# Full counting statistics of the photocurrent through a double quantum dot
embedded in a driven microwave resonator

Drilon Zenelaj  Physics Department and NanoLund, Lund University, Box 118,
22100 Lund, Sweden Patrick P. Potts  Department of Physics, University of
Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland Peter Samuelsson
Physics Department and NanoLund, Lund University, Box 118, 22100 Lund, Sweden

###### Abstract

Detection of single, itinerant microwave photons is an important functionality
for emerging quantum technology applications as well as of fundamental
interest in quantum thermodynamics experiments on heat transport. In a recent
experiment [W. Khan et al., Nat. Commun. 12, 5130 (2021)], it was demonstrated
that a double quantum dot (DQD) coupled to a microwave resonator can act as an
efficient and continuous photodetector by converting an incoming stream of
photons to an electrical photocurrent. In the experiment, average photon and
electron flows were analyzed. Here we theoretically investigate, in the same
system, the fluctuations of the photocurrent through the DQD for a coherent
microwave drive of the resonator. We consider both the low frequency full
counting statistics as well as the finite-frequency noise (FFN) of the
photocurrent. Numerical results and analytical expressions in limiting cases
are complemented by a mean-field approach neglecting dot-resonator
correlations, providing a compelling and physically transparent picture of the
photocurrent statistics. We find that for ideal, unity efficiency detection,
the fluctuations of the charge current reproduce the Poisson statistics of the
incoming photons, while the statistics for non-ideal detection is sub-
Poissonian. Moreover, the FFN provides information of the system parameter
dependence of detector short-time properties. Our results give novel insight
into microwave photon-electron interactions in hybrid dot-resonator systems
and provide guidance for further experiments on continuous detection of single
microwave photons.

##  I Introduction

Photons are the elementary quanta of the electromagnetic field and have been a
central concept in physics for over a century. The ability to experimentally
create and detect individual photons Eisaman  _et al._ (2011) has found
applications in areas ranging from particle physics and astronomy Huber  _et
al._ (2013) to imaging Ntziachristos  _et al._ (2005) and spectroscopy
Michalet  _et al._ (2007). During the last few decades, single-photon
detectors in the optical regime have attracted increasing interest due to
their potential applicability in emerging quantum technologies, such as linear
optics quantum computation Knill  _et al._ (2001), quantum random number
generation Herrero-Collantes and Garcia-Escartin (2017) and quantum
cryptography and key distribution Pirandola  _et al._ (2020). In state-of-the-
art photodetectors, key properties such as near-unity detection efficiency
Pernice  _et al._ (2012), low dark-count rates Hadfield (2009), photon-number
resolved detection Kardynał  _et al._ (2008) and high operation speed Ackert
_et al._ (2015) have been demonstrated.

In the microwave regime, photons have an energy which is four to five orders
of magnitude smaller than in the optical regime. This puts very different
requirements for experimental investigations and applications based on
microwave photons Casariego  _et al._ (2022). During the last decade, a large
number of theoretical proposals for single-microwave-photon detectors have
been presented, see e.g. Romero  _et al._ (2009); Chen  _et al._ (2011); Fan
_et al._ (2014); Sathyamoorthy  _et al._ (2014); Kyriienko and Sørensen
(2016); Gu  _et al._ (2017). In recent years, single-microwave-photon
detection has also been demonstrated experimentally Inomata  _et al._ (2016);
Narla  _et al._ (2016); Opremcak  _et al._ (2018); Besse  _et al._ (2018);
Kono  _et al._ (2018); Dassonneville  _et al._ (2020); Essig  _et al._ (2021).
Functionalities such as close-to-unit efficiency photodetectors Narla  _et
al._ (2016); Opremcak  _et al._ (2018); Dassonneville  _et al._ (2020),
quantum non-demolition measurements Besse  _et al._ (2018); Kono  _et al._
(2018) of itinerant photons and few-photon number resolution Dassonneville
_et al._ (2020); Essig  _et al._ (2021) have been realized in experiments
based on superconducting qubits. A common property of the superconducting
qubit detectors Narla  _et al._ (2016); Opremcak  _et al._ (2018);
Dassonneville  _et al._ (2020); Besse  _et al._ (2018); Kono  _et al._ (2018)
is that they require synchronization with the photon signal and typically
involve advanced qubit pulsing and read-out schemes.

Very recently, taking an altogether different approach, continuous microwave
photodetection was demonstrated in a semiconductor double quantum dot (DQD)
resonantly coupled to a driven superconducting resonator Khan  _et al._
(2021). The operation principle is a direct microwave analog of the
photocarrier generation in conventional optical photo diodes Pearsall (2003),
since incident microwave photons generate an electron current through the DQD
by inducing tunneling of electrons from the ground to the excited DQD state.
The quantum efficiency of the photon-to-electron conversion in the experiment
reached 6$\%$ Khan  _et al._ (2021), several orders of magnitude higher than
previous experiments based on similar approaches Gustavsson  _et al._ (2007a);
Liu  _et al._ (2014).

The prospect of experimentally reaching even higher values, approaching the
theoretically predicted unity efficiency Wong and Vavilov (2017), motivates
further theoretical investigations of the photodetector system. On the
fundamental side, the efficient conversion of a stream of photons, bosonic
particles, into a current of electrons, fermionic particles, raises
interesting questions on the relation of the statistical properties of the two
flows. From a more applied perspective, the short time properties of the
photodetector provide information on dead-times and detection speed, important
figures-of-merit for time resolved, single-photon detection.

In this work, we address these questions theoretically by analyzing both the
statistical distribution, or full counting statistics (FCS) Belzig (2005);
Kießlich  _et al._ (2006); Nazarov and Blanter (2009); Xu and Vavilov (2013a);
Schaller (2014), of the number of photoelectrons transported through the DQD
during a measurement as well as the finite-frequency noise (FFN) of the
photocurrent. We provide a numerical analysis, complemented by analytical
results in the low and large drive limits, based on the solution of the
appropriate generalized quantum master equation. Moreover, we present a mean-
field formalism for both the FCS and the FFN, considerably simplifying the
analysis, and discuss its limits of applicability. As key results, we show
that at unit quantum efficiency, the photoelectrons inherit the Poissonian
statistics of the incoming photons. For lower efficiencies, the electron
statistics is typically sub-Poissonian. We moreover, from the FFN, identify
the system properties governing the detector dead time. Our results will
arguably stimulate further experiments on hybrid semiconductor-resonator
photodetectors. It will moreover provide the framework for further theoretical
investigations into photodetection with different microwave sources, e.g. non-
classical ones, as well as functionalities such as single- and photon-number
resolved detection.

This paper is organized as follows. In Sec. II, we present our theoretical
model to describe the DQD-resonator system and show how it can be used as an
efficient photodetector. We further review the conditions that need to be met
in order to reach near unit photon-to-electron conversion efficiency. In Sec.
III, we analyze the FCS and the FFN in the limiting cases. In Sec. IV, we
present the results for the FCS and the FFN using a mean-field approach and
compare them against the solutions of Sec. III and numerical calculations. We
conclude and give a brief outlook in Sec. V.

##  II System & Model

###  II.1 Hamiltonian for the DQD-resonator system

Figure 1: a) A coherent microwave drive with frequency $\omega_{\text{l}}$ and
incoming photon rate $\dot{N}$ feeds in photons to a superconducting
transmission line resonator with resonance frequency $\omega_{\text{r}}$. The
coupling strength between drive and resonator is given by
$\kappa_{\text{in}}$. The resonator is capacitively coupled to a double
quantum dot (DQD) with coupling strength $g_{0}$. b) The DQD is tunnel coupled
to a source (S) and drain (D) lead with zero applied potential bias and the
levels of the DQD are tuned such that the chemical potentials of the leads lie
in between the DQD ground $\ket{g}$ and excited $\ket{e}$ state, which are
depicted by full black lines. The energy splitting between $\ket{g}$ and
$\ket{e}$ is given by $\Omega$. The charge states of the left $\ket{L}$ and
right $\ket{R}$ dot are depicted by grey dashed lines and the detuning between
those charge states is given by $\epsilon$. Electrons from the leads can
tunnel into $\ket{g}$ from the left and right with respective rates
$\Gamma_{\text{L,in}}$ and $\Gamma_{\text{R,in}}$. If the DQD and the
resonator are in resonance, transitions from $\ket{g}\to\ket{e}$ are enhanced.
Electrons tunnel out from $\ket{e}$ into the leads to the left and right with
respective rates $\Gamma_{\text{L,out}}$ and $\Gamma_{\text{R,out}}$.

We consider a DQD which is capacitively coupled to a driven superconducting
transmission line resonator Cottet  _et al._ (2017), a system investigated
both theoretically Childress  _et al._ (2004); Bergenfeldt and Samuelsson
(2013); Lambert  _et al._ (2013); Contreras-Pulido  _et al._ (2013); Xu and
Vavilov (2013a); Bergenfeldt  _et al._ (2014) and experimentally Frey  _et
al._ (2012); Petersson  _et al._ (2012); Viennot  _et al._ (2015); Liu  _et
al._ (2015); Mi  _et al._ (2017); Stockklauser  _et al._ (2017); Bruhat  _et
al._ (2018) in recent years, see Fig. 1. The total system is described by the
Hamiltonian

$H=H_{\text{DQD}}+H_{\text{r}}+H_{\text{JC}}+H_{\text{d}},$ (1)

where $H_{\text{DQD}}$ describes the DQD, $H_{\text{r}}$ the resonator,
$H_{\text{JC}}$ the coupling between the DQD and the resonator and
$H_{\text{d}}$ accounts for the applied drive.

The DQD is operated in the Coulomb blockade regime, where only one excess
electron is energetically allowed to reside on the DQD. The Hilbert space of
the DQD is thus spanned by the states with one excess electron on the left dot
$\ket{L}$, one on the right dot $\ket{R}$ and no excess electron $\ket{0}$. In
this basis, the Hamiltonian of the DQD is given by (setting $\hbar=1$)

$H_{\text{DQD}}=\frac{\epsilon}{2}(\outerproduct{R}{R}-\outerproduct{L}{L})+t_{\text{c}}(\outerproduct{L}{R}+\outerproduct{R}{L}),$
(2)

where $\epsilon$ is the detuning between the two charge states $\ket{L}$ and
$\ket{R}$ and $t_{\text{c}}$ is the tunnel coupling between them. For the
further analysis it is more convenient to work in the eigenstate basis of the
DQD spanned by the ground state $\ket{g}$ and excited state $\ket{e}$ which
are defined by

$\begin{pmatrix}\ket{g}\\\
\ket{e}\end{pmatrix}=\begin{pmatrix}\cos(\theta/2)&&-\sin(\theta/2)\\\
\sin(\theta/2)&&\cos(\theta/2)\end{pmatrix}\begin{pmatrix}\ket{L}\\\
\ket{R}\end{pmatrix},$ (3)

where the mixing angle $\theta$ is given by $\cos(\theta)=-\epsilon/\Omega$
and $\Omega=\sqrt{4t_{c}^{2}+\epsilon^{2}}$ gives the energy splitting between
the DQD eigenstates. In this basis, the Hamiltonian of the DQD can be written
as

$H_{\text{DQD}}=\frac{\Omega}{2}\sigma_{z},$ (4)

where $\sigma_{z}=\outerproduct{e}{e}-\outerproduct{g}{g}$.

The Hamiltonian for the microwave resonator is given by

$H_{\text{r}}=\omega_{\text{r}}a^{\dagger}a,$ (5)

where $\omega_{\text{r}}$ is the the resonator’s characteristic frequency and
the operator $a^{\dagger}$ ($a$) describes the creation (annihilation) of a
microwave photon in the resonator.

The interaction between the DQD and the resonator is described by a standard
Jaynes-Cummings interaction Hamiltonian Jaynes and Cummings (1963)

$H_{\text{JC}}=g\left(a^{\dagger}\sigma_{-}+a\sigma_{+}\right),$ (6)

where $g=g_{0}\sin(\theta)$ gives the interaction strength between the DQD and
the microwave photons inside the resonator with $g_{0}$ being the bare
coupling strength and $\sigma_{+}=\outerproduct{e}{g}=\sigma_{-}^{\dagger}$ is
the DQD raising operator.

The drive is a coherent source of monochromatic microwave radiation feeding in
photons into the resonator and is described by Khan  _et al._ (2021); Clerk
_et al._ (2010)

$H_{\text{d}}=\sqrt{\kappa_{\text{in}}\dot{N}}\left(a^{\dagger}e^{i\omega_{\text{l}}t}+ae^{-i\omega_{\text{l}}t}\right),$
(7)

where $\kappa_{\text{in}}$ gives the coupling strength between the input port
and the resonator, $\dot{N}$ is the rate of impinging photons and
$\omega_{\text{l}}$ is the frequency of the microwave drive.

Moving to the rotating frame of the incoming microwave radiation gives us Jin
_et al._ (2011); Xu and Vavilov (2013b)

$\begin{split}\tilde{H}&=\Delta_{\text{d}}\frac{\sigma_{z}}{2}+\Delta_{\text{r}}a^{\dagger}a+g\left(a^{\dagger}\sigma_{-}+a\sigma_{+}\right)\\\
&+\sqrt{\kappa_{\text{in}}\dot{N}}\left(a^{\dagger}+a\right),\end{split}$ (8)

where we have used

$\tilde{H}=U^{\dagger}HU+i(\partial_{t}U)U^{\dagger},$ (9)

with $U=\exp\left[i\omega_{\text{l}}t(a^{\dagger}a+\sigma_{z}/2)\right]$ and
$\Delta_{\text{d}}=\Omega-\omega_{\text{l}}$
($\Delta_{\text{r}}=\omega_{\text{r}}-\omega_{\text{l}}$) gives the detuning
between the DQD (resonator) and the drive.

The DQD is also tunnel coupled to two fermionic leads called source and drain,
respectively. There is no applied bias between source and drain and the
chemical potential as well as the energy of the empty state $\ket{0}$ are set
to zero without loss of generality. The temperature of the leads is negligible
compared to the DQD level spacing $k_{B}T\ll\Omega$, such that electrons can
only enter into the ground state with rate

$\Gamma_{g0}=\Gamma_{\text{L,in}}+\Gamma_{\text{R,in}},$ (10)

where

$\begin{split}\Gamma_{\text{L,in}}&=\Gamma_{\text{L}}\cos^{2}(\theta/2),\\\\[1.99997pt]
\Gamma_{\text{R,in}}&=\Gamma_{\text{R}}\sin^{2}(\theta/2).\end{split}$ (11)

Here, $\Gamma_{\text{L}}$ ($\Gamma_{\text{R}}$) is the rate for electron
tunneling events into or out of the left (right) dot.

Electrons in the excited state can tunnel back into the leads with rate

$\Gamma_{0e}=\Gamma_{\text{L,out}}+\Gamma_{\text{R,out}},$ (12)

where

$\begin{split}\Gamma_{\text{L,out}}&=\Gamma_{\text{L}}\sin^{2}(\theta/2),\\\\[1.99997pt]
\Gamma_{\text{R,out}}&=\Gamma_{\text{R}}\cos^{2}(\theta/2).\end{split}$ (13)

The DQD level spacing $\Omega$ is tuned to be in resonance with the
resonator’s characteristic frequency $\omega_{\text{r}}$ (meaning that we can
set $\Delta_{\text{d}}=\Delta_{\text{r}}\equiv\Delta$). In that case,
transitions from the ground to the excited state are enhanced.

###  II.2 Master equation for the DQD-resonator system

The dynamics of the DQD-resonator is modeled by the following Lindblad master
equation Lindblad (1976); Breuer and Petruccione (2007)

$\begin{split}\partial_{t}\rho(t)&=-i[\tilde{H},\rho(t)]+\Gamma_{g0}\mathcal{D}[s_{g}^{\dagger}]\rho(t)+\Gamma_{0e}\mathcal{D}[s_{e}]\rho(t)\\\\[2.5pt]
&+\frac{\gamma_{\phi}}{2}\mathcal{D}[\sigma_{z}]\rho(t)+\gamma_{-}\mathcal{D}[\sigma_{-}]\rho(t)+\kappa\mathcal{D}[a]\rho(t),\end{split}$
(14)

with the Lindblad superoperator
$\mathcal{D}[x]\rho(t)=x\rho(t)x^{\dagger}-\frac{1}{2}\\{x^{\dagger}x,\rho(t)\\}$,
where $\rho(t)$ is the density matrix of the DQD-resonator system.

The first term in Eq. (14) describes the unitary time evolution of the system.
The second term describes tunneling of electrons from the leads into the
ground state of the DQD, where $s_{g}^{\dagger}=\outerproduct{g}{0}$. The
third term describes tunneling of electrons out of the excited state of the
DQD into the leads, where $s_{e}=\outerproduct{0}{e}$. The fourth term
describes dephasing of the state of the DQD, which is typically caused by
voltage fluctuations in the electromagnetic environment Büttiker (2001), and
is quantified by the rate $\gamma_{\phi}$. The fifth term describes relaxation
of an electron from the excited into the ground state and is typically due to
coupling of electrons to phonons in the solid state environment Grodecka-Grad
and Förstner (2010); Gawarecki and Machnikowski (2021). Relaxation is
quantified by the rate $\gamma_{-}$. Thermal excitations can be neglected
because $k_{B}T\ll\Omega$. The resonator is subject to losses due to coupling
to the input port of the resonator and also internal losses due to coupling to
the substrate, leading to a total loss rate given by $\kappa$.

###  II.3 Ideal photodetection

Figure 2: Schematic of an ideal photodetection cycle using a DQD coupled to a
microwave resonator. The DQD starts in its empty state $\ket{0}$ where no
excess electron resides on it. An electron from the source can tunnel into the
ground state $\ket{g}$ with rate $\Gamma_{\text{L,in}}$. The DQD is in
resonance with the resonator’s characteristic frequency $\omega_{\text{r}}$,
such that the electron in the ground state moves to the excited state
$\ket{e}$ by absorbing a microwave photon. It will then tunnel out to the
drain with rate $\Gamma_{\text{R,out}}$, which brings the DQD back to its
empty state and closes the cycle.

The operation of the DQD-resonator as an ideal photodetector was described in
Ref. Wong and Vavilov (2017) and is briefly discussed here for completeness.
The photodetection scheme is illustrated in Fig. 2. In the ideal case, each
photon arriving at the resonator is absorbed by the DQD, giving rise to a
photocurrent from source to drain.

In practice, the photodetector will not be ideal. A relevant figure of merit
to quantify how close the DQD-resonator system is to the ideal photo-detector
is the photon-to-electron conversion efficiency, which is given by

$\eta=\frac{I/e}{\dot{N}},$ (15)

where $I$ is the generated photocurrent through the DQD.

In Ref. Wong and Vavilov (2017), certain conditions for the system have been
derived in order to reach $\eta\approx 1$. First, the total photonic loss rate
needs to be dominated by the coupling to the input port of the resonator, i.e.
$\kappa=\kappa_{\text{in}}$. Second, the frequency of the drive should match
the frequency of the resonator, such that $\Delta=0$. Third, the tunneling
rates to the left and right dot need to be symmetric, i.e.
$\Gamma_{\text{L}}=\Gamma_{\text{R}}\equiv\Gamma$ and occur on a faster time
scale than the decoherence rates, i.e. $\Gamma\gg\gamma_{-},\gamma_{\phi}$.
Fourth, the tunnel coupling strength between the dots needs to be small
compared to the resonator frequency, i.e. $t_{\text{c}}\ll\omega_{\text{r}}$.
Fifth, the rate at which photons get absorbed by the DQD, which is given by
$4g^{2}/\Gamma$, should match the photonic loss rate $\kappa$. Lastly, the
generated photocurrent needs to be linearly proportional to the incoming
photon flux, i.e. $I\propto\dot{N}$. This last point reflects that the
probability to have every single photon absorbed by the DQD is maximized in
the limit of weak microwave drive.

##  III Full counting statistics

Due to the quantum statistical nature of the photon absorption and electron
tunneling processes, the photocurrent will be subject to fluctuations in time.
The statistical properties are analyzed within the framework of the FCS, which
allows us to infer the probability distribution $p(n,t)$ of $n$ electrons
having passed to the drain in a time span $t$. From $p(n,t)$ all the cumulants
of the distribution can be obtained.

We calculate the FCS using a master equation approach developed in Ref.
Bagrets and Nazarov (2003). Introducing the counting field $\chi$, the
variable conjugate to $n$, we can write a master equation like the one in Eq.
(14) using the $\chi-$dependent density matrix $\rho(\chi,t)$ as

$\begin{split}\partial_{t}\rho&=-i[\tilde{H},\rho]+\Gamma_{\text{L,in}}\mathcal{D}[s_{g}^{\dagger}]\rho+\Gamma_{\text{L,out}}\mathcal{D}[s_{e}]\rho\\\
&+\Gamma_{\text{R,in}}(e^{-i\chi}s_{g}^{\dagger}\rho
s_{g}-\tfrac{1}{2}\\{s_{g}s_{g}^{\dagger},\rho\\})\\\
&+\Gamma_{\text{R,out}}(e^{i\chi}s_{e}\rho
s_{e}^{\dagger}-\tfrac{1}{2}\\{s_{e}^{\dagger}s_{e},\rho\\})\\\
&+\frac{\gamma_{\phi}}{2}\mathcal{D}[\sigma_{z}]\rho+\gamma_{-}\mathcal{D}[\sigma_{-}]\rho+\kappa\mathcal{D}[a]\rho\\\
&=L(\chi)\rho,\end{split}$ (16)

where $L(\chi)=L_{0}+e^{i\chi}J_{\text{R}}+e^{-i\chi}J_{\text{L}}$ is the
$\chi-$dependent Liouvillian. The term $L_{0}$ contains all terms in the
Liouvillian which leave the number of electrons in the drain unchanged and
$J_{\text{R(L)}}$ contains the terms which increase (decrease) the number of
electrons by one. For a system described by $\rho(t_{0})$ at time $t_{0}$, the
general solution to Eq. (16) is

$\rho(\chi,t)=e^{L(\chi)(t-t_{0})}\rho(t_{0}).$ (17)

Moving forward, we will set $t_{0}=0$ and assume that the system at that time
is in its steady state, i.e. $\rho(t_{0}=0)\equiv\rho_{\text{st}}$.

Instead of calculating the probability distribution $p(n,t)$ directly, it is
more convenient to look at the cumulant-generating function (CGF) $C(\chi,t)$
of the distribution. It can readily be obtained from the $\chi-$dependent
density matrix as

$e^{C(\chi,t)}=\text{Tr}\\{\rho(\chi,t)\\}=\text{Tr}\\{e^{L(\chi)t}\rho_{\text{st}}\\},$
(18)

The cumulants can then be found by differentiation with respect to the
counting field

$\expectationvalue{\expectationvalue{n^{k}}}=\frac{\partial_{k}}{\partial(i\chi)^{k}}C(\chi,t)|_{\chi=0},$
(19)

where $\expectationvalue{\expectationvalue{n^{k}}}$ gives the $k-$th cumulant
of the distribution. The CGF is related to $p(n,t)$ by inverse Fourier
transformation

$p(n,t)=\int_{-\pi}^{\pi}\frac{d\chi}{2\pi}e^{C(\chi,t)-in\chi}.$ (20)

Below we will employ this recipe to compute the cumulants of the photocurrent
through the DQD.

###  III.1 Zero-frequency FCS

We will first investigate the FCS in the long-time, or zero-frequency, limit.
In this limit, the CGF is Schaller (2014)

$C(\chi,t)=\lambda_{0}(\chi)t,$ (21)

where $\lambda_{0}(\chi)$ is the eigenvalue of $L(\chi)$ with the largest real
part. Finding the cumulant generating function thus becomes an eigenvalue
problem in an infinite-dimensional Hilbert space. A direct analytical analysis
of $\lambda_{0}(\chi)$, providing a clear physical picture of the photocurrent
statistics, will therefore only be possible in certain limiting cases.

####  III.1.1 Low-drive limit

Of particular interest is the regime of low-microwave drive, pointed out above
as a prerequisite for ideal photodetection. In this regime, the photocurrent
statistics can be obtained by perturbation theory Gómez  _et al._ (2018),
treating the drive Hamiltonian $H_{\text{d}}$ as a small perturbation to the
DQD-resonator system. The parameter in which we make the perturbative
expansion is the effective drive amplitude [cf. Eq. (7)], which we abbreviate
by $f\equiv\sqrt{\kappa_{\text{in}}\dot{N}}$.

To make progress, we note that we can write the $\chi-$dependent density
matrix in the long-time limit as
$\lim_{t\to\infty}\rho(\chi,t)=\exp[\lambda_{0}(\chi)t]\mu_{0}(\chi)$ (see
Appendix A for details). This allows us to write the master equation (16) as

$\lambda_{0}(\chi)\tilde{\rho}(\chi)=L(\chi)\tilde{\rho}(\chi),$ (22)

where
$\tilde{\rho}(\chi)\equiv\lim_{t\to\infty}[\rho(\chi,t)/\text{Tr}\\{\rho(\chi,t)\\}]$.
Expanding Eq. (22) up to lowest order in $f$ (which is the second one) using
$\lambda_{0}(\chi)=\lambda_{0}^{(0)}(\chi)+\lambda^{(1)}_{0}(\chi)+\lambda^{(2)}_{0}(\chi)$,
$\tilde{\rho}(\chi)=\tilde{\rho}^{(0)}(\chi)+\tilde{\rho}^{(1)}(\chi)+\tilde{\rho}^{(2)}(\chi)$
and $L(\chi)=L^{(0)}(\chi)+L^{(1)}(\chi)$ gives the following equations

$\displaystyle\lambda_{0}^{(0)}\tilde{\rho}^{(0)}=L^{(0)}\tilde{\rho}^{(0)},$
(23)
$\displaystyle\lambda_{0}^{(0)}\tilde{\rho}^{(1)}+\lambda_{0}^{(1)}\tilde{\rho}^{(0)}=L^{(1)}\tilde{\rho}^{(0)}+L^{(0)}\tilde{\rho}^{(1)},$
(24)
$\displaystyle\lambda_{0}^{(2)}\tilde{\rho}^{(0)}+\lambda_{0}^{(1)}\tilde{\rho}^{(1)}+\lambda_{0}^{(0)}\tilde{\rho}^{(2)}=L^{(1)}\tilde{\rho}^{(1)}+L^{(0)}\tilde{\rho}^{(2)}.$
(25)

For the expansion of $\tilde{\rho}(\chi)$ we make an ansatz where we keep
terms with up to one excitation, either in the resonator or the DQD. Details
on the explicit form of the normalized, $\chi-$dependent density matrix to
different orders in $f$ can be found in Appendix B.

Solving Eqs. (23)-(25) gives
$\lambda_{0}^{(0)}(\chi)=\lambda_{0}^{(1)}(\chi)=0$ and

$\lambda_{0}(\chi)=\Gamma_{\text{LR}}(e^{i\chi}-1)+\Gamma_{\text{RL}}(e^{-i\chi}-1),$
(26)

with

$\Gamma_{\alpha\beta}=\frac{16f^{2}g^{2}(\tilde{\Gamma}+\kappa)(4g^{2}+\tilde{\Gamma}\kappa)\Gamma_{\alpha,\text{in}}\Gamma_{\beta,\text{out}}}{\Gamma_{g0}[4g^{2}(\Gamma_{0e}+\gamma_{-}+\kappa)+\kappa(\Gamma_{0e}+\gamma_{-})(\tilde{\Gamma}+\kappa)][(\tilde{\Gamma}^{2}+4\Delta^{2})(\kappa^{2}+4\Delta^{2})+8g^{2}(\tilde{\Gamma}\kappa-4\Delta^{2})+16g^{4}]},$
(27)

where $\tilde{\Gamma}\equiv\Gamma_{0e}+\gamma_{-}+2\gamma_{\phi}$ is the
effective dephasing rate and $\alpha,\beta\in\\{\text{L, R}\\}$. Equation (26)
describes a bi-directional Poisson process. The corresponding low-drive
probability distribution is given by [cf. Eq. (20)]

$p(n,t)=\left(\sqrt{\frac{\Gamma_{\text{LR}}}{\Gamma_{\text{RL}}}}\right)^{n}I_{n}(2\sqrt{\Gamma_{\text{LR}}\Gamma_{\text{RL}}}t)e^{-(\Gamma_{\text{LR}}+\Gamma_{\text{RL}})t},$
(28)

where $I_{n}(x)$ is the modified Bessel function of the first kind Temme
(1996).

Applying the conditions for ideal photodetection introduced in section II to
Eq. (27), the rate $\Gamma_{\text{RL}}$ will go to zero and
$\Gamma_{\text{LR}}$ will reduce to $\dot{N}$. Thus, Eq. (26) reduces to

$\lambda_{0}(\chi)=\dot{N}(e^{i\chi}-1),$ (29)

with a corresponding uni-directional Poissonian probability distribution

$p(n,t)=e^{-\dot{N}t}\frac{(\dot{N}t)^{n}}{n!}.$ (30)

This shows that the statistics of the electrons arriving to the drain in the
limit of ideal photodetection is equivalent to the statistics of photons from
a coherent light source Walls and Milburn (2008). The ideal photodetector does
therefore not only detect each single photon but also preserves their long-
time statistics.

####  III.1.2 Large-drive limit

We also find analytical results in the regime where the microwave drive is
large. In this regime, the resonator photon state has negligible quantum
fluctuations and is independent on the presence of the DQD. We can then
replace the photonic annihilation operator $a$ with a c-number given by
$-2if/(\kappa+2i\Delta)$. This replacement reduces the Liouvillian to a
$5\times 5$ matrix in the basis of the DQD
$\rho_{\text{DQD}}=(\rho_{0},\rho_{g},\rho_{e},\rho_{eg},\rho_{ge})^{T}$,
where $\rho_{i}=\expectationvalue{\rho}{i}$ and
$\rho_{ij}=\matrixelement{i}{\rho}{j}$ (see Appendix for C details). We derive
an analytical expression for $\lambda_{0}(\chi)$ by finding the eigenvalue of
the corresponding $\chi$-dependent Liouvillian in the limit $f\to\infty$

$\lambda_{0}(\chi)=\frac{1}{4}\left\\{-\Gamma_{0e}-2\Gamma_{g0}+\sqrt{8[(e^{i\chi}-1)\Gamma_{\text{L,in}}\Gamma_{\text{R,out}}+(e^{-i\chi}-1)\Gamma_{\text{R,in}}\Gamma_{\text{L,out}}]+(\Gamma_{0e}+2\Gamma_{g0})^{2}}\right\\}.$
(31)

We see that in the limit of large drive, only the in- and out-tunneling rates
of the electrons are relevant for the statistics of the photocurrent. The
resonator essentially works as a mediator for transitions between the ground
and excited sate of the DQD, which are so fast that the corresponding rates
disappear from the FCS.

###  III.2 Finite-frequency noise

The zero-frequency cumulants give us insight into many of the statistical
properties of the electron transport through our system. However, a full
picture of the relevant correlations and time scales of said properties
requires an analysis over the full frequency range Ubbelohde  _et al._ (2012).
We will thus turn to the FFN, which can be calculated by employing MacDonald’s
formula Macdonald (1949)

$S(\omega)=\omega\int_{0}^{\infty}dt\sin(\omega
t)\frac{d}{dt}\expectationvalue{\expectationvalue{n^{2}}}(t).$ (32)

Following Ref. Flindt  _et al._ (2005a), we write Eq. (32) by moving to
Laplace space as

$S(\omega)=-\omega^{2}\mathfrak{Re}\left\\{\expectationvalue{\expectationvalue{n^{2}}}(z=-i\omega)\right\\}.$
(33)

To find $\expectationvalue{\expectationvalue{n^{2}}}(z)$, we use the
$\chi-$dependent density matrix in Laplace space

$\rho(\chi,z)=\sum_{n=0}^{\infty}\\{\Omega_{z}[(e^{i\chi}-1)J_{\text{R}}+(e^{-i\chi}-1)J_{\text{L}}]\\}^{n}\Omega_{z}\rho_{\text{st}},$
(34)

where the propagator in Laplace space is defined by
$\Omega_{z}=[z-L(\chi=0)]^{-1}$. The second cumulant is found by expanding Eq.
(34) in $\chi$ and keeping only the second-order terms

$\begin{split}\expectationvalue{\expectationvalue{n^{2}}}(z)&=\partial_{i\chi}^{2}\text{Tr}\\{\rho(\chi,z)\\}|_{\chi\to
0}\\\\[1.99997pt]
&=\text{Tr}\\{(\Omega_{z}\mathcal{J}\Omega_{z}+2\Omega_{z}\mathcal{I}\Omega_{z}\mathcal{I}\Omega_{z})\rho_{\text{st}}\\},\end{split}$
(35)

where $\mathcal{J}=J_{\text{R}}+J_{\text{L}}$ and
$\mathcal{I}=J_{\text{R}}-J_{\text{L}}$, where $J_{\text{L}}$ and
$J_{\text{R}}$ are defined below Eq. (16).

####  III.2.1 Low-drive limit

We obtain an analytical expression for the FFN in the low-drive limit. The
steady state is given from Eqs. (23)-(25) by putting $\chi=0$, resulting in
the FFN

$S(\omega)=\Gamma_{\text{LR}}+\Gamma_{\text{RL}}+\frac{2\Gamma_{\text{RR}}\omega^{2}}{\Gamma_{g0}^{2}+\omega^{2}}.$
(36)

The frequency dependence is governed by the rate $\Gamma_{g0}$. This can be
understood in the following way: In the low-drive limit, the system is almost
always in its ground state - the event of an electron getting excited and
jumping out of the DQD occurs with small probability. If it does, another
electron will tunnel into the ground state with rate $\Gamma_{g0}$. Tunneling
events thus occur in pairs. Events within one pair are correlated and
separated by the time-scale $1/\Gamma_{g0}$, which can thus be interpreted as
the dead-time of the photodetector. Tunneling events from different pairs are
uncorrelated, as the pairs are separated by a large time-scale proportional to
$1/f^{2}$ [cf. Eq. (27)].

We note that in the limit of ideal photodetection, the FFN reduces to
$S(\omega)=\dot{N}$, which is equivalent to the noise of photons emitted from
a coherent light source.

####  III.2.2 Large-drive limit

It was pointed out above that the Liouvillian reduces to a $5\times 5$ matrix
in the regime where the drive is strong. Inserting the Liouvillian into the
formula for the propagator $\Omega_{z}$ and taking the limit $f\to\infty$ in
Eq. (35) for both the propagator and the steady state, we find for the FFN

$\begin{split}S(\omega)&=\frac{1}{\Gamma_{0e}+2\Gamma_{g0}}\Big{[}\Gamma_{0e}\Gamma_{\text{R,in}}+\Gamma_{g0}\Gamma_{\text{R,out}}\\\
&-\frac{2(2\Gamma_{\text{R,in}}+\Gamma_{\text{R,out}})(\Gamma_{0e}^{2}\Gamma_{\text{R,in}}+2\Gamma_{g0}^{2}\Gamma_{\text{R,out}})}{(\Gamma_{0e}+2\Gamma_{g0})^{2}+4\omega^{2}}\Big{]}.\end{split}$
(37)

The frequency dependence is thus governed by both the in- and outgoing rate
from the ground and excited state of the DQD. In direct analogy to the low-
drive case we interpret the time-scale $1/(\Gamma_{0e}+2\Gamma_{g0})$ as the
dead-time of the photodetector. We note however that in the large drive limit,
the detector is oversaturated and cannot detect every single photon.

##  IV Mean-field approach

To obtain analytical expressions for the different cumulants of the
photocurrent beyond the low- and large-drive limit, we employ a mean field
approach Plankensteiner  _et al._ (2022); Prataviera and Mizrahi (2014).
Albeit being an approximation, we find that the mean-field solutions reproduce
the exact results to a large degree and in a wide range of parameters. We note
that this approach has been used in a similar system to obtain the average
current through the DQD Wong and Vavilov (2017). Here we extend that work by
analyzing the zero-frequency FCS and the FFN.

Our approach relies on the assumption that correlations between the states of
the DQD and the resonator can be neglected, such that at all times we can
approximate the system’s density matrix as
$\rho(t)=\rho_{\text{DQD}}(t)\otimes\rho_{\text{r}}(t)$, a product between the
reduced density matrix for the DQD and the resonator, respectively. Inserting
this factorization of the density matrix into Eq. (14) and tracing over the
degrees of freedom of the resonator, we get a master equation for the DQD,
where the properties of the resonator enter only via their average values. We
thereafter include the counting fields into this master equation, as we want
to count electron-tunneling events through the DQD, and get

$\displaystyle\partial_{t}\rho_{\text{DQD}}$ $\displaystyle\\!\begin{aligned}
&=-i[\bar{H}_{\text{DQD}},\rho_{\text{DQD}}]\\\\[1.99997pt]
&+\Gamma_{\text{L,in}}\mathcal{D}[s_{g}^{\dagger}]\rho_{\text{DQD}}+\Gamma_{\text{L,out}}\mathcal{D}[s_{e}]\rho_{\text{DQD}}\\\\[1.99997pt]
&+\Gamma_{\text{R,in}}(e^{-i\chi}s_{g}^{\dagger}\rho_{\text{DQD}}s_{g}-\tfrac{1}{2}\\{s_{g}s_{g}^{\dagger},\rho_{\text{DQD}}\\})\\\\[1.99997pt]
&+\Gamma_{\text{R,out}}(e^{i\chi}s_{e}\rho_{\text{DQD}}s_{e}^{\dagger}-\tfrac{1}{2}\\{s_{e}^{\dagger}s_{e},\rho_{\text{DQD}}\\})\\\\[1.99997pt]
&+\frac{\gamma_{\phi}}{2}\mathcal{D}[\sigma_{z}]\rho_{\text{DQD}}+\gamma_{-}\mathcal{D}[\sigma_{-}]\rho_{\text{DQD}},\end{aligned}$
(38) $\displaystyle\bar{H}_{\text{DQD}}$ $\displaystyle\\!\begin{aligned}
&=\Delta\frac{\sigma_{z}}{2}+g\left(\expectationvalue{a^{\dagger}}\sigma_{-}+\expectationvalue{a}\sigma_{+}\right).\end{aligned}$
(39)

Along the same lines, tracing over the degrees of freedom of the DQD gives a
master equation for the resonator

$\displaystyle\partial_{t}\rho_{\text{r}}$ $\displaystyle\\!\begin{aligned}
&=-i[\bar{H}_{\text{r}},\rho_{\text{r}}]+\kappa\mathcal{D}[a]\rho_{\text{r}},\end{aligned}$
(40) $\displaystyle\bar{H}_{\text{r}}$ $\displaystyle\\!\begin{aligned}
&=\Delta
a^{\dagger}a+g\left(a^{\dagger}\expectationvalue{\sigma_{-}}_{\chi=0}+a\expectationvalue{\sigma_{+}}_{\chi=0}\right)\\\
&+f\left(a^{\dagger}+a\right).\end{aligned}$ (41)

We use Eqs. (38)-(40) to find equations of motions for operators which act on
the subspace of the DQD and the resonator, respectively.

###  IV.1 Mean-field FCS

In the long-time limit, we write the reduced, $\chi-$dependent density matrix
for the DQD as

$\tilde{\rho}_{\text{DQD}}(\chi)\equiv\lim_{t\to\infty}\frac{\rho_{\text{DQD}}(\chi,t)}{\text{Tr}\\{\rho_{\text{DQD}}(\chi,t)\\}}.$
(42)

With this at hand, we obtain the following set of equations for the DQD [cf.
Eq. (22)]

$\displaystyle\lambda_{0}(\chi)\tilde{p}_{0}$ $\displaystyle\\!\begin{aligned}
&=-\Gamma_{g0}\tilde{p}_{0}+(\Gamma_{\text{L,out}}+e^{i\chi}\Gamma_{\text{R,out}})\tilde{p}_{e},\end{aligned}$
(43) $\displaystyle\lambda_{0}(\chi)\tilde{p}_{e}$
$\displaystyle\\!\begin{aligned}
&=-(\Gamma_{0e}+\gamma_{-})\tilde{p}_{e}\\\\[1.99997pt]
&-ig(\expectationvalue{a}\expectationvalue{\sigma_{+}}-\expectationvalue{a^{\dagger}}\expectationvalue{\sigma_{-}}),\end{aligned}$
(44) $\displaystyle\lambda_{0}(\chi)\tilde{p}_{g}$
$\displaystyle\\!\begin{aligned}
&=(\Gamma_{\text{L,in}}+e^{-i\chi}\Gamma_{\text{R,in}})\tilde{p}_{0}+\gamma_{-}\tilde{p}_{e}\\\\[1.99997pt]
&+ig(\expectationvalue{a}\expectationvalue{\sigma_{+}}-\expectationvalue{a^{\dagger}}\expectationvalue{\sigma_{-}}),\end{aligned}$
(45) $\displaystyle\lambda_{0}(\chi)\expectationvalue{\sigma_{+}}$
$\displaystyle\\!\begin{aligned}
&=-\left(\frac{\tilde{\Gamma}}{2}-i\Delta\right)\expectationvalue{\sigma_{+}}-ig\expectationvalue{a^{\dagger}}\expectationvalue{\sigma_{z}},\end{aligned}$
(46)

where
$\tilde{p}_{i}=\text{Tr}\\{\outerproduct{i}{i}\tilde{\rho}_{\text{DQD}}(\chi)\\}$
and
$\expectationvalue{\sigma_{j}}=\text{Tr}\\{\sigma_{j}\tilde{\rho}_{\text{DQD}}(\chi)\\}$
for $j\in\\{+,-,z\\}$. For the resonator, we have

$0=-\left(\frac{\kappa}{2}-i\Delta\right)\expectationvalue{a^{\dagger}}+if+ig\expectationvalue{\sigma_{+}}_{\chi=0}.$
(47)

Using Eqs. (43)-(47), together with the conservation of the effective
probabilities $\tilde{p}_{0}+\tilde{p}_{e}+\tilde{p}_{g}=1$, we find two
coupled equations for $\tilde{p}_{e}$ and $\lambda_{0}(\chi)$

$\tilde{p}_{e}=\frac{-16f^{2}g^{2}(\tilde{\Gamma}^{2}+4\Delta^{2})(\tilde{\Gamma}+2\lambda_{0})(\Gamma_{g0}+\lambda_{0})\Lambda}{(\Gamma_{0e}+\gamma_{-}+\lambda_{0})[(\tilde{\Gamma}+2\lambda_{0})^{2}+4\Delta^{2}][(\Gamma_{g0}+\lambda_{0})^{2}(\tilde{\Gamma}^{2}+4\Delta^{2})(\kappa^{2}+4\Delta^{2})+8g^{2}\Lambda(\Gamma_{g0}+\lambda_{0})(4\Delta^{2}-\tilde{\Gamma}\kappa)+16g^{4}\Lambda^{2}]},$
(48)
$\lambda_{0}=\frac{\tilde{p}_{e}}{\Gamma_{g0}+\lambda_{0}}[(e^{i\chi}-1)(\Gamma_{\text{L,in}}\Gamma_{\text{R,out}}+\Gamma_{\text{R,out}}\lambda_{0})+(e^{-i\chi}-1)\Gamma_{\text{R,in}}\Gamma_{\text{L,out}}],$
(49)

where
$\Lambda\equiv(2\tilde{p}_{e}-1)\Gamma_{g0}-\lambda_{0}+\tilde{p}_{e}(\Gamma_{\text{L,out}}+e^{i\chi}\Gamma_{\text{R,out}}+2\lambda_{0})$.

The solution of Eq. (49) which goes to zero for $\chi\to 0$ is the one of
interest. It is given by

$\lambda_{0}(\chi)=\frac{1}{2}\Big{\\{}-\Gamma_{g0}+(e^{i\chi}-1)\tilde{p}_{e}\Gamma_{\text{R,out}}+\sqrt{4\tilde{p}_{e}[(e^{i\chi}-1)\Gamma_{\text{L,in}}\Gamma_{\text{R,out}}+(e^{-i\chi}-1)\Gamma_{\text{R,in}}\Gamma_{\text{L,out}}]+[\Gamma_{g0}-(e^{i\chi}-1)\tilde{p}_{e}\Gamma_{\text{R,out}}]^{2}}\Big{\\}}.$
(50)

By expanding Eq. (50) in $\chi$ using $\lambda_{0}(\chi)\approx
i\chi\expectationvalue{\expectationvalue{n^{1}}}-\frac{\chi^{2}}{2}\expectationvalue{\expectationvalue{n^{2}}}+\mathcal{O}[\chi^{3}]$
and
$\tilde{p}_{e}\approx\tilde{p}_{e}^{(0)}+i\chi\tilde{p}_{e}^{(1)}+\mathcal{O}[\chi^{2}]$,
the first and second cumulant, which give the mean current and the shot-noise,
respectively, are

$\displaystyle\expectationvalue{\expectationvalue{n^{1}}}$
$\displaystyle\\!\begin{aligned}
&=\frac{\tilde{p}_{e}^{(0)}}{\Gamma_{g0}}(\Gamma_{\text{L,in}}\Gamma_{\text{R,out}}-\Gamma_{\text{R,in}}\Gamma_{\text{L,out}}),\end{aligned}$
(51) $\displaystyle\expectationvalue{\expectationvalue{n^{2}}}$
$\displaystyle\\!\begin{aligned}
&=\frac{2\tilde{p}_{e}^{(1)}}{\Gamma_{g0}}(\Gamma_{\text{L,in}}\Gamma_{\text{R,out}}-\Gamma_{\text{R,in}}\Gamma_{\text{L,out}})\\\\[1.99997pt]
&+\frac{2\tilde{p}_{e}^{(0)2}}{\Gamma_{g0}^{3}}[\Gamma_{0e}\Gamma_{\text{R,in}}(\Gamma_{\text{L,in}}\Gamma_{\text{R,out}}-\Gamma_{\text{R,in}}\Gamma_{\text{L,out}})]\\\\[1.99997pt]
&+\frac{\tilde{p}_{e}^{(0)}}{\Gamma_{g0}}(\Gamma_{\text{L,in}}\Gamma_{\text{R,out}}+\Gamma_{\text{R,in}}\Gamma_{\text{L,out}}).\end{aligned}$
(52)

Third and higher-order cumulants could be found along the same lines. The
different orders of $\tilde{p}_{e}$ are found by expanding Eq. (48) to zeroth
and first order in $\chi$, respectively. The full expressions for
$\tilde{p}_{e}^{(0)}$ and $\tilde{p}_{e}^{(1)}$ are given in Appendix D.

In the low-drive regime we can again obtain a closed analytical expression for
the FCS by expanding Eqs. (48)-(49) to lowest order in $f$. This gives

$\lambda^{\text{mf}}_{0}(\chi)=\Gamma_{\text{LR}}^{\text{mf}}(e^{i\chi}-1)+\Gamma_{\text{RL}}^{\text{mf}}(e^{-i\chi}-1),$
(53)

where

$\frac{\Gamma_{\alpha\beta}^{\text{mf}}-\Gamma_{\alpha\beta}}{\Gamma_{\alpha\beta}}=\frac{8g^{2}\kappa\gamma_{\phi}}{(\Gamma_{0e}+\gamma_{-})(\tilde{\Gamma}+\kappa)(4g^{2}+\tilde{\Gamma}\kappa)},$
(54)

where $\alpha,\beta\in\\{\text{L, R}\\}$. We again get a bi-directional
Poisson process describing the statistics of the photocurrent in the low-drive
regime but with different rates compared to the perturbative result in Eq.
(26).

The FCS in the large-drive regime is found by expanding Eqs. (48)-(49) to
zeroth order in $1/f$. This gives exactly the same result as in Eq. (31), i.e.
the mean-field approach is exact in the large-drive limit.

In Fig. 3a) we plot the first two cumulants in Eqs. (51)-(52) and compare them
to the low-drive result in Eq. (53) and the large-drive result in Eq. (31). We
can clearly see that the cumulants match the low-drive limit in the regime
where the drive is weak and approach the large-drive limit asymptotically.

In addition, we look at the non-Gaussian properties of the probability
distribution $p(n,t)$ by plotting the third cumulant, see Fig. 3b). We see
that the third cumulant is non-zero and is monotonically increasing as a
function of the microwave drive. This shows a non-Gaussian behavior of the
distribution. In the inset we plot the logarithm of the probability
distribution in the saddle-point approximation (see Appendix E for details)
for the low- and large-drive limit and compare them to the logarithm of a
Gaussian distribution. For the chosen parameters, we see that both the low-
and large-drive limits are skewed and hence deviate from a Gaussian
distribution.

###  IV.2 Mean-field FFN

The mean-field approach also allows us to find an analytical solution for the
FFN. The Liouvillian for the DQD is a $5\times 5$ matrix in the basis of the
DQD. The full expression for the noise is, however, too lengthy to be given
here. We find a closed analytical expression in the low-drive regime by
expanding Eq.(35) to lowest order in $f$

$S^{\text{mf}}(\omega)=\Gamma^{\text{mf}}_{\text{LR}}+\Gamma^{\text{mf}}_{\text{RL}}+\frac{2\Gamma^{\text{mf}}_{\text{RR}}\omega^{2}}{\Gamma_{g0}^{2}+\omega^{2}}.$
(55)

Similar to the mean-field zero-frequency FCS in Eq. (53), the form of the
expression for the FFN is the same as in Eq. (36), however with the rates
given by their mean-field expression in Eq. (54). In the large-drive limit,
the FFN coincides with the exact result in Eq. (37).

Figure 3: a) The first two zero-frequency cumulants within the mean-field
approximation (full lines) normalized with $\Gamma t$ and plotted as a
function of the normalized drive amplitude $f/\Gamma$, where
$\Gamma\equiv\Gamma_{\text{L}}=\Gamma_{\text{R}}$. The cumulants are compared
to the analytical results in the low- and large-drive limits (dotted lines).
b) The third cumulant within the mean-field approximation (full line)
normalized with $\Gamma t$ and plotted as a function of $f/\Gamma$. The third
cumulant is also compared to the low- and large-drive limits (dotted lines).
Inset: Probability distribution in the saddle-point approximation for the low-
drive (red-dashed line) and the large-drive limit (blue dotted line) as a
function of
$(n-\expectationvalue{\expectationvalue{n^{1}}})/\expectationvalue{\expectationvalue{n^{2}}}$.
A Gaussian is shown for comparison (gray full line). The distributions are
normalized by their second cumulant
$\expectationvalue{\expectationvalue{n^{2}}}$. c) The FFN within the mean-
field approximation (full lines) normalized with its value at zero frequency
$S(0)$ and plotted as a function of the normalized frequency $\omega/\Gamma$
for different values of $f/\Gamma$. The FFN is compared to the analytical
results in low- and large-drive limit (dotted lines). The chosen system
parameters are $g/\Gamma=0.457$, $\kappa_{\text{in}}/\Gamma=0.094$,
$\kappa/\Gamma=0.337$, $\epsilon/\Gamma=-17.57$, $t_{c}/\Gamma=6.78$,
$\gamma_{-}/\Gamma=0.5$, $\gamma_{\phi}/\Gamma=3.92$, and $\Delta=0$. The
numerical values for the rates are taken from Ref. Khan  _et al._ (2021).

In Fig. 3c) we plot the noise for different values of $f$ and compare them to
the low-drive result in Eq. (55) and the large-drive result in Eq. (37). We
can see that the mean-field FFN matches the low-drive limit well for small
values of $f$ and approaches the large-drive limit for larger values of $f$.

###  IV.3 Validity of the mean-field approach

It is a priori not clear to what extent the assumption to neglect correlations
between the DQD and the resonator is justified. We therefore check the
validity of the mean-field approach by first comparing the result in Eq. (53)
to the perturbative one in in Eq. (26). We have seen that the mean-field
results are exact in the large-drive limit. From Eq. (54) it is clear that the
mean-field result in the low-drive limit becomes exact if either
$\gamma_{\phi}=0$, $g=0$ or $\kappa=0$. However, the photo-detector requires a
non-zero $g$ and $\kappa$ to operate. Importantly, in the limit of ideal
photodetection, where $\Gamma\gg\gamma_{-},\gamma_{\phi}$, the difference in
Eq. (54) becomes vanishingly small.

Outside the low- and large-drive limit, the mean field solutions can be
benchmarked against an exact, numerical result for the cumulants. Details on
the numerical calculations can be found in Appendix F. In Fig. 4, we compare
the first two zero-frequency cumulants in Eqs. (51) and (52) and the FFN from
the mean-field calculations with exact numerics. We can clearly see that, in
the chosen parameter regime, the mean-field approach fits the numerics very
well.

Figure 4: a) Comparisons between analytical (dashed lines) and numerical (full
lines) calculations of the normalized, zero-frequency cumulants
$\expectationvalue{\expectationvalue{n^{k}}}/(\Gamma t)$ for $k=1,2$, where
$\Gamma\equiv\Gamma_{\text{L}}=\Gamma_{\text{R}}$. The cumulants are plotted
as a function of the normalized drive amplitude $f/\Gamma$. b) Comparison
between analytical (dotted line) and numerical (full line) calculations of the
normalized FFN $S(\omega)/\Gamma$ plotted as a function of the normalized
frequency $\omega/\Gamma$. The analytical solutions of the cumulants and the
noise were found using a mean-field approach, and the numerical solutions were
found using QuTip Johansson  _et al._ (2012). The chosen system parameters are
$\Gamma t=2.9\cdot 10^{5}$, $g/\Gamma=0.457$,
$\kappa_{\text{in}}/\Gamma=0.094$, $\kappa/\Gamma=0.337$,
$\epsilon/\Gamma=-17.57$, $t_{c}/\Gamma=6.78$, $\gamma_{-}/\Gamma=0.5$,
$\gamma_{\phi}/\Gamma=3.92$, $\Delta=0$ for a) and also $f/\Gamma=0.036$ for
b). The numerical values for the rates are taken from Ref. Khan  _et al._
(2021).

##  V Conclusions and outlook

To summarize, we have theoretically investigated the FCS of the photocurrent
through a DQD which is coupled to a driven microwave resonator. The zero-
frequency FCS and the FFN show that the detector preserves the statistics of
the incoming microwave photons in the parameter regime where the system was
found to be an ideal photodetector. The statistics in that case can be
described by a Poisson process. We have found analytical solutions for the
zero-frequency FCS and the FFN in the limit of low and large applied drive,
respectively. We have also shown how a mean-field approximation can be used to
compute the zero-frequency cumulants and the FFN. Analysis of the third
cumulant showed that the probability distribution $p(n,t)$ exhibits a non-
Gaussian behavior. Comparison of the mean-field approach to exact numerics
showed excellent agreement in the parameter regime of interest.

Our results pave the way for a better theoretical understanding of the
statistical properties of electrons and photons inside a DQD-resonator hybrid
structure used as an efficient and continuous single-photon detector. To
achieve single-photon detection, real-time charge sensing techniques could be
employed Gustavsson  _et al._ (2007b); Ouyang  _et al._ (2010). Such
techniques typically induce back-action from the measurement device Ouyang
_et al._ (2010), affecting the statistics of the photocurrent. A theoretical
investigation into a specific architecture allowing for single-photon
detection provides an interesting avenue for future research. Other
interesting directions include the investigation of different drives, for
instance single-photon sources, as well as the possibility of creating non-
classical states of light and the study of their statistical properties using
the present architecture Walls and Milburn (2008); Liu  _et al._ (2014);
Stockklauser  _et al._ (2015); Liu  _et al._ (2017); van den Berg and
Samuelsson (2019); Agarwalla  _et al._ (2019).

## Acknowledgments

This work was supported by the Knut and Alice Wallenberg Foundation through
the Wallenberg Center for Quantum Technology (WACQT). P. P. P. acknowledges
funding from the Swiss National Science Foundation (Eccellenza Professorial
Fellowship PCEFP2_194268).

##  Appendix A Expression for the density matrix in the long-time limit

In order to derive the expression for the $\chi-$dependent density matrix in
the long-time limit and find the explicit expression for $\mu_{0}(\chi)$, we
have to start by giving the eigendecomposition of the the Liouvillian
$L(\chi)$. Since the Liouvillian is a superoperator, its eigenvectors will be
operators. We represent these eigenvectors using a special Dirac notation and
write $\ket{\psi_{k}(\chi)}\rangle$ for the right and
$\langle\bra{\psi_{k}(\chi)}$ for the left eigenvector, respectively Flindt
_et al._ (2005b). Note that, in general,
$\ket{\psi_{k}(\chi)}\rangle^{\dagger}\neq\langle\bra{\psi_{k}(\chi)}$. Using
this, the eigendecomposition of $L(\chi)$ is given by

$L(\chi)=\sum_{k=0}^{N-1}\lambda_{k}(\chi)\ket{\psi_{k}(\chi)}\rangle\langle\bra{\psi_{k}(\chi)},$
(56)

where $\lambda_{k}(\chi)$ are the eigenvalues of $L(\chi)$. Using Eq. (56), we
can write Eq. (17) of the main text as

$\rho(\chi,t)=\sum_{k=0}^{N-1}e^{\lambda_{k}(\chi)t}\ket{\psi_{k}(\chi)}\rangle\langle\bra{\psi_{k}(\chi)}\rho_{\text{st}}.$
(57)

We have seen that, in the long-time limit, only the eigenvalue
$\lambda_{0}(\chi)$ will be of importance, since it yields the smallest
exponentially damped contribution to the density matrix. Using Eq. (57)
together with Eq. (17) of the main text, we can write the $\chi-$dependent
density matrix in the long-time limit as

$\lim_{t\to\infty}\rho(\chi,t)=e^{\lambda_{0}(\chi)t}\mu_{0}(\chi),$ (58)

where

$\mu_{0}(\chi)=\ket{\psi_{0}(\chi)}\rangle\langle\bra{\psi_{0}(\chi)}\rho_{\text{st}}.$
(59)

##  Appendix B Perturbative low-drive expansion

In this Appendix, we will give the explicit expressions for the normalized
$\chi-$dependent density matrix in the long-time limit $\tilde{\rho}(\chi)$ to
different orders in $f$. To zeroth order, there are no photons arriving in the
resonator, such that the electron on the DQD will remain in the ground state
for ever. We can thus make the following ansatz

$\tilde{\rho}^{(0)}(\chi)=\outerproduct{g,0_{\text{r}}}{g,0_{\text{r}}},$ (60)

where $\ket{0_{\text{r}}}$ describes the state of the resonator with zero
photons. We have added the subscript in order to differentiate the empty-
resonator state from the empty-dot state $\ket{0}$.

To first order, we consider superpositions of states having one and zero
excitations, either in the resonator or in the DQD. We thus make the following
ansatz

$\begin{split}\tilde{\rho}^{(1)}(\chi)&=\alpha_{1}\outerproduct{g,1}{g,0_{\text{r}}}+\alpha_{2}\outerproduct{g,0_{\text{r}}}{g,1}\\\
&+\alpha_{3}\outerproduct{e,0_{\text{r}}}{g,0_{\text{r}}}+\alpha_{4}\outerproduct{g,0_{\text{r}}}{e,0_{\text{r}}}.\end{split}$
(61)

The prefactors $\alpha_{1-4}$ can be found by plugging Eqs. (61) and (60) into
Eq. (24) of the main text.

To second order, we similarly include all density matrix elements where the
number of excitations in the bra and the ket sum to two, resulting in the
following ansatz

$\begin{split}\tilde{\rho}^{(2)}(\chi)&=\beta_{1}\outerproduct{g,1}{g,1}+\beta_{2}\outerproduct{e,0_{\text{r}}}{e,0_{\text{r}}}+\beta_{3}\outerproduct{g,1}{e,0_{\text{r}}}\\\
&+\beta_{4}\outerproduct{e,0_{\text{r}}}{g,1}+\beta_{5}\outerproduct{g,2}{g,0_{\text{r}}}+\beta_{6}\outerproduct{g,0_{\text{r}}}{g,2}\\\
&+\beta_{7}\outerproduct{e,1}{g,0_{\text{r}}}+\beta_{8}\outerproduct{g,0_{\text{r}}}{e,1}+\beta_{9}\outerproduct{g,0_{\text{r}}}{g,0_{\text{r}}}\\\
&+\beta_{10}\outerproduct{0,0_{\text{r}}}{0,0_{\text{r}}}.\end{split}$ (62)

Note that we included the last two terms since they are coupled to the first
and second term by the process of a photon or electron leaving the system,
respectively. Plugging Eq. (62) into Eq. (25) of the main text and using the
fact that $\text{Tr}\\{\tilde{\rho}^{(0)}(\chi)\\}=1$ and
$\text{Tr}\\{\tilde{\rho}^{(1)}(\chi)\\}=\text{Tr}\\{\tilde{\rho}^{(2)}(\chi)\\}=0$
gives us a closed set of equations for the prefactors $\beta_{1-10}$.

##  Appendix C Large-drive limit

We start by giving the equation of motion for the photonic annihilation
operator in steady state [cf. Eq. (47)]

$0=-\left(\frac{\kappa}{2}+i\Delta\right)\expectationvalue{a}-if-
ig\expectationvalue{\sigma_{+}}.$ (63)

For large drives, fluctuations of the resonator photon state and the
backaction from the DQD can be neglected. We thus make the replacement

$a=-\frac{2if}{\kappa+2i\Delta}.$ (64)

Inserting this replacement in to Eq. (16) results in the following,
$\chi-$dependent master equation

$\begin{split}\partial_{t}\rho&=-i[\tilde{H}^{\prime},\rho]+\Gamma_{\text{L,in}}\mathcal{D}[s_{g}^{\dagger}]\rho+\Gamma_{\text{L,out}}\mathcal{D}[s_{e}]\rho\\\
&+\Gamma_{\text{R,in}}(e^{-i\chi}s_{g}^{\dagger}\rho
s_{g}-\tfrac{1}{2}\\{s_{g}s_{g}^{\dagger},\rho\\})\\\
&+\Gamma_{\text{R,out}}(e^{i\chi}s_{e}\rho
s_{e}^{\dagger}-\tfrac{1}{2}\\{s_{e}^{\dagger}s_{e},\rho\\})\\\
&+\frac{\gamma_{\phi}}{2}\mathcal{D}[\sigma_{z}]\rho+\gamma_{-}\mathcal{D}[\sigma_{-}]\rho,\end{split}$
(65)

where

$\tilde{H}^{\prime}=\Delta\frac{\sigma_{z}}{2}+2ifg\left(\frac{\sigma_{-}}{\kappa-2i\Delta}-\frac{\sigma_{+}}{\kappa+2i\Delta}\right).$
(66)

##  Appendix D Analytical expressions for $\tilde{p}_{e}^{(0)}$ and
$\tilde{p}_{e}^{(1)}$.

Expanding Eq. (48) to zeroth order in $\chi$ gives the following cubic
equation for $\tilde{p}_{e}^{(0)}$

$A\tilde{p}_{e}^{(0)3}+B\tilde{p}_{e}^{(0)2}+C\tilde{p}_{e}^{(0)}+D=0,$ (67)

where

$A=-16g^{4}(\Gamma_{0e}+\gamma_{-})(\Gamma_{0e}+2\Gamma_{g0})^{2},$ (68)
$B=8g^{2}\Gamma_{g0}(\Gamma_{0e}+\gamma_{-})(\Gamma_{0e}+2\Gamma_{g0})(\tilde{\Gamma}\kappa+4g^{2}-4\Delta^{2}),$
(69)
$C=-\Gamma_{g0}\\{\Gamma_{g0}(\Gamma_{0e}+\gamma_{-})[(4g^{2}+\kappa\tilde{\Gamma}-4\Delta^{2})^{2}+4\Delta^{2}(\kappa+\tilde{\Gamma})^{2}]+16g^{2}f^{2}\tilde{\Gamma}(\Gamma_{0e}+2\Gamma_{g0})\\},$
(70) $D=16f^{2}g^{2}\tilde{\Gamma}\Gamma_{g0}^{2}.$ (71)

A general solution to Eq. (67) is given by

$p_{e}^{(0)}=\\{q+[q^{2}+(r-p^{2})^{3}]^{\frac{1}{2}}\\}^{\frac{1}{3}}+\\{q-[q^{2}+(r-p^{2})^{3}]^{\frac{1}{2}}\\}^{\frac{1}{3}}+p,$
(72)

where

$\displaystyle p$ $\displaystyle=-\frac{B}{3A},$ (73) $\displaystyle q$
$\displaystyle=p^{3}+\frac{BC-3AD}{6A^{2}},$ (74) $\displaystyle r$
$\displaystyle=\frac{C}{3A}.$ (75)

Two out of the three solutions in Eq. (72) are complex and thus non-physical.
We keep the real solution to compute the cumulants in Eqs. (51)-(52).

Expanding Eq. (48) to first order in $\chi$ gives the following linear
equation for $\tilde{p}_{e}^{(1)}$

$\tilde{p}_{e}^{(1)}=-\frac{G\tilde{p}_{e}^{(0)}\expectationvalue{\expectationvalue{n^{1}}}(\tilde{\Gamma}^{2}+4\Delta^{2})+E}{F(\tilde{\Gamma}^{2}+4\Delta^{2})},$
(76)

where $E$, $F$, and $G$ are polynomials

$E=16f^{2}g^{2}(A_{1}\tilde{p}_{e}^{(0)3}+B_{1}\tilde{p}_{e}^{(0)2}+C_{1}\tilde{p}_{e}^{(0)}+D_{1}),$
(77)

with coefficients

$A_{1}=16g^{4}(\Gamma_{0e}+2\Gamma_{g0})^{2}\\{\tilde{\Gamma}^{3}(\expectationvalue{\expectationvalue{n^{1}}}\Gamma_{0e}-\Gamma_{\text{R,out}}\Gamma_{g0})+4\Delta^{2}\\{\expectationvalue{\expectationvalue{n^{1}}}[2\Gamma_{g0}(\Gamma_{0e}+2\Gamma_{g0})+\Gamma_{0e}\tilde{\Gamma}]-\Gamma_{\text{R,out}}\Gamma_{g0}\tilde{\Gamma}\\}\\},$
(78)
$\begin{split}B_{1}&=32g^{2}\Gamma_{g0}^{2}(\Gamma_{0e}+2\Gamma_{g0})\\{g^{2}\Gamma_{\text{R,out}}\Gamma_{g0}\tilde{\Gamma}(\tilde{\Gamma}^{2}+4\Delta^{2})+\expectationvalue{\expectationvalue{n^{1}}}\\{2\Gamma_{g0}\Delta^{2}(\Gamma_{0e}+2\Gamma_{g0})(4\Delta^{2}-\tilde{\Gamma}\kappa)\\\
&-g^{2}\\{\Gamma_{0e}\tilde{\Gamma}^{3}+4\Delta^{2}[3\Gamma_{g0}(\Gamma_{0e}+2\Gamma_{g0})+\Gamma_{0e}\tilde{\Gamma}]\\}\\}\\},\end{split}$
(79)
$\begin{split}C_{1}&=\Gamma_{g0}^{2}\\{\Gamma_{\text{R,out}}\Gamma_{g0}\tilde{\Gamma}(\tilde{\Gamma}^{2}+4\Delta^{2})[(\tilde{\Gamma}^{2}+4\Delta^{2})(\kappa^{2}+4\Delta^{2})-16g^{4}]+\expectationvalue{\expectationvalue{n^{1}}}\\{16g^{4}\\{\Gamma_{0e}\tilde{\Gamma}^{3}+4\Delta^{2}\\\
&\times[6\Gamma_{g0}(\Gamma_{0e}+2\Gamma_{g0})+\Gamma_{0e}\tilde{\Gamma}]\\}-128g^{2}\Delta^{2}(\Gamma_{0e}+2\Gamma_{g0})(4\Delta^{2}-\tilde{\Gamma}\kappa)-(\tilde{\Gamma}^{2}+4\Delta^{2})(\kappa^{2}+4\Delta^{2})\\\
&\times\\{\Gamma_{0e}\tilde{\Gamma}^{3}+4\Delta^{2}[\Gamma_{0e}\tilde{\Gamma}-2\Gamma_{g0}(\Gamma_{0e}+2\Gamma_{g0})]\\}\\}\\},\end{split}$
(80)
$D_{1}=-8\expectationvalue{\expectationvalue{n^{1}}}\Gamma_{g0}^{4}\Delta^{2}[16g^{4}-8g^{2}(4\Delta^{2}-\tilde{\Gamma}\kappa)+(\tilde{\Gamma}^{2}+4\Delta^{2})(\kappa^{2}+4\Delta^{2})],$
(81)

and

$F=A_{2}\tilde{p}_{e}^{(0)4}+B_{2}\tilde{p}_{e}^{(0)3}+C_{2}\tilde{p}_{e}^{(0)2}+D_{2}\tilde{p}_{e}^{(0)}+E_{2},$
(82)

with

$A_{2}=-16g^{4}(\Gamma_{0e}+2\Gamma_{g0})^{2}A,$ (83)
$B_{2}=-32g^{4}(\Gamma_{0e}+2\Gamma_{g0})^{2}B,$ (84)
$\begin{split}C_{2}&=32g^{4}\Gamma_{g0}(\Gamma_{0e}+2\Gamma_{g0})^{2}\\{\Gamma_{g0}(\Gamma_{0e}+\gamma_{-})[48g^{4}+24g^{2}(\tilde{\Gamma}\kappa-4\Delta^{2})+\tilde{\Gamma}^{2}(3\kappa^{2}+4\Delta^{2})+4\Delta^{2}(\kappa^{2}+12\Delta^{2})\\\
&-16\tilde{\Gamma}\kappa\Delta^{2}]-8f^{2}g^{2}\tilde{\Gamma}(\Gamma_{0e}+2\Gamma_{g0})\\},\end{split}$
(85)
$\begin{split}D_{2}&=-16g^{2}\Gamma_{g0}^{2}(\Gamma_{0e}+2\Gamma_{g0})^{2}\\{\Gamma_{g0}(\Gamma_{0e}+\gamma_{-})\\{64g^{4}+48g^{4}(\tilde{\Gamma}\kappa-4\Delta^{2})+(\tilde{\Gamma}^{2}+4\Delta^{2})(\tilde{\Gamma}\kappa-4\Delta^{2})(\kappa^{2}+4\Delta^{2})\\\
&-4g^{2}[16\tilde{\Gamma}\kappa\Delta^{2}-4\Delta^{2}(\kappa^{2}+12\Delta^{2})-\tilde{\Gamma}^{2}(3\kappa^{2}+4\Delta^{2})]\\}-32f^{2}g^{4}\tilde{\Gamma}(\Gamma_{0e}+2\Gamma_{g0})\\},\end{split}$
(86)
$\begin{split}E_{2}&=16f^{2}g^{2}\tilde{\Gamma}(\Gamma_{0e}+2\Gamma_{g0})[(\tilde{\Gamma}^{2}+4\Delta^{2})(\kappa^{2}+4\Delta^{2})-16g^{4}]+\Gamma_{g0}(\Gamma_{0e}+\gamma_{-})\\{16g^{2}\\{16g^{6}-16g^{4}(4\Delta^{2}-\tilde{\Gamma}\kappa)\\\
&+(\tilde{\Gamma}^{2}+4\Delta^{2})(\kappa^{2}+4\Delta^{2})(\tilde{\Gamma}\kappa-4\Delta^{2})+2g^{2}[\tilde{\Gamma}^{2}(3\kappa^{2}+4\Delta^{2})+4\Delta^{2}(\kappa^{2}+12\Delta^{2})-16\tilde{\Gamma}\kappa\Delta^{2}]\\}\\\
&+[(\tilde{\Gamma}^{2}+4\Delta^{2})(\kappa^{2}+4\Delta^{2})]^{2}\\}\end{split}$
(87)

and

$G=A_{3}\tilde{p}_{e}^{(0)4}+B_{3}\tilde{p}_{e}^{(0)3}+C_{3}\tilde{p}_{e}^{(0)2}+D_{3}\tilde{p}_{e}^{(0)}+E_{3},$
(88)

with

$A_{3}=(2g)^{8}(\Gamma_{0e}+2\Gamma_{g0})^{4},$ (89)
$B_{3}=-2^{8}g^{6}\Gamma_{g0}(\Gamma_{0e}+2\Gamma_{g0})^{3}(\tilde{\Gamma}\kappa+4g^{2}-4\Delta^{2}),$
(90)
$C_{3}=32g^{4}\Gamma_{g0}^{2}(\Gamma_{0e}+2\Gamma_{g0})^{2}[48g^{4}+24g^{2}(\tilde{\Gamma}\kappa-4\Delta^{2})+2(\tilde{\Gamma}\kappa-4\Delta^{2})^{2}+(\tilde{\Gamma}^{2}+4\Delta^{2})(\kappa^{2}+4\Delta^{2})],$
(91)
$D_{3}=-16g^{2}\Gamma_{g0}^{3}(\Gamma_{0e}+2\Gamma_{g0})(\tilde{\Gamma}\kappa+4g^{2}-4\Delta^{2})[(4g^{2}+\kappa\tilde{\Gamma}-4\Delta^{2})^{2}+4\Delta^{2}(\kappa+\tilde{\Gamma})^{2}],$
(92)
$E_{3}=\Gamma_{g0}^{4}[(4g^{2}+\kappa\tilde{\Gamma}-4\Delta^{2})^{2}+4\Delta^{2}(\kappa+\tilde{\Gamma})^{2}]^{2}.$
(93)

##  Appendix E Saddle-point approximation

The saddle-point approximation is a widely used method providing an
approximation to a probability distribution using the CGF Butler (2007). It
states that, given the CGF $C(\chi,t)$, the probability distribution can be
approximated by [cf. Eq. (20)]

$p(n,t)\approx\frac{1}{\sqrt{2\pi
C^{\prime\prime}(\chi,t)|_{\chi=\chi^{*}}}}\exp[C(\chi^{*},t)-in\chi^{*}],$
(94)

where $\chi^{*}$ is the solution to the saddle-point equation

$C^{\prime}(\chi^{*},t)-n=0.$ (95)

In Fig. 3b), we plot the logarithm of the different distributions. Within the
saddle-point approximation, and to exponential accuracy, this logarithm can be
approximated as

$\log[p(n,t)]\approx C(\chi^{*},t)-in\chi^{*}.$ (96)

##  Appendix F Numerics

For the numerical calculations of the different cumulants, we follow Refs.
Flindt  _et al._ (2005b) and Marcos  _et al._ (2010), where the authors give a
method for the evaluation of the zero-frequency cumulants and FFN,
respectively. These methods rely solely on matrix multiplications and are,
therefore, ideal for numerical implementation. We briefly review the methods
for completeness. We write the steady state using the special Dirac notation
defined in Appendix A as $\rho_{\text{st}}=\ket{\psi_{0}}\rangle$ and the
corresponding left eigenvector as $\langle\bra{\psi_{0}}$. The inner product
between these objects is given by
$\langle\innerproduct{\psi_{0}}{\psi_{0}}\rangle=\text{Tr}\\{\rho_{\text{st}}\\}=1$.
Next, we give the projector onto the steady state
$\mathcal{P}=\mathcal{P}^{2}=\ket{\psi_{0}}\rangle\langle\bra{\psi_{0}}$ as
well as its complement $\mathcal{Q}=1-\mathcal{P}$. Note that the following
relations hold $L\mathcal{P}=\mathcal{P}L=0$ and therefore
$L=\mathcal{Q}L\mathcal{Q}$. With this at hand, we can define the pseudo-
inverse of the Liouvillian $\mathcal{R}=\mathcal{Q}L^{-1}\mathcal{Q}$ in the
subspace where $L$ is regular. Having defined the necessary quantities, we
find the first two zero-frequency cumulants

$\displaystyle\expectationvalue{\expectationvalue{n^{1}}}$
$\displaystyle=\expectationvalue{\matrixelement{\psi_{0}}{\mathcal{I}}{\psi_{0}}},$
(97) $\displaystyle\expectationvalue{\expectationvalue{n^{2}}}$
$\displaystyle=\expectationvalue{\matrixelement{\psi_{0}}{\mathcal{J}}{\psi_{0}}}-2\expectationvalue{\matrixelement{\psi_{0}}{\mathcal{I}\mathcal{R}\mathcal{I}}{\psi_{0}}},$
(98)

and the FFN is given by

$S(\omega)=\expectationvalue{\matrixelement{\psi_{0}}{\mathcal{J}}{\psi_{0}}}-2\mathfrak{Re}\left\\{\expectationvalue{\matrixelement{\psi_{0}}{\mathcal{I}\mathcal{R}(\omega)\mathcal{I}}{\psi_{0}}}\right\\},$
(99)

where $\mathcal{R}(\omega)=\mathcal{Q}(L+i\omega)^{-1}\mathcal{Q}$ is the
frequency-dependent pseudo-inverse. We have computed the different cumulants
using QuTip Johansson  _et al._ (2012).

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