Dissipative dynamics of quantum correlations in the strong-coupling regime
aa r X i v : . [ qu a n t - ph ] M a r Dissipative dynamics of quantum correlations in the strong-coupling regime
Ferdi Altintas ∗ and Resul Eryigit † Department of Physics, Abant Izzet Baysal University, Bolu, 14280, Turkey.
The dynamics of entanglement and quantum discord between two identical qubits strongly interacting with acommon single mode leaky cavity field have been investigated beyond the rotating wave approximation (RWA)by using recently derived Lindblad type quantum optical master equation [F. Beaudoin, J.M. Gambetta, A. Blais,Phys. Rev. A , 043832 (2011)] that can describe the losses of the cavity field in a strong atom-field couplingregime. Contrary to previous investigations of the same model in the dissipative regime by using the standardLindblad quantum optical master equation in a strong-coupling regime, the atom-field steady states are foundto be cavity decay rate independent and have a very simple structure determined solely by the overlap of initialatomic state with the subradiant state which is valid for all coupling regimes. Non-RWA dynamics are foundto have remarkable effects on the steady state quantum discord and entanglement that cannot be achieved underRWA conditions, for instance, they can induce steady state entanglement even for the initial states that haveno overlap with the subradiant state. Moreover, the non-RWA dynamics are found to reverse the initial statedependence of steady state entanglement and quantum discord contrary to the RWA case. PACS numbers: 42.50.Pq, 03.65.Yz, 03.65.Ud
I. INTRODUCTION
The description of light-matter interaction is the fundamen-tal question investigated in circuit and cavity quantum elec-trodynamics (QED). The single-mode spin-boson model, theso-called quantum Rabi model, is the simplest possible physi-cal model that describes the interaction of a two-level atom (aqubit) with a quantized single mode electromagnetic field (aharmonic oscillator) [1]. Indeed, this simple model has beenused to understand a wide variety of phenomena in quantumoptics and condensed matter systems, such as quantum dots,trapped ions, superconducting (artificial) qubits, optical andmicrowave cavity QED, among others [2–4]. Despite its oldage and numerous investigations over the last few decades, thespectrum of the Rabi model has been given only recently in aclosed form as the solution of a transcendental equation witha single variable [5]. On the other hand, the Rabi model isquite simplified and can be solved analytically when the ex-citation non-conserving terms, the so-called counter-rotatingterms that are responsible for simultaneously exciting or de-exciting the atom and field, are ignored [6]. This approxi-mation is known as the rotating wave approximation (RWA)which reduces the Rabi model to the Jaynes-Cummings modeland has been widely used in quantum optical settings. TheRWA is valid only at weak atom-field coupling strengths andnearly resonant conditions. Indeed, the typical quantum opticsexperiments are performed at microwave to visible light rangeof electromagnetic wave spectrum where the atom-field cou-pling strength is several orders of magnitude smaller than thetransition frequencies of the atom and the field, so that RWAis inherently justified for those parameters. Despite being anapproximation, the analytical achievements of RWA have suc-cessfully explained many novel phenomena and experimen-tal results, such as Rabi oscillations [7], squeezing [8], non-classical [9] and Fock [10] states, collapse and revivals ofatomic inversion [11], and entanglement between atoms aswell as atom and field systems [12]. However, with the re- ∗ [email protected] † [email protected] cent developments in the area of circuit and cavity QED sys-tems [3, 4], ultra and deep strong light-matter couplings be-came experimentally achievable which makes it necessary touse full the Rabi model to explain all the dynamical and sta-tistical properties of these systems.The discussion of the role of counter-rotating terms onthe atomic dynamics has received plenty of attention, re-cently [13–18]. Although the effects of these terms are ex-tremely small and averages to zero over a very short time scalefor the weak coupling regime, they, nonetheless, cause signif-icant shifts in the dynamics of the system at intermediate todeep interaction strength region. Generation of photons [13]and entanglement [14] from zero excitation initial state, bifur-cation in the phase space [15], a fine structure in the opticalStern-Gerlach effect [16], Bloch-Siegert shift [17] and chaosin the Dicke model [18] are the remarkable examples of thesenovel effects and none of them are observed under RWA. Onthe other hand, the significance of the counter-rotating terms,especially for multipartite and dissipative systems, is still un-der debate and needs further investigation.Any realistic quantum system inevitably interacts with theoutside world which leads to decoherence. In circuit and cav-ity QED systems with atom-field interactions, this unavoid-able interaction leads to qubit relaxation and dephasing as wellas cavity relaxation. The standard Lindblad quantum opticalmaster equation [19] is the general form of a Markovian mas-ter equation that can describe the irreversible and non-unitaryprocesses of the atom-field system { see Eq. (14) in Ref. [20] } .While deriving the standard dissipators in this master equa-tion, the qubit-field interaction is neglected and therefore sep-arate interactions of the involved parts with the environmentare taken into account. For the weak atom-field couplingregime where RWA can be safely applied, this master equationworks well and can be used to predict the experimental resultsin cavity and circuit QED systems [21]. On the other hand, itis controversial to use the standard Lindblad master equationwithout any formal proof to describe the dephasing and relax-ation processes in strong coupling regime where RWA is nolonger valid. In fact, qubit-field coupling becomes influentialand leads to unphysical effects [20]; even at zero tempera-ture, the standard master equation with the Rabi model pro-duces spurious qubit flipping and creates excess photons inthe atom-field system. It was shown in Ref. [20] that it is pos-sible to correct the predictions of the Rabi model in the dissi-pative case by carefully deriving a new Lindblad type dissipa-tor which involves transitions between the eigenstates of theRabi model. The improved Lindblad master equation givenin Ref. [20] is expected to be a new avenue in the studies ofstrong light-matter interactions including dissipative and mul-tipartite systems and to help understanding the role of counter-rotating terms under dissipation and dephasing processes.In the present study, we extend the above ideas by study-ing a model of two cold-trapped atoms resonantly interactingwith a single-mode leaky cavity field. This model is quiteversatile and can be directly verifiable in circuit and cavityQED setups. Under RWA, the dynamics of the atom-fieldsystem under losses have been investigated thoroughly andthe atom-field steady states have a very interesting structuredetermined solely by the overlap between the initial state andthe decoherence free state [22–25]. On the other hand, thesame model has also been investigated under non-RWA con-ditions by using the standard Lindblad master equation [26–29]. Since the master equation with standard dissipators can-not explain properly the effect of dissipation in the strong cou-pling regime, the reported role of the counter-rotating termsshould be reconsidered. By using the improved Lindblad mas-ter equation given in Ref. [20] and the structure of atom-fieldsteady states under RWA, we provide a clear picture for therole of counter-rotating terms on atom-field steady states andalso give their analytical expressions for the general class oftwo-qubit initial states, the so-called ”extended” Werner-likestates [30] which may reduce to the Werner-like mixed statesor to the Bell-like pure states for certain conditions. We alsoreexamined some of the main results obtained by the standardLindblad master equation in the strong atom-field couplingregime and revised them by using the results obtained by theimproved Lindblad master equation. Beside this, we also an-alyzed the role of counter-rotating terms on the dynamics ofquantum correlations, such as entanglement and quantum dis-cord between two atoms and found that the counter-rotatingterms can, peculiarly, reverse the initial state dependence ofthe steady state entanglement and quantum discord contraryto the RWA case. For RWA dynamics, the overlap of ini-tial state with the decoherence free state is crucial to obtaina long-lived quantum correlated state under losses [22–25].We showed that the counter-rotating term contributions breakdown this description and lead to long-lived quantum corre-lated state without this overlap.The paper is structured as follow. In Sec. II, we present thesystem Hamiltonians for both RWA and non-RWA conditionsand the master equations that describe the losses of the cavityfield. The correlation measures, quantum discord and entan-glement, are also briefly discussed in this section. In Sec. III,we present our results. We conclude with a brief summary ofimportant results in Sec. IV. II. HAMILTONIAN AND THE MASTER EQUATION
The Rabi model describes the interaction of a two-levelatom with a single mode electromagnetic field. The Hamil- tonian can be given as (with ~ = 1 throughout) [1]: H R = ω a σ z + ω r a † a + g ( σ + + σ − )( a + a † ) , (1)where ω a ( ω r ) is the transition frequency for the atom (cavityfield), σ z is the qubit energy operator, σ ± are the spin-flipoperators and g is the atom-field coupling constant.The terms a † σ + and aσ − that appear in the interaction partof the Rabi Hamiltonian are called counter-rotating terms andthey describe ”virtual processes”; aσ − describes a process inwhich a photon is annihilated in the cavity mode as the atommakes a transition from its excited to the ground state, while a † σ + describes the creation of a photon in the cavity mode asthe atom makes an upward transition. So, the counter-rotatingterms do not conserve the total number of excitations in thesystem.The virtual processes have observable effects when g iscomparable with ω a and ω r , i.e., in strong atom-field cou-pling regime, however when the weak coupling, g << ω a , ω r ,and the nearly resonant, | ∆ | = | ω a − ω r | << ω a , ω r , condi-tions are simultaneously satisfied, these effects become negli-gible and the counter-rotating terms can be neglected in theinteraction part of the Rabi Hamiltonian. This approxima-tion is known as the rotating wave approximation (RWA) andthe Rabi model reduces to the well known Jaynes-Cummingsmodel given by the Hamiltonian [6]: H JC = ω a σ z + ω r a † a + g ( σ + a + σ − a † ) . (2)In opposition to the Rabi Hamiltonian, the total number ofexcitations in H JC is a conserved quantity which makes H JC analytically accessible.In cavity QED systems, there would be photon loss fromthe cavity due to the imperfections in the cavity mirrors. Atabsolute zero temperature ( T = 0 ) and under Markov ap-proximation, this can be represented by the standard Lindbladquantum optical master equation [19]: ˙ ρ = − i [ H, ρ ] + κD [ a ] ρ, (3)where D [ m ] ρ = (cid:0) mρm † − ρm † m − m † mρ (cid:1) , κ is thephoton leakage rate, m is the appropriate dissipator opera-tor, and ρ is the atom-field density matrix. The first term inEq. (3) describes the unitary Hamiltonian dynamics, while thesecond part accounts for the dissipation. At the weak-couplingregime, where RWA can be applied, the dissipator part of themaster equation is well known and involves field annihilationoperator, a , which is responsible for the leakage of photonsfrom the cavity. Indeed, at weak-coupling regime where RWAcan be applied, this master equation can be safely used and thedissipation can bring the system to the ground state, | g i , of H JC .Although it has been widely investigated by many groups,recently [13, 26–29, 31–34], using the above dissipator in thedissipative Rabi model in an ad hoc manner has been shownto lead to unphysical consequences [20]; at T = 0 , the dissi-pation will generate excess excitations when even no energyis added to the system and consequently the dissipation willdrive the system out of the ground state of H R . In fact, themain failure of this master equation in the strong-couplingregime is to neglect the qubit-field coupling when derivingthe dissipator part of the master equation. In Ref. [20], thequbit-field coupling have been included in the derivation ofthe master equation and the new Lindblad master equation thatdescribes the losses of the cavity field at T = 0 and strongcoupling regime under Markov approximation can be givenas: ˙ ρ = − i [ H R , ρ ] + X j,k>j Γ jkκ D [ | j i h k | ] ρ, (4)where Γ jkκ are the relaxation coefficients and are related tothe spectral density of the bath and the system-bath cou-pling constant [20]. It can be simplified as [35], Γ jkκ = κ ω k − ω j ω r | h j | ( a + a † ) | k i | . Here | j i and ω j are the eigen-states (dressed basis) and the eigenvalues of the Rabi Hamil-tonian, respectively, i.e., H R | j i = ω j | j i and the eigenstatesare labeled according to increasing energy, i.e., label | j i suchthat ω k > ω j for k > j . Indeed, as shown in Ref. [20], thenew dissipator drives the atom-field system to the true groundstate. Also, note that the standard dissipator of Eq. (3) can beobtained from that of Eq. (4) in the limit g → .Since the strong atom-field coupling regime became rele-vant with the development of the recent technology [3, 4] anda number of investigations of the dissipative Rabi model withstandard dissipator have been reported in the literature in thestrong-coupling regime [26–29, 31–34], a reexamination ofthese studies in the light of the improved master equation (4)is due. Toward that goal, in the present study, we will analyzethe dynamics of entanglement and quantum discord betweentwo identical qubits (A and B) interacting with a common sin-gle mode leaky cavity field. The Hamiltonians for non-RWAand RWA cases can be found by just replacing σ z → σ Az + σ Bz and σ ± → σ A ± + σ B ± in Eqs. (1) and (2), respectively. We willsolve the master equations (3) and (4) numerically for the con-sidered Hamiltonians and the initial states where the atoms areone of the four extended Werner-like states [30] and the cavityfield is in its vacuum: ρ Ψ ± α (0) = (cid:20) − r I AB + r (cid:12)(cid:12) Ψ ± α (cid:11) (cid:10) Ψ ± α (cid:12)(cid:12)(cid:21) ⊗ | i h | ,ρ Φ ± α (0) = (cid:20) − r I AB + r (cid:12)(cid:12) Φ ± α (cid:11) (cid:10) Φ ± α (cid:12)(cid:12)(cid:21) ⊗ | i h | , (5)where I AB is the × identity, r (0 ≤ r ≤ is the purity ofthe initial states, | Φ ± α i = α | eg i ± √ − α | ge i and | Ψ ± α i = α | ee i ± √ − α | gg i are the Bell-like states, and α (0 ≤ α ≤ is called the degree of correlations. The extendedWerner-like states are mixed and reduce to the well knownWerner-like mixed states at α = 1 / √ and to the Bell-likepure states at r = 1 . The atomic reduced density matrix can beobtained by taking a partial trace of atom-field density matrixover the cavity degrees of freedom. The above initial states inthe standard basis {| i ≡ | ee i , | i ≡ | eg i , | i ≡ | ge i , | i ≡| gg i} have an X structure in the atomic space with non-zeroelements only in its main- and anti-diagonals: ρ AB = ρ ρ ρ ρ ρ ρ ρ ρ . (6)It was shown that the standard Lindblad master equation (3)for both RWA and non-RWA Hamiltonians preserves the X structure of the density matrix [26, 28, 29]. We have checkedthat the X structure of the atomic reduced density matrix re-mains intact also under the improved master equation (4).Since the excitation number in the system is not conservedfor the Rabi model, it is worth to explain briefly the numericaltechnique to solve the master equations (3) and (4) for the two-qubit Rabi Hamiltonian. In the present study, we have care-fully checked the convergence of computed properties versusthe dimension of the cavity Fock field. We have consideredbasis vectors of type | i, j, n i , where i, j = e, g and n =0 , , , . . . M to obtain a set of differential equations for thedensity matrix elements. It is found that M ≈ gives well-converged results for the highest interaction strength consid-ered in the present work. Moreover, depending on the inter-action strength, 100-150 dressed basis of the two-qubit RabiHamiltonian are used to calculate the time evolution. Dur-ing the numerical calculations, the basic properties of densitymatrix, such as positivity, hermiticity, and trace preservationproperties are all monitored.Our next step is to determine the correlations between theatoms. The entanglement, a kind of quantum correlation,determines whether or not a given bipartite state is separa-ble. Entangled states are broadly accepted as a necessaryresource for a set of quantum tasks in quantum informationtheory [36, 37], such as quantum key distribution and tele-portation. Entanglement can be calculated through entangle-ment of formation (EoF) which is the function of the entangle-ment monotone, the so-called concurrence, for bipartite statesas [38]: C EoF ( ρ AB ) = − η log η − (1 − η ) log (1 − η ) , (7)where η = 1 / (cid:0) √ − C (cid:1) with C = 2 max { , | ρ | −√ ρ ρ , | ρ |−√ ρ ρ } being the concurrence for X states.On the other hand, quantum discord (QD) emerged as a newfundamental type of quantum correlation beyond entangle-ment [39–41]. QD as a fundamental resource is shown tobe useful in practical quantum information tasks, such asDQC1 (deterministic quantum computation with one quan-tum bit) [42], Grover search [43] algorithms and remote statepreparation [44]. Its definition is based on the difference be-tween quantum versions of two classically equivalent defini-tions of mutual information. Non-zero QD signifies that it isimpossible to extract all information about one subsystem byperforming a set of measurements on the other subsystem. QDcan also be calculated, analytically, for X states as [45]: C QD ( ρ AB ) = min { Q , Q } , (8)where Q j = h [ ρ + ρ ] + P k =1 λ k log λ k + D j with λ k being the eigenvalues of ρ AB and h [ x ] = − x log x − (1 − x ) log (1 − x ) is the binary entropy. Here D = h [ τ ] , where τ = (cid:16) p [1 − ρ + ρ )] + 4( | ρ | + | ρ | ) (cid:17) / and D = − P k =1 ρ kk log ρ kk − h [ ρ + ρ ] . QD and EoF areequal for pure states, but the relation is complicated for themixed states; there are separable mixed states with non-zeroQD [46]. Recently, the comparison attempts for the dynamicsof quantum discord and entanglement have received a greatdeal of attention for open quantum systems [47–57]. Thestudies show that QD is much more robust compared to en-tanglement under dissipative and dephasing processes whereentanglement can suffer sudden death.In the following, our main concerns will be threefold. Sincenumerous studies have been recently based on the standardmaster equation (3) in the strong-coupling regime [26–29, 31–34], our first thought is to reexamine some of the main resultsobtained in these studies by using the new Lindblad masterequation (4). Since the standard dissipator cannot describethe losses in the strong atom-field coupling regime, the roleof counter-rotating terms on the dissipative dynamics of theatom-field system are far from being understood. As the sec-ond issue, we will provide a comprehensive picture for theatom-field steady states and give their analytical expressionsfor the extended Werner-like initial states. Our final focus isto analyze the role of counter-rotating terms on the dynamicsof entanglement and quantum discord. In the following, wewill restrict ourselves to the resonant case, ω a = ω r = ω . III. RESULTS
We start our qualitative analysis by examining the main dis-agreements between the two master equations in the strong-coupling regime. In Fig. 1, we display the effect of cavitydecay rate, κ , on the dynamics of QD and EoF for the vac-uum ( | gg i ) initial state at g = 0 . ω . The results in thetop plots are obtained by the use of the standard master equa-tion [Eq. (3)], while the bottom plots display the results for thenew dissipator [Eq. (4)] in the case of two-qubit Rabi Hamilto-nian. Since the vacuum state contains no initial excitations tostart with and the Jaynes-Cummings Hamiltonian conservesthe total excitation number in the atom-field system, no typeof correlations will be induced in the time evolution underRWA. On the other hand, the counter-rotating terms in theRabi Hamiltonian will generate virtual excitations and conse-quently non-zero correlations between the atoms which can betrapped at a non-zero value under cavity losses for both cases.If we study the effect of cavity losses on the dynamics of cor-relations by using the standard dissipator [Figs. 1(a) and 1(b)]as was done in Refs. [29, 31–33], the results seem to be quitesurprising; as κ increases, although the disentanglement timeand the maximum of EoF decreases, the steady state QD, pe-culiarly, increases with κ and is saturated in the maximumpossible value ( C QD max = 1 / ) in the absence of entangle-ment for a very bad quality cavity case ( κ/ω > ) [29].Although these results seem to indicate a surprising possibil-ity of dissipation enhancement of quantum discord [29] (aswell as atomic energy as found in Refs. [31–33]) to its max-imally possible value for a separable state, its source is theerror in the dissipator which creates excess excitations in thesteady state. Considering the role of κ on the dynamics ofQD [Fig. 1(c)] and EoF [Fig. 1(d)] for the new dissipatordemonstrates that the steady states are κ independent; QDand EoF reaches 0.069 and 0.0086 in the long time limit, re-spectively, regardless of the magnitude of κ . In fact, κ onlydetermines how fast the correlations reach their steady statevalues. This is quite understandable, since κ is just a pa-rameter that determines how fast the photons can escape fromthe cavity mirrors, it should control the speed of reaching thesteady state rather than its magnitude in the long time limit.Bath decay dependent steady state shifts of several proper-ties of the system in the strong-coupling regime have beenrecently investigated by using the standard dissipator [29, 31– 34]. Moreover, Ma et al. in Ref. [58] studied the dynam-ics of entanglement between two qubits strongly interactingwith a common Lorentz-broadened cavity mode at zero tem-perature by using a more general master equation approach,the so-called hierarchical equation method without employ-ing rotating wave, Born, and Markovian approximations. Oneshould note that the approach in Ref. [58] is quite differentcompared to the standard Lindblad master equation case, butstill the asymptotic values of entanglement are found to be dis-sipation strength dependent, which is in contradiction to ourfindings. The simple example shown in Fig. 1 indicates thatsome of the previously obtained results with the Rabi modelneed to be reexamined. As shown in Fig. 1(b) for the standarddissipator case, the atomic steady states become separable as κ > . ω [29], while the steady states can posses entangle-ment for the new dissipator case [Fig. 1(d)].Now, we will consider the steady states of RWA and non-RWA dynamics. The steady state density matrix under cav-ity losses and RWA dynamics investigated by the standardmaster equation (3) are known to be κ -independent [22, 23].As shown in the bottom plots of Fig. 1, they are also κ -independent in the case of non-RWA dynamics with the im-proved dissipator. Therefore, in the following, we will set κ = 0 . ω .The dynamics under RWA with losses given by Eq. (3) havebeen investigated by many groups, which indicate that theatom-field steady states have a simple structure. We followthe ideas developed in Refs. [22–25] to determine those states.The cavity decay tends to throw out any initial excitationsfrom the cavity mirror and consequently it drives the atom-field system to the ground state ( | gg i ) of two-qubit H JC .On the other hand, there exist a highly entangled subradiantstate (decoherence-free state) for this model due to the inter-action of the atoms with a common environment which leadsto an indirect qubit-qubit interaction [22, 23]. This subradiantstate for this model can be simply determined by finding theeigenvector of two-qubit H JC for zero eigenvalue, which isthe maximally entangled (Bell) state in the atomic subsystemgiven as | Φ − i = √ ( | eg i − | ge i ) . If an initial atomicstate has a non-vanishing overlap with the subradiant state, b = h Φ − | ρ AB (0) | Φ − i , the amount of overlap determined by b would be trapped in the subradiant state, while the remainingpart, − b , would decay to the ground state. Therefore, de-pending on the specific initial state [59], the atom-field steadystates would be a combination of | gg i and | Φ − i which canbe written as: ρ SS = (1 − b ) | gg i h gg | + b (cid:12)(cid:12) Φ − (cid:11) (cid:10) Φ − (cid:12)(cid:12) . (9)This, indeed, explains why the steady states under RWA withlosses are κ and g independent. By using Eq. (9) or by study-ing the solution of the master equation (3), for example by themethod of solving first order differential equations via eigen-values and eigenvectors or the pseudomode approach [23], wecan determine the analytic form of the steady states for the ex-tended Werner-like initial states (5). For ρ Ψ ± α (0) type initialstates, the steady states can be found as: ρ SS Ψ ± α = (cid:18) r (cid:19) | gg i h gg | + (cid:18) − r (cid:19) (cid:12)(cid:12) Φ − (cid:11) (cid:10) Φ − (cid:12)(cid:12) . (10) Κ= ΩΚ=
ΩΚ= ΩΚ= Ω H a L Ω t QD H b L Ω t E o F H c L Ω t QD H d L Ω t E o F FIG. 1. (Color online) The effect of cavity decay rate κ (a) and (c) on quantum discord and (b) and (d) entanglement versus ωt for | gg i initial state, κ = 0 . ω (black, solid), κ = 0 . ω (red, dashed), κ = 2 ω (blue, dotted), and κ = 20 ω (green, dot-dashed) in strong couplingregime with g = 0 . ω . The top figures are obtained by using the standard dissipator Eq. (3), while the bottom plots are for the improveddissipator Eq. (4). For r = 1 , i.e., for initially pure Bell-like states, | Ψ ± α i , ρ Ψ ± α (0) has no overlap with the subradiant state and all theexcitations are lost in time evolution regardless of what α is,while for ≤ r < , there would be a non-vanishing overlapterm with the subradiant state determined by r and the atom-field system will decay to a trapping state which is superposi-tion of | gg i and | Φ − i . On the other hand, for ρ Φ ± α (0) typeinitial states, the steady states can be determined as: ρ SS Φ ± α = (cid:18) − r ± rα p − α (cid:19) | gg i h gg | + (cid:18) r ∓ rα p − α (cid:19) (cid:12)(cid:12) Φ − (cid:11) (cid:10) Φ − (cid:12)(cid:12) . (11)One should note that ρ Φ ± α (0) have always a non-vanishingterm [except for ρ Φ + α (0) at α = √ , r = 1 ] with the subradi-ant state, so the steady states are highly α and r dependent. | Φ − i is also the subradiant state for two-qubit H R , whichcan be proven by noting that the Hilbert space H of theRabi model can be separated into two uncoupled subspaces H and H with H = √ ( | eg i − | ge i ) ⊗ H cavity where theseparation is also valid under decoherence mechanism and atthe long time limit the state in H becomes the dark state, √ ( | eg i − | ge i ) ⊗ | i , independent of the coupling strength.Since H JC is the limiting case of H R and the master equa-tion (4) can describe the cavity losses in the strong-couplingregime, we conjecture that such a simple structure for theatom-field steady states as in Eqs. (10) and (11) should ex-ist in the strong-coupling regime also. Below we will provide numerical evidence for our conjecture.We start to analyze the dynamics of correlations in thestrong-coupling regime by using the improved master equa-tion (4) for the type of initial states which have no overlapwith the subradiant state. Thus, in Fig. 2 we display QDand EoF versus ωt for the initial state ρ Ψ + α (0) with r = 1 , g = 0 . ω , and κ = 0 . ω and for several α values. In the caseof RWA, the steady state is the vacuum which has no typesof correlations. On the other hand, as shown in Fig. 2, thecavity decay in the strong-coupling regime drives the atom-field to a state which can carry appreciably high quantum dis-cord ( C QD = 0 . ) and entanglement ( C EoF = 0 . ) in-dependent of what α is. In fact, we have observed that forthe ρ Ψ ± α (0) type of initial states with r = 1 , the atom-fieldsteady state is the ground state of two-qubit H R regardlessof g . One can observe that the most important difference be-tween the RWA and non-RWA dynamics at the long-time limitis the existence of non-zero QD and EoF created by the virtualprocesses; for RWA dynamics the necessary condition for thelong-lived correlated state is the overlap of initial state withthe subradiant state [22–25], while this can be achieved with-out the overlap under non-RWA conditions because of the en-tangled nature of the ground state. In the insets of Fig. 2 wedisplay the g dependence of the ground state QD and EoF.Both QD and EoF as a function of g/ω display a peculiarstructure; they have a single maximum at the intermediatevalues of g/ω . Furthermore, g/ω dependence of EoF is al-most Gaussian. For small g/ω , where the RWA regime isvalid, the QD and EoF are nearly zero, since the ground stateis close to the vacuum state, while the correlations are high H a L (cid:144) Ω QD Ω t QD H b L (cid:144) Ω E o F Ω t E o F FIG. 2. (Color online) Effects of degree of correlations α on (a) QDand (b) EoF versus ωt for ρ Ψ + α (0) initial state with r = 1 , g =0 . ω , κ = 0 . ω , and α = 0 (black, solid), α = 0 . (red, dashed), α = 1 / √ (blue, dotted), α = 0 . (green, dot-dashed) and α =1 (yellow, dot-dot-dashed). Here the results are obtained by usingtwo-qubit H R and the master equation, Eq. (4). Inset: QD and EoFof the ground state versus g/ω . Around g/ω ≈ . , the groundstate of the Rabi model becomes almost twofold degenerate. So, weconsider only g/ω ∈ [0 , . range in the insets. between . < g/ω < . . Note that for the g/ω regions,where EoF is absent, non-zero QD still exists especially for g/ω > . . Also note that the ground state QD and EoFshow peaks at different g/ω values; QD is maximum nearly at g/ω ≈ . , while EoF is at g/ω ≈ . . On the other hand,the decrease in both QD and EoF as a function of g/ω at verystrong-coupling regime might be an indication of the inabilityof the Rabi model to account for the possible nonlinear effectsat large g values, which may require higher order terms in thedescription of atom-field interactions [60].Special attention should also be given to the initial Bell | Φ + i = √ ( | eg i + | ge i ) and Bell-like | Ψ ± α i = α | ee i ±√ − α | gg i states in the strong-coupling regime, since theentanglement dynamics have been extensively analyzed byusing the standard dissipator (3) under losses [26–29]. For κ ≥ . ω , entanglement between atoms becomes zero or use-less (smaller than − ) for a wide range of g , especially be-tween . < g/ω < under the standard dissipator case [26–29]. Contrary to those findings, one can see that such statescan contain appreciably high EoF for . < g/ω < . inde-pendent of κ as shown in the inset of Fig. 2(b), when one usesthe improved master equation. H a L RWA g = Ω g = Ω g = Ω g = Ω g = Ω r QD H b L RWAg = Ω g = Ω g = Ω g = Ω g = Ω r E o F FIG. 3. (Color online) Steady state (a) QD and (b) EoF versusinitial purity r for ρ Ψ ± α (0) initial states under RWA (red, dashed)and non-RWA dynamics with g = 0 . ω (yellow, dot-dot-dashed), g = 0 . ω (black, solid), g = 0 . ω (blue, dotted), g = 0 . ω (green,dot-dashed) and g = 0 . ω (black, long-dotted). Note that for g < . ω , the results obtained by the new master equation (4) co-incide with the RWA case (red, dashed) given by Eq. (10) and thesteady states for both cases are α independent. Now we consider the dynamics of initial states that havenon-zero overlap with the dark state, | Φ − i . The effect of r for ρ Ψ ± α (0) initial states on the dynamics of QD and EoF un-der losses have been investigated and the r dependence of thesteady state correlations are plotted in Fig. 3 for both RWAand non-RWA cases. Before discussing the role of counter-rotating terms on the steady state quantum correlations as afunction of r , we should stress that a detailed analysis demon-strates that the atom-field steady states in the strong-couplingregime are in the form of Eq. (10) with just | gg i replaced bythe ground state, (cid:12)(cid:12)(cid:12)g gg E , of two-qubit H R and can be writtenas: ρ SS Ψ ± α = (cid:18) r (cid:19) (cid:12)(cid:12)(cid:12)g gg E Dg gg (cid:12)(cid:12)(cid:12) + (cid:18) − r (cid:19) (cid:12)(cid:12) Φ − (cid:11) (cid:10) Φ − (cid:12)(cid:12) . (12)Since the dark state of two-qubit H JC is also the dark stateof two-qubit H R , the form of the atom-field steady state fornon-RWA Hamiltonian [Eq. (12)] is in expected form basedon the same arguments that lead to Eq. (10). More precisely,the overlap of the atomic initial state with the subradiant stateis trapped in the subradiant state, while the remaining partdecays to the ground state of the open quantum system un-der consideration. Also note that in the small g limit (cid:12)(cid:12)(cid:12)g gg E reduces to | gg i . Therefore, the overall role of the counter-rotating term contributions in the steady states stems fromthe ground state of the two-qubit Rabi Hamiltonian. We havechecked the form of Eq. (12) extensively by numerical meansand found that it holds for all the initial states considered inthe present work.The initial purity dependence of steady state QD and EoFdisplayed in Fig. 3 for the RWA case shows an interesting fea-ture; steady state values of both correlation measures showan inverse dependence on r . Since r is a measure of initialmixedness of the state and the dissipation dynamics lead toan increase of the mixedness of the initial state, this findingmight seem to be contradictory, but it stems from the fact thatas r decreases the overlap between the initial state and themaximally correlated dark state increases and the final steadystate would contain a larger fraction of | Φ − i which leads tohigher QD and EoF. In the strong-coupling regime, the groundstate is g dependent, which makes QD and EoF dependenton g also. In fact, the counter-rotating contributions compli-cate the simple RWA picture and can induce separable atomicsteady states for a wide range of r . Similar to the RWA case,the steady state correlations can display inverse r dependencefor a wide range, but contrary to the RWA case, the correla-tions can also be linearly r dependent in the strong-couplingregime. Moreover, a generic result can be obtained from thecomparison of the magnitudes of QD and EoF in Fig. 3; for theregions where EoF is zero (especially for strong couplings),appreciably high QD can still exist in the steady states.To further elucidate the role of initial state and virtual pro-cesses on the steady state quantum correlations, we displayQD versus α for ρ Φ ± α (0) with r = 1 in Fig. 4 and QD ver-sus r for the same initial states with α = 0 . in the insetsof Fig. 4. Similar to the ρ Ψ ± α (0) case, our detailed numericalanalysis shows that the atom-field steady states under non-RWA dynamics for ρ Φ ± α (0) type initial states are in the formof Eq. (11) with the vacuum state replaced by the ground stateof the Rabi Hamiltonian, i.e., ρ SS Φ ± α = (cid:18) − r ± rα p − α (cid:19) (cid:12)(cid:12)(cid:12)g gg E Dg gg (cid:12)(cid:12)(cid:12) + (cid:18) r ∓ rα p − α (cid:19) (cid:12)(cid:12) Φ − (cid:11) (cid:10) Φ − (cid:12)(cid:12) . (13)Concisely analyzing the steady state QD dependence on theinitial state parameters under RWA (red, dashed line) signifiesthat the magnitude of QD directly depends on the amount ofoverlap with the subradiant state. On the other hand, strongvirtual processes can stir the simple RWA picture and inter-estingly can reverse the initial state parameter dependence ofthe steady QD compared to the RWA case. The α and r de-pendence of steady state EoF is qualitatively similar to thatof QD with a difference as previously indicated in Fig. 3(b);EoF can suffer death in the steady states for a wide range of α and r under non-RWA dynamics, especially at g = 0 . ω and g = 0 . ω . Thus, they are not plotted here. QD H a L Ρ F Α+ H L RWA g = Ω g = Ω g = Ω g = Ω g = Ω Α QD QD H b L Ρ F Α- H L Α QD FIG. 4. (Color online) Steady state QD versus degree of correlations α for (a) ρ Φ + α (0) and (b) ρ Φ − α (0) initial states with r = 1 underRWA (red, dashed) and non-RWA dynamics with g = 0 . ω (yel-low, dot-dot-dashed), g = 0 . ω (black, solid), g = 0 . ω (blue,dotted), g = 0 . ω (green, dot-dashed), and g = 0 . ω (black, long-dotted). Inset: Steady state QD versus r for (a) ρ Φ + α (0) and (b) ρ Φ − α (0) initial states with α = 0 . and the same g values used inthe subfigures. Note that for g < . ω , the results obtained by thenew master equation (4) coincide with the RWA case (red, dashed)given by Eq. (11). IV. CONCLUSION
We have studied a model of two cold-trapped two-levelatoms strongly coupled to a single-mode leaky cavity fieldand initially prepared in extended Werner-like states. The dy-namics of the atom-field system and quantum correlations be-tween atoms, such as quantum discord and entanglement inthe strong-coupling regime, have been investigated by usingthe recently derived Lindblad type master equation which of-fers quantum jumps among the eigenstates of the Hamiltonianof the open quantum system under consideration. Our resultsdemonstrate that for extended Werner-like initial states, theamount of overlap between initial and dark states remains in-tact in the time evolution, while the other part decays to theground state of the atom-field system. This result is foundto hold for all atom-field coupling strengths and coincides atthe weak-coupling regime with the results under RWA stud-ied by the standard Lindblad master equation. The presence ofcounter-rotating terms are found to break down some of the re-cently predicted results for RWA dynamics, for example, theylead to quantum-correlated atomic steady states without theoverlap of an initial state with the dark state which is crucialfor RWA conditions. Moreover, the virtual processes can re-verse the initial state parameter dependence of steady state QDand EoF contrary to the RWA case which is strongly initial-dark state overlap dependent. However, we should stress herethat the strong coupling regime seems to have quantum advan-tages compared to the RWA case only for initial states whichlead to low ( C QD < . ) and zero discordant atomic steadystates for the RWA case. Specifically, the strong virtual pro-cesses can induce separable atomic steady states.The results obtained in the present study are quite simpleand general. 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