Evolution of the entanglement of the N00N -type of states in a coupled two cavity system via an adiabatic approximation
EEvolution of the entanglement of the N N -type of states in acoupled two cavity system via an adiabatic approximation R. Chakrabarti † , G. Sreekumari , ‡ and V. Yogesh § The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600 113, India. Department of Theoretical Physics, University of Madras, Maraimalai Campus,Guindy, Chennai 600 025, India. Department of Physics, Loyola College, Chennai 600 034, India.
Abstract
We study a system of two cavities each encapsulating a qubit and an oscillator degrees offreedom. An ultrastrong interaction strength between the qubit and the oscillator is assumed, andthe photons are allowed to hop between the cavities. A partition of the time scale between thefast moving oscillator and the slow moving qubit allows us to set up an adiabatic approximationprocedure where we employ the delocalized degrees of freedom to diagonalize the Hamiltonian.The time evolution of the N N -type initial states now furnishes, for instance, the reduced densitymatrix of a bipartite system of two qubits. For a macroscopic size of the N N component ofthe initial state the sudden death of the entanglement between the qubits and its continued nullvalue are prominently manifest as the information percolates to the qubits after long intervals. Forthe low photon numbers of the initial states the dynamics produces almost maximally entangledtwo-qubit states, which by utilizing the Hilbert-Schmidt distance between the density matrices, areobserved to be nearly pure generalized Bell states. † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected] a r X i v : . [ qu a n t - ph ] D ec Introduction
The physical structure of the cavity and circuit quantum electrodynamics is represented by the lo-calized oscillator modes interacting with the two level systems. Models involving coupled arrays ofthe qubit-oscillator degrees of freedom, where the photons are permitted to hop between the cavities,recently attracted much experimental and theoretical attention. Various experimental advances inareas such as the photonic crystals [1], the optical microcavities containing the highly localized defectmodes within the photonic band gap [2], and superconducting devices [3, 4] triggered many studiesof these arrays. Such formations have been recently considered for providing a framework for thedistributed quantum computation [5], the generation of entanglement [6], the transport of a quantumstate [7-9], and the cluster state quantum computation that uses the polaritonic excitations[10, 11].The qubit-oscillator interaction has been studied extensively under the Jaynes-Cummings model[12] that employs the rotating wave approximation holding good for the regime characterized by aweak coupling as well as a small detuning between the qubit and the oscillator frequencies. Recent ex-periments, however, probe the ultrastrong coupling domain, where the rotating wave approximationis not valid. Experimental realizations such as a metal-dielectric-metal microcavity combined withquantum well intersubband transitions generating the cavity polariton states in the terahertz region[13, 14], a quantum semiconductor microcavity displaying specific signatures of the ultrastrong cou-pling regime of the light-matter interaction [15, 16], a nanoelectromechanical resonator capacitivelycoupled to a Cooper-pair box driven by the microwave currents [3, 4, 17], a flux-biased quantum circuitthat utilizes the large inductance of a Josephson junction to produce an ultrastrong coupling with acoplanar waveguide resonator [18, 19] fall in this group. In particular, the superconducting qubitsand circuits facilitate wide range of variability of the parameters, and, consequently, may be chosenas the preferred building blocks for the quantum simulators [20-22]. Moreover, the integrated hybridquantum circuits involving the atoms, spins, cavity photons and the superconducting qubits with thenanomechanical resonators may significantly contribute towards the fabrication of interfaces [23] inthe quantum communication network.The Hamiltonian of the strongly coupled qubit-oscillator system embodies terms that do not pre-serve the total excitation number. To analyze them in the regime where the high oscillator frequencydominates over the low qubit frequency, the authors of [24, 25] have advanced an adiabatic approxi-mation scheme that exploits the separation of the slow and the fast changing degrees of freedom. Thisvalidates the decoupling of the full bipartite Hamiltonian into components related to each time scale,and permits its approximate diagonalization [24]. Utilizing the adiabatic approximation the energyeigenvalues and the eigenstates of the physical systems comprising of two [26, 27] and three [28] qubitscoupled with a single oscillator degree of freedom have been studied.In another development, much notice is devoted [29,30] to a bipartite, path-entangled, Schr¨odingercat type discrete photon number state, commonly called the N N state, where a fixed finite numberof photons are all in either of the two available modes. These states endowed with the multiphotonexcitations possess the same degree of entanglement as the Bell states. High precision phase measure-ment may be accomplished by harnessing the multiphoton entangled N N states, where a higherphoton number leads to increased advantage. In particular, these states facilitate [29-31] achieving theoptimal accuracy permitted by the Heisenberg uncertainty principle. The enhanced phase sensitiv-ity of these states is employed towards reaching the sub-Rayleigh resolution in quantum lithography[32]. Additionally, utilizing the N N component of an entangled four-photon state precise opticalphase measurement with a visibility that surpasses the accuracy limit obtainable with the unentangledphotons has been experimentally realized [33]. The optical N N states with high photon numbers,which, therefore, tend towards the macroscopic entangled states, have recently been generated [34]using the multiphoton interference of quantum down-converted light with a classical coherent state.2mploying a superconducting quantum circuit that includes the Josephson qubits coupled with the twoindependent microwave resonators the entangled multiphoton N N states have also been achieved[35].In the overall context it is important to study the evolution of various nonclassical states in acluster of coupled cavity systems. Recently the authors of Ref. [8] considered the dynamics of atwo-site coupled cavity model over a large range of values of the qubit-cavity detuning and the photontunneling strength. Describing the electromagnetic fields and the spin operators in the system viathe delocalized modes they investigated the atomic state transfer. The qubit-oscillator interaction,however, is characterized [8] by the rotating wave approximation [12] that preserves the total number ofexcitations. Making a departure, we, in this work admit strong coupling of the qubit-oscillator hybridsystem where the conservation of the total number of excitations is not assumed. For specificity,we consider the evolution of a N N -type of state in a structure comprising of two coupled qubit-oscillator systems. In Sec. II we enlist the delocalized coordinates in conjunction with the adiabaticapproximation procedure to diagonalize the Hamiltonian. The evolution of the N N -type of statedescribed in Sec. III allows us to construct (Sec. IV) the bipartite reduced density operator for thequbits. The time-variation of the entanglement of the qubits is studied utilizing the concurrence [36]as the measure. We conclude in Sec. V. II Diagonalizing the Hamiltonian via the adiabatic approximation
We consider two identical cavities each containing a two-level atom that is strongly coupled to alocalized oscillator degree of freedom, where the Hamiltonian in natural units ( (cid:126) = 1) reads H = (cid:88) =0 , (cid:18) − ∆2 σ x + ω a † a + λ σ z (cid:16) a + a † (cid:17)(cid:19) + ν (cid:16) a † a + a a † (cid:17) . (2.1)The harmonic oscillator modes { a , a † , ˆ n j ≡ a † a | ∈ (0 , } are characterized by the frequency ω ,and the qubit variables described by the Pauli spin operators { σ X | ∈ (0 , , X = x , y , z } possess theenergy splitting parameter ∆. The qubit-oscillator coupling strength is denoted by λ , whereas the twocavities are interlinked via the photon hopping parameter ν . To facilitate our analysis we now recastthe Hamiltonian (2.1) using the delocalized field and atomic modes, which are given by the symmetricand the antisymmetric linear combinations of their local analogs pertaining to a cavity: A = 1 √ a + a ) , A = 1 √ a − a ) , S X = 1 √ (cid:0) σ X + σ X (cid:1) , S X = 1 √ (cid:0) σ X − σ X (cid:1) , X = x , y , z . (2.2)The delocalized oscillator modes obeying the commutation relation (cid:8)(cid:2) A , A † (cid:96) (cid:3) = δ (cid:96) ; , (cid:96) = 0 , (cid:9) , andthe corresponding spin variables introduced above transform the Hamiltonian (2.1) as follows: H = H Q + Ω A † A + Ω A † A + λ (cid:16) S z0 (cid:16) A + A † (cid:17) + S z1 (cid:16) A + A † (cid:17)(cid:17) , H Q = − ∆ √ S x0 , (2.3)where the tunneling of the photons between the cavities lifts the degeneracy of the frequencies of thedelocalized quanta: Ω = ω + ν, Ω = ω − ν . It has been noted [24] that the parity operator conservesthe qubit-oscillator Hamiltonian. For the two-cavity example (2.1) studied here the parity operatorassumes the form expressed via the localized and the delocalized variables, respectively, as P = exp (cid:16) iπ ( a † a + a † a ) + i π σ x0 + σ x1 ) (cid:17) = ⇒ P = exp (cid:18) iπ ( A † A + A † A ) + i π √ S x0 (cid:19) , (2.4)3here the commutation property [ P, H ] = 0 is preserved.To implement our construction of the evolution of the N N -type states, we, following [24, 25], nowproceed towards the diagonalization process of the Hamiltonian (2.3) in the adiabatic approximationscheme that has been found to be appropriate in the large detuning limit (∆ (cid:28) ω ) as it utilizesthe difference between the time scales of the slow-moving atomic modes and that of the fast-movingoscillators. The high-frequency oscillators are assumed to instantaneously adjust to the slow-changingstate of the qubit observables { σ z | ∈ (0 , } so that the construction permits, in the course ofdiagonalization of the oscillator modes, replacing the spin-variables with the corresponding eigenvalues: {(cid:104) σ z (cid:105) = m = ± | ∈ (0 , } . The delocalized spin variables introduced in (2.2) now admit thesubstitution (cid:104) S z0 (cid:105) = 1 √ (cid:104) ( σ z0 + σ z1 ) (cid:105) = 1 √ m + m ) , (cid:104) S z1 (cid:105) = 1 √ (cid:104) ( σ z0 − σ z1 ) (cid:105) = 1 √ m − m ) (2.5)that expresses the effective Hamiltonian H O of the oscillator degrees of freedom in the following form: H O = Ω (cid:16) A † A + µ (cid:16) A + A † (cid:17)(cid:17) + Ω (cid:16) A † A + µ (cid:16) A + A † (cid:17)(cid:17) , (2.6)where the coefficients read µ = λ √ ( m + m ) , µ = λ √ ( m − m ). The displacement operators D ( µ ) = exp (cid:0) µ (cid:0) A † − A (cid:1)(cid:1) , D ( µ ) = exp (cid:0) µ (cid:0) A † − A (cid:1)(cid:1) acting on the phase space of the oscillatorvariables now facilitates the recasting of the effective Hamiltonian (2.6) as H O = Ω D ( µ ) † A † A D ( µ ) + Ω D ( µ ) † A † A D ( µ ) − Ω µ − Ω µ . (2.7)The eigenstates of the oscillator component (2.7) of the Hamiltonian now readily follows as thedisplaced number states corresponding to the delocalized degrees of freedom: ˆ N ≡ A † A , ˆ N ≡ A † A , ˆ N | N (cid:105) = N | N (cid:105) , ˆ N | N (cid:105) = N | N (cid:105) . The eigenstates of the Hamiltonian (2.7) explicitly read D ( µ ) † D ( µ ) † | N , N (cid:105) = | N ,m + m , N ,m − m (cid:105) , | N (cid:105) = ( A † ) N √ N ! | (cid:105) , | N (cid:105) = ( A † ) N √ N ! | (cid:105) . (2.8)After completing the approximate diagonalization the high frequency oscillator components of theHamiltonian (2.3), we now attend to the corresponding low frequency qubit parts. A tensor productof the qubit states with the displaced oscillator basis states | Ψ (cid:104) N (cid:105)(cid:104) m (cid:105) (cid:105) = | N ,m + m , N ,m − m ; m , m (cid:105) , (cid:104) N (cid:105) ≡ ( N , N ) , (cid:104) m (cid:105) ≡ ( m , m ) (2.9)provides the construction of the relevant matrix elements of the Hamiltonian (2.3). For a dominantoscillator frequency ∆ (cid:28) ω one may neglect [24] the matrix elements that mix the oscillator stateswith different eigenvalues ( N , N ) of its number operators. In other words, the separation of oscillatorenergy levels is much larger than that of the two-level system, and, consequently, transitions in thetwo-level system can never trigger an excitation of the oscillator. This approximation truncates theHamiltonian to a block-diagonal form where each block mixes the displaced oscillator states with iden-tical ( N , N ) eigenstates of photons. The Hamiltonian for the ( N , N )-th block may be representedas follows: H (cid:104) N (cid:105) = N − λ Ω H H H N − λ Ω H H N − λ Ω H H H N − λ Ω , (2.10)4here N = Ω N + Ω N . The minor diagonal elements of the block Hamiltonian H (cid:104) N (cid:105) vanish asthe qubit component H Q produces only a single spin flip at the order considered here. The remainingoff-diagonal elements are real, and may be evaluated as projections in the Hilbert space: H = H = (cid:104) N , − , N ; − , − | H Q | N , N , − ; − , (cid:105) = − ∆2 (cid:104) N , − | N (cid:105) (cid:104) N | N , − (cid:105) ,H = H = (cid:104) N , − , N ; − , − | H Q | N , N , ; 1 , − (cid:105) = − ∆2 (cid:104) N | N , − (cid:105) (cid:104) N , | N (cid:105) ,H = H = (cid:104) N , N , − ; − , | H Q | N , , N ; 1 , (cid:105) = − ∆2 (cid:104) N | N , (cid:105) (cid:104) N , − | N (cid:105) ,H = H = (cid:104) N , N , ; 1 , − | H Q | N , , N ; 1 , (cid:105) = − ∆2 (cid:104) N | N , (cid:105) (cid:104) N , | N (cid:105) . (2.11)The reflection property (cid:104) N , | N (cid:105) = (cid:104) N , − | N (cid:105) , (cid:104) N , | N (cid:105) = (cid:104) N , − | N (cid:105) ensures the equality of the off-diagonal elements in (2.10): H = H = H = H ≡ Λ (cid:104) N (cid:105) = − ∆2 exp ( − Γ + ) L N (cid:16) λ Ω (cid:17) L N (cid:16) λ Ω (cid:17) ,where the parameters read Γ ± = λ (cid:16) ± (cid:17) and the Laguerre polynomial follows the usual expan-sion: L n ( x ) = (cid:80) nk =0 ( − k (cid:0) nk (cid:1) x k k ! .The energy eigenvalues of the block Hamiltonian (2.10) may now be listed as E (cid:104) N (cid:105) = N − λ Ω , E (cid:104) N (cid:105) = N − λ Ω , E (cid:104) N (cid:105)± = N − Γ + ± χ (cid:104) N (cid:105) , χ (cid:104) N (cid:105) = (cid:113) (cid:104) N (cid:105) + Γ − (2.12)and the corresponding eigenstates assume the form |E (cid:104) N (cid:105) (cid:105) = 1 √ | N , , N ; 1 , (cid:105) − | N , − , N ; − , − (cid:105) ) , |E (cid:104) N (cid:105) (cid:105) = 1 √ | N , N , ; 1 , − (cid:105) − | N , N , − ; − , (cid:105) ) , |E (cid:104) N (cid:105)± (cid:105) = 12 (cid:34)(cid:115) χ (cid:104) N (cid:105) ∓ Γ − χ (cid:104) N (cid:105) ( | N , , N ; 1 , (cid:105) + | N , − , N ; − , − (cid:105) ) ± Λ (cid:104) N (cid:105) (cid:12)(cid:12) Λ (cid:104) N (cid:105) (cid:12)(cid:12) (cid:115) χ (cid:104) N (cid:105) ± Γ − χ (cid:104) N (cid:105) ( | N , N , ; 1 , − (cid:105) + | N , N , − ; − , (cid:105) ) (cid:35) , (2.13)The above eigenstates of the block-diagonalized Hamiltonian (2.10) obey the orthonormality property: (cid:104)E (cid:104) N (cid:105) |E (cid:104) N (cid:48) (cid:105) (cid:96) (cid:105) = δ ,(cid:96) δ N ,N (cid:48) δ N ,N (cid:48) , , (cid:96) ∈ { , , ±} . (2.14)Moreover, as the eigenstates (2.13) conform to the following requirement ∞ (cid:88) N ,N =0 (cid:104) |E (cid:104) N (cid:105) (cid:105) (cid:104)E (cid:104) N (cid:105) | + |E (cid:104) N (cid:105) (cid:105) (cid:104)E (cid:104) N (cid:105) | + |E (cid:104) N (cid:105) + (cid:105) (cid:104)E (cid:104) N (cid:105) + | + |E (cid:104) N (cid:105)− (cid:105) (cid:104)E (cid:104) N (cid:105)− | (cid:105) = I (cid:104) N (cid:105) I (cid:104) m (cid:105) , (2.15)the set (cid:8) |E (cid:104) N (cid:105) (cid:105) | ∈ { , , ±} , ( N , N ) ∈ (0 , , . . . ∞ ) (cid:9) provides a complete basis in the Hilbert space.In (2.15) the unit operators for the oscillator and the spin basis states, respectively, read ∞ (cid:88) N ,N =0 | N , N (cid:105) (cid:104) N , N | = I (cid:104) N (cid:105) , (cid:88) m ,m = ± | m , m (cid:105) (cid:104) m , m | = I (cid:104) m (cid:105) . (2.16)5he parity quantum numbers of the energy eigenstates (2.13) under the adiabatic approximationare observed as follows. The transformation properties of the basis vectors (2.9) P | N , ± , N ; ± , ± (cid:105) = ( − N + N +1 | N , ∓ , N ; ∓ , ∓ (cid:105) ,P | N , N , ± ; ± , ∓ (cid:105) = ( − N + N +1 | N , N , ∓ ; ∓ , ± (cid:105) (2.17)impart the parity eigenvalues to the energy states (2.13). We notice that the states (cid:8) |E (cid:104) N (cid:105) (cid:105) | ∈ (0 , (cid:9) have opposite parity compared to their partners |E (cid:104) N (cid:105)± (cid:105) as these two sets comprise of the antisymmetricand symmetric linear combinations of the vectors (2.9), respectively: P |E (cid:104) N (cid:105) (cid:105) = ( − N + N |E (cid:104) N (cid:105) (cid:105) , ∈ (0 , , P |E (cid:104) N (cid:105)± (cid:105) = ( − N + N +1 |E (cid:104) N (cid:105)± (cid:105) . (2.18)In Fig. 1 we plot the energy levels (2.12) with varying coupling strength λ for different choices ofthe oscillator quantum numbers ( N , N ). As we have retained the photon tunneling constant to bepositive ν >
0, the energy eigenvalues consistent with the parametric range studied here maintain thehierarchy: E (cid:104) N (cid:105) + ≥ E (cid:104) N (cid:105) ≥ E (cid:104) N (cid:105) ≥ E (cid:104) N (cid:105)− . The undulations in the diagrams for E (cid:104) N (cid:105)± are manifest dueto the presence of the Laguerre polynomials in the corresponding expressions. In the strong couplinglimit ( λ (cid:46) ω ) the energy eigenvalues satisfy E (cid:104) N (cid:105) + → E (cid:104) N (cid:105) , E (cid:104) N (cid:105)− → E (cid:104) N (cid:105) . Corresponding to the zerosof the Laguerre polynomials, the energies of the opposite parity states {|E (cid:104) N (cid:105) + (cid:105) , |E (cid:104) N (cid:105) (cid:105)} , as well as {|E (cid:104) N (cid:105)− (cid:105) , |E (cid:104) N (cid:105) (cid:105)} , become identical. The degeneracy of the above two pairs of energy levels are realizedin the examples ( N = 5 , N = 6) , ( N = 6 , N = 9) and ( N = 8 , N = 8) for the coupling strength λ equaling 0 . . . L ( x ) , L ( x ) and L ( x ), where x = λ Ω .Figure 1: The variations of the energy eigenvalues (cid:8) E (cid:104) N (cid:105) (blue) , E (cid:104) N (cid:105) (red) , E (cid:104) N (cid:105) + (orange) , E (cid:104) N (cid:105)− (green) (cid:9) with respect to the coupling strength λ are plotted corresponding to the parametric choices ω =1 , ∆ = 0 . , ν = 0 .
5. The eigenenergies corresponding to the ordered quantum numbers ( N = 5 , N =6) , ( N = 6 , N = 9) and ( N = 8 , N = 8) of the delocalized oscillator degrees of freedom arerepresented via the dotted, dashed and the solid lines, respectively. The coupling strength λ regimeswhere the energy levels become pairwise identical are enlarged on the side of the respective energybands (Figs. ( a, b, c )). 6 II Evolution of a localized N N -type of state Towards constructing the time evolution of a N N -type state we first project, in the basis set(2.9), an arbitrary localized number state | n , m ; n , m (cid:105) obeying the property ˆ n | n , m ; n , m (cid:105) = n | n , m ; n , m (cid:105) , ∈ (0 , (cid:104) N , ± , N ; ± , ± | n , m ; n , m (cid:105) = ( − n + n + N C (cid:104) n ; N (cid:105) ( ± λ ) F (cid:104) n ; N (cid:105) δ m , ± δ m , ± , (cid:104) N , N , ± ; ± , ∓ | n , m ; n , m (cid:105) = ( − n + N C (cid:104) n ; N (cid:105) ( ± λ ) F (cid:104) n ; N (cid:105) δ m , ± δ m , ∓ , (3.1)where the coefficients are given by C (cid:104) n ; N (cid:105) ( ± λ ) = exp (cid:18) − λ Ω (cid:19) (cid:18) ± λ Ω (cid:19) n + n + N + N ( N N √ n ! n ! N ! N ! , (cid:104) n ; N (cid:105) ≡ ( n , n ; N , N ) , F (cid:104) n ; N (cid:105) = N (cid:88) k =0 N (cid:88) (cid:96) =0 ( − (cid:96) (cid:18) N k (cid:19)(cid:18) N (cid:96) (cid:19) F (cid:16) − n , − k − (cid:96) ; − ; − Ω λ (cid:17) ×× F (cid:16) − n , − N − N + k + (cid:96) ; − ; − Ω λ (cid:17) , F (cid:104) n ; N (cid:105) = N (cid:88) k =0 N (cid:88) (cid:96) =0 ( − k (cid:18) N k (cid:19)(cid:18) N (cid:96) (cid:19) F (cid:16) − n , − k − (cid:96) ; − ; − Ω λ (cid:17) ×× F (cid:16) − n , − N − N + k + (cid:96) ; − ; − Ω λ (cid:17) . (3.2)The hypergeometric sum in (3.2) is defined as F ( x , y ; − ; τ ) = (cid:80) ∞ (cid:96) =0 ( x ) (cid:96) ( y ) (cid:96) τ (cid:96) (cid:96) ! , where the Pochhammersymbol reads ( x ) (cid:96) = (cid:81) (cid:96) − j =0 ( x + j ). The projections (3.1) of the state | n , m ; n , m (cid:105) facilitate itsexpansion in the complete orthonormal basis set {|E (cid:104) N (cid:105) (cid:105) | ∈ (0 , , ± ); N , N ∈ (0 , , . . . , ∞ ) } : | n , m ; n , m (cid:105) = ∞ (cid:88) N ,N (cid:88) ∈{ , , ±} C ( { n, m ; N } ) |E (cid:104) N (cid:105) (cid:105) , (3.3)where the coefficients may be expressed as C ( { n, m ; N } ) ≡ (cid:104)E (cid:104) N (cid:105) | n , m ; n , m (cid:105) = 1 √ (cid:16) ( − n + n + N δ m , δ m , − ( − N δ m , − δ m , − (cid:17) C (cid:104) n ; N (cid:105) ( λ ) F (cid:104) n ; N (cid:105) , C ( { n, m ; N } ) ≡ (cid:104)E (cid:104) N (cid:105) | n , m ; n , m (cid:105) = 1 √ (cid:16) ( − n + N δ m , δ m , − − ( − n + N δ m , − δ m , (cid:17) C (cid:104) n ; N (cid:105) ( λ ) F (cid:104) n ; N (cid:105) , C ± ( { n, m ; N } ) ≡ (cid:104)E (cid:104) N (cid:105)± | n , m ; n , m (cid:105) = 12 (cid:115) χ (cid:104) N (cid:105) ∓ Γ − χ (cid:104) N (cid:105) (cid:16) ( − n + n + N δ m , δ m , + ( − N δ m , − δ m , − (cid:17) ×× C (cid:104) n ; N (cid:105) ( λ ) F (cid:104) n ; N (cid:105) ±
12 Λ (cid:104) N (cid:105) (cid:12)(cid:12) Λ (cid:104) N (cid:105) (cid:12)(cid:12) (cid:115) χ (cid:104) N (cid:105) ± Γ − χ (cid:104) N (cid:105) (cid:16) ( − n + N δ m , δ m , − +( − n + N δ m , − δ m , (cid:17) C (cid:104) n ; N (cid:105) ( λ ) F (cid:104) n ; N (cid:105) , (3.4)7he orthonormality of the state (3.3) expressed in the basis set of the approximate energy eigenstates (cid:8) |E (cid:104) N (cid:105) (cid:105) | ∈ { , , ±} , ( N , N ) ∈ (0 , , . . . ∞ ) (cid:9) is confirmed via the hypergeometric identity ∞ (cid:88) N ,N =0 N ! N ! S n ,n ( N , N ) S n (cid:48) ,n (cid:48) ( N , N ) (cid:16) x (cid:17) N + N = n ! n ! x n + n exp(2 x ) δ n ,n (cid:48) δ n ,n (cid:48) , (3.5)where the bipartite weight functions read S n ,n ( N , N ) = N (cid:88) k =0 N (cid:88) (cid:96) =0 ( − k (cid:18) N k (cid:19)(cid:18) N (cid:96) (cid:19) F (cid:16) − n , − k − (cid:96) ; − ; − x (cid:17) ×× F (cid:16) − n , − N − N + k + (cid:96) ; − ; − x (cid:17) . (3.6)The expansion (3.3) employing a complete set of energy eigenstates now readily yields the time evo-lution of the localized cavity states as follows: | n , m ; n , m (cid:105) −−→ t | ψ (cid:104) n,m (cid:105) ( t ) (cid:105) = ∞ (cid:88) N ,N =0 (cid:88) ∈{ , , ±} C ( { n, m ; N } ) exp (cid:16) − i E (cid:104) N (cid:105) t (cid:17) |E (cid:104) N (cid:105) (cid:105) . (3.7)The setting specified above permits us now to explore the time evolution of a localized N N -typeof state residing in two cavities: | ψ (0) (cid:105) = 1 (cid:112) | c | ( | n, −
1; 0 , − (cid:105) + c | , − n, − (cid:105) ) , c ∈ C . (3.8)The construction (3.7) immediately provides the subsequent transformation of the initial N N -typeof state (3.8): | ψ ( t ) (cid:105) = 1 (cid:112) | c | ∞ (cid:88) N ,N =0 ( − N C ( n, N ,N )0 ( λ ) (cid:16) F ( n, N ,N )0 + c F (0 ,n ; N ,N )0 (cid:17) ×× (cid:32) − √ − i E (cid:104) N (cid:105) t ) |E (cid:104) N (cid:105) (cid:105) + 12 (cid:115) χ (cid:104) N (cid:105) − Γ − χ (cid:104) N (cid:105) exp( − i E (cid:104) N (cid:105) + t ) |E (cid:104) N (cid:105) + (cid:105) + 12 (cid:115) χ (cid:104) N (cid:105) + Γ − χ (cid:104) N (cid:105) exp( − i E (cid:104) N (cid:105)− t ) |E (cid:104) N (cid:105)− (cid:105) (cid:33) . (3.9)The corresponding pure state density matrix is given by the usual tensorized prescription: ρ ( t ) = | ψ ( t ) (cid:105) (cid:104) ψ ( t ) | . IV The qubit reduced density matrix and its entanglement
In order to utilize the construction of the time-dependent state (3.9) in studying the evolution of theentanglement between, say, the two qubits, we need to compose the relevant reduced density matrixusing a partial tracing on the oscillator states: ρ Q ( t ) = Tr O ρ ( t ) ≡ (cid:88) ı,,k,(cid:96) ∈{± } ρ ı, ; k,(cid:96) ( t ) | ı (cid:105) (cid:104) k(cid:96) | , (4.1)8here the projection operators provide a complete basis set for the two qubit tensor product space.The elements ρ ı, ; k,(cid:96) are conveniently expressed via a kernel whose structure follows from (3.9): G ( (cid:104) N (cid:105) ; (cid:104) N (cid:48) (cid:105) ) = 11 + | c | ( − N + N (cid:48) C ( n, N ,N )0 ( λ ) C (0 ,n ; N (cid:48) ,N (cid:48) )0 ( λ ) ×× (cid:16) F ( n, N ,N )0 + c F (0 ,n ; N ,N )0 (cid:17) (cid:16) F ( n, N (cid:48) ,N (cid:48) )0 + c ∗ F (0 ,n ; N (cid:48) ,N (cid:48) )0 (cid:17) . (4.2)It also obeys the appropriate Hermiticity and normalization properties: G ( (cid:104) N (cid:48) (cid:105) ; (cid:104) N (cid:105) ) ∗ = G ( (cid:104) N (cid:105) ; (cid:104) N (cid:48) (cid:105) ) , ∞ (cid:88) N ,N G ( (cid:104) N (cid:105) ; (cid:104) N (cid:105) ) = 1 . (4.3)The tools developed above now allow us to procure the time-dependent elements of the reduceddensity matrix of the qubits. We first enlist the real diagonal elements: ρ − , − − , − ( t ) = 38 + 18 ∞ (cid:88) N ,N =0 G ( (cid:104) N (cid:105) , (cid:104) N (cid:105) ) χ (cid:104) N (cid:105) (cid:32) Γ − + 4Λ (cid:104) N (cid:105) cos (cid:16)(cid:16) E (cid:104) N (cid:105) + − E (cid:104) N (cid:105)− (cid:17) t (cid:17) (cid:33) + 2 χ (cid:104) N (cid:105) (cid:32) (cid:16) χ (cid:104) N (cid:105) − Γ − (cid:17) cos (cid:16)(cid:16) E (cid:104) N (cid:105) − E (cid:104) N (cid:105) + (cid:17) t (cid:17) + (cid:16) χ (cid:104) N (cid:105) + Γ − (cid:17) cos (cid:16)(cid:16) E (cid:104) N (cid:105) − E (cid:104) N (cid:105)− (cid:17) t (cid:17) ,ρ − , − , ( t ) = 12 ∞ (cid:88) N ,N =0 G ( (cid:104) N (cid:105) , (cid:104) N (cid:105) ) (cid:32) Λ (cid:104) N (cid:105) χ (cid:104) N (cid:105) (cid:33) (cid:32) − cos (cid:16)(cid:16) E (cid:104) N (cid:105) + − E (cid:104) N (cid:105)− (cid:17) t (cid:17) (cid:33) ,ρ , − , − ( t ) = 12 ∞ (cid:88) N ,N =0 G ( (cid:104) N (cid:105) , (cid:104) N (cid:105) ) (cid:32) Λ (cid:104) N (cid:105) χ (cid:104) N (cid:105) (cid:33) (cid:32) − cos (cid:16)(cid:16) E (cid:104) N (cid:105) + − E (cid:104) N (cid:105)− (cid:17) t (cid:17) (cid:33) ,ρ , , ( t ) = 38 + 18 ∞ (cid:88) N N =0 G ( (cid:104) N (cid:105) , (cid:104) N (cid:105) ) χ (cid:104) N (cid:105) (cid:32) Γ − + 4Λ (cid:104) N (cid:105) cos (cid:16)(cid:16) E (cid:104) N (cid:105) + − E (cid:104) N (cid:105)− (cid:17) t (cid:17) (cid:33) − χ (cid:104) N (cid:105) (cid:32) (cid:16) χ (cid:104) N (cid:105) − Γ − (cid:17) cos (cid:16)(cid:16) E (cid:104) N (cid:105) − E (cid:104) N (cid:105) + (cid:17) t (cid:17) + (cid:16) χ (cid:104) N (cid:105) + Γ − (cid:17) cos (cid:16)(cid:16) E (cid:104) N (cid:105) − E (cid:104) N (cid:105)− (cid:17) t (cid:17) . (4.4)The diagonal elements (4.4) ensure that trace of the qubit reduced density matrix is conserved:9r ρ Q ( t ) = 1. The evolution of the off-diagonal elements reflecting Hermiticity are entered below: ρ − , − − , ( t ) = 18 ∞ (cid:88) N ,N =0 ∞ (cid:88) N (cid:48) ,N (cid:48) =0 G ( (cid:104) N (cid:105) ; (cid:104) N (cid:48) (cid:105) ) (cid:32) Λ (cid:104) N (cid:48) (cid:105) χ (cid:104) N (cid:48)(cid:105) (cid:33) (cid:104) N (cid:48) | N , − (cid:105) (cid:104) N (cid:48) , − | N (cid:105) ×× χ (cid:104) N (cid:105) − Γ − χ (cid:104) N (cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) + − E (cid:104) N (cid:48) (cid:105) + (cid:17) t (cid:17) − χ (cid:104) N (cid:105) + Γ − χ (cid:104) N (cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105)− − E (cid:104) N (cid:48) (cid:105)− (cid:17) t (cid:17) + 2 exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) − E (cid:104) N (cid:48) (cid:105) + (cid:17) t (cid:17) − (cid:16) − i ( E (cid:104) N (cid:105) − E (cid:104) N (cid:48) (cid:105)− ) t (cid:17) − χ (cid:104) N (cid:105) − Γ − χ (cid:104) N (cid:105) ×× exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) + − E (cid:104) N (cid:48) (cid:105)− (cid:17) t (cid:17) + χ (cid:104) N (cid:105) + Γ − χ (cid:104) N (cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105)− − E (cid:104) N (cid:48) (cid:105) + (cid:17) t (cid:17) ,ρ − , − , − ( t ) = 18 ∞ (cid:88) N ,N =0 ∞ (cid:88) N (cid:48) ,N (cid:48) =0 G ( (cid:104) N (cid:105) ; (cid:104) N (cid:48) (cid:105) ) (cid:32) Λ (cid:104) N (cid:48) (cid:105) χ (cid:104) N (cid:48)(cid:105) (cid:33) (cid:104) N (cid:48) | N , − (cid:105) (cid:104) N (cid:48) , | N (cid:105) ×× χ (cid:104) N (cid:105) − Γ − χ (cid:104) N (cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) + − E (cid:104) N (cid:48) (cid:105) + (cid:17) t (cid:17) − χ (cid:104) N (cid:105) + Γ − χ (cid:104) N (cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105)− − E (cid:104) N (cid:48) (cid:105)− (cid:17) t (cid:17) + 2 exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) − E (cid:104) N (cid:48) (cid:105) + (cid:17) t (cid:17) − (cid:16) − i (cid:16) E (cid:104) N (cid:105) − E (cid:104) N (cid:48) (cid:105)− (cid:17) t (cid:17) − χ (cid:104) N (cid:105) − Γ − χ (cid:104) N (cid:105) ×× exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) + − E (cid:104) N (cid:48) (cid:105)− (cid:17) t (cid:17) + χ (cid:104) N (cid:105) + Γ − χ (cid:104) N (cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105)− − E (cid:104) N (cid:48) (cid:105) + (cid:17) t (cid:17) ,ρ − , − , ( t ) = − ∞ (cid:88) N ,N ,N (cid:48) =0 G ( N , N ; N (cid:48) , N ) (cid:104) N (cid:48) , | N , − (cid:105) exp (cid:0) − i Ω ( N − N (cid:48) ) t (cid:1) + χ (cid:104) N (cid:105) + Γ − χ (cid:104) N (cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105)− − E N (cid:48) ,N (cid:17) t (cid:17) + χ (cid:104) N (cid:105) − Γ − χ (cid:104) N (cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) + − E N (cid:48) ,N (cid:17) t (cid:17) − χ N (cid:48) ,N +Γ − χ N (cid:48) ,N exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) − E N (cid:48) ,N − (cid:17) t (cid:17) − χ N (cid:48) ,N − Γ − χ N (cid:48) ,N exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) − E N (cid:48) ,N + (cid:17) t (cid:17) − χ (cid:104) N (cid:105) + Γ − χ (cid:104) N (cid:105) χ N (cid:48) ,N + Γ − χ N (cid:48) ,N exp (cid:16) − i (cid:16) E (cid:104) N (cid:105)− − E N (cid:48) ,N − (cid:17) t (cid:17) − χ (cid:104) N (cid:105) + Γ − χ (cid:104) N (cid:105) χ N (cid:48) ,N − Γ − χ N (cid:48) ,N ×× exp (cid:16) − i (cid:16) E (cid:104) N (cid:105)− − E N (cid:48) ,N + (cid:17) t (cid:17) − χ (cid:104) N (cid:105) − Γ − χ (cid:104) N (cid:105) χ N (cid:48) ,N + Γ − χ N (cid:48) ,N exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) + − E N (cid:48) ,N − ) t (cid:17) − χ (cid:104) N (cid:105) − Γ − χ (cid:104) N (cid:105) χ N (cid:48) ,N − Γ − χ N (cid:48) ,N exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) + − E N (cid:48) ,N + (cid:17) t (cid:17) , − , , − ( t ) = 14 ∞ (cid:88) N ,N ,N (cid:48) =0 Λ (cid:104) N (cid:105) Λ N ,N (cid:48) χ (cid:104) N (cid:105) χ N ,N (cid:48) G ( N , N ; N , N (cid:48) ) (cid:104) N (cid:48) , | N , − (cid:105) ×× exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) + − E N ,N (cid:48) + (cid:17) t (cid:17) + exp (cid:16) − i (cid:16) E (cid:104) N (cid:105)− − E N ,N (cid:48) + (cid:17) t (cid:17) − exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) + − E N ,N (cid:48) − (cid:17) t (cid:17) − exp (cid:16) − i (cid:16) E (cid:104) N (cid:105)− − E N ,N (cid:48) + (cid:17) t (cid:17) ,ρ − , , ( t ) = 18 ∞ (cid:88) N ,N =0 ∞ (cid:88) N (cid:48) ,N (cid:48) =0 G ( (cid:104) N (cid:105) ; (cid:104) N (cid:48) (cid:105) ) (cid:32) Λ (cid:104) N (cid:105) χ (cid:104) N (cid:105) (cid:33) (cid:104) N (cid:48) , | N (cid:105) (cid:104) N (cid:48) | N , − (cid:105) ×× χ (cid:104) N (cid:48)(cid:105) − Γ − χ (cid:104) N (cid:48)(cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) + − E (cid:104) N (cid:48) (cid:105) + (cid:17) t (cid:17) − χ (cid:104) N (cid:48)(cid:105) + Γ − χ (cid:104) N (cid:48)(cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105)− − E (cid:104) N (cid:48) (cid:105)− (cid:17) t (cid:17) − (cid:16) − i (cid:16) E (cid:104) N (cid:105) + − E (cid:104) N (cid:48) (cid:105) (cid:17) t (cid:17) + 2 exp (cid:16) − i (cid:16) E (cid:104) N (cid:105)− − E (cid:104) N (cid:48) (cid:105) (cid:17) t (cid:17) + χ (cid:104) N (cid:48)(cid:105) + Γ − χ (cid:104) N (cid:48)(cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) + − E (cid:104) N (cid:48) (cid:105)− (cid:17) t (cid:17) − χ (cid:104) N (cid:48)(cid:105) − Γ − χ (cid:104) N (cid:48)(cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105)− − E (cid:104) N (cid:48) (cid:105) + (cid:17) t (cid:17) ,ρ , − , ( t ) = 18 ∞ (cid:88) N ,N =0 ∞ (cid:88) N (cid:48) ,N (cid:48) =0 G ( (cid:104) N (cid:105) ; (cid:104) N (cid:48) (cid:105) ) (cid:32) Λ (cid:104) N (cid:105) χ (cid:104) N (cid:105) (cid:33) (cid:104) N (cid:48) , | N (cid:105) (cid:104) N (cid:48) | N , (cid:105) ×× χ (cid:104) N (cid:48)(cid:105) − Γ − χ (cid:104) N (cid:48)(cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) + − E (cid:104) N (cid:48) (cid:105) + (cid:17) t (cid:17) − χ (cid:104) N (cid:48)(cid:105) + Γ − χ (cid:104) N (cid:48)(cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105)− − E (cid:104) N (cid:48) (cid:105)− (cid:17) t (cid:17) − (cid:16) − i (cid:16) E (cid:104) N (cid:105) + − E (cid:104) N (cid:48) (cid:105) (cid:17) t (cid:17) + 2 exp (cid:16) − i (cid:16) E (cid:104) N (cid:105)− − E (cid:104) N (cid:48) (cid:105) (cid:17) t (cid:17) + χ (cid:104) N (cid:48)(cid:105) + Γ − χ (cid:104) N (cid:48)(cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105) + − E (cid:104) N (cid:48) (cid:105)− (cid:17) t (cid:17) − χ (cid:104) N (cid:48)(cid:105) − Γ − χ (cid:104) N (cid:48)(cid:105) exp (cid:16) − i (cid:16) E (cid:104) N (cid:105)− − E (cid:104) N (cid:48) (cid:105) + (cid:17) t (cid:17) . (4.5)Our evaluation (4.5) of the off-diagonal elements of the reduced density matrix employs the followingscalar products of the shifted number states of the delocalized oscillators: (cid:104) M , − | N , (cid:105) = ( − N + M (cid:104) M , | N , − (cid:105) = ( − M √ M ! N ! (cid:32) √ λ Ω (cid:33) M + N exp (cid:18) − λ Ω (cid:19) F (cid:16) − M, − N ; − ; − Ω λ (cid:17) , (cid:104) M | N , (cid:105) = (cid:104) M , − | N (cid:105) = ( − N + M (cid:104) M , | N (cid:105) = ( − N + M (cid:104) M | N , − (cid:105) = ( − M √ M ! N ! (cid:32) √ λ Ω (cid:33) M + N exp (cid:18) − λ Ω (cid:19) F (cid:16) − M, − N ; − ; − Ω λ (cid:17) . (4.6)With the explicit description of the reduced density matrix of the two-qubit system in hand wenow turn towards studying its entanglement properties. To determine the extent of entanglementbetween the qubits we use the concurrence which is widely accepted as its measure for the bipartitemixed states. The concurrence introduced by Wootters [36] is defined as C ( t ) = max (cid:8) , (cid:112) λ − (cid:112) λ − (cid:112) λ − (cid:112) λ (cid:9) , (4.7)11here (cid:8) λ ı | ı = (1 , . . . , (cid:9) are the eigenvalues, ordered in the descending sequence, of the matrix R ( t ) = ρ Q ( t ) (cid:101) (cid:37) ( t ) , (cid:101) (cid:37) ( t ) = ( σ y ⊗ σ y ) ρ ∗ Q ( t ) ( σ y ⊗ σ y ) . (4.8)The matrix (cid:101) (cid:37) ( t ) results from the spin-flip operation on the reduced qubit density matrix ρ Q ( t ). Thetwo-qubit system remains entangled for C ( t ) >
0. The maximum possible entanglement is achievedat the limiting value C ( t ) = 1, while C ( t ) = 0 implies separability. We now examine the evolutionof the entanglement of the bipartite reduced density matrix (4.1) as quantified by the measure (4.7).We notice that for the higher values of the localized photon numbers ( n ) of the N N -type states(3.8) the variation of the concurrence of the spin degrees of freedom with the scaled time (Fig. 2 a, b, c ) depicts the sudden death [37, 38], and the disappearance of the two-qubit entanglement fora comparatively longer period. The qubit-oscillator interaction ensures that informations carried bythe phase correlation between the qubits passes away to the oscillator degrees of freedom. The largerthe size of the N N -type of the oscillator state, the information will take a more prolonged time toreappear in the qubit subsystem and rejuvenate the entanglement between the two qubits. ( a) ( b) ( c) Figure 2: For the parametric choices c = i, ω = 1 , λ = 0 . , ∆ = 0 . , ν = 0 .
5, the figures ( a, b, c )refer to the ascending eigenvalues of the localized number operator n = 4 , ,
10, respectively. Suddendisappearance of the entanglement between the qubits, while observed for all values of n , becomesmore prolonged with the increasing photon number.Lastly, we use our evaluation of the concurrence C ( t ) (4.7) to produce two-qubit states which are inthe close neighborhood of the maximally entangled generalized Bell states. Towards this, we consider(Figs. 3 ( a, b )) the qubit reduced density matrix ρ Q ( t ) at instants corresponding to the dominantvalues of the concurrence C ( t ) (cid:46)
1, and minimize its Hilbert-Schmidt distance [39] from a pure statedensity matrix | Φ (cid:105) (cid:104) Φ | :d HS = (cid:113) Tr ( ρ Q − ρ | Φ (cid:105) ) , ρ | Φ (cid:105) = | Φ (cid:105) (cid:104) Φ | , | Φ (cid:105) = α | φ + (cid:105) + β | φ − (cid:105) + γ | ϕ + (cid:105) + δ | ϕ − (cid:105) , (4.9)where the generalized Bell basis states read | φ ± (cid:105) = 1 √ | , (cid:105) ± i |− , − (cid:105) ) , | ϕ ± (cid:105) = 1 √ | , − (cid:105) ± i |− , (cid:105) ) . (4.10)The above coefficients (( α, β, γ, δ ) ∈ C ) maintaining the normalization ( | α | + | β | + | γ | + | δ | = 1) arevaried to detect a linear combination of the generalized Bell states (4.10) that minimizes the distance(4.9). To emphasize the dynamical effects that produce the entangled two-qubit almost pure states,we, in this instance (Fig. 3), adopt the choice c = 0 in the initial state (3.8) imparting an unentangledfactorized structure to it. 12s high concurrence (4.7) limits are evinced more frequently for low-lying photon number states,we set the values n = 1 and n = 2 for the initial state (3.8) in the description of the Figs. 3 ( a ) and ( b ),consecutively. Diagrams ( a , a ) and ( b , b ) specify the time slices in Figs. 3 ( a ) and ( b ), respectively,at which the local peaks in the concurrence C ( t ) are studied. The Hilbert-Schmidt distance d HS (4.9)is minimized over the ensemble of states {| Φ (cid:105) | ( α, β, γ, δ ) ∈ C } obtained via the variations in thesaid complex coefficients. The relevant quantities and the characterization of the states engenderingminimum distance d HS are registered in Table 1. For the parametric range considered here we noticethat at the instants, when the local maxima of the concurrence C ( t ) are realized, the resultant qubitreduced density matrices are predominantly majorized by the pure generalized Bell state densitymatrix ρ Q ∼ | φ ± (cid:105) (cid:104) φ ± | . It is interesting to note that in the two-qubit states observed in Figs. 3 ( a,b ), the relative phases equaling ± π appear between the components | , (cid:105) and |− , − (cid:105) signifying theintroduction of an effective magnetic field. ( a) ( b ) Figure 3: For the present set of diagrams we make the following parametric choices: c = 0 , ω = 1 , λ =0 . , ∆ = 0 . , ν = 0 .
5. The graphs ( a , a , a ) plot the concurrence C ( t ) for the value n = 1 in theinitial state (3.8), whereas the illustrations ( b , b , b ) involve the corresponding selection n = 2. Thevalues of the concurrence C ( t ) at its local maxima, and the pertinent construction of the nearly purestate two-qubit density matrices | Φ (cid:105) (cid:104) Φ | described earlier are reported in Table 1. n = 1 n = 2 ωt
592 24152 598 24390 C ( t ) 0 . . . . HS | min . . . . | Φ (cid:105) . | φ + (cid:105) − . | φ − (cid:105) +0 . | ϕ − (cid:105)≈ | φ + (cid:105) . | φ + (cid:105) +0 . | φ − (cid:105)≈ | φ − (cid:105) . | φ + (cid:105) − . | φ − (cid:105) − . | ϕ − (cid:105)≈ | φ + (cid:105) . | φ − (cid:105) − . | ϕ + (cid:105)≈ | φ − (cid:105) Table 1
V Conclusion
We considered a two cavity system where each cavity included a qubit and an oscillator degreesof freedom strongly interacting with each other. The tunneling of photons between the cavities ispermitted. The dominant oscillator frequency controls the slow moving qubit allowing us to use anadiabatic approximation. Setting up the delocalized variables for the oscillators and the qubits weapproximately diagonalize the Hamiltonian. Starting with a N N -type initial state we examine its13volution and construct, for instance, the reduced density matrix of the bipartite system of two qubits.Utilizing the concurrence the entanglement of the two-qubit reduced density is measured. For the high N N -type initial states the phenomenon of sudden death and the continued absence of entanglementbetween the qubits are increasingly visible as the macroscopic size of the entangled photonic statesprevents the information passing to the qubits for a longer time. On the other hand for a low value ofthe photon quantum number of the initial state nearly maximal entanglement is reached, at certaintimes, for the two-qubit states that behave as almost pure generalized Bell states. Our analysis of thetime-development of the initial state makes it possible to extract the reduced density matrix for thetwo oscillator degrees of freedom, and thereby study the multidimensional phase space quasiprobabilitydistributions. This will be pursued elsewhere. Acknowledgements
One of us (RC) wishes to thank the Quantum Optics and Quantum Information Theory group inthe Institute of Mathematical Sciences and the Department of Nuclear Physics, University of Madrasfor kind hospitality. Another author (VY) acknowledges the support from Department of Scienceand Technology (India) under the INSPIRE Fellowship scheme. We are also indebted for generouscomputational help from the Department of Central Instrumentation and Service Laboratory, and theDepartment of Nuclear Physics, University of Madras. We are happy to express our sincere gratitudeto B. Virgin Jenisha for numerous discussions.
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