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Journal of Physics A Mathematical and Theoretical

Time-dependent pointer states and decoherence of the quantized atom-field model in a nonresonant regime

Hoofar Daneshvar, G. W. F. Drake

Published Feb 1, 2012• 1 citation
Publisher

Abstract

We consider the quantized atom-field model and for the regime ; where , and respectively represent the self-Hamiltonians of the environment, the system, and the interaction between the system and the environment. Considering a single-mode quantized field we obtain the time evolution operator for the model. Using our time evolution operator we calculate the time-dependent pointer states of the system and the environment, which are characterized by their property of not entangling with states of another subsystem. We obtain a closed form expression for the off-diagonal 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 where the system initially is not prepared in one of its pointer states, the off-diagonal element of the reduced density matrix of the system decays with a decoherence time that is inversely proportional to the square root of the mass of the bosonic field particles.

DOI
10.1088/1751-8113/45/7/075305

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We consider the quantized atom-field model and for the regime ; where , and respectively represent the self-Hamiltonians of the environment, the system, and the interaction between the system and the environment. Considering a single-mode quantized field we obtain the time evolution operator for the model. Using our time evolution operator we calculate the time-dependent pointer states of the system and the environment, which are characterized by their property of not entangling with states of another subsystem. We obtain a closed form expression for the off-diagonal 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 where the system initially is not prepared in one of its pointer states, the off-diagonal element of the reduced density matrix of the system decays with a decoherence time that is inversely proportional to the square root of the mass of the bosonic field particles.
We consider the quantized atom-field model and for the regime that H^E≪H^S≪H′^\hat{H}_{\cal E}\ll\hat{H}_{\cal S}\ll\hat{H'}H^E​≪H^S​≪H′^ (but H^E≠0\hat{H}_{\cal E}\neq0H^E​=0 and H^S≠0\hat{H}_{\cal S}\neq0H^S​=0); where H^E\hat{H}_{\cal E}H^E​, H^S\hat{H}_{\cal S}H^S​ and H′^\hat{H'}H′^ respectively represent the self Hamiltonians of the environment and the system, and the interaction between the system and the environment. Considering a single-mode quantized field we obtain the time-evolution operator for the 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 states of another subsystem) by assuming an initial state of the environment in the form of a Gaussian package in position space. We 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 will show that for the case that the system initially is not prepared in one of its pointer states, the offdiagonal element of the reduced density matrix of the system will decay with a decoherence time which is inversely proportional to the square root of the mass of the bosonic field particles.
We consider a spin-boson Hamiltonian which is generalized such that the Hamiltonians for the system (H^S\hat{H}_{\cal S}H^S​) and the interaction with the environment (H^int\hat{H}_{\rm int}H^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 nˉ\bar{n}nˉ, 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 nˉ\bar{n}nˉ; while it will experience a much faster decoherence, when the system initially is \emph{not} prepared in one of its initial pointer states.
We present a general analytic method for evaluating the generally time-dependent pointer states of a subsystem, which are defined by their capability not to entangle with the states of another subsystem. We explore the conditions under which the pointer states of the system become independent of time; so that a preferred basis of measurement can be realized. We relate the mathematical conditions for having time-independent pointer states to some classes of possible symmetries in the Hamiltonian of the total composite system. Indeed, our theory would serve as a generalization of the existing theory for determination of the preferred basis of measurement. By exploiting this new theory we can obtain those regimes of the parameter space for a given total Hamiltonian defining our system-environment model for which a preferred basis of measurement can be realized. Moreover, we can predict the corresponding preferred basis of measurement for each regime. We can also obtain the time-dependent pointer states of the system and the environment in most of the other regimes where the pointer states of the system are time-dependent and a preferred basis of measurement cannot be realized at all. This ability to obtain time-dependent pointer states is specifically important in decoherence studies; as these pointer states, although they evolve with time and cannot represent the preferred basis of measurement, they correspond to those initial conditions for the state of the system and the environment for which we can have longer decoherence times. In the next step, we consider a spin-boson Hamiltonian which is generalized such that the Hamiltonians for the system and the interaction with the environment do not commute with each other. Considering a single-mode quantized field in exact resonance with the tunneling matrix element of the system, we calculate the time-dependent pointer states of the system and the environment for the case that the environment initially is prepared in the coherent state. 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 will 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 sinusoidal function with a slow decaying envelop which is characterized by a decay time proportional to n-bar.
We present a comprehensive analytical and numerical study of time-dependent decoherence in open quantum systems, focusing on a single-qubit model with time-varying Lindblad dissipation rates. Employing a master equation framework, we derive exact solutions for the evolution of off-diagonal density matrix elements and characterize the emergence of robust pointer states. Our findings reveal an initial rapid coherence decay followed by a gradual slowdown, mirroring experimental trends observed in superconducting qubits influenced by quasiparticle dynamics and in photonic cavities subject to fluctuating environments. By incorporating real experimental parameters and validating the model through quantum simulations using QuTiP, we provide a robust quantitative tool for analyzing decoherence in quantum technologies. This extended model addresses limitations in static-rate approximations, offering insights into non-Markovian effects without resolving the measurement problem, and suggests extensions to multi-qubit systems.
We report and give an alternative derivation of some results on a model for a quantum measurement studied in [1]. The measured microscopic system is coupled to the position of a macroscopic pointer, which itself interacts with its environment via its momentum. The entanglement between the system and the pointer produced by their mutual interaction is simultaneous with the decoherence of distinct pointer readings resulting from leakage of information to the environment. After a discussion on the various time scales in the model we calculate the matrix elements of the system-pointer density operator between eigenstates of the measured observable with distinct eigenvalues. In general, the decay with time of these coherences is neither exponential nor gaussian. We determine the decoherence (decay) time in terms of the strength of the system-pointer and pointer-environment couplings. This decoherence time does not depend upon the details of the pointer-bath coupling as soon as it is smaller than the bath correlation time (non-Markov regime). In contrast, in the Markov regime it depends strongly on whether this coupling is Ohmic or super-Ohmic.