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

Dissipative quantum systems with a potential barrier. II. Dynamics near the barrier top

Joachim Ankerhold, Hermann Grabert

Published Nov 1, 1995• 13 citations
Publisher

Abstract

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 ${\mathit{T}}_{\mathit{c}}$, 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.

DOI
10.1103/physreve.52.4704

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•Jan 1, 2007•1 citation
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.
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 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 nonlinear barrier potential with bound states is presented. Theequilibrium density matrix of a particle moving at temperature T in thisnonlinear barrier potential field is determined. The exact density matrix iscompared with the result of the path integral approach in the semiclassicalapproximation is found to be sufficient at high temperatures while at lowtemperatures the fluctuation paths may have a caustic depending ontemperature and end points. Near the caustics, the divergence of the simplesemicalssical approximation of the density matrix is removed by a nonlinearfluctuation potential. For opaque barriers, the improved semiclassicalapproximation is compulsory required.
We present a barrier potential with bound states that is exactly solvable and determine the eigenfunctions and eigenvalues of the Hamiltonian. The equilibrium density matrix of a particle moving at temperature T in this nonlinear barrier potential field is determined. The exact density matrix is compared with the result of the path integral approach in the semiclassical approximation. For opaque barriers the simple semiclassical approximation is found to be sufficient at high temperatures while at low temperatures the fluctuation paths may have a caustic depending on temperature and endpoints. Near the caustics the divergence of the simple semiclassical approximation of the density matrix is removed by a nonlinear fluctuation potential. For opaque barriers the improved semiclassical approximation is again in agreement with the exact result. In particular, bound states and the form of resonance states are described accurately by the semiclassical approach.
In this PhD thesis we have been concerned with dissipative and decoherence processes which take place specifically for quantum systems in a dissipative bistable potential. This potential describes a Hamiltonian system whose spectrum is composed by a ladder of levels, grouped as doublets under the potential barrier and being almost equally spaced above it. The system interacts with a thermal environment, fully characterized by the so-called bath spectral density. The dynamical quantity we looked at is the so-called reduced density matrix (RDM), obtained by tracing out the full density matrix the environmental degrees of freedom. The matrix elements for the RDM are exactly evaluated by means of the Feynman-Vernon real-time path-integral technique. Depending on which regime of parameters we are interested in, i.e. temperature and coupling strength with the reservoir, a few or several energy levels contribute to the dynamics. In particular, a two-level system (TLS) is first analyzed, corresponding to the regime where only the lowest doublet is energetically accessible. We discussed in the beginning the case of a TLS interacting with Ohmic and super-Ohmic baths and then the situation in which a symmetric TLS is assumed to be coupled with an external detector, like a dc-SQUID, the latter being coupled to an Ohmic environment. This model provides a better description of a real qubit, as frequently discussed in various experiments. In the former case, we developed a theory called Weakly-Interacting Blip Approximation (WIBA), which is nonperturbative in the coupling strength as well as in the tunneling frequency and gives reliable results over a wide range of temperatures and coupling strength to the thermal environment. In the latter case, the dynamics was evaluated upon starting from the Non-Interacting Blip Approximation (NIBA) and approximating the NIBA kernels up to first order in the detector-bath coupling strength. Analytical results, valid also at finite detuning, were obtained and we showed that our theory correctly reproduces the onset of multiple oscillation frequencies in the dynamics due to the entanglement between TLS and detector. Finally, we discussed the more general case of an M-level system coupled with an Ohmic environment. This situation describes the case of a bistable potential where the temperature is high enough that higher-lying energy levels are being populated and the system cannot be reduced to a simple TLS. Many are the experimental situations which can be described by our theory. We developed a novel approximation called Weakly-Interacting VR-Blip Approximation (VR-WIBA) for the dynamics in a driven dissipative bistable potential. In the regimes of low to moderate temperatures and coupling strengths, we exploited a separation of time scales between tunneling and vibrational relaxation events and we could derive analytical kernels for the intra- and inter-well dynamics.