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arXiv:1007.4755 · Physical Review A

Generalized uncertainty relations and entanglement dynamics in quantum Brownian motion models

Charis Anastopoulos, S. Kechribaris, D. Mylonas

Published Oct 29, 2010• 11 citations
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Abstract

We study entanglement dynamics in quantum Brownian motion (QBM) models. Our main tool is the Wigner function propagator. Time evolution in the Wigner picture is physically intuitive and it leads to a simple derivation of a master equation for any number of system harmonic oscillators and spectral density of the environment. It also provides generalized uncertainty relations, valid for any initial state, that allow a characterization of the environment in terms of the modifications it causes to the system's dynamics. In particular, the uncertainty relations are very informative about the entanglement dynamics of Gaussian states, and to a lesser extent for other families of states. For concreteness, we apply these techniques to a bipartite QBM model, describing the processes of entanglement creation, disentanglement, and decoherence at all temperatures and time scales.

DOI
10.1103/physreva.82.042119

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We study entanglement dynamics in quantum Brownian motion (QBM) models. Our main tool is the Wigner function propagator. Time evolution in the Wigner picture is physically intuitive and it leads to a simple derivation of a master equation for any number of system harmonic oscillators and spectral density of the environment. It also provides generalized uncertainty relations, valid for any initial state, that allow a characterization of the environment in terms of the modifications it causes to the system's dynamics. In particular, the uncertainty relations are very informative about the entanglement dynamics of Gaussian states, and to a lesser extent for other families of states. For concreteness, we apply these techniques to a bipartite QBM model, describing the processes of entanglement creation, disentanglement, and decoherence at all temperatures and time scales.
In this paper we generalize the quantum Brownian motion (QBM) to include the momentum-dependent system-environment couplings. The resulting Hamiltonian in the particle number representation is given by ${H}_{t}ot=\ensuremath{\hbar}{\ensuremath{\omega}}_{\mathrm{S}}{a}^{\ifmmode\dagger\else\textdagger\fi{}}a+{\ensuremath{\sum}}_{k}\ensuremath{\hbar}{\ensuremath{\omega}}_{k}{b}_{k}^{\ifmmode\dagger\else\textdagger\fi{}}{b}_{k}+{\ensuremath{\sum}}_{k}\ensuremath{\hbar}({V}_{k}{a}^{\ifmmode\dagger\else\textdagger\fi{}}{b}_{k}+{V}_{k}^{*}{b}_{k}^{\ifmmode\dagger\else\textdagger\fi{}}a)+{\ensuremath{\sum}}_{k}\ensuremath{\hbar}({W}_{k}{a}^{\ifmmode\dagger\else\textdagger\fi{}}{b}_{k}^{\ifmmode\dagger\else\textdagger\fi{}}+{W}_{k}^{*}{b}_{k}a)$. The conventional QBM model corresponds to the special case Wk=Vk{W}_{k}={V}_{k}Wk​=Vk​. The generalized QBM can capture more physical applications because the single-particle transition and the two-particle pair production between the system and the environment represent two very different physical processes that usually do not have the same coupling strengths. We discuss the physical realizations of the generalized QBM in different physical systems, and derive its exact master equation for both the initial decoupled states and initial correlated states. The Hu-Paz-Zhang master equation of the conventional QBM model is reproduced as a special case. We find that the renormalized system Hamiltonian after tracing out all the environmental states induced naturally a momentum-dependent potential, which shows the consistency of including the momentum-dependent coupling in the QBM Hamiltonian. In the Hu-Paz-Zhang master equation, such a renormalized potential is misplaced so that the correct renormalization Hamiltonian has not been found. With the exact master equation for both the initial decoupled and initial correlated states, the issues about the initial jolt are also reexamined. We find that the so-called ``initial jolt,'' which has been thought to be an artificial effect due to the use of the initial decoupled system-environment states, has nothing to do with the initial decoupled state. The exact master equation for the generalized QBM also has potential applications to photonics quantum computing.
The dynamical aspects of quantum Brownian motion at low temperatures are investigated. Exact calculations of quantum entanglement among two Brownian oscillators are given without invoking the Born-Markov or Born approximation widely used for the study of open systems. Nonlocality due to finite oscillator separation is studied. Our approach is applicable for arbitrary time scale, in particular suitable to probe the short-time regime at cold temperatures where many experiments on quantum information processing are performed. We study separability criteria based on the uncertainty relation, negativity, and entanglement of formation. We found a crossover behavior in a disentanglement process between quantum- and thermal-fluctuation-dominated regimes. The fluctuation-dissipation theorem is violated for two Brownian particles interacting with a common environment. The deviation from the relation originates in the interaction mediated by the environment, which drives two particles into a steady oscillatory state, preventing the system from thermalizing. Consequently there is a residual entanglement at low temperatures for arbitrary coupling strength. This entanglement is generated from the environment nonperturbatively even when two modes are not entangled initially. When the distance between two oscillators is varied, competition between entanglement induced from the environment and modified decoherence due to finite separation leads to the characteristic distance where entanglement is minimized.
We solve the model of N quantum Brownian oscillators linearly coupled to an environment of quantum oscillators at finite temperature, with no extra assumptions about the structure of the system-environment coupling. Using a compact phase-space formalism, we give a rather quick and direct derivation of the master equation and its solutions for general spectral functions and arbitrary temperatures. Since our framework is intrinsically nonperturbative, we are able to analyze the entanglement dynamics of two oscillators coupled to a common scalar field in previously unexplored regimes, such as off resonance and strong coupling.
Quantum decoherence is a proposed mechanism for the emergence of classical physics from the quantum mechanics. It has been developed extensively in recent years, but is sufficiently technically complicated to discourage widespread understanding. In this paper we provide a gentle introduction to quantum decoherence. We introduce state operators and their density matrix representations to describe composite systems, such as an experiment and its environment. We illustrate how the loss of information about a subsystem can cause a quantum system to appear classical. We first analyze a discrete example of phase randomization, then a Bell state, and finally a continuous system. In the final case we provide an accessible derivation of a major early result of decoherence theory, the master equation of quantum Brownian motion. We conclude by applying the master equation to the decoherence of a simple harmonic oscillator, with results reminiscent of our earlier discrete examples.
For the purpose of understanding the quantum behavior such as quantum decoherence, fluctuations, dissipation, entanglement and teleportation of a mesoscopic or macroscopic object interacting with a general environment, we derive here a set of exact master equations for the reduced density matrix of N interacting harmonic oscillators in a heat bath with arbitrary spectral density and temperature. For the case N = 2, we also give an influence functional treatment of this model and provide explicit expressions for the evolution operator of the reduced density matrix which are useful for the study of decoherence and disentanglement issues. We show a simple application of this master equation by examining the decoherence and disentanglement of two harmonic oscillators due to their interaction with a common environment under Markovian approximation. Two classes of problems of interest to us which these equations can be usefully applied to are that of the quantum dynamics of nanoelectromechanical oscillators and the entanglement evolution of multi-partite macroscopic states such as quantum superposition of mirrors in a high Q cavity. To address a key conceptual issue for macroscopic quantum phenomena we examine the conditions for an assumption often implicitly made in these studies to be valid, namely, that the quantum behavior of a macroscopic object in an environment can be accurately represented by only treating the dynamics of its center-of-mass variable. We also mention how these results can be used to calculate the uncertainty principle governing a macroscopic object at finite temperature.