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PHYSICAL REVIEW LETTERS

Negative Full Counting Statistics Arise from Interference Effects
1

Patrick P. Hofer1,2 and A. A. Clerk1

Department of Physics, McGill University, Montreal, Quebec, Canada H3A 2T8
Département de Physique Théorique, Université de Genève, 1211 Genève, Switzerland(Received 15 October 2015; published 7 January 2016)

2

The Keldysh-ordered full counting statistics is a quasiprobability distribution describing the fluctuationsof a time-integrated quantum observable. While it is well known that this distribution can fail to be positive,
the interpretation and origin of this negativity has been somewhat unclear. Here, we show how the fullcounting statistics can be tied to trajectories through Hilbert space, and how this directly connects negative
quasiprobabilities to an unusual interference effect. Our findings are illustrated with the example ofenergy fluctuations in a driven bosonic resonator; we discuss how negative quasiprobability here could
be detected experimentally using superconducting microwave circuits.
DOI: 10.1103/PhysRevLett.116.013603

Introduction.—Quasiprobability distributions such as the
Wigner function [1] are powerful tools that allow one tovisualize quantum states in phase space. They have played
a seminal role in quantum mechanics since the early
beginnings of the theory. Among their many uses are theidentification of nonclassical states: these are states where
the Wigner function (or some other related distribution)
fails to be positive definite (see, e.g., Ref. [2]). Suchnonclassicality can constitute a resource for quantum
information processing [3,4].
Recently, a very different kind of quasiprobability
distribution has found widespread utility, the so-calledfull counting statistics (FCS) [5–7]. Unlike the Wigner
function, the FCS does not describe the instantaneous stateof a quantum system, but rather describes its time history
and dynamics: it characterizes the fluctuations of a timeintegrated quantum observable. As has been discussed
extensively, the FCS distribution describes the “intrinsic”
fluctuations of the system absent any coupling to ameasurement device [7,8]. Nonetheless, it can be used todirectly predict the outcome of realistic measurement
setups, where the added noise of the measurement combines with the intrinsic system fluctuations to determine thefinal measured distribution [7–10]. FCS first arose in the
study of current fluctuations in quantum electronic conductors, where the transmitted charge is the time integral of
the current operator [5,6,11]; it continues to be a crucial
tool in quantum transport, and has also been used tocharacterize cold atom systems [12], work [13] and heatfluctuations [14], dynamical phase transitions of classicalsystems [15,16], and quantum-optical systems [8]. FCS
have also recently been connected to weak measurement
theory [9,17].
Similar to conventional quasiprobability distributions,
the FCS distribution can fail to be positive definite. As
the FCS describes the time history of a system, negativity
here is indicative of the presence of nonclassical temporal
0031-9007=16=116(1)=013603(5)correlations and/or dynamics which render a backactionfree measurement impossible [8,9]. Largely because manyof the most studied systems are immune to backaction (e.g.,gauge invariant electronic transport at long times), and thus
described by a positive-definite FCS [7], very little work
has been undertaken on the meaning, origin, or utility ofnegative FCS; notable exceptions are Refs. [8–10,18–20].
Considering the utility of negativities in more conventional
quasiprobabilities, it is desirable to obtain a better understanding of negative FCS.
In this work we present a clear physical picture for how
negative FCS emerge. We connect the FCS distribution
to trajectories the system takes through Hilbert space. The
resulting expression gives an intuitive understanding of
the microscopic processes that contribute to the FCS andallows us to show that negative FCS are the direct resultof an unusual interference phenomena: the interference of
amplitudes associated with two trajectories can contributeto the quasiprobability, even though the classical probabilities for each trajectory do not contribute. Our approachalso demonstrates why negative FCS in general requires
systems where a few degrees of freedom are relatively
isolated. We stress that, in contrast with Refs. [7,21–23],
our main focus is to understand the negativity in the FCS,
and not on how the inclusion of detectors modifies the FCS
and restores positivity in the final measured distribution.
Nonetheless, our approach also gives an intuitive pictureof this process [see Supplemental Material [24]]. Our
approach is particularly well suited to investigating theshort-time FCS, a regime that is relevant to fast experimental protocols but that has received only limited
attention.
To make the utility of our approach clear, we focus on aparticularly simple system that exhibits negative FCS: thetime-integrated energy fluctuations in a coherently driven
bosonic single-mode resonator. The FCS here are particularly amenable to experimental measurement, and their013603-1© 2016 American Physical Society

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negativity was recently discussed as a potentially powerful
way to detect nonclassical behavior in an optomechanicalsystem [20]. Finally, we analyze a realistic circuit quantum
electrodynamics (cQED) measurement setup for detecting
negative FCS.
Definition of FCS.—We consider an observable n̂ðtÞ inthe Heisenberg picture, and are interested Rin characterizing
the fluctuations of its time integral m̂ ¼ 0t dt0 n̂ðt0 Þ. Sincen̂ðtÞ does not necessarily commute with itself at different
times, the higher moments of m̂ will be contingent on
how one chooses to time order the various factors of n̂.
The well-developed field of FCS resolves this ambiguity
by considering how one would measure m̂; guided bythis, the appropriate moment generating function for m is
[5–7] (ℏ ¼ 1)ZΛðλÞ ≡ dmPðmÞe−iλm ≡ Trfe−iĤλ t ρ̂eiĤ−λ t g; ð1Þ

Unravelling the FCS.—In the spirit of Feynman’s pathintegral approach, we now divide the time evolution in
Eq. (1) into N infinitesimal steps of duration δt; between
these partitions, we introduce resolved identity operators.
We start with the simplest case, where n̂ has a discrete
spectrum, and further, where our system
has no additionalP
quantum numbers, such that I ¼ n jnihnj is the identity
operator. Inserting the identities allows us to replace theoperator n̂ by its eigenvalues. The FCS can then be
obtained by Fourier transforming Eq. (1),


X
1
1
PðmÞ ¼
δnLf ;nRf δ m − mL − mR22n~ ;~nwhere PðmÞ is the quasiprobability distribution of interest(the FCS), ρ̂ is the system density matrix at t ¼ 0, andĤλ ¼ Ĥþλn̂=2, with Ĥ being the Hamiltonian of the system.
A simple way to motivate Eq. (1) is to consider an
idealized measurement where an auxiliary qubit couples
to n̂ via Ĥ c ¼ λn̂σ̂ z =2 [5]. If n̂ were a classical stochastic
variableR nðtÞ, the qubit would precess by an angle
λm ¼ λ 0t dt0 nðt0 Þ, and the off-diagonal reduced density
matrix element would directly yield the average ofexpð−iλmÞ, i.e., the moment generating function. This thenmotivates Eq. (1) in the quantum case. This is only oneof several idealized measurement schemes that lead toEq. (1) [7,8,20]. Equation (1) can also be motivated by the
Keldysh path-integral approach [25]. The time orderingof n̂ðtÞ, which ultimately leads to the negativities in the
FCS, is thus dictated by the fact that the FCS is ameasurement-independent quantity.

Here the nαj denote the states inserted at the jth time slice
either on the left (α ¼ L) or on the right side of the density
matrix (α ¼ R) in Eq. (1). The quantity Að~nα Þ gives theamplitude for a trajectory through Hilbert space, definedby the vector n~ α ¼ ðnα1 ; …; nαN ; nαf Þ. The time integral of theobservableP n̂ over such a discrete trajectory is given bymα ¼ j nαj δt. Examples of such trajectories are illustrated
in Figs. 1(a) and 1(b). Finally, as we are interested in theδt → 0 limit, we neglect terms that are order ðδtÞ2 and higher.
Each term in Eq. (2) describes the contribution to PðmÞfrom a pair of trajectories n~ L and n~ R ; the second line is the
product of probability amplitudes for each of the trajectories, weighted by the density matrix element corresponding
to the initial “position” of each trajectory. The trajectories
are summed over, given the constraints on the first line. The
Kronecker delta enforces the two trajectories to end at theLR× hnL1 jρ̂jnR1 iAð~nL ÞA ð~nR Þ;ð2Þwith the amplitudes
Að~nα Þ ¼ hnαf je−iĤδt jnαN i    hnα2 je−iĤδt jnα1 i:ð3ÞFIG. 1. FCS for a bosonic resonator and contributing pairs of trajectories. (a) Illustration of a pair of trajectories contributing to Pðm0 Þ,
where the m value of each trajectory is the same: mL ¼ mR ¼ m0 . Such pairs yield a positive contribution. (b) Illustration of a pair oftrajectories with mL ≠ mR , but ðmL þ mR Þ=2 ¼ m0 . As discussed in the text, such a pair can yield a negative contribution to Pðm0 Þ.
(c) FCS for a cavity initially prepared in the n0 ¼ 2 Fock state. The analytical result (black, solid line) consists of a contribution where
the jumps are located on different trajectories (blue, dotted line) and a contribution where the jumps are located on the same trajectory(red, dashed line). The singular, zero-jump contribution of Eq. (5) is omitted. A Monte Carlo simulation (gray line) using 50 000
trajectories is in good agreement with the analytical results. Parameters are time t ¼ 4=Δ, drive strength f ¼ Δ=16, where Δ is the drive
detuning.

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same position and is a consequence of the trace in Eq. (1).
The Dirac delta tells us that a pair of trajectories contributes
to PðmÞ when m is equal to the average of mL and mR .
While Eq. (2) is just a direct representation of thestandard FCS PðmÞ distribution, we immediately noticea rather strange feature: for a given particular value m0 , theinterference terms between two trajectories can contributeto Pðm0 Þ even though the corresponding classical probabilities do not. To be explicit, suppose we have a pair oftrajectories with m values mL and mR . The classicalprobability from each trajectory, i.e., the terms proportional
to jAð~nL Þj2 and jAð~nR Þj2 , contribute to PðmL Þ and PðmR Þ,
respectively. Their interference terms, i.e., the terms proportional to Að~nL ÞA ð~nR Þ and Að~nR ÞA ð~nL Þ, contribute
instead to PðmL =2 þ mR =2Þ. If mL ≠ mR , the interferenceterms are thus separated from their classical probabilities,
allowing the quasiprobability distribution PðmÞ to become
negative. We thus have one of the key conclusions of ourapproach: negativity in the distribution PðmÞ is directly
and necessarily connected to a kind of anomalously strong
influence of interferences between pairs of trajectories.
This motivates us to separate the contributions to thesum in Eq. (2) into two generic kinds. Terms with mL ¼ mR
are denoted “classical” contributions. These yield a total
contribution to PðmÞ that is positive definite. Terms withmL ≠ mR are denoted “interference” contributions. These
are the interference terms that are separated from theirclassical probabilities and responsible for any negativities
in the FCS. Examples of pairs of trajectories yieldingclassical and interference contributions to PðmÞ are illustrated in Figs. 1(a) and 1(b).
Driven cavity.—We now illustrate our trajectory
approach to FCS by considering a coherently driven
bosonic single-mode resonator, first in the absence ofany dissipation. This constitutes a simple system that is
amenable to cQED [26–28] and optomechanical [29,30]
experiments. In the frame rotating at the driving frequency,
the Hamiltonian of the system readsĤ ¼ Δâ† â − fðâ† þ âÞ;

ð4Þ

where Δ denotes the detuning of the drive and f the drive
strength (which we take to be real without loss of generality). We are interested in the photon number fluctuations,n̂ ¼ â† â. Despite the seemingly trivial nature of the system
and its linear dynamics, we are measuring a nonlinear
observable, and the integrated-energy fluctuations aredescribed by negative FCS [8].
The coherent drive can induce jumps in the trajectories(i.e., from one Fock state to another), whereas the detuning
introduces a phase factor in Að~nL ÞA ð~nR Þ whenevermL ≠ mR . For ft ≪ 1, only pairs of trajectories with alow number of jumps will contribute to the FCS, and we canmake some analytical progress. To this end, we consider an
initial Fock state ρ̂ ¼ jn0 ihn0 j and pairs of trajectories
including a total of up to two jumps. The contribution frompairs exhibiting no jumps at all is given by

P0 ðmÞ ¼ δðm − n0 tÞ:

ð5Þ

The zero jump contribution thus reflects the initial distribution and does not decay with time. To ensure thenormalization of PðmÞ, all contributions with a higher
number of jumps must thus average to zero: this ensuresnegativities in the FCS as long as the dynamics of the system
is nontrivial. These considerations remain valid for an
arbitrary initial state.
For an initial Fock state, there is no contribution frompairs of trajectories exhibiting a single jump in total becauseof the Kronecker delta in Eq. (2). The two jump contribution
is discussed in the Supplemental Material [24] and plotted inFig. 1(c) together with a Monte Carlo simulation of the FCS.
As illustrated in Fig. 1(c), the distribution PðmÞ shows ahighly nontrivial behavior and becomes negative over a
substantial range of its argument. The jump at m=t ¼ n0 as
well as the kinks at m=t ¼ n0  1=2 are a consequence ofthe discreteness of photon numbers and can be well understood in terms of the few trajectories that contribute at shorttimes (see Supplemental Material [24]).
We stress that these unusual short-time features also
occur for different choices of initial states, including a
coherent state; in that case, our calculations agree withthe approach used in Refs. [8,31] (see also Fig. 2). The
presence of negative FCS is thus not a function of the initialstate, but rather reflects the nonclassicality of the system
dynamics; this is in stark contrast to the Wigner function(where coherent states exhibit no negativity). We thus
conclude that even systems that remain in a seemingly nearclassical state at all times can exhibit extremely nonclassical behavior in their dynamics.
Additional degrees of freedom.—Equation (2) (and thesingle resonator example) discussed so far are somewhat
special cases, in that the relevant dynamics only involves asingle degree of freedom. As we now show, if the dynamics
starts to couple to additional degrees of freedom, negativity
can be rapidly lost, as there is a strong suppression of therequired “interference” contributions.
Consider first the situation where the additional degrees of
freedom correspond to a dissipative environment; forconcreteness, we return to our example of a driven resonator,
and add a coupling to a Markovian bath. In sucha situation, the contribution of the bath to the dynamics
can be modeled in terms of dissipative quantum jumps, incomplete analogy to how they are treated in the standard
quantum trajectory approach of quantum optics [32].
These dissipation-induced jumps are described by thesuperoperatorsJ ↓ ρ̂ ¼ κðnB þ 1Þâ ρ̂ â† ;J ↑ ρ̂ ¼ κnB â† ρ̂ â;

ð6Þ

where κ is the energy damping rate and nB is the thermal
occupation number of the bath at the cavity frequency. The
first (second) term describes photons that are lost to (gainedfrom) the bath.
We can again incorporate these jump operators into apath-integral expression for the FCS distribution function

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FIG. 2. Time evolution of the integrated-energy FCS for a damped
pﬃﬃﬃ cavity initially prepared in a coherent state. The blue curve shows
the FCS in the presence of a coherent drive with strength f=κ ¼ 5=2, where κ is the energy damping rate. The green curve shows the
FCS describing the dissipative emptying of the cavity (f ¼ 0). The resulting distribution is fully positive and can be described with aclassical model. (a) At very short times, the FCS is dominated by peaks at integer m=t reflecting the initial photon distribution.
(b) The exchange of photons with the coherent drive and the dissipative bath leads to features in between the peaks which can beunderstood in terms of the few-jump trajectories. An experimental reconstruction of the FCS (red, dash-dotted line) using an auxiliary
qubit detector is feasible using 500 measurements covering a range of λ up to λmax ¼ 418.9κ (see main text). (c) At long times, the
FCS is a continuous function peaked around the mean photon number in the cavity. Except for the reconstructed FCS, all distributions
are convolved with a sharply peaked Gaussian (width σ ¼ tκ=10) to resolve the Dirac deltas. For all panels, the drive is onresonance Δ ¼ 0.

PðmÞ; see Supplemental Material [24]. Similar to standard
quantum trajectory theory, the dissipation correlates the
behavior of the left and right trajectories, thus suppressing
the negativity-induced interference contributions (which
require distinct trajectories on the left and on the right). Fora purely dissipative process, the left and the right trajectories are always identical and the FCS always positive, beinga simple sum of classical probabilities. In this case, the FCS
recovers the results obtained by classical master equations(see Fig. 2). Details on the dissipative FCS calculation areprovided in the Supplemental Material [24] along with a
discussion on how coupling coherently to an additional
degree of freedom also suppresses negativity.
Time evolution of the FCS.—To stress the utility of
our approach we consider the time evolution of the FCS
in an experimentally relevant system. To this end, we add
dissipation to our driven cavity system resulting in the
Lindblad master equationdρ̂κ¼ −i½Ĥ; ρ̂ − fâ† â; ρ̂g þ κ â ρ̂ â† ;
dt
2

ð7Þ

where Ĥ is given in Eq. (4). As an initial state, we take thesteady-state solution which is given by a coherent state withan average photon number, nD ¼ 4f 2 =ðκ2 þ 4Δ2 Þ. Since inthis case the Wigner function is Gaussian at all times, we
use the method described in detail in Refs. [8,31] to
calculate the moment generating function. The time evolution of the resulting FCS is illustrated in Fig. 2. At very
short times [cf. Fig. 2(a)], the FCS is dominated by sharp
peaks at integer m=t corresponding to trajectories where
the photon number remains constant. At large times[cf. Fig. 2(c)], the FCS is a smooth function centeredaround the mean photon number nD. At times where thetrajectories with few jumps dominate [cf. Fig. 2(b)], the
FCS exhibits features in between the peaks at integer m=t.
For a purely dissipative process, the FCS is continuous

in between the peaks and can be captured by a classicalcalculation involving only occupation probabilities. In thepresence of a coherent drive, the FCS exhibits a surprisingshape with discontinuities at half-integer m=t. In complete
analogy to Fig. 1(c), this can be well understood in termsof the few-jump trajectories and ultimately results fromthe discreteness of the number of photons. The jumps at
half-integer m=t are a consequence of the coherences in theinitial state. Our approach thus allows for a quantitative
understanding of the nontrivial short-time FCS.
Reconstructing the FCS.—As discussed in detail in the
Supplemental Material [24], measurement noise (uncertainty and backaction) will often mask the sharp featuresthat are characteristic for the short-time regime. Motivatedby the exceptional quality of cQED experiments [26–28],
we thus dedicate the remainder of this Letter to thereconstruction of the FCS by coupling a qubit dispersively
to the observable of interest. As discussed above, the(unperturbed) moment generating function Eq. (1) canbe accessed through the off-diagonal density matrix
element of a qubit which couples to the observable ofinterest with the coupling Hamiltonian, Ĥc ¼ λn̂σ̂ z =2.
Since the FCS is given by the Fourier transform of the
moment generating function, the latter would have to bemeasured for all possible values of the coupling strength λin order to faithfully reconstruct the FCS. Here we areinterested in how well this reconstruction performs if themeasurements are limited in number and the couplingstrength cannot exceed a maximal value.
As shown in Fig. 2(b), a maximal coupling strength of
λmax ≈ 420κ with 500 equally spaced measurement points
is sufficient to reconstruct most features of the FCS.
As discussed in the Supplemental Material [24], thisprocedure is robust against uncertainties in the couplingstrength up to a magnitude of ∼κ=2. However, some care

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has to be taken in the choice of λmax and the postprocessingof the measured values.
Conclusions.—By unraveling the FCS in terms of trajectories through Hilbert space, we demonstrated thatnegative FCS arise from a peculiar interference effect, where
the interference contribution from a pair of trajectories cancontribute without the corresponding classical probabilities.
Our approach highlights how negative FCS are directly tied
to nonclassical dynamics, in contrast to standard quasiprobabilities which characterize nonclassical states. We hopethat the understanding of negative FCS presented here will
inspire further work on nonclassical dynamical processes, aswell as experiments to measure these effects.
We acknowledge useful comments from D. Dasenbrook,
Y. Nazarov, P. Solinas, and S. Gasparinetti. This work was
supported by NSERC. P. P. H. acknowledges funding fromthe Swiss NSF.

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Supplement: Negative Full Counting Statistics Arise From Interference Effects
Throughout the supplement, equations, figures, and citations without the prefix ’S’ refer to the main text.

I. ADDITIONAL DEGREES OF FREEDOMIn this section, we show how additional degrees of freedom which couple to the operator of interest suppress theinterference terms and thus reduce the negativity in the FCS.

A. One additional bosonic mode

We first investigate the effect of one additional bosonic mode (described by the operator âφ ) on the FCS of thecounted bosonic mode (described by the operator ân ). We split the Hamiltonian in three parts,
Ĥ = Ĥn + Ĥφ + Ĥnφ ,(S1)

where the first two terms only act on the counted and the additional mode respectively. Similarly to the derivationof Eq. (2), we now insert identities into the definition of the moment generating function [cf. Eq. (1)]. Unlike thecounted mode, we resolve the additional mode in terms coherent states
I=

X

Z|nihn| ⊗|φ|2|φi = e− 2dφ̄dφ|φihφ|,

n

∞
X
φn
√ |nφ i.
n!
n=0

(S2)

We find that Eq. (2) has to be replaced with a similar expression which includes an average over the additional mode
D

E
X
1
1
L
R
L
R
L
L
R
R
~ L )A∗ (~nR , φ~ R ) , (S3)m−m
Pφ (m) =δ(φδnLf ,nR
−φ
)δ(
φ̄
−
φ̄
)hn
|ρ̂(
φ̄
,
φ
)|n
iA(~
n
,
φ
δ
m
−
L
R
L
f
f
f
f
1
1
1
1
f
2
2
φ
~
nL ,~
nR

with the amplitudes
α

α

α

α

~ α ) = hnα |e−iĤn δt e−iĤnφ (φ̄f ,φN )δt |nα i · · · hnα |e−iĤn δt e−iĤnφ (φ̄2 ,φ1 )δt |nα i.
A(~nα , φ
f
N
2
1

(S4)

Here O(φ̄, φ0 ) = hφ|Ô|φ0 i and the average over the fields readsD(· · · )

E

=

φ

Z 2N
+2
Y

dφ̄j dφj eiG[φ̄,φ] (· · · ).

(S5)

j=1

The index j is an index which follows the Keldysh contour
L
L
R
R
R
φ 1 = φL
1 , · · · , φN = φN , φN +1 = φf , φN +2 = φN , · · · , φ2N +1 = φ1 , φ2N +2 = φf ,(S6)and the (discrete) Greens function for the additional cavity fields reads
2N
+1
X



φj − φj−1− Hφ (φ̄j , φj−1 ) + i|φ1 |2 ,G[φ̄, φ] =
δtj iφ̄j
δt
j
j=2

(S7)

where δtj≤N +1 = δt and δtj>N +1 = −δt.
To illustrate the effect of the additional mode, we discuss two examples for the coupling Hamiltonian. For simplicity,
we assume that the additional cavity has no internal dynamics and is initially in a coherent stateρ̂ = ρ̂n ⊗ |φ0 ihφ0 |.

Ĥφ = 0,(S8)

For the optomechanical coupling HamiltonianĤnφ = χn̂ â†φ + âφ ,

(S9)

2
we findPφ (m) =

X
~
nL ,~
nR




2
1
1
−iχ(φ̄0 +φ0 )(mL −mR ) − χ2 (mL −mR )2Rδm−δnLf ,nR
m
−
m
e
e
hnL
nL )A∗ (~nR ),LR
1 |ρ̂|n1 iA(~
f
2
2

(S10)

where A(~nα ) is given in Eq. (3). The first exponential is the average detuning induced by the additional bosonicmode while the second exponential decreases the weight of the non-classical trajectories due to the uncertainty in the
induced detuning. This reduction is analogous to the backaction of a heavy mass detector [cf. Eq. (S22)]. In bothcases, the detuning is described by a Gaussian random variable which remains constant in time due to the lack ofdynamics of the additional degree of freedom coupling to n̂.
For the coupling Hamiltonian (i.e. a cross-Kerr interaction)Ĥnφ = χn̂â†φ âφ ,(S11)

we find




h
i
1
1
2
−iχ(mL −mR )
R
δ
m
−
e
m
−
m
exp
|φ
|
−
1
hnL
nL )A∗ (~nR )δnLf ,nRPφ (m) =LR
01 |ρ̂|n1 iA(~
f
2
2
~
nL ,~
nR


∞
X
X
1
1
|φ0 |2p −ipχ(mL −mR ) L−|φ0 |2δm−=δnLf ,nRm−meehn1 |ρ̂|nR
nL )A∗ (~nR ).
L
R
1 iA(~
f
2
2
p!
p=0
X

(S12)

~
nL ,~
nR

Again, the additional mode has the effect of an induced, random detuning. Here the different number states which
contribute to the coherent state |φ0 i lead to different detuning strengths.
Analogous to the above discussion, we would expect a coupling Hamiltonian of the form Ĥnφ = χ(â†n âφ + â†φ ân ) tohave the same effect as a fluctuating driving force. Additional degrees of freedom can thus introduce uncertainties
in different parameters. The distribution of these parameters depends on the initial state and the dynamics of theadditional degrees of freedom.
B. FCS for a time evolution governed by a Lindblad master equation

We now consider the FCS in the presence of a Markovian bath. The time evolution is governed by the masterequationdρ̂= −i[Ĥ, ρ̂] − κ (nB + 1/2) â† â, ρ̂ − κnB ρ̂ + κ (nB + 1) âρ̂â† + κnB â† ρ̂â = Lρ̂,
dt

(S13)

where nB denotes the occupation number of the bath at the cavity frequency. We now split the superoperator L
into two parts: a Hamiltonian evolution with a non-hermitian Hamiltonian and a dissipative jump superoperator
L = LH + J . These superoperators act on the density matrix as†LH ρ̂ = −i[Ĥ, ρ̂] − κ (nB + 1/2) â† â, ρ̂ − κnB ρ̂
⇔
eLH t ρ̂ = e−iĤt ρ̂eiĤ t ,(S14)J ρ̂ = κ (nB + 1) κâρ̂â† + κnB â† ρ̂â,
with the non-hermitian HamiltonianκĤ = Ĥ − iκ (nB + 1/2) â† â − i nB .
2

(S15)

We can then unravel the evolution of the master equation times ti [32]Lt

ρ̂(t) = e ρ̂(0) =

∞ Z
X
p=0 0

Ztp

t

Zt2
dtp−1 · · ·dtp
0

dt1 eLH (t−tp ) J eLH (tp −tp−1 ) · · · J eLH t1 ρ̂(0).

(S16)

0

This corresponds to a Hamiltonian evolution with a non-unitary Hamiltonian which is interrupted by p dissipative
quantum jumps occurring at times tj which consist of a photon entering or leaving the cavity. We can express the
Hamiltonian evolution in terms of trajectories as we did in the derivation of Eq. (2)X
1RR
eLH t ρ̂ =e−κ(nB + 2 )(mL +mR ) e−κnB t hnLnL )A∗ (~nR )|nL
(S17)
1 |ρ̂|n1 iA(~f ihnf |,
~
nL ,~
nR

3

FIG. S1. Effect of dissipation on the FCS for an initial Fock state n0 = 2.(a) Purely dissipative process (f = 0) for different
damping parameters κ. For small κ only one photon leaks out of the cavity. As κ increases, the probability of both photons
leaking out increases. (b) Combined effect of coherent drive and dissipation for κt = 0.25 f t = 0.25, ∆t = 5. As a comparison,
the FCS for κ = 0 is plotted in blue (dotted) and the FCS for f = 0 is plotted in red (dashed). Noisy lines are calculated usinga Monte Carlo simulation with 50, 000 trajectories, steady lines are analytical results (omitting singular contributions).

where again mα =

N
P
j=1

nα
j δt. The dissipation thus introduces dissipative jumps which happen on the left and on theright trajectory simultaneously and it reduces the weight of terms with a high m.
As discussed in the main text, the moment generating function is given by the off-diagonal element of the reduced
density matrix of a qubit that couples linearly to the operator n̂. Replacing Ĥ with Ĥ ± λn̂/2 on the left/righttrajectory, we can derive a quasi-probability analogously to Eq. (2)


X
1
1
1
(S18)
P (m) =
δnLf ,nR
δ m − mL − mR e−κ(nB + 2 )(mL +mR ) e−κnB t Tκ (~nL , ~nR ),
f
2
2
~
nL ,~
nR

with
Tκ (~nL , ~nR ) =

∞ Z
X
p=0 0

Ztp

t

Zt2
dtp−1 · · ·

dtp
0

P(~
nL ,~
nR ;t,tp )dt1 hnL
J eP(~nL ,~nR ;tp ,tp−1 ) · · · J eP(~nL ,~nR ;t1 ,0) ρ̂(0)|nR
f |e
f i.

(S19)

0

Here the superoperator appearing between the dissipative jumps evolves the density matrix along a given pair oftrajectories determined by the elements of the vectors ~nα which correspond to times between the two time arguments(where tp+1 = t and t0 = 0). For l < k, we haveL
−iĤδt L−iĤδt L
eP(~nL ,~nR ;kδt,lδt) ρ̂ =|nL
|nk i · · · hnL|nl+1 i
k+1 ihnk+1 |e
l+2 |eRR
iĤδt RiĤδt R×hnL
|nl+2 i · · · hnR|nk+1 ihnR
l+1 |ρ̂|nl+1 ihnl+1 |ek |e
k+1 |,

(S20)

and
R
R
eP(~nL ,~nR ;t,0) ρ̂ = hnLnL )A∗ (~nR )|nL
1 |ρ̂|n1 iA(~f ihnf |,(S21)

The effect of dissipation is illustrated in Fig. S1.

II. ADDED NOISE OF THE MEASUREMENTAs mentioned in the introduction, the FCS distribution P (m) represents the intrinsic fluctuations of the system.
It can be used to predict the distribution of outcomes in an actual measurement setup, but one must convolve-in theeffects of the added noise of the measurement apparatus (both imprecision noise and backaction noise). While this has
been discussed in the past, our path-integral approach gives a particularly intuitive way to understand measurementnoise effects. We consider the generic heavy-mass detector used in Ref. [7]. In this setup, the observable to bemeasured n̂ is linearly coupled to the position x̂ of an infinite mass via the coupling Hamiltonian Hc = −Ax̂n̂. Theinteraction between the system and the detector is then turned on for a time t and the FCS can be inferred from the

4
momentum distribution of the detector after the measurement. The initial momentum uncertainty σp of the detector
gives rise to measurement imprecision, while the uncertainty in its position σx induces backaction. The measurement
imprecision has the effect of convolving the FCS with a Gaussian of width σimp = σp /A. The backaction can beincorporated by introducing an additional term HBA = ∆0 n̂ in the Hamiltonian, where ∆0 is a random variable witha Gaussian distribution of width σBA = Aσx [8]. Note that although random, ∆0 is constant in time because thedetector remains at a fixed position (due to its infinite mass). The measured probability distribution reads2(m− 2 mL − 2 mR )
X
σ2
−
2
1
2σ 2
− BA
imp
2 (mL −mR ) hnL |ρ̂|nR iA(~e
δnLf ,nRenL )A∗ (~nR ),Pd (m) = √11f2πσimp ~n ,~n
1

L

1

(S22)

R

where the subscript d reminds us of the presence of a detector. Measurement imprecision replaces the Dirac deltain Eq. (2) with a Gaussian, smearing out any sharp features in the FCS. The backaction exponentially reduces
interference contributions with mL 6= mR . The Heisenberg uncertainty principle applied to the detector implies therelation σBA σimp ≥ 1/2.
In the case of a quantum limited measurement, where σBA σimp = 1/2, the probability distribution reduces to
r
X
2
2 2
2
R −(m−mL )2 σBAhnL
e−(m−mR ) σBA A(~nL )A∗ (~nR ).
(S23)
σBA
δnLf ,nR
Pd (m) =1 |ρ̂|n1 ie
f
π
~
nL ,~
nR

Using δnLf ,nR
=
f

P

R
L
n hnf |nihn|nf i and ρ̂ =P

k pk |ψk ihψk |, we find2r
Pd (m) =X
X
2
2
L −(m−mL )2 σBA
σBA
pk
hnL
A(~nL ) ,1 |ψk ihn|nf ie
π
n,k

(S24)

~
nL

which is evidently positive.

III. TWO JUMP CONTRIBUTION TO THE FCS OF AN INITIAL FOCK STATE

Here we explicitly calculate the two jump contribution to the FCS displayed in Fig. 1 (c). The term where thetwo jumps happen on different trajectories can be divided into two terms. One where the jumps increase the photon
number (|n0 i → |n0 + 1i) and one where the jumps reduce the photon number (|n0 i → |n0 − 1i). Starting from
Eq. (2) with the Hamiltonian given in Eq. (4) and an initial Fock state ρ̂ = |n0 ihn0 |, we findP̄2↑ (m) =

N
X
l,r=0



1
δ m − (n0 + 1)t + (lδt + rδt) hn0 + 1|e−i∆n̂(t−lδt) |n0 + 1ihn0 + 1|if δtâ† |n0 ihn0 |e−i∆n̂lδt |n0 i2× hn0 |ei∆n̂rδt |n0 ihn0 |(−i)f δtâ|n0 + 1ihn0 + 1|ei∆n̂(t−rδt) |n + 1i


N
N
X
X
1
2
=(n0 + 1)f
δt
δtδ m − (n0 + 1)t + (lδt + rδt) ei∆(lδt−rδt) ,
2
r=0
l=0

(S25)
where the bar denotes that the two jumps happen on different trajectories and the subscript denotes that the term
includes two jumps which increase the photon number by one. The sum is over all possible times at which the jumps
may occur. In the continuum limit, we findP̄2↑ (m) = (n0 + 1)f

2

Zt

Zt
dtL

0



1
dtR δ m − (n0 + 1)t + (tL + tR ) ei∆(tL −tR ) .
2

(S26)

0

Evaluating the integral and including the term where the jumps decrease the photon number, we find the contribution
where the two jumps happen on different trajectories


 
k
2f 2 j m
m 1
P̄2 (m) =+ 1 sin 2∆ m −+t
for m ∈ [(n0 − 1)t, (n0 + 1)t],
(S27)
∆
t
t
2

5
and zero otherwise. Here b·c denotes the floor function. There are kinks whenever m/t is a multiple of 1/2. This isa consequence of the fact that a single trajectory with one upward/downward jump contributes to m with mα /2 ∈[n0 /2, n0 /2 ± 1/2].
A similar calculation yields the contribution where the two jumps happen on the same trajectoryP̃2 (m) = −8f 2

jm
t

+1

k

jm
k
1
m+ t−
+ 1 t cos [2∆(m − n0 t)]
2t

for m ∈ [(n0 − 1/2)t, (n0 + 1/2)t],(S28)

and zero otherwise. There is a jump at m/t = n0 reflecting the fact that for higher (lower) m the photon number israised (lowered) in between the two jumps. Since one of the trajectories does not exhibit any jumps, the last term is
non-zero only if |m − n0 t| < 1/2. The above equations are plotted in Fig. (1) (c).
IV. RECONSTRUCTING THE FCS: POSTPROCESSING THE DATA

FIG. S2. Possible issues when reconstructing the FCS. All panels correspond to a zoom in of Fig. (2) (b) in the main text withthe full FCS (without the convolution with a Gaussian) given by the black (solid) line and the reconstructed FCS given by
the blue (dotted) line. (a) Sensitivity of the reconstructed FCS to the choice of λmax . (b) Red (dashed) line is obtained byevaluating Eq. (S29) at all values of m. (c) Reconstruction of the FCS if the coupling strengths are normally distributed withwidth σ. In terms of the energy damping rate κ, the observation time is tκ = 0.15.

To reconstruct the FCS with only a limited number of measurements N up to a maximal coupling strength λmax ,we approximate the Fourier transform in Eq. (1) by(N −1 
)

X
λmax
nλmax
2πi nNm̃
P (m) ≈
<
Λ λ=
e
,
(S29)
Nπ
N
n=0
where we made use of the relation Λ(λ) = Λ∗ (−λ), which guarantees the FCS to be real, and introduced m̃ = λmax
2π m.
The discontinuous features at multiples of half-integer values for m/t suggest that the moment generating function has
terms that oscillate with λt/2 which decay very slowly. To properly take into account these oscillations, we need tomake sure that our window of integration (here summation) includes an integer number of oscillations. We therefore
choose λmax
2 t = n2π with n ∈ N. That this condition is indeed important is illustrated in Figure S2 (a).
In Eq. (S29), only integer values of m̃ have been used to create Fig. (2) (b) which allows us to use the fast Fouriertransform. Naively, one might expect that an additional evaluation of Eq. (S29) for non-integer m̃ would lead to abetter reconstruction of the FCS without the need of more measurement points. However, similarly to the above
condition for λmax , the summand in Eq. (S29) would then include frequencies which fit a non-integer number ofoscillations in the window of summation. This itself leads to an oscillatory behavior as illustrated in Fig. S2 (b).
These observations might give the impression that a high degree of control over the coupling strength is necessary andthat a small deviation from the desired value could make a faithful reconstruction of the FCS impossible. Fortunately,
this is not the case. This precision is only required for the post processing of the data. In Fig. S2 (c), we reconstruct
the FCS using coupling strengths that are normally distributed around their desired values with a width σ. A valueof σ = κ/2 still yields a good reconstruction of the FCS.
It therefore seems achievable to reconstruct the FCS by a finite number of feasible measurements. In order toreduce λmax , κ should be chosen as small as possible. However, if it becomes to small, the uncertainty in the actual
coupling strength can complicate the reconstruction.