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Journal of Physics Communications

The entanglement evolution of the quantum optical systems

Urjjarani Patel, K. V. S. Shiv Chaitanya

Published Jan 28, 2026• 0 citations
Publisher

Abstract

Abstract In this paper, we apply the Thermofield Dynamics (TFD) method to transform a quantum optical master equation into a Schrödinger-like equation for an arbitrary initial condition, making it a preferred approach for solving open quantum system problems. Entanglement and quantum mutual information are analyzed for each case using the symplectic eigenvalues of the corresponding covariance matrix, followed by two-mode and single-mode squeezed states.

DOI
10.1088/2399-6528/ae3eca

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Abstract In this paper, we apply the Thermofield Dynamics (TFD) method to transform a quantum optical master equation into a Schrödinger-like equation for an arbitrary initial condition, making it a preferred approach for solving open quantum system problems. Entanglement and quantum mutual information are analyzed for each case using the symplectic eigenvalues of the corresponding covariance matrix, followed by two-mode and single-mode squeezed states.
In this paper, Thermofield Dynamics (TFD) is applied to map a quantum optics nonlinear master equation into a Schrodinger-like equation for any arbitrary initial condition. This formalism provides a more efficient way for solving open quantum system problems. Then we use the Hartree-Fock approximation to solve the master equations of two separate noisy quantum systems analytically, which allows us to analyze the entanglement and quantum mutual information in each case using the eigenvalues of a covariance matrix, followed by two-mode and single-mode squeezed states.
We use the symplectic transformation of the known linear optical elements to investigate the dynamical evolution of a three-mode cavityless optomechanical system and give explicit formulas for the input and output covariance matrices of the three mode fields when the initial state is Gaussian. Unlike the conventional approaches, the present one does not necessitate solution of the equation of motion of such a system. We study the dynamical behavior of bipartite entanglement in this system when the mirror vibrational mode is initially in a single-mode squeezed vacuum state and each of the two reflected optical sideband modes is in the vacuum state. It is found that the entanglement configuration alternates between a fully separable state and a continuous-variable entangled state during evolution.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we study the continuous variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment. We solve the Kossakowski-Lindblad master equation for the time evolution of the considered system and describe the entanglement in terms of the covariance matrix for an arbitrary Gaussian input state. Using Peres–Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we show that for certain values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a periodic collapse and revival of entanglement take place. We analyze also the time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state.
In the framework of the theory of open systems based on quantum dynamical semigroups, we solve the master equation for two independent bosonic oscillators interacting with an environment in the asymptotic long-time regime. We give a description of the continuous-variable entanglement in terms of the covariance matrix of the quantum states of the considered system for an arbitrary Gaussian input state. Using the Peres-Simon necessary and sufficient condition for separability of two-mode Gaussian states, we show that the two non-interacting systems immersed in a common environment and evolving under a Markovian, completely positive dynamics become asymptotically entangled for certain environments, so that their non-local quantum correlations exist in the long-time regime.
We study the entanglement dynamics of multi-qubit systems coupled to a common dissipative environment, focusing on systems with one or two initially excited qubits. Using the Lindblad master equation, we derive the time evolution of the density matrix and analyze the entanglement between qubit pairs via the concurrence measure. The solution method involves applying the Lindblad super-operator to the initial density matrix, generating a subspace of non-duplicate states. We extend our analysis to nnn-qubit systems, including three-qubit and four-qubit configurations, and explore the effects of thermal noise on entanglement dynamics. Our results demonstrate how initial conditions, system size, and environmental interactions shape entanglement, providing valuable insights for quantum information processing applications.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous variable entanglement for a system consisting of two uncoupled modes interacting with a thermal environment. Using Peres‐Simon necessary and sufficient criterion for separability of two‐mode Gaussian states, we describe the generation and evolution of entanglement in terms of the covariance matrix for a Gaussian input state. For some values of the temperature of environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a periodic collapse and revival of entanglement take place. We analyze also the logarithmic negativity, which characterizes the degree of entanglement of the quantum state.