Front. Phys. 11(4), 110309 (2016)DOI 10.1007/s11467-016-0570-9

Review article

Stochastic description of quantum Brownian dynamics

Yun-An Yan1,∗, Jiushu Shao2,†

1Guizhou Provincial Key Laboratory of Computational Nano-Material Science,Guizhou Education University, Guizhou 550018, China

2Center of Advanced Quantum Studies and College of Chemistry,Beijing Normal University, Beijing 100875, China

Corresponding authors. E-mail: ∗yunan@gznc.edu.cn, †jiushu@bnu.edu.cnReceived February 20, 2016; Accepted March 14, 2016

Classical Brownian motion has well been investigated since the pioneering work of Einstein, whichinspired mathematicians to lay the theoretical foundation of stochastic processes. A stochastic for-mulation for quantum dynamics of dissipative systems described by the system-plus-bath modelhas been developed and found many applications in chemical dynamics, spectroscopy, quantumtransport, and other fields. This article provides a tutorial review of the stochastic formulation forquantum dissipative dynamics. The key idea is to decouple the interaction between the system andthe bath by virtue of the Hubbard-Stratonovich transformation or Itô calculus so that the systemand the bath are not directly entangled during evolution, rather they are correlated due to thecomplex white noises introduced. The influence of the bath on the system is thereby defined by aninduced stochastic field, which leads to the stochastic Liouville equation for the system. The exactreduced density matrix can be calculated as the stochastic average in the presence of bath-inducedfields. In general, the plain implementation of the stochastic formulation is only useful for short-timedynamics, but not efficient for long-time dynamics as the statistical errors go very fast. For linearand other specific systems, the stochastic Liouville equation is a good starting point to derive themaster equation. For general systems with decomposable bath-induced processes, the hierarchicalapproach in the form of a set of deterministic equations of motion is derived based on the stochasticformulation and provides an effective means for simulating the dissipative dynamics. A combinationof the stochastic simulation and the hierarchical approach is suggested to solve the zero-temperaturedynamics of the spin-boson model. This scheme correctly describes the coherent-incoherent tran-sition (Toulouse limit) at moderate dissipation and predicts a rate dynamics in the overdampedregime. Challenging problems such as the dynamical description of quantum phase transition (local-ization) and the numerical stability of the trace-conserving, nonlinear stochastic Liouville equationare outlined.

Keywords stochastic description, quantum dissipation, spin-boson model, hierarchical approach

PACS numbers 03.65.Yz, 02.70.-c, 05.30.-d, 05.10.Gg

Contents

1 Introduction 12 Decoupling system-bath interaction 3

2.1 Heuristic decoupling via stochastic fields 32.2 Decoupling via Itô calculus 42.3 Bath-induced stochastic field 52.4 Hermitian stochastic Liouville equation 7

3 Stochastic simulations of spin-boson model 84 From stochastic to deterministic: Master

equation 114.1 High-temperature approximations 124.2 Master equation for dissipative harmonic

oscillators 13

∗Special Topic: Progress in Open Quantum Systems: Fundamen-tals and Applications.

5 Numerical algorithms based on stochasticformulation 145.1 Decomposition of bath response function 145.2 Hierarchical approach 145.3 Performance of hierarchical approach 16

6 Hybrid stochastic-hierarchical equation 177 Summary and perspectives 20

Acknowledgements 21References 21

1 Introduction

If one would calculate the dynamics of the universe mi-croscopically by either classical or quantum mechanics,one would have an exact solution to the evolution ofthe universe as a whole and also for any physical vari-ables. This is, of course, an impossible task. Exact sim-ulation with quantum dynamics is subject to the curse

c⃝ Higher Education Press and Springer-Verlag Berlin Heidelberg 2016

Review article

of dimensionality, a term borrowed from Bellman [1]. Infact, as the computer memory required to store the wavefunction in the Schrödinger picture or the CPU time topropagate the wavefunction by virtue of the path in-tegral grow exponentially with the size of the system,the curse of dimensionality is unavoidable and the exactsimulation with quantum dynamics is limited to systemswith few degrees of freedom. By contrast, the computa-tional cost with molecular dynamics based on classicalmechanics is linearly proportional to the size of the sys-tem. Consequently, the molecular dynamics simulationhas become a popular and powerful tool for studyingphysical properties of complex molecular systems whenthe quantum effect is negligible [2]. However, even lin-ear scaling is too expansive for the simulation of thewhole universe or any macroscopic system. Fortunately,we are in general interested in the dynamical behavior of“few” variables instead of the whole universe [3]. Thesevariables, therefore, define the system of interest andthe rest of the universe coupled to the system is usuallycalled environment or heat bath.

When a system of interest is coupled to the other per-haps infinite variables, it is a dissipative or open system.Open systems are ubiquitous in the real world becauseinnumerable dynamical processes are mediated by theenvironmental condensed phases [4–7]. Of most impor-tance are diversified transport phenomena in gases, liq-uids and solids, chemical reactions in solutions, surfacesor interfaces, biological processes in cells and so on.

A paradigm of open systems is the Brownian parti-cle. In his miracle year of 1905, Einstein laid the phys-ical foundation of the Brownian motion in terms ofthe Einstein–Smoluchowski relation [8, 9], the precur-sor of the more general fluctuation-dissipation theorem[10–12]. After the pioneering work of Einstein manyscientists have made significant contributions to thisfield and the culminating results include the Johnson-Nyquist theorem [13, 14], the Langevin equation [15,16], the Fokker-Planck equation [17–20], the Ornstein–Uhlenbeck process [21], the Kramers problem [22, 23],and the phenomenological stochastic model for line-shape in spectroscopy [24].

Although various open system dynamics have beenextensively investigated at different levels, an apparentobstacle in developing a microscopic description is thequantum effect, which becomes even dominant whenthe temperature of the environment is lower than thescale of the intrinsic energy. The proof of the quantumfluctuation-dissipation theorem was a milestone in thestudy of quantum Brownian motion [25, 26]. Up to date,there are still no reliable approaches for simulating thequantum dynamics of general open systems. Formally,one can employ the projection operator technique tosolve the motion of the bath. By doing so, one obtainsthe Nakajima-Zwanzig equation which is a generaliza-tion of the master equation and contains an extremelycomplicated retarded time integration over the history

of the reduced system [27, 28]. The Langevin equationcan be also generalized to deal with the quantum dis-sipation when the quantum noise is used instead of theclassical c-number noise [29]. There have been severalexcellent monographs and reviews covering quantumBrownian motion [30–34].

A widely used, generic model for describing open sys-tems has been proposed by Caldeira and Leggett [35].In this model the bath consists of infinite “effective” har-monic oscillators that are linearly coupled to the systemand the impact of the bath on the system is fully char-acterized by its spectral density function. To study thequantum dissipative dynamics, Caldeira and Leggett ex-plored and popularized the path integral influence func-tional technique originally suggested by Feynman andVernon [35, 36]. For the Caldeira–Leggett model the in-fluence functional assumes a Gaussian functional formand Feynman first recognized that its real part can beconstructed with a Gaussian colored noise [36]. With thesame reasoning, Cao and co-workers introduced that theaverage over the degrees of freedom of the bath can beperformed by directly sampling paths of the discretizedharmonic modes and then propagating the system un-der the influence of random Gaussian field [37]. As such,the quantum dissipative dynamics can be convenientlysimulated with the stochastic differential equation. Fol-lowing the interpretation of Feynman, Stockburger andGrabert resolved the influence functional with Gaussiancolored noises and derived a stochastic Liouville equa-tion for the reduced density matrix [38–40]. Strunz, Yuand coworkers presented the non-Markovian quantumstate diffusion method to describe the dynamics of theopen system in terms of the stochastic Schrödinger’sequation [41–46]. Alternatively, Breuer suggested thequantum jump approach and transferred the full non-Markovian reduced dynamics into Markovian randomjump processes [47]. A stochastic description of a linearopen quantum system was also developed by Calzettaand coworkers [48].

Our group put forward a stochastic Liouville equationfor the dissipative dynamics of the system by decouplingthe system-bath interaction in the equation of motiondirectly [49]. In this way the system-bath correlation isrepresented with the common random fields exerting onboth the system and the bath. It is general and appli-cable for arbitrary types of bath and system-bath cou-pling. For the linear dissipation, it can be recast into thesame form suggested by Stockburger and Grabert [50].

This review will focus on the stochastic description ofdissipative systems. Special emphasis is placed on nu-merical simulations of dissipated two-level systems. Sec-tion 2 presents a detailed derivation of the stochastic Li-ouville equation via a stochastic decoupling of the quan-tum dissipative interaction. Both non-Hermitian andHermitian schemes are derived and their numerical per-formance is discussed. Section 3 gives numerical exam-ples for dissipated two-level systems with the stochas-

110309-2Yun-An Yan and Jiushu Shao, Front. Phys. 11(4), 110309 (2016)

Review article

tic schemes. In Section 4 we demonstrate that the ex-act master equation for analytically solvable models canbe conveniently worked out, starting from the stochas-tic equation. For general systems with a decomposablebath response function we show that the hierarchicalapproach can be achieved and is efficient at interme-diate to high temperatures in Section 5. For dynamicsat zero temperature or low temperatures, we suggestthe hybrid stochastic-hierarchical equation in Section 6.The last section gives a brief summary and discusses theremaining challenges.

2 Decoupling system-bath interaction

Because of the couplings between the system and thebath, one has to take into account of the influence ofthe heat bath in order to describe the dynamics of thesystem exactly. For the description of the dissipativedynamics, we start with the system-plus-model [34],

H = Hs + Hb + f(s)g(b), (1)

where Hs and Hb specify the relevant system and thebath, respectively. The system operator s describes thecoupling to the bath with an arbitrary function g(b) ofthe bath operator b. In quantum mechanics the generalinteraction is a sum of force-field terms

∑j fj(s)gj(b).

To simplify the formalism, we first investigate the abovefactorized form and discuss the extension to multipleterms in Section 2.2. Eq. (1) is for general bath andsystem-bath coupling. It becomes the Caldeira–Leggettmodel if the bath consists of an infinite independentquantum harmonic oscillators with linear system-bathcoupling, i.e., g(b) =

∑j cj xj and a proper counterterm

is included in the system Hamiltonian Hs [35].The dynamics of the whole system is determined by

the density matrix ρ(t) which satisfies the Liouville–vonNeumann equation

ihdρ/dt = [H, ρ(t)], (2)

with an initial condition ρ(0) at t = 0. Because we areonly interested in the dynamical behavior of the sys-tem, the reduced density matrix ρs(t) = trb[ρ(t)] con-tains sufficient information for the reduced dynamics.Therefore, it is desired that the influence of the bathshould be incorporated without the explicit considera-tion of the bath. This could be done by stochasticallydecoupling the dissipation interaction.

2.1 Heuristic decoupling via stochastic fields

The propagation of the density matrix, in turn, the in-trinsic dynamical feature of the system, is dictated bythe time evolution operator U(t) ≡ exp(−iHt/h). Thepropagator can be split along the time axis with Suzuki–

Trotter expansion [51]

U(t) =

N∏j=1

U(∆t), ∆t = t/N, (3)

where the propagator for a short time interval ∆t isapproximated as

U(∆t) ≈ e−iHs∆t/he−iHb∆t/he−if(s)g(b)∆t/h. (4)

Here, the first two factors are due to the contributions ofthe system and the bath, respectively, whereas the lastone originates from the system-bath coupling. Becausethe operators f(s) and g(b) commutate, one can invokethe Hubbard–Stratonovich transformation to decouplethe last factor [49]

e−i∆tf(s)g(b)/h =∆t

2π

∫dυ1,jdυ2,je

−(υ21,j+υ2

2,j)∆t/2

× ei∆t(υ1,j+iυ2,j)f(s)/√2h−∆t(υ1,j−iυ2,j)g(b)/

√2h. (5)

As the decoupling procedure is applied to every timestep, the time evolution operator U(t) is thus expressedas

U(t) =(∆t

2π

)N∫ ∞

−∞dυ1,1 · · ·dυ1,Ndυ2,1 · · ·dυ2,N

×N∏j=1

(e−iHs∆t/he−iHb∆t/he(υ

21,j+υ2

2,j)∆t/2

× ei∆t(υ1,j+iυ2,j)f(s)/√2h−∆t(υ1,j−iυ2,j)g(b)/

√2h).(6)

In the limit of N → ∞, the integration becomes a pathintegral

U(t) =

∫D[υ1]D[υ2]W [υ1, υ2]Us[υ1, υ2]Ub[υ1, υ2], (7)

where υ1(t) and υ2(t) now are two time-dependent ex-ternal fields acting on the two subsystems and Us and Ub

are the propagators dictated by the “decoupled” systemand the bath described respectively by the HamiltonianHs(t) = Hs +

√h[υ1(t) + iυ2(t)]f(s)/

√2 and Hb(t) =

Hb +√h[υ2(t) + iυ1(t)]g(b)/

√2. In Eq. (7), the weight

functional W [υ1, υ2] = exp[−∫ t

0dτ(υ2

1,τ +υ22,τ )/2] is the

probability of the path. Due to the Gaussian functionalform of the weight W [υ1, υ2], υ1(t) and υ2(t) are twoGaussian white noises with mean M⟨υj(t)⟩ = 0 andcorrelation M⟨υj(t)υk(t′)⟩ = δjkδ(t− t′). Therefore, theexact evolution operator is the stochastic average of thecombined propagator Us[υ1, υ2]Ub[υ1, υ2] with respect totwo noises υ1(t) and υ2(t),

U(t) ≡ M⟨Us[υ1, υ2]Ub[υ1, υ2]⟩, (8)

where M denotes the stochastic averaging. Apparently,upon introducing common random fields, there is nodirect interaction between the system and the bath. As

Yun-An Yan and Jiushu Shao, Front. Phys. 11(4), 110309 (2016)110309-3

Review article

a consequence, the stochastic system and bath evolve“independently”. Note that the exact propagator is theaverage of the product of the two stochastic propagators.The decoupling scheme essentially converts the quantuminteraction to the stochastic correlation manifested inthe average. In this sense the system and the bath aredecoupled. Now, the evolution of the system and thebath is separate for a factorized initial density matrixρ(0) = ρs(0)ρb(0), that is,

ρs(t) = Us[υ1, υ2]ρs(0)U†s [υ3, υ4], (9)

ρb(t) = Ub[υ1, υ2]ρb(0)U†b [υ3, υ4], (10)

where two more white noise fields υ3(t) and υ4(t) areintroduced to decouple the dissipative interaction forthe backward propagation. Then the exact density ma-trix is obtained when taking the stochastic average of aproduct of the system and the bath density operators,ρ(t) = M⟨ρs(t)ρb(t)⟩.

One could start from Eqs. (9) and (10) to derive theequations of motion for the stochastic densities ρs(t) andρb(t). However, the results would be lengthy. To havea compact form, we define two complex-valued Wienerprocesses

w1,t =

∫ t

0

dt′[υ1(t′) + υ3(t

′) + iυ2(t′)− iυ4(t

′)]/√2,

w2,t =

∫ t

0

dt′[υ2(t′) + υ4(t

′) + iυ1(t′)− iυ3(t

′)]/√2,

which have the properties

M⟨dwj,t⟩ = M⟨dwj,tdwk,t⟩ = 0,

M⟨dwj,tdw∗k,t⟩ = 2δjkdt. (11)

The equations of motion for ρs(t) and ρb(t) now become

ihdρs = [Hs, ρs(t)]dt+√h/2[f(s), ρs(t)]dw1,t

+ i√h/2f(s), ρs(t)dw∗

2,t, (12)

ihdρb = [Hb, ρb(t)]dt+√h/2[g(b), ρb(t)]dw2,t

+ i√h/2g(b), ρb(t)dw∗

1,t. (13)

Here, both the stochastic differential equation and thestochastic integral are in Itô form.

2.2 Decoupling via Itô calculus

Alternatively, one may decouple the dissipative inter-action via Itô calculus. For this purpose, we may in-troduce white noises as common forces exerting on thesystem and the bath, write down a general linear formof the stochastic differential equations with indetermi-nate coefficients for the independent evolution of thedensity matrices ρs/b(t), require the stochastic aver-age M⟨ρs(t)ρb(t)⟩ satisfying the quantum Liouville-vonNeumann equation, and solve the indeterminate coeffi-cients in the stochastic differential equations.

We have to recognize two key properties of Itô cal-culus. One is the differential equation of the product oftwo Itô processes Xt and Yt

d(XtYt) = XtdYt + (dXt)Yt + (dXt)dYt, (14)

which is the differential form for the integration by partsin Itô calculus. The last term in Eq. (14) has to be in-cluded because dwj,t is of the order

√dt. The other prop-

erty is the nonanticipating rule

M⟨Xtdwj,t⟩ = M⟨Xtdwj,tdwk,t⟩ = 0, (15a)

M⟨Xtdwj,tdw∗k,t⟩ = 2δjkM⟨Xt⟩dt, (15b)

which is analogous to Eq. (11). It holds because the Itôprocess Xt is independent of the noise increments dwj,t

and dw∗j,t.

To illustrate the procedure of the decoupling, we willuse the general Hamiltonian for the dissipated system

H = Hs + Hb +

NI∑j

fj(s)gj(b), (16)

where NI is the number of interaction terms. Basingon the heuristic derivation, we propose the followingansatz for the stochastic differential equations of thetwo subsystems

ihdρs = [Hs, ρs]dt+√h

NI∑j

[fj(s), ρs]sj,1dwj,1

+ i√h

NI∑j

fj(s), ρssj,2dwj,2, (17a)

ihdρb = [Hb, ρb]dt+√h

NI∑j

[gj(b), ρb]bj,1dwj,3

+ i√h

NI∑j

gj(b), ρbbj,2dwj,4, (17b)

where sj,a and bj,a are coefficients to be determined andthe Wiener processes wj,a could be real or complex.

We readily find the differential equation for the den-sity matrix ρ(t) = M⟨ρs(t)ρb(t)⟩,

ihdρ = M⟨ih(dρs)ρb + ihρsdρb + ih(dρs)dρb⟩

= M⟨[Hs, ρs]ρbdt+ [Hb, ρb]ρsdt

− i

NI∑j,k

[fj(s), ρs][gk(b), ρb]sj,1bk,1dwj,1dwk,3

+

NI∑j,k

[fj(s), ρs]gk(b), ρbsj,1bk,2dwj,1dwk,4

+

NI∑j,k

fj(s), ρs[gk(b), ρb]sj,2bk,1dwj,2dwk,3

110309-4Yun-An Yan and Jiushu Shao, Front. Phys. 11(4), 110309 (2016)

Review article

+i

NI∑j,k

fj(s), ρsgk(b), ρbsj,2bk,2dwj,2dwk,4⟩.

(18)

When the following requirements are satisfied

sj,1bk,1M⟨dwj,1dwk,3⟩ = 0,

sj,1bk,2M⟨dwj,1dwk,4⟩ = δjkdt/2,

sj,2bk,1M⟨dwj,2dwk,3⟩ = δjkdt/2,

sj,2bk,2M⟨dwj,2dwk,4⟩ = 0, (19)

Eq. (18) will reproduce the Liouville–von Neumannequation. Although there are many possible choices forsj,a and bk,a, we only consider the following simple set-tings,

M⟨dwj,adwk,c⟩ = 0, M⟨dwj,adw∗k,c⟩ = δjkδac,

wj,3 = w∗j,2, wj,4 = w∗

j,1, (20)

where a, c = 1, 2. When there is only one couplingterm in Eq. (16), Eqs. (17a) and (17b) are nothingbut Eqs. (12) and (13), respectively. Note that the so-obtained stochastic differential equation for the system,Eq. (17a) is non-Hermitian. We will discuss how to de-rive a Hermitian stochastic differential equation for thesystem using different settings of sj,a, bk,a and wj,a inSection 2.4.

Once the decoupling scheme is chosen, one must pro-ceed to establish the stochastic Liouville equation of thesystem by taking into account the influence of the bath.We will show that the impact of the bath can be char-acterized by the induced stochastic field.

2.3 Bath-induced stochastic field

Although the system and the bath are decoupled, theyare still correlated because the real physical quantitiesare averages over the common stochastic fields thatthey are subject to. For instance, the reduced den-sity matrix is obtained as ρs(t) = trbM⟨ρs(t)ρb(t)⟩ =M⟨ρs(t)trbρb(t)⟩. Once the trace of the bath is solved,the effect of the bath on the evolution of the system canbe taken into account. This implies that the trace of therandom bath includes the effect of the bath on the re-duced dynamics. One may take the trace of Eq. (13) tofind

trbρb(t) = exp

[∫ t

0

g(t′)dw∗1,t′/

√h

], (21)

where g(t) = trbρb(t)g(b)/trbρb(t).In performing the stochastic average to calculate the

reduced density matrix, it is more convenient to includethe contribution from the bath by absorbing the trace ofρb(t) into the probability measure of the random fields.This can be achieved by applying the Girsanov trans-

formation [52]. To illustrate the transformation clearly,we resort to the path integral form of the stochastic av-erage,

ρs(t) =

∫D[w1]D[w∗

1 ]D[w2]D[w∗2 ]W [w1, w2]

× ρs[dw1,dw∗2 ; t] exp

[ ∫ t

0

dw∗1,τ g(τ)/

√h].(22)

Here the notation ρs[dw1, dw∗2 ; t] is used to refer to the

functional dependence of the density matrix ρs(t) on thestochastic processes. As the exponential factor formallyassumes a Gaussian form, one may employ the Girsanovtransformation, namely,

w1,t = w1,t + 2

∫ t

0

dt′g(t′)/√h, (23)

to absorb the contribution from the bath. This transfor-mation leaves g(t) invariant because ρb(t), hence g(t),depends on only w∗

1 and w2, but not w1 and w∗2 . As a

result, Eq. (22) becomes

ρs(t) =

∫D[w1]D[w∗

1 ]D[w2]D[w∗2 ]W [w1, w2]

× ρs[dw1 + 2gdt/√h, dw∗

2 ; t]

= M⟨ρs[dw1 + 2gdt/√h,dw∗

2 ; t]⟩. (24)

This equation shows that the measure of the stochasticprocesses w1,t and w2,t keep unchanged while the differ-ential equation for the stochastic density matrix of thesystem is transformed to

ihdρs = [Hs + g(t)f(s), ρs]dt+√h/2 [f(s), ρs] dw1,t

+ i√h/2 f(s), ρs dw∗

2,t. (25)

Now the reduced density matrix is yielded by directlytaking the stochastic average of ρs(t) over the noisesw1,t and w2,t, that is, ρs(t) = M⟨ρs(t)⟩. In words, oncethe function g(t) is known, the reduced dynamics can besolved without the need of explicit treatment of the bathat all. The role that g(t) plays becomes obvious: likethe influence functional in the path integral treatment[53], it is the bath-induced field fully representing theinfluence of the environment.

So far the stochastic formalism is general, which issuitable for arbitrary baths. Using this formalism asa working procedure, we will focus on the Caldeira-Leggett model. In this case, because the dynamics ofthe bath is analytically solvable, one readily obtainsthe bath-induced stochastic field g(t). As usual, we as-sume that the bath starts from a thermal equilibriumstate of noninteracting harmonic oscillators ρb(0) =

exp(−βHb)/trb[exp(−βHb)], where β = 1/(kBT ) is theinverse of the absolute temperature T scaled by theBoltzmann constant kB. The trace for the stochastic

Yun-An Yan and Jiushu Shao, Front. Phys. 11(4), 110309 (2016)110309-5

Review article

density matrix of the bath in Eq. (13) can be conve-niently worked out in the interaction picture,

ρb(t) = U(b)0 (t)U

(b)+ (t)ρb(0)U

(b)− (t)U

(b)†0 (t). (26)

Here U(b)0 (t) = exp(−iHbt/h) is the propagator of the

free bath and U(b)± (t) are the forward/backward propa-

gators in the interaction picture satisfying the Itô dif-ferential equations

ihdU(b)+ (t) = g(t)U

(b)+ (t)dχ+

t , (27a)

−ihdU(b)− (t) = U

(b)− (t)g(t)dχ−

t , (27b)

where g(t) = U(b)†0 (t)g(b)U

(b)0 (t) is the interaction pre-

sentation of the operator g(b), and χ±t are the Wiener

processes involved in the forward/backward propaga-tion,

χ+t =

√h/2(w2,t + iw∗

1,t),

χ−t =

√h/2(w2,t − iw∗

1,t).

Due to the unitarity of U (b)0 (t) and the cyclic permuta-

tion invariance of the trace operation, the trace of thebath is expressed as

trb[ρb(t)] = trb[U

(b)− (t)U

(b)+ (t)ρb(0)

]. (28)

Note that we are using factorized initial condition. Be-cause harmonic oscillators of the bath are noninteract-ing, the propagators U

(b)0 (t) and U

(b)± (t), and the trace

of the density matrix ρb(t) are products of that of all theindividual oscillators. The standard way to work out theoperator g(t) is the using of the operator algebra method[54]. Alternatively, it may be calculated by solving theequations of motion for the interaction representationfor the bath coordinate operators xj(t) and momentumoperators pj(t)

d

dtxj(t) = pj(t)/mj ,

d

dtpj(t) = −mjω

2j xj(t). (29)

Combining the initial condition xj(0) = xj and pj(0) =pj , one straightforwardly solves these equations and ob-tains the result

g(t) =∑j

cj [xj cos(ωjt) + pj/(mjωj) sin(ωjt)]. (30)

Eq. (30) is actually a sum of two time-dependent op-erators. Consequently, one may resort to the Baker-Campbell–Hausdorff formula [55–59] to solve Eq. (27),yielding

U(b)+ (t) = exp

(− i

∫ t

0

dχ+τ g(τ)

− 1

2

∫ t

0

dχ+τ

∫ τ

0

dχ+s [g(τ), g(s)]

), (31a)

U(b)− (t) = exp

(+ i

∫ t

0

dχ−τ g(τ)

− 1

2

∫ t

0

dχ−τ

∫ t

τ

dχ−s [g(τ), g(s)]

). (31b)

For linearly driven harmonic oscillators, the formula ter-minates at the second order because the commutator[g(τ), g(s)] = ih

∑j c

2j/(mjωj) sin[ωj(τ − s)] is a scalar

function.The product U

(b)− (t)U

(b)+ (t) can also be expressed

as a single exponential function by using the Baker-Campbell–Hausdorff formula. Again, the first- andsecond-order terms yield the exact result for the linearbath,

U(b)− (t)U

(b)+ (t) = exp

(− i

∫ t

0

dχ+τ g(τ) + i

∫ t

0

dχ−τ g(τ)

+1

2

∫ t

0

dχ−τ

∫ t

0

dχ+s [g(τ), g(s)]

− 1

2

∫ t

0

dχ+τ

∫ τ

0

dχ+s [g(τ), g(s)]

− 1

2

∫ t

0

dχ−τ

∫ t

τ

dχ−s [g(τ), g(s)]

).(32)

Calculating the quantum expectation and doing somerearrangements, we finally find the result for the traceof the bath,

trb[ρb(t)] = e∫ t0dw∗

1,τ

∫ τ0[dw∗

1,sαr(τ−s)+dw2,sαi(τ−s)], (33)

where αr(t) and αi(t) are the real and imaginary partsof the equilibrium correlation function α(t − s) =trb[g(t)g(s)ρb(0)], respectively. For the linear dissipa-tion model the correlation function becomes

α(t)=1

π

∫ ∞

0

dωJ(ω)

[coth

(βhω2

)cos(ωt)−i sin(ωt)

],

(34)

where J(ω) is the spectral density of the bath

J(ω) =π

2

∑j

c2jmjωj

δ(ω − ωj). (35)

A comparison between Eq. (21) and Eq. (33) yields thebath-induced field [49],

g(t) =√h

∫ t

0

[dw∗

1,t′αr(t−t′)+dw2,t′αi(t− t′)]. (36)

One observes that the correlation function of the bathserves as a memory kernel in the bath-induced field.

The stochastic description is flexible because it is al-lowed to make a free combination of the white noise andthe bath-induced stochastic field or noise g(t) expressed

110309-6Yun-An Yan and Jiushu Shao, Front. Phys. 11(4), 110309 (2016)

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by Eq. (25). Doing so, one may obtain different compos-ite Gaussian noises that affects the feasibility and effi-ciency of the numerical implementation. For instance,when the involved noises are rearranged such that

ξ1,t =g(t)√h

+1

2

dw1,t

dt,

ξ2,t =1

2

dw∗2,t

dt,

one recovers the stochastic formulation put forward byStockburger and Grabert [50],

ihρs = [Hs, ρs(t)] +√h [f(s), ρs(t)] ξ1,t

+ i√h f(s), ρs(t) ξ2,t. (37)

Here, the stochastic variables ξ1(2),t are Gaussian col-ored noises with means and correlations

M⟨ξ1,t⟩ = M⟨ξ1,t⟩ = 0,

M⟨ξ1,tξ1,τ ⟩ = αr(t− τ),

M⟨ξ1,tξ2,τ ⟩ = Θ(t− τ)αi(t− τ),

M⟨ξ2,tξ2,τ ⟩ = 0, (38)

where δ is the Dirac δ-function and Θ is the Heavisidestep function

Θ(x) =

0, if x ≤ 0,1, if x > 0.

(39)

2.4 Hermitian stochastic Liouville equation

In general there are two major factors that influencethe numerical performance of the stochastic formula-tion. One factor is the number of involved noises andthe other is the intrinsic property of the stochastic dif-ferential equation, which may cause a stochastic real-ization dramatically different from the exact average.For the former, because the stochastic averaging overa larger number of noises corresponds to higher dimen-sional functional integration, the convergence normallybecomes slower. Note that, although in our case thereare four real noises in Eqs. (25) and (37), one can simplychoose ξ1,t as a real noise to reduce the total number ofnoises to three [39]. For the latter, the stochastic den-sity matrices Eqs. (25) and (37) do not preserve norm-conserving, hermicity and positivity that the exact onemust do. Generally speaking, the more properties thestochastic density matrix possesses like the exact one,the better numerical performance the stochastic methodwill have. It is expected that a Hermitian stochasticscheme could be more efficient in the numerical conver-gence. We then suggest the following stochastic scheme[60]

ihdρs = [Hs, ρs]dt+√h/2[f(s), ρs]dµ2

+ i√h/2f(s), ρsdµ1, (40)

ihdρb = [Hb, ρb]dt+√

h/2[g(b), ρb](dµ1 + idµ4)

+ i√h/2g(b), ρb(dµ2 + idµ3). (41)

This equation corresponds to the stochastic decouplingwith Eq. (17) upon using the following parameters

s1,1 = s1,2 = b1,1 = b1,2 = 1/√2,

w1,1 = µ2, w1,2 = µ1,

w2,1 = µ1 + iµ4, w2,2 = µ2 + iµ3, (42)

for solving Eq. (19).Knowing the trace of ρb(t), one can solve the re-

duced dynamics by taking a two-step stochastic averageρs(t) = Mµ1,µ2⟨ρs(t)Mµ3,µ4trb[ρb(t)]⟩ as the evolu-tion of the random system does not depend on the noisesµ3(t) and µ4(t).

Because Eq. (41) assumes the same form as Eq. (13),the trace of the random density matrix ρb(t) is given byEq. (33) with the replacement dw∗

1 →√2(dµ1 + idµ3)

and dw2 →√2(dµ2 + idµ4). The stochastic average

with respect to the Wiener processes µ3(t) and µ4(t)is a Gaussian type functional integration, which can besolved analytically

Mµ3,µ4⟨tr[ρb(t)]⟩ = exp1

2

2∑a,b=1

∫ t

0

dµa,τ

∫ t

0

dµb,s

×[δabδ(τ − s)− 1

2Gab(τ − s)

], (43)

where G(t − s) is the Green function of the correlationmatrix function Γ(t− τ)

Γ(t) =

(cδ(t) + αr(t) Θ(t)αi(t)Θ(−t)αi(−t) cδ(t)

), (44)

with c = 1/2. The right hand side of Eq. (43) assumes aGaussian functional form and serves as a multiplicativefactor in the stochastic average of the random densitymatrix ρs(t). Consequently, it can be combined with theWiener measure of the noises µ1(t) and µ2(t) in takingthe stochastic average

ρs(t) =

∫D[µ1]D[µ2]W [µ1, µ2]

× ρs(t)Mµ3,µ4⟨tr[ρb(t)]⟩

=

∫D[µ1]D[µ2]e

− 14

2∑a,b=1

∫ t0dµa,τ

∫ t0dµb,sGab(τ−s)

ρs(t).

The exponential part is still of the Gaussian func-tional form and can be treated as the measure of newGaussian noises. With the change of variables accord-ing to µ1,t =

√2∫ t

0dτξ1,τ and µ2,t =

√2∫ t

0dτξ2,τ ,

the stochastic differential equation for the density ma-trix assumes the same form as Eq. (37) but with noiseaverages M[ξa,t] = 0 and new correlation functionsM[ξa,tξb,τ ] = Γab(t− τ).

Yun-An Yan and Jiushu Shao, Front. Phys. 11(4), 110309 (2016)110309-7

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It is expected that the Hermitian formulation willshow a better numerical efficiency because of the her-micity and the smaller number of noises involved. More-over, it was proven that the constant c in Eq. (44) canbe arbitrarily chosen as long as the correlation functionis semi-positive definite [60]. Simulations with the mini-mum allowed c value will further improve the numericalperformance.

3 Stochastic simulations of spin-boson model

Given the initial state of the system and the bath-induced stochastic field g(t), one can solve Eq. (25) orEq. (37) to obtain a stochastic trajectory. In general, thesystem of interest is a small one and the calculation ofa single stochastic trajectory can be easily carried out.Note, however, that in the stochastic scheme the systemis subject to the stochastic field characterizing the en-vironment effect and one has to calculate many randomtrajectories to reach the convergent statistical expecta-tion of the reduced density matrix. Roughly speaking,it is the statistics that makes the problem difficult.

We use the dissipated two-level system to demon-strate the implementation and efficiency of methodsbased on the stochastic formalism. Why are we inter-ested in the spin-boson model? Well, as the two-levelsystem is the simplest model characterizing quantumcoherence, the unique feature in the microscopic world,the spin-boson model would be the most basic defin-ing the fundamental paradigms of quantum dissipation.Moreover, the dynamics of the spin-boson model hasbeen a challenge for many years. It is appropriate to citeWeiss here, “Despite its apparent simplicity, the spin-boson model cannot be solved exactly by any knownmethod (apart from some limited regimes of the param-eter space). Not only is the spin-boson model nontrivialmathematically, it is also nontrivial physically.” [34]

We now consider the numerical simulation of thestochastic density matrix of the spin-boson model. Tothis end, we discretize the time using a uniform grid0, t1, · · · , tk, · · · , (tk = k∆t) with a fixed step-size ∆t.Then, utilizing the conventional splitting operator tech-nique, we obtain the iteration relation for propagation

ρ(tk+1) = e−iHs∆t/h−if(s)[gk∆t/h+√h/2(∆w1,k+i∆w∗

2,k)]

×ρ(tk)eiHs∆t/h+if(s)[gk∆t/h+

√h/2(∆w1,k−i∆w∗

2,k)]. (45)

The quantity ∆wa,k is the change of the Wiener pathin the time interval [tk, tk+1]. In the fixed step-sizestochastic integration, one can simply put ∆wa,k =√∆t(Ba,1,k + iBa,2,k) with Ba,j,k being independent

standard normal random numbers. The vector gk is thediscretized version for the bath-induced random field.Note that a direct summation

gk=√h

k−1∑j=0

[αr((k−j)∆t)∆w2,k+αi((k−j)∆t)∆w∗

1,k

]takes O(N2) multiplication and addition operations forN -step propagation which becomes expensive for thelong time dynamics. Fortunately, it is a convolution in-tegral and can be generated with an O(N logN) algo-rithm by using the fast Fourier transform [61].

One may generate the Gaussian colored noises withcirculant embedding of the correlation Eq. (38) [62, 63].We will exemplify the generation of a stochastic fieldζt with mean M[ζt] = 0 and correlation M[ζtζs] =αr(t−s). Upon discretization, one may build a 2N×2Ncirculant matrix embedding the correlation matrix

Cjk =

αr(|j − k|∆t), if |j − k| < Nαr(|2(N−1)−|j−k||∆t), if |j − k| ≥ N

.

The two principal submatrices with the first and thelast N columns and rows are the original covariance ma-trix. The circulant matrix is diagonalized by the discreteFourier transform F †CF = Λ, where Λ is the diago-nal matrix taking the eigenvalues of the circulant ma-trix and F is the 2N × 2N discrete Fourier transformmatrix. If all the eigenvalues are non-negative, one canform a 2N -dimensional complex vector with standardGaussian random numbers, multiply it with Λ1/2, andcarry out the discrete Fourier transform to obtain tworeal random vectors x and y, i.e., x + iy = FΛ1/2z.Now x and y are uncorrelated but have identical cor-relation matrix C. Thus their first half and second halfare four colored noises with zero mean and autocorre-lation function αr(t). This procedure can be extendedto generate multiple correlated Gaussian random fieldsand the interested readers can refer to Refs. [64] and[65] for further details.

To show how the stochastic formulation works, wetake the quantum nondemolition measurement of thetwo-level system as the first numerical example. Specif-ically, Hs =

12σz, f(s) = σz, and α(t) = exp(it− 2|t|)/2

are adopted with an initial condition ρs(0) = 0.5I +

0.5σx + 0.6σy. In this case the system Hamiltonian Hs

commutates with the coupling operator f(s) and there isno energy exchanging between the system and the bath[49, 66–68]. Accordingly, the diagonal elements in theeigen-energy representation do not change with time butthe off-diagonal matrix elements are damped to zero,which is pure dephasing caused by the bath. The modelis exactly solvable and the master equation reads

ih ˙ρ = [Hs, ρ]− iCr(t)[f(s), [f(s), ρ]] + Ci(t)[f(s)2, ρ],

where Cr/i =∫ t

0dταr/i(t − τ). The results of the

stochastic simulations with noises generated from circu-lant embedding of Eq. (38) are depicted in Fig. 1. Oneclearly sees that the random average with 256 stochas-tic trajectories already produces the reduced density

110309-8Yun-An Yan and Jiushu Shao, Front. Phys. 11(4), 110309 (2016)

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Fig. 1 The stochastic simulation for the coherence in aquantum nondemolition measurement. (a) The real and (b)the imaginary part of ρ01 with 256 stochastic trajectories arecompared with the exact results.

matrix up to time t = 2 quite well for both the realand imaginary parts of the off-diagonal matrix elementρ01(t).

The second example is the spontaneous decay of atwo-state atom at zero temperature. As its exact quan-tum master equation is known [31, 43, 47], the exact dy-namics can be calculated numerically. It thus provides agood model for numerical tests of the stochastic scheme.The Hamiltonian for the dissipative system reads

Hs =h∆

2σz, Hb =

∑j

hωjb†kbk,

Hsb = σ−g1 + σ+g2, g1 = h∑j

cjb†j = g†2.

Here there are two coupling terms in the Hamiltonian,we will introduce two pairs of complex-valued Wienerprocesses and use parameters given in Eq. (20) for thestochastic Liouville equation Eq. (17). Following a simi-lar procedure for Eqs. (26)–(22), we can solve the equa-tion of motion for the bath to obtain the bath-inducedrandom fields

g1(t) =

√h

2

∫ t

0

(dw∗21,τ − idw∗

22,τ )α(t− τ), (46)

g2(t) =

√h

2

∫ t

0

(dw∗11(4),τ + idw∗

21,τ )α∗(t− τ), (47)

with the response function of the bath α(t) =∑j |cj |2eiωjt. To simplify the notations we define

µ1 = (w11 − iw12)/√2, µ2 = (w21 + iw22)/

√2,

µ3 = (w11 + iw12)/√2, µ4 = (w21 − iw22)/

√2.

Then the bath-induced fields are transformed to

g1(t) =

√h

2

∫ t

0

dµ∗2,τα(t− τ),

g2(t) =

√h

2

∫ t

0

dµ∗1,τα

∗(t− τ).

As a consequence, the stochastic differential equation(17a) becomes

ihdρs = [Hs+σ−g1(t)+σ+g2(t), ρs]dt−√

h

2ρsσ

−dµ1

+

√h

2σ+ρsdµ2 +

√h

2σ−ρsdµ3 −

√h

2ρsσ

+dµ4. (48)

Note that the bath-induced fields g1(t) and g2(t) areindependent of the noises µ3(t) and µ4(t). We first takethe average over µ3(t) and µ4(t) to obtain

ihdρs = [Hs + σ−g1(t) + σ+g2(t), ρs]dt

−√

h

2ρsσ

−dµ1 +

√h

2σ+ρsdµ2. (49)

Although ρs(t) in Eq. (49) is not the same as that inEq. (48), their stochastic averages are identical. Eq. (49)assumes better numerical performance because it in-volves a smaller number of stochastic fields.

We generate the bath-induced random fields by theconvolution method [61] and carry out the stochasticsimulations with Eq. (49) using α(t) = γ

2 e−γ|t|+it(γ =

0.1). The excited state population follows a non-exponential decay. As shown in Fig. 2, the results av-eraged with 224 trajectories reproduce the damped os-cillation very well and the maximum statistical error isless the 0.006 for all times.

In the third example we investigate the dynamicsof the dissipated two-level system using the Hermitianscheme. The Hamiltonian and the coupling operator forthe two-level system are

Hs = h∆ σx/2, f(s) = σz/2, (50)

where ∆ is the tunneling matrix element. The spectraldensity is assumed to take an Ohmic form with the De-bye regulation

J(ω) = ηωω2c/(ω

2c + ω2), (51)

where η is the dissipation strength and ωc is the cut-offfrequency. We will use a relatively large cut-off ωc = 4∆with a moderate dissipation strength η = 0.4 for the

Yun-An Yan and Jiushu Shao, Front. Phys. 11(4), 110309 (2016)110309-9

Review article

Fig. 2 Spontaneous decay of the excited state populationof a two-state atom. Solid line: The exact results. Bullets:The stochastic simulations of Eq. (49) with 224 realizations.

stochastic simulations. The corresponding correlationfunction can be expressed in terms of exponentials [69]

α(t) =ηω2

c

2[cot (βhωc/2)− i] e−ωct

+2ηω2

c

hβ

∞∑k=1

νke−νkt

ν2k − ω2c

, (52)

where νk = 2πk/(hβ) is the kth Matsubara frequencyof the bath.

The Hermitian scheme is integrated with the order 2weak Runge–Kutta approximation [70]. The two corre-lated stochastic fields are generated with the circulantembedding approach. The number of time steps of thestochastic integration is 8192 with a step size of 0.001/∆and the number of trajectories is 40 million for all sim-ulations at different temperatures. With these settings,it takes about 20 CPU hours with 16 Intel(R) Xeon(R)E5620 CPUs having a clock speed of 2.40 GHz.

Stochastic simulations with Eqs. (25) and (44) at tem-peratures of 0.001, 0.5, 1.0, 2.0, 3.0, 4.0 and 5.0 h∆/kBare performed. Numerical simulations based on the hier-archical approach outlined in Section 5 are carried outat temperatures 1.0–5.0 h∆/kB to serve as numericalbenchmarks because this model is not exactly solvable.

The expectation σz(t) = ⟨σz ρs(t)⟩, the populationdifference between the left and right states is a keyquantity for understanding the dynamics of the dissi-pated two-level system. The temperature dependence ofthe evolution of the population difference is depicted inFig. 3. The dominant feature is the decrease of coher-ence at higher temperatures. Further calculations showthat the population difference decrease monotonicallywhen temperature is higher than 6 h∆/kB. Another in-teresting observation is that the line for the positionsof the minimum is not a straight line perpendicular tox-axis. This fact is the reflection of the mechanism forthe temperature-dependent dephasing [71].

Figure 3(b) displays the deviations from the numeri-cally exact results simulated with hierarchical approach.

Fig. 3 (a) The temperature dependence of the time evo-lution for the population difference between the left and theright states. The line connecting the minimum of each curveis also shown. (b) The deviations of the stochastic simu-lations from the numerically exact hierarchical results. Theerrors are defined as (⟨σz⟩SDE−⟨σz⟩HE)/(1+|⟨σz⟩HE |) with⟨σz⟩SDE being the results from the stochastic simulation and⟨σz⟩HE that from the hierarchical approach. The tempera-ture corresponding to the curves increases from the bottomto the top at the intersections with the connecting curve in(a) and the indicated arrow in (b).

One finds that the maximum errors are less than 0.004up to the time t = 16/∆. If the results from the stochas-tic simulation are plotted on top of the hierarchicalones, the differences cannot be recognized with nakedeye. Thus the non-Markovian non-perturbative stochas-tic master equation in Eq. (38) produces reliable results.The other main feature is that bigger errors occur atlower temperatures, as already noticed in the Monte-Carlo simulations.

Define the trace distance D between two quantumstates ρ1,2 [72]

D(ρ1, ρ2) =1

2tr√(ρ†1 − ρ†2)(ρ1 − ρ2), (53)

which is a natural metric on the state space basedon the Schatten p-norm with p = 1 [73]. It satisfies0 ≤ D ≤ 1 and is invariant under unitary transforma-tion, non-increasing for completely positive and trace-preserving quantum maps. Therefore, the trace distanceis often interpreted as a measure for the distinguishabil-ity of quantum states. The quantity D is always decreas-

110309-10Yun-An Yan and Jiushu Shao, Front. Phys. 11(4), 110309 (2016)

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Fig. 4 The trace distance between the left and right statesat different temperatures.

ing with time for Markovian propagation but sometimesincreasing for non-Markovian processes. Breuer, Laineand Piilo suggested a measure for the non-Markovianitybased on the increase of D [74]. Interested readers mayrefer to Ref. [75] for a recent review on the characteri-zation and quantification of the non-Markovianity.

Note that the parity symmetry is preserved in the ex-act reduced dynamics of the model. The symmetry im-plies that the condition ρ2(t) = σxρ1(t)σx holds for anytime if it is satisfied at the initial time for two quantumstates ρ1(t) and ρ2(t). The parity symmetry allows us toexplore the non-Markovianity of the model by calculat-ing the trace distance between a quantum state and itssymmetry image. Fig. 4 depicts the changes of the tracedistance between the left state and its symmetry im-age (the right state) at different temperatures. It clearlydepicts the trend that the trace distance approacheszero at long times. For all the simulated temperaturesthere exist time ranges with increasing distance, whichis a clear sign of non-Markovianity. However, the in-crease becomes weaker at higher temperatures, whichagrees with the intuition that the non-Markovianity be-comes weaker as the temperature goes higher. We havealso carried out the propagation of quantum trajecto-ries starting from the right state to check the reliabilityof the symmetry based method. The differences betweenthe two sets of results are always less than 5% regardlessof time and temperature.

4 From stochastic to deterministic: Masterequation

The direct application of the stochastic differential equa-tion is only suitable for weak to intermediate dissipation.By contrast, deterministic methods are far more efficientto study moderate to strong dissipation. Unfortunately,exact deterministic method is not available for generalsystems and thus accurate approximations are requiredfor the study of quantum dissipative dynamics. In devel-oping approximations, analytically solvable models will

be appreciated because they not only provide insightfulunderstanding of quantum dissipation but also serve asbenchmarks.

The stochastic scheme outlined in previous sectionsis useful to derive deterministic equations for certaindissipative systems. To this end, one should first take thestochastic average of Eq. (25) to obtain the deterministicequation of motion for the reduced density matrix,

ihdρs/dt = [Hs, ρs(t)] + [f(s),M⟨g(t)ρs(t)⟩]. (54)

In the above derivation the nonanticipating propertyEq. (15) is used. Unfortunately, Eq. (54) is not in closedform because of the correlation between the stochasticdensity matrix ρs(t) and the random field g(t). To workout a master equation one has to express the statisticalaverage M⟨g(t)ρs(t)⟩ explicitly in terms of the reduceddensity matrix and other known operators of the sys-tem. It is a challenging task to derive a deterministicmethod from a stochastic one for arbitrary noise [12,76–78].

For the Caldeira–Leggett model, the stochastic aver-age M⟨g(t)ρs(t)⟩ in Eq. (54) can be formally expressedas follows by recognizing the expression of the bath-induced field

M⟨g(t)ρs(t)⟩ = M⟨OR(t)⟩+M⟨OI(t)⟩, (55)

OR(I)(t) =

∫ t

0

dt′αr(i)(t− t′)ρs,1(2)(t, t′), (56)

where ρs,1(t, t′) =

√h [υ1(t

′)− iυ4(t′)] ρs(t) and

ρs,2(t, t′) =

√h [υ2(t

′) + iυ3(t′)] ρs(t). Here the real

Gaussian white noises υj are connected to theWiener processes w1/2 through the relations w1,t =∫ t

0ds[υ1(s) + iυ4(s)] and w2,t =

∫ t

0ds[υ2(s) + iυ3(s)].

To obtain the expressions of M⟨ρs,1(2)(t, t′)⟩ we resortto the Furutsu-Novikov theorem [79], that is,

M⟨υ(t′)F [υ]⟩ = M⟨δF [υ]

δυ(t′)

⟩(57)

for a white noise υ and an arbitrary functional F [υ].Thanks to this theorem we are allowed to express therandom average M⟨ρs,1(2)(t, t′)⟩ in terms of functionalderivatives

M⟨ρs,1(t, t′)⟩ =√hM

⟨δρs(t)

δυ1(t′)− i

δρs(t)

δυ4(t′)

⟩≡ Os,1(t, t

′), (58)

M⟨ρs,2(t, t′)⟩ =√hM

⟨δρs(t)

δυ2(t′)+ i

δρs(t)

δυ3(t′)

⟩≡ Os,2(t, t

′). (59)

Also by applying the Furutsu–Novikov theorem and rec-ognizing the formal solution of ρs(t), we find the resultsfor the functional derivatives with another type of noisecombination

Yun-An Yan and Jiushu Shao, Front. Phys. 11(4), 110309 (2016)110309-11

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M δρs(t)

δυ1(t′)+i

δρs(t)

δυ4(t′)

=

2√h

∫ t

t′dsαr(s−t′)Os,1(t, s),

M δρs(t)

δυ2(t′)− i

δρs(t)

δυ3(t′)

=

2√h

∫ t

t′dsαi(s−t′)Os,2(t, s).

(60)

To proceed, we invoke the formal solution of Eq. (25)

ρs(t) = U+(t, 0)ρs(0)U−(0, t), (61)

where U±(t, 0) are the forward/backward propagator as-sociated with the time-dependent stochastic Hamilto-nian

H+(t) = Hs + [g(t) +√hη+(t)/2]f(s), (62)

and

H−(t) = Hs + [g(t) +√hη−(t)/2]f(s), (63)

respectively. Here η±(t) = υ1,t + iυ4,t ± iυ2,t ± υ3,tare the noises ascociated with the forward/backwardpropagation. By virtue of the stochastic propaga-tor, one defines the Heisenberg operators x±(t, t

′) =

U±(t, t′)f(s)U±(t

′, t). Then the operators Os,1(2)(t, t′)

corresponding to the functional derivatives are re-expressed as

Os,1(t, t′) = −iM

⟨x+(t, t

′)ρs(t)− ρs(t)x−(t, t′)⟩,

Os,2(t, t′) = M

⟨x+(t, t

′)ρs(t) + ρs(t)x−(t, t′)⟩. (64)

Eq. (64) converts the task from solving Mg(t)ρs(t) totackling Os,1(2)(t, t

′) in terms of ρs(t) and other knownoperators.

For arbitrary systems, deriving the master equationrelies on whether the explicit expressions of dissipativeoperators Os,1(2)(t, t

′) can be worked out or not. Unfor-tunately, it cannot be analytically solved for general sys-tems. However, the master equation can be constructedwhen either the Hamiltonian of the system or the re-sponse function of the bath assumes special properties.Here we will present two examples, one with a specialresponse function and the other with the harmonic os-cillator Hamiltonian.

4.1 High-temperature approximations

First, we consider dissipative dynamics at the high-temperature limit. This is the classical regime, in whichthe bath response is more local in time. Thus one mayresort to the Markovian approximation. We again con-sider the Caldeira–Leggett model, see Eq. (36). Notethat in the bath-induced field the imaginary part of thememory function is independent of temperature, whichis of a genuine quantum effect. At high temperatures(small β) we use the Taylor expansion of coth(hβω) inthe real part of the memory function to obtain the dom-inant contribution

αr(t) =2

π

∫ ∞

0

dωJ(ω)

hβωcos(ωt).

In the strict Ohmic dissipation J(ω) = ηω, the memorytime of the bath response becomes zero, that is

αr(t) =2η

hβδ(t)

and

αi(t) = ηδ′(t).

It is a bit subtle to use the δ-functions because they areinvolved in the integral from 0 to t. In other words, onlyhalf of the distribution contributes to the integration.Without a rigorous proof we simply put down as∫ t

0

dt′δ(t′)f(t′) =1

2f(0),∫ t

0

dt′δ′(t′)f(t′) = −1

2f ′(0).

These expressions can be substituted into Eq. (55) toobtain

OR(t) =η√hβ

ρs,1(t, t), (65)

OI(t) =η

2

√h

∂

∂t′ρs,2(t, t

′)

∣∣∣∣t′=t

. (66)

With the help of Eq. (25) we readily acquire

M⟨OR(t)⟩ = − iη

hβ[f(s), ρs(t)], (67)

M⟨OI(t)⟩ =iη

2h[Hs, f(s)] , ρs(t) . (68)

Inserting into Eq. (55), we obtain the semiclassical mas-ter equation

ih∂ρs∂t

= [Hs, ρs]−iη

hβ[f(s), [f(s), ρs]]

+iη

2h[f(s), [Hs, f(s)] , ρs] . (69)

For conventional systems Hs = p2/(2M) + VR(x) andf(s) = x there is [Hs, f(s)] = −ihp/M . Note thatthe potential operator here includes the renormalizationcontribution 1/2Mw2x2 with

ω2 =∑j

c2jmjω2

j

=2

π

∫ ∞

0

dωJ(ω)

ω. (70)

Substituting into Eq. (69) leads to the well-knownCaldeira–Leggett master equation

ih∂ρs∂t

= [Hs, ρs]−iη

hβ[x, [x, ρs]]

+η

2M[x, p, ρs] . (71)

This equation has been frequently discussed from dif-ferent perspectives in the literature, for instance, in

110309-12Yun-An Yan and Jiushu Shao, Front. Phys. 11(4), 110309 (2016)

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Ref. [80]. In this approximation the system followsthe exact quantum dynamics while the environmentis rather “classical” and Markovian as it is assumedto be at high temperatures. At low temperatures, thenon-Markovian effect cannot be neglected and rigorousmethods are required [40].

4.2 Master equation for dissipative harmonicoscillators

We consider a dissipative quantum harmonic oscillatoras the second example. The dissipative harmonic oscil-lator is one among the few exactly solvable model fortheoretical analysis and has been extensively studied inthe literature [45, 48, 81–101]. Here we will follow Ref.[101] to present a brief derivation in a straightforwardand elementary way through the stochastic decouplingapproach.

Suppose the renormalized frequency of the quantumharmonic oscillator is ω when the contribution due toEq. (70) is accounted. For such a linear system, thestochastic propagators U±(t, 0) can be solved by usingEqs. (27)–(31) and the stochastic Heisenberg operatorsx±(t, t

′) in Eq. (64) can be calculated correspondingly

x±(t, t′) = cosω(t− t′)x− sinω(t− t′)

Mωp

− 1

Mω

∫ t

t′dt1 sinω(t1 − t′)

[g(t1) +

√h

2η±(t1)

].

Inserting into Eq. (64) and carrying out the involvedstochastic averaging, we obtain the following integralequations for the operators Os,1 and Os,2

Os,1(t, t′) = −i cosω(t− t′)

[x, ρs(t)

]+

i

Mωsinω(t− t′)

[p, ρs(t)

]+

2

Mω

∫ t

t′dt1

∫ t

t1

dt2 sinω(t1 − t′)

× αi(t1 − t2)Os,1(t, t2), (72)

Os,2(t, t′) = cosω(t− t′)

x, ρs(t)

− sinω(t− t′)

Mω

p, ρs(t)

− 2

Mω

∫ t

t′dt1

∫ t

0

dt2 sinω(t1 − t′)

× αr(t1 − t2)Os,1(t, t2)

− 2

Mω

∫ t

t′dt1

∫ t1

0

dt2 sinω(t1 − t′)

× αi(t1 − t2)Os,2(t, t2). (73)

Here, Eq. (60) is used to obtain these equations.The linear integral equation Eq. (72) for Os,1 does not

depend on Os,2 and can be solved at first. Taking intoaccount the inhomogeneous term, a reasonable guess forthe solution is

Os,1(t, t′) = x11(t, t

′) [p, ρs(t)] + x12(t, t′) [x, ρs(t)] ,

(74)

which can be verified by iteration. With the same rea-soning, the expression of Os,2(t, t

′) possesses the follow-ing form

Os,2(t, t′) = x21(t, t

′) x, ρs(t)+ x22(t, t

′)p, ρs(t)+ x23(t, t

′) [p, ρs(t)] + x24(t, t′) [x, ρs(t)] . (75)

The coefficient functions xjk(t, t′) are determined upon

substituting Eqs. (74) and (75) into the integral equa-tions. The procedure results in the following integralequations

x11(t, t′) =

i

Mωsinω(t− t′)

+2

Mω

∫ t

t′dt1

∫ t

t1

dt2 sinω(t1 − t′)

× αi(t1 − t2)x11(t, t2),

x12(t, t′) = −i cosω(t− t′)

+2

Mω

∫ t

t′dt1

∫ t

t1

dt2 sinω(t1 − t′)

× αi(t1 − t2)x12(t, t2),

x21(t, t′) = cosω(t− t′)

− 2

Mω

∫ t

t′dt1

∫ t1

0

dt2 sinω(t1 − t′)

× αi(t1 − t2)x21(t, t2),

x22(t, t′) = − sinω(t− t′)

Mω

− 2

Mω

∫ t

t′dt1

∫ t1

0

dt2 sinω(t1 − t′)

× αi(t1 − t2)x22(t, t2),

x23(t, t′) = − 2

Mω

∫ t

t′dt1 sinω(t1 − t′)

×[∫ t

0

dt2αr(t1 − t2)x11(t, t2)

+

∫ t1

0

dt2αi(t1 − t2)x23(t, t2)

],

x24(t, t′) = − 2

Mω

∫ t

t′dt1 sinω(t1 − t′)

×[∫ t

0

dt2αr(t1 − t2)x12(t, t2)

+

∫ t1

0

dt2αi(t1 − t2)x24(t, t2)

].

Yun-An Yan and Jiushu Shao, Front. Phys. 11(4), 110309 (2016)110309-13

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Finally one obtains the desired master equation in termsof the above expressions,

ihdρs(t)

dt= [Hs, ρs(t)] +A1(t) [x, x, ρs(t)]

+A2(t) [x, p, ρs(t)] +A3(t) [x, [p, ρs(t)]]

+A4(t) [x, [x, ρs(t)]] , (76)

where the coefficient functions are

A1(t) =

∫ t

0

dt′αi(t− t′)x21(t, t′),

A2(t) =

∫ t

0

dt′αi(t− t′)x22(t, t′),

A3(t) =

∫ t

0

dt′ [αr(t−t′)x11(t, t′)+αi(t−t′)x23(t, t

′)] ,

A4(t) =

∫ t

0

dt′ [αr(t−t′)x12(t, t′)+αi(t−t′)x24(t, t

′)] .

The master equation Eq. (76) agrees with that derivedby Hu, Paz, and Zhang by virtue of path integral tech-nique [88, 89]. As clarified in Ref. [89], A1(t) is the co-efficient for the frequency shift because [x, x, ρs(t)] =[x2, ρs(t)], A2(t) is a quantum dissipation term, A3(t)reflects the anomalous quantum diffusion, and A4(t) isthe coefficient for the normal quantum diffusion.

Furthermore, the exact master equation for a dis-sipative harmonic oscillator driven by external time-dependent fields was also conveniently solved with aboveprocedure [101]. In this case, the driving fields medi-ate an interplay between the system and the bath sothat the system is dressed by both the driving and thestochastic fields.

5 Numerical algorithms based on stochasticformulation

5.1 Decomposition of bath response function

When a simple master equation exists, the calculation ofthe reduced density matrix is straightforward. By con-trast, the stochastic simulation is deemed to be less ef-ficient than solving the master equation due to the fluc-tuations of the stochastic density matrix. Therefore, adeterministic approach is always preferred. However, fora general system a closed set of equations like Eq. (75)cannot be reached because the stochastic average relatedto the bath-induced noises cannot be solved exactly dueto the correlation between the stochastic density matrixρs(t) and random field g(t). Any attempt to derive de-terministic differential equations based on the stochasticLiouville equation has to find a strategy to represent theaverage involving the bath-induced noises in Eq. (54).

In this section we show that deterministic differentialequations can still be achieved if the response function

of the bath can be expanded as

α(t) =∑j

bjtnje−γjt. (77)

Here γj may be complex with positive real part and nj

are non-negative integers. Eq. (77) can be viewed as anexpansion of α(t) in terms of the basis functions ϕj(t) =tnje−γjt, which are related to the orthonormal Laguerrepolynomials when the constants γj are universal. Theresponse function in Eq. (77) is decomposable such thatit can be recast as

α(t− s) =∑j,k

cj,kϕj(t)ϕk(s). (78)

Furthermore, an arbitrary order derivative of the ex-pansion can also be expressed in terms of these basisfunctions

dn

dtnα(t) =

∑j

d(n)j ϕj(t), (79)

where d(n)j are constants of time. The properties of

Eqs. (78) and (79) greatly simplify the derivation of thedeterministic equation for the reduced density matrix.

It could be pointed out that Eq. (77) is a generalform of the response function for a wide range of modelspectral density functions. In many models for the bath,the spectral density function assumes the form of theodd rational function,

J(ω) = ωPK(ω2)

QM (ω2), (80)

where PK(QM ) is a polynomial of order K(M). Thespectral density functions for Ohmic dissipation withthe Debye [Eq. (51)] and the algebraic [Eq. (89)] [102]regulation as well as the multiple-mode Brownian os-cillator model [85] assume this form. Eq. (34) then canbe integrated with contour integration and the contri-bution of the poles of the spectral density function andthe hyperbolic cotangent function results in a form ofEq. (77). Note that when the denominator polynomialQM (x) has only simple roots, Eq. (77) will reduce to asummation of exponentials. The integers nj in Eq. (77)will become non-zero when the polynomial QM (x) hasmultiple roots. Furthermore, the spectral density func-tion may well be approximated with rational functionseven if it does not assume the form of odd rational func-tions. For example, the Ohmic spectral density can bedecomposed into shifted Drude–Lorentzian terms [103,104]. In this case, the corresponding response functionis indeed a sum of exponentials.

5.2 Hierarchical approach

In 1978 Shapiro and Loginov proposed the formulae ofdifferentiation to calculate the first order momentum fora stochastic process driven by an Ornstein-Uhlenbeck

110309-14Yun-An Yan and Jiushu Shao, Front. Phys. 11(4), 110309 (2016)

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noise with a correlation function of a single exponential[105]. Tanimura and Kubo also obtained the Shapiro-Loginov equations when employing the semiclassical ap-proximation for the dissipative dynamics at high tem-peratures, where the response function of the bath isa single exponential [106]. The low temperature correc-tions were included soon by Tanimura [107]. Later, ourgroup gave a rigorous derivation of the hierarchical ap-proach through the stochastic formulation.

To derive the hierarchical approach, one again needsto deal with Eq. (54). For a general system the aver-age related to the bath-induced random field M⟨g(t)ρs⟩cannot be solved analytically and one has to resort tothe differential equation for its time evolution,

ihd

dtM⟨g(t)ρs⟩

= ihM⟨g(t)dρs⟩/dt+ ihM⟨[dg(t)]ρs⟩/dt+ ihM⟨[dg(t)]dρs⟩/dt

= [Hs,M⟨g(t)ρs⟩] + iM⟨g1(t)ρs⟩+ [f(s),M⟨g2(t)ρs⟩] + αr(0)[f(s), ρs]

+ iαi(0)f(s), ρs. (81)

Here g1(t) is a random field arising from the differenti-ation of g(t)

g1(t) =

∫ t

0

[(dµ1,τ − idµ4,τ )αr(t− τ)

+ (dµ2,τ + idµ3,τ )αi(t− τ)]. (82)

The second identity in Eq. (81) is obtained by using thestochastic Liouville equation Eq. (25) and recognizingthe nonanticipating rule Eq. (15). New density matricesM⟨g2(t)ρs(t)⟩ and M⟨g1(t)ρs(t)⟩ appear on the righthand side of this equation. The former originates fromthe bath-induced field term in the stochastic Liouvilleequation and the latter comes from the differentiationof the bath-induced field.

To work out a deterministic equation forM⟨g(t)ρs(t)⟩, one needs to find the differentialequations for the two density matrices M⟨g2(t)ρs(t)⟩and M⟨g1(t)ρs(t)⟩. This procedure generates newdensity matrices whose equations of motion involvemore terms. However, it will be greatly simplified ifthe response function can be decomposed in termsof Eq. (77). Originally, only the sum-of-exponentialapproximation for the bath response function was usedfor the hierarchical approach. In Ref. [108], we at thefirst time went beyond this approximation by using aresponse function related to the Laguerre polynomials

α(t) = ibte−Ωt, (83)

where b and Ω are real numbers. Very recently, Tang andco-workers suggested an expansion of the bath responsefunction with a complete set of orthonormal functionsfor the hierarchical scheme [109].

In the following, we will illustrate the derivation ofthe hierarchical approach with Eq. (83). To include allterms generated in the procedure of differentiation, weintroduce the auxiliary density matrices

ρm,n(t) = Mgm(t)hn(t)ρs(t)

, (84)

with the root entry ρ0,0(t) being the reduced density ma-trix. Here, the auxiliary stochastic process h(t), whichaccounts the new term other than g(t) when substitut-ing Eq. (83) into Eq. (82), is defined as

h(t) =

∫ t

0

(dµ2,τ + idµ3,τ )be−Ω(t−τ). (85)

The differential equations of the auxiliary density matri-ces are obtained through the fundamental Itô calculus

ihρm,n(t) = −ih(m+ n)Ωρm,n(t) + [Hs, ρm,n(t)]

+ [f(s), ρm+1,n(t)] + inhbf(s), ρm,n−1(t)

+ imhρm−1,n+1(t). (86)

By definition the initial condition is ρ0,0(0) = ρs(0) andρm,n(0) = 0 (m+ n = 0).

Eq. (86) forms a hierarchical structure for the evolu-tion of the auxiliary density matrices. The evolution ofthe reduced density matrix (ρ0,0) involves the densitymatrix in the first layer, i.e., ρ1,0. The auxiliary densitymatrix in the first layer is coupled to root and those inthe second layer, and so forth. Here the auxiliary den-sity matrices with the same value of m+2n form a layerof the hierarchy with the value m + 2n indicating thetimes of differentiation required for the root to accessthis layer. Thus the value m + 2n can be understoodas the depth of the auxiliary density matrix ρm,n andreflects the order of correlation of the noise in the prop-agation of the reduced density matrix.

The derivation can be extended to a response func-tion with arbitrary number of terms in Eq. (77). Forexample, with a response function containing three ex-ponentials, one can use three integers to identify theauxiliary density matrices ρm,n,k(t). Then as depictedin Fig. 5, all the density matrices in the hierarchy arelinked to the previous and the next layers through apyramid-like structure.

Actually the hierarchy approach is a scheme to em-bed a non-Markovian system in a larger, possibly infiniteMarkovian system and the memory effects are character-ized by the auxiliary terms. In other words, it convertsan integro-differential equation with a degenerate ker-nel to infinite number of hierarchy equations. We wouldlike to point out that the auxiliary density matrices alsocarry the information for the bath and have physicalmeaning instead of being pure mathematical intermedi-ate entities. For example, the auxiliary density matricesin the first layer are directly related to the transport

Yun-An Yan and Jiushu Shao, Front. Phys. 11(4), 110309 (2016)110309-15

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Fig. 5 The pyramid-like structure of the auxiliary densitymatrices in the hierarchical evolution for response functioncontaining three exponentials.

current in fermionic systems [110].

5.3 Performance of hierarchical approach

The hierarchy equation Eq. (86) is an infinite-dimensional ordinary differential equation. It is anunattainable yet unnecessary work to handle an infinitesystem exactly in numerical simulations. Normally, thedeeper the layer is, the less impact it has on the evolu-tion of the reduced density matrix. In practice, one cansafely truncate the hierarchy equation to the Kth layerwith a sufficiently large integer K. That is, all termsρm,n,k(t) with m+ n+ k > K are set to zero. The levelof truncation K depends on the system-bath couplingstrength. The stronger the coupling is, the larger valueof K has to be used for the truncation. In the numeri-cal simulations we will choose the minimum value Kmin

such that the differences between the results with trun-cation of Kmin and Kmin + 1 are smaller than a pres-elected criteria but those between Kmin and Kmin − 1are not.

To verify this point we will check the performance ofthe hierarchical approach for the dissipated symmetrictwo-level system Eq. (50). For the clarity of showing theimpact of parameters on the performance of the hierar-chy equation, the response function is assumed to takea single exponential,

α(t) = 0.5ηω2c (0.5 cot(βωc/2)− i) exp(−ωc|t|), (87)

which is essentially the first term in the response func-tion Eq. (52) by ignoring contributions from all Matsub-ara frequencies. Here we set β = 0.5 and explore how thelevel of truncation depends on the dissipation strengthη with a cut-off ωc = 4. Figure 6(a) shows the dissipa-tion strength dependence of the truncation level Kmin

with tolerances of 2 × 10−3 and 10−4. The differencesin the results with a maximum deviation of 2 × 10−3

cannot be recognized with naked eye if they are plottedon top of each other. It verifies that in general the hier-archy equation can converge when we include more andmore layers. When the tolerance decreases by a factor of

Fig. 6 The relation between the truncation level Kmin and(a) the dissipation strength η with ωc = 4∆ for the toleranceof 2×10−3 and 10−4, and (b) the memory time with η = 0.5for the TLS with response function Eq. (87) at temperatureT = 2∆.

20, usually we have to increase the truncation level byone or two. With a properly chosen truncation level, weeven can obtain numerically exact results with machineaccuracy of the double precision.

Because the hierarchy approach gives the finite mem-ory time corrections to the Markovian approximation,we expect that the truncation level would be small forshort bath memory times. Again, we will use the sym-metric two-level system Eq. (50) and the single exponen-tial response function Eq. (87) to illustrate this point.Here β = 2∆, η = 0.5 are fixed and ωc varies from 0.1to 70. The results are plotted in Fig. 6(b). It shows anoverall monotonic increase of the truncation level withrespect to the memory time (measured by ∆/ωc). Whenthe memory time is around 0.15, Kmin = 3 is alreadysufficient to produce the accuracy of 0.002. The trun-cation level gradually increases up to seven when thememory goes to 10.

In the hierarchy equation, the auxiliary density ma-trix ρm,n is damped by the factor (m + n)Ω . The ma-trices with large damping factors will decay fast in timepropagation. There exist alternative truncation schemesbased on this fact. Shi and co-workers introduced a fil-tering scheme after rescaling the auxiliary density ma-trices that automatically truncates the hierarchy witha preselected tolerance [111]. This truncation schememay significantly reduce the number of auxiliary den-

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sity matrices and speed up the numerical simulation ofthe hierarchical approach.

The computational effort, such as computer mem-ory and CPU time required for the numerical simu-lation, directly depends on the number of auxiliarydensity matrices in the hierarchy. The jth layer has(j +N − 1)!/(j!(N − 1)!) auxiliary density matrices forN exponentials. The total number of density matricesfor K layers is

L(K,N) =

K∑j=0

(j +N − 1)!

j!(N − 1)!=

(K +N)!

K!N !. (88)

Therefore, the computational effort scales exponentiallywith respect to the number of terms approximating theresponse function. It is appreciated to reduce the num-ber of exponentials decomposing the response functionwithout loss of accuracy. For this purpose, Yan and co-workers used the Padé approximant of the Bose/Fermifunction to obtain accurate approximation for the re-sponse function with a few exponentials [112, 113].

Here the hierarchy equation is derived based on thestochastic decoupling for the reduced density matrix.Similar hierarchy equations were derived for multi-timecorrelation functions [114] and for diffusively drivenquantum systems [115]. Very recently, de Vega suggestedanother form of hierarchical equations [116]. Zhou andco-workers also derived non-Markovian quantum Blochequations in the hierarchical form [117].

Now the hierarchical approach has become a standardtool for the simulation of the non-Markovian dynamics.It has been applied to the study of quantum coherencein a photosynthetic system [118], the dissipative two-exciton dynamics in molecular aggregate [119], the ex-citon Seebeck effect in molecular systems [120], the exci-ton interference [121], two-dimensional electronic spec-tra [122], Kondo effects in driven strongly correlatedquantum dots [123], to name but only a few.

As numerical examples, the hierarchical simulationsof the spin-boson model Eq. (50) are performed assum-ing the Ohmic spectral density function with the Debyeregulation Eq. (51). We will explore the dynamics witha broad range of the dissipation strength η and the fre-quency cut-off ωc. As discussed above, only the first Ne

exponentials in the response function Eq. (52) need tobe incorporated into the hierarchy with a finite level oftruncation Kmin.

Figure 7 displays the dynamics of the TLS with afast bath (ωc = 5∆) at temperature T = 2∆. It showsthat for weak dissipation with η = 0.4, clear quantumcoherence is manifested by the oscillation of the pop-ulation difference. The calculation converges with fourexponentials up to the seventh order. When the dissi-pation is strong, i.e., η = 8, the system returns to theequilibrium without oscillation, but with a longer re-laxation time. In this case, we have to incorporate six

Fig. 7 The time evolution of the population difference forthe TLS with a relative large cut-off ωc = 5∆ at temperatureT = 2∆ for (a) η = 0.4 and (b) η = 8. The hierarchyparameters are Ne = 4, Kmin = 7 for (a) and Ne = 6,Kmin = 9 for (b).

Fig. 8 The time evolution of the population difference forthe TLS with a small cut-off ωc = 0.25∆ at temperatureT = 0.5∆ for (a) η = 4, Ne = 4, Kmin = 6; (b) η = 8,Ne = 6, Kmin = 9; and (c) η = 80, Ne = 7, Kmin = 25.

exponentials and truncate the hierarchy at the ninth or-der.

The scenario changes for a TLS with a slow bath(ωc = 0.25∆) at a lower temperature T = 0.5∆. Asdepicted in Fig. 8, the system is slightly damped for arather large coupling constant η = 4. The coherence isstill observable with a strength as large as η = 80.

6 Hybrid stochastic-hierarchical equation

For the dissipative harmonic oscillator, either classicalor quantum, as the dissipation increases, the motionwill change from coherence to exponential decay. For theundissipated two-level system, the dynamics exhibits co-herence as the system changes from one localized stateto the other periodically. Under weak to intermediatedissipation, the coherence between the two localizedstates will be destroyed and the zero-temperature dy-

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namical behavior of the spin-boson model is expectedto exhibit the similar coherence-to-decoherence transi-tion. However, the dissipated two-level system is nonlin-ear and there exists a critical value for the dissipationstrength beyond which the dynamics is frozen. In thiscase the dynamical phase transition occurs for the spin-boson model, which is nothing but the localization [124].

It remains a challenge for theorists to simulate non-Markovian quantum dissipative dynamics under strongdissipation at zero temperature. Both the stochastic andhierarchical approaches have disadvantages in numeri-cal simulations of the low temperature dynamics. Onone side, the stochastic Liouville equation is suitablefor the dynamics at arbitrary temperatures under weakto intermediate dissipation. The performance is mainlydegraded by the imaginary part of the response func-tion which makes the stochastic density matrix non-Hermitian and non-norm-conserving. On the other side,hierarchical approach is numerically efficient for inter-mediate to strong dissipation. However, when the tem-perature becomes low, many terms have to be includedin the expansion of the real part of the response func-tion. Therefore, the treatment of the real part of theresponse function makes the hierarchical approach onlysuitable for medium to high temperatures. Based onthe above analysis, a natural question arises: Can wecombine the hierarchical and stochastic schemes to usetheir advantages and circumvent their disadvantages toachieve an efficient numerical method for dynamics un-der strong dissipation at low temperatures or even atzero temperature?

This question led to the hybrid stochastic-hierarchicalequations [108, 125]. The original idea was to use themixed random-deterministic (MRD) scheme to dealwith the imaginary part of the response function by thehierarchical approach and to handle the real part withthe stochastic method [108]. To be specific, the stochas-tic average of Eq. (25) with respect to w2 is tackledwith the hierarchical approach and that with respectto w1 is maintained with the stochastic scheme. It issuccessful to calculate long-time dynamics for weak tomoderate dissipation at zero temperature but fails forstrong dissipation. A natural remedy is the introduc-tion of the flexible random-deterministic (FRD) schemewhere a few exponentials are extracted from the realpart of the response function and tackled with the hi-erarchical approach [125]. By doing so, the rest of theresponse function which needs to be dealt with by therandom scheme is much smaller. Consequently, the flex-ible scheme is efficient both in producing the randomaverage and in representing the response function. Itsnumerical performance is greatly improved and suitablefor the zero temperature dynamics under strong dissi-pation.

To explore the performance of these methods we usethe dissipated symmetric two-level system under Ohmicdissipation with a polynomial cut-off [102]

J(ω) = 2πηω/[1 + (ω/ωc)

2]2

, (89)

where η is the Kondo parameter characterizing the dis-sipation strength. The imaginary part of the responsefunction is an exponentially weighted polynomial

αi(t) = −1

2πηω3

c te−ωct, (90)

which can be effectively tackled with the hierarchicalapproach.

The dynamics of the dissipative system at zero tem-perature for the Kondo parameter α = 0.1–0.5 andthe cut-off ωc = 10∆ were simulated with the mixedrandom-deterministic scheme. The results shown inFig. 9 display a clear pattern of the transition fromcoherent to incoherent motion. The population differ-ences exhibit obvious coherence when the dissipationstrength is less than 0.5 and follow an exponential de-cay for α = 0.5.

To scrutinize the dynamics at zero temperature, wealso carried out the simulations with the mixed random-deterministic scheme for the population differences withcut-offs of 20∆ and 40∆. Here we present the results forα = 0.1 as an example. As shown in Fig. 10, the resultsfor different frequency cut-offs fit together once the timeaxis is rescaled by the renormalization frequency

∆r = ∆

(∆

ωc

)α/(1−α)

. (91)

This feature implies that the frequency cut-off ωc onlycomes into the time scale of the dynamics through ∆r

for large cut-offs. In this case, the scaling limit is al-most reached when ωc = 10. The results based onthe noninteracting-blip approximation (NIBA) are alsoplotted in this figure. For such weak dissipation thenoninteracting-blip approximation gives almost exactresults. It produces noticeable and even significant er-rors for stronger dissipation, except for the case withα = 0.5 [125].

Fig. 9 The coherent dynamics of the spin-boson modelat zero temperature for various dissipation strength α sim-ulated with the mixed random-deterministic scheme. Thefrequency cut-off is ωc = 10∆.

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Fig. 10 The scaling limit of the spin-boson model at zerotemperature for dissipation strength α = 0.1 simulated withthe mixed random-deterministic scheme. NIBA gives almostexact results for this case.

The results for α = 0.2 − 0.5 with different cut-offsconfirm the scaling behavior. For α = 0.5, the popu-lation difference at long times exhibits an exponentialdecay with respect to the time rescaled by the renor-malization frequency ∆r. This exponential decay agreeswith the theoretical prediction from the noninteracting-blip approximation [32], which becomes exact for an in-finite frequency cut-off for α = 0.5.

The mixed random-deterministic scheme becomes in-efficient under strong dissipation and one has to resortto the flexible scheme. Fig. 11 presents a comparison ofthe efficiency between the mixed and flexible random-deterministic schemes for the case with α = 0.5. It takesthe mixed scheme more than one day to obtain the con-verged results with a common desktop computer. Butthe flexible scheme yields reasonable results within fiveseconds with the same computer. For a clear compari-son the results based on the mixed random-deterministicscheme within five seconds are also demonstrated, whichdisplay a dramatic deviation from the ensemble average.

The flexible random-deterministic scheme allows theexploration of the dynamics for α > 0.5 within areasonably short amount of time. It will take severaldays to several weeks to simulate the dynamics with0.5 < α ≤ 0.8. Figure 12 depicts the results for α = 0.6.Other cases assume similar features and are not shownhere. Again the results based on the noninteracting-blipapproximation are compared, which are only good atshort times and become qualitatively correct for longtimes for such strong dissipations. In these cases thepopulation differences follow an exponential decay af-ter an initial quick drop. But now the frequency cut-offdependency is different to that in the weak dissipationcases with α ≤ 0.5. As revealed by panel (b), the quan-tities ln[σz(t)] with different frequency cut-offs fit to-gether if the time is rescaled with 1/ωc for large cut-off.For such strong dissipation, the scaling limit is reachedaround ωc = 20∆, which is later than ωc = 10∆ for aweak dissipation with α = 0.1. Once the scaling limit ismaintained, the quantity ln[σz(t)] shows a linear depen-

Fig. 11 The dynamics of the spin-boson model at zerotemperature for dissipation strength α = 0.5. The frequencycut-off is ωc = 10∆. Solid line shows the converged resultswith the mixed random-deterministic scheme. The resultsfor the flexible scheme were calculated with a common desk-top within five seconds. A 5-seconds simulation of the mixedrandom-deterministic scheme is also shown for the compari-son of the numerical efficiency.

Fig. 12 The dynamics of the spin-boson model at zerotemperature for dissipation strength α = 0.6 with differentfrequency cut-off. (a) The comparison between the FRD andNIBA; (b) The scaling limit for the FRD simulations.

dence on ∆(∆/ωc)t. It strongly suggests that there isa phase transition at α = 0.5 for the spin-boson modelat zero temperature. In the scaling limit ωc/∆ → ∞,the dynamics changes from coherent motion for α < 0.5to exponential decay for 0.5 ≤ α < 1.0. Accordingly,the scaling factor changes from ∆r to ∆(∆/ωc). The

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transition to localization at α = 1.0 is also predicted byvarious analytic theories [32, 34] in the limit ωc/∆ → ∞.These results are summarized in Fig. 13.

One may introduce a decay rate as

κ = −ωc limt→tc

d ln[σz(t)]

dt, (92)

with a sufficiently large time tc. The results of the de-cay rates for the Kondo parameters 0.5 ≤ α ≤ 0.8 arepresented in Fig. 14. Though the existence of the expo-nential decay could be explained by conformal field the-ory [34, 126], the predicted results are different from thecurrent simulations. The decay rate decreases as the dis-sipation becomes stronger. Interestingly, the logarithmof the decay rate shows a linear dependence on the dis-sipation strength.

7 Summary and perspectives

The past two decades have witnessed an increasing pop-ularity of stochastic simulations of quantum dissipativedynamics. Particularly, the influence functional withinthe framework of linear dissipation can be naturally re-constructed with the random average by introducingGaussian random forces to the system. Thanks to theHubbard–Stratonovich transformation, we have decou-pled the system and the bath. The original system-bathcoupling is converted into the correlation induced bythe motion of the stochastic system and the bath un-

Fig. 13 The phase diagram for the spin-boson model atzero temperature in the scaling limit ωc/∆ → ∞.

Fig. 14 The logarithm of the decay rate as the functionof the dissipation strength for the spin-boson model at zerotemperature. Symbols are calculated results and the line isfor the guide of the eye with a linear function lnκ = 4.14−7.66α.

der the influence of the common white noise forces. Thesole effect of the bath is to provide a random field onthe evolution of the system. For the linear dissipationmodel, the motion of the bath is exactly known and thebath-induced random field can be worked out. Thus astochastic Liouville equation can be derived for the re-duced dynamics without explicit treatment of the bath.

The final stochastic Liouville equation does not pre-serve hermicity, norm-conserving, or positivity, whichaffects its numerical performance. To have a better nu-merical performance, a Hermitian scheme has also beendeveloped. The stochastic simulation is simple, easy tobe implemented and suitable for numerical simulationsof dissipative dynamics with weak to intermediate cou-pling strength at arbitrary temperatures.

The stochastic scheme can also serve as a useful theo-retical tool to derive deterministic master equation. Forthe analytically solvable models and for the memorylessbath, we can carry out the random average analyticallyto achieve the exact master equation. The procedure forgeneral systems with arbitrary bath response function isnot closed. However, if the response function is a degen-erate kernel, i.e., decomposable in terms of exponentialsor exponential-weighted Laguerre polynomials, an effi-cient hierarchy approach can be developed, where thememory effects is characterized by the auxiliary densitymatrices.

But at low temperatures, the response function of thebath must be approximated with many terms and thehierarchical simulations may become prohibitively ex-pansive. In these cases, we can combine the determin-istic and stochastic schemes to handle part of the re-sponse function with the hierarchical approach and therest with the stochastic scheme. Such hybrid stochastic-hierarchical equations exploit the advantages of thestochastic and hierarchical schemes and are suitable forthe lower temperature dynamics with strong dissipation.

As described above, the Hubbard-Stratonovich trans-formation provides us a machinery to decouple the in-teraction between different physical variables via intro-ducing white noises. It is, in principle, a (at least par-tial) cure for the curse of dimensionality, yet would notbecome powerful unless the average of the concomitantstochastic evolution is feasibly obtained. We see that itworks well for a bosonic bath with linear couplings tothe system and the bath-induced stochastic field can bederived in a closed form. For a general bath, however,no analytic results are available. It is no doubt that ob-taining a reasonably accurate bath-induced stochasticfield is a challenging task.

It is really ideal if the master equation that a dissipa-tive system obeys can be derived. Then it would be eas-ier to use such a deterministic equation to reveal the dy-namical features of quantum dissipation. Unfortunately,even if the induced stochastic field is known for the lin-ear bath and the stochastic Liouville equation has beendetermined, the master equation for the nonlinear dis-

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sipative system is elusive. The difficulty is rooted at thecorrelation between the bath-induced stochastic fieldand the evolution of the system. Except for the case ofweak dissipation in which a reliable perturbation treat-ment may lead a master equation, it is hardly possible tofind a deterministic equation of motion with general dis-sipation. Although numerical simulations based on thestochastic Liouville equations are applicable, the com-putational cost would increase rapidly as the dissipationgets stronger and the statistical error would eventuallybe uncontrollable. This problem may be alleviated byusing a nonlinear but norm-conserved stochastic Liou-ville equation instead of the linear one, but nonlinearitymay also cause numerical instability.

The lesson we have learnt is that stochastic simula-tion is a general approach with low efficiency while de-terministic method is highly effective but limited to spe-cific systems. The mixed stochastic-deterministic tech-nique enjoys some success in solving the dynamics ofthe spin-boson model, yet more effective numerical al-gorithm with more flexibility and feasibility, addressinggeneral quantum dissipative dynamics is yet to be de-veloped.

From a fundamental perspective, because manymacroscopic effects ranging from pure physical suchas turbulence, phase transition to biological such asmemory and so on are only possible within dissipativesystems, to reveal the underlying mechanism of theseemergent effects needs conceptual as well as theoreticaladvances. In a recent article Whitesides listed howdissipative systems work as the third new class ofproblems for chemists [127]. As he pointed, “Studyingdissipative systems has, of course, been a subject ofphysical science for decades, but, unlike equilibriumsystems, understanding dissipative systems—boththeoretically and empirically—is still at the verybeginning”. More fruitful emergent phenomena suchas quantum phase transition, a behavior change fromdelocalized quantum world to the localized classicalone, may undergo in quantum dissipative systems andstill await for exploration [128].

Acknowledgements The authors thank Yun Zhou for his gener-ous help with figure plotting. This work was supported by the Na-tional Natural Science Foundation of China (Grant Nos. 21421003and 21373064) and the 973 Program of the Ministry of Science andTechnology of China (Grant No. 2013CB834606).

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