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arXiv:2007.00288 · Journal of Physics B Atomic Molecular and Optical Physics

Entanglement of quantum oscillators coupled to different heat baths

Wei-Can Syu, Da-Shin Lee, Chen-Pin Yeh

Published Jan 21, 2021• 5 citations
arXivHTMLPDF

Abstract

Abstract We study the non-equilibrium dynamics of two coupled oscillators interacting with their own heat baths of quantum scalar fields at different temperature T 1 and T 2 with bilinear couplings between them. We particularly focus on the entanglement or inseparability property of their quantum states. The critical temperatures of two respective oscillators, T 1c and T 2c , higher than which the entanglement disappears, can be determined. It is found that when two damping parameters are largely different, say γ 1 ≪ γ 2 , the critical temperature T 1c with respect to the frequency Ω + , the higher frequency among two normal modes frequencies, can be very large, T 1c ≫ Ω + , while T 2c ∝ Ω + with the possibility of hot entanglement. The entanglement of two oscillators with the temperature-dependent damping parameters γ 1;2, T from heat baths is also discussed.

DOI
10.1088/1361-6455/abde53

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Abstract We study the non-equilibrium dynamics of two coupled oscillators interacting with their own heat baths of quantum scalar fields at different temperature T 1 and T 2 with bilinear couplings between them. We particularly focus on the entanglement or inseparability property of their quantum states. The critical temperatures of two respective oscillators, T 1c and T 2c , higher than which the entanglement disappears, can be determined. It is found that when two damping parameters are largely different, say γ 1 ≪ γ 2 , the critical temperature T 1c with respect to the frequency Ω + , the higher frequency among two normal modes frequencies, can be very large, T 1c ≫ Ω + , while T 2c ∝ Ω + with the possibility of hot entanglement. The entanglement of two oscillators with the temperature-dependent damping parameters γ 1;2, T from heat baths is also discussed.
This is the second of a series of three papers examining how viable it is for entanglement to be sustained at high temperatures for quantum systems in thermal equilibrium (Case A), in nonequilibrium (Case B) and in nonequilibrium steady state (NESS) conditions (Case C). The system we analyze here consists of two coupled quantum harmonic oscillators each interacting with its own bath described by a scalar field, set at temperatures T 1 > T 2. For constant bilinear inter-oscillator coupling studied here (Case C1) owing to the Gaussian nature, the problem can be solved exactly at arbitrary temperatures even for strong coupling. We find that the valid entanglement criterion in general is not a function of the bath temperature difference, in contrast to thermal transport in the same NESS setting [1]. Thus lowering the temperature of one of the thermal baths does not necessarily help to safeguard the entanglement between the oscillators. Indeed, quantum entanglement will disappear if any one of the thermal baths has a temperature higher than the critical temperature T c, defined as the temperature above which quantum entanglement vanishes. With the Langevin equations derived we give a full display of how entanglement dynamics in this system depends on T 1, T 2, the inter-oscillator coupling and the system-bath coupling strengths. For weak oscillator-bath coupling the critical temperature T c is about the order of the inverse oscillator frequency, but for strong oscillator-bath coupling it will depend on the bath cutoff frequency. We conclude that in most realistic circumstances, for bosonic systems in NESS with constant bilinear coupling, ‘hot entanglement’ is largely a fiction.
The possibility of maintaining entanglement in a quantum system at finite, even high, temperatures – the so-called ‘hot entanglement’ – has obvious practical interest, but also requires closer theoretical scrutiny. Since quantum entanglement in a system evolves in time and is continuously subjected to environmental degradation, a nonequilibrium description by way of open quantum systems is called for. To identify the key issues and the contributing factors that may permit ‘hot entanglement’ to exist, or the lack thereof, we carry out a model study of two spatially-separated, coupled oscillators in a shared bath depicted by a finite-temperature scalar field . From the Langevin equations we derived for the normal modes and the entanglement measure constructed from the covariance matrix we examine the interplay between direct coupling, field-induced interaction and finite separation on the structure of late-time entanglement. We show that the coupling between oscillators plays a crucial role in sustaining entanglement at intermediate temperatures and over finite separations. In contrast, the field-induced interaction between the oscillators which is a non-Markovian effect becomes very ineffective at high temperature. We determine the critical temperature above which entanglement disappears to be bounded in the leading order by the inverse frequency of the center-of-mass mode of the reduced oscillator system, a result not unexpected, which rules out hot entanglement in such settings.
The non-equilibrium state of two oscillators with a mutual interaction and coupled to separate heat baths is discussed. Bosonic baths are considered, and an exact spectral representation for the elements of the covariance matrix is provided analytically. A wide class of spectral densities for the relevant bath modes is allowed for. The validity of the fluctuation-dissipation relation is established for global equilibrium (both baths at the same temperature) in the stationary state. Spectral measures of entanglement are suggested by comparing to the equilibrium spectrum of zero-point fluctuations. No rotating-wave approximation is applied, and anomalous heat transport from cold to hot bath, as reported in earlier work, is demonstrated not to occur.
Abstract The non‐equilibrium state of two oscillators with a mutual interaction and coupled to separate heat baths is discussed. Bosonic baths are considered, and an exact spectral representation for the elements of the covariance matrix is provided analytically. A wide class of spectral densities for the relevant bath modes is allowed for. The validity of the fluctuation‐dissipation relation is established for global equilibrium (both baths at the same temperature) in the stationary state. Spectral measures of entanglement are suggested by comparing to the equilibrium spectrum of zero‐point fluctuations. No rotating‐wave approximation is applied, and anomalous heat transport from cold to hot bath, as reported in earlier work, is demonstrated not to occur.
This article surveys quantum correlations dynamics, in the Markovian and non-Markovian regimes, in a system of two harmonic oscillators connected by a time-dependent coupling and in contact with a common heat bath. The results show the survival of the quantum correlations, including entanglement, even at very high temperatures, as well as a remarkable relation between entanglement and the instability of the system. The results also show that the indirect interaction of the oscillators via a bath significantly enhances the quantum correlations and that quantum correlations are much more sensitive to the parameters of the oscillators than the temperature of the bath.
Abstract Entanglement being a foundational cornerstone of quantum sciences and the primary resource in quantum information processing, understanding its dynamical evolution in realistic conditions is essential. Unfortunately, numerous model studies show that degradation of entanglement from a quantum system’s environment, especially thermal noise, is almost unavoidable. Thus the appellation ‘hot entanglement’ appears like a contradiction, until Galve et al. [Phys. Rev. Lett. 105, 180501 (2010)] announced that entanglement can be kept at high temperatures if one considers a quantum system with time-dependent coupling between the two parties, each interacting with its individual bath. With the goal of understanding the sustenance of entanglement at high temperatures, working with the same model and set up as Galve et al, namely, parametrically-driven coupled harmonic oscillators interacting with their own Markovian baths, this work probes into the feasibility of ‘hot entanglement’ from three aspects listed in the subtitle. Our findings show that 1) hot entanglement functions only in the unstable regimes, 2) instability is a necessary but not sufficient condition, and 3) the power intake required by the drive operating in the unstable regime to sustain entanglement increases exponentially. The last factor indicates that hot entanglement under this modeling is theoretically untenable and its actual implementation likely unattainable.