Open System Dynamics with Non-Markovian Quantum Trajectories

Open System Dynamics with Non-Markovian Quantum Trajectories W. T. Strunz 1,,L.Diósi,anN.Gisin 3 1 Fachbereich Physik, Universität GH Essen, 45117 Essen, Germany Research Institute for Particle an Nuclear Physics, 155 Buapest 114, POB 49, Hungary 3 Group of Applie Physics, University of Geneva, 111 Geneva 4, Switzerlan (March 31, 1998) A non-markovian stochastic Schröinger equation for a quantum system couple to an environment of harmonic oscillators is presente. Its solutions, when average over the noise, reprouce the stanar reuce ensity matrix without any approximation. We illustrate the power of this approach with several examples, incluing exponentially ecaying memory correlations an extreme non-markovian perioic cases, where the environment consists of only a single oscillator. The latter case shows the ecay an revival of a Schröinger cat state. For strong coupling to a issipative environment with memory, the asymptotic state can be reache in a finite time. Our escription of open systems is compatible with ifferent positions of the Heisenberg cut between system an environment. The ynamics of open quantum systems is a very timely problem, both to aress funamental questions (quantum ecoherence, measurement problem) as well as to tackle the more practical problems of engineering the quantum evices necessary for the emerging fiels of nanotechnology an quantum computing. So far, the true ynamics of open systems has almost always been simplifie by the Markov approximation: environmental correlation times are assume negligibly short compare to the system s characteristic time scale. For the numerical solution of Markovian open systems, escribe by a master equation of Linbla form t ρt = i[h, ρt]+1 ( [Lmρ t,l m]+[l m,ρ tl m] ) (1) m (where ρ t enotes the ensity matrix, H the system s Hamiltonian an the operators L m escribe the effect of the environment in the Markov approximation), a breakthrough was achieve through the iscovery of stochastic unravellings [1,]. These are stochastic Schröinger equations for states ψ t(z), riven by a certain classical noise z t with istribution functional P (z). Crucially, the ensemble mean M[...] overthe noise recovers the ensity operator, [ ] ρ t = M ψ t(z) ψ t(z). () Hence, the solution of eq.(1) is reuce from a problem in the matrix space of ρ to a much simpler Monte Carlo simulation of quantum trajectories ψ t(z) in the state space. For the Markov master eq.(1), several such unravellings are known. Some involve jumps at ranom times [1], others have continuous, iffusive solutions []. They have been use extensively over recent years, as they provie useful insight into the ynamics of continuously monitore (iniviual) quantum processes [3], or into the mechanism of ecoherence [4]. In aition, they provie an efficient tool for the numerical solution of the master eq.(1). It is thus esirable to exten the powerful concept of stochastic unravellings to the more general case of non-markovian evolution. The simplest unravellings are linear stochastic Schröinger equations. In the Markov case (1), for a single L, the linear equation t ψt = ihψt + Lψt zt 1 L Lψ t, (3) provies such an unravelling, where, z t is a complex-value Wiener process of zero mean an correlations M[zt z s] = δ(t s), M[z tz s] =, an where enotes the Stratonovich prouct [5]. However, eq.(3) is of limite value, since the norm ψ t(z) of its solutions tens to with probability 1 an to infinity with probability, such that the mean square norm is constant. To be really useful, one shoul fin unravellings in terms of the normalize states ψ t(z) = ψt(z) ψ, (4) t(z) which requires a reefinition of the istribution of the noise P (z) P t(z) ψ t(z) P(z) [6] so that eq.() remains vali for the normalize solutions: ] ρ t = M t [ ψ t(z) ψ t(z). (5) Now (5) can be interprete as an unravelling of the mixe state ρ t into an ensemble of pure states. For the Markov unravelling (3), the normalize states ψ t satisfy the non-linear Quantum State Diffusion (QSD) equation []: t ψ t = ih ψ t +(L L t) ψ t (z t + L t) (6) 1 (L L L L t) ψ t, where L t ψ t L ψ t. Contrary to eq.(3), eq.(6) provies an efficient Monte-Carlo algorithm for the numerical solution of (1) [1,]. In this Letter we present for the first time a nonlinear non- Markovian stochastic Schröinger equation that unravels the ynamics of a system interacting with an arbitrary environment of a finite or infinite number of harmonic oscillators, without any approximation. In the Markov limit, this unravelling reuces to QSD (6) an will therefore be referre to as non-markovian Quantum State Diffusion. Other authors have treate non-markovian open systems effectively with Markovian unravellings: either the system has in fact been influence by a secon Markovian environment in aition to the 1

original non-markovian one or, alternatively, fictious moes have been ae to the system [7]. In our approach, the system remains unaltere, an the unravelling is genuinely non-markovian. Below we summarize the general theory, which will be presente in etail elsewhere [8], an we present four examples: First, we consier a measurement-like environment. Then, a issipative environment with exponentially ecaying environment correlations is iscusse. Remarkably, here the asymptotic state can be reache in a finite time. In the thir example we consier an environment consisting of only a single oscillator. This example is thus perioic, that is extremely non- Markovian. It shows the ecay an revival of a Schröinger cat state. Finally, the fourth example shows that the escription of a subsystem in terms of non-markovian QSD is inepenent of the Heisenberg cut, that is inepenent of where precisely the bounary between system an environment is set. Our starting point is the non-markovian generalization of the linear stochastic equation (3), erive in [9], t ψt = ihψt + Lψtzt L α(t, s) δψt s, (7) δz s which unravels the exact reuce ynamics of a system couple to an environment of harmonic oscillators. Here, z t is colore complex Gaussian noise of zero mean an correlations M [zt z s]=α(t, s), M [z tz s]=. (8) The Hermitian α(t, s) =α (s, t) is the environment correlation function [8,9]. The functional erivative uner the memory integral in (7) inicates that the evolution of the state ψ t at time t is influence by its epenence on the noise z s at earlier times s. In [8] we show that it amounts to applying an operator to the state, δ ψ t Ô(s, t, z)ψt, (9) δz s where the explicit expression of Ô(s, t, z) canbeetermine consistently from eq.(7). Just as in the Markov limit, to be really useful, one has to fin the corresponing non-linear non-markovian QSD equation for the normalize states (4). This quite elaborate erivation can be foun in [8] an leas to t ψ t = ih ψ t +(L L t) ψ t z t (1) α(t, s) ( L Ô(s, t, z) L Ô(s, t, z) t ) s ψt, which is the basic equation of non-markovian QSD. Here, z t is the shifte noise z t = z t + α (t, s) L ss, anforbrevity we use L = L L t. Let s turn to concrete examples of non-markovian QSD (1). First, we consier an environment moeling energy measurement: L = L = H. It is easy to prove that Ô = H in (9), an hence (1) reas ψ t t = ih ψ t (H H t) ψ t α(t, s)s (11) ) +(H H t) ψ t (z t + α(t, s) H ss + α(t, s)s H t. Notice that inee, (11) reuces to the Markov QSD equation (6) for α(t, s) δ(t s). If the correlation α(t, s) ecreases fast enough, the asymptotic solution of (11) is an eigenstate φ n of H, reache with the expecte quantum probability φ n ψ. Numerical solutions of (11) for the -imensional case H = ω σz an exponentially ecaying correlation are shown in Fig.1a (soli lines). The asymptotic state is either the up or the own state. The ensemble mean M[ σ z ] remains constant (ashe line) as expecte from the analytical solution (ot-ashe line). Note, however, that if the environment consists of a finite number of oscillators, represente by a quasi-perioic correlation function α(t, s), such a reuction to an eigenstate will not occur (see our thir example). As a secon example, we consier a issipative spin with H = ω σz, anl=λσ. We choose exponentially ecaying correlations α(t, s) = γ e γ t s iω(t s) with an environmental central frequency Ω an memory time γ 1. The non- Markovian QSD equation (1) reas [8] t ψ t = i ω σz ψ t λf (t)(σ +σ σ +σ t) ψ t (1) ) + λ(σ σ t) ψ t (z t+λ α(t, s) σ + ss + σ + tf (t) with F (t) etermine from t F (t) = γf(t)+i(ω Ω)F (t)+λf (t) + λγ (13) an initial conition F () =. The equation for F (t) can be solve analytically [8]. It is worth mentioning the case of exact resonance, ω = Ω. Two regimes shoul be istinguishe. First, when γ > λ (short memory compare to coupling strength), F (t) tens to (γ ) γ γλ /(λ). Hence, for large γ, one recovers Markov QSD (6). For longer memory times or stronger coupling, γ<λ, things are very ifferent: ( F (t) iverges to infinity when ) the time t approaches t c = π + arctan(γ/ λ γ γ ) / λ γ γ. All realizations ψ t(z) reach the own state in a finite time an remain there! In Fig.1b we show quantum trajectories from (1) (soli lines), their ensemble mean value M[ σ z t](ashe line), an the analytical mean value (ot-ashe), which is almost inistinguishable. The reuction time in this case is ωt c = 3 π 4.71. This is the first example of a continuous quantum state iffusion that reaches its asymptotic state in a finite time, which was proven impossible for Markovian iffusions [1]. Our thir example is a harmonic oscillator couple to a finite or infinite number of oscillators initially in their groun states. Here, the non-markovian QSD eq.(1) takes the same form (1), where the Hamiltonian is ωa a an where σ (σ +) has to be replace by the annihilation (creation) operator a (a ). The resulting equation preserves coherent states. More interesting is the case of an initial superposition of two symmetric coherent states, known as a Schröinger cat [11]. If the environment correlation α(t, s) ecays, so oes the cat. If, however, the environment consists of only a finite number of oscillators, then the cat will first ecay, ue to the localization property of QSD, but since the entire system is quasiperioic, the cat will then revive! In Fig. we show contour

plots of the evolution of the Q-function of such a cat, in the extreme case where the environment consists of a single oscillator (α(t, s) =e iω(t s) ). Apart from an overall spiraling motion ue to the system Hamiltonian, the cat state first ecays but later revives. Our non-markovian QSD equation thus provies a nice illustration of propose experiments on reversible ecoherence [1]. As a last example we consier a case where the split between system an environment can be shifte naturally between two positions, see Fig.3. A spin (Hamiltonian H 1)an a istinguishe harmonic oscillator (H ) are linearly couple (H 1). Moreover, the spin is couple (H I) to a heat bath (H env) at zero temperature. We can either consier the quantum state of the spin-oscillator system couple to a heat bath, or the quantum state of the spin couple to a heat bath an couple to the istinguishe oscillator. In the first case, we can apply the Markov QSD escription, ie a family of spinoscillator states ψ t(ξ) inexe by a complex Wiener process ξ t. In the secon case, using non-markovian QSD, we have a family of spin states φ t(ξ, z) inexe by the same ξ t an also by the noise z t, ue to the istinguishe oscillator environment with correlation M[z t z s]=e iω (t s). Let us stuy a shift of the Heisenberg cut : compare the states φ t(ξ,z) of the spin average over the noise z t with the mixe state obtaine by tracing out (Tr ) the oscillator from the spin-oscillator states ψ t(ξ). We prove in [8] that the states corresponing to both escriptions are equal: M z[ φ t(ξ, z) φ t(ξ, z) ] =Tr ( ψ t(ξ) ψ t(ξ) ). (14) This illustrates the general fact that non-markovian QSD attributes stochastic pure states to a system in a way which epens on the position of the Heisenberg cut, but which is consistent for all possible choices of the cut. In conclusion, we present the first non-markovian unravelling of the ynamics of a quantum system couple to an environment of harmonic oscillators, which can thus be simulate by classical complex noise. In the Markov limit, stanar Quantum State Diffusion is recovere. We emphasize that non-markovian QSD (1) reprouces the true evolution of the system taking into account the exact unitary ynamics of system an environment [8,9]. The power of this new approach to open quantum systems is illustrate with four examples. For measurement-like interactions, reuction to eigenstates takes place whenever the environment correlation function ecreases fast enough. For issipative interaction with a heat bath at zero temperature, the groun state may be reache in a finite time. The thir example is an application to the most extreme non-markovian case: two linearly couple oscillators, one of them playing the role of the environment. We see the ecay an revival of a Schröinger cat state. Finally, the last example illustrates that unravellings corresponing to ifferent positions of the Heisenberg cut between system an environment are mutually compatible. Most of these features are entirely new an have no counterpart in any Markov unravelling. Hence, non-markovian unravellings represent a promising route to open systems, as for instance to quantum Brownian motion [8]. Moreover, our approach represents a new efficient tool for the numerical simulation of quantum evices, whenever non-markovian effects are relevant [7,13]. WethankICPercivalforhelpfulcommentsantheUniversity of Geneva where part of the work was one. WTS woul like to thank the Deutsche Forschungsgemeinschaft for support through the SFB 37 Unornung un große Fluktuationen. LD is supporte by the Hungarian Scientific Research Fun through grant T1647. NG thanks the Swiss National Science Founation. Electronic aresses: walter.strunz@uni-essen.e, iosi@rmki.kfki.hu, Nicolas.Gisin@physics.unige.ch [1] J. Dalibar, Y. 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Strunz, (to be publishe), see quant-ph/9836. [9] L. Diósi an W. T. Strunz, Phys. Lett. A. 35, 569 (1997); L. Diósi, Quantum Semiclass. Opt. 8, 39 (1996); W. T. Strunz, Phys. Lett. A. 4, 5 (1996). [1] P. Pearle, Phys. Rev. Lett. 53 1775 (1984); N. Gisin, Phys. Rev. Lett. 53 1776 (1984); P. Pearle, Phys. Rev. D 33, 4 (1986); N. Gisin, Helv. Phys. Acta 6, 363 (199); P. Pearle, in Experimental Metaphysics, es. R.S. Cohen, M. Horne, an J. Stachel, (Kluwer Aca. Press), 143 (1997). [11] B. Yurke an D. Stoler, Phys. Rev. Lett. 57 13 (1986); R. Glauber in Frontiers in Quantum Optics, (es. Pike an Sarkar) (1986); B. W. Shore an P. L. Knight, J. Mo. Opt. 4 1195 (1993); B. M. Garraway an P. L. Knight, Phys. Rev. A 5, 548 (1994). [1] J.M. Raimon, M. Brune, an S. Haroche, Phys. Rev. Lett. 79, 1964 (1997). [13] A. Imamoglu an Y. Yamamoto, Phys. Lett. A 191, 45 (1994); G. M. Moy, J. J. Hope, an C. M. Savage, see quant-ph/98146 (1998). 3

FIGURE CAPTIONS FIG. 1. Non-Markovian quantum trajectories (soli lines) for a spin 1 system H = ω σz with an exponentially ecaying environment correlation α(t, s) = γ exp ( γ t s iω(t s)), where we choose γ = ω. The ensemble mean value over 1 runs (ashe line) is in very goo agreement with the analytical result (ot-ashe line). We show (a) a measurement-like interaction L = λσ z with Ω =, an (b) a issipative interaction L = λσ on resonance Ω=ω, where each trajectory reaches the groun state in a finite time ωt c = 3 π 4.71. In both cases we choose a coupling strength λ = ω an an initial state ψ =3 +. FIG.. Reversible ecoherence of an initial symmetric Schröinger cat state ψ = α + α with α =. The contour plots show the Q-function of a non-markovian quantum trajectory of a harmonic oscillator (ω) system, couple to just a single environment oscillator (Ω =.5ω). The coupling strength between the two oscillators is.1ω, anthe time step between two successive plots is.47/ω. FIG. 3. Spin - single oscillator - heat bath system. First, we consier the spin - single oscillator as the system with state ψ t(ξ), couple to the heat bath with noise ξ t. Alternatively, we can consier the spin as the system φ t(ξ, z), couple to the single oscillator + heat bath environment (noises (ξ t,z t)). In non-markovian QSD, both escriptions are possible an lea to the same reuce spin state. 4

1 M NMQSD [NMQSD] ρ(t) <σ > z z <σ > -1 1 (a) NMQSD [NMQSD] ρ(t) M (b) -1 1 3 4 ωtc 5 ωt

1 3 4 5 6 7 8 9 1 11

H 1 H I spin φ (ξ,z) t H env H 1 ψ (ξ) t single oscillator bath of oscillators ξ t H z t