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arXiv:1111.5531

Entanglement of remote quantum systems by environmental modes

Friedemann Queißer, Thomas Zell, Rochus Klesse

Published Nov 23, 2011• 1 citation
arXivHTMLPDF

Abstract

We investigate the generation of quantum mechanical entanglement of two remote oscillators that are locally coupled to a common bosonic bath. Starting with a Lagrangian formulation of a suitable model, we derive two coupled Quantum Langevin Equations that exactly describe the time evolution of the two local oscillators in presence of the coupling to the bosonic bath. Numerically obtained solutions of the Langevin Equations allow us to study the entanglement generation between the oscillators in terms of the time evolution of the logarithmic negativity. Our results confirm and extend our previously obtained findings, namely that significant entanglement between oscillators embedded in a free bosonic bath can only be achieved if the system are within a microscopic distance. We also consider the case where the bosonic spectral density is substantially modified by imposing boundary conditions on the bath modes. For boundary conditions corresponding to a wave-guide like geometry of the bath we find significantly enlarged entanglement generation. This phenomenon is additionally illustrated within an approximative model that allows for an analytical treatment.

DOI
10.48550/arxiv.1111.5531

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We investigate the generation of quantum mechanical entanglement of two remote oscillators that are locally coupled to a common bosonic bath. Starting with a Lagrangian formulation of a suitable model, we derive two coupled Quantum Langevin Equations that exactly describe the time evolution of the two local oscillators in presence of the coupling to the bosonic bath. Numerically obtained solutions of the Langevin Equations allow us to study the entanglement generation between the oscillators in terms of the time evolution of the logarithmic negativity. Our results confirm and extend our previously obtained findings, namely that significant entanglement between oscillators embedded in a free bosonic bath can only be achieved if the system are within a microscopic distance. We also consider the case where the bosonic spectral density is substantially modified by imposing boundary conditions on the bath modes. For boundary conditions corresponding to a wave-guide like geometry of the bath we find significantly enlarged entanglement generation. This phenomenon is additionally illustrated within an approximative model that allows for an analytical treatment.
The search for methods to create and maintain entanglement has led to the idea of environmentally induced entanglement. Roughly speaking, the usually detrimental effect of coupling a non-interacting bipartite system to an environment is turned into an advantage by using the environment to mediate an indirect interaction, which can result in entanglement of the two parts of the system under certain conditions. Of course, care has to be taken to properly evaluate the conflicting influences of the environment. Only if the indirect interaction overcompensates for the decoherence, entanglement creation can be expected. \n \nIt has been suggested that entanglement creation can be achieved in bosonic heat baths even over finite spatial separations with only a moderate polynomial decay of entanglement with distance. In this work, we look more closely at the distance dependence, for the first time employing an oscillator model that is both exactly solvable and includes dissipation. We numerically prove that entanglement creation is, in fact, extremely distance-sensitive and it is not possible to entangle objects which are further apart than approximately their own size. \n \nAdditionally, we suggest an approach how to mitigate the distance dependence. It comes at the cost of geometrically modifying the bath modes by imposing physical boundary conditions resulting in a gap in the spectrum. This is implemented by placing the system inside of an infinitely long superconducting cavity. An experimental implementation of this could be feasible.
Within a generalized Caldeira-Leggett model, we analyze the conditions under which a bosonic heat bath can entangle two microscopic quantum systems at a distance r. We find that the attainable entanglement is extremely distance-sensitive. Significant entanglement can only be achieved if the systems are within a microscopic distance that is of order of the cutoff wavelength lambda of the system-bath interaction. At larger distances, the maximal entanglement is exponentially suppressed with a decay length of order lambda. We conclude that entanglement generation via a heat bath is not suitable for entangling remote objects.
We describe the evolution of the quantum entanglement of an open system consisting of two bosonic modes interacting with a common thermal environment, described by two different models. The initial state of the system is taken of Gaussian form. In the case of a thermal bath, characterized by temperature and dissipation constant which correspond to an asymptotic Gibbs state of the system, we show that for a zero temperature of the thermal bath an initial entangled Gaussian state remains entangled for all finite times. For an entangled initial squeezed thermal state, the phenomenon of entanglement sudden death takes place and we calculate the survival time of entanglement. For the second model of the environment, corresponding to a non-Gibbs asymptotic state, we study the possibility of generating entanglement. We show that the generation of the entanglement between two uncoupled bosonic modes is possible only for definite values of the temperature and dissipation constant, which characterize the thermal environment.
Abstract We present an exact analytical solution of the Hu–Paz–Zhang master equation in a precise Markovian limit for a system of two harmonically coupled harmonic oscillators interacting with a common thermal bath of harmonic oscillators. The thermal bath is initially considered to be at arbitrary temperatures and characterized by an Ohmic Lorentz–Drude spectral density. In the examined system, couplings between the two harmonic oscillators and the environment ensure a complete decoupling of the center-of-mass and relative degrees of freedom, resulting in undamped dynamics in the relative coordinate. The exact time evolution is used to analyze the system’s entanglement dynamics, quantified through logarithmic negativity and quantum mutual information, while ensuring the positivity of the density operator to confirm the physical validity of the results. We demonstrate that, under certain parameter regimes and initial conditions, the asymptotic dynamics can give rise to periodic entanglement-disentanglement behavior. Furthermore, numerical simulations reveal that for negative values of the direct coupling between the oscillators, which are sufficiently close to a critical lower bound beyond which the system becomes unstable, the system can maintain entanglement across a broad temperature range and for arbitrarily long durations.
We study two continuous variable systems (or two harmonic oscillators) and investigate their entanglement evolution under the influence of non-Markovian thermal environments. The continuous variable systems could be two modes of electromagnetic fields or two nanomechanical oscillators in the quantum domain. We use the quantum open system method to derive the non-Markovian master equations of the reduced density matrix for two different but related models of the continuous variable systems. The two models both consist of two interacting harmonic oscillators. In model A, each of the two oscillators is coupled to its own independent thermal reservoir, while in model B the two oscillators are coupled to a common reservoir. To quantify the degrees of entanglement for bipartite continuous variable systems in Gaussian states, logarithmic negativity is used. We find that the dynamics of the quantum entanglement is sensitive to the initial states, the oscillator-oscillator interaction, the oscillator-environment interaction and the coupling to a common bath or to different, independent baths.
We study the evolution of entanglement for a pair of coupled nonidentical harmonic oscillators in contact with an environment. For both cases of a common bath and of two separate baths for each of the oscillators, a full master equation is provided without rotating-wave approximation. The entanglement dynamics is analyzed as a function of the diversity between the oscillators' frequencies and their positive or negative mutual coupling and also the correlation between the occupation numbers. The singular effect of the resonance condition (identical oscillators) and its relationship with the possibility of preserving asymptotic entanglement are discussed. The importance of the bath's memory properties is investigated by comparing Markovian and non-Markovian evolutions.