Mapping generalized Jaynes-Cummings interaction into correlated finite-sized systems
aa r X i v : . [ qu a n t - ph ] A p r Mapping generalized Jaynes-Cummings interactioninto correlated finite-sized systems
Himadri Shekhar Dhar , Arpita Chatterjee and RupamanjariGhosh , School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India School of Natural Sciences, Shiv Nadar University, Gautam Buddha Nagar, UP201314, IndiaE-mail: [email protected]
Abstract.
We consider a generalized Jaynes-Cummings model of a two-level atominteracting with a multimode nondegenerate coherent field. The sum of the modefrequencies is equal to the two-level transition frequency, creating the resonancecondition. The intermediate levels associated with the multi-photon process areadiabatically eliminated using the non-resonant conditions for these transitions. Undersuch general conditions, the infinite atom-multiphoton interaction is effectively mappedonto an equivalent reduced × bipartite qubit system that facilitates the study of thenonclassical features of the interaction using known information-theoretic measures.We observe that the bipartite pure system is highly entangled as quantified by itsentanglement of formation. Further, it is shown that the dynamics of the mappedsystem can be generated using optically truncated, quantum scissored states thatreduce the infinite atom-multiphoton interaction to a finite × k system, where k is asuitable truncation number. This allows us to introduce atomic dephasing and studythe mixed state dynamics, characterized by the decay of quantum correlations suchas quantum discord, which is observed to be more robust than entanglement. Thequantum correlation dynamics of the dissipative system qualitatively complementsthe behavior of collapse and revival of the Rabi oscillations in the system. Theeffective mapping of the composite system proves to be an efficient tool for measuringinformation-theoretic properties. (Some figures may appear in color only in the online journal) PACS numbers: 42.50.-p, 42.50.Ct, 03.67.-a, 03.67.Bg
Keywords : generalized Jaynes-Cummings model, atom-photon interaction, reducedinteraction dynamics, entanglement, quantum discord
Submitted to:
J. Phys. B: At. Mol. Phys. apping generalized Jaynes-Cummings interaction...
1. Introduction
The archetypal work horse representing the simplest quantum interaction, that of asingle atom and a single cavity mode, is the Jaynes-Cummings (JC) model [1]. Themodel was first proposed to study spontaneous emission and examine the underlyingsemiclassical behavior of quantum radiation [2]. The periodic collapse and subsequentrevival of Rabi oscillations [3] in the temporal dynamics of the JC model later proved thediscrete and quantum nature of photons. In the following years, the JC interaction hasbeen generalized to dissipative and perturbative models [4] to study various fundamentalaspects of the manifested quantum atom-photon interaction [5]. Experimentally, theadvances in cavity quantum electrodynamics involving Rydberg atoms in single-modecavities [6, 7] have allowed investigation of quantum features and possible extension ofthe properties of atom-field interaction for the implementation of quantum tasks [8].The JC model is an extremely purposeful model from the perspective of quantuminformation theory (QIT) and quantum computation. The nonclassical interactionbetween the atomic levels and the cavity modes gives rise to entanglement [9], which isthe primary resource for performing various QIT tasks such as superdense coding [10],quantum teleportation [11], and quantum key distribution [12]. Coupled arrays of JCsystems have been considered to generate and study correlated many-body entangledstates [13], which can be applied in quantum computing and networks [14]. Further, themanipulation and control of entanglement in different generalized versions of JC modelhave received a lot of attention (for a review, see [15]). However, the quest for suitablequantum systems to operationalize various aspects of QIT for practical purposes is stillfar from over. The main obstacles are the difficulty in the manipulation of quantumstates and the irreversible decoherence present in most interacting quantum systems[16]. With the advent of innovative experiments with ultracold atoms [17] and opticallattices [18], it has been possible to increase interaction and manipulability of many-body quantum systems.The most pervasive feature of the JC interaction is the nonclassical nature ofthe field and its correlation with the atomic levels. The dynamics of these atom-cavity correlations is the key to the interesting properties of all generalized JCmodels. For QIT applications, the quantum correlations need to be studied from therelevant perspective. From information-theoretic arguments, the concept of nonclassicalcorrelations is neither limited to nor described fully by entanglement. The conceptof measurement-based, information-theoretic quantum correlation measures, such asquantum discord (QD) [19] and quantum information deficit (QID) [20] use the factthat physical measurements disturb the noncommutative nature of quantum systems andthus erase quantum correlation [21], and are, in general, non-monotonic with measuresof entanglement such as entanglement of formation [22] and logarithmic negativity [23].These information-theoretic measures are known to be non-zero for separable mixedstates. There exist non-trivial operational relations between entanglement and otherquantum correlation measures such as QD and QID [24, 25], however, in general there apping generalized Jaynes-Cummings interaction... × system. Such a mapping is known for single-modeinput fields [33]. Thus the infinite-dimensional problem gets mapped into a bipartitequbit system that simplifies the correlation dynamics and allows the investigation ofimportant characteristics of the quantum system in a reduced Hilbert space. Further,for dissipative, nonunitary evolution of the model under the influence of environment,the dynamics of the system can be studied using optically truncated, quantum scissoredstates [34]. For unitary dynamics, we show that an optically truncated, reduced × k quantum system can simulate the pure state dynamics of the mapped two-qubit system.Under dissipative interactions in the atom-photon interaction, the optically truncated,finite-size system can be used to study the behavior of quantum correlation emergingfrom the mixed state dynamics of the composite open system.Our general mapping scheme leading to the reduced dynamical structure of theproblem can be used for studying important correlation properties or photon statisticsin similar model systems with multimode fields. Using single- and two-mode fields,we observe that the reduced bipartite system is highly entangled, as quantified by itsentanglement of formation. Hence, the interaction can be used to generate maximallyentangled bipartite systems for information processing and communication. Underdissipative atomic dephasing, we study the temporal decay of the correlations, andQD turns out to be more robust than entanglement. Additionally, we observe thatthe quantum correlation dynamics complements the collapse and revival of the Rabioscillations and its subsequent decay in the damped atom-field interaction. For thefeatures in our investigation, the use of a multimode field gives qualitatively similarresults.This paper is organized as follows. We begin by defining the various quantumcorrelation measures in section 2. We describe our generalized JC model of an effectivetwo-level atom interacting with a nondegenerate N -mode coherent field in section 3. apping generalized Jaynes-Cummings interaction... × system, and provide an alternate description using optically truncated, quantumscissored × k state. Section 5 gives the dynamics of entanglement for the purecomposite atom-field system with single- and two-mode fields. In section 6, we discussthe effect of nonunitary atomic dephasing on quantum correlations. Section 7 containsdiscussions of our main results.
2. Quantum correlation measures
For pure states, most entropic quantum correlation measures such as entanglement offormation and quantum discord reduce to the entropy of entanglement [35] whereas log-arithmic negativity gives an upper bound. For mixed states, the operational relationbetween these measures is non-trivial [24, 25], and there is no unique characterizationor hierarchy [26]. In this section, we briefly define two entanglement measures, viz.entanglement of formation and logarithmic negativity, and the information-theoreticquantum correlation measure of quantum discord.
Entanglement : The entanglement of formation (EOF) of a bipartite quantum state[22] is defined as the average entropy of entanglement of its pure state decomposition,optimized over all such possible decompositions. For a bipartite quantum state, ρ = P i p i | ψ i ih ψ i | , the entanglement of formation is defined as E ( ρ ) = min { φ } X i p i E ′ ( | ψ i i ) , (1)where E ′ ( | ψ i ) is the entropy of entanglement, defined as the von Neumann entropy ofthe subsystem of any bipartite pure state, | ψ i . The minimization is over all possiblepure state decompositions ( p i , | ψ i i ) of the bipartite state, ρ . For pure states, the EOFreduces to the entropy of entanglement. The optimization or convex-roof constructionto obtain the mixed state EOF is not easily tractable. For a two-qubit system, theEOF has a closed analytical form defined in terms of its concurrence [36]. For arbitrarydimensions, the convex-roof construction for mixed state EOF is not very well defined,though non-trivial generalizations of concurrence have been explored [37].A computable measure of entanglement for mixed systems is the logarithmicnegativity (LN) [23]. The definition of LN stems from the fact that the negativity of thepartial transpose of a bipartite quantum system is a sufficient condition for entanglement[38]. For two-qubit and × systems, this is also a necessary condition [39]. The LNof an arbitrary bipartite state, ρ , is defined as L ( ρ ) = log [2 N ( ρ ) + 1] , (2)where N ( ρ ) is called the negativity. It is defined as N ( ρ ) = ( (cid:13)(cid:13) ρ T A (cid:13)(cid:13) − (cid:13)(cid:13) ρ T A (cid:13)(cid:13) is the trace norm of the partially transposed state, ρ T A , of ρ . For pure states, the LNdoes not reduce to the entropy of entanglement and can be zero for bound entangledstates in bipartite systems with dimensions higher than × . However, non-zero values apping generalized Jaynes-Cummings interaction... Quantum discord : The definition of quantum discord [19] is borne from theintricate nature of measurement in quantum mechanics and its role in disturbing thenoncommutative nature of quantum operators [21]. In classical information theory, thetotal correlation between two random variables is contained in the mutual informationbetween these variables. The mutual information can be arrived at using the classicalShannon entropy via two distinct methods: first, using the idea of joint probabilitydistribution, and secondly, using the idea of conditional entropy. Extending the conceptto the quantum regime and replacing Shannon entropy with the von Neumann entropy,we obtain two distinct, inequivalent but classically equal definitions of mutual entropy: I ( ρ ( t )) ≡ S ( ρ a ) + S ( ρ f ) − S ( ρ ( t )) , and (3)˜ I ( ρ ( t )) ≡ S ( ρ f ) − S ( ρ f | a ) , (4)where S ( ρ ) = − tr( ρ log ρ ) is the von Neumann entropy of the quantum state ρ . ρ a ( t )and ρ f ( t ) are the reduced density operators of the atomic and the field subsystems,respectively. I ( ρ ( t )) is the quantum mutual information [40] and S ( ρ f | a ) is the quantumconditional entropy [41]. Quantum discord is defined as the difference, Q ( ρ ( t )) = I ( ρ ( t )) − ˜ I ( ρ ( t )). In the classical regime, I ( ρ ( t )) ≡ ˜ I ( ρ ( t )).The inequivalence of the two expressions in the quantum regime is due to thequantum conditional entropy, S ( ρ f | a ), which is defined as the lack of information aboutone subsystem (say, ρ f ) when that of the other ( ρ a ) is known. The complete knowledgeof a subsystem invokes a measurement on the subsystem. Let {F ia } ( i = 1 ,
2) bea basis set of one-dimensional projectors (with F ia F ja = δ ij F ia , P i F ia = I a ) actingon the two-dimensional atomic subsystem ( ρ a ). I a ( I f ) is the identity operator actingon the Hilbert space on which that atomic (field) subsystem, ρ a ( ρ f ), is defined. Ifone makes a projective measurement on the atomic subsystem, the post-measurementdensity matrix can be written as ρ iaf ( t ) = p i ( t ) ( F ia ⊗ I f ρ ( t ) F ia ⊗ I f ), where p i ( t ) =tr af ( F ia ⊗ I f ρ ( t ) F ia ⊗ I f ). The quantum conditional entropy is then given by S ( ρ f | a ) = min {F ia } X i p i ( t ) S ( ρ iaf ( t )) , (5)where the minimization is over all possible sets of rank-1 projective measurements, {F ia } .˜ I ( ρ ( t )) and QD can then be defined as˜ I ( ρ ( t )) = S ( ρ f ) − min {F ia } X i p i ( t ) S ( ρ iaf ( t )) , (6) Q ( ρ ( t )) = I ( ρ ( t )) − ˜ I ( ρ ( t )) . (7)The measurement involved in ˜ I ( ρ ( t )) ensures that I ( ρ ( t )) ≥ ˜ I ( ρ ( t )). Hence, Q ( ρ ( t )) ≥ apping generalized Jaynes-Cummings interaction...
3. Generalized interaction of a two-level atom with a multimode field
We consider the interaction between an effective two-level atom and a nondegenerate N -mode coherent field. The excited state and the ground state of the two-level atomare denoted by | e i and | g i , respectively. The transition frequency between the levels | e i and | g i is denoted by ω . ω , ω , ..., and ω N are the frequencies of the N modes ofthe field, where ω + ω + ... + ω N = ω . The intermediate states associated with anymulti-photon process can be adiabatically removed for the considered frequencies withthe additional assumption that the intermediate transition frequencies are not resonantwith the chosen field modes [42]. In the rotating-wave approximation, the effectiveHamiltonian is described by [43] ( ~ = 1) H = H + H int = ω σ z + Σ Nj =1 ω j a † j a j + ˜g( A α σ + + A † α σ − ) , (8)where A α = Q Nj =1 a j , A † α = Q Nj =1 a † j . Further, σ z = | e ih e | − | g ih g | , σ + = | e ih g | , σ − = | g ih e | , a † j and a j are the creation and annihilation operators, respectively, of the fieldmodes j = 1 , , ..., N , and ˜g is the electric dipole coupling constant. The unitary time-evolution operator derived from the above Hamiltonian [43] is U ( t ) = exp( − iH int t )= cos( ˆ N ˜g t ) | e ih e | − i A α sin( ˆ N ˜g t )ˆ N | e ih g | + cos( ˆ N ˜g t ) | g ih g | − i A † α sin( ˆ N ˜g t )ˆ N | g ih e | , (9)where ˆ N = qQ Nj =1 a j a † j , and ˆ N = qQ Nj =1 a † j a j .The initial ( t = 0) nondegenerate multimode infinite-dimensional coherent field canbe written as [44] | ψ f (0) i = N Y j =1 | α j i = ∞ X n ,n ,...,n N =0 C αn j N Y j =1 | n j i , C αn j = exp − P Nj =1 | α j | ! N Y j =1 α n j j p n j ! , (10)where | α j i is a single-mode coherent state with complex amplitude α j [44, 45].The initial state of the two-level atom is a generalized superposition of the excitedstate ( | e i ) and the ground state ( | g i ), | ψ a (0) i = cos θ | e i + sin θ | g i , where 0 ≤ θ ≤ π . θ = π/ θ = 0 apping generalized Jaynes-Cummings interaction... | e i . The initial state of thetwo-level atom interacting with the coherent field can be written in the following way, | ψ (0) i = | ψ a (0) i ⊗ | ψ f (0) i = cos θ | e, α N i + sin θ | g, α N i , (11)where | α N i = Q Nj =1 | α j i .At any time t , the state vector of the total atom-field system evolves from theinitial state according to (9). If we consider the initial atomic state to be | e i ( θ = 0),the unitary time evolved state is | ψ ( t ) i = U ( t ) | ψ (0) i = X C αn j { cos (˜g t N Y j =1 p n j + 1) | e, N i = | e, ǫ + ( t ) i + | g, ǫ − ( t ) i , (12)where |N i = Q Nj =1 | n j i , |N + 1 i = Q Nj =1 | n j + 1 i , and | ǫ + ( t ) i = X C αn j cos (˜g t N Y j =1 p n j + 1) |N i , | ǫ − ( t ) i = − i X C αn j sin (˜g t N Y j =1 p n j + 1) |N + 1 i . (13)
4. Reduced density operator for the atom and the field
The time-evolved state of the two-level atom–multimode field interaction considered insection 3 is a dynamical quantum state (12) in a ×∞ Hilbert space. However, theatom-field interaction is such that the dynamics of the composite pure system can bereduced onto an effective × mapped state. This is due to the fact that the atomictransitions in the two-level system reduces the interacting field states to superpositionsof the Fock states |N i and |N + 1 i . We introduce two orthonormal basis states, | ξ i = (1 /δ ) | ǫ + ( t ) i , (14)and | ξ i = 1 p − | A | (cid:20) | ǫ − ( t ) i δ − A | ǫ + ( t ) i δ (cid:21) , (15)where δ = p h ǫ + ( t ) | ǫ + ( t ) i , δ = p h ǫ − ( t ) | ǫ − ( t ) i , and A = h ǫ + ( t ) | ǫ − ( t ) i /δ δ .The time-evolved state in the orthonormal × Hilbert space is given by | ψ ( t ) i = δ | e i| ξ i + δ p − | A | | g i| ξ i + δ A | g i| ξ i . (16)The density operator corresponding to the state (12) using (16) is ρ ( t ) = | ψ ( t ) ih ψ ( t ) | apping generalized Jaynes-Cummings interaction... δ δ δ A δ δ β δ δ A ∗ δ | A | δ A βδ δ β δ A ∗ β δ β , (17)where β = p − | A | , δ i = δ ∗ i (i = 1, 2), and A = − A ∗ . The density matrix can benumerically calculated by evaluating δ i (i=1,2) and A , provided the properties of theinteracting coherent field are known. The × mapped density matrix has rank 3.Similar rank-3 density matrices are also seen in other forms of atom-photon interactionwhere there is a mapping of states from the atomic to the photonic subsystem [46]. Theatomic and the field subsystems at time t can be obtained as reduced density matricesby tracing the correlated density matrix, ρ ( t ) over the field and the atomic variables,respectively.The mapping of the ( ×∞ ) infinite level system onto an effective × system ispossible for closed pure state dynamics of the atom-photon system, where the effectiveSchmidt rank of the composite system is governed by the lower dimension of the twosubsystems. For dissipating open systems, the mixed state dynamics can be reduced byoptical truncation of the infinite dimensional field using quantum engineered operationscalled quantum scissors [34]. Let us consider a dissipative nonunitary evolution of thedensity matrix under phase damping of the atomic system. The environment-induceddecohering open system can be written in its Lindblad form [47] as˙ ρ ( t ) = − i ~ [ H, ρ ( t )] + γ σ z ρ ( t ) σ z − ρ ( t )) , (18)where γ is the phase decoherence parameter, and σ z = | e ih e | − | g ih g | . The nonunitary,mixed state evolution represented by equation (18) does not allow solutions that can bemapped from the infinite level atom-photon interaction to an equivalent × state. Theinfinite-level problem for mixed states can be reduced using suitable optical truncationof the interacting field. Using quantum scissors, the N -mode input coherent state canbe written as | ψ ′ f (0) i = 1 N N Y j =1 | α ′ j i = 1 N k X n ,n ,...,n N =0 C αn j N Y j =1 | n j i , (19)where the infinite summation over all number states has been truncated to summationover k states. N is the new normalization constant. This reduces the infinite dimensionalatom-photon interaction to a reduced × k composite state problem. For non-dissipativepure state dynamics, it can be shown that, for a suitable choice of k , the dynamics ofthe truncated × k state is equivalent to that of the mapped × system, derived in(16). For a single-mode field, truncated to k states, the following simplifications can bemade: [ H int , ρ ( t )] = k − X i =0 √ i + 1 (˜g | e, i ih g, i + 1 | + H.c.) , (20) apping generalized Jaynes-Cummings interaction... γ σ z ρ ( t ) σ z − ρ ( t )) = k X m = k/ k/ X l =0 − γ ( | e, l ih g, k | + H.c.) . (21)In the composite atom–number state basis, the density matrix ρ ( t ) can be written as ρ ( t ) = k/ X i =0 k/ X j =0 c ij ( t ) | e, i ih e, j | + k X i = k/ k X j = k/ c ij ( t ) | g, i ih g, j | + k/ X i =0 k X j = k/ c ij ( t ) | e, i ih g, j | + H.c. (22)Using the form of the density matrix (22) in the atom–number state representation, theLindblad equation (18) can be written in the form˙ ρ ( t ) = L ρ ( t ) , (23)where L is the superoperator of the dynamic nonunitary mapping. Solving equation (23)for a known initial state ρ (0), and a suitable truncation number k , we can obtain thefinal dephased state of the atom-photon interaction (with dissipation). This final state, ρ ( t ), can then be used to calculate the quantum correlation properties of the compositesystem.
5. Correlation dynamics of the atom-field system
The quantum correlation properties of the reduced × atom-field system (16),obtained by mapping the infinite-dimensional pure state interaction, can be seen bystudying the dynamics of the entanglement of formation. All entropic measures such asquantum discord reduce to EOF for pure quantum states. For consistency in comparingwith mixed state dynamics, we also consider the evolution of the logarithmic negativity,which is an upper bound for the EOF for pure states. Using single- and two-mode fields,we check and compare the dynamics of EOF [ E ( ρ )] and LN [ L ( ρ )], defined in section 2.Figure 1 shows the evolution of the EOF and LN as a function of the scaled evolutiontime ( τ ≡ ˜g t ), for a single-mode initial coherent field, with average photon number, ¯ n = 10. We observe that reduced atom-photon composite system is highly correlated. Attimes, τ ≈ n = ¯ n = 10, as a function of the scaledevolution time ( τ ). The EOF, as expected, is bounded above by LN, similar to thecorrelation bound observed for the case of initial single-mode field. The oscillatory anddiscontinuous nature of the quantum correlations in the two-mode case is due to the factthat the collapse and revival of Rabi oscillations occur rapidly in short intervals [48]. QD apping generalized Jaynes-Cummings interaction... τ E( ρ ) L( ρ) Figure 1.
Time evolution of quantum correlations, EOF [ E ( ρ )] and LN [ L ( ρ )], inthe two-level atom interacting with a single-mode field, with average photon number,¯ n = 10. τ ≡ ˜g t is the scaled dimensionless time. The dynamics of EOF (blackcontinuous line) and LN (red dashed line) show that the system is highly correlatedand is maximally entangled around τ ≈ τ E( ρ) L( ρ) Figure 2.
Time evolution of EOF [ E ( ρ )] and LN [ L ( ρ )] in the two-level atominteracting with a two-mode field, with average photon numbers, ¯ n = 10, and ¯ n = 10.LN (red dashed line) forms an upper bound on EOF (black continuous line), which isequal at all times to QD. is equal to EOF at all times since the dynamics is unitary (pure). Hence, the increasein the number of modes does not qualitatively affect the dynamics of entanglement inthe atom-photon system.We next consider the pure state dynamics of the optically truncated, quantumscissored state given by relation (22), with γ = 0. We probe different values of k forwhich the pure, optically truncated × k states attempt to simulate the dynamics of themapped × system given by relation (16). Figure 3 shows the evolution of the EOF forthe infinite optical field ( k = ∞ ) mapped to the reduced × state (16) in comparisonto the optically truncated × k systems (22), corresponding to k = 12, 16, and 20, fora single-mode field with average photon number, ¯ n = 10. We observe that at k = 20,the optically truncated, quantum scissored state effectively simulates the correlationdynamics of the reduced infinite system. For calculations involving dissipative mixedstate dynamics in the following section, we have used k = 30. apping generalized Jaynes-Cummings interaction... τ E( ρ ) k = ∞ k = 12k = 16k = 20 Figure 3.
Dynamics of EOF in the two-level atom interacting with a single-modefield, with average photon number ¯ n = 10, for different values of optical truncationnumber k . The nontruncated mapped system corresponds to k = ∞ (red continuousline). The truncated states are shown by blue dotted ( k = 12), black dashed ( k = 16),and green dot-dashed ( k = 20) lines. We observe that the evolution of EOF for k = 20truncated state overlaps with that for the nontruncated mapped system.
6. Quantum correlation dynamics under atomic phase damping
The atomic phase damping in the atom-photon interaction can be suitably studied bythe dissipative nonunitary transformation of the composite density matrix governed bythe Lindblad equation (18). Such phase damping of the atomic subsystem may arise dueto elastic collisions in atomic vapor in trapped systems [49]. The evolution of the dampedsystem under nonunitary transformation is mixed, and the infinite-dimensional atom-photon interaction is reduced to a × k state using optical truncation (22). For × k -dimensional mixed states, logarithmic negativity is a computable measure of distillableentanglement. In this regime, information-theoretic measures such as quantum discordare no longer equal to bipartite entanglement. In this section, we study the effect ofatomic phase damping on the evolution of LN and QD, in the optically truncated,finite-dimensional states. The robustness and sudden death of distillable entanglementin arbitrary dimensional mixed states have been partly investigated in [50].Figure 4 shows the dynamics of the quantum correlation measures, LN and QD,under the effect of atomic dephasing. QD is numerically evaluated using relation (7) withprojective measurement done on the atomic subsystem. We observe that the quantumcorrelations steadily decay with increase in the scaled dephasing parameter ˜ γ (= γ/ ˜g) asit evolves in time τ . Further, we observe that as time is increased, the LN in the systemdecreases to values close to zero (around τ ≈
10) for values of dephasing parameter ˜ γ = 0.5 and 1.0. This implies that the distillable entanglement in the system vanishes attimes ( τ ≥
10) for high values of ˜ γ . However, the information-theoretic measure, QD,does not vanish with increasing time, converging at a low value. Further, for zero or lowdamping, there is distinct oscillatory characteristic for all the measures. This propertyis reminiscent of the Rabi oscillations in the system.To study the robustness of the correlation measures, we study the decay of quantum apping generalized Jaynes-Cummings interaction... τ L( ρ) γ∼ = 1.0γ∼ = 0.5γ∼ = 0.2γ∼ = 0.0 (a) τ Q( ρ) γ∼ = 1.0γ∼ = 0.5γ∼ = 0.2γ∼ = 0.0 (b) Figure 4.
Dynamics of (a) LN [ L ( ρ )] and (b) QD [ Q ( ρ )], for a two-level atominteracting with a single-mode field, with average photon number ¯ n = 10, for differentvalues of the atomic phase damping: ˜ γ = 1.0 (green down triangles), ˜ γ = 0.5 (blue uptriangles), ˜ γ = 0.2 (red circles), and ˜ γ = 0.0 (black squares). ˜ γ = γ/ ˜g, and τ = ˜gtare the scaled, dimensionless dephasing parameter and time, respectively. We observethat with increase in the scaled dephasing parameter ˜ γ , the quantum correlations decayfaster with increase in the scaled time ( τ ). correlations for highly correlated initial states. From figures 1 and 3, we see thatthe composite atom-photon system is close to maximal entanglement at τ ≈ γ = 0.0). We apply a protocol whereby the decayterm leading to nonunitary evolution is activated at τ = 2.0. Hence, the system evolvesunitarily to a near maximally entangled state till τ = 2.0, when the atomic phasedamping is applied ,and the system nonunitarily decays to lower correlated states. Thedecay of the different measures reflects the robustness of the quantum correlations underatomic phase damping. τ L( ρ) γ∼ = 1.0γ∼ = 0.5γ∼ = 0.2γ∼ = 0.0 (a) τ Q( ρ) γ∼ = 1.0γ∼ = 0.5γ∼ = 0.2γ∼ = 0.0 (b) Figure 5.
The same as figure 4, but now with atomic dephasing applied at time, τ =2.0 (maroon dotted vertical line) onwards. From figure 5, we observe that even for highly entangled initial states, the decayof quantum correlation possesses certain characteristic features. For high values ofscaled atomic dephasing (˜ γ = 1.0), the LN reduces to values close to zero. Hence, thesystem possesses no distillable entanglement. However, the decay of QD ensures thatsome residual amount of quantum correlations always remains between the atomic and apping generalized Jaynes-Cummings interaction... γ = 0.0) and a damped (˜ γ = 1.0) two-level atom interactingwith a single-mode coherent input with average photon number, ¯ n = 10. The atomicdephasing is applied at time, τ = 2.0. Hence, the effective initial field at the onset ofdamping is almost maximally entangled. For the undamped system, the evolution isunitary and the system is always pure during the dynamics. Hence, QD is equal to theentanglement of formation and LN is an upper bound on these entropic measures. In caseof the damped system, the system evolution is nonunitary, dissipative and mixed due toatomic dephasing induced by the environment. Hence, there is imminent decay of thequantum correlations. Since QD is no longer entropically equivalent to entanglement,the tight bound of LN over QD vanishes. Figure 6 shows that QD is more robust todephasing than LN. We observe that LN decreases at a greater rate and is quenchedwith increase in time. QD decreases at a relatively slower late and is never completelyquenched in the dynamics. This shows that QD is robust even at dissipation rates thatlead to sudden death of distillable entanglement.In figure 7, we show the effect of atomic phase damping on the collapse andrevival of Rabi oscillations in the system. It is known that the undamped JC modelexhibits distinct revival after the collapse of the initial system, which is the hallmarkof the nonclassical nature of photons. We observe that the dynamics of the quantumcorrelations in the damped atom-photon system complement the collapse and revival ofthe Rabi oscillations for a single-mode initial coherent field with photon number, ¯ n =10. From figure 4, it is evident that the oscillation in the correlation of the undamped(˜ γ = 0.0) system, with increasing scaled time ( τ ), fades for higher dephasing parameter,˜ γ > γ . For ˜ γ = 1.0, the revival of Rabi oscillations is almost absent, and this ismanifested in the close to zero values of LN and QD for increasing time τ . This is an τ L( ρ) Q( ρ) γ∼ = 1.0 γ∼ = 0.0 Figure 6.
Comparison of LN (black squares) and QD (red circles) for the undamped(˜ γ = 0.0) and the damped (˜ γ = 1.0) two-level atom interacting with a single-modefield, with average photon number ¯ n = 10. The atomic dephasing is applied for times, τ > τ in the presence of atomic dephasing. apping generalized Jaynes-Cummings interaction... τ -0.75-0.5-0.2500.250.50.75 τ -0.75-0.5-0.2500.250.50.750 10 20 30 40 50 τ -0.6-0.4-0.200.20.40.6 0 10 20 30 40 50 τ -0.6-0.4-0.200.20.40.6 γ∼ = 0.0γ∼ = 0.5 γ∼ = 1.0γ∼ = 0.2 Figure 7.
The collapse and revival of Rabi oscillations in a two-level atom interactingwith a single-mode field, with average photon number ¯ n = 10, for different values ofatomic dephasing (˜ γ ). We observe that for greater damping, there is no revival, whichis reminiscent of the behavior of the quantum correlations. interesting qualitative relation as the Rabi oscillations correspond to the difference inatomic level populations whereas the quantum correlations correspond to the coherenceelements of the density matrix. The nonclassical feature of the interaction contained inthe density matrix elements is highlighted in the qualitatively similar oscillations of thequantum correlations and population inversion. Hence, the nonclassical revival of Rabioscillations is ingrained in the atom-field quantum correlations.
7. Discussions
A generalized characterization and study of atom-field interaction is of great importancefor the practical implementation of the field of quantum information and computation.A simple generalized JC model interacting with multimode coherent state input is ableto generate highly correlated quantum systems that may be put to use for performingbasic quantum protocols. The important qualification for such a model is the fact thatan interaction between a discrete two-level atomic system and a continuous variablemultimode quantum field can be mapped into an effective finite-dimensional bipartitesystem interaction that contains the dynamics of the highly correlated atom-field system.Such systems can also be suitably engineered using optical truncation to simulatelow-dimensional dynamics. This enables one to study dissipation induced by atomic apping generalized Jaynes-Cummings interaction...
Acknowledgments
HSD thanks University Grants Commission, India, and AC thanks National Board ofHigher Mathematics, Department of Atomic Energy, Government of India, for financialsupport.
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