Time-local unraveling of non-Markovian stochastic
Schr
¨
odinger equations
Antoine Tilloy
Max-Planck-Institut f
¨
ur Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
September 19, 2017
Non-Markovian stochastic Schr
¨
odinger
equations (NMSSE) are important tools in
quantum mechanics, from the theory of open
systems to foundations. Yet, in general, they
are but formal objects: their solution can be
computed numerically only in some specific
cases or perturbatively. This article is focused
on the NMSSE themselves rather than on
the open-system evolution they unravel and
aims at making them less abstract. Namely,
we propose to write the stochastic realiza-
tions of linear NMSSE as averages over the
solutions of an auxiliary equation with an
additional random field. Our method yields
a non-perturbative numerical simulation al-
gorithm for generic linear NMSSE that can
be made arbitrarily accurate for reasonably
short times. For isotropic complex noises,
the method extends from linear to non-linear
NMSSE and allows to sample the solutions of
norm-preserving NMSSE directly.
1 Introduction
Stochastic pure state representations have a wide
range of applications in quantum mechanics: they
serve as computational tools to unravel open-system
evolutions [4, 19, 32], as modelling tools to describe
continuous measurement situations [22, 36] and as
foundational tools to solve the measurement problem
in models where the superposition principle breaks
down [2, 3]. In the Markovian limit, the solutions
of stochastic Schr¨odinger equations can be efficiently
computed numerically, justifying their wide use in the
three aforementioned fields. In the non-Markovian
case however, there exists no general purpose method
to compute a realization of the random pure state pro-
cess: plotting a single trajectory is in general impossi-
ble. In this article, we propose to write the solutions of
general stochastic Schr¨odinger equations as averages
over the solutions of time-local stochastic Schr¨odinger
equations with an auxiliary noise. We shall use the
same trick that usually allows to replace openness by
stochasticity, this time to get rid of non-Markovianity.
Antoine Tilloy: antoine.tilloy@mpq.mpg.de
Although stochastic Schr¨odinger equations have a
long history dating back to the eighties [8, 18], the
field took off when their potential in numerics was
understood by Dalibard, Castin and Mølmer [4, 26]
(see also Dum, Zoller and Ritsch [12]) in the jump
case and by Gisin and Percival [19] in the diffusive
case. The connection with actual quantum trajec-
tories coming from realistic measurement setups was
understood roughly at the same time by Milburn and
Wiseman [35] (see e.g. [22] for an introduction). In-
terestingly, the introduction of continuous stochastic
pure state equations in foundations is slightly anterior
with the Continuous Spontaneous Localization (CSL)
model of Pearle, Ghirardi and Rimini [17, 27] and the
gravity related collapse model of Di´osi [6], both aimed
at solving the measurement problem (see [2, 3] for a
review). The generalization of the formalism to the
non-Markovian realm was carried out by Di´osi and
Strunz [11, 30]. In that case, the measurement in-
terpretation exists but is admittedly more subtle: a
given trajectory only has a local point by point mea-
surement interpretation [15] but no real time mean-
ing [9, 10, 36], at least within orthodox interpreta-
tions of the formalism [16]. Non-Markovian stochastic
approaches have also infected foundations with vari-
ous extensions of the initial CSL proposal [1, 13]. In
all these applications, the practical problem is always
the same: the equations are very formal, involving
functional derivatives, and plotting their solution is
typically as hard as solving a general non-Markovian
open-system evolution. There are, of course, quite
a few cases where the equations can be solved [23–
25, 31, 37] as well as perturbative methods [5] to deal
with a broader class of problems, but general non-
perturbative approaches have so far been lacking.
In the following, we will introduce stochas-
tic Schr¨odinger equations as unravelings of non-
Markovian open-system dynamics making an exten-
sive use of the general framework introduced recently
by Di´osi and Ferialdi [7]. Yet, our objective will not
primarily be to find a numerically efficient way to
compute the open-system evolution: we will be inter-
ested in the stochastic pure state trajectories them-
selves, having in mind their broader range of applica-
tions.
Accepted in Quantum 2017-09-16, click title to verify 1
arXiv:1701.01948v3 [quant-ph] 18 Sep 2017
2 General framework
2.1 Gaussian master equations
We consider an open quantum system evolving ac-
cording to a general Gaussian Master Equation
(GME), one of the broadest non-Markovian gener-
alizations of the Lindblad equation that is analyti-
cally tractable. We suppose the system has a proper
Hamiltonian H
0
and will use the interaction picture
for all operators
ˆ
O(t) = e
iH
0
t
ˆ
Oe
−iH
0
t
. The GME for
the density matrix in interaction picture can then be
written as a time ordered exponential [7]:
ρ(t) =Φ
t
· ρ(0)
=T exp
Z
t
0
Z
t
0
dτ ds D
ij
(τ, s)
h
A
j
L
(s)A
i
R
(τ) (1)
− θ
τ s
A
i
L
(τ)A
j
L
(s) − θ
sτ
A
j
R
(s)A
i
R
(τ)
i
ρ(0)
where T is the time ordering operator, {
ˆ
A
k
}
1≤k≤n
are
arbitrary system Hermitian operators, D
ij
(τ, s) is a
complex positive semi-definite kernel, θ
τ s
= θ(τ −s) is
the Heaviside function, and we have used summation
on repeated indices as well as the left-right notation
for superoperators:
A
L
· ρ =
ˆ
Aρ ; A
R
· ρ = ρ
ˆ
A
.
Equation (1) is a direct non-Markovian generalization
of the Lindblad master equation and reduces to it in
the limit D
ij
(τ, s) → d
ij
δ(τ − s). It is also an equiv-
alent operator rewriting of the well known quadratic
Feynman-Vernon influence functional for systems lin-
early coupled to harmonic oscillators [14], a represen-
tation used e.g. by Hu, Paz and Zhang [20, 21] for the
Quantum Brownian Motion and which can be derived
from a wide class of microscopic models [34].
At least if D is time translation invariant, equation
(1) can be obtained from the linear coupling of the sys-
tem of interest with a general bosonic bath (without
the Born-Markov approximation). To avoid being ex-
cessively abstract, let us recall the explicit derivation
of [7]. We consider a bath constituted of a contin-
uum of frequency of n types of harmonic oscillators.
We write the corresponding creation and annihilation
operators a
k
(ω), a
†
k
(ω) which verify [a
i
(ω), a
†
j
(ω
0
)] =
δ
ij
δ(ω − ω
0
). We then couple our system of interest
with the bath through the interaction Hamiltonian:
H
int
=
ˆ
A
k
⊗
Z
dω κ
`
k
a
`
(ω) + κ
`∗
k
a
†
`
(ω), (2)
where κ
`
k
(ω) is an arbitrary matrix of complex coef-
ficients. If the initial state of the total system is a
product with the bath ground state, the reduced sys-
tem density matrix can be expressed exactly using
Wick’s theorem [7] and is given by formula (1). In
that case, D is given by:
D
ij
(τ, s) =
Z
dω κ
k
i
(ω)κ
`∗
j
(ω)e
−iω(τ−s)
. (3)
Here, we are not interested in the open system dynam-
ics per se and thus step back from its implementation
details to take the very general equation (1) as our
starting point.
2.2 Stochastic unraveling
A stochastic unraveling of equation (1) is a set of pure
state trajectories |ψ
ξ
(t)i indexed by a set of random
fields ξ = {ξ
k
}
1≤k≤n
such that:
ρ(t) = E
ξ
|ψ
ξ
(t)ihψ
ξ
(t)|
, (4)
where E
ξ
[·] =
R
·dµ
◦
(ξ) denotes averaging over the
set of complex stochastic fields ξ. The interest of
this rewriting is that it decouples the left and right
parts of the time ordered exponential Φ
t
of equation
(1), a trick which is sometimes called a Hubbard-
Stratonovich transformation. Let us now find an ex-
plicit candidate for |ψ
ξ
(t)i. Following e.g. [7], we first
introduce a set of Gaussian complex fields of zero av-
erage and thus fully characterized by its following two
point correlation functions:
E
ξ
ξ
i
(τ)ξ
∗
j
(s)
= C
ij
(τ, s)
E
ξ
ξ
i
(τ)ξ
j
(s)
= S
ij
(τ, s)
(5)
where S is the relation function, a symmetric complex
kernel which can be chosen freely provided the full
correlation kernel is positive semi-definite.
For two sets of fields a
k
t
and b
k
t
, the following generalized characteristic function is obtained by Gaussian
integration:
E
ξ
exp
−i
Z
t
0
ds(a
k
s
ξ
k
(s) − b
k
s
ξ
∗
k
(s))
= exp
(
Z
t
0
Z
t
0
dτds C
ij
(τ, s)a
i
τ
b
j
s
−
S
ij
(τ, s)a
i
τ
a
j
s
+ S
∗
ij
(τ, s)b
i
τ
b
j
s
2
)
, (6)
an equality that can be further generalized for a’s and b’s promoted to operators provided both sides are time
ordered:
E
ξ
T exp
−i
Z
t
0
ds(ˆa
k
s
ξ
k
(s) −
ˆ
b
k
s
ξ
∗
k
(s))
= T exp
(
Z
t
0
Z
t
0
dτds C
ij
(τ, s)ˆa
i
τ
ˆ
b
j
s
−
S
ij
(τ, s)ˆa
i
τ
ˆa
j
s
+ S
∗
ij
(τ, s)
ˆ
b
i
τ
ˆ
b
j
s
2
)
.
(7)
Accepted in Quantum 2017-09-16, click title to verify 2
The similarity between the r.h.s of (7) and (1) sug-
gests to identify E
ξ
[|ψ
ξ
ihψ
ξ
|] with the l.h.s of (7) with
ˆa
k
s
=
ˆ
A
k
L
(s),
ˆ
b
k
s
=
ˆ
A
k
R
(s) and C
ij
(τ, s) = D
ij
(τ, s).
Actually, this does not fully do the trick, a counter
term remains, and one sees following [7] that the cor-
rect prescription is:
|ψ
ξ
(t)i =
ˆ
G
ξ
(t)|ψ
0
i
:=T exp
− i
Z
t
0
ds
ˆ
A
k
(s)ξ
k
(s) (8)
−
Z
t
0
Z
t
0
dτ ds θ
τ s
[D − S]
ij
(τ, s)
ˆ
A
i
(τ)
ˆ
A
j
(s)
|ψ
0
i,
which fulfills the unraveling condition (4). Equation
(8) can subsequently be written in differential form to
yield a non-Markovian stochastic Schr¨odinger equa-
tion:
d
dt
|ψ
ξ
(t)i = − i
ˆ
A
i
(t)
ξ
i
(t)
+
Z
t
0
ds [D − S]
ij
(t, s)
δ
δξ
j
(s)
|ψ
ξ
(t)i.
(9)
To our knowledge, this equation is the most general
linear non-Markovian stochastic Schr¨odinger equa-
tion ever proposed [7]. Unfortunately, equation (9)
is quite formal unless it is possible to write the func-
tional derivative
1
as a simple local operator acting on
|ψ
ξ
(t)i.
3 Main result
The core objective of this article is to show how the
solutions of (9) can nonetheless be computed in the
general case by reusing a stochastic unraveling trick,
i.e. by writing this time:
|ψ
ξ
(t)i = E
η
|ψ
ξ,η
(t)i
(10)
where η is an auxiliary set of classical Gaussian fields.
More precisely, we can try to write the evolution op-
erator of (8) in the following way:
ˆ
G
ξ
(t) = E
η
T exp
−i
Z
t
0
ds
ˆ
A
k
(s)[ξ
k
(s) + η
k
(s)]
.
(11)
Using again the Gaussian integration formula (7), one
sees that the new unraveling condition (10) is satisfied
provided η is a set of Gaussian fields with zero mean
and two-point correlation functions given by:
E
η
[η
i
(τ) η
j
(s)] = K
ij
(τ, s)
E
η
[η
i
(τ) η
∗
j
(s)] = J
ij
(τ, s)
(12)
1
The reader puzzled by the rigorous definition of the func-
tional derivative can as well use (8) in all situations.
with:
K
ij
(τ, s) = θ
τ s
[D − S]
ij
(τ, s) + θ
sτ
[D − S]
ji
(s, τ)
(13)
and where the other correlation function J can be
chosen freely provided the total kernel Γ:
Γ =
J K
K
∗
J
∗
(14)
is positive semi-definite
2
.
Equation (11) can be written in an explicit linear
differential form free of functional derivatives:
d
dt
|ψ
ξ,η
(t)i = −i
ˆ
A
k
(t)[ξ
k
(t) + η
k
(t)] |ψ
ξ,η
(t)i. (15)
This latter equation, together with its unraveling in-
terpretation (10) is the main result of this article. The
evolution given by equation (15) is time-local in the
sense that once the random fields ξ and η are fixed,
the state can be evolved without reference to the past.
Although our interest was primarily in the stochastic
pure state |ψ
ξ
i itself, we can write the open-system
density matrix as a double average over our successive
unravelings (4) and (10):
ρ(t) = E
ξ
h
E
η
|ψ
ξ,η
(t)i
E
η
hψ
ξ,η
(t)|
i
. (16)
This can be written equivalently:
ρ(t) = E
ξ,η
(1)
,η
(2)
|ψ
ξ,η
(1)
(t)ihψ
ξ,η
(2)
(t)|
, (17)
where η
(1)
and η
(2)
are independent. This is the two
state unraveling proposal of Stockburger and Grabert
[28, 29]. We have thus found a connection between
the standard stochastic pure state representations and
the more exotic two state vector method used in nu-
merical approaches to open-systems. This allows us
to identify at least part of the freedom in the noise
present in this latter method as coming from the ker-
nel S which encodes different stochastic Schr¨odinger
evolutions corresponding to the same open-system
evolution. We can further risk the following heuristic
interpretation of the two noises. The noise ξ is classi-
cal in the sense that it is averaged over at the density
matrix level. The noise η, on the other hand, is quan-
tum or coherent as all the possible contributions are
summed over coherently at the pure state level.
4 Norm preserving unravelings
4.1 Context and objectives
The linear stochastic differential equation (9) does not
preserve the norm of the state vector |ψ
ξ
i. Normal-
ized pure states are usually preferred in foundations
2
For a kernel K given by equation (13), there always exists
a kernel J such that Γ is positive semi-definite. The existence
of a random field η verifying (12) is then a consequence of the
Bochner-Minlos theorem.
Accepted in Quantum 2017-09-16, click title to verify 3
but also for numerics: the norm of |ψ
ξ
i typically dif-
fuses and for large times, most trajectories give a van-
ishing contribution to the average (4) while nearly all
the weight is provided by rare events.
To obtain a norm preserving evolution from the lin-
ear one, one needs to define a normalized state |
e
ψ
ξ
i:
|
e
ψ
ξ
(t)i =
|ψ
ξ
(t)i
p
hψ
ξ
(t)|ψ
ξ
(t)i
(18)
and also to introduce a new “cooked” probability mea-
sure:
dµ
t
(ξ) = hψ
ξ
(t)|ψ
ξ
(t)idµ
◦
(ξ) (19)
which insures that the unraveling condition ρ(t) =
E
t
ξ
h
|
e
ψ
ξ
(t)ih
e
ψ
ξ
(t)|
i
–where E
t
ξ
is the expectation value
taken with the new measure µ
t
– is still valid. It
is important to note at that stage that the state
|
e
ψ
ξ
(t)i with the probability measure dµ
t
(ξ) unravels
the open-system evolution only for a single instant
t. If one wants a trajectory that unravels the open-
system evolution at all times, it is necessary to dy-
namically redefine the field variable ξ. It can be done
by introducing ξ
[t]
, a complete random field trajectory
depending on ξ such that for any functional f:
E
t
ξ
[f(ξ)] = E
ξ
h
f
ξ
[t]
(ξ)
i
. (20)
This way, starting from a field realization ξ drawn
from the Gaussian measure dµ
◦
, one can compute a
trajectory t 7→ |
e
ψ
ξ
[t]
(t)i that unravels the open evo-
lution for all times. By “computing the norm pre-
serving unraveling” one may thus want to sample two
different things:
1. the state |
e
ψ
ξ
[t]
(t)i for a single time,
2. the full trajectory t 7→ |
e
ψ
ξ
[t]
(t)i.
While knowing the state at a single time may be
enough in most cases, it is not sufficient to compute
correlation functions of the stochastic states at differ-
ent times (something one might want to do for col-
lapse models in foundations). In any case, the ideal
solution would be to find a way to write an explicit
stochastic differential equation sampling the states of
interest directly. This is unfortunately not a trivial
endeavour and the method we shall propose is ar-
guably less appealing than in the linear case.
4.2 Computation of the full trajectory for S=0
When S = 0, there exists an efficient way to construct
the full norm preserving trajectory t 7→ |
e
ψ
ξ
[t]
(t)i, ex-
ploiting the fact that ξ and ξ
[t]
, defined implicitly in
(20), can be connected explicitly in that case. Indeed,
after relatively painful computations (see [19, 33, 35])
one can show that for S = 0:
ξ
[t]
k
(u) = ξ
k
(u) + i
Z
t
0
ds D
k`
(u, s)hA
`
(s)i
s
, (21)
Figure 1: Schematic representation of the procedure de-
scribed in 4.2. The states and fields are computed from
bottom to top for increasing values of v. The final product,
the norm-preserving trajectory, is obtained for u = v.
with hA
`
(s)i
s
= h
e
ψ
ξ
[s]
(s)|A
`
(s)|
e
ψ
ξ
[s]
(s)i. Using this
result, it is possible to compute |
e
ψ
ξ
[t]
(t)i inductively.
We start from ξ
[0]
= ξ sampled from the Gaussian
measure dµ
◦
. At a time v ≤ t we assume that:
– ∀u ≤ v, |
e
ψ
ξ
[u]
(u)i is known,
– ∀u ≤ t, ξ
[v]
u
is known.
We first compute ξ
[v+dv]
for u ≤ t using (21):
ξ
[v+dv]
k
(u) = ξ
[v]
k
(u) + iD
k,`
(u, v)hA
`
(v)i
v
dv + O(dv
2
),
(22)
We then compute |
e
ψ
ξ
[v+dv]
(v + dv)i solving the linear
equation (9) up to time v + dv with the new complete
noise trajectory ξ
[v+dv]
and normalizing the result.
This is done using the unraveling (15). By induction,
we thus obtain |
e
ψ
ξ
[t]
(t)i for all t starting from a sin-
gle realization of the noise ξ (see figure 1). For that
matter, it is necessary to solve the linear stochastic
Schr¨odinger equation from the beginning with a new
field at every time step. Hence this adds a linear a
overhead ∝ t/dt to the time needed to compute lin-
ear trajectories. As the errors coming from the dis-
cretization are typically much smaller than the statis-
tical errors coming from the averaging of (15), there
is however no loss in precision.
4.3 General case, single time
In the general case, the previous method cannot be
used as we do not know of a generalization of formula
(20) for S 6= 0. Finding a way to sample from norm-
preserving trajectories has eluded us. In most cases
however, one is only interested in the properties of the
stochastic state at a single time. In such a situation,
the linear unraveling is sufficient to compute every-
thing. Indeed, let us consider an arbitrary function ϕ
of the norm-preserving trajectory |
e
ψ
ξ
[t]
(t)i at a fixed
Accepted in Quantum 2017-09-16, click title to verify 4
0 1 2 3 4
t
0.00
0.05
0.10
0.15
0.20
0.25
hψ
ξ
(t)|
ˆ
A|ψ
ξ
(t)i
approximation with 10
3
terms
exact
0 1 2 3 4
t
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
h
˜
ψ
ξ
[t]
(t)|
ˆ
A|
˜
ψ
ξ
[t]
(t)i
approximation with 10
3
terms
exact
Figure 2: Left: Snapshot of a linear trajectory |ψ
ξ
i for a random realization of ξ. The exact trajectory is obtained from
equation (26) while the approximate trajectory is obtained from (15) –integrated with dt = 10
−2
–, and the estimation
|ψ
ξ
i '
1
N
P
N
n=1
|ψ
ξ,η
i with N = 10
3
independent realizations of η and the same fixed ξ. Right: Snapshot of a non-linear
trajectory |
e
ψ
ξ
[t]
(t)i for the same realization ξ, and computed with the method of 4.2. In the algorithm, the linear NMSSE is
solved by integrating (15) with dt = 10
−2
and averaging over N = 10
3
independent realizations of η.
The shaded areas indicate the zones where 50% (' 0.5σ) and 85% (' 1σ) of the approximate trajectories would lie (obtained
by computing 100 samples for the same ξ). In both cases, number of realizations is voluntarily left small to make the estimate
of the error visible as the latter decreases ∝ N
−1/2
.
time t. We have:
E
ϕ(|
e
ψ
ξ
[t]
(t)i)
= E
t
ξ
ϕ(|
e
ψ
ξ
(t)i)
(23)
= E
t
ξ
"
ϕ
|ψ
ξ
(t)i
p
hψ
ξ
(t)|ψ
ξ
(t)i
!#
(24)
= E
ξ
"
ϕ
|ψ
ξ
(t)i
p
hψ
ξ
(t)|ψ
ξ
(t)i
!
hψ
ξ
(t)|ψ
ξ
(t)i
#
. (25)
That is, expectation values of all time-local function-
als of the norm preserving trajectory can be computed
by averaging over the solutions of the linear NMSSE.
5 Example
5.1 Linear evolution
We now illustrate our method on an example where
the stochastic Schr¨odinger equation is fully explicit.
For that matter, we consider a single Hermitian oper-
ator
ˆ
A commuting with H
0
(s.t.
ˆ
A(t) =
ˆ
A) and purely
isotropic noise, i.e. S = 0, which gives the following
stochastic Schr¨odinger equation:
d
dt
|ψ
ξ
(t)i = −i
ˆ
A
ξ(t) +
Z
t
0
ds D(t, s)
δ
δξ(s)
|ψ
ξ
(t)i.
(26)
For a given realization of ξ, |ψ
ξ
i obeys an explicit
differential equation. Indeed, using the integral form
(8), we can replace the functional derivative:
δ
δξ(s)
|ψ
ξ
(t)i = −i
ˆ
A|ψ
ξ
(t)i. (27)
This example thus offers a nice test bed for our
stochastic approach. In this example, the relax-
ation timescale responsible for the non-Markovianity
and the timescale of the system-bath coupling are
both of order 1 and we would thus be deep in the
non-Markovian regime had the system Hamiltonian
not been trivial. To compute the linear trajecto-
ries, we average over the solutions of (15) with η
a real noise such that E[η
t
η
s
] = D(t, s). Numeri-
cal results for
ˆ
A = diag(1, 0, −1), D(t, 0) = e
−t
, and
|ψ(0)i = (1, 1, 1)/
√
3 are shown in figure 2.
Our method is efficient for evolution times of the
order of the the system-bath coupling timescale but
becomes expensive for larger times. Indeed, we are
summing states |ψ
ξ,η
(t)i with a phase with respect to
|ψ
ξ
(t)i that is more and more random as time grows.
This is an important limit of this scheme for large
time.
5.2 Norm preserving evolution
We also sample the norm-preserving non-linear trajec-
tories using the method of section 4.2. The precision
for the sampling of a single trajectory |ψ
ξ
[t]
(t)i is very
similar to that of the linear case (see figure 2).
To test the robustness of the method, we also com-
pute the probability distribution of hψ
ξ
[t]
(t)|
ˆ
A|ψ
ξ
[t]
(t)i
for two different times using norm-preserving trajec-
tories computed with the numerical method of 4.2 and
using exact linear trajectories and the re-weighting of
4.3. Up to statistical and discretization errors, the
second method is exact. The results are displayed in
figure 3 and show an excellent agreement between the
Accepted in Quantum 2017-09-16, click title to verify 5
Figure 3: Top: Snapshot of 80 normalized trajectories com-
puted with the method of 4.2. In the algorithm, the linear
equation is solved by integrating (15) with dt = 10
−2
and
averaging over N = 10
3
independent realizations of η.
Bottom: Empirical probability densities; green histogram
computed by sampling 10
3
non-linear trajectories (each com-
puted with the same method as in the snapshot), brown line
computed by sampling 10
5
exact linear trajectories and re-
weighting a posteriori as in 4.3
two methods.
6 Conclusion
In this article, we have introduced a rewriting (or “un-
raveling”) of the solutions of formal NMSSE as aver-
ages over the solutions of explicit time-local stochas-
tic differential equations. This new formulation may
provide a basis for further analytical studies, notably
as it gives an extremely simple new way to define
NMSSE. More importantly, it immediately offers a
method to compute numerically their solutions. Al-
though we have proposed a heuristic interpretation
of the quantum and classical character of the two
noises involved, the auxiliary state |ψ
ξ,η
i does not yet
have a clear operational characterization. The latter
would certainly help understand the freedom in the
noise kernel J. We have also extended our method to
non-linear norm-preserving NMSSE in the case where
S = 0, i.e. where the complex noise ξ is isotropic.
Computing expectation values of time non-local func-
tionals of norm-preserving trajectories when S 6= 0 is
still an open problem. Our work has been focused on
the NMSSE as an end in themselves rather than on
the open-system evolution they unravel. It is how-
ever possible that our methods, especially the one to
sample non-linear NMSSE directly, might ultimately
help improve our ability to deal with non-Markovian
open-systems numerically.
Acknowledgments
I thank Lajos Di´osi and Luca Ferialdi for helpful dis-
cussions. Part of this work was supported by the
Alexander von Humboldt foundation and the Agence
Nationale de la Recherche (ANR) contract ANR-14-
CE25-0003-01.
References
[1] S. L. Adler and A. Bassi. Collapse models with
non-white noises. Journal of Physics A: Math-
ematical and Theoretical, 40(50):15083, 2007.
DOI: 10.1088/1751-8113/40/50/012.
[2] A. Bassi and G. Ghirardi. Dynamical reduc-
tion models. Physics Reports, 379(56):257 – 426,
2003. DOI: 10.1016/S0370-1573(03)00103-0.
[3] A. Bassi, K. Lochan, S. Satin, T. P. Singh,
and H. Ulbricht. Models of wave-function col-
lapse, underlying theories, and experimental
tests. Rev. Mod. Phys., 85:471–527, Apr 2013.
DOI: 10.1103/RevModPhys.85.471.
[4] J. Dalibard, Y. Castin, and K. Mølmer. Wave-
function approach to dissipative processes in
quantum optics. Phys. Rev. Lett., 68:580–583,
Feb 1992. DOI: 10.1103/PhysRevLett.68.580.
[5] I. de Vega, D. Alonso, and P. Gaspard. Two-level
system immersed in a photonic band-gap ma-
terial: A non-markovian stochastic schr¨odinger-
equation approach. Phys. Rev. A, 71:023812, Feb
2005. DOI: 10.1103/PhysRevA.71.023812.
[6] L. Di´osi. Models for universal reduction of macro-
scopic quantum fluctuations. Phys. Rev. A,
40:1165–1174, Aug 1989. DOI: 10.1103/Phys-
RevA.40.1165.
[7] L. Di´osi and L. Ferialdi. General non-
markovian structure of gaussian master and
stochastic schr¨odinger equations. Phys. Rev.
Lett., 113:200403, Nov 2014. DOI: 10.1103/Phys-
RevLett.113.200403.
[8] L. Di´osi. Stochastic pure state representation for
open quantum systems. Phys. Lett. A, 114(8):451
– 454, 1986. DOI: 10.1016/0375-9601(86)90692-
4.
[9] L. Di´osi. Non-markovian continuous quantum
measurement of retarded observables. Phys. Rev.
Lett., 100:080401, Feb 2008. DOI: 10.1103/Phys-
RevLett.100.080401.
[10] L. Di´osi. Erratum: Non-markovian continuous
quantum measurement of retarded observables
[phys. rev. lett. 100 , 080401 (2008)]. Phys. Rev.
Lett., 101:149902, Oct 2008. DOI: 10.1103/Phys-
RevLett.101.149902.
[11] L. Di´osi and W. T. Strunz. The non-markovian
stochastic schr¨odinger equation for open sys-
tems. Phys. Lett. A, 235(6):569–573, 1997. DOI:
10.1016/S0375-9601(97)00717-2.
Accepted in Quantum 2017-09-16, click title to verify 6
[12] R. Dum, P. Zoller, and H. Ritsch. Monte
carlo simulation of the atomic master equa-
tion for spontaneous emission. Phys. Rev. A,
45:4879–4887, Apr 1992. DOI: 10.1103/Phys-
RevA.45.4879.
[13] L. Ferialdi and A. Bassi. Dissipative col-
lapse models with nonwhite noises. Phys. Rev.
A, 86:022108, Aug 2012. DOI: 10.1103/Phys-
RevA.86.022108.
[14] R. P. Feynman and F. L. Vernon. The theory of a
general quantum system interacting with a linear
dissipative system. Annals of physics, 24:118–
173, 1963. DOI: 10.1016/0003-4916(63)90068-X.
[15] J. Gambetta and H. M. Wiseman. Non-
markovian stochastic schr¨odinger equations:
Generalization to real-valued noise using
quantum-measurement theory. Phys. Rev.
A, 66:012108, Jul 2002. DOI: 10.1103/Phys-
RevA.66.012108.
[16] J. Gambetta and H. M. Wiseman. Interpretation
of non-markovian stochastic schr¨odinger equa-
tions as a hidden-variable theory. Phys. Rev.
A, 68:062104, Dec 2003. DOI: 10.1103/Phys-
RevA.68.062104.
[17] G. C. Ghirardi, P. Pearle, and A. Rimini. Markov
processes in hilbert space and continuous spon-
taneous localization of systems of identical par-
ticles. Phys. Rev. A, 42:78–89, Jul 1990. DOI:
10.1103/PhysRevA.42.78.
[18] N. Gisin. Quantum measurements and stochastic
processes. Phys. Rev. Lett., 52:1657–1660, May
1984. DOI: 10.1103/PhysRevLett.52.1657.
[19] N. Gisin and I. C. Percival. The quantum-state
diffusion model applied to open systems. J.
Phys. A: Math. Gen., 25(21):5677, 1992. DOI:
10.1088/0305-4470/25/21/023.
[20] B. L. Hu, J. P. Paz, and Y. Zhang. Quan-
tum brownian motion in a general environment:
Exact master equation with nonlocal dissipation
and colored noise. Phys. Rev. D, 45:2843–2861,
Apr 1992. DOI: 10.1103/PhysRevD.45.2843.
[21] B. L. Hu, J. P. Paz, and Y. Zhang. Quantum
brownian motion in a general environment. ii.
nonlinear coupling and perturbative approach.
Phys. Rev. D, 47:1576–1594, Feb 1993. DOI:
10.1103/PhysRevD.47.1576.
[22] K. Jacobs and D. A. Steck. A straightfor-
ward introduction to continuous quantum mea-
surement. Contemporary Physics, 47(5):279–303,
2006. DOI: 10.1080/00107510601101934.
[23] J. Jing and T. Yu. Non-markovian relaxation
of a three-level system: Quantum trajectory ap-
proach. Phys. Rev. Lett., 105:240403, Dec 2010.
DOI: 10.1103/PhysRevLett.105.240403.
[24] J. Jing, X. Zhao, J. Q. You, and T. Yu. Time-
local quantum-state-diffusion equation for multi-
level quantum systems. Phys. Rev. A, 85:042106,
Apr 2012. DOI: 10.1103/PhysRevA.85.042106.
[25] J. Jing, X. Zhao, J. Q. You, W. T. Strunz,
and T. Yu. Many-body quantum trajectories
of non-markovian open systems. Phys. Rev.
A, 88:052122, Nov 2013. DOI: 10.1103/Phys-
RevA.88.052122.
[26] K. Mølmer, Y. Castin, and J. Dalibard. Monte
carlo wave-function method in quantum optics.
J. Opt. Soc. Am. B, 10(3):524–538, Mar 1993.
DOI: 10.1364/JOSAB.10.000524.
[27] P. Pearle. Combining stochastic dynamical state-
vector reduction with spontaneous localization.
Phys. Rev. A, 39:2277–2289, Mar 1989. DOI:
10.1103/PhysRevA.39.2277.
[28] J. T. Stockburger. Simulating spin-boson dynam-
ics with stochastic liouville–von neumann equa-
tions. Chemical physics, 296(2):159–169, 2004.
DOI: 10.1016/j.chemphys.2003.09.014.
[29] J. T. Stockburger and H. Grabert. Exact c-
number representation of non-markovian quan-
tum dissipation. Phys. Rev. Lett., 88:170407, Apr
2002. DOI: 10.1103/PhysRevLett.88.170407.
[30] W. T. Strunz. Linear quantum state diffu-
sion for non-markovian open quantum systems.
Phys. Lett. A, 224(1):25 – 30, 1996. DOI:
10.1016/S0375-9601(96)00805-5.
[31] W. T. Strunz and T. Yu. Convolutionless
non-markovian master equations and quantum
trajectories: Brownian motion. Phys. Rev.
A, 69:052115, May 2004. DOI: 10.1103/Phys-
RevA.69.052115.
[32] W. T. Strunz, L. Di´osi, N. Gisin, and T.
Yu. Quantum trajectories for brownian motion.
Phys. Rev. Lett., 83:4909–4913, Dec 1999. DOI:
10.1103/PhysRevLett.83.4909.
[33] A. Tilloy. Interacting quantum field theories as
relativistic statistical field theories of local be-
ables. arXiv:1702.06325, 2017.
[34] U. Weiss. Quantum dissipative systems, vol-
ume 10. World Scientific, 1999.
[35] H. M. Wiseman and G. J. Milburn. Quan-
tum theory of field-quadrature measurements.
Phys. Rev. A, 47:642–662, Jan 1993. DOI:
10.1103/PhysRevA.47.642.
[36] H. M. Wiseman and J. M. Gambetta. Pure-
state quantum trajectories for general non-
markovian systems do not exist. Phys. Rev.
Lett., 101:140401, Sep 2008. DOI: 10.1103/Phys-
RevLett.101.140401.
[37] X. Zhao, J. Jing, B. Corn, and T. Yu. Dynam-
ics of interacting qubits coupled to a common
bath: Non-markovian quantum-state-diffusion
approach. Phys. Rev. A, 84:032101, Sep 2011.
DOI: 10.1103/PhysRevA.84.032101.
Accepted in Quantum 2017-09-16, click title to verify 7