Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0907.2298 · Journal of Physics B Atomic Molecular and Optical Physics

Multi-mode entanglement ofNharmonic oscillators coupled to a non-Markovian reservoir

Gao-xiang Li, Li-hui Sun, Z. Ficek

Published Jun 14, 2010• 20 citations
arXivHTMLPDF

Abstract

Multi-mode entanglement is investigated in a system composed of N coupled identical harmonic oscillators interacting with a common environment. We treat the problem generally by working with the Hamiltonian without the rotating-wave approximation and by considering the environment as a non-Markovian reservoir to the oscillators. We invoke an N -mode unitary transformation of the position and momentum operators and find that in the transformed basis the system is represented by a set of independent harmonic oscillators with only one of them coupled to the environment. Working in the Wigner representation of the density operator, we find that the covariance matrix has a block diagonal form that can be expressed in terms of multiples of 3 × 3 and 4 × 4 matrices. This simple property allows us to treat the problem to some extent analytically. We illustrate the advantage of working in the transformed basis using a simple example of three harmonic oscillators and find that the entanglement can persist for long times due to the presence of constants of motion for the covariance matrix elements. We find that, in contrast to what one would expect, a strong damping of the oscillators leads to a better stationary entanglement than in the case of a weak damping.

DOI
10.1088/0953-4075/43/13/135501

Related papers

Continuous variables approach to entanglement creation and processing

Lihui Sun, Gao‐xiang Li, Z. Ficek

We apply the continuous variable approach to study entangled dynamics of coupled harmonic oscillators interacting with a thermal reservoir and to a deterministic creation of entanglement in an atomic ensemble located inside a high-Q ring cavity. In the case of harmonic oscillators, we show that a suitable unitary transformation of the position and momentum operators transforms the system to a set of independent harmonic oscillators with only one of them coupled to the reservoir. Working in the Wigner representation of the density operator, we find that the covariance matrix has a block diagonal form of smaller size matrices. This simple property allows to treat the problem to some extend analytically. We analyze the time evolution of an initial entanglement and find that the entanglement can persists for long times due to presence of constants of motion for the covariance matrix elements. In the case of an atomic ensemble located inside the cavity, the attention is focused on creation of one and two-mode continuous variable entangled states from the vacuum by applying laser pulses of a suitably adjusted amplitudes and phases. The pulses together with the cavity dissipation prepare the collective modes of the atomic ensemble in a desired entangled state.

arXiv:1010.1817•Oct 9, 2010•13 citations

Entanglement dynamics of nonidentical oscillators under decohering environments

Fernando Galve, Gian Luca Giorgi, Roberta Zambrini

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.

arXiv:1002.1927•Physical Review A•Jun 17, 2010•47 citations

Entanglement dynamics of quantum oscillators nonlinearly coupled to thermal environments

Aurora Voje, Alexander Croy, Andreas Isacsson

We study the asymptotic entanglement of two quantum harmonic oscillators nonlinearly coupled to an environment. Coupling to independent baths and a common bath are investigated. Numerical results obtained using the Wangsness--Bloch--Redfield method are supplemented by analytical results in the rotating wave approximation. The asymptotic negativity as function of temperature, initial squeezing, and coupling strength, is compared to results for systems with linear system-reservoir coupling. We find that, due to the parity-conserving nature of the coupling, the asymptotic entanglement is considerably more robust than for the linearly damped cases. In contrast to linearly damped systems, the asymptotic behavior of entanglement is similar for the two bath configurations in the nonlinearly damped case. This is due to the two-phonon system-bath exchange causing a suppression of information exchange between the oscillators via the bath in the common-bath configuration at low temperatures.

arXiv:1412.1999•Physical Review A•Jul 10, 2015•3 citations

Entanglement dynamics for two harmonic oscillators coupled to independent environments

Ruggero Vasile

We study the entanglement evolution between two harmonic oscillators having different free frequencies each leaking into an independent bath. We use an exact solution valid in the weak coupling limit and in the short time non-Markovian regime. The reservoirs are identical and characterized by an Ohmic spectral distribution with Lorents-Drude cut-off. This work is an extension of the case reported in [Phys. Rev. A 80, 062324 (2009)] where the oscillators have the same free frequency.

arXiv:1002.0973•Physica Scripta•Sep 1, 2010•1 citation

Dynamics of interacting oscillators in the presence of environments

Juan Nicolas Moreno Tarquino

This thesis explores the dynamics of linearly coupled quantum and classical oscillators interacting with environments. In the quantum regime, such dynamics is described in terms of the Liouvillian superoperator, which leads to non-unitary time evolution of the system, specifically of the density matrix. In particular, when the correlation time of the environment is significantly shorter than the characteristic timescale of the system, the dynamics are classified as Markovian. Therefore, the equation of motion for the density matrix that fulfils this property is known as the Lindblad master equation. Systems of coupled cavities and waveguides in the presence of gain and loss can be described by this equation and, more importantly, by non-Hermitian Hamiltonians. In particular, when two coupled cavities exhibit balance between gain and loss, the system is PT -symmetric. However, for certain values in the parameter space, this symmetry is broken and the non-Hermitian Hamiltonian is no longer diagonalizable. Such values are denoted as exceptional points. In the case of large arrays of coupled quantum oscillators, the dynamics description becomes more challenging since the dimensions of the density matrix grow exponentially. To address this, we study the dynamics of these arrays via the quantum state diffusion approach. In such approach, the Lindblad master equation is unraveled by a stochastic Schrödinger equation. Then, we decouple this equation by diagonalizing the non-Hermitian Hamiltonian that describes the system. Such diagonalization leads to a transformation of a system of coupled oscillators into a decoupled one, offering a great reduction in the basis size. For systems of classical oscillators with dissipative nearest-neighbor coupling, the dynamics leads to synchronized motion. We study how an array of such oscillators reaches the synchronized state. This behavior is found when the system is dissipation free, which implies that all the eigenvectors of the dynamical matrix are damped, except one. To measure the degree of synchronization, we introduce the inverse of participation ratio, whose maximum value is associated with a delocalized state and, therefore, fully synchronized motion. For larger arrays of oscillators, we examine the case where the frequency fluctuations of each oscillator are obtained from a Gaussian distribution with zero mean and a given standard deviation. Then, we derive a scaling law for the largest value of the mentioned standard deviation that still allows the oscillators fully synchronize their motion.:1 Introduction 2 Open quantum systems 3 Dynamics of two coupled cavities in the presence of losses 4 Unraveling of Lindblad equation for system of oscillators 5 Synchronization of dissipatively coupled harmonic oscillators 6 Conclusions and outlook A Derivation of equations of motion for the amplitudes B Quantum state diffusion unraveling for N decoupled oscillators C Model for disspative coupling D Perturbation theory for disorder in the frequencies References

•Oct 23, 2024•0 citations

Non-Markovian entanglement dynamics of quantum continuous variable systems in thermal environments

Kuan-Liang Liu, Hsi‐Sheng Goan

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.

arXiv:0706.0996•Physical Review A•Aug 14, 2007•130 citations
Multi-mode entanglement is investigated in a system composed of N coupled identical harmonic oscillators interacting with a common environment. We treat the problem generally by working with the Hamiltonian without the rotating-wave approximation and by considering the environment as a non-Markovian reservoir to the oscillators. We invoke an N -mode unitary transformation of the position and momentum operators and find that in the transformed basis the system is represented by a set of independent harmonic oscillators with only one of them coupled to the environment. Working in the Wigner representation of the density operator, we find that the covariance matrix has a block diagonal form that can be expressed in terms of multiples of 3 × 3 and 4 × 4 matrices. This simple property allows us to treat the problem to some extent analytically. We illustrate the advantage of working in the transformed basis using a simple example of three harmonic oscillators and find that the entanglement can persist for long times due to the presence of constants of motion for the covariance matrix elements. We find that, in contrast to what one would expect, a strong damping of the oscillators leads to a better stationary entanglement than in the case of a weak damping.
We apply the continuous variable approach to study entangled dynamics of coupled harmonic oscillators interacting with a thermal reservoir and to a deterministic creation of entanglement in an atomic ensemble located inside a high-Q ring cavity. In the case of harmonic oscillators, we show that a suitable unitary transformation of the position and momentum operators transforms the system to a set of independent harmonic oscillators with only one of them coupled to the reservoir. Working in the Wigner representation of the density operator, we find that the covariance matrix has a block diagonal form of smaller size matrices. This simple property allows to treat the problem to some extend analytically. We analyze the time evolution of an initial entanglement and find that the entanglement can persists for long times due to presence of constants of motion for the covariance matrix elements. In the case of an atomic ensemble located inside the cavity, the attention is focused on creation of one and two-mode continuous variable entangled states from the vacuum by applying laser pulses of a suitably adjusted amplitudes and phases. The pulses together with the cavity dissipation prepare the collective modes of the atomic ensemble in a desired entangled state.
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.
We study the asymptotic entanglement of two quantum harmonic oscillators nonlinearly coupled to an environment. Coupling to independent baths and a common bath are investigated. Numerical results obtained using the Wangsness--Bloch--Redfield method are supplemented by analytical results in the rotating wave approximation. The asymptotic negativity as function of temperature, initial squeezing, and coupling strength, is compared to results for systems with linear system-reservoir coupling. We find that, due to the parity-conserving nature of the coupling, the asymptotic entanglement is considerably more robust than for the linearly damped cases. In contrast to linearly damped systems, the asymptotic behavior of entanglement is similar for the two bath configurations in the nonlinearly damped case. This is due to the two-phonon system-bath exchange causing a suppression of information exchange between the oscillators via the bath in the common-bath configuration at low temperatures.
This thesis explores the dynamics of linearly coupled quantum and classical oscillators interacting with environments. In the quantum regime, such dynamics is described in terms of the Liouvillian superoperator, which leads to non-unitary time evolution of the system, specifically of the density matrix. In particular, when the correlation time of the environment is significantly shorter than the characteristic timescale of the system, the dynamics are classified as Markovian. Therefore, the equation of motion for the density matrix that fulfils this property is known as the Lindblad master equation. Systems of coupled cavities and waveguides in the presence of gain and loss can be described by this equation and, more importantly, by non-Hermitian Hamiltonians. In particular, when two coupled cavities exhibit balance between gain and loss, the system is PT -symmetric. However, for certain values in the parameter space, this symmetry is broken and the non-Hermitian Hamiltonian is no longer diagonalizable. Such values are denoted as exceptional points. In the case of large arrays of coupled quantum oscillators, the dynamics description becomes more challenging since the dimensions of the density matrix grow exponentially. To address this, we study the dynamics of these arrays via the quantum state diffusion approach. In such approach, the Lindblad master equation is unraveled by a stochastic Schrödinger equation. Then, we decouple this equation by diagonalizing the non-Hermitian Hamiltonian that describes the system. Such diagonalization leads to a transformation of a system of coupled oscillators into a decoupled one, offering a great reduction in the basis size. For systems of classical oscillators with dissipative nearest-neighbor coupling, the dynamics leads to synchronized motion. We study how an array of such oscillators reaches the synchronized state. This behavior is found when the system is dissipation free, which implies that all the eigenvectors of the dynamical matrix are damped, except one. To measure the degree of synchronization, we introduce the inverse of participation ratio, whose maximum value is associated with a delocalized state and, therefore, fully synchronized motion. For larger arrays of oscillators, we examine the case where the frequency fluctuations of each oscillator are obtained from a Gaussian distribution with zero mean and a given standard deviation. Then, we derive a scaling law for the largest value of the mentioned standard deviation that still allows the oscillators fully synchronize their motion.:1 Introduction 2 Open quantum systems 3 Dynamics of two coupled cavities in the presence of losses 4 Unraveling of Lindblad equation for system of oscillators 5 Synchronization of dissipatively coupled harmonic oscillators 6 Conclusions and outlook A Derivation of equations of motion for the amplitudes B Quantum state diffusion unraveling for N decoupled oscillators C Model for disspative coupling D Perturbation theory for disorder in the frequencies References
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.