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Laser Physics

Exact dynamics of three Tavis–Cummings atoms interacting with a coherent resonator field

Башкиров Евгений Константинович, Alexander R. Bagrov

Published Feb 27, 2026• 1 citation
Publisher

Abstract

Abstract In this paper, we demonstrate the exact solution of the nonstationary Schrodinger equation for the Tavis–Cummings model, in which three identical two-level atoms (qubits) are prepared in an arbitrary superposition of 8 Hilbert space vectors and which resonantly interact with a coherent resonator field mode of any intensity. The exact solution was used to calculate the parameters of atom–atom and atom-field entanglement: linear entropy, pairwise and three-particle negativity, and population inversion of three atoms. The results of computer modeling have shown that the degree of correlation between the atoms and the resonator field can significantly depend on the initial state of the atoms. Separable, biseparable, and genuine entangled W -type states demonstrate a high rate of energy exchange between atoms and the resonator field mode and the effect of sudden death of pairwise entanglement of atoms. On the contrary, all GHZ-states exhibit stable pairwise and three-particle entanglement between atoms and a slow rate of energy exchange between atoms and the resonator field mode. We also demonstrate the effect of the sudden death of three-particle entanglement for various states of atoms, the collapse and revival of the Rabi oscillations of the population inversion of three atoms, and the effect of the transition of entanglement from a subsystem of atoms to the total system. These results can be used for optimal operation of quantum networks.

DOI
10.1088/1555-6611/ae4196

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Abstract In this paper, we demonstrate the exact solution of the nonstationary Schrodinger equation for the Tavis–Cummings model, in which three identical two-level atoms (qubits) are prepared in an arbitrary superposition of 8 Hilbert space vectors and which resonantly interact with a coherent resonator field mode of any intensity. The exact solution was used to calculate the parameters of atom–atom and atom-field entanglement: linear entropy, pairwise and three-particle negativity, and population inversion of three atoms. The results of computer modeling have shown that the degree of correlation between the atoms and the resonator field can significantly depend on the initial state of the atoms. Separable, biseparable, and genuine entangled W -type states demonstrate a high rate of energy exchange between atoms and the resonator field mode and the effect of sudden death of pairwise entanglement of atoms. On the contrary, all GHZ-states exhibit stable pairwise and three-particle entanglement between atoms and a slow rate of energy exchange between atoms and the resonator field mode. We also demonstrate the effect of the sudden death of three-particle entanglement for various states of atoms, the collapse and revival of the Rabi oscillations of the population inversion of three atoms, and the effect of the transition of entanglement from a subsystem of atoms to the total system. These results can be used for optimal operation of quantum networks.
The article studies the dynamics of pairwise entanglement of three qubits, two of which are trapped in a resonator and interact with a single-mode ideal resonator through single-photon transitions, and the third qubit is outside the resonator. This takes into account the dipole-dipole coupling between the isolated qubit and the qubit in the resonator. We have found a solution to the quantum nonstationary Schrodinger equation for the total wave function of the system for the initial separable and biseparable states of qubits and the thermal initial state of the resonator field. Using these solutions, the criterion of entanglement of qubit pairs - negativity is calculated. The results of numerical simulation of the negativity criterion have shown that the including of a small dipole-dipole coupling between an isolated and one of the trapped qubits can lead to significant entanglement of qubit pairs for all initial states. There is a transition of entanglement from one pair of atoms to other pairs of atoms during the evolution of the system. It is also shown that for some separable and biseparable states, the dipole-dipole interaction can suppress the effect of sudden death of entanglement.
In this paper we consider a system of identical three two-level atoms interacting at resonance with a single-mode of the quantized field in a lossless cavity. The initial cavity field is prepared in the coherent state while the atoms are taken initially to be either in the uppermost excited state "∣eee>|eee>∣eee>" or The GHZ\textmd{GHZ}GHZ-state or the W\textmd{W}W-state. For this system we investigate different kinds of atomic inversion and entanglement, which arise between the different parts of the system due to the interaction. Also the relationship, between entanglement and some other nonclassical effects in the statistical properties, such as collapses and revivals in the atomic inversion where superharmonic effects appear, is discussed. The QQQ-functions for different cases are discussed. Most remarkably it is found that the GHZ\textmd{GHZ}GHZ-state is more robust against energy losses, showing almost coherent trapping and Schrödinger-cat states can not be produced from such state. Also the entanglement of GHZ\textmd{GHZ}GHZ-state is more robust than the W\textmd{W}W-state. Another interesting feature found is that the state which has no pairwise entanglement initially will have a much improvement of such pairwise entanglement through the evolution. Sudden death and sudden revival of atoms-pairwise entanglement are produced with the W\textmd{W}W-state.
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 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 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.
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.