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Physical Review A

Dynamical generation of maximally entangled states in two identical cavities

Moorad Alexanian

Published Nov 3, 2011• 7 citations
Publisher

Abstract

The generation of entanglement between two identical coupled cavities, each containing a single three-level atom, is studied when the cavities exchange two coherent photons and are in the $N=2,4$ manifolds, where $N$ represents the maximum number of photons possible in either cavity. The atom-photon state of each cavity is described by a qutrit for $N=2$ and a five-dimensional qudit for $N=4$. However, the conservation of the total value of $N$ for the interacting two-cavity system limits the total number of states to only 4 states for $N=2$ and 8 states for $N=4$, rather than the usual 9 for two qutrits and 25 for two five-dimensional qudits. In the $N=2$ manifold, two-qutrit states dynamically generate four maximally entangled Bell states from initially unentangled states. In the $N=4$ manifold, two-qudit states dynamically generate maximally entangled states involving three or four states. The generation of these maximally entangled states occurs rather rapidly for large hopping strengths. The cavities function as a storage of periodically generated maximally entangled states.

DOI
10.1103/physreva.84.052302

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The generation of entanglement between two identical coupled cavities, each containing a single three-level atom, is studied when the cavities exchange two coherent photons and are in the N=2,4N=2,4N=2,4 manifolds, where NNN represents the maximum number of photons possible in either cavity. The atom-photon state of each cavity is described by a qutrit for N=2N=2N=2 and a five-dimensional qudit for N=4N=4N=4. However, the conservation of the total value of NNN for the interacting two-cavity system limits the total number of states to only 4 states for N=2N=2N=2 and 8 states for N=4N=4N=4, rather than the usual 9 for two qutrits and 25 for two five-dimensional qudits. In the N=2N=2N=2 manifold, two-qutrit states dynamically generate four maximally entangled Bell states from initially unentangled states. In the N=4N=4N=4 manifold, two-qudit states dynamically generate maximally entangled states involving three or four states. The generation of these maximally entangled states occurs rather rapidly for large hopping strengths. The cavities function as a storage of periodically generated maximally entangled states.
The generation of entanglement between three identical coupled cavities, each containing a single three-level atom, is studied when the cavities exchange two coherent photons and are in the N=2, 4, and 6 manifolds, where NNN represents the maximum number of photons possible in any one cavity. The combined states of the atom and the photon in a cavity is given by a qutrit for N=2, a five-dimensional qudit for N=4, and a seven-dimensional qudit for N=6. The conservation of the operator N^\hat{N}N^ for the interacting three-cavity system limits the total number of tripartite states to only 6, 18, and 38, rather than the usual 33=273^3=2733=27, 53=1255^3=12553=125, and 73=3437^3 =34373=343 states for N=2, 4, and 6, respectively. The deterministic generation of entanglement from general initially unentangled tripartite states is studied in the limit of large hopping strength, where all the solutions are analytic and given in terms of exponential functions. Several types of resulting tripartite entanglement are analyzed in order to obtain maximally entangled states.
The generation of entanglement between three identical coupled cavities, eachcontaining a single three-level atom, is studied when the cavities exchange two coherent photonsand are in the N =2, 4, and 6 manifolds, where N represents the maximum number of photonspossible in any one cavity. The combined states of the atom and the photon in a cavity is given bya qutrit for N = 2, a five-dimensional qudit for N = 4, and a seven-dimensional qudit for N = 6. Theconservation of the operator Nˆ for the interacting three-cavity system limits the total number oftripartite states to only 6, 18, and 38, rather than the usual 33 = 27, 53 = 125, and 73 = 343 states for N =2, 4, and 6, respectively. The deterministic generation of entanglement from general initiallyunentangled tripartite states is studied in the limit of large hopping strength, where all the solutionsare analytic and given in terms of exponential functions. Several types of resulting tripartiteentanglement are analyzed in order to obtain maximally entangled states.
Based on two-photon exchange interaction between nnn coupled optical cavities each of them containing a single three level atom, the nnn-qubit and nnn-photonic state transfer is investigated. In fact, following the approach of Ref.\cite{Alex1}, we consider nnn coupled cavities instead of two cavities and generalize the discussions about quantum state transfer, photon transition between cavities and entanglement generations between nnn atoms. More clearly, by employing the consistency of number of photons (the symmetry of Hamiltonian), the hamiltonian of the system is reduced from 3n3^n3n dimensional space into 2n dimensional one. Moreover, by introducing suitable basis for the atom-cavity state space based on Fourier transform, the reduced Hamiltonian is block-diagonalized, with 2 dimensional blocks. Then, the initial state of the system is evolved under the corresponding Hamiltonian and the suitable times TTT at which the initially unentangled atoms, become maximally entangled, are determined in terms of the hopping strength ξξξ between cavities.
The temporal evolution of two coupled cavities, each containing a single three-level atom, is studied when the cavities exchange two coherent photons. The general state of the system is a linear superposition of symmetric and antisymmetric states with the symmetric states controlled by two of the four eigenfrequencies and the antisymmetric states by the other two. The system undergoes Rabi oscillations between the two symmetric (antisymmetric) states. There is state transfer between the cavities when both atoms are in the ground state and two photons are exchanged. In addition, there is also Rabi ``flopping'' whereby one atom is in the excited state and the other in the ground state and the roles are reversed in a periodic fashion by the exchange of two photons. The generation of entanglement can be explicitly given as a function of time. Models of coupled cavities are of interest in distributed quantum information and computation.
The generation of entanglement between two identical, interacting quantum dots---initially in ground states---by a coherent field and the subsequent time evolution of the entanglement are studied by calculating the concurrence between the two dots. The results predict that while it is possible to generate entanglement (or entanglement of formation, as defined for a mixed state) between the two dots, at no time do the dots become fully entangled to each other or is a maximally entangled Bell state ever achieved. We also observe that the degree of entanglement increases with an increase in the photon number inside the cavity and a decrease in the dot-photon coupling. The behavior of the two-dot system, initially prepared in an entangled state and interacting with thermal light, is also studied.
We present a scheme for generating entangled states of two qutrits, by using a 4-level system and a 3-level system coupled to a single cavity. Because of only employing resonant interactions, the entangled state can be fast created. This scheme is easy to implement in experiments since only a single cavity is needed and none of measurement, auxiliary system and identical system-cavity coupling constant is required. Furthermore, two-qutrit partially or maximally entangled states can be created by using this scheme. The proposal is quite general, which can be used to prepare entangled states of two qutrits in various physical systems, such as two natural or artificial atoms of the Λ-type or Δ-type three lowest levels, coupled to an optical or microwave cavity.