Generation of two-mode entangled states by quantum reservoir engineering
Christian Arenz, Cecilia Cormick, David Vitali, Giovanna Morigi
GGeneration of two-mode entangled states by quantum reservoir engineering
Christian Arenz, Cecilia Cormick,
1, 2
David Vitali, and Giovanna Morigi Theoretische Physik, Universit¨at des Saarlandes, D 66123 Saarbr¨ucken, Germany Institute for Theoretical Physics, Universit¨at Ulm, D 89081 Ulm, Germany School of Science and Technology, Physics Division, University of Camerino, Camerino (MC), Italy (Dated: October 31, 2018)A method for generating entangled cat states of two modes of a microwave cavity field is proposed.Entanglement results from the interaction of the field with a beam of atoms crossing the microwaveresonator, giving rise to non-unitary dynamics of which the target entangled state is a fixed point. Weanalyse the robustness of the generated two-mode photonic “cat state” against dephasing and lossesby means of numerical simulation. This proposal is an instance of quantum reservoir engineering ofphotonic systems.
PACS numbers: 42.50.Dv, 03.67.Bg, 03.65.Ud, 42.50.Pq
I. INTRODUCTION
Quantum reservoir engineering generally labels a strat-egy at the basis of protocols which make use of the non-unitary evolution of a system in order to generate robustquantum coherent states and dynamics [1]. The idea isin some respect challenging the naive expectation, thatin order to obtain quantum coherent dynamics one shallwarrant that the evolution is unitary at all stages. Due tothe stochastic nature of the processes which generate thetarget dynamics, strategies based on quantum reservoirengineering are in general more robust against variationsof the parameters than protocols solely based on uni-tary evolution [1–3]. A prominent example of quantumreservoir engineering is laser cooling, achieving prepara-tion of atoms and molecules at ultralow temperatures bymeans of an optical excitation followed by radiative de-cay [4]. The concept of quantum reservoir engineeringand its application for quantum information processinghas been formulated in Refs. [5, 6], and further pursuedin Refs. [7–9]. Proposals for quantum reservoir engineer-ing of quantum states in cavity quantum electrodynamics[10–15] and many-body systems [1, 2, 16, 17] have beenrecently discussed in the literature and first experimentalrealizations have been reported [18–20]. Applications forquantum technologies are being pursued [20–23].In this article we propose a protocol based on quantumreservoir engineering for preparing a cavity in a highlynonclassical entangled “cat-like” state. This protocol isapplicable to the experimental setup realized in [24, 25],which is pumped by a beam of atoms with random ar-rival times. In this setup the system dynamics intrinsi-cally stochastic due to the impossibility of controlling thearrival times of the atoms, but only their rate of injec-tion, and the finite detection efficiency. The protocol wediscuss allows one to generate and stabilize an entangledstate of two modes of a microwave resonator, by means ofan effective environment constituted by the atoms. Weshow that when the internal state of the atoms enter-ing the cavity is suitably prepared and external classicalfields couple the atomic transitions, then the asymptotic
FIG. 1. A high-finesse microwave resonator is pumped bya beam of atoms with random arrival times. Two modes ofthe cavity are coupled to two atomic transitions, which aredriven by external lasers while interacting with the fields. Thefields undergo non-unitary dynamics, whose asymptotic stateis an entangled state as in Eq. (1). These dynamics could beimplemented in the experimental setup of Ref. [27]. state of the cavity modes takes the form | ψ ∞ (cid:105) = ( | α (cid:105) A | α (cid:105) B + | − α (cid:105) A | − α (cid:105) B ) / N , (1)where | α (cid:105) j denotes a coherent state of mode j = A, B with complex amplitude α and N = (cid:112) − | α | )]is the normalization constant.Our proposal extends previous works of some of us,which are focussed on generating two-mode squeezing ina microwave cavity [10] and entangling two distant cav-ities using a beam of atoms [11]. The state of Eq. (1)whose robust generation is proposed here is not simplyentangled but possesses strongly nonclassical features,being a nonlocal macroscopic superposition state simi-lar to those discussed in Ref. [26]. The setup we consideris sketched in Fig. 1, and is similar to the one realized inRef. [25, 27].This work is structured as follows. In Sec. II we sketchthe general features of our proposal. Section III presentsa method to engineer each of the target dynamics startingfrom the Hamiltonian of an atom of the beam, whichinteracts with the cavity for a finite time. Results fromnumerical simulations are reported and discussed in Sec.IV. The conclusions are drawn in Sec. V. a r X i v : . [ qu a n t - ph ] M a r II. TARGET MASTER EQUATION ANDASYMPTOTIC STATE
Let ρ be the density matrix for the degrees of freedomof the two cavity modes and ρ ∞ = | ψ ∞ (cid:105)(cid:104) ψ ∞ | the targetstate we want to generate with | ψ ∞ (cid:105) in Eq. (1). Thepurpose of this section is to derive the master equation ∂∂t ρ = L ρ , (2)for which ρ ∞ is a fixed point, namely, L ρ ∞ = 0 . (3)In order to determine the form of the Lindbladian L wefirst introduce the operators a and b which annihilate aphoton of the cavity mode A and B, respectively. It issimple to show that ρ ∞ is a simultaneous right eigenop-erator at eigenvalue zero of the Liouvillians L j ρ = γ j (2 C j ρC † j − { C † j C j , ρ } ) , j = 1 , γ j rates which are model-dependent and where theoperators C j read C = a − b √ , C = 2( ab − α ) . (5)In fact, | ψ j (cid:105) is eigenstate of C and C with eigenvalue 0, C j | ψ ∞ (cid:105) = 0. The procedure we will follow aims at con-structing effective dynamics described by the Liouvillian L = L + L (6)by making use of the interaction with a beam of atoms.Before we start, we shall remark on two importantpoints. In first place, the state ρ ∞ is not the uniquesolution of Eq. (3) when L = L + L . Indeed, states | α (cid:105) A | α (cid:105) B and | − α (cid:105) A | − α (cid:105) B , and any superposition ofthese two states, are also eigenstates of both C and C ateigenvalue zero. We denote the corresponding eigenspaceby H d , which is a subspace of the Hilbert space of allstates of the two cavity modes. The most general sta-tionary state of L can be written as a statistical mixture, ρ ss = (cid:80) d p d | ψ d (cid:105)(cid:104) ψ d | [28], where the sum spans over allthe states | ψ d (cid:105) ∈ H d , and p d are real and positive scalarssuch that (cid:80) d p d = 1.Nevertheless, for the evolution determined by the Lind-bladian of Eq. (6) the state ρ ∞ is the unique asymptoticstate provided that the initial state is the vacuum statefor both cavity modes, ρ = | A , B (cid:105)(cid:104) A , B | . This canbe shown using the parity operator defined asΠ + = ( − c † + c + (7)with c ± = ( a ± b ) / √
2. Operator Π + commutes with theoperators C and C , since C = c − , C = c − c − − α . (8) Therefore, if the initial state can be written as statisti-cal mixture of eigenstates of Π + with eigenvalue +1, thetime-evolved state will also be a statistical mixture ofeigenstates with eigenvalue +1, and so will be the steadystate. In particular, | ψ ∞ (cid:105) is the only state of subspace H d which is eigenstate of Π + with eigenvalue +1, namely,Π + | ψ ∞ (cid:105) = | ψ ∞ (cid:105) , and thus, under this condition, theasymptotic state will be pure and given by ρ ∞ . Here wewill assume just this situation, i.e., that the cavity modesare initially prepared in the vacuum state, which is aneven eigenvalue of operator Π + , and which represents avery natural initial condition.These considerations are so far applied to the idealcase in which the dynamics of the cavity modes densitymatrix are solely determined by Liouvillian L in Eq. (6).In this article we will construct the dynamics in Eq. (6)using a beam of atoms crossing with the resonator, as itis usual in microwave cavity quantum electrodynamics.We will then analyze the efficiency of generating state ρ ∞ at the asymptotics of the interaction of the cavity withthe beam of atoms, taking also into account experimentallimitations. III. ENGINEERING DISSIPATIVE PROCESSES
Our starting point is the Hamiltonian for the coher-ent dynamics of an atom whose selected Rydberg transi-tions quasi-resonantly couple with the cavity modes. Theatoms form a beam with statistical Poissonian distribu-tion in the arrival times. The mean velocity determinesthe average interaction time τ during which each atominteracts with the cavity field, while the arrival rate r is such to warrant that rτ (cid:28)
1, namely, the probabilitythat two atoms interact simultaneously with the cavityis strongly suppressed. The master equation for the den-sity matrix χ describing the dynamics of the cavity modescoupled with one atom reads ∂∂t χ = 1i (cid:126) [ H, χ ] + κ K χ , (9)with H the Hamiltonian governing the coherent dynamicsand K χ = 2 aχa † + 2 bχb † − { a † a, χ } − { b † b, χ } (10)the superoperator describing decay of the cavity modes atrate κ . The field density matrix is found after tracing outthe atomic degrees of freedom, and formally reads ρ ( t ) =Tr at { χ ( t ) } . In the following we will specify the form ofHamiltonian H and derive an effective master equationfor the density matrix ρ of the cavity field interactingwith a beam of atoms, which approximates the dynamicsgoverned the Liouvillian L in Eq. (6).In the following we shall analyze separately each of theprocesses corresponding to the two types of Lindblad su-peroperators composing the sum in Eq. (6). Note thatcavity losses are detrimental, as they do not preserve theparity Π + of the state of the cavity. In the rest of thissection they will be neglected, their effect will be consid-ered when calculating numerically the efficiency of theprotocol. A. Realization of the Lindblad superoperator L . We now show how to implement the dynamics de-scribed by the Lindblad superoperator L . For this pur-pose, we assume that the atomic transitions effectivelycoupling with the cavity modes form a Λ-type configu-ration of levels, as schematically represented in Fig. 2.The interaction of a single atom with the cavity modesis governed by the Hamiltonian H = (cid:126) ω a a † a + (cid:126) ω b b † b + (cid:126) ω σ , + (cid:126) ω σ , (11)+ (cid:126) ( g a a † σ , + g b b † σ , + H . c . ) , where ω a and ω b are the frequencies of the cavity modes, ω ( ω ) is the energy of level | (cid:105) ( | (cid:105) ), here setting theenergy of level | (cid:105) to zero, g a and g b are the vacuumRabi frequencies characterizing the strength of the cou-pling of the dipolar transitions | (cid:105) → | (cid:105) and | (cid:105) → | (cid:105) ,respectively, with the corresponding cavity mode, and σ j,k = | j (cid:105)(cid:104) k | is the spin-flip operator. In the following weassume that the transitions are resonant, i.e. ω a = ω and ω b = ω − ω . FIG. 2. Relevant atomic levels and couplings leading to thedynamics which realizes the Lindblad superoperator L . Theatom is prepared in state |−(cid:105) , Eq. (13). In the reference frame rotating with the cavity modes,the Hamiltonian can be rewritten as H = (cid:126) √ g a g b g ( c †− σ − , + c (cid:48)† + σ + , + H . c . ) , (12)where g = (cid:112) g a + g b , c − is defined in Eq. (8) and σ ± , = |±(cid:105)(cid:104) | , with |−(cid:105) = g b | (cid:105) − g a | (cid:105) g , | + (cid:105) = g a | (cid:105) + g b | (cid:105) g , (13)while c (cid:48) + is a superposition of modes a and b . This repre-sentation clearly shows that, if the atoms are injected inthe state |−(cid:105) and interact with the resonator for a time τ such that gτ (cid:28)
1, they may only absorb photons ofthe “odd” mode c − . More precisely, the condition to befulfilled is gτ (cid:112) N − + 1 / (cid:28)
1, where N − is the meannumber of photons in the odd mode, N − = (cid:104) c †− c − (cid:105) . Inthis case, if ρ ( t ) is the state of the field at the instant inwhich an atom in state |−(cid:105) is injected, the state of the field ρ at time t + τ reads ρ ( t + τ ) = ρ ( t ) + g a g b g τ (cid:104) c − ρ ( t ) c †− − { c †− c − , ρ ( t ) } (cid:105) . (14)This corresponds to the desired process, which drives theodd mode into the vacuum state. Here, we neglect correc-tions that are smaller by a factor of order g τ ( N − +1 / |−(cid:105) are injected atrate r with r τ (cid:28)
1, the probability of having twoatoms simultaneously inside the cavity can be neglected.In this case the field evolution can be analysed on acoarsed-grained time scale ∆ t such that ∆ t (cid:29) τ and r ∆ t (cid:28)
1. After expressing the differential quotient[ ρ ( t + ∆ t ) − ρ ( t )] / ∆ t as a derivative with respect to timeone recovers the master equation [11] ∂∂t ρ ( t ) (cid:39) γ (cid:104) c − ρ ( t ) c †− − { c †− c − , ρ ( t ) } (cid:105) , (15)which corresponds to the dynamics governed by super-operator L in Eq. (4). Here, γ = r g a g b g τ . (16)We note that Eq. (15) is valid as long as higher or-der corrections are negligible. This condition providesan upper bound to the rate γ , i.e., γ (cid:28) r . However, itis not strictly necessary that the dynamics take place inthis specific limit: One can indeed speed up the processof photon absorption from the odd mode taking longerinteraction times between the atom and the cavity. Inthis case, the form of the master equation is different,but one could obtain absorption of photons from the oddmode. We refer the reader to Ref. [11], where the re-quired time has been characterized for a similar proposalin the different regimes. B. Realization of the Lindblad superoperator L . The dynamics described by the Lindblad operator L ,Eq. (6), can be realized using a level scheme as shown inFig. 3. We denote by ω (cid:48) j the frequency of the atomic state | j = 2 , (cid:105) , such that ω (cid:48) > ω (cid:48) > ω (cid:48) = 0. The transition issuch that ω (cid:48) = ω a + ω b .A laser drives resonantly the transition | (cid:48) (cid:105) → | (cid:48) (cid:105) , sothat the frequency ω L = ω (cid:48) = ω a + ω b . In the framerotating at the frequency of the cavity modes the Hamil-tonian governing the coherent dynamics reads H = (cid:126) ∆ σ (cid:48) (cid:48) + (cid:126) ( g (cid:48) a a † σ (cid:48) (cid:48) + g (cid:48) b b † σ (cid:48) (cid:48) +Ω σ (cid:48) (cid:48) +H . c . ) , (17)where ∆ = ω (cid:48) − ω a . We assume that g (cid:48) a (cid:112) (cid:104) n a (cid:105) , g (cid:48) b (cid:112) (cid:104) n a (cid:105) (cid:28) | ∆ | , with (cid:104) n j (cid:105) the mean numberof photons in the cavity mode j = A, B , and analyzethe state of the cavity field after it has interacted with
FIG. 3. Relevant atomic levels and couplings leading to thedynamics which approximates the Lindblad superoperator L .A classical field of amplitude Ω drives resonantly the transi-tion | (cid:48) (cid:105) → | (cid:48) (cid:105) . This transition is also resonantly driven bytwo-photon processes, in which a photon of cavity mode Aand a photon of cavity mode B are simultaneously absorbedor emitted. These dynamics dominate over one-photon pro-cesses by choosing the detuning | ∆ | sufficiently larger thanthe coupling strengths g (cid:48) a , g (cid:48) b . an atom which is injected in state | (cid:48) (cid:105) . The interactiontime is denoted by τ and is chosen such that | ∆ | τ (cid:29) g (cid:48) j (cid:104) n j (cid:105) τ / | ∆ | (cid:28)
1. The density matrix for the cavityfield at time t + τ can be cast in the form [11] ρ ( t + τ ) = ρ ( t ) + 18 (cid:18) g (cid:48) a g (cid:48) b τ ∆ (cid:19) (cid:34) C ρC † − (cid:110) C † C , ρ f (cid:111) (cid:35) + i g (cid:48) a ∆ (∆ τ − sin ∆ τ ) [ a † a, ρ ]+2 g (cid:48) a ∆ sin (cid:18) ∆ τ (cid:19) (cid:0) aρa † − { a † a, ρ } (cid:1) − (cid:18) g (cid:48) a τ ∆ (cid:19) [ a † a, [ a † a, ρ ]] , (18)where ρ ( t ) is the density matrix before the interactionand C = 2( ab − α ), Eq. (5). Here, α = Ω∆ / ( g (cid:48) a g (cid:48) b ),showing that the number of photons at the asymptoticsis determined by Ω. Equation (18) has been derived inperturbation theory and by tracing out the degrees offreedom of the atom after the interaction. The first lineof Eq. (18) describes two-photon processes leading to thetarget dynamics at a rate determined by the frequency γ (0)2 = 18 (cid:18) g (cid:48) a g (cid:48) b τ ∆ (cid:19) , while the terms in the other lines are unwanted processes,which occur at comparable rates and therefore lead tosignificant deviations from the ideal behaviour. The sec-ond line of Eq. (18), in particular, corresponds to one-photon processes on the transition | (cid:48) (cid:105) → | (cid:48) (cid:105) , leading tophase fluctuations of the cavity mode A. The third linedescribes losses of mode A due to one-photon processes,and the last line gives dephasing effects of cavity mode Aassociated with two-photon processes. Other detrimentalprocesses, leading to dephasing and amplification of thefield of cavity mode B, have been discarded under the as-sumption that the corresponding amplitude is of higher order. This assumption is correct as long as the ampli-tude Ω, determining the number of photons, is chosento be of the order of g (cid:48) j / ∆ and fulfills the inequalities( | ∆ | τ )(Ω τ ) (cid:29) τ (cid:28)
1. This is therefore a re-striction over the size of the cat state one can realize bymeans of this procedure.Let us now discuss possible strategies in order to com-pensate the effect of the unwanted terms in Eq. (18).We first consider the term in the second line. This termscales with g (cid:48) a τ / ∆ and is larger than γ (0)2 . It can becompensated by means of a term of the same ampli-tude and opposite sign. This can be realized by con-sidering another atomic transition which is quasi reso-nant with the same cavity field, say, a third transition | aux (cid:105) → | aux (cid:105) such that cavity mode A couples withstrength g aux and detuning ∆ aux with the dipolar tran-sition with | ∆ aux | (cid:29) g aux . If the atom is prepared inthe superposition cos( ϕ ) | (cid:48) (cid:105) + sin( ϕ ) | aux (cid:105) before beinginjected into the cavity, then the coherent dynamics aregoverned by Hamiltonian H (cid:48) = H + h aux , with h aux = (cid:126) ∆ (cid:48) σ aux aux + (cid:126) g aux ( a † σ aux aux + H . c . ) , (19)which is reported apart for a global energy shift ofthe auxiliary levels. It is thus sufficient to selectthe parameters so that the condition cos ( ϕ ) g (cid:48) a / ∆ +sin ( ϕ ) g / ∆ aux = 0 is fulfilled, requiring that ∆ and∆ aux have opposite signs.This operation does cancel part of the dephasing dueto the dynamical Stark shift of cavity mode A. It doesnot compensate, however, the dephasing and dissipa-tion terms due to one-photon processes and scaling with g (cid:48) a ∆ sin ∆ τ and g (cid:48) a / ∆ sin (∆ τ / g (cid:48) a τ / ∆) / g (cid:48) a ∆ ) /γ (0)2 ∼ ( g (cid:48) b τ ) − and we choose g (cid:48) b τ (cid:29) γ (0)2 when( g (cid:48) b /g (cid:48) a ) (cid:29)
1. Nevertheless, this ratio cannot be in-creased arbitrarily, since the model we consider is validas long as Ω τ (cid:28)
1. This term can be identically canceledout when specific configurations can be realized, like theone shown in Fig. 4: In this configuration state | (cid:48) (cid:105) cou-ples simultaneously with the excited states | (cid:48) (cid:105) and | e (cid:105) by absorption of a photon of mode A. The coherent dy-namics are now described by Hamiltonian H (cid:48) = H + h (cid:48) with h (cid:48) = (cid:126) ∆ (cid:48) σ ee + (cid:126) g (cid:48)(cid:48) a ( a † σ (cid:48) e + H . c . ) , (20)If the coupling strengths and detunings are such that g (cid:48) a / ∆ = − g (cid:48)(cid:48) a / ∆ (cid:48) , then not only the dynamical Starkshift cancels out, but interference in two-photon pro-cesses lead to the disappearance of the last line in Eq.(18). Under this condition, the resulting master equa-tion is obtained in a coarse-grained time scale ∆ t assum-ing the atoms are injected in state | (cid:48) (cid:105) at rate r with avelocity distribution leading to a normalized distribution p ( τ ) over the interaction times τ , with mean value τ andvariance δτ such that ∆ t > τ + δτ . For r ∆ t (cid:28) ∂∂t ρ = γ (cid:34) C ρC † − (cid:110) C † C , ρ f (cid:111) (cid:35) (21) − i f [ a † a, ρ ] + f (cid:0) aρa † − { a † a, ρ } (cid:1) , with γ = ( r / g (cid:48) a g (cid:48) b / ∆) ( τ + δτ ) , and f = r g (cid:48) a ∆ (cid:90) ∆ t d τ p ( τ ) sin(∆ τ ) , (22) f = r g (cid:48) a ∆ (cid:90) ∆ t d τ p ( τ ) sin (cid:18) ∆ τ (cid:19) . (23)When p ( τ ) is a Dirac- δ function, namely, δτ →
0, and τ ∆ = 2 nπ with n ∈ N , then f and f vanish identicallyand the dynamics describes the target Liouville operator.Under the condition that δτ (cid:54) = 0, but (cid:15) ≡ ∆ δτ (cid:28) π ,then f = O( (cid:15) ) while f = (cid:15) /
4. In the other limit, inwhich p ( τ ) is a flat distribution over [0 , π/ ∆], then f vanishes while f → / FIG. 4. Level scheme leading to the master equation (21).The coupling to the additional level | e (cid:105) allows one to can-cel out dephasing due to one-photon processes on transition | (cid:48) (cid:105) → | (cid:48) (cid:105) . C. Discussion
In this section we have shown how to generate the tar-get dynamics by identifying atomic transitions and ini-tial states for which the desired multiphoton processesare driven. The level schemes we consider could be theeffective transitions tailored by means of lasers. If thecavity modes to entangle have the same polarization butdifferent frequencies, the levels which are coupled canbe circular Rydberg states, while the coupling strengths g j can be effective transition amplitudes, involving cav-ity and/or laser photons. The scheme then requires theability to tune external fields so as to address resonantlytwo or more levels, together with the ability to preparethe internal state of the atoms entering the resonator.Depending on the initial atomic state, then, the dynam-ics can follow either the one described by superoperator L or L . An important condition is that no more than asingle atom is present inside the resonator, which sets thebound over the total injection rate, ( r + r )∆ t (cid:28)
1. Theother important condition is that the dynamics are fasterthan the decay rate of the cavity. For the experimentalparameters we choose, this imposes a limit, among oth-ers, on the choice of the ratio g j / | ∆ | , determining boththe rate for reaching the ideal steady state as well as themean number of photons per each mode, i.e., the size ofthe cat. IV. RESULTS
We now evaluate the efficiency of the scheme, im-plementing the dynamics given by Eq. (9) with H = H + H (cid:48) , where H is given in Eq. (12) and H (cid:48) = H + h ,with H given in Eq. (17) while h depends on the addi-tional levels which are included in the dynamics in orderto optimize it. The initial state of the cavity is the vac-uum, and the atoms are injected with rate r in state | (cid:105) (thus undergoing the coherent dynamics governed by H ) and with rate r in state | ˜1 (cid:105) , which depending onthe considered scheme can be either (i) | (cid:48) (cid:105) when h = h (cid:48) ,or (ii) cos( ϕ ) | (cid:48) (cid:105) + sin( ϕ ) | (cid:48)(cid:48) (cid:105) , when h = h aux . The case h = 0 is not reported, since the corresponding efficiency issignificantly smaller than the one achievable in the othertwo cases. In order to determine the efficiency of thescheme we display the fidelity, namely, the overlap be-tween the density matrix χ ( t ) and the target state | ψ ∞ (cid:105) as a function of the elapsed time. This is defined as F ( t ) = (cid:104) ψ ∞ | Tr at { χ ( t ) }| ψ ∞ (cid:105) , where χ ( t ) is the density matrix of the whole system,composed by cavity modes and atoms of the beam whichhave interacted with the cavity at time t , and Tr at de-notes the trace over all atomic degrees of freedom.For the purpose of identifying the best parameterregimes, we first analyze the dynamics neglecting theeffect of cavity losses. Figure 5 displays the fidelity asa function of time when the dynamics are governed byHamiltonian H = H + H (cid:48) for different realizations of H (cid:48) and for different parameter choices, when the ampli-tude of the coherent state α = 1. Values of F (cid:39) .
99 arereached when H (cid:48) = H + h (cid:48) is implemented. The fidelitythen slowly decays due to higher order effects, which be-come relevant at longer times. The effect of two-photonprocesses involving mode A (which identically vanish for H (cid:48) = H + h (cid:48) ) is visible in the two other curves, whichcorrespond to the dynamics governed by H (cid:48) = H + h aux when g (cid:48) b = 10 g (cid:48) a (blue curve) and g (cid:48) b = 3 g (cid:48) a (red curve). Acomparison between these two curves shows that detri-mental two-photon processes can be partially suppressedby choosing the coupling rate g (cid:48) a sufficiently smaller than g (cid:48) b . F i d e li t y t r FIG. 5. Fidelity as a function of time (in units of the in-jection rate r = r = r ) for α = 1, obtained by integratingnumerically Eq. (9) after setting the cavity losses to zero, κ = 0. The other parameters are g a τ = g b τ = 0 . , g (cid:48) b τ =10 , g (cid:48) b / ∆ = 10 − , Ω τ = 0 .
1. From top to bottom: Theblack curve refers to H (cid:48) = H + h (cid:48) with g (cid:48) a = g (cid:48) b , the othercurves to H (cid:48) = H + h aux with g (cid:48) b = 10 g (cid:48) a (blue) and g (cid:48) b = 3 g (cid:48) a (red). Figure 6 displays in detail the optimal case where H (cid:48) = H + h (cid:48) . The fidelity for the parameter choices g (cid:48) b / ∆ = 10 − and g (cid:48) b / ∆ = 10 − are reported, showingthat a smaller ratio leads to larger fidelity in absence ofcavity decay. The inset shows the corresponding fidelitywhen α = 0 .
5, which is notably larger: Reaching thistarget state starting from the vacuum, in fact, requires ashorter time, for which higher-order corrections are stillirrelevant.The effect of cavity losses is accounted for in Fig. 7,where the full dynamics of master equation (9) is simu-lated when H (cid:48) = H + h (cid:48) and for different choices of theratio κ/r . One clearly observes that the effect of cavitylosses can be neglected over time scales of the order of10 − /κ , so that correspondingly larger rates γ and γ are required. Considered the parameter choice, this ispossible only by increasing the injection rate r . How-ever, this comes at the price of increasing the probabilitythat more than one atom is simultaneously inside theresonator, thus giving rise to further sources of deviationfrom the ideal dynamics.These results show that degradation due to photonlosses poses in general a problem to attain the target state(1): the rate of photon losses sets a maximum achievablefidelity, and also determines a time window during whichthe fidelity is close to the maximum, after which the en-tanglement is gradually lost. The effect of the photonlosses is twofold: it leads to a decrease in the mean pho-ton number, and also breaks the symmetry preservation F i d e li t y tr FIG. 6. (a) Fidelity as a function of time (in units of theinjection rate r = r = r ) for α = 1, obtained by integratingnumerically Eq. (9) after setting the cavity losses to zero, κ = 0. The other parameters are Ω τ = 0 . g a τ = g b τ =0 .
1, whereby the black curve is evaluated for g (cid:48) b τ = g (cid:48) a τ =10 and g (cid:48) b / ∆ = 10 − , while the red curve corresponds to g (cid:48) b τ = g (cid:48) a τ = 10 and g (cid:48) b / ∆ = 10 − (from top to bottom).The inset has been evaluated for the same parameters exceptfor Ω τ = 0 .
05, leading to α = 0 . in the evolution. The decrease in the mean photon num-ber can be compensated by increasing the strength Ω ofthe pumping in the implementation of the second Lind-blad operator, as long as the approximations made inSection III B are still valid. V. CONCLUDING REMARKS
A strategy has been discussed which implements non-unitary dynamics for preparing a cavity in an entangledstate. It is based on injecting a beam of atoms into acavity, where the coherent interaction of the atoms withthe cavity is a multiphoton process pumping in phasephotons, so that the cavity modes approach asymptot-ically the entangled state of Eq. (1). The procedureis robust against fluctuations of the number of atomsand interaction times. It is however sensitive againstcavity losses: the protocol is efficient, in fact, as longas the time scale needed in order to realize the targetstate is faster than cavity decay. The effect of the pho-ton losses is twofold: it damps the mean photon numberand also changes the parity of the state. It could bepossible to partially revert the process by measuring theparity of the total photon number and then performinga feedback mechanism, similar to the one proposed inRefs. [30, 31] and which has been partially implementedin Refs. [32, 33]. Alternatively, one can find a dissipativeway to stabilize a unique entangled target state withoutthe need for feedback. This would require a process thatcan stabilize the parity of the photon number in the evenmode. First studies have been performed showing some (a) F i d e t li y tr (b) F i d e li t y tr FIG. 7. Fidelity as a function of time for (a) α = 1 and(b) α = 0 .
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