Environment induced entanglement in a refined weak-coupling limit
aa r X i v : . [ qu a n t - ph ] O c t Environment induced entanglementin a refined weak-coupling limit
F. Benatti a,b , R. Floreanini b and U. Marzolino a,b a Dipartimento di Fisica Teorica, Universit`a di Trieste, 34014 Trieste, Italy b Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, 34014 Trieste, Italy
Abstract
Two non-directly interacting qubits with equal frequencies can become entangled viaa Markovian, dissipative dynamics through the action of a weakly coupled Ohmic heatbath. In the standard weak-coupling limit derivation, this purely dissipative effectdisappears if the frequencies are different because of the “ergodic average” used by thisapproach. However, there are physical situations where this technique is too rough tocapture all the relevant aspects of the dissipative dynamics. In these cases, in order tobetter describe the physical behavior of the open system, it is necessary to go beyondthe “ergodic average”. We show that, in this more refined framework, the entanglementcapability of the environment persists also in the case of different frequencies.
Quantum systems are usually treated as isolated: this is just an approximation, justifiedfor vanishingly small couplings with the external environment. When the interaction with theenvironment is weak but not negligible, a reduced dynamics can be obtained by eliminatingthe environment degrees of freedom and by subsequently performing a so-called Markovianapproximation [1]-[5]. These systems are known as open quantum systems and their reduceddynamics is irreversible and satisfies a forward-in-time composition law: it is describedby a so-called quantum dynamical semigroup that incorporates the dissipative and noisyeffects due to the environment. The latter acts as a source of decoherence: in general,the corresponding reduced dynamics irreversibly transforms pure states (one-dimensionalprojections) into statistical mixtures (density matrices).One of the most intriguing aspects of quantum coherence is entanglement [6], that is theexistence of purely quantum mechanical correlations, which has become a central topic inquantum information for its many applications as a physical resource enabling otherwiseimpossible information processing protocols. With reference to the entanglement contentof a state of two qubits embedded in a same heat bath, it is generally expected that itwould be depleted by decoherence effects. However, this is not the only possibility: if suit-ably engineered, the environment can entangle an initial separable state of two dynamically1ndependent systems; the reason is that, although not directly interacting between them-selves, there can be an environment mediated generation of quantum correlations betweentwo systems immersed in it.This possibility has been demonstrated analytically for two qubits with a same oscillationfrequency [8, 7, 9] and two identical harmonic oscillators [10] evolving according to a reducedmaster equation of the typical Lindblad form [11, 12], obtained via the so-called weak-coupling limit. This technique is based on the fact that the dissipative effects are visibleonly over a coarse-grained time-scale ∆ t , so large that the free dynamics of the embeddedsystems can be averaged out over ∆ t [2]-[5]. The resulting elimination of too rapid oscillationsis mathematically implemented through a time “ergodic average”.Noticeably, such a prescription guarantees that, unlike for reduced dynamics of Redfieldtype (see [14, 5]), the resulting quantum dynamical semigroups consist of completely positivemaps [1, 2, 3, 11]. Complete positivity ensures that the open quantum evolution is consistentwith entanglement in the sense that not only the positivity of any initial density matrix of theopen system is preserved in time, but also that of any initial state of the open system coupledto any other possible ancillary system. Indeed, only complete positivity can guarantee thefull physical consistency of the Markovian approximations that one uses to describe an openquantum dynamics; in other words, without complete positivity, it always occurs that atleast one initial state carrying entanglement between the given system and an ancilla willassume negative eigenvalues in the course of time [5].It turns out that, in the standard weak-coupling limit approximation, when two qubitsor two harmonic oscillators embedded in a same environment have different oscillation fre-quencies ω = ω , no matter how small the difference ω − ω is, the elimination of too rapidoscillations destroys the generation capability that the environment possesses when ω = ω .In this paper, we study this behavior in the case of two qubits weakly interacting, via aOhmic coupling, with a heat bath made of free bosons at high temperature. We shall firstrelate the sharp dependence of the entanglement capability of the environment to the oscil-lation frequencies of the two qubits on the drastic elimination of too fast oscillations throughthe “ergodic average”. This procedure is only allowed when the coupling to the environmentis such that the coarse-grained time-scale ∆ t can effectively be considered infinitely large.However, there are situations where this approximation is not physically meaningful and ∆ t should be kept finite. In these cases, to better capture the effects of the open system dynam-ics, a less rough time coarse-graining is needed; in the following, we use a master equationderived without recourse to the ergodic average that nevertheless generates a completelypositive dynamical semigroup. We show that, in this refined framework, the entanglementgeneration capability of the environment is preserved even when ω = ω .The problem we will address in the following regards whether two non-interacting qubits For the system composed by two harmonic oscillators immersed in a heat bath of other oscillators similarresults have been obtained by studying numerically their exact time evolution [13]. H S = ω σ (1)3 + ω σ (2)3 , (1)can become entangled when weakly coupled to free spinless Bosons in thermal equilibriumvia a (finite volume) interaction of the form H I = λ X ( f ) (cid:16) σ (1)1 + σ (2)1 (cid:17) , X ( f ) = X k (cid:16) f ( k ) a † k + f ∗ ( k ) a k (cid:17) , (2)where a † k and a k denote the creation and annihilation operators of Bose modes with momen-tum k and energy ω ( k ), f ( k ) is a one-particle Bose state in momentum representation, λ isa small, dimensionless coupling constant, while σ (1)1 , = σ , ⊗ σ (2)1 , = 1 ⊗ σ , representthe first and third Pauli matrices for the two qubits. The total system comprising the twoqubits and the thermal bosons will thus be described by the Hamiltonian H = H S + H B + H I , H B = X k ω ( k ) a † k a k . (3)In general, the state of the compound system S + B at time t > ρ SB ( t ) from which the state of the two qubits can be extracted as ρ ( t ) = Tr B ( ρ SB ( t )) bytracing out the environment degrees of freedom; however, the time-evolution equation for ρ ( t ) is quite complicated. In order to arrive at a memoryless master equation, one starts withthe physically acceptable hypothesis that the initial state of open system plus environmentbe of the uncorrelated form ρ ⊗ ρ β , where ρ β ∝ exp ( − βH B ) is the Boson thermal equilibriumstate at inverse temperature β . Then, one goes to the interaction representation by replacing ρ ( t ) with e ρ SB ( t ) = e it ( H S + H B ) ρ SB ( t ) e − it ( H S + H B ) and looks at the time-evolution over time-intervals ∆ t by stopping at the first significative order in λ in the time-ordered expansion of e ρ ( t + ∆ t ). Then, e ρ ( t + ∆ t ) − e ρ ( t )∆ t ≃ − λ ∆ t Z t +∆ tt d t Z t t d t Tr h H I ( t ) , h H I ( t ) , e ρ SB ( t ) ii! , (4)where H I ( t ) = X t ( f ) (cid:16) σ (1)1 ( t ) + σ (2)1 ( t ) (cid:17) , X t ( f ) = X k (cid:16) f ( k )e itω ( k ) a † k + f ∗ ( k )e − itω ( k ) a k (cid:17) , (5)with σ ( a )1 ( t ) = cos( ω a t ) σ ( a )1 − sin( ω a t ) σ ( a )2 , a = 1 , e ρ ( t ) is of order λ , namely it is non-negligible only over times t = τ /λ ; if λ ≪ t is such that λ ∆ t is small on the scale of τ , one may reasonablysubstitute in (4) the finite ratio with a time-derivative: e ρ (cid:0) ( τ + λ ∆ t ) /λ (cid:1) − e ρ (cid:0) τ /λ (cid:1) ∆ t ≃ ∂ t e ρ ( t ) . (6) For sake of simplicity, we consider the two qubits located at a same point in space. For the effects onentanglement creation that result when the two qubits are spatially separated we refer to [7, 9] for equalfrequencies and to [15] in the case of ω = ω . λ ∆ t . Further, if ∆ t is chosen much larger than the decaytime τ B of the environment two-point time-correlation functions, one may approximate e ρ SB ( t )with the uncorrelated state ρ ( t ) ⊗ ρ β as it was at t = 0 [2, 4]. Therefore, if τ B ≪ ∆ t , onegets the following approximated master equation (in interaction representation): ∂ t e ρ ( t ) = − λ ∆ t Z t +∆ tt d t Z t t d t Tr h H I ( t ) , h H I ( t ) , e ρ ( t ) ⊗ ρ β ii! . (7)The meaning of ∆ t is that of a time-coarse graining parameter naturally associated tothe slow dissipative time-scale τ = tλ . More precisely, significant variations of the systemdensity matrix due to the presence of the environment can only be seen after a time ∆ τ = λ ∆ t has elapsed. Given the coupling constant λ ≪
1, the dissipative time-scale is set andactual experiments cannot access faster time-scales; furthermore, if ∆ τ ≪ τ , then the experimental evidences are consistently described by a master equation asin (7). The weak-coupling limit consists in letting λ → t → + ∞ . This allows one toeliminate all fast oscillations through a time “ergodic average” as in the standard weak-coupling limit approach [3, 2]. Instead, if the system-environment coupling λ is small, butnot vanishingly small, then ∆ t cannot be taken infinitely large. In such cases, the usualweak-coupling limit techniques provide an approximation which is too rough to properlydescribe the dissipative time-evolution and one needs a more refined approach in order to beable to keep contributions that would otherwise be washed out. Returning to the Schr¨odinger representation, (7) yields the following memoryless masterequation (see [16, 17, 18]) of Kossakowski-Lindblad type [2, 4]: ∂ t ρ ( t ) = − i h H S + λ H ∆ t , ρ ( t ) i + D ∆ t [ ρ ( t )] . (8)The environment contributes to the generator of the reduced dynamics with an Hamiltonian H ∆ t and a purely dissipative term D ∆ t [ ρ ( t )]; both of them depend on the environmentthrough the two-point time-correlation functions G ( t ) = Tr (cid:16) ρ β X ( f ) X t ( f ) (cid:17) = G ( − t ) ∗ . (9)The bath-induced Hamiltonian H ∆ t contains a bath-mediated qubit-qubit interaction H int ∆ t = X i,j =1 , h ij (∆ t ) σ (1) i σ (2) j , (10) A different approach is developed in [16]: there, a non-Markovian weak coupling approximation ofthe reduced dynamics is introduced, leading to a two-parameter family of dynamical maps, with a time-dependent generator [2]. We stress that this treatment is completely different from the one discussed below,which instead describes the reduced two-atom dynamics in terms of a Markovian, one parameter semigroup. × h ∆ t = [ h ij (∆ t )] is real, but not necessarily Hermitian. It is convenientto introduce the following matrices [Ψ εj ] = 12 (cid:18) i − i (cid:19) , ε = ±
1; then h ∆ t = Ψ † H (12)∆ t Ψ + (cid:16) Ψ † H (21)∆ t Ψ (cid:17) T , (11)where H ( ab )∆ t = [ H ( ab ) εε ′ (∆ t )], a, b = 1 ,
2, is explicitly given by H ( ab ) εε ′ (∆ t ) = − i t Z ∆ t d t Z ∆ t d t e i ( εω a t − ε ′ ω b t ) sign( t − t ) G ( t − t ) , (12)where X T denotes matrix transposition. Instead, the purely dissipative part can be writtenas D ∆ t [ ρ ( t )] = X a,b =1 , i,j =1 , C ( ab ) ij (∆ t ) σ ( a ) i ρ ( t ) σ ( b ) j − n σ ( b ) j σ ( a ) i , ρ ( t ) o! , (13)where the 2 × C ( ab )∆ t = [ C ( ab ) ij (∆ t )] read C ( ab )∆ t = Ψ † D ( ab )∆ t Ψ , (14)in terms of the matrices D ( ab )∆ t = [ D ( ab ) εε ′ (∆ t )] with entries D ( ab ) εε ′ (∆ t ) = 1∆ t Z ∆ t d t Z ∆ t d t e i ( εω a t − ε ′ ω b t ) G ( t − t ) . (15)The 4 × C ∆ t formed by the 2 × C ( ab )∆ t , a, b = 1 ,
2, involves theFourier transforms of the correlation functions (9) and is automatically positive definite; thisfact guarantees that the master equation (8) generates a semigroup of dynamical maps γ ∆ tt on the two-quibit density matrices which are completely positive [1]-[5], whence, as discussedin the introduction, fully physically consistent.Given two qubits weakly interacting with their environment, a sufficient condition forthem to get entangled at small times by the completely positive reduced dynamics has beenderived in [19, 20] and is based on the properties of the generator in (8), that is on theinteraction Hamiltonian (10) and the dissipative contribution (13). We shall focus on aninitial two qubit state of the form | ↓i ⊗ | ↑i , where | ↓i , | ↑i are the eigenstates of σ ; then,the condition is as follows: δ = D (11) −− (∆ t ) D (22)++ (∆ t ) − (cid:12)(cid:12)(cid:12) D (12)∆ t + i H (12)∆ t (cid:12)(cid:12)(cid:12) < , (16)where, from (12) and (15), D (12)∆ t = D (12) −− (∆ t ) + ( D (12) −− (∆ t )) ∗ H (12)∆ t = H (12) −− (∆ t ) + H (21)++ (∆ t ) . (18)5f δ >
0, such a separable state cannot get entangled by the environment at small times [20].We are interested in the capacity of the environment to generate entanglement, at leastat small times with respect to the dissipative time-scale. Because of the positivity of theKossakowski matrix C ∆ t the diagonal block matrices C ( aa )∆ t and D ( aa )∆ t , a = 1 ,
2, are positivedefinite; therefore, the dissipative entanglement generation depends on the quantity D (12)∆ t ,but also on how it interferes with H (12)∆ t in (16). In the case at hand, it turns out that (cid:12)(cid:12)(cid:12) D (12)∆ t + i H (12)∆ t (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) D (12)∆ t (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) H (12)∆ t (cid:12)(cid:12)(cid:12) . (19)Indeed, using (9), one can check by explicit computation that the quantities e − i ( ω − ω )∆ t/ D (12)∆ t and e − i ( ω − ω )∆ t/ H (12)∆ t are both real. We shall thus concentrate on the purely dissipativeentanglement generation, that is on the difference˜ δ = D (11) −− (∆ t ) D (22)++ (∆ t ) − (cid:12)(cid:12)(cid:12) D (12)∆ t (cid:12)(cid:12)(cid:12) . (20)If ˜ δ is negative, a fortiori also δ in (16) is negative.We shall consider (infinite volume) Ohmic correlation functions [21] G ( t ) = Z + ∞ d ω e − ω/ω c ω (cid:16) coth βω ωt − i sin ωt (cid:17) , (21)where ω c is a Debye cut-off; then, setting sinc( x ) = sin xx and ( a, ε ) = (1 , − ) , (2 , +), oneexplicitly gets D ( aa ) εε (∆ t ) = ε ∆ t Z + ∞−∞ d ω ω e −| ω | /ω c e εβω − (cid:20) ( ω − ω a )∆ t (cid:21) , (22) (cid:12)(cid:12)(cid:12) D (12)∆ t (cid:12)(cid:12)(cid:12) = ∆ t Z + ∞−∞ d ω ω e −| ω | /ω c coth ( βω/
2) sinc (cid:20) ( ω − ω )∆ t (cid:21) sinc (cid:20) ( ω − ω )∆ t (cid:21) . (23)The weak-coupling limit amounts to ∆ t → + ∞ ; since lim α → + ∞ α π sinc( α ( x − x )) = δ ( x − x ),one finds lim ∆ t → + ∞ D ( aa ) εε (∆ t ) = 4 πε ω a e − ω a /ω c e εβω a − ∆ t → + ∞ (cid:12)(cid:12)(cid:12) D (12)∆ t (cid:12)(cid:12)(cid:12) = 2 πδ ω ω ω e − ω /ω c coth βω . (25)If ω = ω , the difference (20) is always negative: ˜ δ = − π ω e − ω /ω c ; instead, if ω = ω ,˜ δ = D (11) −− (∆ t ) D (22)++ (∆ t ) ≥
0. Thus, in the latter case, the initial separable state | ↓i ⊗ | ↑i cannot get dissipatively entangled by the environment at small times, unless ω = ω .6his behavior is characteristic of physical situations where an infinitely large coarse-grained time-scale ∆ t is justified by a vanishingly small coupling λ to the environment.However, if λ is small, but not negligibly small, the qubit density matrix effectively variesover large but finite ∆ t . Then, terms of order 1 / ∆ t like (cid:12)(cid:12)(cid:12) D (12)∆ t (cid:12)(cid:12)(cid:12) ≃ π sinc( δω ∆ t ) X a =1 ω a e − ω a /ω c coth( βω a /
2) (26)can make ˜ δ negative even when δω = ( ω − ω ) / >
0. Indeed, for high temperatures βω , ≪ ω , /ω c ≪
1, expanding (24) and (26) yields ˜ δ ≃ π β (cid:16) − βδω − sinc ( δω ∆ t ) (cid:17) . (27)If the qubit frequency difference δω and the coupling strength λ are such that δω ∆ t ≪ δ ≃ − π β (cid:16) βδω − ( δω ) (∆ t ) (cid:17) , (28)which is negative when 1 ω , ≫ β > δω (∆ t ) λ → t over which the qubit density matrix effectively changes due to the presence ofthe environment become so large that all off-resonant phenomena are averaged out. Instead,if the physical conditions ask for a coupling constant λ that is small but not vanishingly so,a consistent description of the open dynamics requires ∆ t finite; in this way, one may keeptrack of finer effects and save the possibility of dissipative entanglement generation even if ω = ω . The high temperature hypothesis has been made in order to permit an analytical study of the behaviorof ˜ δ . This is the worst possible scenario; indeed, in general, lower temperatures favor the entanglementcreation capability of the environment [15]. eferences [1] H. Spohn, Rev. Mod. Phys. (1980) 569[2] R. Alicki, K. Lendi, Lect. Notes Phys. , Springer, Berlin (1987)[3] E.B. Davies, Comm. Math. Phys. (1974) 91[4] H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford UniversityPress, Oxford 2002)[5] F. Benatti, R. Floreanini, Int. J. Mod. Phys. B (2005) 3063[6] R. Horodecki et al. , Rev. Mod. Phys. (2009)865[7] F. Benatti, R. Floreanini, J. Opt. B. (2005) S429[8] D. Braun, Phys. Rev. Lett. 89, (2002) 277901[9] D.P.S. McCutcheon et al. , Phys. Rev. A (2009) 022337[10] F. Benatti, R. Floreanini, J. Phys. A (2006) 2689[11] V. Gorini et al. , Rep. Math. Phys. (1978) 149[12] G. Lindblad, Comm. Math. Phys. (1976) 119[13] J. P. Paz, A. J. Roncaglia, Phys. Rev. Lett. (2008) 220401[14] R. D¨umcke, H. Spohn, Z. Phys. B34 (1979) 419[15] F. Benatti et al. , Entangling two unequal atoms through a common bath, preprint, 2009[16] R. Alicki,