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Physical Review Letters

Dynamical Hysteresis in Bistable Quantum Systems

Michael Thorwart, Peter Jung

Published Mar 31, 1997• 45 citations
Publisher

Abstract

We are studying bistable quantum systems coupled to a thermal environment in the presence of an external controlling force. The impact of thermal and quantum effects on the switching hysteresis is studied by evaluating a time-dependent real-time double path integral with the recently developed iterative tensor multiplication scheme for the quasiadiabatic path integral propagator method for temperatures ranging from well above to well below quantum-classical crossover temperature. The switching dynamics and its intimate connection to quantum stochastic resonance is studied here for the first time without limitation to a two-state description and/or small external forces.

DOI
10.1103/physrevlett.78.2503

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We are studying bistable quantum systems coupled to a thermal environment in the presence of an external controlling force. The impact of thermal and quantum effects on the switching hysteresis is studied by evaluating a time-dependent real-time double path integral with the recently developed iterative tensor multiplication scheme for the quasiadiabatic path integral propagator method for temperatures ranging from well above to well below quantum-classical crossover temperature. The switching dynamics and its intimate connection to quantum stochastic resonance is studied here for the first time without limitation to a two-state description and/or small external forces.
We are studying bistable quantum systems coupled to a thermal environment and driven by a strong external force. Thermal and quantum effects on the switching hysteresis are studied numerically by evaluating a time-dependent real-time double path-integral with the recently developed iterative tensor multiplication scheme for the Quasi-adiabatic path-integral propagator method. For temperatures below the quantum-classical crossover temperature, the response of the system exhibits steps for slow external driving. We show that these steps are due to dissipative resonant tunneling and appear in the vicinity of avoided level-crossings of the spectrum of the undamped Hamiltonian. We discuss possible relations to the recently discovered hysteresis steps in organic high-spin molecules.
The dissipative dynamics of a quantum bistable system coupled to a Ohmic heat bath is investigated beyond the spin-boson approximation. Within the path-integral approach to quantum dissipation, we propose an approximation scheme which exploits the separation of time scales between intra- and interwell (tunneling) dynamics. The resulting generalized master equation for the populations in a space localized basis enables us to investigate a wide range of temperatures and system-environment coupling strengths. A phase diagram in the coupling-temperature space is provided to give a comprehensive account of the different dynamical regimes.
The stabilizing effect of quantum fluctuations on the escape process and the relaxation dynamics from a quantum metastable state are investigated. Specifically, the quantum dynamics of a multilevel bistable system coupled to a bosonic Ohmic thermal bath in strong dissipation regime is analyzed. The study is performed by a non-perturbative method based on the real-time path integral approach of the Feynman-Vernon influence functional. We consider a strongly asymmetric double well potential with and without a monochromatic external driving, and with an out-of-equilibrium initial condition. In the absence of driving we observe a nonmonotonic behavior of the escape time from the metastable region, as a function both of the system-bath coupling coefficient and the temperature. This indicates a stabilizing effect of the quantum fluctuations. In the presence of driving our findings indicate that, as the coupling coefficient γ increases, the escape time, initially controlled by the external driving, shows resonant peaks and dips, becoming frequency-independent for higher γ values. Moreover, the escape time from the metastable state displays a nonmonotonic behavior as a function of the temperature, the frequency of the driving, and the thermal-bath coupling, which indicates the presence of a quantum noise enhanced stability phenomenon. Finally, we investigate the role of different spectral densities, both in sub-Ohmic and super-Ohmic dissipation regime and for different cutoff frequencies, on the relaxation dynamics from the quantum metastable state. The results obtained indicate that, in the crossover dynamical regime characterized by damped intrawell oscillations and incoherent tunneling, the spectral properties of the thermal bath influence non-trivially the short time behavior and the time scales of the relaxation dynamics from the metastable state.
For common condensed phase problems described by a low-dimensional system coupled to a harmonic bath, Feynman’s path integral formulation of time-dependent quantum mechanics leads to expressions for the reduced density matrix of the system where the effects of the harmonic environment enter through an influence functional that is nonlocal in time. In a recent Letter [Chem. Phys. Lett. 221, 482 (1994)], we demonstrated that the range of the nonlocal interactions is finite even at zero temperature, such that the nonlocal kernel extends over only a few time steps if the path integral is expressed in terms of accurate quasiadiabatic propagators. This feature arises from disruption of phase coherence in macroscopic environments and leads to Markovian dynamics for an augmented reduced density tensor, permitting iterative time evolution schemes. In the present paper we analyze the structure and properties of the relevant tensor propagator. Specifically, we show that the tensor multiplication scheme rigorously conserves the trace of the reduced density matrix, and that in cases of short-range nonlocality it leads to Redfield-type equations which are correct to all orders in perturbation theory and which take into account memory effects. We also argue that a simple eigenvector analysis reveals (without actual iteration) the nature of the dynamics and of the equilibrium state, and directly yields quantum reaction or relaxation rates.
We describe an efficient path integral scheme for calculating the propagator of an arbitrary quantum system, as well as that of a stochastic system in special cases where the Fokker–Planck equation obeys strict detailed balance. The basic idea is to split the respective Hamiltonian into two exactly solvable parts and then to employ a symmetric decomposition of the time evolution operator, which is exact up to a high order in the time step. The resulting single step propagator allows rather large time steps in a path integral and leads to convergence with fewer time slices. Because it involves no system-specific reference system, the algorithm is amenable to all known numerical schemes available for evaluating quantum path integrals. In this way one obtains a highly accurate method, which is simultaneously fast, stable, and computationally simple. Numerical applications to the real time quantum dynamics in a double well and to the stochastic dynamics of a bistable system coupled to a harmonic mode show our method to be superior over the approach developed by the Makri group in their quasiadiabatic propagator representation, to say nothing about the propagation scheme based on the standard Trotter splitting.
We study the dissipative quantum dynamics and the asymptotic behavior of a particle in a bistable potential interacting with a sub-Ohmic broadband environment. The reduced dynamics, in the intermediate to strong dissipation regime, is obtained beyond the two-level system approximation by using a real-time path integral approach. We find a crossover dynamic regime with damped intra-well oscillations and incoherent tunneling and a completely incoherent regime at strong damping. Moreover, a nonmonotonic behavior of the left/right well population difference is found as a function of the damping strength.