Time-dependent pointer states of the generalized spin-boson model and consequences regarding the decoherence of the central system

TL;DR

This paper analyzes a generalized spin-boson model with non-commuting Hamiltonians, deriving time-dependent pointer states and examining how initial states influence decoherence, with results applicable over timescales proportional to the environment's average boson number.

Contribution

It provides an exact solution for pointer states in a generalized spin-boson model with non-commuting Hamiltonians, highlighting their time evolution and decoherence behavior.

Findings

Pointer states are valid over times proportional to the average boson number.

Decoherence time depends on initial state preparation, with slower decay when starting in a pointer state.

Off-diagonal density matrix elements exhibit sinusoidal decay with a slow envelope.

Abstract

We consider a spin-boson Hamiltonian which is generalized such that the Hamiltonians for the system ($\hat{H}_{\cal S}$) and the interaction with the environment ($\hat{H}_{\rm int}$) do not commute with each other. Considering a single-mode quantized field in exact resonance with the tunneling matrix element of the system, we obtain the time-evolution operator for our model. Using our time-evolution operator we calculate the time-dependent pointer states of the system and the environment (which are characterized by their ability not to entangle with each other) for the case that the environment initially is prepared in the coherent state. We show that our solution for the pointer states of the system and the environment is valid over a length of time which is proportional to $\bar{n}$, the average number of bosons in the environment. We also obtain a closed form for the offdiagonal element of the reduced density matrix of the system and study the decoherence of the central system in our model. We show that for the case that the system initially is prepared in one of its pointer states, the offdiagonal element of the reduced density matrix of the system will be a \emph{sinusoidal function} with a slow decaying envelope which is characterized by a decay time proportional to $\bar{n}$; while it will experience a much faster decoherence, when the system initially is \emph{not} prepared in one of its initial pointer states.