Time-local unraveling of non-Markovian stochastic Schr\"odinger equations

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by  Antoine Tilloy

Time-local unraveling of non-Markovian stochastic Schr¨odinger equations
Antoine Tilloy∗
Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
(Dated: March 7, 2017)
Non-Markovian stochastic Schro¨dinger equations (NMSSE) are an important tool in quantum
mechanics, from the theory of open systems to foundations. Yet, in general, they are but formal
objects: theirsolutioncanbecomputednumericallyonlyforsomeclassesofnon-trivialsystems. We
proposetowritethestochasticrealizationsofNMSSEasaveragesoverthesolutionsofanauxiliary
equationwithanadditionalrandomfield. Ourmethodyieldsnon-perturbativenumericalsimulation
algorithm for generic NMSSE that can be made arbitrarily accurate for reasonably short times.
I. INTRODUCTION within orthodox interpretations of the formalism [20].
7
Non-Markovian stochastic approaches have also infected
1
0 Stochasticpurestaterepresentationshaveawiderange foundations with various extensions of the initial CSL
2 ofapplicationsinquantummechanics: theyserveascom- proposal [21, 22]. In all these applications, the practical
putational tools to unravel open-system evolutions, as problem is always the same: the equations are very for-
r
a modelling tools to describe continuous measurement sit- mal, involving functional derivatives, and plotting their
M uations and as foundational tools to solve the measure- solution is typically as hard as solving a general non-
ment problem in models where the superposition princi- Markovian open-system evolution. There are, of course,
6 plebreaksdown. IntheMarkovianlimit,thesolutionsof quite a few cases where the equations can be solved [23–
stochastic Schr¨odinger equations can be efficiently com- 27] as well as perturbative methods [28] to deal with a
]
h puted numerically, justifying their wide use in the three broader class of problems, but general non-perturbative
p aforementioned fields. In the non-Markovian case how- approaches have so far been lacking.
- ever, thereexistsnogeneralpurposemethodtocompute In the following, we will introduce stochastic
t
n arealizationoftherandompurestateprocess: plottinga Schr¨odinger equations as unravelings of non-Markovian
a single trajectory is in general impossible. In this article, open-system dynamics making an extensive use of the
u
we propose to write the solutions of general stochastic general framework introduced recently by Di´osi and Fe-
q
[ Schr¨odinger equations as averages over the solutions of rialdi [29]. Yet, our objective will not primarily be to
time-localstochasticSchr¨odingerequationswithanaux- find a numerically efficient way to compute the open-
2 iliary noise. We shall use the same trick that usually system evolution: we will be interested in the stochastic
v allows to replace openness by stochasticity, this time to pure state trajectories themselves, having in mind their
8
4 get rid of non-Markovianity. broader range of applications.
9 AlthoughstochasticSchr¨odingerequationshavealong
1 history dating back to the eighties [1, 2], the field took
0 off when their potential in numerics was understood by II. GENERAL FRAMEWORK
1. Dalibard,CastinandMølmer[3,4](seealsoDum,Zoller
0 and Ritsch [5]) in the jump case and by Gisin and Perci- Weconsideranopenquantumsystemevolvingaccord-
7 val [6] in the diffusive case. The connection with actual ingtoageneralGaussianMasterEquation(GME),oneof
1 quantum trajectories coming from realistic measurement the broadest non-Markovian generalization of the Lind-
v: setups was understood roughly at the same time by Mil- blad equation that is analytically tractable. We suppose
i burn and Wiseman [7] (see e.g. [8] for an introduction). thesystemhasaproperHamiltonianH0 andwillusethe
X Interestingly, the introduction of continuous stochastic interactionpictureforalloperatorsOˆ(t)=eiH0tOˆe−iH0t.
r pure state equations in foundations is slightly anterior The GME for the density matrix in interaction picture
a
with the Continuous Spontaneous Localization (CSL) can then be written as a time ordered exponential [29]:
model of Pearle, Ghirardi and Rimini [9, 10] and the
gravity related collapse model of Di´osi [11], both aimed ρ(t)=Φt·ρ(0)
at solving the measurement problem (see [12, 13] for a (cid:26)Z tZ t h
review). The generalization of the formalism to the non- =T exp dτdsDij(τ,s) AˆjL(s)AˆiR(τ) (1)
Markovian realm was carried out by Di´osi and Strunz 0 0
i(cid:27)
[14, 15]. In that case, the measurement interpretation −θ Aˆi (τ)Aˆj(s)−θ Aˆj (s)Aˆi (τ) ρ(0)
τs L L sτ R R
exists but is admittedly more subtle: a given trajectory
only has a local point by point measurement interpre-
tation [16] but no real time meaning [17–19], at least where T is the time ordering operator, {Aˆk}1≤k≤n are
arbitrarysystemHermitianoperators,D (τ,s)isacom-
ij
plex positive semi-definite kernel, θ = θ(τ −s) is the
τs
Heaviside function, and we have used summation on re-
∗ [email protected] peatedindicesaswellastheleft-rightnotationforsuper-
2
operators: ξ ={ξ } such that:
k 1≤k≤n
ρ(t)=E (cid:2)|ψ (t)ihψ (t)|(cid:3), (2)
Aˆ ρ=Aˆρ; Aˆ ρ=ρAˆ. ξ ξ ξ
L R
whereE [·]=R ·dµ (ξ)denotesaveragingoverthesetof
ξ ◦
At least if D is time translation invariant, equation (1) complex stochastic fields ξ. The interest of this rewrit-
can be obtained from the linear coupling of the system ing is that it decouples the left and right parts of the
with a general bosonic bath, in which case D can be re- timeorderedexponentialΦ ofequation(1),atrickwhich
t
lated to the field two-point correlation functions [29]. It is sometimes called a Hubbard-Stratonovich transforma-
isalsoanequivalentoperatorrewritingofthewellknown tion. Let us now find an explicit candidate for |ψ (t)i.
ξ
quadratic Feynman-Vernon influence functional for sys- Following e.g. [29], we first introduce a set of Gaussian
tems linearly coupled to harmonic oscillators [30], a rep- complex fields of zero average and thus fully character-
resentation used e.g. by Hu, Paz and Zhang [31, 32] for ized by its following two point correlation functions:
the Quantum Brownian Motion and which can be de-
rived from a vast class of microscopic models [33]. Here, E (cid:2)ξ (τ)ξ∗(s)(cid:3)=C (τ,s)
ξ i j ij
(3)
we are not interested in the open system dynamics per E (cid:2)ξ (τ)ξ (s)(cid:3)=S (τ,s)
ξ i j ij
se and thus step back from its implementation details to
take the very general equation (1) as our starting point. where S is the relation function, a symmetric complex
A stochastic unraveling of equation (1) is a set of pure kernel which can be chosen freely provided the full cor-
statetrajectories|ψ (t)iindexedbyasetofrandomfields relation kernel is positive semi-definite.
ξ
For two sets of fields ak and bk, the following generalized characteristic function is obtained by Gaussian integration:
t t
(cid:20) (cid:26) Z t (cid:27)(cid:21) (Z tZ t S (τ,s)aiaj +S∗(τ,s)bibj)
E exp −i ds(akξ (s)−bkξ∗(s)) =exp dτdsC (τ,s)aibj − ij τ s ij τ s , (4)
ξ s k s k ij τ s 2
0 0 0
anequalitythatcanbefurthergeneralizedfora’sandb’spromotedtooperatorsprovidedbothsidesaretimeordered:
(cid:20) (cid:26) Z t (cid:27)(cid:21) (Z tZ t S (τ,s)aˆiaˆj +S∗(τ,s)ˆbiˆbj)
E T exp −i ds(aˆkξ (s)−ˆbkξ∗(s)) =T exp dτdsC (τ,s)aˆiˆbj − ij τ s ij τ s . (5)
ξ s k s k ij τ s 2
0 0 0
The similarity between the r.h.s of (5) and (1) suggests proposed [29]. Unfortunately, equation (7) is quite for-
to identify E [|ψ ihψ |] with the l.h.s of (5) with aˆk = malunlessitispossibletowritethefunctionalderivative
ξ ξ ξ s
AˆkL(s), ˆbks = AˆkR(s) and Cij(τ,s) = Dij(τ,s). Actually, 1 as a simple local operator acting on |ψξ(t)i.
this does not fully do the trick, a counter term remains,
and one sees following [29] that the correct prescription
is: III. MAIN RESULT
|ψ (t)i=Gˆ (t)|ψ i
ξ ξ 0 The core objective of this article is to show how the
(cid:26) Z t solutions of (7) can nonetheless be computed in the gen-
:=T exp −i dsAˆk(s)ξ (s) (6)
k eral case by reusing a stochastic unraveling trick, i.e. by
0
writing this time:
Z tZ t (cid:27)
− dτdsθτs[D−S]ij(τ,s)Aˆi(τ)Aˆj(s) |ψ0i, |ψ (t)i=E (cid:2)|ψ (t)i(cid:3) (8)
ξ η ξ,η
0 0
where η is an auxiliary set of classical Gaussian fields.
which fulfills the unraveling condition (2). Equation (6)
Moreprecisely,wecantrytowritetheevolutionoperator
can subsequently be written in differential form to yield
of (6) in the following way:
a non-Markovian stochastic Schr¨odinger equation:
(cid:20) (cid:20) (cid:26) Z t (cid:27)(cid:21)
ddt|ψξ(t)i=−iAˆi(t) ξi(t) Gˆξ(t)=Eη T exp −i 0dsAˆk(s)[ξk(s)+ηk(s)] .
(7) (9)
Z t δ (cid:21)
+ ds[D−S] (t,s) |ψ (t)i.
ij δξ (s) ξ
0 j
To our knowledge, this equation is the most general lin- 1 The reader puzzled by the rigorous definition of the functional
ear non-Markovian stochastic Schr¨odinger equation ever derivativecanaswelluse(6)inallsituations.
3
Using again the Gaussian integration formula (5), one IV. NORM PRESERVING UNRAVELINGS
sees that the new unraveling condition (8) is satisfied
providedη isasetofGaussianfieldswithzeromeanand Thelinearstochasticdifferentialequation (7)doesnot
two-point correlation functions given by: preserve the norm of the state vector |ψ i. Normalized
ξ
E [η (τ)η (s)]=K (τ,s) pure states are usually prefered in foundations but also
η i j ij
E [η (τ)η∗(s)]=J (τ,s) (10) for numerics: the norm of |ψξi typically diffuses and for
η i j ij large times, most trajectories give a vanishing contribu-
with: tion to the average (2) while nearly all the weight is pro-
vided by rare events. Here, we shall not be primarily
K (τ,s)=θ [D−S] (τ,s)+θ [D−S] (s,τ) (11)
ij τs ij sτ ji
concerned by numerical efficiency but would neverthe-
andwheretheothercorrelationfunctionJ canbechosen less like to be able to compute the solutions of normal-
freely provided the total kernel Γ: ized NMSSE for their other uses in measurement theory
(cid:18) J K (cid:19) and in foundations.
Γ= (12)
K∗ J∗ To obtain a norm preserving evolution from the linear
is positive semi-definite 2. one, one needs to define a normalized state |ψeξi:
Equation (9) can be written in an explicit linear dif- |ψ (t)i
ferential form free of functional derivatives: |ψeξi(t)= p ξ (16)
hψ (t)|ψ (t)i
ξ ξ
d
|ψ (t)i=−iAˆk(t)[ξ (t)+η (t)]|ψ (t)i. (13)
dt ξ,η k k ξ,η butalsotointroduceanew“cooked”(orGirsanovtrans-
formed) probability measure:
This latter equation, together with its unraveling inter-
pretation(8)isthemainresultofthisarticle. Theevolu- dµ (ξ)=hψ (t)|ψ (t)idµ (ξ) (17)
t ξ ξ ◦
tiongivenbyequation(13)istime-local inthesensethat
once the random fields ξ and η are fixed, the state can which insures that the unraveling condition ρ(t) =
h i
be evolved without reference to the past. Although our Etξ |ψeξ(t)ihψeξ(t)| –where Etξ is the expectation value
interest was primarily in the stochastic pure state |ψ i
ξ taken with the new measure µ – is still valid. The ideal
itself, we can write the open-system density matrix as a t
double average over our successive unravelings (2) and solution would be to find a way to sample |ψeξi directly.
This is unfortunately not a trivial endeavour and the
(8):
method we shall propose is arguably less appealing than
ρ(t)=E hE (cid:2)|ψ (t)i(cid:3) E (cid:2)hψ (t)|(cid:3)i. (14) in the linear case.
ξ η ξ,η η ξ,η
Let us first notice that, in most cases, being able to
This can be written equivalently: find solutions to the linear NMSSE is good enough. In
ρ(t)=E (cid:2)|ψ (t)ihψ (t)|(cid:3), (15) the measurement context, the stochastic realization ξ
ξ,η(1),η(2) ξ,η(1) ξ,η(2) is given: it is the measurement result3, and the fact
where η(1) and η(2) are independent. This is the two that its probability distribution is non-Gaussian is irrel-
state unraveling proposal of Stockburger and Grabert evant. One can simply use the linear equation to re-
[34, 35]. We have thus found a connection between the construct the state from the measurement record and
standard stochastic pure state representations and the normalize at the very end. Another thing one may
more exotic two state vector method used in numerical want to do is to compute certain expectation values
approaches to open-systems. This allows us to identify with the Girsanov transformed probability measure dµ .
t
at least part of the freedom in the noise present in this Again, this only requires to be able to solve the linear
lattermethodascomingfromthekernelS whichencodes equation: one just needs to use the defining equation
differentstochasticSchr¨odingerevolutionscorresponding Et[·] = E◦[· hψ (t)|ψ (t)i]. Consequently, for all prac-
ξ ξ ξ ξ
to the same open-system evolution. We can further risk tical purposes, being able to solve the linear NMSSE is
the following heuristic interpretation of the two noises. good enough, even if it may be numerically inefficient.
The noise ξ is classical in the sense that it is averaged Still, one may wish to be able to sample from the
overatthedensitymatrixlevel. Itisalsoclassicalinthe norm preserving evolution, if only to plot “typical” tra-
sense that it can be measured, each realization of a tra- jectories and get an intuition of the physics. Because
jectory corresponding to a possible measurement result. the probability distribution of ξ is non-Gaussian in the
The noise η, on the other hand, is quantum or coherent normalized case, the task is not trivial. Yet, when di-
as all the possible contributions are summed over coher- rect sampling seems too hard, it is always possible to
ently at the pure state level. use indirect sampling methods [36], which in the sit-
uation we are interested in are quite straightforward.
2 For a kernel K given by equation (11), there always exists a
kernel J such that Γ is positive semi-definite. The existence
of a random field η verifying (10) is then a consequence of the 3 Thelatterisunderstoodagainasasingleshotrecordandnotas
Bochner-Minlostheorem. areal-timemeasurementtrajectory[19].
4
We would typically like to do rejection sampling, i.e.
keep a given trajectory |ψξi, initially generated with the approximationwith103terms
measure dµ◦, if a random number in [0,1] is smaller exact
than hψ (t)|ψ (t)imax [hψ (t)|ψ (t)i]−1. Unfortunately,
ξ ξ ζ ζ ζ 0
the latter maximum does not exist, hψ (t)|ψ (t)i is un-
ξ ξ fi
)
bounded. Wethusneedtouseaslightextensionofrejec- (t
tion sampling which is but a trivial instantiation of the ψξ
ˆA| 0.1
Metropolis algorithm. )|−
t
WedefineaMarkovchainXn =ξ(n) ofcompletenoise ψ(ξ
trajectories (and implicitly associate their corresponding ›
0.2
pure states). Assuming the chain is defined at step n −
(unrelated to the physical time t), we draw a new noise
trajectory ξ independently from the Gaussian measure
dµ , and compute the corresponding state |ψ i with the
◦ ξ 0 1 2 3
linearequation. WethenupdatetheMarkovchaininthe
t
following way:
(cid:26)
ξ with probability p FIG. 1. Snapshot of a linear trajectory |ψ i for a random
X = n+1 (18) ξ
n+1 X with probability 1−p realization of ξ. The shaded areas indicate the zones where
n n+1
50%(’0.5σ)and85%(’1σ)oftheapproximatetrajectories
wouldlie. Theexacttrajectoryisobtainedfromequation(20)
with p = min(1,hψ (t)|ψ (t)i/hψ (t)|ψ (t)i).
n+1 ξ ξ ξ(n) ξ(n) whiletheapproximatetrajectoryisobtainedfrom(8)andthe
This gives a Markov chain with the transition probabili-
estimation |ψ i’ 1 PN |ψ i with N =103 independent
ties: ξ N n=1 ξ,η
realizations of η and the same fixed ξ. The number of real-
(cid:18) (cid:19) izations is voluntarily left small to make the estimate of the
hψ (t)|ψ (t)i
dP[ξ →ζ]=min 1, ζ ζ dµ (ζ), (19) error visible as the latter decreases ∝N−1/2.
hψ (t)|ψ (t)i ◦
ξ ξ
fromwhichitistheneasytoreadthattheinvariantmea- can replace the functional derivative:
sure is dµ . Hence, the Markov chain X samples the
t n
noise trajectories with the correct non-Gaussian proba- δ |ψ (t)i=−iAˆ|ψ (t)i. (21)
bility measure. The method is not too inefficient in this δξ(s) ξ ξ
context because the samples are independent before the
This example thus offers a nice testbed for our stochas-
rejectionstep. Thisfinalstepisindeedtheonlyonethat
tic approach. Numerical results for Aˆ = diag(1,0,−1)
correlatesthefinaloutcomesthroughthepossiblerepeti-
and D(t,0) = e−t are shown in figure 1. In this ex-
tion of the same realization. In practice, it is even possi-
ample, the relaxation timescale responsible for the non-
bletosavesomecomputingtimeateachstepbydrawing
Markovianity and the timescale of the system-bath cou-
a random real number r uniformly distributed between
plingarebothoforder1andwewouldthusbedeepinthe
0 and 1 before computing |ψ i. As the approximation
ξ
non-Markovian regime had the system Hamiltonian not
of |ψ i is progressively refined through the addition of
ξ
been trivial. Our method is efficient for evolution times
more |ψ i, one can compare the approximated norm to
ξ,η
of the order of the this second timescale and becomes
q ×r and stop further refining if it is already clear that
n
exponentiallyexpensiveforlargertimesasonewouldex-
the sample is unlikely to be kept.
pect from the fact that the norm of |ψ i diffuses and
ξ,η
converges almost surely to zero.
We also test our Metropolis method to sample solu-
V. EXAMPLE
tions of the norm-preserving non-linear NMSSE (see fig-
ure 2). We observe that the typical trajectories for the
Wenowillustrateourmethodonanexamplewherethe initialGaussianandMetropolissampledmeasureswidely
stochasticSchr¨odingerequationisfullyexplicit. Forthat differ. This means that sampling from the correct non-
matter, we consider a single Hermitian operator Aˆ com- Gaussian measure is the only way to get a reasonably
muting with H0 (s.t. Aˆ(t) = Aˆ) and purely imaginary goodintuitionofthephysics(although,asweargued,all
noise, i.e. S = −D, which gives the following stochastic measurable quantities can be computed with the linear
Schr¨odinger equation: NMSSE).Thisisalsothesign,ratherunfortunately,that
ourrejectionsamplingmethodisexpensiveasitrequires
d|ψ (t)i=−iAˆ(cid:20)ξ(t)+2Z tdsD(t,s) δ (cid:21)|ψ (t)i. finding and giving weight to trajectories that are atypi-
dt ξ δξ(s) ξ calwiththeGaussianmeasure. Again,thiscostincreases
0
(20) exponentially with time as it is related to the fact that
For a given realization of ξ, |ψ i obeys an explicit differ- thenormof|ψ (t)igoestozeroexponentiallyquicklyfor
ξ ξ
ential equation. Indeed, using the integral form (6), we typical trajectories. Even if direct sampling remains out
5
ofreachinthefuture,onemayhopethatnewMetropolis
sampling methods, smarter than the naive one we have
proposed, could yield dramatic speed-ups.
VI. CONCLUSION
Inthisarticle,wehaveintroducedarewriting(or“un-
raveling”) of the solutions of formal NMSSE as averages
over the solutions of explicit time-local stochastic differ-
entialequations. Thisnewformulationmayprovideaba-
sis for further analytical studies, notably as it provides
an extremely simple new way to define NMSSE. More
importantly, it immediately offers a method to compute
numerically their solutions. Although we have proposed
a heuristic interpretation of the quantum and classical
character of the two noises involved, the auxiliary state
|ψ i does not yet have a clear operational characteri-
ξ,η
zation. The latter would certainly help understand the
freedom in the noise kernel J. An important limitation
ofthisworkisthelackofadirectextensiontonon-linear
NMSSE (although we have presented a Metropolis sam-
plingwayout). Anyadvanceinthisdirectionwoulddra-
maticallyimproveourabilitytodealwithnon-Markovian
FIG. 2. Illustration of the change of measure: 103 normal-
open-systems numerically.
ized quantum trajectories |ψeξi where ξ is sampled with the
(incorrect) raw Gaussianmeasure dµ on theupper plotand
◦
(correct) non-Gaussian measure dµ on the lower plot. The
t
latter is obtained using the Metropolis Markov chain, with a
relaxation time of 103 iterations between each sample. The ACKNOWLEDGMENTS
initial condition is |ψeξ(0)i = 3−1/2(1,1,1) such that one ex-
pects hψeξ(t)|Aˆ|ψeξ(t)i to converge to −1, 0, or +1 for large I thank Lajos Di´osi and Luca Ferialdi for helpful dis-
time with equal probability. cussions.
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