Open quantum systems that interact with structured reservoirs exhibit
non-Markovian dynamics. We present a quantum jump method for treating the
dynamics of such systems. This approach is a generalization of the standard
Monte Carlo Wave Function (MCWF) method for Markovian dynamics. The MCWF method
identifies decay rates with jump probabilities and fails for non-Markovian
systems where the time-dependent rates become temporarily negative. Our
non-Markovian quantum jump (NMQJ) approach circumvents this problem and
provides an efficient unravelling of the ensemble dynamics.Comment: 4 pages, 2 figures.V2: rewritten abstract and introduction, title
  modified. V3: published version, new example case with photonic band ga

Harkonen, K.

Maniscalco, S.

Piilo, J.

Suominen, K. -A.

English

arXiv

Open quantum systems that interact with structured reservoirs exhibit non-Markovian dynamics. We present a quantum jump method for treating the dynamics of such systems. This approach is a generalization of the standard Monte Carlo wave function (MCWF) method for Markovian dynamics. The MCWF method identifies decay rates with jump probabilities and fails for non-Markovian systems where the time-dependent rates become temporarily negative. Our non-Markovian quantum jump approach circumvents this problem and provides an efficient unraveling of the ensemble dynamics

Piilo, Jyrki

Maniscalco, Sabrina

Harkonen, Kari

Suominen, Kalle-Antti

Heriot Watt Pure

Non-Markovian Quantum JumpsJyrki Piilo, Sabrina Maniscalco, Kari Ha¨rko¨nen, and Kalle-Antti SuominenDepartment of Physics, University of Turku, FI-20014 Turun yliopisto, Finland(Received 11 March 2008; published 5 May 2008)Open quantum systems that interact with structured reservoirs exhibit non-Markovian dynamics. Wepresent a quantum jump method for treating the dynamics of such systems. This approach is a general-ization of the standard Monte Carlo wave function (MCWF) method for Markovian dynamics. TheMCWF method identifies decay rates with jump probabilities and fails for non-Markovian systems wherethe time-dependent rates become temporarily negative. Our non-Markovian quantum jump approachcircumvents this problem and provides an efficient unraveling of the ensemble dynamics.DOI: 10.1103/PhysRevLett.100.180402 PACS numbers: 03.65.Yz, 42.50.LcIntroduction.—When an open quantum system interactswith a reservoir having nontrivial structure, the systemdynamics exhibits non-Markovian memory effects [1].The information on the state of the open system is con-tained in the density matrix whose time evolution is gov-erned by a master equation consisting of two parts. Thesystem Hamiltonian induces unitary evolution of the den-sity matrix, while the dissipative part, which includes theinformation on the properties of the environment in theform of decay rates, induces nonunitary effects via thejump operators. Already for Markovian systems, whichdo not have memory, finding the solution of the masterequation may be very complicated. The task gets evenmore challenging with non-Markovian systems and struc-tured reservoirs. Such systems display modified decaydynamics paving the way to new types of quantum controltechniques [2].Non-Markovian systems appear in many branches ofphysics, such as quantum optics [1–3], solid state physics[4], and quantum chemistry [5]. In quantum informationprocessing [6], the non-Markovian character of decoher-ence has to be accounted for, and it leads to the concept ofnon-Markovian quantum channels [7]. Decoherence alsoplays a central role in the transition from quantum toclassical world [8]. In fact, non-Markovianity has beenrecently proposed as a means to manipulate the quantum-classical border [9]. Since it is elusive to solve the opensystem dynamics, new methods for non-Markovian sys-tems are highly desired.In this Letter we provide an efficient way to unravel ageneral non-Markovian master equation. The differentways to build an ensemble of stochastic wave functionsdescribing the density matrix fall roughly into two catego-ries [10]: time evolution including (i) discontinuouschanges (quantum jumps), e.g., the Monte Carlo wavefunction (MCWF) method [11]; (ii) continuous stochasticchanges, e.g., the Quantum State Diffusion (QSD) method[12,13]. Our non-Markovian quantum jump (NMQJ)method generalizes the widely used Markovian MCWFinto the field of non-Markovian systems, and thus belongsto the first of the two categories.There exists a non-Markovian variant of QSD [12] and asomewhat related formulation [14]. These methods, how-ever, are difficult to implement beyond very simple ex-amples. Other unravelings of non-Markovian masterequations contain fictitious harmonic oscillator modes[15] and pseudomodes [16], or some other forms of ex-tensions of the system Hilbert space [17,18]. One formu-lation, using quantum jumps, exploits an analogue to thehidden variable theory [19]. The use of extended Hilbertspaces comes always with an added cost for computationalefficiency.Our formulation avoids the use of Hilbert space exten-sions and is based on the following observation. Theinformation, which the system loses to the environmentat the time of the jump, can be later recovered by thesystem due to non-Markovian memory. We show explicitlyhow this happens on the level of single realizations. Beforediscussing the insight and benefits that our NMQJ methodprovides, we first introduce the master equation and themethod and present a case study with an atom in a photonicband gap.Non-Markovian master equation.—The non-Markoviandynamics of the reduced system density matrix t isgiven by the master equation [1] _t  1i@Hs; t XjjtCjttCyj t 12Xjjtft; Cyj tCjtg: (1)Above, Hs is the system Hamiltonian and Cjt are thejump operators describing changes in the system due tointeraction with the reservoir. jt is the decay rate ofchannel j. It can be shown that the most general masterequations local in time for non-Markovian systems can becast in the form of Eq. (1) [18]. In the Markovian case allj are positive constants. In the non-Markovian case therates may oscillate and take negative values for finite timeintervals. This is a sign of the non-Markovian memoryPRL 100, 180402 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending9 MAY 20080031-9007=08=100(18)=180402(4) 180402-1 © 2008 The American Physical Societyeffects and reflects the exchange of information back andforth between the system and the reservoir.MCWF and NMQJ methods.—The system properties arecalculated as an average over the state vector ensemble ofsize N, and we follow closely the MCWF method [11]. Ageneric way to write the density matrix is t XNtNj tih tj; (2)where Nt is the number of ensemble members in thestate j ti at time t. The deterministic evolution of agiven state vector j ti, for small enough time steps tand before the renormalization, is given by jt ti 1 iHt@j ti; (3)where the non-Hermitian Monte Carlo Hamiltonian isH  Hs  i@Pj12 jtCjtyCjt and the renormalizedstate is j t ti  jt ti=jjjt tijj. Forpositive decay channels j, jt> 0, the deterministicevolution is interrupted by jumps j ti !Cjtj ti=jjCjtj tijj which occur with probabil-ity Pj t  jtth tjCyjtCjtj ti (4)during time step t [11]. The Markovian MCWF methodcan be extended to the situations where the rates becometime dependent, but this is limited to positive decay ratesonly.In our approach the non-Markovian quantum jumps fornegative channels j, jt< 0 have the form Dj!0 t  j 0 tih tj; (5)where the source state of the jump is j ti Cjtj 0 ti=jjCjtj 0 tijj. This transition for a givenstate vector j i in the ensemble (2) occurs with theprobability Pj!0 t N0 tNt jjtjt h 0 tjCyjtCjtj 0 ti: (6)Note that the probability of the non-Markovian jump isgiven by the target state of the jump. The sign of the decayrate jt can be understood in the following way. First,when for a given channel j, jt> 0 the process goes asj 0 i ! j i  Cjj 0 i=jjCjj 0 ijj. Later on, when thedecay rate becomes negative, jt< 0, the direction ofthis process is reversed and the jump occurs to oppositedirection j 0 i  j i.The proof of our NMQJ method goes in a very similarway to that of the Markovian MCWF method [11]. Byweighting the deterministic and jump paths over time stept with the appropriate probabilities we should obtain themaster Eq. (1). Calculating the average  of the evolutionof the ensemble (2) over t gives t t XNtN1XjPj t Xj;0Pj!0 t jt tiht tjjjjt tijj2XjPj tCjtj tih tjCyjtjjCjtj tijj2 Xj;0Pj!0 tDj!0 tj ti h tjDjy!0 t: (7)Here, the summations  and 0 run over the ensemble, andthe summations over j and j cover the positive andnegative channels, respectively. The first term on theright-hand side for the summation over  is the productof the no-jump probability and the deterministic evolutionof the state vector; the second and third terms describe thepositive and negative channel jumps, respectively, with thecorresponding probabilities. By using Eq. (5), the last termcan also be written asPj;0Pj!0 tj 0 tih 0 tj. Ingeneral, using Eqs. (2)–(7) in Eq. (7), and keeping inmind the form of the reversed jump j i ! j 0 i withj i  Cjj 0 i=jjCjj 0 ijj gives the master Eq. (1).Example: Photonic band gap.—To illustrate the NMQJmethod we choose a two-level atom inside a photonic bandgap (PBG) [2,20]. Fictitious and pseudomode methods[15,16] do not work for this system since the typicalreservoir distribution function for PBG is not a meromor-phic function due to the band edge. Moreover, an earlierattempt to develop a specific jump approach for this system[21] has been shown to be correct only in the Born-Markovlimit [22]. One of the reasons for this is that the method ofRef. [21] fails to describe the reabsorption of photons bythe atoms [22]. Our method succeeds in this by using non-Markovian quantum jumps [cf. Eq. (5)]. This example alsoshows that local-in-time master equations can be used todescribe non-Markovian dynamics for strong system-reservoir interactions.The master equation for the density matrix of the two-level system takes the form [1] _t  1i@St2; ttt  12 f; tg; (8)where St is the Lamb shift, t the decay rate,  jgihej, and   y. Here, jgi denotes the ground state ofthe two-level atom, jei the excited state, and there is onedecay channel taking the atom from jei to jgi. We calculatethe Lamb shift and the decay rates by using Eq. (2.21) ofPRL 100, 180402 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending9 MAY 2008180402-2Ref. [20] and Eqs. (10.22) and (10.23) from Ref. [1]. Theoscillatory behavior and negative values of the decay rateare displayed in Fig. 1(a).Figure 1(b) shows the match between the exact result{cf. Eq. (2.21) of Ref. [20]} and the simulation results withN  105 realizations for two different initial states. Wehave chosen parameters which correspond to Fig. 1 ofRef. [20] with the detuning    from the edge ofthe gap. Here,   !7=20 d2=60@c32=3, where !0 isthe Bohr frequency and d the absolute value of dipolemoment of the atom. The results illustrate a typical featureof PBG: atom-photon bound state and population trapping.Figure 2 displays an example of non-Markovian quantumjump in a single realization of the process for the case ofinitial superposition state. First, during the positive decay,a jump takes the atom to its ground state and the excitationresides in the environment. Later on with negative rate, thesuperposition state is restored by a non-Markovian quan-tum jump, and the photonic component is reabsorbed bythe atom.Insight by NMQJ.—In the PBG example above, the keyingredient to describe non-Markovian memory is the vir-tual photon emission reabsorption cycle on the level ofsingle realization. The physical state of the system is givenby the density matrix, i.e., the ensemble of state vectors.This illustrates an interesting aspect of our method: it ispossible to describe the effects of non-Markovian memorywithout extending the Hilbert space of the reduced system,which is a trait used in the previously developed jumpmethods [15–18]. In the NMQJ method, the memory ofthe ensemble member j i, i.e., the information about thestate before the positive rate jump to the state j i oc-curred, is carried by the other ensemble member j 0 i.Consequently, the density matrix and the correspondingensemble indeed carry information on the earlier state ofthe system.Negative decay rates, which occur in non-Markoviansystems, can be interpreted in the following way. Duringthe initial period of positive decay, the correspondingjumps distribute the state vector probability over theHilbert space accordingly; the number of terms in thesummation of Eq. (2) increases. When the decay rate lateron becomes negative, which indicates the memory effects,the direction of the probability flow is reversed. This meansthat a process j i ! j 0i with negative rate corresponds toj i  j 0i. From the classical perspective, it seems ratherusual that changing the sign of the rate of the processmeans that the process occurs to the opposite direction.In the quantum world with superpositions, probabilityamplitudes, and coherences the issue is less straightfor-ward. In our method, this appears as a restoration ofseemingly lost superpositions and subsequent revival ofcoherences.The algorithm and numerical efficiency.—Since in theNMQJ method the realizations depend on each other due tomemory effects [cf. Eq. (6)], it seems at first sight that allthe N ensemble members have to be evolved simulta-neously. However, according to Eq. (2), the ensembleconsists of several copies of each j ti. Obviously, thereis no need to have on a computer several copies of the samestate vector. It is sufficient to have one copy and thecorresponding integer number N. Any number N of therealizations of the process can be done by making Neff 	N state vector evolutions where Neff is equal to the numberof terms in the summation N  PN. When the realiza-tions of the process are generated on a computer, a jumpmeans changing the integer numbers Nt accordingly inEq. (2). A considerable saving in CPU time is achieved00.51 (a)0 5 10 1500.20.40.60.81TIME  [1/β]EXCITED STATE   DECAY  PROBABILITY  RATE  [β]Example 1Example 2(b)NMQJ simulationanalyticalFIG. 1 (color online). (a) The decay rate for a two-level atomin photonic band gap as a function of time. (b) NMQJ and exactresults. In (a) the decay rate displays oscillatory behavior withtemporary negative values. In (b) we plot the excited stateprobability of the atom, and the results show the match betweenthe exact and simulation results. The initial pure states inexamples 1 and 2 are jei and jgi  jei= 2p , respectively.0 2 4 6 8 1000.20.40.60.81TIME  [1/β]GROUND AND EXCITED STATE PROBABILITIESexcited stateground stateFIG. 2 (color online). An example realization with a jump—reverse jump cycle. The ground and excited state probabilitiesare given as function of time. The first jump at time t ’ 0:8occurs at the positive decay rate region and destroys the super-position state. The second jump at t ’ 5:0 occurs at the negativedecay region and recreates the superposition. The dotted linesshow the evolution without any jumps.PRL 100, 180402 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending9 MAY 2008180402-3since it is not necessary at each point of time to evolve Nstate vectors; instead, it is enough to decide N times if thejumps occurred or not.Let us illustrate this with an example. In the PBG caseabove, we have Neff  2 and the corresponding state vec-tors are j 0ti and j 1ti  jgi for all t. These are theinitial state affected by the deterministic evolution and theground state, respectively. In the positive decay region thejumps occur as j 0ti ! j 1ti  jgi; each jump reducesN0 by 1 and increases N1 by 1. In the negative decay regionthe process goes to opposite direction j 1ti  jgi !j 0ti. In the optimized simulation to have 105 realiza-tions we need to generate only one deterministic evolutionfor j 0ti and then decide the jumps as described above.In QSD [12], the stochastic change of the state vectors iscontinuous, which leads in practice to Neff  N. For thedoubled Hilbert space (DHS) method [17], the norm of thestate vectors increases in the negative decay region. As aconsequence, the norm of a given state vector depends onthe point of time where the DHS jump happens during thenegative decay. In the ensemble, the jumps occur at eachtime point and Neff becomes large compared to the NMQJmethod. Moreover, the DHS state vectors are evolved inthe Hilbert space twice as large as in NMQJ.In contrast to the DHS method, the triple Hilbert space(THS) method preserves the norm of state vectors [18].However, in the most general case, when the jump opera-tors depend on time in the master Eq. (1), the jumps withthe extended THS operators increase Neff at each point oftime during the negative decay. Consequently, the THSmethod cannot use the built-in optimization of the NMQJmethod. Moreover, the THS method has two other ingre-dients which have an impact on its numerical performance:(i) the need for 4 times larger number of decay channelsthan NMQJ uses {see the text below Eq. (57) in Ref. [18]};(ii) the state vectors live in the space which is 3 times largerthan the original one {see Eqs. (27)–(29) in Ref. [18]}. Theconsequent complications of the THS method make itdifficult to make a general statement on its numericalperformance. However, all the facts above lead to theconclusion that even the most cautious estimate wouldgive roughly an order of magnitude difference in thenumerical efficiency between the NMQJ and the THSmethods.Conclusions.—The quantum jump description forMarkovian systems (MCWF) is widely accepted due toits straightforward nature and the simple physical picturethat it provides. For non-Markovian systems, the NMQJmethod maps memory into reverse jumps that restorequantum superpositions. Furthermore, our approach be-comes equivalent to the standard MCWF method in theMarkovian limit. In a broader view, the continuously grow-ing interest in quantum information [6] and nanophysics[23] emphasizes the need to consider single quantum sys-tems at diminishing time scales and in tailored and finiteenvironments. This development provides the backgroundfor the NMQJ approach.This work has been supported by the Academy ofFinland (Projects No. 108699, No. 115682, andNo. 115982) and the Magnus Ehrnrooth Foundation. Wethank H.-P. Breuer and B. Garraway for stimulatingdiscussions.[1] H.-P. Breuer and F. Petruccione, The Theory of OpenQuantum Systems (Oxford University Press, Oxford,2002).[2] P. Lambropoulos et al., Rep. Prog. Phys. 63, 455 (2000).[3] C. W. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, Berlin, 1999).[4] See, e.g., C. W. Lai et al., Phys. Rev. Lett. 96, 167403(2006) and references therein.[5] J. Shao, J. Chem. Phys. 120, 5053 (2004); A. Pomyalovand D. J. Tannor, J. Chem. Phys. 123, 204111 (2005) andreferences therein.[6] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge University Press,Cambridge, England, 2000); S. Stenholm and K.-A.Suominen, Quantum Approach to Informatics (JohnWiley & Sons, Hoboken, New Jersey, 2005).[7] See, e.g., D. 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Non-Markovian quantum jumps

Non-Markovian quantum jumps

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Heriot Watt Pure

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