Quantum effects in barrier dynamics (original) (raw)
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Physical review, 1997
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 T 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 ͑1995͔͒ on the quantum generalization of the Kramers flux state to the region about T 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 T c. ͓S1063-651X͑97͒01002-7͔
Dissipative quantum systems with a potential barrier. II. Dynamics near the barrier top
Physical review, 1995
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 T, large quantum Quctuations render the harmonic approximation near the barrier top insufBcient 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.
Quantum versus Classical Dynamics in a driven barrier: the role of kinematic effects
Arxiv preprint arXiv: …, 2010
We study the dynamics of the classical and quantum mechanical scattering of a wave packet from an oscillating barrier. Our main focus is on the dependence of the transmission coefficient on the initial energy of the wave packet for a wide range of oscillation frequencies. The behavior of the quantum transmission coefficient is affected by tunneling phenomena, resonances and kinematic effects emanating from the time dependence of the potential. We show that when kinematic effects dominate (mainly in intermediate frequencies), classical mechanics provides very good approximation of quantum results. In that frequency region, the classical and quantum transmission coefficients are in optimal agreement. Moreover, the transmission threshold, i.e. the energy above which the transmission coefficient becomes larger than a specific small threshold value, is found to exhibit a minimum. We also consider the form of the transmitted wave packet and we find that for low values of the frequency the incoming classical and quantum wave packet can be split into a train of well separated coherent pulses, a phenomenon which admits purely classical kinematic interpretation.
Dissipative quantum systems with a potential barrier: General theory and the parabolic barrier
Physical review, 1995
We study the real time dynamics of a quantum system with a potential barrier coupled to a heat-bath environment. Employing the path integral approach, an evolution equation for the time dependent density matrix is derived. The time evolution is evaluated explicitly near the barrier top in the temperature region where quantum efFects become important. It is shown that there exists a quasistationary state with a constant Qux across the potential barrier. This state generalizes the Kramers Qux solution of the classical Fokker-Planck equation to the quantum regime. In the temperature range explored the quantum Qux state depends only on the parabolic approximation of the anharmonic barrier potential near the top. The parameter range within which the solution is valid is investigated in detail. In particular, by matching the Qux state onto the equilibrium state on one side of the barrier we gain a condition on the minimal damping strength. For very high temperatures this condition reduces to a known result from classical rate theory. Within the specified parameter range the decay rate out of a metastable state is calculated from the Qux solution. The rate is shown to coincide with the result of purely thermodynamic methods. The real time approach presented can be extended to lower temperatures and smaller damping.
Quantum escape kinetics over a fluctuating barrier [J. Chem. Phys. 123, 224104 (2005);]
http://jcp.aip.org/resource/1/jcpsa6/v123/i22/p224104\_s1 . The escape rate of a particle over a fluctuating barrier in a double-well potential exhibits resonance at an optimum value of correlation time of fluctuation. This has been shown to be important in several variants of kinetic model of chemical reactions. We extend the analysis of this phenomenon of resonant activation to quantum domain to show how quantization significantly enhances resonant activation at low temperature due to tunneling. ©2005 American Institute of Physics
Semiclassical density matrix near the top of a potential barrier
Physica A: Statistical Mechanics and its Applications, 1996
Employing the path integral approach, we calculate the semiclassical equilibrium density matrix of a particle moving in a nonlinear potential field for coordinates near the top of a potential barrier. As the temperature is decreased, near a critical temperature T c the harmonic approximation for the fluctuation path integral fails. This is due to a caustic arising at a bifurcation point of the classical paths. We provide a selfconsistent scheme to treat the large quantum fluctuations leading to a nonlinear fluctuation potential. The procedure differs from methods used near caustics of the real time propagator. The semiclassical density matrix is determined explicitly for the case of asymmetric barriers from high temperatures down to temperatures somewhat below T c .
Quantum escape kinetics over a fluctuating barrier
The Journal of Chemical Physics, 2005
The escape rate of a particle over a fluctuating barrier in a double well potential exhibits resonance at an optimum value of correlation time of fluctuation. This has been shown to be important in several variants of kinetic model of chemical reactions. We extend the analysis of this phenomenon of resonant activation to quantum domain to show how quantization significantly enhances resonant activation at low temperature due to tunneling.
Quantum-wave evolution in a step potential barrier
2002
By using an exact solution to the time-dependent Schrödinger equation with a point source initial condition, we investigate both the time and spatial dependence of quantum waves in a step potential barrier. We find that for a source with energy below the barrier height, and for distances larger than the penetration length, the probability density exhibits a forerunner associated with a non-tunneling process, which propagates in space at exactly the semiclassical group velocity. We show that the time of arrival of the maximum of the forerunner at a given fixed position inside the potential is exactly the traversal time, τ. We also show that the spatial evolution of this transient pulse exhibits an invariant behavior under a rescaling process. This analytic property is used to characterize the evolution of the forerunner, and to analyze the role played by the time of arrival, 3^-1/2τ, found recently by Muga and Büttiker [Phys. Rev. A 62, 023808 (2000)].
Approach to quantum Kramers’ equation and barrier crossing dynamics
Physical Review E, 2002
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 ccc-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 temperature and friction. While almost all the earlier theories rest on quasi-probability distribution functions (like Wigner function) and path integral methods, the present work is based on {\it 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 thermal activated processes and tunneling.
2021
In this paper we have presented the mechanism of the barrier crossing dynamics of a Brownian particle which is coupled to a thermal bath in the presence of both time independent and fluctuating magnetic fields. Here the following three aspects are important in addition to the role of the thermal bath on the barrier crossing dynamics. Magnetic field induced coupling may introduce a resonance like effect. Another role of the field is that enhancement of its strength reduces the frequency factor of the barrier crossing rate constant. Finally, the fluctuating magnetic field introduces an induced electric field which activates the Brownian particle to cross the energy barrier. As a result of interplay among these aspects versatile non-monotonic behavior may appear in the variation of the rate constant as a function of the strength of the time independent magnetic field. PACS numbers: 05.40.Jc,05.20.-y,89.70.Cf ∗ Author for correspondence, e-mail:bidhanchandra.bag@visva-bharati.ac.in