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Progress of Theoretical Physics

Quantum Langevin Dynamics of a Two-State System Coupled to a Bosonic Bath

T. Tsuzuki

Published Apr 1, 1989• 16 citations
Publisher

Abstract

It is pointed out that the simultaneous inclusion of the reaction field in addition to the direct field of the bath is essentially important to describe the dynamics of the two-state system coupled to a bosonic bath. This is formulated in the framework of the quantum Langevin equation where the direct field of the bath is treated as the quantum mechanical random force. We study in detail the quantum-mechanical and statistical-mechanical notions of the reaction field as well as the direct field in the Ohm-Drude case of the spectral intensity of the bath. It is shown that the non-interacting blip approximation of Leggett et al. is equivalent to the approximation of the complete neglect of the reaction field. We propose a fundamental equation of the quantum Langevin dynamics, accompanied by a brief discussion about the feature of the equation.

DOI
10.1143/ptp.81.770

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It is pointed out that the simultaneous inclusion of the reaction field in addition to the direct field of the bath is essentially important to describe the dynamics of the two-state system coupled to a bosonic bath. This is formulated in the framework of the quantum Langevin equation where the direct field of the bath is treated as the quantum mechanical random force. We study in detail the quantum-mechanical and statistical-mechanical notions of the reaction field as well as the direct field in the Ohm-Drude case of the spectral intensity of the bath. It is shown that the non-interacting blip approximation of Leggett et al. is equivalent to the approximation of the complete neglect of the reaction field. We propose a fundamental equation of the quantum Langevin dynamics, accompanied by a brief discussion about the feature of the equation.
The quantum Langevin equation of a two-state system coupled to a bosonic bath is studied on the basis of the bare boson representation. We can realize the role of the reaction field more explicitly in this case than in the case of displaced boson representation studied in I. The characteristic frequency of variable which mediates the interaction between the slow variable of the two-state system and the bath is determined quantum-mechanically by the commutation relation whose non-commutability is governed by the reaction field.
The spin-boson model, describing a two-level system coupled to a bosonic bath, is extensively studied as a paradigmatic dissipative quantum system, exhibiting rich dynamical behavior and even a localization transition in the strong coupling regime. Here, we additionally consider dephasing as a source of Markovian dissipation on top of the non-Markovian dynamics due to an Ohmic bath, and investigate the dynamics of the spin. We show that the characteristic frequency of the spin dynamics, while strongly renormalized by the bosonic bath, changes in a simple fashion (or does not change at all) with dephasing. To obtain these results, we develop an exact nonperturbative method known as the stochastic Schr\"odinger equation, mimicking the Ohmic bath via a stochastic magnetic field combined with the Lindblad quantum master equation due to dephasing, which allows us to numerically compute the dynamics. Furthermore, we derive weak-coupling analytic results utilizing the well-known noninteracting blip approximation. Our findings are relevant to quantum simulation of the spin-boson model in the regime of strong coupling in trapped ions and circuit QED architectures among others.
Starting with a quantum Langevin equation describing in the Heisenberg representation a quantum system coupled to a quantum bath, the Markov approximation and, further, the closure approximation are applied to derive a semiclassical Langevin equation for the second-order quantized Hamilton dynamics (QHD) coupled to a classical bath. The expectation values of the system operators are decomposed into products of the first and second moments of the position and momentum operators that incorporate zero-point energy and moderate tunneling effects. The random force and friction as well as the system-bath coupling are decomposed to the lowest classical level. The resulting Langevin equation describing QHD-2 coupled to classical bath is analyzed and applied to free particle, harmonic oscillator, and the Morse potential representing the OH stretch of the SPC-flexible water model.
We obtain the quantum Langevin equation (QLE) of a charged quantum particle moving in a harmonic potential in the presence of a uniform external magnetic field and linearly coupled to a quantum heat bath through momentum variables. The bath is modeled as a collection of independent quantum harmonic oscillators. The QLE involves a random force which does not depend on the magnetic field, and a quantum-generalized classical Lorentz force. These features are also present in the QLE for the case of particle-bath coupling through coordinate variables. However, significant differences are also observed. For example, the mean force in the QLE is characterized by a memory function that depends explicitly on the magnetic field. The random force has a modified form with correlation and commutator different from those in the case of coordinate-coordinate coupling. Moreover, the coupling constants, in addition to appearing in the random force and in the mean force, also renormalize the inertial term and the harmonic potential term in the QLE.
The quantum dynamics of a particle coupled to the dissipative degrees of freedom of a generalized spin bath in the presence of an external force field is presented. Based on Holstein-Primakoff transformation, which sets up a mapping between boson and spin operators, we show that the spin-1/2 bath and the harmonic bath can be realized as two special limits of the generalized bath. In the large spin limit, the Hamiltonian reduces to well-known Zwanzig Hamiltonian. Making use of spin coherent states and canonical thermal distributions of the associated c numbers, we derive and explore the quantum Langevin equation and the properties of the spin and harmonic baths in the context of quantum dissipation of the system.
A temperature-dependent Schrödinger equation is proposed for the study of quantum dynamics in systems which interact with an external bath. The derivation is based on a time-dependent self consistent field (TDSCF) approximation for a system-bath wave function. As previously shown by Miller, this approximation leads to a quantum mechanical analog to the classical generalized Langevin equation. By replacing the time-evolution of the quantum mechanical bath observables with the corresponding classical trajectories, the fluctuating force and the nonlinear friction kernel in the Langevin-Schrödinger equation become temperature-dependent where the fluctuations intensity is proportional to T. Application of the new equation to the spin-boson model shows agreement with numerically exact path integral calculations for T>0. We relate the success of the TDSCF approximation in this case to the bath-induced noise which diminishes the importance of quantum mechanical system-bath correlations.