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Izvestiya vysshikh uchebnykh zavedenii Fizika

Dynamics of the fidelity and the negativity criterion for three-qubit Jaynes–Cummings model

Alexander R. Bagrov, Eugene K. Bashkirov

Published Jan 1, 2025• 0 citations
Publisher

Abstract

This paper presents an exact solution for the evolution operator of the three-qubit Jaynes-Cummings model. It is assumed that the qubits are prepared at the initial moment of time in genuine entangled W- and GHZ-type states, and the field is in a vacuum state. Based on the exact solution, wave functions, density matrix elements, and qubit entanglement parameters - the pairwise negativity criterion and the fidelity - were calculated. Using the entanglement parameters, the full analysis of the qubit entanglement dynamics was performed.

DOI
10.17223/00213411/68/10/13

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Background. The study of entangled states in Jaynes–Cummings models and its multiqubit generalizations and extensions remains an important and relevant topic in modern quantum optics, quantum information science and solid state physics. This is due to the fact that such models naturally describe the dynamics of interaction of logical elements of quantum devices, such as quantum computers or quantum networks (qubits) with microwave fields of resonators used to control the states of qubits. In this regard, it is very important to search for the most efficient schemes of generation, control and monitoring of entangled qubit states within the framework of Jaynes–Cummings type models. Aim. To investigate the entanglement dynamics of a pair of qubits, each of which is locked in a single-mode resonator and interacts non-resonantly with the vacuum field mode, in the presence of detuning between frequencies of transitions in the qubits and frequencies of resonator modes and dipole-dipole interaction of the qubits. Methods. To analyze the dynamics of the considered system, the solution of the time-dependent quantum Schrödinger equation is investigated. The exact solution of the above equation in the case of entangled Bell-type initial qubits states is found. This is used to calculate the criterion of entanglement of a pair of qubits – negativity. Numerical modeling of the time dependence of negativity for different values of the parameters of the considered model has been carried out. Results. It is shown that the presence of detuning, the difference in the qubit-photon couplings and the intensity of the dipole-dipole interaction significantly affect the maximum degree of entanglement of the qubit subsystem during its evolution. It is found that for certain parameters of the model under consideration, the initial Bell entangled states of qubits can be considered as long-lived stable states. Conclusion. The possibility of realization of long-lived and robust two-qubit entangled states in the system under consideration has been established. The obtained results can be used for effective control and manipulation of the degree of entanglement of qubits interacting with microwave fields of resonators.

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In this paper, we study the dynamics of a system consisting of two identical qubits interacting non-resonantly with a single-mode field of an ideal resonator in the presence of direct dipole-dipole and Ising interactions between the qubits. In the representation of "dressed" states, we find an exact solution of the quantum time Liouville equation for the full density matrix of the system under consideration for the initial separable and entangled states of the qubits and the thermal initial state of the resonator field. Based on the exact solution, the criterion for qubit entanglement, negativity, is calculated. The results of numerical modeling of negativity show that the inclusion of dipole-dipole and Ising coupling in the case of separable initial states of qubits can lead to a significant increase in the maximum degree of their entanglement, and in the case of entangled initial states of qubits, to stabilization of the initial entanglement. For the model with resonant interaction of qubits and the resonator field, direct interaction of qubits can also lead to the disappearance of the sudden death of entanglement effect.

Vestnik of Samara University Natural Science Series•Jan 15, 2026•0 citations
This paper presents an exact solution for the evolution operator of the three-qubit Jaynes-Cummings model. It is assumed that the qubits are prepared at the initial moment of time in genuine entangled W- and GHZ-type states, and the field is in a vacuum state. Based on the exact solution, wave functions, density matrix elements, and qubit entanglement parameters - the pairwise negativity criterion and the fidelity - were calculated. Using the entanglement parameters, the full analysis of the qubit entanglement dynamics was performed.
Background. To operate a quantum computer, a set of universal gates must be implemented, such as a two-cubit gate of the controlled negation type plus one-cubit spins. As a universal alternative, three-cubic-bit gates may be used. In this regard, it seems very relevant to investigate the dynamics of three-qubit systems in microwave resonators, in particular to study the most efficient schemes for generating, controlling, and monitoring entangled qubit states. Aim. To investigate the features of the dynamics of entangled pairs of qubits for a system in which two qubits are locked in a single-mode resonator and interact with the thermal field mode, and the third qubit is in a free state. Methods. To analyze the dynamics of the considered system, the solution of the quantum Liouville equation for the full density matrix is investigated. The exact solution of the above equation in the case of initial biseparable states of the qubits is found. The exact solution of the evolution equation is used to calculate the criterion of entanglement of qubit pairs – negativity. Numerical simulations of negativity for biseparable qubit states as well as different values of the thermal field intensity of the resonator have been carried out. Results. It is shown that for intense thermal fields of the resonator the effect of instantaneous death of entanglement is observed, while the time intervals between death and revival of entanglement of qubits depend essentially on the choice of their initial biseparable state. It is found that for one of the biseparable states, entanglement of qubits trapped in the resonator does not occur at any field intensities of the resonator. Conclusion. It is found that the peculiarities of the dynamics of entanglement of qubits, in particular the time intervals between the death and birth of entangled qubits, are determined by the choice of the initial biseparable state of qubits, as well as by the values of the field intensity of the resonator. The results obtained can be used to effectively control and manage the degree of qubit entanglement in three-qubit systems in microwave resonators.
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 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 paper, we analyze in detail the dynamics of the negativity criterion, fidelity and concurrence fill for the Tavis-Cummings model in which three qubits are in a common single-mode ideal resonator and interact resonantly with the field mode via multiphoton processes. By solving the unsteady Schrödinger equation, we give analytic solutions for genuine entangled Werner- and Greenberger-Horn-Zeilinger states. It is shown that the multiplicity of photon transitions can significantly stabilize the genuine entangled states of both types. Key features in the dynamics pattern of the entanglement parameters are highlighted. We conclude that the negativity criterion is not suitable for describing the dynamics of entanglement of qubits in the case of three-qubit Greenberger-Horn-Zeilinger states. Main advantages of the concurrence fill over the negativity criterion are shown.
Background. The study of entangled states in Jaynes–Cummings models and its multiqubit generalizations and extensions remains an important and relevant topic in modern quantum optics, quantum information science and solid state physics. This is due to the fact that such models naturally describe the dynamics of interaction of logical elements of quantum devices, such as quantum computers or quantum networks (qubits) with microwave fields of resonators used to control the states of qubits. In this regard, it is very important to search for the most efficient schemes of generation, control and monitoring of entangled qubit states within the framework of Jaynes–Cummings type models. Aim. To investigate the entanglement dynamics of a pair of qubits, each of which is locked in a single-mode resonator and interacts non-resonantly with the vacuum field mode, in the presence of detuning between frequencies of transitions in the qubits and frequencies of resonator modes and dipole-dipole interaction of the qubits. Methods. To analyze the dynamics of the considered system, the solution of the time-dependent quantum Schrödinger equation is investigated. The exact solution of the above equation in the case of entangled Bell-type initial qubits states is found. This is used to calculate the criterion of entanglement of a pair of qubits – negativity. Numerical modeling of the time dependence of negativity for different values of the parameters of the considered model has been carried out. Results. It is shown that the presence of detuning, the difference in the qubit-photon couplings and the intensity of the dipole-dipole interaction significantly affect the maximum degree of entanglement of the qubit subsystem during its evolution. It is found that for certain parameters of the model under consideration, the initial Bell entangled states of qubits can be considered as long-lived stable states. Conclusion. The possibility of realization of long-lived and robust two-qubit entangled states in the system under consideration has been established. The obtained results can be used for effective control and manipulation of the degree of entanglement of qubits interacting with microwave fields of resonators.
In this paper, we study the dynamics of a system consisting of two identical qubits interacting non-resonantly with a single-mode field of an ideal resonator in the presence of direct dipole-dipole and Ising interactions between the qubits. In the representation of "dressed" states, we find an exact solution of the quantum time Liouville equation for the full density matrix of the system under consideration for the initial separable and entangled states of the qubits and the thermal initial state of the resonator field. Based on the exact solution, the criterion for qubit entanglement, negativity, is calculated. The results of numerical modeling of negativity show that the inclusion of dipole-dipole and Ising coupling in the case of separable initial states of qubits can lead to a significant increase in the maximum degree of their entanglement, and in the case of entangled initial states of qubits, to stabilization of the initial entanglement. For the model with resonant interaction of qubits and the resonator field, direct interaction of qubits can also lead to the disappearance of the sudden death of entanglement effect.