Table Of ContentNon-Markovian dynamics without using quantum trajectory
Chengjun Wu, Yang Li, Mingyi Zhu, and Hong Guo∗
CREAM Group, State Key Laboratory of Advanced Optical Communication Systems and Networks (Peking University)
School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China
(Dated: January 25, 2011)
Open quantumsystem interacting with structured environment is important and manifests non-
Markovianbehavior,whichwasconventionallystudiedusingquantumtrajectorystochasticmethod.
In this paper, by dividing the effects of the environment into two parts, we propose a determin-
istic method without using quantum trajectory. This method is more efficient and accurate than
stochasticmethodinmostMarkovianandnon-Markoviancases. Wealsoextendthismethodtothe
generalized Lindblad master equation.
1
1 PACSnumbers: 03.65.Yz,42.50.Lc
0
2
Whenanopenquantumsysteminteractswithenviron- unitary evolution of the states and the probability flow
n
ment, it experiences decoherence and dissipation which between these states. Moreover, we also extend this
a
J leadtolossofinformation. Suchopenquantumsystemis approach to the generalized Lindblad master equation
depicted by a reduced density matrix which shows non- whichcandealwithsomestrongcouplingcases[19]. The
3
2 unitaryevolution. Ontheotherhand,theenvironmentis algorithmandnumericalefficiencyaregiven,whichshow
classified as Markovianwith no memory effect, and non- thatourmethodismoreefficientandaccuratethanthose
] Markovianwithmemoryeffect. InMarkoviancase,since based on stochastic simulation in most Markovian and
h
there is no memoryeffect, the quantumtrajectorybased non-Markoviancases.
p
Monte Carlo wave function (MCWF) method [1–3] and The dynamics of the non-Markovian system is gov-
-
t quantumstatediffusion(QSD)method[4,5]areapplied. erned by the following master equation [8]
n
a However, in non-Markovian case, due to memory effect, 1
u theinformationlostbythesystemduringtheinteraction ρ˙(t)= [H ,ρ(t)]+ γ (t)C (t)ρ(t)C†(t)
q with the environment will come back to the system in a i~ s Xj j j j
2 [ liaotresrthtiamneMaanrdkosvoiashnocwassem. uch more complicated behav- − 21 γj(t){ρ(t),Cj†(t)Cj(t)}, (1)
v Non-Markovian systems are important for their ap- Xj
1
plications to many fields of physics, such as quan- where H is the systemHamiltonian including the Lamb
7 s
tum information processing [6, 7], quantum optics [8], shift, C (t)arethe jumpoperatorswhichinduce changes
7 j
0 solid state physics [9], and chemical physics [10]. Re- [e.g., jump from state ψα(t) to ψα′(t) i.e., |ψα′(t)i =
. cently, non-Markovian behaviors have also been studied Cj(t)|ψα(t)i/||Cj(t)|ψα(t)i|| ] in the system, and γj(t)
2
in biomolecules where the molecules are embedded in are the decay rates which may take negative values for
1
a solvent and/or in a protein environment [11]. Since some time intervals. The reduced density matrix can be
9
0 there is no true pure state quantum trajectory due to written as [17]
: the memory effect [12], the quantum trajectory based
v Markovianmethods do not work. Thus, doubled Hilbert Neff
i ρ(t)= p (t)|ψ (t)ihψ (t)|, (2)
X space (DHS) method [13], triple Hilbert space (THS) α α α
r method [14], non-Markovian QSD method [15, 16], and αX=1
a non-Markovianquantum jump (NMQJ) method [17, 18] where p (t) is the probability of the system being in the
α
areproposedtosolvethenon-Markoviandynamicsofthe state |ψ (t)i at time t. Further, it should be pointed
α
system where the memory effect is taken into account. out that the effective number of the states N is de-
eff
However,inordertoobtainhighaccuracy,allthesemeth- termined by C (t)’s [18], Neff p (t) = 1 and that the
j α=1 α
ods, which are based on stochastic simulations, need to
state |ψα(t)i is normalizedP.
fulfill a large number of realizations and is very time-
To solve the dynamics ofthe system, one shouldknow
consuming. So, new methods which are more efficient
thetimeevolutionof|ψ (t)ianditsprobabilityp (t). In
α α
and accurate are highly desired.
ourmethod,thetimeevolutionofthestate|ψ (t)iisthe
α
In this paper, a deterministic method without us- same as that in NMQJ [17]. In NMQJ, the probability
ing quantum trajectory is proposed to solve the non- p (t) is calculated in a stochastic way by using quan-
α
Markovian dynamics. The influence of the environment tum trajectory to N ensemble members. In our method,
on the system is divided into two parts, i.e., the non- however, the evolution of probability p (t) is given in a
α
deterministic way:
′
p˙α(t)=− Γjα(t)pα(t)+ Γjα′(t)pα′(t), (3)
∗Correspondenceauthor:hongguo@pku.edu.cn Xj (Xα′,j)
2
where Γj(t) =γ (t)kC (t)|ψ (t)ik2 and ′represents where Ni is determined in the same way in Eq. (2)
α j j α eff
(αP′,j) by taking all the jump operators Rij,s and all the states
the summation over all the pairs (α′,j) satisfying ψα(t),s in each ρ (t) into consideraνtion.
|ψα(t)i = Cj(t)|ψα′(t)i/kCj(t)|ψα′(t)ik. One finds that jThe evolutionjof state |ψα(t)i is governedby the non-
the probabilityofthe state, p (t), changesvia the mech- i
α linear differential equation[21]
anism of jumps for “out” (α → α′) and “in” (α′ → α),
respectively. d
i |ψα(t)i=Gˆ(ψα)(t)|ψα(t)i, (8)
The numerical simulation corresponding to Eq. (3) is dt i i i
straightforward:
where Gˆ(ψα)(t) = Hi − i Rji†Rji +
i 2 ν ν
pα(t+δt)=pα(t)−δt Γjα(t)pα(t)+δt ′Γjα′(t)pα′(t). i Rji|ψα(t)i 2. By combining EqPjsν. (6), (7),
Xj (Xα′,j) 2 ν i
(4) (8P)jνa(cid:13)(cid:13)nd noting th(cid:13)(cid:13)at ψ1(t) ψ1(t) , ψ2(t) ψ2(t) , ···,
Note that there is no stochastic noise and no need to i i i i
Ni Ni (cid:12) (cid:11)(cid:10) (cid:12) (cid:12) (cid:11)(cid:10) (cid:12)
consider the sign of the decay rate during the simula- ψ eff(t) ψ eff(t(cid:12)) are linear(cid:12)ly(cid:12) independent(cid:12), the
(cid:12) i (cid:29)(cid:28) i (cid:12)
tion. Additionally, the pα(t)’s in our method do repre- (cid:12)(cid:12)evolution of pα(t) is g(cid:12)(cid:12)iven by
sent the probability of the system actually being in the (cid:12) i (cid:12)
corresponding pure state ensemble. p˙α(t)=− Γji pα(t)+ ′Γij pα′(t), (9)
Consider a particular transition: |ψα′(t)i = i να i να′ j
C (t)|ψ (t)i/||C (t)|ψ (t)i||, then the corresponding Xjν (jX,ν,α′)
j α j α
probability change takes the form:
where Γij = Rij ψα(t) 2 and ′ represents the
να ν j
p (t+δt)=p (t)−δtp (t)Γj(t), (cid:13) (cid:12) (cid:11)(cid:13) (j,Pν,α′)
α α α α (5) summationove(cid:13)rallt(cid:12)hepair(cid:13)s(j,ν,α′)satisfying|ψiα(t)i=
pα′(t+δt)=pα′(t)+δtpα(t)Γjα(t). Rij ψα′(t) Rij ψα′(t) . It can be easily seen that
ν j ν j
(cid:12) E.(cid:13) (cid:12) E(cid:13)
by s(cid:12)etting n=(cid:13)1an(cid:12)dtaking(cid:13)the decayratesγ(t) into the
When the decay rate γj(t) is positive or negative, the (cid:12) (cid:13) (cid:12) (cid:13)
equation, Eq. (9) degenerates to Eq. (3).
probability flow is from |ψα(t)i to |ψα′(t)i or reversed.
Example 1: Detuned Jaynes-Cummings model.–
ThishasbeenmentionedinRef.[17]. However,itismore
Consider a system with a two-level atom in a detuned
explicit in our method. From Eq. (5), it is clear that, in
damped cavity, which is governed by the time convolu-
thenegativedecayregion,theamountofprobabilityflow
tionless master equation [8]
only depends on the target state and the probability of
the system being in the target state. This is similar to
i
the situation in NMQJ [17], where the jump probability ρ˙(t)=− S(t){σ+σ−,ρ(t)}
2
in the negative decay region is proportional to the num-
1 1
ber of particles in the target state. These indicate that +γ(t){σ ρ(t)σ − σ σ ρ(t)− ρ(t)σ σ }.
− + + − + −
2 2
the trajectory of a particle in NMQJ can not be inter-
(10)
pretedastrue trajectorysince thejump processdepends
on the status of other particles in the system. Because The spectral density of the cavity is supposed to be of
true pure state quantum trajectories do not exist in the Lorentzian profile, i.e., J(ω) = γ0λ2 , where
non-Markoviandynamics [12], it is not necessary to cal- 2π[(ω0−∆−ω)2+λ2]
∆ = ω −ω is the detuning between the cavity mode
culate p (t) in a stochastic way. 0 c
α and the atom. To second order approximation, the
Next, we extend our method to the recently proposed
Lamb shift and the decay rate take the form [8] S(t) =
generalizedLindbladmasterequationwhichcansolvethe γ0λ∆ {1−e−λt[cos(∆t)+ λ sin(∆t)]},γ(t)= γ0λ2 {1−
dynamics of some highly non-Markoviansystems[19], λ2+∆2 ∆ λ2+∆2
e−λt[cos(∆t)− ∆sin(∆t)]}. In this model, there is only
λ
d ρ =−i[H ,ρ ]+ Rijρ Rij†− 1{Rji†Rji,ρ } , one jump operator C = σ− = |gihe|, which is a lower-
dt i i i (cid:18) λ j λ 2 λ λ i (cid:19) ing operator. We assume thatρ(0)=|ψ1(0)ihψ1(0)| and
Xjλ choose|ψ (0)i=(4|ei+3|gi)/5. Actingthejumpopera-
1
(6)
tor on the state |ψ (0)i, we get |ψ (0)i=|gi. According
1 2
where i,j =1,2,··· ,n, H are any Hermitian operators,
i to Eq. (4), at time t+δt, the probabilities become
andRij areanysystemoperators. Itshouldbeindicated
λ
n p (t+δt)=p (t)−δtp (t)Γ1(t),
that ρ(t)= ρi(t). p1(t+δt)=p1(t)+δtp1(t)Γ11(t), (11)
iP=1 2 2 1 1
The ith density matrix is decomposed as:
where Γ1(t)=γ(t)|he|ψ (t)i|2.
1 1
Neiff Inthis example, ρee(t)is proportionaltothe energyof
ρi(t)= pαi(t)|ψiα(t)ihψiα(t)|, (7) the system and p2(t) represents the probability for one
αX=1 photon being in the environment. Although p1(t) and
3
p (t) canbe solvedanalytically,inorderto illustrateour
2
method, we use Eq. (11) to do the simulation. The pa-
rameters are chosen as ∆ = 12λ,γ λ = 4,λδt = 0.005.
0
Figure 1 (a) shows explicitly the reversal of the proba-
FIG.2: (coloronline)Atwo-statesystemcoupledtoanenvi-
ronment consisting of two energy bands. Comparison of our
method(bluelong-dashedcurve)andMonteCarlosimulation
4
(with N =10 trajectories, green dash-dot curve) to analyt-
FIG. 1: (color online) Dynamics of detuned Jaynes-
ical result (red solid curve). The parameters are δǫ = 0.31,
Cummings model. The initial state is |ψ1(0)i = (4|ei +
γ1=γ2 =1 and time step δt=0.01.
3|gi)/5andtheparametersare∆=12λ,γλ=4,λδt=0.005.
(a) The probabilities for the system in states |ψ1(t)i and
|ψ2(t)i. (b) The population of the excited state ρee (ini-
tiallyhigherline)andtheabsolutevalueofthecoherenceρeg than Monte Carlo simulation method.
(initially lower line) with three methods: analytic (red solid According to Eqs. (4), we only need to calculate N
eff
curve),ourmethod(bluelong-dashedcurve)andNMQJ(with statesandchangetheprobabilitiesdeterministically. The
N =104 particles in the system, green dash-dot curve).
timecostisalmostdeterminedbythecalculationofN
eff
states. However,the evolutionofN statesis indepen-
eff
bility flow. We can see from Fig. 1 (a) and (b) that dentwitheachother,sowecancalculatethemparallelly.
when the probability flow gets reversed, the energy and In addition, if the jump operatorscan be representedby
coherenceoftheatomincrease. Theseshowexplicitlythe sparse matrixes, we only need to calculate the evolution
memory effect that the reduced system restores the in- of the states appearing in the decomposition of ρ(0) and
formation lost earlier. In Fig. 1 (b), the result of NMQJ usethejumpoperatorstoobtainotherstates. Moreover,
(with N = 104 particles in the system) is also given, since the sign of the decay rate makes no difference dur-
which shows that our method is more accurate. ing the simulation, in non-Markovian case, our method
Example 2: Application to generalized Lindblad mas- is as efficient as it behaves in Markoviancase.
ter equation.– To illustrate our method for this kind of
Similar to our method, the NMQJ method [17, 18]
equation, we consider a two-state system coupled to an
needs to calculate N states. However, in addition to
eff
environment consisting of two energy bands, each with
that, NMQJ has to consider the sign of the decay rates
a finite number of evenly spaced levels. This may be
and generate N random numbers (N ≫N ) to decide
eff
viewed as a spin coupled to a single molecule or a single
the jump process at each time step δt. Apparently, our
particlequantumdot[20]. Byusingtime-convolutionless
method is more efficient than NMQJ in any case.
projection operator technique, to the second order, the
InMarkoviancase,theMCWF[1]andQSD[4]method
generalizedLindbladmasterequationtakestheform[21]
need to realize a large number of trajectories for every
state appearing in the decomposition of ρ(0). When
d t
ρ = dt h(t−t )[2γ σ+ρ σ−−γ {σ+σ−,ρ }], the number of these trajectories is larger than N ,
1 1 1 1 2 2 1 eff
dt Z
0 which is always the case, our method is more efficient
d t
ρ = dt h(t−t )[2γ σ−ρ σ+−γ {σ−σ+,ρ }], than them. In non-Markovian case, the DHS method
2 1 1 2 1 1 2
dt Z0 [13], THS method [14] and non-MarkovianQSD method
(12)
[15, 16] all introduce additional cost for computational
where γ h(t − t ),(i = 1,2), is the environment corre-
i 1 efficiencycomparedtoMCWForQSD.However,innon-
lation function with h(t) = δεsin2(δεt/2) where δε is the Markoviancase,ourmethodisasefficientasitbehavesin
2π(δεt/2)2
widthoftheupperandlowerenergybands. Thereduced Markoviancase. Thus,when the number ofthese trajec-
density matrix for the system is given by ρ = ρ1 +ρ2. tories is larger than Neff, our method is obviously more
We assume that ρ (0) = |eihe| and ρ (0) = 0. The pa- efficient than them, too.
1 2
rameters are chosen as δǫ = 0.31 and γ = γ = 1. In As for the accuracy, since there is no statistical noise
1 2
Fig. 2 we compare the results of our method, analytical in our method and the error caused by finite time step
solution and Monte Carlo simulation which is based on δt is the same, compared with all the methods based
theunravelingofthemasterequation(withN =104tra- on stochastic simulation, our method is more accurate.
jectories) [21]. Apparently, our method is more accurate Actually, our method is the limit case when the number
4
of realizations in the stochastic based methods tends to tages in efficiency and accuracy. Additionally, we ex-
infinite. tendedthis approachto the generalizedLindbladmaster
Inconclusion,bydividingthe influenceofthe environ- equation , which is useful to solve the dynamics of some
ment on the system into two parts, i.e., the non-unitary highly non-Markoviansystems.
evolutionofthesestatesandtheprobabilityflowbetween
them, we propose a deterministic method to solve the This work is supported by the Key Project of the Na-
non-Makovian dynamics. Compared with the method tional Natural Science Foundation of China (Grant No.
based on stochastic simulation, our method has advan- 60837004).
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