# Quantum polyspectra for modeling and evaluating quantum transport
measurements: A unifying approach to the strong and weak measurement regime

M. Sifft  Ruhr University Bochum, Faculty of Physics and Astronomy,
Experimental Physics VI (AG), Germany  A. Kurzmann  Faculty of Physics and
CENIDE, University of Duisburg-Essen, Lotharstraße 1, 47057 Duisburg, Germany
J. Kerski  Faculty of Physics and CENIDE, University of Duisburg-Essen,
Lotharstraße 1, 47057 Duisburg, Germany  R. Schott  Lehrstuhl für Angewandte
Festkörperphysik, Ruhr-Universität Bochum, Universitätsstraße 150, D-44780
Bochum, Germany  A. Ludwig  Lehrstuhl für Angewandte Festkörperphysik, Ruhr-
Universität Bochum, Universitätsstraße 150, D-44780 Bochum, Germany  A. D.
Wieck  Lehrstuhl für Angewandte Festkörperphysik, Ruhr-Universität Bochum,
Universitätsstraße 150, D-44780 Bochum, Germany  A. Lorke  Faculty of Physics
and CENIDE, University of Duisburg-Essen, Lotharstraße 1, 47057 Duisburg,
Germany  M. Geller  Faculty of Physics and CENIDE, University of Duisburg-
Essen, Lotharstraße 1, 47057 Duisburg, Germany  D. Hägele  Ruhr University
Bochum, Faculty of Physics and Astronomy, Experimental Physics VI (AG),
Germany

###### Abstract

Quantum polyspectra of up to fourth order are introduced for modeling and
evaluating quantum transport measurements offering a powerful alternative to
methods of the traditional full counting statistics. Experimental time-traces
of the occupation dynamics of a single quantum dot are evaluated via
simultaneously fitting their 2nd-, 3rd-, and 4th-order spectra. The scheme
recovers the same electron tunneling and spin relaxation rates as previously
obtained from an analysis of the same data in terms of factorial cumulants of
the full counting statistics and waiting-time distributions. Moreover, the
evaluation of time-traces via quantum polyspectra is demonstrated to be
feasible also in the weak measurement regime even when quantum jumps can no
longer be identified from time-traces and methods related to the full counting
statistics cease to be applicable. A numerical study of a double dot system
shows strongly changing features in the quantum polyspectra for the transition
from the weak measurement regime to the Zeno-regime where coherent tunneling
dynamics is suppressed. Quantum polyspectra thus constitute a general unifying
approach to the strong and weak regime of quantum measurements with possible
applications in diverse fields as nano-electronics, circuit quantum
electrodynamics, spin noise spectroscopy, or quantum optics.

##  I Introduction

Quantum measurements are at the heart of many fields in physics like quantum
electronics, quantum optics, circuit quantum electrodynamics [1], or the
quickly developing field of quantum sensing [2]. In many cases, the detector
output of a measurement scheme results in stochastic time-traces with
information on the measured quantum system hidden in the data. Various schemes
for recovering that information are employed depending on the specifics of
both the quantum system and the measurement setup. In the field of quantum
electronics, the dynamics of electron occupation of semiconductor quantum dots
can, e.g., be measured via a so-called quantum point contact (QPC) in the
vicinity of the quantum dot. The charge state of the quantum dot is
immediately revealed by the strength of the probe current [3]. Alternatively,
the occupation of an illuminated quantum dot has been measured via its
resonance fluorescence [4] (see Fig. 1). The resulting time traces $z(t)$ of
the detector output exhibit for both schemes telegraph noise due to quantum
jumps in the occupation dynamics (see inset of Fig. 2). Jumps relating to an
electron leaving the dot are then often analyzed via the so-called full
counting statistics (FCS) $p(N,t)$, where $p$ is the probability that $N$
electrons have left the quantum dot in the time interval $t$ [5, 6]. The
counting statistic $p(n,t)$ is given for a simple tunnel-barrier and fixed $t$
by a Poisson-distribution. A deviating super-Poisson behavior has been
reported e.g. in single-electron tunneling through quantum dots [7] and in
single-electron tunneling at high magnetic fields [4]. Early theory for the
full counting statistics in the case of incoherent tunneling and without any
coherent quantum dynamics was given by Belzig [8]. These examples considered
the property of only one observable. The correlation of two observables,
namely the single electron dynamics in a quantum dot and a probe current
through an adjacent quantum point contact, was reported in 2007 [9].

Figure 1: Tunnelung events of electrons between the semiconductor QD and the
charge reservoir are monitored via the resonance fluorescence of the exciton
transition. Figure 2: Power spectra $S^{(2)}_{z}(\omega)$ of experimental
fluorescence time traces $z(t)$ (inset) for a single quantum dot at 10 T and
different gate voltages. Different regimes for tunnel rates
$\gamma_{\text{in}}$ and $\gamma_{\text{out}}$ are observed: Case a:
$\gamma_{\text{in}}\ll\gamma_{\text{out}}$; case b:
$\gamma_{\text{in}}\gtrapprox\gamma_{\text{out}}$; and case c:
$\gamma_{\text{in}}>\gamma_{\text{out}}$.

Depending on the problem, classical rate equations or the so-called
$n$-resolved master equation have been used to calculate cumulants of the
counting statistics [10, 11], factorial cumulants [12, 13], or second- and
third-order spectra of the frequency-resolved counting statistics [14]. All
these approaches to characterizing quantum transport dynamics assume and
require a strong continuous quantum measurement where the quantum system is
immediately forced to reveal its state of occupation. Consequently, coherent
quantum mechanical superpositions of the two alternatives of an occupied and
unoccupied quantum dot are always destroyed by the measurement.

The other limit of a quantum measurement, a weak continuous measurement, is,
e.g., realized by spin noise spectroscopy which has been demonstrated on
ensembles of spins in gases, semiconductors, and even on single spins [15, 16,
17]. Here, the Faraday-rotation of a probe laser beam is measured to reveal
spin fluctuations [18]. Owing to the weak measurement, the spins are not
projected onto spin eigenstates but may coherently process in an external
magnetic field. The power spectrum of the time-resolved Faraday-signal $z(t)$
reveals a peak at the precession frequency and a broad background due to
Gaussian shot noise of the probe laser. Spin noise theories of the power
spectrum have been given in terms of the spin-spin correlation function [19],
Langevin approaches [20], or path integral methods for weak quantum
measurements [21, 22].

The purpose of this article is to demonstrate that the evaluation of both
strong and weak measurement regimes can be unified in a common framework also
including the intermediate regime. While transport experiments in the
intermediate regime have been reported, their evaluation had been limited to
second-order spectra and was lacking comparison with theory [23]. The
framework here is based on frequency-resolved higher-order correlation
functions (so-called polyspectra [24]) of the detector output $z(t)$. Only
recently, compact quantum mechanical expressions were found for quantum
polyspectra up to fourth order from continuous measurement theory [25, 26]. We
will treat a real-world example from nano-electronics where we use polyspectra
to characterize the stochastic measurement traces and compare them with
quantum polyspectra calculated from the stochastic master equation [27]. As a
result, we obtain important parameters of the system like tunneling times and
spin relaxation rates. An efficient method for evaluating quantum mechanical
expressions of quantum polyspectra is described in Appendix C. The formulas
required for the calculation of higher-order spectra from experimental time
traces can be found in Appendix B including cumulant estimators and correct
usage of spectral window functions.

The stochastic master equation (SME) is an approach to continuous quantum
measurements that provides a stochastic differential equation for both the
system’s density matrix and the detector output $z(t)$ [28, 29, 30, 27]. The
coherent evolution, environment-induced damping in Markov approximation, the
detector output, a stochastic measurement-induced backaction on the system,
and a measurement-induced damping (Zeno-effect) are modeled. Thus, the SME is
able to unify the full regime from weak to strong measurements. We, therefore,
consider the SME a very general most direct link between the measurable
quantity $z(t)$ and the properties of the quantum system which enter the
master equation. As $z(t)$ can in principle be fully characterized in terms of
multi-time moments $\langle z(t_{n})..z(t_{1})\rangle$, an uncompromising
approach to its evaluation requires quantum mechanical expressions for such
moments. While multi-time correlators of detector output had been derived in
many different contexts of varying generality [see e.g. [31, 22] and
discussion after Eq. (6)], a direct derivation of multi-time moments from the
very general SME had been found by three different groups independently only
in 2018 [32, 33, 25]. This paved the way for finding compact expressions for
second-, third-, and fourth-order cumulants as well as their corresponding
quantum polyspectra and developing recipes for an efficient numerical
evaluation [25, 26]. We use the term ”quantum polyspectra” as recently
introduced by Wang for polyspectra of the detector output of continuous
quantum measurements [34]. Roughly speaking, the polyspectra of $z(t)$ can be
interpreted as $n$th order correlators of its Fourier-coefficients
$a_{\omega}$ (see App. B for a strict definition). The usual powerspectrum
$S^{(2)}(\omega)$ is then given by the expectation value $\langle
a^{*}_{\omega}a_{\omega}\rangle$ and thus by the average intensity of $z(t)$
at frequency $\omega$. The third-order spectrum
$S^{(3)}(\omega_{1},\omega_{2})$ (often called the bispectrum) is strongly
related to $\langle
a_{\omega_{1}}a_{\omega_{2}}a^{*}_{\omega_{1}+\omega_{2}}\rangle$ and is
sensitive to time-inversion (while $S^{(2)}$ is not) [35]. The fourth-order
spectrum (trispectrum) usually depends on three frequencies. Below, we will
only consider a two-dimensional cut which is related to $\langle
a^{*}_{\omega_{1}}a_{\omega_{1}}a^{*}_{\omega_{2}}a_{\omega_{2}}\rangle-\langle
a^{*}_{\omega_{1}}a_{\omega_{1}}\rangle\langle
a^{*}_{\omega_{2}}a_{\omega_{2}}\rangle$. The spectrum
$S^{(4)}(\omega_{1},\omega_{2})$ may therefore be interpreted as a frequency-
dependent intensity-intensity correlation. Emary et al. gave an early example
of a bispectrum related to transport theory of quantum dots [14]. Their
bispectrum for the current through a quantum dot follows from the $n$-resolved
master equation which requires the strong measurement limit. Moreover, most
experiments do not access the current from the quantum dot but its occupation.
A recent example of a measured bispectrum of a current was therefore
reconstructed from an occupation measurement [3]. Here, we directly evaluate
occupation measurements via their polyspectra.

##  II Telegraph Signal from a Single Quantum Dot

The time traces we are going to model and evaluate were recorded in an
experiment by Kurzmann et al. [4]. A single InAs quantum dot within an
electrically biased quantum dot layer in a GaAs-based p-i-n diode structure is
optically read out via resonance fluorescence (see Fig. 1). The fluorescence
time traces $z(t)$ exhibit telegraph noise due to the electron occupation
dynamics of the quantum dot. The inset of Fig. 2 shows traces for an external
magnetic field of 10 T and different gate voltages labeled with a) (360 mV),
b) (380 mV), and c) (382 mV). The gate voltages shift the chemical potential
of the electron reservoirs with respect to the single-electron level resulting
in voltage-dependent tunnel rates. Corresponding power spectra
$S_{z}^{(2)}(\omega)$ were calculated from time traces of 6 minutes duration
each (evaluation scheme see App. B). While the traces clearly show
quantitative differences in, e.g., typical up- and down-times, the power
spectra are all Lorentzian-shaped and differ only weakly in widths and overall
height (see Fig. 2). This changes dramatically for the bispectrum
$S^{(3)}(\omega_{1},\omega_{2})$ and the fourth-order correlation spectrum
$S^{(4)}(\omega_{1},\omega_{2})$ (definitions see App. B). Now, clear
differences become visible (see Fig. 3): The bispectrum $S^{\rm(3)}$ is
completely positive for case a) while it is negative for cases b) and c)
showing a similar overall structure. Clear differences between cases b) and c)
are, however, found in the trispectrum which exhibits a negative peak for case
b) but an almost flat negative structure for case c).

Figure 3: (upper two rows) Experimental bispectra
$S^{(3)}_{z}(\omega_{1},\omega_{2})$ at $B=$ 10 T and model fits for the
tunnel regimes a) (360 mV), b) (380 mV), and c) (382 mV) (compare Fig. 2).
(lower two rows) Experimental trispectra $S^{(4)}_{z}(\omega_{1},\omega_{2})$
and model fits. The analytical expressions of the polyspectra reveal that the
bispectra are sensitive to the sign of
$\gamma_{\text{out}}-\gamma_{\text{in}}$, whereas the trispectrum depends
mostly on $(\gamma_{\text{out}}-\gamma_{\text{in}})^{2}$.

In the following, we formulate the SME for the single dot experiment described
by Kurzmann. The QD dynamics is modeled by an electron tunneling rate
$\gamma_{\rm in}$ onto the dot and a rate $\gamma_{\rm out}$ from the dot.
Spin-dependent effects can be neglected at high magnetic fields (see below).
The quantum states related to the occupied and empty dot follow from each
other via a fermionic creation operator $a^{\dagger}$ and an annihilation
operator $a$, respectively. The number operator $n=a^{\dagger}a$ assumes the
eigenvalues 1 and 0, respectively. The SME propagates the density matrix
$\rho(t)$ of the quantum dot while it is constantly monitored for its
occupation by a continuous measurement given by the operator $A=n$. This
simple model will be extended to a spin-dependent version in Sec. III covering
the case of external fields below 10 T. The stochastic master equation (Ito-
calculus)

$\displaystyle\mathrm{d}\rho$ $\displaystyle=$
$\displaystyle\frac{i}{\hbar}[\rho,H]\mathrm{d}t+\frac{\gamma_{\rm
in}}{2}\mathcal{D}[a^{\dagger}](\rho)\mathrm{d}t+\frac{\gamma_{\rm
out}}{2}\mathcal{D}[a](\rho)\mathrm{d}t$ (1) $\displaystyle+$
$\displaystyle\beta^{2}\mathcal{D}[A](\rho)\mathrm{d}t+\beta\mathcal{S}[A](\rho)\mathrm{d}W$
$\displaystyle=$
$\displaystyle\mathcal{L}[\beta](\rho)\textrm{d}t+\beta\mathcal{S}[A](\rho)\textrm{d}W$
(2)

with damping terms

$\mathcal{D}[c](\rho)=c\rho c^{\dagger}-(c^{\dagger}c\rho+\rho
c^{\dagger}c)/2,$ (3)

and backaction term

$\mathcal{S}[c](\rho)=c\rho+\rho
c^{\dagger}-\mathrm{Tr}\left[(c+c^{\dagger})\rho\right]\rho$ (4)

describes the system dynamics $\rho(t)$ and the resulting detector output

$z(t)=\beta^{2}{\rm Tr}[\rho(t)(A+A^{\dagger})/2]+\frac{1}{2}\beta\Gamma(t)$
(5)

as it is monitored for the measurement operator $A$ with measurement strength
$\beta$. We use the notation for the measurement strength $\beta$ from Ref. 25
and for damping ${\cal D}$ and backaction ${\cal S}$ from Ref. 33. The SME has
been derived in various forms and varying generality and was rediscovered
several times in literature [36, 30, 37, 38, 39, 28, 29, 40]. An especially
intuitive way of deriving the SME was given by Gross et al. [41] and similarly
by Atal et al. [42, 43]. They introduce a continuous sequence of two-level
quantum systems (qubits) that each weakly interact for a short period with the
system and are subsequently readout by a projection measurement. The
projection measurements give some information on the system and at the same
time result in Gaussian background noise as the measurement outcome is still
highly stochastic. The interaction with the stream of probe systems also
causes measurement-induced system damping. The first line of Eq. (1) is
identical to a von-Neumann master equation in Lindblad form which correctly
describes incoherent tunneling on and from the quantum dot [44]. The
Hamiltonian $H$ is set to zero since the model here disregards spin dynamics
or other coherent behavior. The second line contains the differential of a
stochastic Wiener process $\mathrm{d}W$ where
$\Gamma(t)=\mathrm{d}W/\mathrm{d}t$ with
$\langle\Gamma(t)\Gamma(t^{\prime})\rangle=\delta(t-t^{\prime})$ is delta-
correlated Gaussian noise. Constant monitoring of occupation $n$ leads to
overall damping towards an eigenstate of $n$ (first term of the second line)
and a stochastic measurement backaction on the system which is correlated with
the detector output $z(t)$ [Eq. (5)] via the common Wiener process. The SME
can in principle be solved numerically to simulate time traces $z(t)$ (see
Fig. 8) which can then be evaluated in terms of polyspectra. Instead, we will
use general analytical expressions for the 2nd- to 4th-order multi-time
cumulants of $z(t)$ for general systems to obtain expressions for higher-order
spectra of the quantum dot dynamics [25, 26]. The expressions are given in
terms of the system Liouvillian ${\cal L}[\beta](\rho)$ where ${\cal
L}[\beta](\rho)$ covers all RHS terms of Eq. (1) but the stochastic backaction
term which is non-linear in $\rho$. We define a system propagator ${\cal
G}(\tau)=e^{{\cal L}\tau}\Theta(\tau)$ with the Heaviside-stepfunction
$\Theta(\tau)$, a steady state $\rho_{0}={\cal G}(\infty)\rho(t)$, a
measurement operator $A$, and its corresponding super operator [45] ${\cal
A}x=(Ax+xA^{\dagger})/2$. These definitions allow for a compact notation of
multi-time moments [32, 33, 25]

$\displaystyle\langle z(t_{n})$ $\displaystyle\cdots z(t_{1})\rangle=$
$\displaystyle\beta^{2n}$ $\displaystyle{\rm Tr}[{\cal A}{\cal
G}(t_{n}-t_{n-1}){\cal A}\cdots{\cal G}(t_{2}-t_{1}){\cal A}\rho_{0}],$ (6)

where time order $t_{n}>t_{n-1}>...>t_{1}$ is required and the system is
assumed to be in its steady state $\rho_{0}$. Consequently, the moments depend
only on time differences but not on absolute times. Quantum mechanical
expressions for multi-time moments in the form of Eq. (6) have been given in
the literature before for several special cases. Zoller and Gardiner discuss
moments of the photon counting statistics [see Ref. 31, Eq. (98)]. Similarly,
already in 1981 Srinivas gives an early theory for photon counting where
expressions with very similar structure compared to Eq. (6) appear [46].
Bednorz et al. derive a moment generating functional within a path integral
theory assuming a weak measurement limit and evaluate the functional to arrive
at Eq. (6) [see Ref. 22, Eq. (17)]. Wang and Clerk find the same functional as
Bednorz via a Keldysh approach and use it to calculate ”Keldysh-ordered”
moments, cumulants, and spectra of quantum noise up to third order [Ref. 34,
Eq. (3)]. Nazarov’s full counting statistics is based on the Keldysh approach
and uses a coupled ancilla system to derive a generating functional for
moments that yields Keldysh order operators in the Heisenberg picture. The
dependence of detector back-action on the measurement strength is however not
regarded in that approach [47, 48].

Jordan and coworkers give a path integral framework for treating continuous
quantum measurements and apply it to the simultaneous continuous measurement
of two non-commuting observables of a single qubit [49, 50]. They, however,
derive ”self-correlators” only up to second order [51]. Jordan and coworkers
also give an early example of a violation of a generalized Leggett-Garg
inequality which is based on second-order moments of weak measurements [52,
53]. Below we will see that a polyspectrum of at least third order is required
to extract tunneling rates from quantum dot measurements. The great advantage
of using the SME is that it provides a solid foundation for deriving
analytical expressions for higher-order multi-time moments without
restrictions on the measurement strength as well as a way to simulate
experiment-like time-traces $z(t)$ (see Sec. IV).

Cumulants instead of moments are often used in statistics since cumulants of
the sum of independent stochastic variables are simply the sum of the
individual cumulants. Additive noise in a measurement can therefore be
subtracted from cumulant-based quantities. Consequently, the cumulant-based
polyspectra are the desired quantities for evaluating quantum noise time
traces (App. B). A modified propagator ${\cal G}^{\prime}(\tau)={\cal
G}(\tau)-{\cal G}(\infty)\Theta(\tau)$ and a modified measurement super
operator ${\cal A}^{\prime}x={\cal A}x-{\rm Tr}({\cal A}\rho_{0})x$ allow for
a compact notation of multi-time cumulants [25, 26] despite their generally
intricate representation in terms of moments (App. A). The expressions

$\displaystyle C_{2}(z(t_{1}),z(t_{2}))$ $\displaystyle=$
$\displaystyle\frac{\beta^{2}}{4}\delta(t_{2}-t_{1})+\beta^{4}\sum_{\text{prm.
$t_{j}$}}{\rm Tr}[{\cal A}^{\prime}{\cal G}^{\prime}(t_{2}-t_{1}){\cal
A}^{\prime}\rho_{0}],$ (7) $\displaystyle C_{3}(z(t_{1}),z(t_{2}),z(t_{3}))$
$\displaystyle=$ $\displaystyle\beta^{6}\sum_{\text{prm. $t_{j}$}}{\rm
Tr}[{\cal A}^{\prime}{\cal G}^{\prime}(t_{3}-t_{2}){\cal A}^{\prime}{\cal
G}^{\prime}(t_{2}-t_{1}){\cal A}^{\prime}\rho_{0}],$ (8) $\displaystyle
C_{4}(z(t_{1}),z(t_{2}),z(t_{3}),z(t_{4}))$ $\displaystyle=$
$\displaystyle\beta^{8}\sum_{\text{prm. $t_{j}$}}{\rm Tr}[{\cal
A}^{\prime}{\cal G}^{\prime}(t_{4}-t_{3}){\cal A}^{\prime}{\cal
G}^{\prime}(t_{3}-t_{2}){\cal A}^{\prime}{\cal G}^{\prime}(t_{2}-t_{1}){\cal
A}^{\prime}\rho_{0}]$ (9) $\displaystyle-$
$\displaystyle\beta^{8}\sum_{\text{prm. $t_{j}$}}{\rm Tr}[{\cal
A}^{\prime}{\cal G}^{\prime}(t_{4}-t_{3}){\cal G}^{\prime}(t_{3}-t_{2}){\cal
A}^{\prime}\rho_{0}]{\rm Tr}[{\cal A}^{\prime}{\cal
G}^{\prime}(t_{3}-t_{2}){\cal G}^{\prime}(t_{2}-t_{1}){\cal
A}^{\prime}\rho_{0}]$ $\displaystyle-$ $\displaystyle\beta^{8}\sum_{\text{prm.
$t_{j}$}}{\rm Tr}[{\cal A}^{\prime}{\cal G}^{\prime}(t_{4}-t_{3}){\cal
G}^{\prime}(t_{3}-t_{2}){\cal G}^{\prime}(t_{2}-t_{1}){\cal
A}^{\prime}\rho_{0}]{\rm Tr}[{\cal A}^{\prime}{\cal
G}^{\prime}(t_{3}-t_{2}){\cal A}^{\prime}\rho_{0}]$

are also valid for equal times and hold without any restrictions on the time
order [25]. The term under the sum yields a contribution if and only if the
correct time order for $t_{1}$ to $t_{4}$ is fulfilled by one of the
permutations. The delta-function in $C_{2}$ appears due to the Gaussian noise
contribution $\Gamma(t)$ to $z(t)$ which is delta-correlated [Eq. (5)]. Unlike
moments, cumulants beyond second order are not sensitive to Gaussian noise
explaining the absence of delta-functions in $C_{3}$ and $C_{4}$. Compact
expressions for cumulants beyond the fourth order are still elusive.

Next, we calculate the multi-time cumulants and quantum polyspectra for the
quantum dot model, Eq. (1). The Liouvillian ${\cal L}[\beta]$ of the quantum
dot system can be represented as a $4\times 4$ matrix with a relatively simple
structure acting on the density matrix which itself can be represented by a
vector with four entries (compare Section XV of Ref. 25). The quantities
$e^{{\cal L}\tau}$ and $\rho_{0}$ can be expressed analytically with the help
of computer algebra. Assuming time order $t_{4}>t_{3}>t_{2}>t_{1}$, we find
the cumulants

$\displaystyle C_{2}(z(t_{1}),z(t_{2}))$ $\displaystyle=$
$\displaystyle\beta^{4}\frac{\gamma_{\text{in}}\gamma_{\text{out}}e^{-\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right)\tau_{1}}}{\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}}+\frac{\beta^{2}}{4}\delta(t_{2}-t_{1}),$
(10) $\displaystyle C_{3}(z(t_{1}),z(t_{2}),z(t_{3}))$ $\displaystyle=$
$\displaystyle\beta^{6}\frac{\gamma_{\text{in}}\gamma_{\text{out}}\left(\gamma_{\text{in}}-\gamma_{\text{out}}\right)e^{-\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right)(\text{$\tau_{1}$}+\text{$\tau_{2}$})}}{\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{3}},$
(11) $\displaystyle C_{4}(z(t_{1}),z(t_{2}),z(t_{3}),z(t_{4}))$
$\displaystyle=$
$\displaystyle\beta^{8}\frac{\gamma_{\text{in}}\gamma_{\text{out}}\left(\left(\gamma_{\text{in}}-\gamma_{\text{out}}\right){}^{2}e^{\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right)\text{$\tau_{2}$}}-2\gamma_{\text{in}}\gamma_{\text{out}}\right)e^{-\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right)((\text{$\tau_{1}$}+2\text{$\tau_{2}$}+\text{$\tau_{3}$}))}}{\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{4}},$
(12)

where we have introduced the positive time differences
$\tau_{i}=t_{i+1}-t_{i}$. Apart from the $\beta$-prefactors and the delta-
function contribution to $C_{2}$, the expressions $C_{2}$ and $C_{3}$ agree
with those derived from a classical rate equation model [54]. The cumulants
$C_{2}$ to $C_{4}$ show no further dependence on $\beta$ despite the fact that
the Liouvillian ${\cal L}[\beta]$ contains a $\beta$-dependent damping term. A
dependence of cumulants on $\beta$ can be found, e.g., in systems where a
large measurement strength $\beta$ leads to suppression of coherent dynamics
(Zeno-effect) [55, 29]. The absence of a $\beta$-dependence in our system is
explained by the absence of coherent dynamics ($H=0$).

The analytical expressions for the polyspectra follow after Fourier
transformation of the cumulants with respect to $t_{j}$ [Eq. (27)]. In Sec.
XIV of Ref. 25 it is shown how in general the time order can be dealt with
when performing a multi-dimensional Fourier transform. We obtain

$\displaystyle S^{(2)}_{z}(\omega)$ $\displaystyle=$
$\displaystyle\beta^{4}\frac{2\gamma_{\text{in}}\gamma_{\text{out}}}{\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right)\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\omega^{2}\right)}+\frac{\beta^{2}}{4},$
(13) $\displaystyle S^{(3)}_{z}(\omega_{1},\omega_{2})$ $\displaystyle=$
$\displaystyle\beta^{6}\frac{2\gamma_{\text{in}}\gamma_{\text{out}}\left(\gamma_{\text{out}}-\gamma_{\text{in}}\right)\left(3\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\omega_{1}^{2}+\omega_{2}^{2}+\omega_{1}\omega_{2}\right)}{\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right)\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\omega_{1}^{2}\right)\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\omega_{2}^{2}\right)\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\left(\omega_{1}+\omega_{2}\right){}^{2}\right)}.$
(14)

A cut through the trispectrum
$S_{z}^{(4)}(\omega_{1},\omega_{2})=S_{z}^{(4)}(\omega_{1},-\omega_{1},\omega_{2})$
is given in the Appendix, Eq. (41). Alternatively, a direct evaluation of the
quantum polyspectra in the frequency domain yields the same results (App. C).
Similar to the cumulant expressions, the measurement strength $\beta$ leads
for the present system only to a prefactor in the polyspectra and a variation
of the spectrally flat background noise in $S_{z}^{(2)}$. In contrast, we will
treat in Section V a double-dot system where an increasing measurement
strength $\beta$ leads to a clear suppression of coherent quantum oscillations
and a corresponding overall change in the polyspectra.

The experimental time traces of the quantum dot occupation dynamics are
evaluated for $\gamma_{\rm in}$ and $\gamma_{\rm out}$ by simultaneously
fitting the analytical expression of the spectra $S_{z}^{(2)}(\omega)$,
$S_{z}^{(3)}(\omega_{1},\omega_{2})$, and $S_{z}^{(4)}(\omega_{1},\omega_{2})$
to the spectra of the measured time traces. A constant background contribution
to the power spectrum $S_{z}^{(2)}(\omega)$ is also regarded separately. Time
traces of 6 min durations with a temporal resolution of 100 $\mu$s were taken
for each gate voltage. The measured traces and $z(t)$ differ by a setup-
dependent scaling factor that is regarded in the fitting procedure. In our
case, the scaling factor is negative since the occupied quantum dot state
results in absent fluorescence.

The polyspectra for cases a), b), and c) obtained from fitting are displayed
for illustration along with the originally measured spectra in Fig. 3. The
three sample traces of Fig. 2 can now be attributed to three regimes of the
tunneling rates: a) $\gamma_{\text{in}}\ll\gamma_{\text{out}}$, b)
$\gamma_{\text{in}}\gtrapprox\gamma_{\text{out}}$, and c)
$\gamma_{\text{in}}>\gamma_{\text{out}}$. Figure 4 compares as an important
result the tunneling rates for different gate voltages and a magnetic field of
10 T obtained from polyspectra with those obtained from a previous analysis of
the waiting-time distribution (WTD), i.e, $p(0,t)$ of the full counting
statistics [4, 56].

Figure 4: Tunneling rates determined from experimental polyspectra of the full
time trace (blue and red dots) in comparison with values obtained from the
waiting-time distribution (open squares and triangles) [4]. Both methods
arrive at the same results. The vertical lines mark the gate voltages
belonging to the time-traces shown in Fig. 2. Figure 5: Three-state model of
the spin-depended quantum dot dynamics.

The excellent agreement demonstrates that polyspectra are a powerful tool for
evaluating transport measurements. The general weak variation in the power
spectra is easily explained by Eq. (13) which does depend only on the sum
$\gamma_{\text{in}}+\gamma_{\text{out}}$ of the tunnel rates. The prefactor
$\gamma_{\text{in}}\gamma_{\text{out}}$ can not be exploited to separate
$\gamma_{\text{in}}$ and $\gamma_{\text{out}}$ since the measurement strength
$\beta$ acts as an overall scaling parameter. In contrast, the prefactor of
the bispectrum, Eq. (14), together with the prefactor of the power spectrum
contain information on $\gamma_{\text{in}}-\gamma_{\text{out}}$ and are in
principle sufficient to extract both parameters. We found more reliable
results by simultaneously fitting also the fourth-order spectrum which is
sensitive to $\gamma_{\text{in}}+\gamma_{\text{out}}$ and
$(\gamma_{\text{in}}-\gamma_{\text{out}})^{2}$ [see Eq. (12)]. We note that
the second- and fourth-order spectrum do not change under exchange of
$\gamma_{\text{in}}$ and $\gamma_{\text{out}}$ making the evaluation of the
bispectrum mandatory.

##  III Spin-dependent quantum dot dynamics

Next, we apply our method to a quantum dot in magnetic fields below 10 T.
Following Kurzmann et al. [4], the Zeeman spin splitting $\Delta$ leads to
spin-dependent tunneling rates

$\displaystyle\gamma_{0\uparrow}$ $\displaystyle=$ $\displaystyle d\Gamma
f\left(\epsilon+\Delta/2\right),$ $\displaystyle\gamma_{0\downarrow}$
$\displaystyle=$ $\displaystyle d\Gamma f\left(\epsilon-\Delta/2\right),$
$\displaystyle\gamma_{\uparrow 0}$ $\displaystyle=$
$\displaystyle\Gamma\left[1-f\left(\epsilon+\Delta/2\right)\right],$
$\displaystyle\gamma_{\downarrow 0}$ $\displaystyle=$
$\displaystyle\Gamma\left[1-f\left(\epsilon-\Delta/2\right)\right].$ (15)

The tunnel-coupling strength $\Gamma$ characterizes the tunnel barrier, $f(x)$
is the Fermi distribution function of the electron reservoir, the quantum dot
level energy is given by $\epsilon$, and the temperature by $T=10$ K. The
prefactor $d=10/11$ regards a reduction of the tunneling due to the presence
of the exciton whose fluorescence is detected by the measurement setup [4].
After introduction of a spin relaxation rate $\gamma_{\uparrow\downarrow}$ to
the down state, the system is fully described by an incoherent transition
dynamics depicted schematically in Fig. 5. Spin flips to the up state are
neglected as the spin down state is energetically favorable in magnetic fields
[4].

Figure 6: Calculated power-, bi- and trispectrum of the spin-dependent quantum
dot dynamics for $\gamma_{0\downarrow}=\gamma_{\downarrow 0}=2.5$
kHz,$\gamma_{0\uparrow}=\gamma_{\uparrow 0}=0.5$ kHz, and
$\gamma_{\uparrow\downarrow}=0$ (from top to bottom). The bispectrum shows
deviations from a simple Lorentzian shape and a dip at
$\omega_{1}=\omega_{2}=0$. The trispectrum strongly deviates from those of the
quantum dot at 10 T (compare Fig. 2 and 3). Figure 7: Comparison between the
measured and fitted polyspectra at $B=$ 2 T and 4 T. Cuts with $\omega_{2}=0$
(upper row) and cuts with $\omega_{1}=\omega_{2}$ (lower row) are shown for
the bi- and trispectrum. The maximum values of all spectra have been
normalized to 1. We can see excellent agreement between the measurement and
the three-state model. The diagonals of the bi- and trispectrum coincidentally
overlap for the quantum dot system.

The model in Kurzmann et al. can be formulated as a stochastic master equation

$\displaystyle\mathrm{d}\rho$ $\displaystyle=$
$\displaystyle\frac{\gamma_{0\uparrow}}{2}\mathcal{D}[a_{\uparrow}^{\dagger}a_{0}](\rho)\mathrm{d}t+\frac{\gamma_{0\downarrow}}{2}\mathcal{D}[a_{\downarrow}^{\dagger}a_{0}](\rho)\mathrm{d}t$
(16) $\displaystyle+$ $\displaystyle\frac{\gamma_{\uparrow
0}}{2}\mathcal{D}[a_{0}^{\dagger}a_{\uparrow}](\rho)\mathrm{d}t+\frac{\gamma_{\downarrow
0}}{2}\mathcal{D}[a_{0}^{\dagger}a_{\downarrow}](\rho)\mathrm{d}t$
$\displaystyle+$
$\displaystyle\frac{\gamma_{\uparrow\downarrow}}{2}\mathcal{D}[a_{\downarrow}^{\dagger}a_{\uparrow}](\rho)\mathrm{d}t+\frac{\beta^{2}}{2}\mathcal{D}[n_{\uparrow}+n_{\downarrow}](\rho)\mathrm{d}t$
$\displaystyle+$
$\displaystyle\beta\mathcal{S}[n_{\uparrow}+n_{\downarrow}](\rho)\mathrm{d}W,$

with detector output

$z(t)=\beta^{2}{\rm
Tr}[\rho(t)(n_{\uparrow}+n_{\downarrow})]+\frac{1}{2}\beta\Gamma(t).$ (17)

The measurement operator appears as a sum $n_{\uparrow}+n_{\downarrow}$ since
the detection scheme does not distinguish between up and down spins, but is
only sensitive to the mere presence of an electron in the quantum dot.

As an example, the power spectrum, bispectrum, and trispectrum were calculated
from Eqs. (38) to (C) for tunneling rates $\gamma_{0\uparrow}=\gamma_{\uparrow
0}=0.5$ kHz, $\gamma_{0\downarrow}=\gamma_{\downarrow 0}=2.5$ kHz, and absent
spin relaxation $\gamma_{\uparrow\downarrow}=0$ (see Fig. 6). Their structure
is clearly different from the spectra of the simple quantum dot model
discussed above. The power spectrum appears to be a superposition of two
Lorentzian peaks. The bispectrum reveals a small dip at zero frequencies and
the trispectrum displays positive maxima on the diagonals that were absent for
the simple quantum dot model. Spin relaxation rates
$\gamma_{\uparrow\downarrow}$ larger than the tunneling rate cause practically
all electrons to tunnel from the QD via the spin-down level at the rate
$\gamma_{\downarrow 0}$ while electrons enter the empty dot at an effective
rate $\gamma_{0\uparrow}+\gamma_{0\downarrow}$. The dot dynamics, therefore,
follow the simple quantum dot model and the spectra resume the appearance of
spectra shown in Fig. 3. The dependence of the spectra on the spin relaxation
rate suggests that the spin relaxation rate can be extracted from data
measured at finite magnetic fields. This dependence will get weaker for
similar tunneling rates of the two spin orientations, i.e.
$\gamma_{0\uparrow}\approx\gamma_{0\downarrow}$ and $\gamma_{\uparrow
0}\approx\gamma_{\downarrow 0}$. For exact agreement, the occupation dynamics
will obviously not depended on the spin orientation and is therefore not
sensitive to the spin relaxation rate.

In contrast to the simple model above, analytical expressions for the spectra
are not available. The fitting procedure, therefore, relies on a numerical
evaluation of the quantum polyspectra via Eqs. (38) to (C). The parameter
space for the five relaxation rates is restricted by their dependence on
$\Gamma$ and $\epsilon$ [compare Eqs. (15)]. Spin relaxation rates
$\gamma_{\uparrow\downarrow}$ were determined for a gate voltage of 371 mV at
a field of 2 T and for 376 mV at 4 T. We obtain an almost perfect agreement
between data and model for both fits. Fig. 7 shows for a quantitative
comparison cuts of all three spectra along the $\omega_{2}=0$ axes and cuts
for the bi- and trispectra along their diagonal $\omega_{1}=\omega_{2}$.

Figure 8: Simulated detector time trace $z(t)$ of a single quantum dot from
solving the stochastic master equation in the weak measurement regime (light
blue line). The electron occupation ${\rm Tr}(\rho(t)n)$ (dark blue line)
follows from the calculation but is not directly accessible in an experiment.
The detector output has been scaled by $\beta^{-2}$ for comparison.

The 2 T-case yields $\gamma_{\uparrow\downarrow}^{\text{2T}}=2.0$ kHz and
tunneling rates $(\gamma_{0\uparrow},\gamma_{0\downarrow},\gamma_{\uparrow
0},\gamma_{\downarrow 0})=(1.18,1.24,0.29,0.22)$ kHz. For 4 T we obtain
$\gamma_{\uparrow\downarrow}^{\text{4T}}=11$ kHz and tunneling rates
$(\gamma_{0\uparrow},\gamma_{0\downarrow},\gamma_{\uparrow
0},\gamma_{\downarrow 0})=(0.89,0.97,0.21,0.13)$ kHz. The discrepancy to the
values given in Kurzmann et al. for 2 T
$\gamma_{\uparrow\downarrow}^{\text{2T}}\approx 0.0$ kHz and 4 T
$\gamma_{\uparrow\downarrow}^{\text{4T}}=3.0$ kHz may be explained by a weak
dependence of the tunneling rates on the spin orientation and a large spin
relaxation (see previous paragraph). In both cases, the spin-relaxation
$\gamma_{\uparrow\downarrow}$ rate has only little influence on the tunneling
dynamics giving rise to large errors in the model fit.

Figure 9: Comparison between the simulated (left) and analytical (right)
polyspectra of the quantum dot system in the weak measurement regime. The
spectra show good agreement and tunneling rates could successfully be
recovered from the numerical spectra by a fitting procedure. The dashed lines
in the power spectra indicate the level of white Gaussian background noise.

##  IV Evaluating transport in the weak measurement regime

The experimental time-traces from above exhibit telegraph noise and a small
amount of additional noise. This allows in principle for the evaluation of
data in terms of waiting-time distributions of the FCS. Continuous measurement
theory, however, states that telegraph noise does disappear for weaker
measurements. Such a disappearance was recently reported for the case of gate-
tunable quantum point contacts [57]. It was shown that a cross-correlation
spectrum of two adjacent QPCs showed similar signatures as the spectra from
the stronger measurement regime [23]. However, an analysis in terms of the
full counting statistics as, e.g., previously required for separating in- from
out-tunneling rates is no longer possible for vanishing telegraph behavior.
Here we show that an evaluation of general time traces in terms of quantum
polyspectra is possible even in the weak measurement regime without any
restrictions.

We simulate a weak measurement ($\beta^{2}=1$ kHz) on a quantum dot, Eq. (1),
with parameters $\gamma_{\text{in}}$ = 1 kHz and $\gamma_{\text{out}}=0.5$
kHz. The integration of the SME with the QuTiP software package [58] yields
the QD occupation ${\rm Tr}[n\rho(t)]$ (see Fig. 8, dark blue curve). It
assumes values in the full regime between 0 and 1. Hence, the system is not
being projected into one of its eigenstates. The actual detector time trace
exhibits large background noise (see Fig. 8, light blue curve) that
dramatically exceeds the interval from 0 to 1, an effect put forward by
Aharonov et al. in their pioneering work on the notion of weak measurements
[59]. Quantum jumps can no longer quantitatively be evaluated from the time
trace and methods related to the FCS cannot be applied. Fig. 9 (left column)
shows polyspectra calculated from the simulated measurement trace with a
duration of 30 minutes. The polyspectra follow from a scheme based on
multivariate cumulant estimators of Fourier coefficients of the time trace
(App. B). We see excellent agreement between numerical spectra and the ones
evaluated from the exact expressions for quantum polyspectra, Eqs. (38), (39),
and (C). For comparison of spectra, no addition normalization was needed as
prefactors arising, e.g., from the spectral window were correctly accounted
for by our formulas. The simulated polyspectra exhibit increasing noise for
increasing order which is a known feature for estimates of cumulant-based
quantities [60]. The negative bispectrum immediately reveals
$\gamma_{\text{in}}>\gamma_{\text{out}}$. A simultaneous fit of all spectra
yields the predefined tunneling rate within an error of 10 %. Traces with even
stronger background noise can be evaluated if spectra are averaged for
sufficiently long measurement times.

Figure 10: Occupation dynamics and quantum polyspectra of a double-dot system:
the occupation dynamics of the second dot and corresponding polyspectra change
for increasing measurement strengths $\beta$ ($\beta$ increases for lower
rows). The coherent oscillations are suppressed for stronger measurements
(quantum Zeno effect). For $\beta=$2\text{\,}\sqrt{kHz}$$ the dynamics has
changed to telegraph-like switching. The peak due to coherent oscillations at
$\omega/2\pi=$0.5\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ in the power
spectrum $S_{z}^{(2)}$ broadens and shifts to zero frequency (lower row). The
signatures in $S_{z}^{(3)}$ and $S_{z}^{(4)}$ are also broadened and move
towards the origin. The detector output $z(t)$ is dominated by strong white
Gaussian noise and therefore not displayed in the first column except for the
largest measurement strength in the last row (light blue line). The units of
$S^{(3)}_{z}$ and $S^{(4)}_{z}$ are kHz-2 and kHz-3, respectively.

Next, we shortly discuss the nature of the SME detector model in comparison
with another model that was recently investigated by Li et al. in the context
of a randomly flipping spin [54]. The time trace in Figure 8 corresponds to
the noisy current that traverses a quantum point contact and thereby probes
the quantum dot occupation. In the above case of weak coupling, the probe
current modulation due to the dot occupation is much weaker than the
omnipresent shot-noise. Nevertheless, cumulants and spectra for order three or
higher exhibit only signatures of the quantum system as they are not sensitive
to Gaussian shot-noise. Li et al. give a different example of a continuous
measurement on a two-level system [54]. Their detector model yields a -1 or 1
for the two levels and 0 if no information could be gathered. For a slow
telegraph dynamics of the system, the jump dynamics can be resolved in the
detector stream and spectra similar to ours [Eqs. (13), (14), and (41)] were
recovered. For system dynamics faster than the average appearance rate of 1 or
-1 (a weak measurement in the terminology of Li) the authors find by a
numerical simulation of the detector output that their fourth-order cumulant
“quickly deteriorates” and no longer bears resemblance with the desired
spectra. A full theory of their detector model that would be capable of
predicting spectra in the spirit of our C1 to C3 for different measurement
strength is, however, still elusive. A separation of detector-related and
system-related contributions to polyspectra remains therefore as a challenge
in the case of Li’s detector model.

##  V Zeno-transition in a double-dot system

The quantum system treated in the preceding section was completely dominated
by incoherent tunneling dynamics and spin relaxation. Consequently, coherent
quantum dynamics was neither expected to leave a signature in the experiment
nor needed to be described by a Hamiltonian (i.e., so far $H=0$).

The SME is, however, capable of describing coherent quantum dynamics and the
effect of a continuous measurement on the dynamics. Here, we investigate the
transition from the weak to the strong measurement regime in a double-dot
system where electrons can coherently oscillate between the dots due to tunnel
coupling. The dynamics follows from the hopping term that quantifies the
coupling of the dots

$\displaystyle H=\hbar
g\left(a_{1}^{\dagger}a_{2}+a_{2}^{\dagger}a_{1}\right),$ (18)

where $g$ is the coupling constant and $a_{1}$ ($a_{2}$) is the annihilation
operator acting on the first (second) QD. We neglect the spin degree of the
electrons. The stream of electrons between the leads through the system is
modeled via incoherent tunneling as presented in Section II. The electrons can
enter the system only via the first dot and leave it only via the second dot.
The Hamiltonian allows for the simultaneous occupation of both dots with one
electron each. A continuous measurement of the occupation of the second
quantum dot is modeled by the operator $n_{2}=a_{2}^{\dagger}a_{2}$, which
corresponds to a measurement via a QPC in the vicinity of the second dot. The
SME becomes

$\displaystyle\mathrm{d}\rho$ $\displaystyle=$
$\displaystyle\frac{i}{\hbar}[\rho,H]\mathrm{d}t+\frac{\gamma_{\rm
in}}{2}\mathcal{D}[a_{1}^{\dagger}](\rho)\mathrm{d}t+\frac{\gamma_{\rm
out}}{2}\mathcal{D}[a_{2}](\rho)\mathrm{d}t$ (19) $\displaystyle+$
$\displaystyle\beta^{2}\mathcal{D}[n_{2}](\rho)\mathrm{d}t+\beta\mathcal{S}[n_{2}](\rho)\mathrm{d}W$

with the detector output

$\displaystyle z(t)=\beta^{2}{\rm
Tr}[\rho(t)n_{2}]+\frac{1}{2}\beta\Gamma(t).$ (20)

The parameters $g=$\pi\text{/}2\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$,
$\gamma_{\text{in}}=$0.071\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ and
$\gamma_{\text{out}}=$0.069\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ are used
for the calculation of system dynamics and quantum polyspectra. Fig. 10 shows
the transition of the system from the weak measurement regime to the quantum
Zeno regime. Each row depicts the occupation dynamics ${\rm Tr}[\rho(t)n_{2}]$
and the quantum polyspectra $S^{(2)}$, $S^{(3)}$, and $S^{(4)}$ for increasing
measurement strengths $\beta$ [calculated from Eq. (38), (39), (C)]. The
transients of the occupation dynamics start at $t=0$ with the first dot fully
occupied. In the weak measurement case, the system relaxes towards the steady
state where $n_{1}\approx n_{2}\approx 0.5$. The weak measurement is however
causing non-vanishing oscillations in ${\rm Tr}[\rho(t)n_{2}]$ as the system
is forced to reveal partly whether the second dot is empty or full. A
subsequent oscillation dynamics towards or from the dot, respectively, needs
then to occur. The polyspectra thus reveal a signature at frequency
$g/2\pi=$0.5\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$. The occupation number
$n_{2}$ oscillates around a value that itself exhibits slow fluctuations in
time. The reason being that the number of electrons in the system is not
strictly conserved due to random tunneling events from and into the leads.
Consequently, a clear peak at zero frequency appears in all spectra.

For low measurement strength (first and second row), we find sharp spectral
lines in $S^{(2)}$, $S_{z}^{(3)}$, and $S_{z}^{(4)}$ . The lines in
$S_{z}^{(4)}$ exhibits the same characteristic butterfly signature as display
in Figure 14 of Smirnov et al. [61]. For increasing measurement strengths
(third and fourth row) the oscillating dynamics become more and more
suppressed. Telegraph-like switching is observed for the strongest measurement
strength where all coherent dynamics is suppressed (so-called quantum Zeno
regime). Consequently, all peaks in the spectra move towards zero frequency.
The incomplete switching events seen in transient for the largest measurement
strength are known as spikes and have been investigated in detail by Tilloy
[62]. The white background noise in $S_{z}^{(2)}$ decreases as the expectation
value in Eq. (20) dominates over the Gaussian noise term for large $\beta$.
The signatures in $S_{z}^{(3)}$ and $S_{z}^{(4)}$ are equally broadened as
signals from a larger range of frequencies are getting correlated. The quantum
Zeno effect had been previously studied theoretically by Korotkov in a double-
dot system without leads. He found a damping of oscillation dynamics and a
correspondingly broadened spectral line in his $S_{z}^{(2)}$ [29]. As a
consequence of leads, our $S_{z}^{(2)}$ exhibits an additional peak at zero
frequency (see above). Korotkov finds an unrelated peak at zero frequency only
after introducing an energy imbalance $\varepsilon$ of the two dots.
Suppression of coherent dynamics was recently observed in experiments on a
superconducting qubit in a GHz-cavity (circuit quantum electrodynamics) where
measurement strengths and observation operators were highly controllable [63].
The dynamics of the system was obtained from averaging many noisy time-traces
for an identically prepared initial state, while here steady state
fluctuations are observed and analyzed without the need for state preparation.
While we are currently not fully able to interpret all features in the higher-
order spectra, it is clear that they contain more information than usual power
spectra. Our analysis of a real-world experiment in Sections II and III
demonstrates that higher-order spectra can contribute substantially to the
analysis of quantum systems.

##  VI conclusion

We presented quantum polyspectra within a stochastic master equation approach
as a viable alternative to full counting statistics approaches for evaluating
time-traces of transport measurements. The framework is applicable to general
systems where both coherent evolution and incoherent coupling to the
environment are important [27]. The stochastic master equation has previously
been shown to cover the whole regime from weak to strong measurements and
allows for modeling of the weak measurement regime as well as investigating
the transition to the Zeno regime [29]. In quantum electronics, quantum
polyspectra can in the future be used for evaluating weak quantum point
contact measurements where background noise prevents a traditional analysis in
terms of the full counting statistics, i.e. when quantum jumps can no longer
be identified from time traces. Weakly coupled QPCs and polyspectra may be the
key for fully characterizing coherent dynamics in transport measurements like,
e.g., spin precession or tunnel dynamics between adjacent quantum dots [64].
We also expect applications of quantum polyspectra in circuit quantum
electrodynamics and quantum optics in general [65, 63, 66, 34].

###### Acknowledgements.

We acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) Project No. 278162697— SFB 1242. D. H., M. G., and
A.L. acknowledge financial support by the DFG by the individual research
grants No. HA3003/7-1, No. GE2141/5-1, and No. LU2051/1-1.

##  Appendix A Cumulants

The cumulants can be represented in terms of products of moments as [67, 68]

$\displaystyle C_{2}(x,y)$ $\displaystyle=$ $\displaystyle\langle
yx\rangle-\langle y\rangle\langle x\rangle,$ (21) $\displaystyle C_{3}(x,y,z)$
$\displaystyle=$ $\displaystyle\langle zyx\rangle-\langle yx\rangle\langle
z\rangle$ (22) $\displaystyle-$ $\displaystyle\langle zx\rangle\langle
y\rangle-\langle zy\rangle\langle x\rangle+2\langle z\rangle\langle
y\rangle\langle x\rangle,$ $\displaystyle C_{4}(x,y,z,w)$ $\displaystyle=$
$\displaystyle\langle wzyx\rangle-\langle wzy\rangle\langle x\rangle-\langle
wyx\rangle\langle z\rangle$ (23) $\displaystyle-$ $\displaystyle\langle
wzx\rangle\langle y\rangle-\langle zyx\rangle\langle w\rangle-\langle
wz\rangle\langle yx\rangle$ $\displaystyle-$ $\displaystyle\langle
wy\rangle\langle zx\rangle-\langle wx\rangle\langle zy\rangle+2\langle
yx\rangle\langle w\rangle\langle z\rangle$ $\displaystyle+$ $\displaystyle
2\langle zx\rangle\langle w\rangle\langle y\rangle+2\langle wx\rangle\langle
y\rangle\langle z\rangle$ $\displaystyle+$ $\displaystyle 2\langle
wy\rangle\langle z\rangle\langle x\rangle+2\langle zy\rangle\langle
w\rangle\langle x\rangle$ $\displaystyle+$ $\displaystyle 2\langle
wz\rangle\langle y\rangle\langle x\rangle-6\langle x\rangle\langle
y\rangle\langle z\rangle\langle w\rangle.$

##  Appendix B Polyspectra and their estimation from time traces

Starting from the auto-correlation function of the detector output $z(t)$

$\displaystyle a(\tau)$ $\displaystyle=$ $\displaystyle C_{2}(z(t),z(t+\tau))$
(24) $\displaystyle=$ $\displaystyle\left\langle
z(t)z(t+\tau)\right\rangle_{t}-\left\langle z(t)\right\rangle_{t}\left\langle
z(t+\tau)\right\rangle_{t},$

where $\left\langle...\right\rangle_{t}$ relates to the ideal infinite time
average with respect to $t$, the power spectrum

$\displaystyle S_{z}^{(2)}(\omega)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}a(\tau)e^{i\omega\tau}\textrm{d}\tau$
(25)

can be defined. Alternatively, the power spectrum can be expressed via the
Fourier transform of the detector output
$z(\omega)=\int_{-\infty}^{\infty}z(t)e^{i\omega t}\textrm{d}t$ as

$\displaystyle 2\pi\delta(\omega+\omega^{\prime})S_{z}^{(2)}(\omega)$
$\displaystyle=$ $\displaystyle C_{2}(z(\omega),z(\omega^{\prime})).$ (26)

Brillinger generalized this expression to define polyspectra of order $n$
(Ref. 24)

$\displaystyle 2\pi\delta(\omega_{1}+...+\omega_{n})$ $\displaystyle
S_{z}^{(n)}$ $\displaystyle(\omega_{1},...,\omega_{n-1})$ (27)
$\displaystyle=$ $\displaystyle C_{n}(z(\omega_{1}),...,z(\omega_{n})).$

Figure 11: Parameter dependent approximate confined Gaussian window function
$g_{j}$ for parameter $s=0.14T$ [69].

Above, the bispectrum

$\displaystyle S_{z}^{(3)}(\omega_{1},\omega_{2}),$ (28)

and a cut through the trispectrum $S_{z}^{(4)}$

$\displaystyle
S_{z}^{(4)}(\omega_{1},\omega_{2})=S_{z}^{(4)}(\omega_{1},-\omega_{1},\omega_{2}).$
(29)

are used for characterizing experimental time traces and comparison with
quantum polyspectra. Polyspectra are estimated from experimental time traces
in the following way: The detector output $z(t)$ is discretized and divided
into time frames leading to arrays $z^{(n)}$ of length $N$ with $0\leq j<N$

$z^{(n)}_{j}=z(jT/N+nT),$ (30)

where $T$ is the temporal length of the time frames. The coefficient
$a^{(n)}_{k}$ of the discrete Fourier transformation (fast Fourier
transformation can be used for evaluation) are obtained after applying a
window function $g_{j}$ to $z_{j}$

$a^{(n)}_{k}=\frac{T}{N}\sum_{j=0}^{N-1}g_{j}z^{(n)}_{j}e^{2\pi ijk/N}.$ (31)

Window functions are routinely used in signal processing for improving the
spectral resolution [70]. Here we appy the approximated confined Gaussian
window with window parameter $s=0.14T$ (Figure 11) for its optimal RMS time-
bandwidth product [69].

The ideal polyspectra are then approximately (finite spectral resolution)
given by [71]

$\displaystyle S^{(2)}_{z}(\omega_{k})$ $\displaystyle\approx$
$\displaystyle\frac{NC_{2}(a_{k},a_{k}^{*})}{T\sum_{j=0}^{N-1}g_{j}^{2}}$ (32)
$\displaystyle S^{(3)}_{z}(\omega_{k},\omega_{l})$ $\displaystyle\approx$
$\displaystyle\frac{NC_{3}(a_{k},a_{l},a_{k+l}^{*})}{T\sum_{j=0}^{N-1}g_{j}^{3}}$
(33) $\displaystyle S^{(4)}_{z}(\omega_{k},\omega_{l})$ $\displaystyle\approx$
$\displaystyle\frac{NC_{4}(a_{k},a_{k}^{*},a_{l},a_{l}^{*})}{T\sum_{j=0}^{N-1}g_{j}^{4}},$
(34)

where $\omega_{k}=2\pi k/T$ for $k<N/2$ and $\omega_{k}=2\pi(k-N)/T$ for
$k\geq N/2$. The cumulants $C_{2}$, $C_{3}$, and $C_{4}$ (see App. A) are
estimated from so-called cumulant estimators [72]

$\displaystyle c_{2}(x,y)=$
$\displaystyle\frac{m}{m-1}\left(\overline{xy}-\overline{x}\,\overline{y}\right)$
(35) $\displaystyle c_{3}(x,y,z)=$ $\displaystyle\frac{m^{2}}{(m-1)(m-2)}$
(36)
$\displaystyle\times\left(\overline{xyz}-\overline{xy}\,\overline{z}-\overline{xz}\,\overline{y}\right.$
$\displaystyle-\left.\overline{yz}\,\overline{x}+2\overline{x}\,\overline{y}\,\overline{z}\right)$
$\displaystyle c_{4}(x,y,z,w)=$ $\displaystyle\frac{m^{2}}{(m-1)(m-2)(m-3)}$
(37)
$\displaystyle\times\Big{[}(m+1)\overline{xyzw}-(m+1)\left(\overline{xyz}\
\overline{w}+\textrm{ 3 o.p.}\right)\Big{.}$
$\displaystyle\Big{.}-(m-1)\left(\overline{xy}\ \overline{zw}+\textrm{2
o.p.}\right)\Big{.}$ $\displaystyle\Big{.}+2m\left(\overline{xy}\
\overline{z}\ \overline{w}+\textrm{5 o.p.}\right)\Big{.}$
$\displaystyle-\Big{.}6m\overline{x}\ \overline{y}\ \overline{z}\
\overline{w}\Big{.}],$

where o.p. means ”other permutations”. The overline $\overline{(...)}$ denotes
an average of $m$ samples. Their structure is similar to that of the cumulants
apart from $m$-dependent prefactors [compare Eqs. (21)-(23)]. The estimators
have the property $\langle c_{j}\rangle=C_{j}$ for finite $m$ (unbiased
estimators) and $c_{j}\rightarrow C_{j}$ for $m\rightarrow\infty$
(consistency). The estimators $c_{2}$, $c_{3}$, and $c_{4}$ are multivariate
versions of the well-known k-statistics [73, 74, 60, 75]. The estimator
$c_{2}(x,x)$ is identical with a frequently used estimator for the variance of
$x$. It exhibits the typical prefactor $m/(m-1)$ which is sometimes called the
Bessel-correction [76].

##  Appendix C Quantum polyspectra

The polyspectra of detector output $z(t)$ of the continuously monitored
quantum system in the steady state follow from the SME without any
approximations as

$\displaystyle S_{z}^{(2)}(\omega)$ $\displaystyle=$
$\displaystyle\beta^{4}({\rm Tr}[{\cal A}^{\prime}{\cal
G}^{\prime}(\omega){\cal A}^{\prime}\rho_{0}]+{\rm Tr}[{\cal A}^{\prime}{\cal
G}^{\prime}(-\omega){\cal A}^{\prime}\rho_{0}])+\beta^{2}/4$ (38)
$\displaystyle
S_{z}^{\rm(3)}(\omega_{1},\omega_{2},\omega_{3}=-\omega_{1}-\omega_{2})$
$\displaystyle=$ $\displaystyle\beta^{6}\sum_{\text{prm.
$\omega_{1},\omega_{2},\omega_{3}$}}{\rm Tr}[{\cal A}^{\prime}{\cal
G}^{\prime}(\omega_{3}){\cal A}^{\prime}{\cal
G}^{\prime}(\omega_{3}+\omega_{2}){\cal A}^{\prime}\rho_{0}].$ (39)
$\displaystyle
S_{z}^{\rm(4)}(\omega_{1},\omega_{2},\omega_{3},\omega_{4}=-\omega_{1}-\omega_{2}-\omega_{3})$
$\displaystyle=$ $\displaystyle\beta^{8}\sum_{\text{prm.
$\omega_{1},\omega_{2},\omega_{3},\omega_{4}$}}\left[{\rm Tr}[{\cal
A}^{\prime}{\cal G}^{\prime}(\omega_{4}){\cal A}^{\prime}{\cal
G}^{\prime}(\omega_{3}+\omega_{4}){\cal A}^{\prime}{\cal
G}^{\prime}(\omega_{2}+\omega_{3}+\omega_{4}){\cal
A}^{\prime}\rho_{0}]\right.$ $\displaystyle-$
$\displaystyle\frac{1}{2\pi}\int{\rm Tr}[{\cal A}^{\prime}{\cal
G}^{\prime}(\omega_{4}){\cal G}^{\prime}(\omega_{3}+\omega_{4}-\omega){\cal
A}^{\prime}\rho_{0}]{\rm Tr}[{\cal A}^{\prime}{\cal G}^{\prime}(\omega){\cal
G}^{\prime}(\omega_{2}+\omega_{3}+\omega_{4}){\cal
A}^{\prime}\rho_{0}]\textrm{d}\omega$ $\displaystyle-$
$\displaystyle\left.\frac{1}{2\pi}\int{\rm Tr}[{\cal A}^{\prime}{\cal
G}^{\prime}(\omega_{4}){\cal
G}^{\prime}(\omega_{2}+\omega_{3}+\omega_{4}){\cal
G}^{\prime}(\omega_{3}+\omega_{4}-\omega){\cal A}^{\prime}\rho_{0}]{\rm
Tr}[{\cal A}^{\prime}{\cal G}^{\prime}(\omega){\cal
A}^{\prime}\rho_{0}]\textrm{d}\omega\right].$

Their derivation via multi-time cumulants of $z(t)$ and an efficient method
for their numerical evaluation are given in Refs. 25, 26. All spectra are free
from delta-function contributions because the time-dependent ${\cal
G}^{\prime}(\tau)$ decays exponentially to zero for increasing $\tau$ as long
as existence of a steady state $\rho_{0}={\cal G}(\tau)\rho$ for
$\tau\rightarrow\infty$ can be assumed.

##  Appendix D The fourth-order trispectrum of the SQD

Analytical expression for the trispectrum of the quantum dot system
(neglecting spin-dependent dynamics):

$\displaystyle S^{(4)}_{z}(\omega_{1},\omega_{2})$ $\displaystyle=$
$\displaystyle
4\gamma_{\text{in}}\gamma_{\text{out}}\left(\gamma_{\text{in}}^{2}\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\omega_{1}^{2}\right)\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\omega_{2}^{2}\right)\left(3\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}\left(2\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}\right.\right.\right.$
(41) $\displaystyle+$
$\displaystyle\left.\left.\left.\omega_{1}^{2}+\omega_{2}^{2}\right)+\left(\omega_{1}^{2}-\omega_{2}^{2}\right){}^{2}\right)-2\gamma_{\text{in}}\gamma_{\text{out}}\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\omega_{1}^{2}\right)\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\omega_{2}^{2}\right)\right.$
$\displaystyle\times$
$\displaystyle\left.\left(3\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}\left(2\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\omega_{1}^{2}+\omega_{2}^{2}\right)+\left(\omega_{1}^{2}-\omega_{2}^{2}\right){}^{2}\right)+2\gamma_{\text{in}}\gamma_{\text{out}}\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\left(\omega_{1}-\omega_{2}\right){}^{2}\right)\right.$
$\displaystyle\times$
$\displaystyle\left.\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\left(\omega_{1}+\omega_{2}\right){}^{2}\right)\left(\omega_{1}^{2}\omega_{2}^{2}-\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}\left(3\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\omega_{1}^{2}+\omega_{2}^{2}\right)\right)\right.$
$\displaystyle+$
$\displaystyle\left.\gamma_{\text{out}}^{2}\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\omega_{1}^{2}\right)\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\omega_{2}^{2}\right)\left(3\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}\right.\right.$
$\displaystyle\times$
$\displaystyle\left.\left.\left(2\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\omega_{1}^{2}+\omega_{2}^{2}\right)+\left(\omega_{1}^{2}-\omega_{2}^{2}\right){}^{2}\right)\right)$
$\displaystyle/$
$\displaystyle\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{3}\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\omega_{1}^{2}\right){}^{2}\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\left(\omega_{1}-\omega_{2}\right){}^{2}\right)$
$\displaystyle/$
$\displaystyle\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\omega_{2}^{2}\right){}^{2}\left(\left(\gamma_{\text{in}}+\gamma_{\text{out}}\right){}^{2}+\left(\omega_{1}+\omega_{2}\right){}^{2}\right)$

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