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Qubit-qubit entanglement in the Tavis–Cummings model with two independent resonators

Alexander R. Bagrov, Eugene K. Bashkirov

Published Jan 1, 2025• 0 citations
Publisher

Abstract

In this paper, a three-qubit Tavis-Cummings model with two independent lossless single-mode resonators is considered. It is assumed that the initial states of the resonator fields are thermal fields and the qubits are in genuine entangled W- and GHZ-type states. For the model under study and the specified initial states of the resonator fields and qubits, we have exactly solved the quantum Liouville equation for the full density matrix. The full density matrix was used to calculate the entanglement parameters - negativity and fidelity. The computer simulation results showed that in the investigated model, the entanglement for all initial qubit states breaks down rapidly with increasing intensity of the thermal fields of the resonators compared to the previously investigated three-qubit models. Moreover, a sudden death of entanglement is observed even for vacuum resonator fields.

DOI
10.18721/jpm.183.162

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In this paper, a three-qubit Tavis-Cummings model with two independent lossless single-mode resonators is considered. It is assumed that the initial states of the resonator fields are thermal fields and the qubits are in genuine entangled W- and GHZ-type states. For the model under study and the specified initial states of the resonator fields and qubits, we have exactly solved the quantum Liouville equation for the full density matrix. The full density matrix was used to calculate the entanglement parameters - negativity and fidelity. The computer simulation results showed that in the investigated model, the entanglement for all initial qubit states breaks down rapidly with increasing intensity of the thermal fields of the resonators compared to the previously investigated three-qubit models. Moreover, a sudden death of entanglement is observed even for vacuum resonator fields.
In this work we study in detail the generalized Tavis — Cummings model, in which three identical qubits are trapped in a lossless single-mode resonator and interact resonantly with the quantized thermal electromagnetic field of this resonator by means of multiphoton processes. An exact solution of the unsteady Schr¨odinger equation for separable qubit states and Fock states of the resonator field was demonstrated for the model under study. The resulting expressions were used to calculate the density matrix in the case of the thermal field of the resonator and to calculate the criterion for pairwise entanglement of qubits — Wootters concurrence. The results showed that the thermal field can induce pairwise entanglement between qubits for all initial separable states for certain system parameters. The calculations also showed that the photon multiple can be considered as an effective mechanism for controlling the entanglement of qubits, since its increase leads to a significant suppression of the effect of the sudden death of entanglement and to an increase in the maximum degree of entanglement for some separable qubit states.
In this article, we have studied the entanglement dynamics of three identical qubits (natural or artificial two-level atoms) resonantly interacting with the one mode of the thermal field of a microwave lossless resonator via one-photon transitions. An exact solution of the quantum time Schrodinger equation is found for the total wave function of the system for the initial separable and entangled states of qubits and the Fock initial state of the resonator. On the basis of this solution, an exact solution of the quantum Liouville equation for the total time-dependent density matrix of the system in the case of a thermal field of the resonator is constructed. The exact solution for the full density matrix is used to calculate the criterion of entanglement of pairs of qubits negativity. The resultsof numerical simulation of the time dependence of the negativity of pairs of qubits showed that with an increase in the intensity of the thermal resonator field, the degree of entanglement of pairs of qubits decreases. It is also shown that In the model under consideration, for any initial states of qubits and intensities of the thermal field of the resonator, the effect of sudden death of entanglement takes place. This behavior of the entanglement parameter in the model under consideration differs from that in the two-qubit model. For two-qubit model, the effect of the sudden death of entanglement takes place only for the initial entangled states of qubits and intense thermal fields of the resonator.
In this article, we consider the entanglement between three identical qubits resonantly interacting with a thermal field of an Q-infinite resonator. We derived the exact solution of the evolution equation for density matrix of the considered system. On it basis we calculate the negativity as a criteria of entanglement between pairs of qubits for biseparable and entangled W-type states of qubits. The results showed that for considered initial qubits states the effect of the entanglement sudden death occurs for all values of the average number of thermal photons.
We obtained a solution of the Liouville equation for a system of three identical qubits interacting without detuning with a thermal field of a cavity with a Kerr medium, for two type of initial pure genuine W-entangled qubits states. On its basis, three entanglement parameters: fidelity, tangle and pairwise negativities were calculated. We obtained that sudden death of entanglement takes place for large intensities of the cavity thermal noise for all considered W-states. We also showed that Kerr nonlinearity prevents the destruction of the initial qubits entanglement induced by the thermal noise of the cavity and eliminates the sudden death of entanglement.
In this paper, we investigated the dynamics of entanglement of two qubits interacting non-resonantly with the thermal field of a one-mode lossless resonator, taking into account the Ising-type direct interaction between qubits. Based on the exact solution of the quantum Liouville equation, we found the density matrix of the system under consideration. With its help, the reduced qubit-qubit density matrix was calculated and the entanglement criteriоn of the two-qubit system, Pres-Horodeсki parameter, was found. It was shown that for the resonance model and separable initial states of qubits, the direct interaction of qubits leads to a significant increase in the maximum degree of their entanglement. It was also found that for the nonresonant interaction between qubits and field, the increase of the maximum degree of entanglement of qubits is much greater than that due to direct interaction. For the original entangled Bell-type state of the qubits, the direct interaction was found to lead to the vanishing of the effect of the sudden death of entanglement in the case of a resonant qubit-field interaction and, conversely, to an increase in this effect for a nonresonant interaction.
In this paper, we investigated the dynamics of systems of two and three identical qubits interacting resonantly with a selected mode of a thermal field of a lossless resonator. We found solutions of the quantum time-dependent Liouville equation for various three- and two-qubit entangled states of qubits. Based on these solutions, we calculated the criterion of the qubit entanglement — fidelity. The results of numerical calculations of the fidelity showed that increasing the average number of photons in a mode leads to a decrease in the maximum degree of entanglement. It is shown that the two-qubit entangled state is more stable with respect to external noise than the three-qubit entangled Greenberger — Horne — Zeilinger states (GHZ). Moreover, a genuine entangled GHZ-state is more stable to noise than a GHZ-like entangled state.