point fluctuating fields. These stochastic effects can be used to give a classical interpretation to some quantum tunneling phenomena. Relevant experimental data are used to illustrate the theoretical results.
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 quantum Brownian motion of a particle in an external potential V(x) is treated using the master equation for the Wigner distribution function W(x, p, t) in phase space (x, p). A heuristic method of determination of diffusion coefficients in the master equation is proposed. The time evolution equation so obtained contains explicit quantum correction terms up to o(4) and in the classical limit, → 0, reduces to the Klein–Kramers equation. For a quantum oscillator, the method yields an evolution equation for W(x, p, t) coinciding with that of Agarwal (1971 Phys. Rev. A 4 739). In the non-inertial regime, by applying the Brinkman expansion of the momentum distribution in Weber functions (Brinkman 1956 Physica 22 29), the corresponding semiclassical Smoluchowski equation is derived.