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arXiv:cond-mat/9509083

Quasi-calssical approach to two-level systems with dissipation

Mo‐Lin Ge, Lei Wang

Published Sep 14, 1995• 0 citations
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Abstract

The quantum dynamics of two-level systems under classical oscillator heat bath is mapped to the classical one of a charged particle under harmonic oscillator potential plus a magnetic field in a plane. The behavior of eigenstates and tunneling and localization are studied in detail. The broken symmetry condition and Langevin-like dissipative equation of motion are obtained. Some special dynamic features are considered.

DOI
10.48550/arxiv.cond-mat/9509083

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The quantum dynamics of two-level systems under classical oscillator heat bath is mapped to the classical one of a charged particle under harmonic oscillator potential plus a magnetic field in a plane. The behavior of eigenstates and tunneling and localization are studied in detail. The broken symmetry condition and Langevin-like dissipative equation of motion are obtained. Some special dynamic features are considered.
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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.
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We obtain the generalized susceptibility for the motion of a charged oscillator in a harmonic potential well in the presence of a uniform external magnetic field, and linearly coupled to a heat bath, by using the generalized quantum Langevin equation we obtained in an earlier paper. The result is then used to calculate the autocorrelation function of the position operator of the oscillator by the fluctuation-dissipation theorem. The free energy of the oscillator is then obtained in terms of the determinant of the susceptibility matrix. Applications are made to two types of heat baths: the Ohmic case and the case of blackbody radiation. As a special case (no heat bath) of our general formalism, a well-known eigenspectrum result is obtained but in a simple and rather novel fashion.
We present a comprehensive study of the effect of a transverse magnetic field on the macroscopic quantum tunneling of two identical coupled particles with dissipation. Two coupled particles of identical masses but opposite charges in the plane, in the presence of a constant transverse external magnetic field and an external potential interacting with a bath of harmonic oscillators are studied. In this case, the problem cannot be mapped to a one-dimensional problem, it strictly remains two-dimensional. We calculate the exact effective action both for the case of linear coupling to the bath and without a linear coupling to the bath using imaginary time path integral and Leggett's prescription at T=0. In the limit of zero magnetic field we recover a two dimensional version of the result found in \cite{em1} for the case of two identical particles. We find that in the limit of weak and strong dissipation, the effective action reduces to a two dimensional version of the Caldeira-Leggett form in terms of the reduced mass and magnetic field. The case of Ohmic dissipation with the motion of the two particles damped by the Ohmic frictional constant η\etaη is studied in detail.