Atom–Atom Entanglement in a Nonresonant Two-Photon Tavis–Cummings Model

Entanglement between two identical two-level atoms (qubits) that interact nonresonantly with the thermal field of a single-mode ideal resonator via effective degenerate two-photon transitions is studied. Based on an exact solution for the time-dependent density matrix, negativity is calculated as a measure of atom entanglement. The effect detuning between the atomic frequencies and the doubled frequency of the resonator field has on the dynamics of qubit entanglement in cases of separable and entangled initial atomic states is investigated.


INTRODUCTION
Quantum nonlocal correlations (entanglement) are of primary importance in quantum information science. In recent years, the entangled states of qubits in different physical systems have been studied in many experimental and theoretical works [1]. The problem of managing and controlling the entangled states of qubits is of key importance in developing effective protocols for the functioning of quantum computers and quantum networks. One of the most promising ways of solving this problem is using qubit interaction with dedicated modes of the resonators in systems of resonator quantum electrodynamics (RQED). A number of RQED experiments have recently been performed to study qubit entanglement for neutral atoms, ions in magnetic traps, superconducting circuits, quantum dots, and impurity spins [1]. Theoretical investigations of these schemes were based on the Jaynes-Cummings model (JCM) and generalizations of it [2]. Different JCM generalizations that considered the possibility of multiphoton transitions, the presence of several atoms and several working levels in atoms, multimode fields, dipole-dipole interaction between atoms, frequency detuning between atoms and a field, time and intensity dependences of the constants of atom-field interaction, the Kerr nonlinear optical effect, and others have been made in recent years. JCM polyatomic generalization is usually referred to as the Tavis-Cummings model (TCM).
In the last few years, special attention in quantum electrodynamics (QED) has been given to the experimental and theoretical study of two-photon TCMs.
This interest in two-photon processes was initiated by the experimental use of a two-photon maser [3]. The theoretical description of such systems requires nonlinear versions of TCM, particularly models with twophoton transitions. These models have been used successfully to describe RQED experiments with ions in Paul traps [4], neutral atoms [5], quantum dots [6], and superconducting circuits [7]. A two-photon Tavis-Cummings model was also used to describe superconducting qubit entanglement in a coplanar resonator [8]. The possibility of using diatomic oneand two-photon TCMs to generate entangled states with a resonator's thermal field was studied in a number of recent works (see references in [9]). In this work, we investigate the dynamics of the entanglement of two natural or artificial atoms that interact via twophoton transitions in the thermal field mode of an ideal resonator within a diatomic two-photon Tavis-Cummings model.

MODEL AND ITS EXACT SOLUTION
Let us consider a system consisting of two identical natural or artificial two-level atoms (qubits) with resonant transition frequency ω 0 , interacting with a quantum electromagnetic field in an ideal resonator via degenerated two-photon transitions. Physically, these could be Rydberg neutral atoms, ions in Paul traps, superconducting Josephson rings, impurity spins, quantum dots, and other two-level systems that interact with the microwave fields of resonators. The effective Hamiltonian of the considered system in a refer- BULLETIN (1) where is the operator of population inversion in the i-th atom (i = 1, 2), are the increasing and decreasing operators in the i-th atom; and are the excited and ground states of the i-th twolevel atom; a + (a) are the operators of creation (annihilation) of photons of the resonator mode; g is the constant of the effective two-photon atom interaction with the resonator field; and is the detuning between the atomic transition frequency and the doubled frequency of the resonator mode We studied the dynamics of the system for both the initial separable (2) and the entangled Bell-type state of the atoms where θ is the parameter that determines the initial degree of atom entanglement ( ). The maximum degree of atom entanglement corresponds to θ = π/4. It was assumed that at the initial moment in time, the resonator field was in a single-mode thermal state with density matrix Here, is the mean number of thermal photons in the resonator: = where is the Boltzmann constant, and T is the temperature of the resonator.
The aim of this work was to investigate the temporal dynamics of atomic entanglement for a diatomic (two-qubit) two-photon Tavis-Cummings model. Many quantitative criteria describing the degree of the two-qubit system entanglement are now being developed in quantum information science. The criteria used most often for calculations are those of Peres and Horodecki (negativity) [11,12] and Wooters (concurrence) [13]. We used negativity for a quantitative assessment of the degree of atom entanglement. This criterion can be defined as where are the negative eigenvalues of reduced atomic density matrix partially transposed in respect to the variables of one qubit. The values of the negativity parameter correspond to the entangled states of qubits in the range of In this work, we found an exact solution to the Liouville quantum equation for the full time density matrix of considered system ρ(t) in the representation of dressed states (i.e., the eigenfunctions of Hamiltonian (1)). After averaging the full density matrix over the field variables, we obtained the reduced time density matrix of qubit subsystem ρ A (t) = Tr F ρ(t) and the reduced atomic density matrix partially transposed over the variables of one qubit Finally, we obtained the analytical expression for negativity.

RESULTS AND DISCUSSION
The results from our numerical modeling of negativity are presented in Figs. 1 and 2. The negativity for separable initial atomic state (2) is shown in Fig. 1 as a function of dimensionless time gt for slight detunings between the atomic frequencies and the doubled frequency of the resonator field, and of the fixed value of average number of photons Figure 1 shows that an increase in dimensionless detuning parameter considerably raises the maximum degree of qubit entanglement. Since we considered the thermal field of a resonator with a low mean number of photons ( ), our result agrees well with the results of [10]. The authors of that work proposed a scheme for the generation of the two-particle maximally entangled state of two atoms prepared initially in separable state and interacting via one-photon transitions n n n n n p n n with the tuned-out mode of the field of the vacuum resonator. A similar effect in a diatomic one-photon Tavis-Cummings model was theoretically predicted in [11] for a resonator's thermal field.
The time dependence of negativity for entangled initial atomic state (3) is presented in Fig. 2

for mean photon number
As is shown in Fig. 2, introducing detuning between the atomic frequencies and the field double frequency under the condition that δ 0 considerably reduces the oscillations of the entanglement parameter; i.e., it stabilizes the initial atomic entanglement in relation to fluctuations caused by the thermal noise. For the given model, under the condition that δ 0 when there is no exchange of energy between the atoms and the field mode, the excited virtual environment of the resonator thus does not destroy the initial atomic quantum correlations or the atoms' state of entanglement. As is shown in [12], a similar effect is observed for Rydberg atoms prepared in Bell entangled states and flying consecutively through the thermal resonator of a monoatomic maser.

CONCLUSIONS
We studied the effect the frequency of atom detuning and the double frequency of a field have on the entanglement of natural or artificial atoms interacting nonresonantly via degenerated two-photon transitions with a thermal single-mode field of an ideal resonator. The Peres-Horodecki parameter (negativity) was = 0.1. n @ @ chosen as the criterion of qubit entanglement. In the representation of dressed states, we determined the precise time density matrix of the considered system for the thermal state of the resonator field and different states of atoms, and calculated the negativity when using it. Our results show that slight detuning between the frequencies of atoms and the doubled frequency of the field mode for separable initial state of atoms (2) can considerably increase the degree of atom entanglement induced by the resonator's thermal field. For a Bell entangled initial atomic state in form (3), the introduction of considerable detuning between the atomic frequencies and the doubled frequency of the field mode substantially lowers the amplitudes of negativity oscillations; i.e., it stabilizes the initial atomic entanglement. The atomic frequencies can be tuned out from those of the resonator field modes for many types of qubits (and superconducting qubits in particular) by modifying the magnetic field. By tuning out the qubit frequencies from that of the resonator field for the initially entangled qubits, we can therefore considerably reduce the oscillations of the entanglement parameter that emerge due to qubits interacting with the thermal photons of the resonator.