# Beyond-adiabatic Quantum Admittance of a Semiconductor Quantum Dot at High
Frequencies: Rethinking Reflectometry as Polaron Dynamics

L. Peri  Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue,
Cambridge CB3 0HE, United Kingdom  Quantum Motion, 9 Sterling Way, London N7
9HJ, United Kingdom  G. A. Oakes  Cavendish Laboratory, University of
Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Quantum
Motion, 9 Sterling Way, London N7 9HJ, United Kingdom  L. Cochrane  Cavendish
Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE,
United Kingdom  Quantum Motion, 9 Sterling Way, London N7 9HJ, United Kingdom
C. J. B. Ford  Cavendish Laboratory, University of Cambridge, J.J. Thomson
Avenue, Cambridge CB3 0HE, United Kingdom  M. F. Gonzalez-Zalba
<EMAIL_ADDRESS>Quantum Motion, 9 Sterling Way, London N7 9HJ,
United Kingdom

###### Abstract

Semiconductor quantum dots operated dynamically are the basis of many quantum
technologies such as quantum sensors and computers. Hence, modelling their
electrical properties at microwave frequencies becomes essential to simulate
their performance in larger electronic circuits. Here, we develop a self-
consistent quantum master equation formalism to obtain the admittance of a
quantum dot tunnel-coupled to a charge reservoir under the effect of a
coherent photon bath. We find a general expression for the admittance that
captures the well-known semiclassical (thermal) limit, along with the
transition to lifetime and power broadening regimes due to the increased
coupling to the reservoir and amplitude of the photonic drive, respectively.
Furthermore, we describe two new photon-mediated regimes Floquet broadening,
determined by the dressing of the QD states, and broadening determined by
photon loss in the system. Our results provide a method to simulate the high-
frequency behaviour of QDs in a wide range of limits, describe past
experiments, and propose novel explorations of QD-photon interactions.

††preprint: APS/123-QED

##  I Introduction

Semiconductor quantum dots (QDs) are a promising platform for developing
solid-state quantum technologies. In the area of quantum computing [1], six-
qubit quantum processors [2] and the fabrication of two-dimensional arrays of
4$\times$4 quantum dots have been shown [3]. In conjunction, demonstrations of
qubit control at the threshold for fault-tolerant computing [4, 5, 6],
advanced manufacturing [7, 8], and integration with classical semiconductor
electronics [9, 10] indicate that QDs are a compelling platform for quantum
computation.

More recently, new electronic applications of QD devices are beginning to
emerge, particularly when manipulated at high frequencies. Their non-linear
admittance, consisting of a combination of circuit equivalents such as the
Sisyphus resistance [11] and quantum and tunnelling capacitances [12], can be
utilised for quantum sensing — in the form of fast electrometers [13, 14, 15]
and local thermometers [16, 17] — and for electronic signal conversion — in
the form of frequency multipliers [18] and quantum amplifiers [19].

As these technologies become established, developing accurate electrical
models of semiconductor QDs at high frequencies becomes essential to simulate
their performance as stand-alone elements in hybrid quantum-classical circuits
or circuit quantum-electrodynamics architectures [20, 21, 22, 23].
Semiclassical models accurately predict the behaviour in the limit of slow and
weak driving along with weak coupling to the environment [24, 25].
Furthermore, quantum models have been developed that further describe the
admittance in the limit of strong coupling to the electron bath [26, 27].
However, a consolidated model that accurately describes the interaction of a
QD with a charge reservoir and the quantum nature of the photonic bath across
multiple regimes is missing.

Here, we present a Markovian Master-Equation (ME) formalism for the charge
dynamics in a driven QD tunnel-coupled to a charge reservoir (CR), hereafter,
a semiconductor-based Single Electron Box (SEB) (see Fig 1a) to obtain a
general form of the effective quantum admittance of the SEB. The entire system
can be described from a quantum mechanical point of view as three interacting
subsystems: a QD with a single discrete electrochemical level, a CR in thermal
equilibrium at temperature $T$, and a Photon Bath (PhB) representing microwave
radiation. The QD can exchange charged particles with the CR, and both the
photon number and frequency can be controlled by external means. We describe
the role of temperature and charge tunnel rate to the reservoir with a smooth
transition from thermal to lifetime broadening. Furthermore, we capture the
impact of increasing the driving amplitude, resulting in power broadening of
the charge transition. Our description of the coupled electron-photon states
in the QD as a polaron will shed new and physically insightful light on this
process. Moreover, we present a novel kind of broadening, Floquet Broadening,
which is dictated by the photonic part of the polaron. The Heisenberg
uncertainty principle predicts this effect, but its intrinsically quantum
nature makes it inaccessible to a simple extension of semiclassical adiabatic
theories. Finally, we describe the effects due to non-idealities in the photon
bath and how this affects the equivalent admittance of the SEB. This result is
the direct analogue of known phenomena in superconductor transmon qubits [28,
29, 30], and it shows how the formalism developed here lays the foundations to
bridge the gap between equivalent circuit simulation and circuit quantum
electrodynamics with mixed bosonic and fermionic states. In particular, we
identify five quantities that mostly dictate the SEB dynamics, which give rise
to the same number of possible regimes. Because of the nature of the SEB
admittance at the fundamental frequency, which takes the form of a single
peak, we distinguish these regimes by the limiting factor that causes a
broadening of the lineshape. (1) Thermal Broadening (TB) occurs when the peak
is limited by the charge temperature of the CR. (2) Lifetime Broadening (LB)
is where the broadening is due to the finite lifetime of the charge in the QD.
(3) Floquet Broadening (FB), a high-frequency broadening caused by the
discrete nature of the photon energy. (4) Power Broadening (PB) is a large-
signal effect where the peak is dictated by a large amplitude of the drive.
(5) Photon Loss Broadening (PLB), where the large-signal response is modified
by a finite rate of photon loss in the photonic bath.

##  II Statement of Key Results

In this section, we introduce the necessary concepts and notation and
summarize the key finding of this work while providing the required context
and presenting physical implications. Section III describes the SEB and its
time evolution by framing the charge dynamics quantum mechanically and proves
that it only depends on the QD-CR tunnel rates. Section IV outlines the tunnel
rates semiclassically and in a fully quantum setting. In the process, we will
develop a formalism for a self-consistent quantum Master Equation of the SEB,
which is general to other quantum systems embedded in classical circuits, for
which it is interesting to consider effective circuit analogues. These results
are used in Section V to derive the equivalent SEB admittance, which is
discussed from an electrical and physical point of view. While the formalism
here developed is capable of capturing strong QD-PhB coupling effects, the
discussion is carried forward only in the weak coupling regime. The strong QD-
CR coupling case is however fully discussed. Finally, Section VI presents the
equivalent SEB admittance and its reflectometry signals, exploring the roles
of the various physical and dynamic properties of the system. The discussion
identifies several regimes of SEB operation, depending on the parameter
dominating the charge dynamics, casting new insight into previous well-known
results and identifying new effects when the semiclassical approximations are
no longer valid.

While Sections III and IV give precise mathematical insight into the results
of this work, especially highlighting the breakdown of the semiclassical
approach, an effort is made from Section V onwards in restating key concepts
to understand the effective SEB admittance. Therefore, the reader not directly
interested in the derivation of the self-consistent quantum Master Equation
are welcome to skip to Section V, where the point of view of equivalent
circuit simulation is adopted. This reader, in particular, may be interested
in Section V.2, which, in light of the rigorous quantum calculation, offers a
more direct physical interpretation to guide the intuition towards the results
in Section VI and the various regimes presented therein. In Section VI we
focus on measurable quantities of experimental relevance, predicting novel
regimes and proposing experiments within reach of the current state of the
art.

###  II.1 Gate Current and Equivalent Quantum Admittance

A SEB consists of a single QD connected via a tunnel barrier to a CR (at
temperature $T$) and whose electrochemical potential can be capacitively
changed via a Gate, see Fig. 1a. We consider the CR to be connected to ground.
In this work, we focus on the case in which a sinusoidally varying gate
potential drives the SEB,

$V_{g}(t)=V_{0}+\delta V_{g}\cos{\omega t},$ (1)

and calculate the resulting Gate current $I_{g}(t)$ due to CR-QD charge
tunnelling events to understand the functional relationship between $V_{g}$
and $I_{g}$ introduced by the SEB.

More specifically, the induced gate charge due to a charge tunnelling event
reads

$Q_{g}(t)=\alpha eP(t),$ (2)

where $P(t)$ is the time-dependent probability of occupation and $\alpha$ is
the lever arm that couples the Gate to the QD. For legibility, in this
Section, we will not distinguish between driving and collection lever arms
(see Appendix C). The Gate current can be expressed as

$I_{g}(t)=\alpha e\frac{d}{dt}P(t),$ (3)

which is the quantity we shall derive in this work. Previous works have
focused on the small-signal regime where the SEB can be replaced by an
equivalent admittance linking $I_{g}$ and $V_{g}$ [12, 25, 24, 31]. However,
such a picture fails outside that limit due to the inherent nonlinearities of
the quantum dynamics in the SEB. In this work, we will prove that the concept
of an equivalent admittance can be extended to arbitrarily sized signals, as
the Gate current can be expressed as a Fourier series containing higher-order
harmonics as

$I_{g}(t)=\sum_{N}I_{N}(t)=\frac{\delta V_{g}}{2}\sum_{N}Y_{N}e^{iN\omega
t}+c.c.$ (4)

where the equivalent quantum admittance at the $N$-th harmonic is

$Y_{N}=\frac{\int_{0}^{\frac{2\pi}{\omega}}e^{iN\omega
t}I_{g}(t)dt}{\int_{0}^{\frac{2\pi}{\omega}}e^{i\omega t}V_{g}(t)dt}.$ (5)

Equation 5 becomes of particular physical interest when the SEB is coupled
with an electrical resonator, for example, in a reflectometry setup, [12, 14,
18], allowing the exploration of the different Fourier terms. An equivalent
circuit is best thought of as in Fig. 1c, with the SEB appearing as a parallel
combination of a nonlinear resistor and capacitor to the input of the setup,
while its effect on the output is best modelled as a parallel of Voltage-
Controlled Current Sources (VCCS) associated to the aforementioned Fourier
terms, whose output current is then filtered by the resonator (Appendix C).
From a quantum perspective, the role of the resonator can be thought of as an
electrodynamic cavity coupled to the SEB, whose role is simultaneously to
amplify the resonant signals and select the desired harmonic.

In Appendix C, we show how, if the $N$-th harmonic is the only one in the
bandwidth of the resonator, one can approximate, in the usual regime where
$|Y_{Res}|\gg|Y_{N}|$,

$V_{out}^{N}=\frac{Y_{N}}{Y_{Res}}V_{in}$ (6)

where $V_{out}^{N}(t)=\Re\left[V_{out}e^{iN\omega t}\right]$ and
$V_{in}(t)=\Re\left[V_{in}e^{i\omega t}\right]$ and $Y_{Res}$ is the resonator
admittance. Therefore, one only needs to determine the equivalent admittance
of the system at the relevant harmonic to determine its reflection and
transmission coefficients through a cavity. We emphasise that this result and
the methodology developed are general for any QD system.

Finally, we shall notice from Eq.(3) that $Y_{N}$ only depends on the
probability of the QD being occupied. In Section III.2 we show how, the time
evolution of the SEB can be generally expressed in a Master Equation of the
form

$\frac{d}{dt}P=-\Gamma P+\Gamma_{-}(t),$ (7)

where $\Gamma$ is the total QD-CR tunnel rate (independent of time because of
conservation of charge) and $\Gamma_{-}(t)$ is the tunnel rate out of the QD.
Thus, the admittance solely depends on the quantity $\Gamma_{-}(t)$ (which, as
well as $\Gamma$, has units Hz throughout this work). In this work, we shall
present a self-consistent fully quantum way of deriving such a rate, which
when compared with the semiclassical result used thus far in the literature,
will shed new light on the high-frequency behaviour of the SEB.

In particular, in Section V we derive a general expression for $Y_{N}$, and in
Section VI we identify different regimes differentiated by the physical
process dominating the charge dynamics. While we present a complete and
thorough discussion of the various regimes, for the benefit of the reader
Tables 2 and 3 present a short summary of the result derived in this work. A
short glossary of the relevant symbols can be found in Tab. 1. In particular,
we emphasise the response of the SEB at its fundamental frequency $N=1$ (Tab.
2), as this is the result of most experimental interest.

Glossary of Symbols
---
$\varepsilon_{0}$ |  Average QD-CR electrochemical potential detuning
$\delta\varepsilon$ |  Amplitude of QD-CR detuning variation
$\omega$ |  Frequency of the PhB
$\Gamma$ |  QD-CR tunnel rate
$T$ |  CR charge temperature
$\kappa$ |  PhB photon loss rate
$g$ |  QD-PhB coherent coupling
$\gamma=\frac{\kappa\omega^{2}}{(2g/\hbar)^{2}+\kappa^{2}}$ |  Photon Loss Broadening Rate
$\alpha$ |  Gate lever arm
$J_{n}$ |  $n$-th Bessel Function of the first kind
$\psi_{0}$ |  Digamma function
$\psi_{1}$ |  Trigamma function
Table 1: Glossary of the symbols present in the effective SEB admittance
(Tabs. 2 and 3) and their physical meaning.

Regimes |  Fundamental Admittance
---|---

Thermal Broadening (TB)
$k_{B}T\gg h\Gamma,\hbar\omega,\delta\varepsilon$ |  $Y_{1}=\displaystyle\frac{(\alpha e)^{2}}{2k_{B}T}\frac{\Gamma\omega}{\omega-i\Gamma}\cosh^{-2}{\left({\frac{\varepsilon_{0}}{2k_{B}T}}\right)}$
---|---
Lifetime Broadening (LB)
$h\Gamma\gtrsim k_{B}T\gg\hbar\omega,\delta\varepsilon$ |  $Y_{1}=\displaystyle\frac{(\alpha e)^{2}}{2\pi^{2}k_{B}T}\frac{\Gamma\omega}{\omega-i\Gamma}\Re\left[\psi_{1}\left(\frac{1}{2}+i\frac{\varepsilon_{0}}{2\pi k_{B}T}+\frac{h\Gamma}{2\pi k_{B}T}\right)\right]$
Floquet Broadening (FB)
$\hbar\omega\gtrsim k_{B}T,h\Gamma\gg\delta\varepsilon$ |  $Y_{1}=\displaystyle\frac{(\alpha e)^{2}}{2\pi\hbar}\frac{\Gamma}{\omega-i\Gamma}\Im\left[\psi_{0}\left(\frac{1}{2}+i\frac{\varepsilon_{0}+\hbar\omega}{2\pi k_{B}T}+\frac{h\Gamma}{2\pi k_{B}T}\right)-\psi_{0}\left(\frac{1}{2}+i\frac{\varepsilon_{0}-\hbar\omega}{2\pi k_{B}T}+\frac{h\Gamma}{2\pi k_{B}T}\right)\right]$
Power Broadening (PB)
$\displaystyle\delta\varepsilon\gg k_{B}T,h\Gamma,\hbar\omega$ |  $Y_{1}=\displaystyle\frac{2\hbar}{\pi}\left(\frac{\alpha e}{\delta\varepsilon}\right)^{2}\frac{\Gamma\omega^{2}}{\omega-i\Gamma}\sum_{m=-\infty}^{+\infty}mJ_{m}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)^{2}\Im\left[\psi_{0}\left(\frac{1}{2}+i\frac{\varepsilon_{0}+m\hbar\omega}{2\pi k_{B}T}+\frac{h\Gamma}{2\pi k_{B}T}\right)\right]$
Photon Loss Broadening (PLB)
$\displaystyle\delta\varepsilon\gtrsim\hbar\omega\sqrt{\Gamma/\gamma}$
$\displaystyle\kappa\ll\Gamma+\gamma(\delta\varepsilon/\hbar\omega)^{2}$ |  $Y_{1}=\displaystyle\frac{2\hbar}{\pi}\left(\frac{\alpha e}{\delta\varepsilon}\right)^{2}\frac{\Gamma\omega^{2}}{\omega-i\Gamma}\sum_{m=-\infty}^{+\infty}mJ_{m}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)^{2}\Im\left[\psi_{0}\left(\frac{1}{2}+i\frac{\varepsilon_{0}+m\hbar\omega}{2\pi k_{B}T}+\frac{\Gamma+\gamma(\delta\varepsilon/\hbar\omega)^{2}}{2\pi k_{B}T}\right)\right]$

Table 2: Quantum Admittance at the fundamental of the driving frequency
$Y_{1}$ of the SEB in the various regimes presented in this work: TB
(Sec.VI.1), LB (Sec.VI.2), FB (Sec.VI.3), PB (Sec.VI.4), and PLB (Sec.VI.5). A
glossary of the symbols and their physical interpretation can be found in Tab.
1. $\Re$/$\Im$ indicate the real and imaginary parts of the complex argument.

Regime |  $N$-th Harmonic Admittance
---|---
Thermal Broadening (TB)
$k_{B}T\gg h\Gamma,\hbar\omega,\delta\varepsilon$ |  $Y_{N}=\displaystyle(\alpha e)^{2}\frac{\Gamma N\omega}{N\omega-i\Gamma}\frac{\partial^{N}}{\partial\varepsilon_{0}^{N}}\left({1+e^{\varepsilon_{0}/k_{B}T}}\right)^{-1}$
---|---
Lifetime Broadening (LB)
$h\Gamma\gtrsim k_{B}T\gg\hbar\omega,\delta\varepsilon$ |  $Y_{N}=\displaystyle\frac{(\alpha e)^{2}}{\pi}\frac{\Gamma N\omega}{N\omega-i\Gamma}\frac{\partial^{N}}{\partial\varepsilon_{0}^{N}}\Im\left[\psi_{0}\left(\frac{1}{2}+i\frac{\varepsilon_{0}}{2\pi k_{B}T}+\frac{h\Gamma}{2\pi k_{B}T}\right)\right]$
Floquet Broadening (FB)
$\hbar\omega\gtrsim k_{B}T,h\Gamma\gg\delta\varepsilon$
|  $\displaystyle\quad\begin{aligned} Y_{N}=\frac{(\alpha
e)^{2}}{\pi\delta\varepsilon}\frac{N\Gamma\omega}{N\omega-i\Gamma}\sum_{m=-\infty}^{+\infty}J_{m}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)\bigg{(}&J_{m+N}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)+J_{m-N}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)\bigg{)}\cdot\\\
&\cdot\Im\left[\psi_{0}\left(\frac{1}{2}+i\frac{\varepsilon_{0}+m\hbar\omega}{2\pi
k_{B}T}+\frac{h\Gamma}{2\pi k_{B}T}\right)\right]\end{aligned}$
Power Broadening (PB)
$\delta\varepsilon\gg k_{B}T,h\Gamma,\hbar\omega$
|
Photon Loss Broadening (PLB)
$\displaystyle\delta\varepsilon\gtrsim\hbar\omega\sqrt{\Gamma/\gamma}$
$\displaystyle\kappa\ll\Gamma+\gamma(\delta\varepsilon/\hbar\omega)^{2}$ |  $\displaystyle\quad\begin{aligned} Y_{N}=\frac{(\alpha e)^{2}}{\pi\delta\varepsilon}\frac{N\Gamma\omega}{N\omega-i\Gamma}\sum_{m=-\infty}^{+\infty}J_{m}&\left(\frac{\delta\varepsilon}{\hbar\omega}\right)\bigg{(}J_{m+N}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)+J_{m-N}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)\bigg{)}\cdot\\\ &\cdot\Im\left[\psi_{0}\left(\frac{1}{2}+i\frac{\varepsilon_{0}+m\hbar\omega}{2\pi k_{B}T}+\frac{\Gamma+\gamma(\delta\varepsilon/\hbar\omega)^{2}}{2\pi k_{B}T}\right)\right]&\end{aligned}$

Table 3: Quantum Admittance at the $N$-th harmonic of the driving frequency
$Y_{N}$ of the SEB in the various regimes presented in this work: TB
(Sec.VI.1), LB (Sec.VI.2), FB (Sec.VI.3), PB (Sec.VI.4), and PLB (Sec.VI.5). A
glossary of the symbols and their physical interpretation can be found in Tab.
1. $\Re$/$\Im$ indicate the real and imaginary parts of the complex argument.
Figure 1: Schematics of the driven single electron box as its low-frequency
circuit equivalent (a) and energy level diagram (b). The AC voltage source is
seen as a Photonic Bath (PhB), and ground is seen as a Charge Reservoir (CR),
connected to the QD via a tunnel barrier. Panel (c) shows an effective high-
frequency model of the system, thought of a parallel combination of the
variable tunneling capacitance and Sisyphus resistance as seen from the input
and as a combination of parallel of Voltage Controlled Current Sources at all
the harmonics of the drive as seen from the output. In the schematic, a
resonant circuit is added as a spectroscopic tool to investigate specific
harmonics.

###  II.2 Reflectometry as Polaron Dynamics

While the main result of this work is the derivation in closed-form of the
effective SEB admittance $Y_{N}$, a prerequisite was to derive a self-
consistent fully quantum formalism for an open quantum system subject to
sinusoidal driving. The results and mathematical toolbox assembled in this
process lay the foundation to go beyond semiclassical master equations in the
description of equivalent electronic circuits. In particular, describing a
driven QD as a polaron sheds new light on the concept of Gate Current and the
process of reflectometry, also serving as a cautionary tale when semiclassical
methods venture beyond the limits imposed by the Heisenberg Uncertainty
Principle.

The quantum description of the SEB begins by considering the semiclassical
drive as the manifestation of (weak) coupling with a (coherent) Photonic Bath
(PhB), which is held by a source in a coherent state. We can now begin by
describing the SEB in a second-quantization formalism, as outlined in Section
III.1, by representing the dynamics via the annihilation of the QD, CR, and
PhB and their respective adjoint operators.

Semiclassically, we would group the systems in Fig. 1a horizontally, i.e.
consider first the QD as coupled with a CR, which preserves the conservation
of charge during the time evolution. However, this leads our intuition to
think of the SEB as a completely incoherent system, where the effect of the
PhB is merely to drive stochastic tunnelling events in and out of the QD.
Formally, as we show in Section IV, this relies on the Instantaneous
Eigenvalues Approximation (IEA) [32], which consists of a secular
approximation of the quantum dynamics, where the driven QD acts as a single
level whose energy periodically oscillates in time. Such an adiabatic
treatment of the problem forgoes the quantum nature of the dynamics. In
particular, while the final ME will be Markovian and thus time-local, the
expression to derive the tunnel rates necessarily contains a time convolution
[33], preventing a simple extension of adiabatic result in the time-dependent
case [34].

We shall see, however, how the Lindblad formalism drastically simplifies when
we group Fig. 1a vertically and consider a mixed electron-photon state (i.e.,
a polaron) whose fermionic part stochastically interacts with the CR. Most
interestingly, in Section IV.2, we show how the CR-polaron interaction reads

$H_{CR-Pol}\propto c^{\dagger}Dd+h.c.,$ (8)

where $c$ and $d$ are the annihilation operators of the CR and QD,
respectively.

$D=\exp\left({-\frac{g}{\hbar\omega}\left({a^{\dagger}-a}\right)}\right)$ (9)

is known as a displacement operator of the PhB, where $a$ is the annihilation
operator of the PhB, $\hbar\omega$ is the photon energy, and $g$ is the
coherent QD-PhB coupling. For the benefit of the reader, we recall that

$D\propto\left({\sum_{k}\frac{1}{k!}\left({-\frac{g}{\hbar\omega}a^{\dagger}}\right)^{k}}\right)\left({\sum_{k}\frac{1}{k!}\left({\frac{g}{\hbar\omega}a}\right)^{k}}\right).$
(10)

Fundamentally, this shows that in a driven QD, a tunnelling event can only
occur through the creation or annihilation of a polaron. This does not alter
the total number of photons but only changes their state. Eq.(8) gives a
direct physical interpretation of the concept of Gate current. In the Lindblad
model, we assume the bath to remain unperturbed by the state of the QD. The
Gate current is a manifestation of those extra photons that change state as a
back-action of charge tunnelling event, and which manifest semiclassically as
an alternating current (AC).

This formalism allows us to go beyond the limits of the semiclassical
approach. In particular, we can describe the effects of a high-frequency drive
or a short lifetime of the charge in the QD, both cases in which the
semiclassical picture breaks the time-energy Uncertainty Principle. By
describing reflectometry as polaron dynamics, we shall see that the Heisenberg
principle is recovered in both limits, which will be reflected in the
reflectometry lineshapes at the fundamental $Y_{1}$. Notably, we stress that
this must be the case. Formally, we can think of the limit of a zero
temperature CR as a spectroscopic probe of the effective density of states of
the SEB. In this case, $Y_{1}$ takes the exact form of the spectrum of the
system [35], and thus, in a rigorous quantum description, the Heisenberg
Uncertainty Principle must be satisfied. More physically, we note that $Y_{1}$
directly derives from the Fourier transform of a physical observable, i.e. the
charge in the QD. Considering that the time-energy Uncertainty Principle
derives from the relation between real and reciprocal space, we expect it to
be conserved.

##  III Quantum Dynamics of The Single Electron Box

In this section, we shall define the SEB in terms of its separate subsystem: a
QD, a CR, and a PhB. We will also show that any ME in Lindblad form can be
written without loss of generality in terms of a single Ordinary Differential
Equation for the probability of occupation of the QD as a function of time.

###  III.1 System Hamiltonian

Regarding its separate subsystems, the complete SEB Hamiltonian reads

$H=H_{QD}+H_{CR}+H_{PhB}+H_{DR}+H_{DP},$ (11)

where $H_{QD}$, $H_{CR}$, and $H_{PhB}$ represent the unperturbed Hamiltonians
of the QD, CR and PhB, respectively. While $H_{DR}$ and $H_{DP}$ describe the
QD-CR and QD-PhB coupling, respectively. In particular, we take the zero of
energy at the Fermi level of the CR and consider Hamiltonians of the form

$\displaystyle H_{QD}=\varepsilon_{0}~{}d^{\dagger}d$ (12) $\displaystyle
H_{R}=\sum_{\epsilon}~{}\epsilon~{}c_{\epsilon}^{\dagger}c_{\epsilon}$ (13)
$\displaystyle H_{PhB}=\hbar\omega~{}a^{\dagger}a,$ (14)

with $d$, $c$, and $a$ representing the destruction operators of the QD, CR,
and PhB, respectively. The parameter $\varepsilon_{0}$ describes the static
energy detuning of the single QD level (Fig. 1a), and $\epsilon$ is the energy
of the (potentially discrete) CR levels. We assume the microwave radiation to
be monochromatic at the single frequency $\omega$. For ease of future
notation, we can define

$H_{0}=H_{QD}+H_{R}+H_{PhB}$ (15)

as the Hamiltonian describing the free evolution of the (uncoupled) quantum
systems. Those are then coupled by interaction Hamiltonians, which in the
second quantization formalism are [36]

$\displaystyle
H_{DR}=\sum_{\epsilon}~{}V_{\epsilon}c_{\epsilon}d^{\dagger}+V_{\epsilon}^{*}c_{\epsilon}^{\dagger}d$
(16) $\displaystyle H_{DP}=g(a+a^{\dagger})d^{\dagger}d,$ (17)

where we have considered the experimentally-relevant case of longitudinal QD-
PhB coupling. The parameter $g$ is the coherent QD-PhB coupling, and
$V_{\epsilon}$ is the CR-QD coupling at energy $\epsilon$.

###  III.2 A General Master Equation

Equation (11) describes the complete quantum dynamics of the QD-CR-PhB system.
In this work, we are interested in the QD charge dynamics. Therefore, in the
subsequent sections, we shall employ the Lindblad formalism to write a ME that
only describes the QD degree of freedom by appropriately tracing over the CR
and PhB. The final Lindblad Master Equation (LME) will take the standard form

$\hbar\frac{d}{dt}\rho(t)=-i[H_{0}(t),\rho(t)]+\sum_{i=+,-}h\Gamma_{i}(t)\mathcal{D}[L_{i}],$
(18)

where

$\mathcal{D}(L_{i})[\rho(t)]=L_{i}^{\dagger}\rho(t)L_{i}-\frac{1}{2}\left\\{L_{i}^{\dagger}L_{i},\rho(t)\right\\}$
(19)

are the dissipation superoperators that account for the non-unitary dynamics
caused by the coupling to the CR. The latter is described by the jump
operators

$\displaystyle L_{+}=d^{\dagger}$ (20) $\displaystyle L_{-}=d$

linked to the corresponding tunnel rates $\Gamma_{\pm}(t)$, which we shall
derive in the subsequent sections.

We can simplify the problem further by formulating it as a two-level system,
where the QD is either occupied ($\left|{o}\right\rangle$) or empty
($\left|{e}\right\rangle$). A closer look at the jump operators in Eq.(20)
leads to identifying $\Gamma_{\pm}(t)$ as the probability per unit time of an
electron entering (exiting) the QD from the CR. Therefore, a simple argument
based on the Fermi Golden Rule leads to the rate of jumping in (out) of the QD
only depending on the number of occupied (empty) states in the CR.
Consequently,

$\Gamma_{+}(t)+\Gamma_{-}(t)=\Gamma,$ (21)

with $\Gamma$ being a constant representing the total charge tunnel rate [37,
35]. Moreover, the previous interpretation of the QD as a two-level system
immediately leads us to write the density matrix as

$\rho=\begin{pmatrix}P&C\\\ C^{*}&1-P\end{pmatrix},$ (22)

where $P(t)$ is the probability of occupation of the QD and
$C=\textnormal{tr}\\{|e\rangle\langle o|\rho\\}$ quantifies the degree of
coherent superposition between the two states. It is now fruitful to examine
the problem in the Fock-Liouville space by considering the flattened density
matrix [33]

$\rho\rightarrow\vec{\rho}_{FL}=\begin{pmatrix}P,&C,&C^{*},&1-P\end{pmatrix}^{T}.$
(23)

This picture allows us to explicitly write the LME in Eq.(18) in its matrix
form, which, after some algebra [38] and making use of Eq.(21), reads

$\frac{d}{dt}\begin{pmatrix}P\\\ C\\\ C^{*}\\\
1-P\end{pmatrix}=\begin{pmatrix}-\Gamma_{+}(t)&0&0&\Gamma_{-}(t)\\\
0&-\Gamma/2-i\varepsilon(t)/\hbar&0&0\\\
0&0&-\Gamma/2+i\varepsilon(t)/\hbar&0\\\
\Gamma_{+}(t)&0&0&-\Gamma_{-}(t)\end{pmatrix}\begin{pmatrix}P\\\ C\\\ C^{*}\\\
1-P\end{pmatrix}.$ (24)

After matrix multiplication, we arrive at an expression for the coherence

$\frac{d}{dt}C=\left(-\frac{\Gamma}{2}-i\frac{\varepsilon(t)}{\hbar}\right)C.$
(25)

Intuitively, Eq.(25) shows how the CR exponentially destroys any possible
coherence between the full and empty QD. The system becomes a statistical
mixture of the two kets because of the stochastic tunnelling events to and
from the CR. Finally, we arrive at the key result that, generally, the charge
dynamics are given by the statistical properties of $P(t)$, which are uniquely
determined by the functional form of $\Gamma_{-}(t)$,

$\frac{d}{dt}P+\Gamma P=\Gamma_{-}(t)$ (26)

In the following sections, we shall derive the expressions for $\Gamma_{-}(t)$
in different limits and levels of complexity: the semiclassical limit (Section
IV.1) and with a fully quantum self-consistent approach (Section IV.2). These
sections give detailed mathematical insight into the results of this work,
especially highlighting the breakdown of the semiclassical approximation and
the importance of quantum theory in describing the SEB. Readers not directly
interested in the derivation of the tunnel rates and the underlying Lindblad
formalism are welcome to proceed to Section V, where the key results are
restated, with additional physical insight regarding their precise functional
form.

##  IV Derivation of the Tunnel Rates

In the previous section, we showed how the tunnelling rate $\Gamma_{-}(t)$
fully defines the charge dynamics. In this section, we derive such tunnel
rates in the semiclassical and self-consistent quantum formalism. In
particular, we highlight the nontrivial approximations made in the
semiclassical model and stress the comparison with the fully quantum
formalism.

###  IV.1 Semiclassical Lindblad Master Equation

A simple method to describe the longitudinal QD-PhB coupling is to write

$H_{0}(t)=\left\langle{H_{QD}+H_{DP}}\right\rangle_{PhB}=\left({\varepsilon_{0}+\delta\varepsilon\cos{\omega
t}}\right)d^{\dagger}d,$ (27)

where $\left\langle{\cdot}\right\rangle_{PhB}$ represents the partial trace
over the PhB, and we have used the fact that

$\left\langle{g\left({a^{\dagger}+a}\right)}\right\rangle_{PhB}=\delta\varepsilon\cos{\omega
t}.$ (28)

We will derive this more thoroughly in the subsequent section. It stems from
the semiclassical insight that $\left({a^{\dagger}+a}\right)$ is proportional
to the electric field operator, which we expect to be monochromatic. We can
extend this parallel more deeply by observing from Eq. (27) that
$\delta\varepsilon$ represents the amplitude of the energy-detuning
oscillation. Therefore, we can rederive the conventional result by identifying

$\delta\varepsilon=\alpha e\delta V_{g},$ (29)

where $\delta V_{g}$ is the peak-to-peak voltage amplitude applied to the QD
gate and $\alpha$ the correspondent gate lever arm (Fig. 1a). Therefore, the
longitudinal coupling is the quantum equivalent of a sinusoidally-oscillating
detuning

$\varepsilon(t)=\varepsilon_{0}+\delta\varepsilon\cos{\omega t}.$ (30)

We shall note that Eq. (27) is formally obtained by considering the PhB at
first order in the Lindblad Theory and describing the system as an interacting
QD-CR pair following the time-dependent Hamiltonian

$H_{SC}=H_{0}(t)+H_{R}+H_{DR}.$ (31)

This is the semiclassical description of an SEB considered thus far in the
literature [24, 25]. In this section, however, we shall briefly retrace the
steps to derive a LME from Eq.(31) to highlight the differences between the
semiclassical result and the new self-consistent ME derived in section IV.2.

By defining the Liouville superoperators as

$\displaystyle\mathcal{L}(t)=\left[H_{0}(t)+H_{R}(t),\cdot\right]$ (32)
$\displaystyle\mathcal{L}^{\prime}(t)=\left[H_{DR}(t),\cdot\right],$ (33)

we can write the semiclassical LME in the known form [39, 36]

$\displaystyle\hbar\frac{d}{dt}\rho$
$\displaystyle=-i\left[H_{0}(t),\rho(t)\right]-$ (34)
$\displaystyle-\int_{0}^{+\infty}d\tau\langle\mathcal{L}^{\prime}(t)\mathcal{G}(t,\tau)\mathcal{L}^{\prime}(\tau)\mathcal{G}^{\dagger}(t,\tau)\rangle_{R}\rho(t),$

where

$\mathcal{G}(t-\tau)=\mathcal{T}e^{-i\int_{t}^{\tau}\mathcal{L}(t^{\prime})dt^{\prime}}$
(35)

is the free propagator of the unperturbed dynamics, and
$\left\langle{\cdot}\right\rangle_{R}$ represents the partial trace over the
CR. The notation $\mathcal{T}$ represents the time-ordered integral.

We shall note that in Eq.(34), we have already assumed Markovian dynamics when
extending the upper bound of the integral to $\tau\rightarrow+\infty$. As
usual in the Lindblad formalism, we can compute the partial trace in Eq.(34)
via the Born approximation by requiring that the dynamics in the reservoir be
faster than the CR-QD interaction timescales, and thus the CR can be
considered constantly in thermal equilibrium. Therefore,

$\displaystyle\left\langle{c_{\epsilon}}\right\rangle_{R}=\left\langle{c_{\epsilon}^{\dagger}}\right\rangle_{R}=0$
(36)
$\displaystyle\left\langle{c_{\epsilon}^{\dagger}c_{\epsilon}}\right\rangle_{R}=f(\epsilon)$
(37)
$\displaystyle\left\langle{c_{\epsilon}c_{\epsilon}^{\dagger}}\right\rangle_{R}=f(-\epsilon)=1-f(\epsilon),$
(38)

where

$\displaystyle f(\epsilon)=\frac{1}{e^{\epsilon/k_{B}T}+1}$ (39)

is the Fermi-Dirac distribution. After making what is commonly referred to as
the Rotating Wave Approximation (RWA) [40, 41, 36, 42, 25, 24, 12, 43, 44] or
the Instantaneous Eigenvalues Approximation (IEA) [32, 45], Eq.(34) becomes
equivalent to Eq.(18), with

$\displaystyle\Gamma_{\pm}(t)=\Re\left[\sum_{\epsilon}|V_{\epsilon}|^{2}f(\pm\epsilon)\int_{0}^{\infty}d\tau
e^{-i(\epsilon+\varepsilon(t))\tau}\right].$ (40)

We can now take the continuous limit of the CR
($\sum_{\epsilon}\rightarrow\int\mathcal{D}(\epsilon)d\epsilon$, with
$\mathcal{D}(\epsilon)$ the density of states in the CR) and take the wide-
band limit of both $\mathcal{D}(\epsilon)$ and $V_{\epsilon}$ being weakly
dependent on $\epsilon$. In this limit, Eq.(40) becomes

$\displaystyle\Gamma_{\pm}(t)=\Gamma f(\mp\varepsilon(t)),$ (41)

which is a well-known result [41, 19, 14, 18, 12]. Notably, this is
effectively equivalent to a stationary phase approximation of Eq.(40), and
would be exact if $\varepsilon$ does not depend on time. In the driven case,
the IEA implicitly assumes an adiabatic (secular) approximation of the
propagator, and thus the evolution of the quantum phase, with $\omega$ much
slower than any other timescale (i.e. $\Gamma$) [46, 47, 32, 43]. This will
become apparent in the further discussion of the effective admittance. Not
only it immediately proves the ansatz in Eq.(21), but it also defines
$\Gamma=|V|^{2}\mathcal{D}$ [36, 42]. It ought to be apparent now how we
should expect Eq.(21) from the fermionic character of the CR and thus the
identity $\\{c,c^{\dagger}\\}=0$.

We shall stress how semiclassically, consistently with the name IEA, the ME
only considers the reservoir states at the instantaneous energy of the QD,
thus treating the level as a Dirac Delta in energy space and neglecting any
broadening that may occur because of the finite coupling to the CR. Moreover,
it highlights the secular nature of the approximation. Therefore, it fails to
consider that the dynamics generated by a periodically-driven Hamiltonian are
far richer than a simple oscillation of the QD energy. Therefore, we expect it
to break down if the driving frequency is comparable with the other energy and
time scales of the dynamics. These concerns will be addressed by the self-
consistent approach discussed in the next section. In the following, we shall
refer to Eq.(26) with the rates defined in Eq.(41) as the Semiclassical Master
Equation (SME) [48].

###  IV.2 Self-Consistent Single Electron Box Master Equation

In this section, we derive a fully quantum ME, now treating the PhB quantum
mechanically. To do so, we begin by re-defining the Liouvile superoperators as

$\displaystyle\mathcal{L}(t)=\left[H_{QD}+H_{R}+H_{PhB},\cdot\right]$ (42)
$\displaystyle\mathcal{L}^{\prime}(t)=\left[H_{DR}+H_{DP},\cdot\right].$ (43)

Therefore, the LME is now written in a similar form to Eq.(34), as

$\displaystyle\hbar\frac{d}{dt}\rho$
$\displaystyle=-i\left[H_{QD}(t),\rho(t)\right]-$ (44)
$\displaystyle-\int_{0}^{+\infty}d\tau\langle\mathcal{L}^{\prime}(t)\mathcal{G}(t,\tau)\mathcal{L}^{\prime}(\tau)\mathcal{G}^{\dagger}(t,\tau)\rangle\rho(t),$

where the partial trace must now be taken over both the CR and the PhB.

####  IV.2.1 Polaron Transformation

Before tackling Eq.(44), we shall consider electron-photon interaction in the
Lang-Firsov formalism and perform a canonical transformation into the polaron
frame of reference[49, 50, 51]. This is achieved by considering the operator

$S=-\frac{g}{\hbar\omega}\left({a^{\dagger}-a}\right)d^{\dagger}d$ (45)

and studying the polaron-transformed Hamiltonian

$\tilde{H}=e^{-S}He^{S}.$ (46)

This is equivalent to applying the displacement operator

$D=\exp\left({-\frac{g}{\hbar\omega}\left({a^{\dagger}-a}\right)}\right),$
(47)

which selectively depends on the state of the QD. In Appendix E, we show how,
in the polaron frame

$\displaystyle e^{-S}\left({H_{PhB}+H_{QD}+H_{DP}}\right)e^{S}=$ (48)
$\displaystyle\left({\varepsilon_{0}+\frac{g^{2}}{\hbar\omega}}\right)d^{\dagger}d+\hbar\omega
a^{\dagger}a,$

where we can see the longitudinal coupling merely becomes a Lamb Shift of the
QD detuning, taking the form

$\tilde{H}_{QD}=\left({\varepsilon_{0}+\frac{g^{2}}{\hbar\omega}}\right)d^{\dagger}d=\tilde{\varepsilon}_{0}d^{\dagger}d.$
(49)

Moreover, in Appendix E, by employing the Baker-Campbell-Hausdroff (BCH)
theorem, we find that [42, 36]

$e^{-S}\left({H_{DR}}\right)e^{S}=\sum_{\epsilon}~{}V_{\epsilon}c_{\epsilon}D^{\dagger}d^{\dagger}+V_{\epsilon}^{*}c_{\epsilon}^{\dagger}Dd.\\\
$ (50)

####  IV.2.2 Self-Consistent Propagator

In Section IV.1, we saw how the SME does not consider level broadening. The
reason for this is the combination of the IEA with Eq.(34) using the free
propagator $\mathcal{G}(t)$ in the Born approximation. We can remedy this by
taking into account the self-consistent Born approximation, where we replace
$\mathcal{G}(t)$ with a self-consistent propagator [52]

$\mathcal{U}(t,\tau)=\mathcal{T}e^{-i\int_{t}^{\tau}(\mathcal{L}+\mathcal{L}^{\prime})(t^{\prime})dt^{\prime}}.$
(51)

We now consider the operators in the Heisenberg Picture [53, 54]

$\mathcal{U}(0,t)[d]=\left\langle{e^{-i\tilde{H}t}de^{-i\tilde{H}t}}\right\rangle,$
(52)

with $\tilde{H}$ representing the full polaron-transformed Hamiltonian and the
partial trace taken over the CR and PhB after the time evolution. Using the
BCH theorem, one can prove that this is equivalent to requiring [36, 42]

$\begin{cases}&\hbar\frac{d}{dt}d(t)=i\tilde{\varepsilon}_{0}d(t)-i\sum_{\epsilon}V_{\epsilon}D^{\dagger}(t)c_{\epsilon}(t)\\\
&\hbar\frac{d}{dt}c_{\epsilon}(t)=i\epsilon
c_{\epsilon}(t)-iV^{*}_{\epsilon}D(t)d(t)\end{cases}$ (53)

and that $D(t)$ evolves normally, as the phonon operators only appear in the
free Hamiltonian $H_{PhB}$. Interestingly, the time evolution of the ladder
operators of the QD and CR are now coupled, reflecting how, for finite
$V_{\epsilon}$, the state in the QD is metastable. Taking the Laplace
transform of Eq.(53) in the wide-band limit of the CR yields the intuitive
result [42, 36]

$\displaystyle\mathcal{U}(0,t)[d]=de^{-i\tilde{\varepsilon}_{0}t}e^{-\Gamma
t}$ (54)
$\displaystyle\mathcal{U}(0,t)[d^{\dagger}]=d^{\dagger}e^{i\tilde{\varepsilon}_{0}t}e^{-\Gamma
t}.$ (55)

####  IV.2.3 Self-Consistent Tunnel Rates

Using the result in Eq.(55), we can write the LME in the polaron frame as

$\displaystyle\hbar$
$\displaystyle\frac{d}{dt}\tilde{\rho}=-i\left[\tilde{H_{S}},\tilde{\rho}(t)\right]-$
(56)
$\displaystyle-\int_{0}^{+\infty}d\tau\left\langle\tilde{\mathcal{L}}^{\prime}(t)\mathcal{U}(t,\tau)\tilde{\mathcal{L}}^{\prime}(\tau)\mathcal{U}^{\dagger}(t,\tau)\right\rangle\tilde{\rho}(t),$

which we can similarly recast in the form of Eq.(18) with the equivalent of
Eq.(40) reading

$\displaystyle\Gamma_{-}(t)=\frac{\Gamma}{\pi}\Re\left[\int d\epsilon
f(\epsilon)\mathcal{K}(\epsilon,t)\right]$ (57)
$\displaystyle\Gamma_{+}(t)=\frac{\Gamma}{\pi}\Re\left[\int
d\epsilon(1-f(\epsilon))\mathcal{K}^{*}(\epsilon,t)\right],$ (58)

where we have defined

$\mathcal{K}(\epsilon,t)=\int_{0}^{\infty}d\tau
e^{-i(\epsilon-\varepsilon_{0})\tau}e^{-\Gamma\tau}\left\langle{D(t)D^{\dagger}(\tau-t)}\right\rangle_{PhB}.$
(59)

To compute the partial trace over the PhB, we notice that the semiclassical
result in Eq.(28) can be obtained simply assuming that the radiation is in a
coherent state $\left|{\delta\varepsilon/2g}\right\rangle$. This view,
although simplistic, can be justified by considering the typical phase noise
characteristic of rf sources, together with the fact that, as made evident in,
Eq.(59), in the self-consistent picture, phase-coherence is only required on
timescales of the order $1/\Gamma$ 111Typical figures for phase noise for a
1GHz Local Oscillator are of -100 dBc at 100 Hz offset and -40dBc at 1Hz
offset, while experimentally $\Gamma$ varies between 10 MHz and 100 GHz. Thus,
the excitation can be considered monochromatic for experimental purposes.. We
take the occasion also to stress that the Markovian assumption becomes natural
in the self-consistent Born approximation, thanks to the self-consistent
propagator decaying exponentially on the same timescales.

In this case, we can use the composition properties of displacement operators
to write

$\displaystyle\left\langle{D(t)D^{\dagger}(\tau-t)}\right\rangle_{PhB}=\exp\left({-\left({\frac{g}{\hbar\omega}}\right)^{2}(1-e^{-i\omega\tau})}\right)\cdot$
(60)
$\displaystyle\cdot\exp\left({-i\left(\frac{\delta\varepsilon}{\hbar\omega}\right)\left({\sin{\omega(\tau-t)}+\sin{\omega
t}}\right)}\right),$

where we have separated the contribution of photon absorption and stimulated
emission, containing both rotating and counter-rotating terms, and spontaneous
emission, containing only counter-rotating terms independent of the microwave
amplitude. We shall note that the spontaneous emission tends to 1 in the
regime $g\ll\hbar\omega$. This is the case of weak QD-PhB coupling, which is
the case of all the results we shall consider in this work. Therefore, for the
sake of clarity and ease of notation, we shall disregard this term from now
on. We can compute the integral in Eq.(59) by making use of the Jacobi-Anger
Identity (JAI)

$\exp\left(-ix\sin{\omega\tau}\right)=\sum_{m=-\infty}^{+\infty}J_{m}\left(x\right)e^{-im\omega\tau},$
(61)

where $J_{m}\left(x\right)$ is the Bessel function of the first kind. This
solution allows us to write, neglecting spontaneous emission,

$\displaystyle\mathcal{K}(\epsilon,t)=e^{i\left(\frac{\delta\varepsilon}{\hbar\omega}\right)\sin{\omega
t}}$
$\displaystyle\sum_{m=-\infty}^{+\infty}J_{m}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)e^{-im\omega
t}\cdot$ (62) $\displaystyle\cdot\int_{0}^{+\infty}d\tau
e^{-i(\epsilon-\varepsilon_{0}-m\omega)\tau}e^{-\Gamma\tau}.$

Notably, modulo the self-consistent term $e^{-\Gamma\tau}$, this expression
resembles previous results obtained in the non-adiabatic quantum ME framework
or Floquet-Lindblad formalisms [47, 32, 43, 55, 34, 56]. However, our
treatment not only allows us to include the effect of the finite QD lifetime
in the dynamics but immediately gives us a physical interpretation of the
coefficients, as determined from the photonic part of the polaron, rather than
having to derive them separately from a Floquet decomposition [47, 55, 34,
57].

Using the Fubini-Tonelli theorem, we can write

$\Gamma_{\pm}(t)=\Gamma\Re\left[e^{i\left(\frac{\delta\varepsilon}{\hbar\omega}\right)\sin{\omega
t}}\sum_{m=-\infty}^{+\infty}J_{m}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)e^{-im\omega
t}\mathcal{F}_{m}^{\pm}(\varepsilon_{0})\right],$ (63)

where we have defined

$\mathcal{F}_{m}^{\pm}(\varepsilon_{0})=\frac{\Gamma}{\pi}\int_{-\infty}^{\infty}~{}~{}~{}\frac{f(\mp\epsilon)}{\Gamma^{2}+((\epsilon-\varepsilon_{0})/\hbar-m\omega)^{2}}~{}~{}~{}d\epsilon$
(64)

being the convolution of the Fermi-Dirac distribution and the effective
Lorentzian density of states of the QD caused by the coupling with the CR.
Here, we have disregarded the well-known diverging Lamb shift arising from
taking the trace of the (unbound) CR number operator [42, 58]. Notably, in
Eq.(63) appear the semiclassical phase of the driven level
$\exp\left(-i\int_{0}^{t}dt^{\prime}\varepsilon(t^{\prime})/\hbar\right)$, as
well as an expansion on the Floquet modes of the system (see Section. V.2),
recalling the functional form expected from Floquet-Lindblad theory [55, 34].

Computationally, it is easier to explicitly write the tunnelling rate in terms
of its Fourier components. Therefore, after another application of the JAI,
Eq.(63) reads

$\displaystyle\Gamma_{-}(t)=\Gamma\sum_{m=-\infty}^{+\infty}\sum_{n=-\infty}^{+\infty}$
$\displaystyle
J_{n}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)J_{m}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)\cdot$
(65) $\displaystyle\cdot\Re\left[e^{-i(m-n)\omega
t}\mathcal{F}_{m}^{-}(\varepsilon_{0})\right]$

Within Appendix D, we showed how this (lifetime-) broadened Fermi-Dirac can be
rewritten analytically as

$\mathcal{F}_{m}^{\pm}(\varepsilon_{0})=\frac{1}{2}-\frac{1}{\pi}\Im\left[\psi_{0}\left(\frac{1}{2}+i\frac{\Theta^{\pm}_{m}}{2\pi}\right)\right],$
(66)

where $\psi_{0}$ is Euler’s digamma function and

$\Theta^{\pm}_{m}=\pm\frac{\varepsilon_{0}+m\hbar\omega}{k_{B}T}-i\frac{h\Gamma}{k_{B}T}.$
(67)

The numerator in Eq.(67) (as we shall see in Section V.2) can be interpreted
as the (complex) quasi-energies of the metastable Floquet Modes of the damped-
driven SEB[40].

We can now use the property that, thanks to the anticommutation relations of
the fermionic CR operators,

$\mathcal{F}_{m}^{-}(\varepsilon_{0})=1-\mathcal{F}_{m}^{+}(\varepsilon_{0})$
(68)

to show that the property in Eq.(21) still holds and, thus, we re-obtain a
Self-Consistent Quantum Master Equation (SCQME) of the form in Eq.(26).

##  V Charge Dynamics and Effective Admittance

So far, with Eq.(26), we have introduced the ME formalism that can calculate
the time-dependent QD occupation probability in different limits by
appropriately selecting the relevant tunnel rate $\Gamma_{-}(t)$. In this
section, we shall explain how to use this to derive an effective SEB
admittance.

###  V.1 Semiclassical and Quantum Admittance

Firstly, we notice Eq.(26) has the steady-state solution

$P(t)=e^{-\Gamma t}\int_{-\infty}^{t}e^{\Gamma\xi}\Gamma_{-}(\xi)~{}d\xi.$
(69)

For a periodic function $\Gamma_{-}(t)$ with frequency omega, this can be
rewritten as [18]

$P(t)=\frac{\omega}{2\pi}\sum_{N}\frac{1}{\Gamma+iN\omega}\int_{0}^{\frac{2\pi}{\omega}}e^{iN\omega
t}\Gamma_{-}(t)~{}~{}dt+c.c.$ (70)

Recalling the definition of Gate current and quantum admittance in Eqs. (3)
and (5), we can therefore write

$\displaystyle Y_{N}$
$\displaystyle=i\frac{(e\alpha)^{2}}{\delta\varepsilon}\frac{N\omega}{\Gamma+iN\omega}\int_{0}^{\frac{2\pi}{\omega}}e^{iN\omega
t}\Gamma_{-}(t)~{}~{}dt=$ (71)
$\displaystyle=\frac{\mathcal{C}_{N}}{\delta\varepsilon}\int_{0}^{\frac{2\pi}{\omega}}e^{iN\omega
t}\frac{\Gamma_{-}(t)}{\Gamma}~{}~{}dt,$

where, for conciseness, we have defined the coefficient

$\mathcal{C}_{N}=i(e\alpha)^{2}\frac{N\omega\Gamma}{\Gamma+iN\omega},$ (72)

which accounts for the semiclassical high-pass filtering effect of the
capacitive coupling of the collection gate.

Semiclassically, we have shown in Section IV.1 that the tunnelling rate reads

$\Gamma_{-}(t)=\Gamma f(\varepsilon_{0}+\delta\varepsilon\cos{\omega t}),$
(73)

where $f(\varepsilon)$ is the Fermi-Dirac distribution. This result allows us
to directly compute the impedance from the Semiclassical Master Equation (SME)
as

$Y^{SME}_{N}=2\frac{\mathcal{C}_{N}}{\delta\varepsilon}\int_{0}^{\frac{2\pi}{\omega}}e^{iN\omega
t}f(\varepsilon_{0}+\delta\varepsilon\cos{\omega t})~{}~{}dt.$ (74)

To compute the admittance for the Self-Consistent Quantum Master Equation
(SCQME), instead, we can use the rate from Eq.(65), which reads

$\displaystyle\Gamma_{-}(t)=\Gamma\sum_{m=-\infty}^{+\infty}\sum_{n=-\infty}^{+\infty}$
$\displaystyle
J_{n}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)J_{m}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)\cdot$
(75) $\displaystyle\cdot\Re\left[e^{-i(m-n)\omega
t}\mathcal{F}_{m}^{-}(\varepsilon_{0})\right],$

where in Section IV.2, we have defined $\mathcal{F}_{m}^{-}$ as the self-
consistent distribution of the SEB is the convolution between the Fermi-Dirac
distribution and a Lorentzian representing the effective density of states of
the metastable QD level, and $J_{n}$ are Bessel functions of the first kind.
We remind the reader that the only approximation in Eq.(75) is to be in the
weak QD-PhB coupling regime.

We can obtain the effective admittance in Eq. Eq.(71) by selecting the correct
Fourier component from Eq.(75), yielding

$\displaystyle Y^{SCQME}_{N}$
$\displaystyle=\frac{\mathcal{C}_{N}}{\delta\varepsilon}\sum_{m=-\infty}^{+\infty}J_{m}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)\cdot$
(76)
$\displaystyle\cdot\left(J_{m+N}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)+J_{m-N}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)\right)\mathcal{F}_{m}^{-}(\varepsilon_{0}).$

It is worth noting that in the case $N=1$, we can use the well-known identity

$J_{m+1}(x)+J_{m-1}(x)=2\frac{m}{x}J_{m}(x)$ (77)

to write for the fundamental

$\displaystyle
Y^{SCQME}_{1}=2\frac{\hbar\omega}{\delta\varepsilon^{2}}\mathcal{C}_{1}{\delta\varepsilon}\sum_{m=-\infty}^{+\infty}mJ_{m}^{2}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)\mathcal{F}_{m}^{-}(\varepsilon_{0})=$
(78)
$\displaystyle=2\frac{\hbar\omega}{\delta\varepsilon}\mathcal{C}_{1}\sum_{m=1}^{+\infty}mJ_{m}^{2}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)\left({\mathcal{F}_{m}^{-}(\varepsilon_{0})-\mathcal{F}_{-m}^{-}(\varepsilon_{0})}\right),$

where we have used the fact that $J_{-m}(x)=(-1)^{m}J_{m}(x)$. Equations (76)
and (78) represent the major result of this work, as they are the most general
expression of the SEB admittance (assuming weak photon coupling). However,
because of their mathematical complexity, Section VI presents them in
different regimes, where approximations can be made to simplify the
expressions and thus highlight the effect of each parameter.

###  V.2 A Floquet Interpretation

Before discussing the impact of the SCQME on the reflectometry lineshapes, we
believe valueable to the reader to physically interpret Eq.(76) in terms of
Floquet Theory.

The first step consists of the solution of a sinusoidally-driven single-level
system whose Hamiltonian reads

$H=\varepsilon_{0}+\delta\varepsilon\cos{\omega t}.$ (79)

This simple system is of interest due to an explicit solution for the Fourier
decomposition of the single level, known as the Tien-Gordon model, which
reads[59, 60]

$\left|{\phi}\right\rangle(t)=e^{i\varepsilon_{0}t/\hbar}\sum_{m=-\infty}^{\infty}J_{m}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)e^{im\omega
t}\left|{\phi_{m}}\right\rangle,$ (80)

where the Fourier components $\left|{\phi_{m}}\right\rangle$ traditionally
take the name of Floquet Modes. This result can be used to highlight key
concepts of the dynamics of a driven quantum system. We can consider adding an
integer offset $n$ to the index $m$, which would not change Eq.(80) apart from
a redefinition of dummy indexes and the transformation
$\varepsilon_{0}\rightarrow\varepsilon_{0}+n\hbar\omega$. Therefore, the
energy of a periodically-driven quantum system is defined up to an integer
multiple of the photon energy. For this reason, they are traditionally named
quasi-energies. However, this simple picture invites us to think of a single
driven level as a ladder of energetically equally-spaced Floquet Modes, whose
phase in time oscillates at the harmonics of the fundamental driving tone.
This is sometimes referred to as the Fourier or Sambe decomposition of the
state [61].

This simple model, nonetheless, predicts an effective density of states of the
driven QD [59]

$\mathcal{D}_{QD}=\sum_{m=-\infty}^{+\infty}J_{m}^{2}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)\frac{\Gamma/\pi}{\Gamma^{2}+(\epsilon-\varepsilon_{0}-m\omega)^{2}},$
(81)

which, despite the mathematical similarity, does not lead to the result in the
SCQME for the simple reason that, without the CR, the occupation of the QD
trivially reads

$P(t)=\sum_{m}J_{m}^{2}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)=1,$
(82)

i.e. the electron is stuck in the QD and therefore, no current can be
generated.

From Eq.(67), however, we can immediately observe the similarity with the
concept of quasi-energies, with the only key difference that CR introduces an
imaginary part. In the semiclassical derivation, we have seen how the Fermi-
Dirac distribution is evaluated at the instantaneous energy of the QD
$\varepsilon(t)$. In the SCQME, we can consider Eq.(66) as its natural
extension to a quantum formalism, which is evaluated at the (complex) energies

$\mathcal{E}_{m}=\varepsilon_{0}+m\hbar\omega-ih\Gamma.$ (83)

This is an obvious consequence of the CR, which makes the state in the QD
metastable, and the eigenstates of the self-consistent time propagator in
Eq.(55) are exponentially decaying.

From classical mechanics, moreover, we know that, at the steady state, an
exponential decay can be modelled as a phase delay compared to driving the
system. This phase lag can be derived directly from the ME in Eq.(26), but it
is also easily obtained by noticing that, for both the SME and SCQME,

$\arg{Y_{N}}=\arctan{\frac{\Gamma}{N\omega}}=\frac{\pi}{2}-\phi_{N}$ (84)

is notably independent of $\varepsilon_{0}$. This last remark is particularly
interesting because, electrically, we can separate real and imaginary parts of
the SEB admittance as

$Y=G_{S}+i\omega C_{T},$ (85)

where $C_{T}$ takes the name of the tunnelling capacitance while
$G_{S}=1/R_{S}$ is the Sisyphus conductance [12, 24]. Equation (84), thus,
indicates that the distinction between the resistive and reactive response of
the SEB is dictated uniquely by its dynamical properties (quasi-energies), and
the detuning only modulates the magnitude of the signal. In Appendix A, we
discuss the significance of this on the energy balance of the system.

Because of the non-unitary interaction with the CR, it is formally impossible
to write the SEB evolution as a single ket. Section III.2 proves that the
dynamics discussed thus far are the only solution to the SEB SCQME that
satisfies the requirement of a density matrix, and therefore, is the steady
state, no matter the initial electronic state [62].

Nonetheless, we believe it valuable to the reader to stress that the
metastable counterpart of the modes in Eq.(80) are the building blocks of the
SEB evolution. The interpretation of the dynamics in terms of damped-driven
Floquet Modes with equally-spaced quasi-energies will lead to a novel
interpretation of Power Broadening as an interference phenomenon and will
serve as an intuitive explanation of a novel effect introduced in this work:
Floquet Broadening.

Figure 2: Pictorial representation of the dressed states in a driven QD. The
single state in the QD whose energy is driven sinusoidally in time can be
equivalently thought of as a ladder of levels equally spaced by one photon
energy, whose occupation is dictated by the strength of the drive.

##  VI Reflectometry Lineshapes in Different Regimes

From Eq.(76), it ought to be clear that four energy scales, namely $k_{B}T$,
$\delta\varepsilon$, $\hbar\omega$, and $h\Gamma$, determine the SEB dynamics.
In this section, therefore, we shall discuss the interplay between them and
the different behaviour of the SEB when one dominates compared to the others.
Finally, we will introduce the effect of photon loss $\kappa$ in the PhB and
determine its repercussions on the effective admittance. Considering the
discussion of Eq.(84), in this section, we shall only discuss the magnitude of
the reflectometry signal $|Y|$. When possible, general properties of $Y_{N}$
are discussed, with a particular focus given to the fundamental frequency,
$Y_{1}$, due to its experimental significance.

In particular, for the small-signal regime, it is possible to expand the
Bessel functions as

$J_{m}(x)\xrightarrow[x\rightarrow 0]{}\frac{x^{m}}{2^{m}m!}.$ (86)

Therefore, when $\delta\varepsilon\ll\hbar\omega$ we can neglect all $N>1$
terms. Physically, this is equivalent to the intuitive fact of considering
only first dressed state in the Floquet ladder.

In this case, it is trivial to show that one always obtains

$Y_{1}=\frac{(\alpha
e)^{2}}{2k_{B}T}\frac{\Gamma\omega}{\omega-i\Gamma}\mathcal{F}^{\prime}(\varepsilon_{0}),$
(87)

with $\mathcal{F}^{\prime}$ a new function defined such that
$\mathcal{F}^{\prime}(\varepsilon_{0})\in[0,1]$ and, more notably,

$\mathcal{F}^{\prime}(\varepsilon_{0}=0)\xrightarrow[h\Gamma/k_{B}T\rightarrow
0]{\hbar\omega/k_{B}T\rightarrow 0}1.$ (88)

Therefore, the different regimes only differ for the shape of
$\mathcal{F}^{\prime}(\varepsilon_{0})$. Interestingly, for a given
temperature, the maximum achievable signal is $\frac{(\alpha
e)^{2}}{2k_{B}T}$, which we will see is the semiclassical prediction [12, 18].
Perhaps unsurprisingly, we will also prove that, in all regimes, the maximum
signal for the fundamental is always obtained at zero detuning.

###  VI.1 Thermal Broadening

The most straightforward regime is when
$k_{B}T\gg\delta\varepsilon,\hbar\omega,h\Gamma$. In this case, the thermal
smearing of the Fermi-Dirac distribution in the CR is the dominant effect.
Therefore, all the assumptions of the SME are valid, and the SEB behaves fully
semiclassically.

In particular, if $k_{B}T\gg\delta\varepsilon$, it is trivial to obtain,
through Taylor expansion for small $\delta\varepsilon$ of the SME in Eq.(74),
the well-known result

$\displaystyle Y_{1}$
$\displaystyle=2\mathcal{C}_{1}\frac{\partial}{\partial\varepsilon_{0}}f(\varepsilon_{0})=$
(89) $\displaystyle=\frac{(\alpha
e)^{2}}{2k_{B}T}\frac{\Gamma\omega}{\omega-i\Gamma}\cosh^{-2}{\left({\frac{\varepsilon_{0}}{2k_{B}T}}\right)}$

as expected from Eq.(87).

From Eq.(76), we can obtain the same result via Eq.(86). Therefore, in the
limit $k_{B}T\gg h\Gamma$, $\mathcal{F}_{-m}^{-}(\varepsilon_{0})\approx
f(\varepsilon_{0}-\hbar m\omega)$, and

$\displaystyle
Y_{1}=\frac{\mathcal{C}_{1}}{\delta\varepsilon}\frac{\delta\varepsilon}{\hbar\omega}\left({f(\varepsilon_{0}+\hbar
m\omega)-f(\varepsilon_{0}-\hbar m\omega)}\right)$ $\displaystyle\approx$ (90)
$\displaystyle\approx
2\mathcal{C}_{1}\frac{\partial}{\partial\varepsilon_{0}}f(\varepsilon_{0}),$

where the latter approximate equality derives from the fact that if
$k_{B}T\gg\hbar\omega$ the difference approaches the definition of a
derivative.

In fact, in the small-signal regime, we can show by Taylor expansion of
Eq.(74) around $\delta\varepsilon=0$ that [18]

$Y_{N}\propto\frac{\partial^{N}}{\partial\varepsilon_{0}^{N}}f(\varepsilon_{0}).$
(91)

We obtain a similar result for the SCQME, whereby recursively using Eq.(77),
$Y_{N}^{SCQME}$ follows the finite difference stencil coefficients for the
$N$-th derivative with second-order accuracy. Thus, in the limit of
$k_{B}T\gg\hbar\omega,\delta\varepsilon$ the SCQME result approaches Eq.(91),
as expected.

###  VI.2 Lifetime Broadening

By remaining in the small-signal regime, it is interesting to compare the two
models when $h\Gamma\gtrsim k_{B}T\gg\delta\varepsilon,\hbar\omega$. In this
case, the QD is strongly coupled to the CR, and the reservoir-induced
broadening of the (metastable) discrete level is comparable with thermal
energy, know as Lifetime Broadening (LB). This effect, however, is not
considered in the SME. This should be apparent as $\Gamma$ and $\omega$ enter
in Eq.(74) only in the prefactor $\mathcal{C}_{N}$. Therefore, in the
semiclassical picture, $\Gamma$ only rescales the signal because of the high-
pass filtering of the gate current from the series quantum capacitance, but
does not influence its lineshape.

Using the mathematical result in Appendix D, we can write the small-signal SEB
admittance in the LB regime as Eq.(87) with

$\mathcal{F}^{\prime}_{LB}(\varepsilon_{0})=\frac{2}{\pi^{2}}\Re\left[\psi_{1}\left(\frac{1}{2}+i\frac{\varepsilon_{0}}{2\pi
k_{B}T}+\frac{h\Gamma}{2\pi k_{B}T}\right)\right],$ (92)

where $\psi_{1}(z)$ is the trigamma function. Notably,

$\displaystyle\mathcal{F}^{\prime}_{LB}(\varepsilon_{0})=\frac{\partial}{\partial\varepsilon}\mathcal{F}_{0}^{-}(\varepsilon_{0})=$
(93)
$\displaystyle=\frac{1}{2k_{B}T}\left({\cosh^{-2}{\left({\frac{\epsilon}{2k_{B}T}}\right)}*\frac{\Gamma/\pi}{\Gamma^{2}+(\epsilon-\varepsilon_{0})^{2}}}\right)$

which is the convolution between the $\cosh^{-2}$ in the TB regime to a
Lorentzian peak in the case $h\Gamma\gg k_{B}T$, interpolating smoothly
between the two. The effect of increasing tunnel rate in the admittance
lineshape is shown in Fig. 3a.

Interestingly, experimentally, it is a known fact that different transitions
may show a different Full-Width-Half-Maximum (FWHM) at the same electron
temperature. It has been shown that the small-signal quantum capacitance of a
DTR in the LB regime is equal to [17, 63]

$\displaystyle C_{Q}=\frac{(e\alpha)^{2}}{2k_{B}T}$
$\displaystyle\frac{1}{1+\omega^{2}/\Gamma^{2}}\cdot$ (94)
$\displaystyle\cdot$
$\displaystyle\left({\cosh^{-2}{\left({\frac{\epsilon}{2k_{B}T}}\right)}*\frac{\Gamma/\pi}{\Gamma^{2}+(\epsilon-\varepsilon_{0})^{2}}}\right).$

Therefore, considering the result in Eq.(93), the self-consistent treatment in
the SCQME naturally extends Eq.(94) to both the Sisyphus resistance and higher
harmonics.

Fig. 3b shows the former effect of LB on the Full-Width Half-Maximum (FWHM) of
the peak, where we see that for $h\Gamma\gtrsim k_{B}T$, the FWHM starts
increasing from the expected TB value of $3.53~{}k_{B}T$, obtained from the
inverse hyperbolic cosine, and becomes linear with the tunnelling rate for
increasing tunnel rates. In particular,

$\textnormal{FWHM}(|Y_{1}|)\xrightarrow[h\Gamma\gg
k_{B}T,\hbar\omega,\delta\varepsilon]{}2h\Gamma.$ (95)

For clarity, in all Figures, the parameters which are not varied are indicated
with a subscript $0$.

Finally, we can consider the effect of $\Gamma$ on the amplitude of SEB
admittance at the fundamental frequency, which is shown in Fig. 3c. As
discussed above, $\Gamma$ enters in $Y_{1}$ both in the prefactor and because
of lifetime broadening. The former is a purely classical effect, and it is due
to the charge requiring a finite time to tunnel out of the QD. In contrast,
the other is purely quantum and arises from the effective density of states of
the electron in the SEB. In Fig. 3a, we show the effect purely of the latter
for increasing $\Gamma/\omega$, which highlights another crucial effect of LB.
As the coupling to the CR increases, the discrete energy level in the QD
becomes increasingly broader, and thus the peak decreases. Therefore, any
spectroscopic broadening of the lineshape inevitably translates into lowering
the maximum signal, as it must be the case because of the properties of the
convolution.

This effect, however, is complemented by the semiclassical contribution in
$\mathcal{C}_{1}$. One could posit that larger coupling to the CR ought to
translate in a higher probability of tunnelling events per rf cycle. This
effect is shown in Fig. 3c, where the inclusion of the prefactor
$\mathcal{C}_{1}$ leads monotonically increasing signal for increasing
$\Gamma$.

However, a Self-Consistent quantum picture clearly shows that this is not the
case. In the limit of $h\Gamma\gg k_{B}T$, the discrete level is so broadly
distributed in energy that no tunnelling events can be observed. Consequently,
the maximum reflectometry signal begins to drop when $h\Gamma$ starts to
become comparable with $k_{B}T$ and completely vanishes for
$h\Gamma\gg\hbar\omega,k_{B}T$. Consistently with Fig. 3a, this is accompanied
by an increase of the FWHM of the peak, as evidenced by Fig. 3b.

More specifically, while there is no simple analytical expression for the
behaviour in Fig. 3c, we can notice that
$\mathcal{F}^{\prime}_{LB}\propto\Gamma^{-1}$ for large $\Gamma$. Therefore,
we similarly expect $\textnormal{max}\left(|Y_{1}|\right)\propto\Gamma^{-1}$
for $h\Gamma\gg\hbar\omega$.

Figure 3: Effect of Lifetime Broadening (LB) on $Y_{1}$. (a) Shape of the
function $\mathcal{F}^{\prime}_{LB}(\varepsilon_{0})$ (Eq.(93)) for varying
$h\Gamma/k_{B}T_{0}$, showing how the broadening of the peak is accompanied by
a lowering of the maximum. (b) FWHM of $|Y_{1}|$ showing the linear increase
when $h\Gamma\gtrsim k_{B}T$. The graph is normalized to $3.53~{}k_{B}T$, the
FWHM of $|Y_{1}|$ in the TB regime (SME). (c) Maximum of $|Y_{1}|$ normalized
to the TB regime (SME). This panel shows the competition between the increase
in signal because of $C_{1}$ and the drop because of the broadening of
$\mathcal{F}^{\prime}_{LB}$ for $h\Gamma\gg k_{B}T$.

###  VI.3 Floquet Broadening

The final small-signal regime presented in this work is a novel kind of
broadening which arises from the rf photon energy being comparable with
thermal and lifetime broadening. Similarly to lifetime broadening, this effect
is purely quantum and cannot be captured semiclassically, as $\omega$ only
appears in the already-discussed prefactor $C_{N}$. Interestingly, however,
neither $\delta\varepsilon$ nor $\omega$ appears separately in the SCQME, but
only in the ratio $\frac{\delta\varepsilon}{\hbar\omega}$. We can gain further
insight by reconsidering the effective density of states in Eq.(81), where the
Floquet modes are equally spaced by the photon energy $\hbar\omega$.
Therefore, the ratio $\frac{\delta\varepsilon}{\hbar\omega}$ determines the
highest dressed state that will be reached by the semiclassical voltage swing.
Moreover, from the discussion in Section V.2, the reflectometry lineshape is
determined by the interference between the different dressed states.
Therefore, if $\hbar\omega\gtrsim h\Gamma$, i.e. the separation between the
Floquet modes is comparable with the broadening of the levels and, thus, the
discrete nature of the dressed states may emerge.

In this work, we present a novel result, which we name Floquet Broadening
(FB), which becomes apparent in the regime where $\hbar\omega\gtrsim
k_{B}T,h\Gamma$, i.e. the discrete nature of the photon energy becomes the
primary source of broadening in the SEB admittance. Here, we take the occasion
to stress that FB is an inherently quantum phenomenon, arising from the
discrete spacing of the Floquet modes, thus, uncapturable with any
semiclassical treatment.

From a mathematical standpoint, FB stems from the fact that, for
$\hbar\omega\sim k_{B}T$, Eq.(90) becomes an increasingly worse approximation
of the derivative. Therefore, we can use the small-signal expansion in Eq.(86)
to write the admittance in the form of Eq.(87) with

$\displaystyle\mathcal{F^{\prime}}_{FB}=\frac{k_{B}T}{\hbar\omega}\Im\bigg{[}$
$\displaystyle\psi_{0}\left(\frac{1}{2}+i\frac{\varepsilon_{0}+\hbar\omega}{2\pi
k_{B}T}+\frac{h\Gamma}{2\pi k_{B}T}\right)-$ (96) $\displaystyle-$
$\displaystyle\psi_{0}\left(\frac{1}{2}+i\frac{\varepsilon_{0}-\hbar\omega}{2\pi
k_{B}T}+\frac{h\Gamma}{2\pi k_{B}T}\right)\bigg{]}.$

Recalling the fact that $\psi_{0}$ is the digamma function, and
$d/dz~{}\psi_{0}(z)=\psi_{1}(z)$, it is clear that for slow frequencies,
$\mathcal{F^{\prime}}_{FB}\approx\mathcal{F^{\prime}}_{LB}$, while for larger
photon energies, it is possible to resolve the two separate terms in Eq.(96),
which correspond to the first rotating and counter-rotating dressed states. We
show the results in Fig. 4, where we present the effect for the cases
$k_{B}T\gg h\Gamma$ (Fig. 4a-c) and $h\Gamma\gg k_{B}T$ (Fig. 4d-f).

In particular, it is interesting to consider the effect of increasing $\omega$
on the FWHM and maximum admittance in Fig. 4b,e and Fig. 4c,f, respectively.
Considering Eq.(96) and ignoring LB for simplicity, we expect because of the
factor $k_{B}T/\hbar\omega$ in Eq.(96)

$|Y_{1}^{SCQME}|\propto\left|\mathcal{C}_{1}\right|\omega^{-1}\propto\frac{\Gamma}{\sqrt{\Gamma^{2}+\omega^{2}}}.$
(97)

In contrast, in the semiclassical case

$|Y_{1}^{SME}|\propto\left|\mathcal{C}_{1}\right|\propto\frac{\Gamma\omega}{\sqrt{\Gamma^{2}+\omega^{2}}}.|Y_{1}^{SME}|\propto\left|\mathcal{C}_{1}\right|\propto\frac{\Gamma\omega}{\sqrt{\Gamma^{2}+\omega^{2}}}.$
(98)

Therefore, contrary to the SME, the SCQME predicts a reduction in the SEB
signal with increasing $\omega$ (Fig. 4c,f). Similarly to LB, this drop in
signal is accompanied by an increase in the FWHM (Fig. 4b,e). In particular,

$\textnormal{FWHM}(|Y_{1}|)\xrightarrow[\hbar\omega\gg
k_{B}T,h\Gamma,\delta\varepsilon]{}2\hbar\omega.$ (99)

There is some interesting physical insight into the process of rf
reflectometry to be gained from this regime. Semiclassically, one can picture
increasing the measurement frequency in the small-signal regime as increasing
the chances per unit time of electron tunnelling. Therefore, this shall
increase the maximum signal until $\omega$ becomes of the order of the
tunnelling rate $\Gamma$. At that point, the probability of a tunnel event per
unit cycle reads

$P_{cycle}=1-e^{-\Gamma\tau_{cycle}}\approx\Gamma\tau_{cycle},$

where $\tau_{cycle}=2\pi/\omega$. Therefore, the average gate current reads

$\left\langle{I_{g}}\right\rangle\propto eP_{cycle}/\tau_{cycle}$

which, as shown in Eq.(98) and Fig. 4f, saturates and becomes independent of
frequency (dashed line).

From a quantum perspective, the tunnelling of the electron in (out) the CR is
more akin to a measurement of the state of the QD. Increasing $\omega$ means
increasing the measurement rate, potentially much faster than the timescales
of the (non-unitary) quantum dynamics. Therefore, the electron becomes trapped
inside (outside) the QD, a process resembling the Quantum Zeno effect [28].

Another interpretation may come from the Heisenberg Uncertainty Principle, as
increasing $\omega$ would attempt to define the energy of a level at time
scales faster than its lifetime. Therefore, we expect a Heisenberg energy
broadening of the order $2\pi/\tau_{cycle}=\omega$, as shown in Fig. 4.

A more detailed discussion of the optimal parameters for the SEB used as a
sensor, taking into account LB and FB, is outlined in Appendix B.

Figure 4: Effect of FB in for varying $\hbar\omega/k_{B}T_{0}$ (a-c) and
$\omega/\Gamma_{0}$ (d-f). Panels (a,d) show how for increasing $\omega$,
$\mathcal{F}^{\prime}_{FB}$ approximates increasingly less a derivative and
the consequent lowering of the peak accompanying the broadening. Panels (b,e)
show the increase of FWHM of $|Y_{1}|$, eventually reaching $2\hbar\omega$,
while panels (c,f) show the dependence of the maximum admittance with drive
frequency.

###  VI.4 Power Broadening

As previously discussed, it is experimentally interesting to compare the SME
and the SCQME in the large-signal regime, where $\delta\varepsilon\gtrsim
k_{B}T,h\Gamma,\hbar\omega$. For reasons that shall become apparent, this is
commonly referred to as the power-broadened (PB) regime, which we will begin
to discuss in the case where $k_{B}T,h\Gamma\gg\hbar\omega$.

In Fig. 5a, we show how the SCQME can reproduce the familiar power-broadening
fan predicted by the SME, where the admittance linewidth increases linearly
with the amplitude of the detuning oscillations [14, 18, 12, 41, 64, 65, 66].
From Eq.(78), however, we can see how the SCQME does so by considering an
infinte summation over dressed states rather than an integral over a smoothly-
varying voltage. We can gain more insight into the physical reasons behind
this via the use of Floquet Theory presented in Section V.2, where we showed
how a single level oscillating with a large amplitude can be modelled as a
ladder of equally-spaced levels.

However, we have already pointed out how a simple application of Eq.(81),
obtained without the CR, does not lead to the SCQME result in Eq.(76) (or
Eq.(78)). The reasoning why is that the Floquet Modes must combine in such a
way to generate a real occupation probability. At any frequency different to
$\omega$, two different Floquet modes will accumulate a different phase each
rf cycle. When averaged out over multiple cycles, any interference becomes
negligible and, therefore, one can separately describe the single dressed
states, as in Eq.(81). If the excitation frequency is the same as the
measurement one (or its integer fraction), the different dressed states have a
defined phase relationship between each other, and will be able to interfere
to generate the various harmonic of the driving. The signal at the $N$-th
harmonic is obtained by combining rotating waves at $N\omega$ and counter-
rotating waves at $-N\omega$. Or, to phrase it differently, reflectometry
techniques are sensitive to changes in AC currents (i.e. emitted and absorbed
photons) and, thus, to transitions between the ladder of dressed states.
Therefore, this result can be considered an AC extension to the Tien-Gordon
model [59], which is only concerned with direct (DC) conductance and, thus,
time-averaged properties of the system [60].

All this is particularly apparent for the fundamental in Eq.(78), where the
lineshape is constructed from the interference of the $m$-th and $-m$-th
Floquet Modes. This equation also highlights the well-known result in Floquet
theory that all the modes enter the definition of the system’s response at
every harmonic[40, 61]. From this discussion, it ought to be clear how PB is
not merely due to incoherence with the CR but rather an interference
phenomenon between the Floquet Modes of the damped-driven quantum system. The
stochastic nature of the QD-CR interaction can thus be reinterpreted as a
scattering process between the dressed states or as a finite lifetime of the
polaron state.

Figure 5: Effect of increasing the amplitude of the driving in the large-
signal regime. (a) Map of the normalized Gate current as a function of
detuning offset and driving amplitude , showing the well-known PB fan. (b)
Line cut of (a) (red dashes) showing the combined effect of PB and LB. (c)
Saturation of the Gate current in the limit of very large drive
($\delta\varepsilon\gg k_{B}T,h\Gamma,\hbar\omega$), and its retardation
because of LB. (d) FWHM of $|Y_{1}|$, showing the convergence to
$\sqrt{3}\delta\varepsilon$ in the limit of very large drive.

Another interesting property of PB is how it is affected by LB (Fig. 5b-d). In
the small-signal regime, LB arises as an additional broadening in addition to
the thermal smearing of the reservoir. If $\delta\varepsilon\gg k_{B}T\gg
h\Gamma$, however, we can imagine that the voltage swing will dominate, and
the QD will be emptied (filled) with probability 1 every cycle [18]. In the
semiclassical picture, this will lead to a saturation of the Gate current.
This trend is clearly shown in Fig. 5c. Notably, the admittance, which is
defined as the Gate current over the excitation voltage, drops as
$\delta\varepsilon^{-1}$. Therefore, for visual clarity, in Fig.5 we plot the
product $\delta\varepsilon|Y_{1}|$.

Lifetime Broadening, on the other hand, is an intrinsic property of the QD and
dictates the lifetime of all dressed states. Therefore we would expect it to
play a role even in the regime where $\delta\varepsilon\gg h\Gamma\gtrsim
k_{B}T$. Indeed, in Fig. 5b, we see that increasing $\Gamma$ leads to a
broadening of the lineshape and the consequent lowering of the peak at
constant power. However, by increasing the rf power, we can still reach
saturation (Fig. 5c). In the semiclassical picture, we can interpret this as a
broadening of the DOS of the QD. Therefore, similarly to the previous argument
regarding temperature, one can imagine the voltage swing to be large enough to
ignore the intrinsic broadening of the level. If this is sufficient to fully
empty or fill the QD every excitation cycle, we expect saturation to be
reached. From a quantum perspective, we notice in Eq.(62) that the evolution
of the polaron phase is proportional to $\delta\varepsilon$, while it decays
with $\Gamma$. Recalling the discussion in Section VI.2 (see also Appendix A),
LB decreases the signal because the polaron dynamics is cut short. Increasing
the number of photons can accelerate the dynamics fast enough to overcome the
finite lifetime. However, it comes at the expense of a broader peak because of
Heisenberg (Fig. 5d).

While a simple analytical solution is not available for the SME in Eq.(74), we
can notice that in the regime $\delta\varepsilon\gg k_{B}T\gg h\Gamma$ the
integrand of the Fourier transform $f(\varepsilon(t))$ tends to a square wave.
Thus, we can immediately write in the very large signal regime [18]

$\begin{cases}&Y_{N}^{SME}=\frac{2}{\pi}\frac{\mathcal{C}_{N}}{N\delta\varepsilon}\sin{\left[N~{}\text{arccos}\left(\frac{\varepsilon_{0}}{\delta\varepsilon}\right)\right]}\hfill~{}~{}~{}~{}if~{}~{}~{}~{}|\varepsilon_{0}|<\delta\varepsilon\\\
&Y_{N}^{SME}=0\hfill~{}~{}~{}~{}if~{}~{}~{}~{}|\varepsilon_{0}|\geq\delta\varepsilon\end{cases}$
(100)

which, for $N=1$, reduces to

$Y_{1}^{SME}\bigg{|}_{\delta\varepsilon\rightarrow\infty}=\frac{2(\alpha
e)^{2}}{\pi\delta\varepsilon}\frac{\Gamma\omega}{\omega-i\Gamma}\sqrt{1+\left(\frac{\varepsilon_{0}}{\delta\varepsilon}\right)^{2}}.$
(101)

For ease of notation, we can therefore define

$\mathcal{M_{I}}=\max\left(|I_{1}|\right)\bigg{|}_{\delta\varepsilon\rightarrow\infty}=\frac{2(\alpha
e)^{2}}{\pi}\left|\frac{\Gamma\omega}{\omega-i\Gamma}\right|,$ (102)

which is the maximum Gate current semiclassically achievable at the
fundamental. Notably, this saturates at very large power, and thus the
admittance, which is defined as the Gate current over the excitation voltage,
drops as $\delta\varepsilon^{-1}$.

Interestingly, we shall note that Eq.(101) (and Eq.(100)) have a discontinuous
derivative at $\varepsilon_{0}=\pm\delta\varepsilon$. This can be understood
by considering that, when thermal broadening is negligible, the Fermi-Dirac
function tends to be a step-function with a discontinuous derivative. The
smearing of the peak caused by LB, however, will smoothen these corners and
keep every derivative continuous, as depicted in Fig. 5b.

Finally, from Fig. 5d, we notice an interesting trend. Because of the
properties of Eq.(78), the LB of the peak happens in such a way as to make
FWHM only dependent on $\delta\varepsilon$. Therefore, we can use Eq.(101) to
derive the fact that, we have

$\textnormal{FWHM}(|Y_{1}|)\xrightarrow[\delta\varepsilon\gg
k_{B}T,h\Gamma,\hbar\omega]{}\sqrt{3}\delta\varepsilon\approx
1.73\delta\varepsilon$ (103)

independently from $k_{B}T$ or $h\Gamma$.

Figure 6: Evolution of the PB fan for increasing $\omega$. As we enter the FB
regime, the effect of the discrete summation in $Y_{1}$ becomes more clear,
revealing the underlying ladder of dressed states. As this occurs, the Gate
current is permitted to go above the semiclassical limit, reaching
$\delta\varepsilon|Y_{1}|\approx 1.02\mathcal{M_{I}}$ for
$\delta\varepsilon/\hbar\omega\approx 2.40$.

Lastly, it is interesting to consider the effect of PB in conjunction with FB,
in the case where $\delta\varepsilon\gg\hbar\omega\gtrsim k_{B}T,h\Gamma$.
Similarly to the small-signal regime, when the photon energy becomes
significant with respect to the other broadenings, we expect to see the
discrete nature of the summation over the dressed states in Eq.(78). We show
the effect of increasing excitation frequency in Fig. 6, where it is clear how
the PB fan becomes a more faithful depiction of the ladder of dressed states.

En passant, it is interesting to note how, in the case of
$\hbar\omega>k_{B}T,h\Gamma$ it is possible to achieve more Gate current that
the semiclassical limit. It is possible to show, in fact, that for
$\delta\varepsilon/\hbar\omega\approx 2.40$, $\delta\varepsilon|Y_{1}|\approx
1.02\mathcal{M_{I}}$. Although not a significant enhancement, it stands as
testament of the coherent interference at the root of the process generating
the reflectometry signals in this regime and points towards new experiments in
the field of coherent hybrid quantum-classical circuits.

###  VI.5 Photon Loss Broadening

Up until this point, we have considered the PhB as a perfect cavity held in a
coherent state. However, since the SEB dynamics are purely dictated by the
interaction between the ladder of dressed states, we must also consider photon
loss, since this will cause the various dressed states to dephase and thus may
strongly affect the signal. We can account for this effect by assuming the
loss rate $\kappa$ to be small compared to the rf frequency $\omega$.
Therefore, redefining the PhB Hamiltonian as

$H_{PhB}=\hbar\omega a^{\dagger}a+\hbar\kappa\mathcal{D}[a],$ (104)

which, in conjunction with a classical drive supplied by the rf source, will
cause the photon number to fluctuate in time as

$n(t)=\bar{n}+\delta n(t).$ (105)

For a damped-driven system, the average number of photons reads [29]

$\bar{n}=\frac{\delta\varepsilon^{2}}{(2g)^{2}+\hbar^{2}\kappa^{2}},$ (106)

while the time correlation of the fluctuations obeys [67]

$\langle\delta n(t)\delta
n(t^{\prime})\rangle=\bar{n}e^{-\frac{\kappa}{2}|t-t^{\prime}|}.$ (107)

With this additional term, we can write the Displacement Operator as [68, 69,
70]

$D_{\alpha}(t)=e^{\frac{|\alpha|^{2}}{2}(1-e^{-\frac{\kappa}{2}t})}e^{-\frac{1}{2}\kappa
a^{\dagger}at}D_{e^{i\omega t}\alpha}.$ (108)

In the Gaussian approximation for the phase, we can modify the kernel in
Eq.(62) as [69, 28, 29]

$\begin{aligned}
&\mathcal{K}(\epsilon,t)=e^{i\left(\frac{\delta\varepsilon}{\hbar\omega}\right)\sin{\omega
t}}\sum_{m=-\infty}^{+\infty}J_{m}\left(\frac{\delta\varepsilon}{\hbar\omega}\right)e^{-im\omega
t}\cdot\\\ &\cdot\int_{0}^{+\infty}d\tau
e^{-i(\epsilon-\varepsilon_{0}-m\omega)\tau}e^{-\Gamma\tau}e^{\bar{n}\left(1-\frac{\kappa}{2}\tau-e^{-\frac{\kappa}{2}\tau}\right)}\end{aligned}.$
(109)

If we assume that $\kappa\gg\Gamma$, the lifetime of an electron in the QD is
much longer than the coherence of the radiation. In this case, we can neglect
the term $e^{-\frac{\kappa}{2}\tau}$, and it ought to be apparent how the
effective lifetime of the ladder of dressed states reads

$\tilde{\Gamma}=\Gamma+\kappa\frac{\delta\varepsilon^{2}}{(2g)^{2}+\hbar^{2}\kappa^{2}}=\Gamma+\gamma\left(\frac{\delta\varepsilon}{\hbar\omega}\right)^{2},$
(110)

which, for large input powers, is dominated by photon dynamics rather than
electronic ones. As seen in the previous sections, it is helpful to describe
the dynamics in terms of the ratio $\delta\varepsilon/\hbar\omega$. Thus, in
Eq.(110), we have defined the rate

$\gamma=\frac{\kappa\omega^{2}}{(2g/\hbar)^{2}+\kappa^{2}}$ (111)

to quantify the effect of photon loss to the CR in the SEB lifetime.

If, on the other hand, $\kappa\ll\Gamma$, photon loss becomes a small
perturbation, and

$e^{\bar{n}\left(1-\frac{\kappa}{2}\tau-e^{-\frac{\kappa}{2}\tau}\right)}\approx
e^{-\frac{\bar{n}\kappa^{2}}{8}\tau^{2}},$ (112)

which means that the spectrum of the SEB becomes the convolution of a
Lorentzian and a Gaussian (i.e. a Voigt profile), the inhomogeneous part
directly arising from the Poisson statistics of photon counts. Thus, this
phenomenon has also been described as photon shot noise in the context of
superconducting flux qubits [71, 72, 73].

We will refer to this novel effect as Photon Loss Broadening (PLB), and it is
the SEB equivalent of the well-known phenomenon in circuit QED of measurement-
induced dephasing [28, 29, 74, 71, 30]. Once again, this result stresses how
PB in a SEB is a coherent process between dressed states in the QD.

We can analytically evaluate the integral in Eq.(109) and expand the resulting
spectrum in an infinite series of Lorentzians [29]. Therefore, we can use the
result in Appendix D to introduce the effect of PLB simply by redefining

$\mathcal{F}_{m}^{\pm}(\varepsilon_{0})=\frac{1}{2}+\frac{e^{\bar{n}}}{2\pi}\sum_{l}\frac{(-\bar{n})^{l}}{l!}\Im\left[\psi_{0}\left(\frac{1}{2}+\tilde{\Theta}^{\pm}_{l,m}\right)\right],$
(113)

where

$\Theta^{\pm}_{l,m}=h\frac{\tilde{\Gamma}+l\kappa}{2\pi k_{B}T}\pm
i\frac{\varepsilon_{0}+m\hbar\omega}{2\pi
k_{B}T}.\Theta^{\pm}_{l,m}=h\frac{\tilde{\Gamma}+l\kappa}{2\pi k_{B}T}\pm
i\frac{\varepsilon_{0}+m\hbar\omega}{2\pi k_{B}T}.$ (114)

In usual experimental settings, we have that $\hbar\kappa\ll
g\ll\hbar\omega\sim h\Gamma$. Therefore, we could approximate

$\tilde{\Theta}^{\pm}_{l,m}\approx\tilde{\Theta}^{\pm}_{m}=\frac{h\tilde{\Gamma}}{2\pi
k_{B}T}\pm i\frac{\varepsilon_{0}+m\omega}{2\pi k_{B}T},$ (115)

which allows us to sum the infinite series as

$\mathcal{F}_{m}^{\pm}(\varepsilon_{0})\approx\frac{1}{2}+\frac{1}{2\pi}\Im\left[\psi_{0}\left(\frac{1}{2}+\tilde{\Theta}^{\pm}_{m}\right)\right],$
(116)

which shows how the main effect of PLB is a superlinear LB and the consequent
reduction in the SEB signal.

We capture this effect in Fig. 7, which shows a comparison of the PB fan with
$\gamma=0$ (Fig. 7a) and with $\gamma=0.2\cdot 10^{-2}\Gamma$ (Fig. 7b).
Perhaps counterintuitively, in the case of a lossy cavity (resonator),
increasing the input power does not always lead to an increase in the
reflectometry signal, as observed experimentally [14, 18, 75]. However, we
shall note that in an experimental setting, the degradation of the signal
because of PLB may be complemented by heating the sample and/or the resonator,
which may also lower the total signal. We highlight the effect of PLB in Figs.
7c-d, where we show maximum signal and its FWHM are shown as a function of
detuning amplitude for varying $\gamma$. For small $\gamma$, the effect of PLB
only manifests at very large detuning amplitudes, where we see a superlinear
broadening for increasing $\delta\varepsilon$ and the consequent reduction in
the peak height. Roughly, we can estimate the threshold for this change in
regime as

$\delta\varepsilon\gtrsim\hbar\omega\sqrt{\frac{\Gamma}{\gamma}},$ (117)

where PLB becomes comparable with the usual LB (which is only caused by the
CR). As shown in Section VI.4, the onset of PB is for

$\delta\varepsilon\gtrsim k_{B}T,h\Gamma.$ (118)

Therefore, depending on the value of $\kappa$, Eq.(117) may be reached before
Eq.(118). Observing Fig. 7c, in this case, the maximum theoretical signal is
never reached. Finally, we see that the FWHM of the peak increases as
$\delta\varepsilon^{2}$ (Fig. 7d).

Figure 7: Effect of PLB. (a,b) Gate current maps showing the alteration of the
PB fans in the presence of photon loss. Both maps are normalized to the
maximum semiclassical Gate current $\mathcal{M_{I}}$, showing how in the case
of PLB this is never reached, and the signal starts decreasing for very large
$\delta\varepsilon$. (c,d) Maximum normalized Gate current and FWHM of the
peak for increasing $\gamma$. The effect of PLB causes panel (c) to not be
monotonically increasing, causing a reduction of signal for increasing power.
This is accompanied by a superlinear increase in FWHM, which eventually become
proportional to $\gamma\delta\varepsilon^{2}$ (panel (d)).

##  VII Conclusions and Outlook

We have presented a new theory to go beyond the semiclassical models for the
development of equivalent circuits of quantum systems and applied it to a
simple yet technologically useful QD system: the SEB. Our results reproduce
the known behaviour of the SEB at low frequency and high temperature [19, 41],
and present an extension beyond the adiabatic regime. The self-consistent
nature of our approach also allows us to include the effect of the finite
lifetime of an electron in a QD when coupled to a reservoir, including the
description of its resistive component, which was missing from the literature
[17, 63, 76].

The mathematical formalism employed in this work sheds new light on the
concept of Gate current and the process of rf reflectometry, which is
interpreted as the dynamics of polaron states in the QD. This allows us to see
the well-known phenomenon of Power Broadening from a new angle, as an
interference effect between dressed states in the QD. Our theory extends the
adiabatic semiclassical SEB description into a novel regime, which we call
Floquet Broadening, where the main source of energy uncertainty comes from the
discrete nature of the photonic part of the polaron. In doing so, our theory
recovers the Heisenberg Uncertainty Principle, which is not respected by the
semiclassical theory. In its discussion, we propose measurements which are
technologically within reach for the experimental validation of our
theoretical results,including a modification of the admittance lineshape and a
drop of its maximum value at high drive frequencies.

The self-consistent quantum formalism here developed is fully general and
applicable to quantum systems such as multi quantum dots or multi-level
systems such as spin qubits. Therefore, it lays the foundations for developing
fully quantum circuit equivalent models for quantum systems. The possibility
of properly accounting for relaxation and dephasing phenomena beyond the
adiabatic limit comes with the enticing perspective of extending such models
to systems conceived for quantum computation, allowing for a more informed
engineering of their design and operation.

Moreover, our quantum description of the photon source allows for the
theoretical treatment of its nonidealities. This possibility has led us to
introduce a new large-signal phenomenon, photon loss broadening, which can
contribute to the experientially observed drop in SEB signal at very large
powers. This is the first description in a QD system and terms of an
equivalent circuit model of a well-known phenomenon observed in
superconducting devices, kown as photon shot noise or measurement induced
dephasing [29, 29, 74, 71, 72, 73, 30]. Also in this case we present a clear
experimental path for the measurement of this effect.

Finally, we have shown how, by exploiting quantum coherence in the system, the
semiclassical limits of the maximum achievable AC currents can be overcome.
While our example only presents a minor improvement (2%), leveraging quantum
phenomena to augment well-known devices may put within technological reach
semiclassically unachievable goals.

##  VIII Acknowledgements

This research was supported by European Union’s Horizon 2020 research and
innovation programme under grant agreement no. 951852 (QLSI), and by the UK’s
Engineering and Physical Sciences Research Council (EPSRC) via the Cambridge
NanoDTC (EP/L015978/1), QUES2T (EP/N015118/1), the Hub in Quantum Computing
and Simulation (EP/T001062/1) and Innovate UK [10000965]. L.P. acknowledges
the Winton Programme for the Physics of Sustainability. M.F.G.Z. acknowledges
a UKRI Future Leaders Fellowship [MR/V023284/1]. The authors acknowledge F.
Martins from Hitachi Cambridge Laboratory for useful discussions.

##  Appendix A Real and Virtual Photon Exchange

In the main text, we have mostly discussed the absolute value of the SEB
admittance, especially the maximum value and FWHM, for they are easily
experimentally-accessible properties. We believe of value, however, to explore
more in depth the real and imaginary part of $Y$, introduced in Section V.2,
for they have very different electrical (and physical) effects [12]. For
simplicity, in this section we shall uniquely discuss the fundamental $Y_{1}$,
dropping the suffix from now on, but this discussion is trivially extendable
to the other harmonics [18]. Moreover, being in the small-signal regime, the
effects of PLB are not considered in this section and the PhB is to be
considered lossless.

To begin with, the presence of only first time derivatives in the LME in
Eq.(34) means that we can, without loss of generality, write

$Y=G_{S}+i\omega C_{T},$ (119)

where $C_{T}$ represents the tunnelling capacitance while $G_{S}=1/R_{S}$ is
the Sisyphus conductance [12, 24]. We shall stress the notable absence of what
is commonly referred to as quantum capacitance [25], for that corresponds to
an adiabatic redistribution of charges along a unitary dynamics because of an
avoided crossing [12, 24]. The level crossing in a SEB at $\varepsilon=0$,
however, is between two states describing the full or empty QD. Therefore,
they formally belong to different canonical single-body Hilbert spaces, and
thus the crossing cannot be avoided.

Redistribution of charge, however, can still happen through incoherent
(stochastic) processes via the CR through the jump operators in the LME and
may occur either elastically or inelastically. In a cyclostationary process,
this results in a gate current either in-phase or out-of-phase with the
driving voltage. In the small-signal regime, this simply reads [25, 18, 24]

$I_{g}=YV_{in}.$ (120)

This semiclassical picture is of particular interest because it clearly shows
how only the former, described by $G_{S}$, can lead to energy dissipation,
while the signal arising from $C_{T}$ conserves energy over an rf cycle.

We can picture these two types of response from a QED perspective as the QD
mediating an interaction between the PhB and the CR. The capacitive response,
thus, would correspond to the QD and the PhB exchanging virtual photons, whose
energy is not accounted for, in the form of creating a polaron and populating
the Floquet ladder, while the resistive components will describe real photons,
which transfer energy to the PhB from the CR and vice versa. Consisting of
fully elastic jumps, the capacitive component is time-reversible. Thus, the
$m$-th and $-m$-th Floquet Modes must share the same occupation (i.e. Eq.(81))
and a precise phase relation. Stocastically, however, an extra real photon can
be emitted (absorbed) resulting in scattering between the Floquet Modes. This
unbalance in the $m$-th and $-m$-th Modes manifests mathematically as a phase
delay in the averaged steady-state response, and thus a resistive behaviour.

In light of this interpretation, it is interesting to consider the behaviour
in Fig. 8, which shows the (maximum of) tunnelling capacitance and Sisyphus
conductance for increasing $\Gamma$ and $\omega$.

Semiclassically, increasing $\Gamma$ always leads to an increased signal
(dashed lines in Figs. 8a-c), as it increases the probability of tunnelling
events per unit time. Moreover, for very large $\Gamma$ the signal is mostly
capacitive, as, in the Instantaneous Eigenvalues Approximation (IEA), $G_{S}$
derives from electrons that failed to tunnel elastically at the CR Fermi
energy, but still have nonzero tunnelling probability because of thermal
smearing. The last point, as shown in Fig. 8c, remains valid in the Floquet
picture, as from the Floquet-Fermi golden rule we expect the CR to efficiently
generate transitions between modes when the energy scale of its interaction
($h\Gamma$) is comparable with the photon energy [61]. For $\Gamma$ largely
detuned from $\omega$ we expect the Floquet dynamics to be mostly unperturbed,
and thus the admittance be mostly reactive. This has a simple interpretation
in light of the time-energy Heisenberg uncertainty principle, recalling that
tunnelling in and out of the QD requires the system to create and destroy
photons. When the lifetime of the polaron becomes shorter than the photon
energy, said photons can avoid being accounted for, and thus this process
becomes dissipationless. Because of LB, however, the Floquet dynamics consists
of decaying states, as obvious by considering the eigenstates of the self-
consistent propagator in Eq.(55). When $\Gamma\gg\omega$, essentially all
dressed states will decay within an rf cycle. Therefore, we expect also the
capacitive component to decay exponentially with the Floquet dynamics.

A similar point can be made about FB when we increase $\omega$ for fixed
$\Gamma$ (Figs. 8b,d). Semiclassically, we expect most of the signal to be
resistive for large $\omega$, for the QD cannot keep up with the rf and thus
the tunnelling phase will be largely randomized. However, we still expect the
Sisyphus conductance to monotonically increase, as already argued in Section
VI.3. This ought to be clear by the fact that a damped system which is driven
too fast classically lags 90° out of phase with the driving force. This,
however, is not allowed by FB, as shown in Fig. 8d. The quantum Zeno argument
made in Section VI.3 finds its natural counterpart in the description Floquet
modes. If the coherent dynamics, in fact, becomes faster than the interaction
with the CR, we expect the Floquet evolution to be mostly unperturbed. Floquet
modes are, however, orthogonal over an rf cycle. Therefore, no transitions
between dressed states becomes allowed, and the resistive signal drops.
Perhaps more simply, this has to occur as for $\omega\gg\Gamma$ the evolution
of the SEB becomes almost unitary, and thus it cannot dissipate any energy. We
can make sense of this also from a polaron perspective, as increasing $\omega$
also increases the energy of the photons that are necessary to create the
polaron. If $\Gamma<\omega$, moreover, those photons live longer than the
Uncertainty Principle allows for virtual photons. Therefore, there must be
energy transferred to and from the CR. If we are in the FB regime, however,
this energy is larger than the average thermal energy $k_{B}T$ of the CR.
This, therefore, will be far less likely to be able to provide real photons to
cause transitions between the modes. Thus the resistive signal will decrease.

Finally, we shall point out how, perhaps unsurprisingly, the SCQME deviates
from the SME when $k_{B}T\approx\hbar\omega,h\Gamma$, while the semiclassical
result is retained in the limit of large temperature.

Figure 8:  Small-signal Tunnelling capacitance ($C_{T}$) and Sisyphus
conductance ($G_{S}$) for varying $\Gamma$ and $\omega$. Panels (a,c) show how
the effect of LB is mostly on the reactive component, while panels (b,d) shows
how the main effect of FB is to lower the Sisyphus conductance. This is to be
expected when considering energy dissipation in the CR.

##  Appendix B Optimal Small-Signal Parameters

In this section, we will briefly discuss the combined effect of LB and FB on
SEB readout. From Section VI.3, in fact, one could assume that increasing
$\omega$ would be detrimental past a certain point because of FB, and
similarly for $\Gamma$ because of LB. While this is certainly true, however,
this assumes that all other parameters are fixed. If both $\Gamma$ and
$\omega$ become free variables which may occur in systems with tunable
couplings and resonators [77].

In Figure 9 we show the impact of the small-signal admittance $|Y_{1}|$ at
constant $k_{B}T$ as a function of $\Gamma$ and $\omega$. As it ought to be
clear, the signal keeps increasing even if $k_{B}T\ll h\Gamma,\hbar\omega$,
well beyond the semiclassical regime.

Despite the dynamics now being fully dominated by quantum degrees of freedom
of the QD dynamics, the semiclassical picture is still insightful. Increasing
the frequency, in fact, means that we can get more tunnelling events per unit
time. This occurs, however, only if the electrons can keep up with the fast
oscillations. Therefore, that the signal is maximized when $\omega\sim\Gamma$,
as transpires from Fig. 9. This fact can also be explained in the quantum
picture by the uncertainty principle. LB, in fact, arises from the finite
lifetime of electrons in the QD. In practice, however, this is not seen by the
system if the QD is, on average, filled and emptied by the PhB faster than the
intrinsic lifetime of the level. A similar argument can be made for FB, which
can essentially be seen as a quantum Zeno phenomenon as the electron gets
trapped in the QD. A short enough lifetime, however, ensures that tunnelling
can still occur. Therefore, these two effects complement each other and the
signal keeps increasing at the onset of LB and FB, just more slowly than it
would have done in the semiclassical model.

Figure 9: Maximum of the small-signal admittance $|Y_{1}|$ when varying
$\Gamma$ and $\omega$. The figure shows how the maximum signal is always
achieved when $\Gamma\approx\omega$, and both are as large as possible.

##  Appendix C From Admittance to Reflectometry Lineshape

Having obtained the effective quantum impedance of the SEB, we need to compute
the system’s transmission coefficient $\mathcal{T}(Y_{N})$ as a function of
$Y_{N}$ when the system is coupled via the Gate to the resonator, whose centre
frequency is $\omega_{res}=N\omega$. It is easier to consider the case of the
setup in Fig. 10a, with the SEB in parallel to the resonator.

We can now define $\mathcal{T}(Y_{N})$ by considering the phasor relation

$V_{out}^{N}=\mathcal{T}(Y_{N})V_{in}.$ (121)

where $V_{out}^{N}(t)=\Re\left[V_{out}e^{iN\omega t}\right]$ and
$V_{in}(t)=\Re\left[V_{in}e^{i\omega t}\right]$. We point out that, for $N=1$,
this is nothing but the standard definition of the transmission coefficient.

To build an equivalent circuit model, we can now consider that the impedance
seen by $V_{in}$ will be much larger than that of the line $Z_{0}=50\Omega$,
of the order of the quantum of resistance $e^{2}/h$. Therefore, most of the
signal will be reflected. For simplicity, we will assume that the drive is
large enough to be able to neglect the self-loading of the collection gate
[41]. In this case, the dynamics of the SEB as seen by the output can be
replaced by a Voltage-Controlled Current Source (VCCS), whose output will be
$I_{N}=\Re\left[V_{in}Y_{N}e^{iN\omega t}\right]$. Considering this equivalent
circuit, we easily obtain

$V_{out}^{N}=-\frac{Y_{N}}{Y_{Res}}~{}V_{in}.$ (122)

We note that the impedance of the resonator, in parallel to the VCCS, will be
significant only at the resonant frequency, while it shall act as a short to
ground at all other frequencies. Hence justifying the formalism of equivalent
admittance, since all the terms not oscillating at the resonant frequency
(here taken $\omega_{Res}=N\omega$) are short to ground. For the resonant
term, however, we see how the low admittance of the resonator amplifies the
resonant signal. Moreover, we can see how the effective admittance of the SEB,
or rather, its transconductance, is directly proportional to the transmitted
signal. This last observation allows us to directly relate Eqs.(74) and (76)
to experimental reflectometry data.

The high admittance of the off-resonance resonator forces us to use 2 separate
gates for the driving. For simplicity, we imagine them to have negligible
crosstalk and the same lever arm. If this is not the case, one can simply
account for this by modifying $\alpha^{2}\rightarrow\alpha_{d}\alpha_{c}$ in
$\mathcal{C}_{N}$ (Eq.(72)), where $\alpha_{d}$ and $\alpha_{c}$ are the lever
arms of the driving and collection gates respectively [18].

Lastly, we point out that, if the measurement is made with a single Gate in
reflection rather than transmission, by a similar argument one can immediately
write

$V_{out}^{\mathcal{R}}=\left(1-\frac{Y_{1}}{Y_{Res}}\right)~{}V_{in}$ (123)

where we highlight that only the fundamental can be observed through the
resonator in this case.

Figure 10: (a) Transmission experiment setup with an SEB connected in parallel
with a resonator. (b) Equivalent circuit schematic in which the SEB is seen by
the resonator as a set of VCCSs in parallel.

##  Appendix D The broadened Fermi-Dirac

In this section, we will briefly discuss the mathematical steps that lead to
the digamma function in Eq.(66). While this result allows for some physical
insights (e.g. the discussion of Eq.(67)), a closed-form equation for
$\mathcal{F}_{m}$ allows for dramatically lower computation times. In
particular, the expansion of the digamma (and trigamma) functions as infinite
series or continued fractions allows for zero-cost inclusion of LB in QD
simulations.

To begin with, we notice the obvious fact that

$\mathcal{F}_{m}^{\pm}(\varepsilon_{0})=\frac{\Gamma}{\pi}\int_{-\infty}^{\infty}~{}~{}~{}\frac{f(\pm\epsilon)}{\Gamma^{2}+((\epsilon-\varepsilon_{0})/\hbar-m\omega)^{2}}~{}~{}~{}d\epsilon$
(124)

is a convolution between the Fermi-Dirac and a Lorentzian, and thus can be
computed as a product in reciprocal space. In the following, we shall for
simplicity only consider the case of
$\mathcal{F}_{0}^{-}(\varepsilon_{0})=\mathcal{F}(\varepsilon_{0})$, and all
other cases trivially follow.

Before tackling the problem, we can simplify the calculation by noticing that

$f(\epsilon)=\frac{1}{2}-\frac{1}{2}\tanh{\left(\frac{\epsilon}{2k_{B}T}\right)}$
(125)

and thus

$\mathcal{F}(\varepsilon_{0})=\frac{1}{2}-\frac{\Gamma}{2\pi}\int_{-\infty}^{\infty}~{}~{}~{}\frac{\tanh{\left(\frac{\epsilon}{2k_{B}T}\right)}}{\Gamma^{2}+(\epsilon-\varepsilon_{0})^{2}/\hbar^{2}}~{}~{}~{}d\epsilon.$
(126)

We can now make use of the fact that the Fourier transform of a Lorentzian is
a decaying exponential, while,

$\mathfrak{F}\left[\tanh{\left(\frac{\epsilon}{2k_{B}T}\right)}\right](\xi)=\frac{i\pi
k_{B}T}{\sinh{\pi k_{B}T\xi}}$ (127)

where $\mathfrak{F}$ indicates the Fourier transform in a distribution sense,
to write

$\mathcal{F}(\varepsilon_{0})=i~{}\mathfrak{F}^{-1}\left[\frac{\pi
k_{B}Te^{-|\Gamma\xi|}}{\sinh{(\pi k_{B}T\xi)}}\right].$ (128)

We can now use the fact that $\mathcal{F}(\varepsilon_{0})-\frac{1}{2}$ is
antisymmetric to write the integrals as

$\mathcal{F}(\varepsilon_{0})-\frac{1}{2}=i\pi
k_{B}T\left(\int_{0}^{\infty}\frac{e^{i\varepsilon_{0}\xi-\Gamma\xi}e^{\pi
k_{B}T\xi}}{1-e^{-2\pi k_{B}T\xi}}+c.c.\right).$ (129)

To get the desired result we must now perform the substitution $t=2\pi
k_{B}T\xi$ and notice that, for $\Re[z]>0$, we can write the digamma function
as[78]

$\psi_{0}(z)=\int_{0}^{+\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-zt}}{1-e^{-t}}\right)dt.$
(130)

Adding and subtracting $\frac{e^{-t}}{t}$ to Eq.(129) therefore, we get

$\displaystyle\mathcal{F}(\varepsilon_{0})-\frac{1}{2}=\frac{1}{2\pi}\bigg{(}\psi_{0}\left(\frac{1}{2}+\frac{h\Gamma+i\varepsilon_{0}}{2\pi
k_{B}T}\right)-$ (131)
$\displaystyle-\psi_{0}\left(\frac{1}{2}+\frac{h\Gamma-i\varepsilon_{0}}{2\pi
k_{B}T}\right)\bigg{)}.$

Finally, we can use the identity

$\psi_{0}(z^{*})=\psi_{0}(z)^{*}$ (132)

to write

$\mathcal{F}(\varepsilon_{0})=\frac{1}{2}-\frac{1}{\pi}\Im\left[\psi_{0}\left(\frac{1}{2}+\frac{h\Gamma+i\varepsilon_{0}}{2\pi
k_{B}T}\right)\right]$ (133)

of which Eq.(66) is an immediate generalization.

As a final remark, an interesting sanity check is that it is a well-known fact
from taking the logarithmic derivative of the Gamma function that, for
$x\in\mathbb{R}$,

$\Im\left[\psi_{0}\left(\frac{1}{2}+x\right)\right]=\frac{\pi}{2}\tanh{(\pi
x)}$ (134)

which immediately shows how Eq.(133) retrieves Eq.(125) in the limit of
$\Gamma=0$.

##  Appendix E Polaron-Transformed Hamiltonian

To discuss the mathematical manipulations presented in Section IV.2.1, we can
begin by noticing that $[d,a]=[c,a]=0$, and similarly for any of the
respective adjoints. Thus, we can simply picture the polaron transformation as
a displacement of the bath, which happens to be dependent on the QD
occupation.

In the main text, we made reference to the BCH theorem, which states for two
matrices $A$ and $B$ that

$e^{A}Ba^{-A}=B+[B,A]$ (135)

if $[B,A]$ is a c-number (i.e. $[B,A]$ commutes with both $A$ and $B$).

Considering this property, for any displacement operator
$D_{\alpha}=\exp(\alpha a^{\dagger}-a^{*}a)$, we have

$D_{\alpha}aD\alpha^{\dagger}=a+\alpha,$ (136)

from which it follows immediately that

$e^{-S}\left({H_{PhB}+H_{DP}}\right)e^{S}=\frac{g^{2}}{\hbar\omega}d^{\dagger}d+\hbar\omega
a^{\dagger}a.$ (137)

This is particularly remarkable because, referencing Section IV.1, the term
$a+a^{\dagger}$, which gives the semiclassically-oscillating QD energy has now
disappeared. Moreover, that term obviously contains rotating and counter-
rotating waves. Thus, especially recalling the discussion in Section VI.4, it
ought to be clear how a Rotating Wave Approximation in the usual Floquet-Rabi
sense is generally not possible if we want to describe effective admittances.

Equation (50) could be derived directly from Eq.(135) using fermionic
commutation relations. However, a simple trick is to note that $a-a^{\dagger}$
is a Grassmann number. Therefore, $e^{S}$ is also a fermionic displacement
operator, for which Eq.(136) is also valid[79].

Therefore, we have shown that the polaron transformation is simultaneously
displacing both the PhB depending on the state of the QD and the QD because of
the PhB electric field. This characteristic of double displacement allows us
to completely remove the semiclassically oscillating field but also invites us
to think of the problem no more as electrons interacting with photons but as a
combined quasiparticle (hence the name polaron). This picture is retrieved if
we consider the QD-CR interaction in the polaron frame (Eq.(50)). An
interaction of the form

$c_{\epsilon}D^{\dagger}d^{\dagger}$ (138)

destroys a fermion in the CR and simultaneously creates an electron in the QD
and displaces the PhB to create a polaron.

Finally, we shall note that, technically, the rates in Eq.(65) are obtained in
a different frame than the semiclassical description. However, from Eq.(24),
we see how, at the steady state, $\rho$ is purely diagonal. Therefore, the
density operator reads

$\rho_{SS}(t)=\frac{1}{2}\mathbb{I}+\left(P(t)-\frac{1}{2}\right)\sigma_{z}$
(139)

where $\mathbb{I}$ is the identity and, obviously, $\sigma_{z}=d^{\dagger}d$
in the two-level picture. Thus, $[\rho_{SS}(t),S]=0$, and the occupation
probability in the polaron frame is the same as in the lab frame. This is an
obvious consequence of $\left|{e}\right\rangle$ and $\left|{o}\right\rangle$
belonging to different canonical Hilbert spaces, and thus they cannot be mixed
by the canonical Lang-Firsov transformation.

## References

  * Loss and DiVincenzo [1998] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998).
  * Philips  _et al._ [2022] S. G. J. Philips, M. T. Ma̧dzik, S. V. Amitonov, S. L. de Snoo, M. Russ, N. Kalhor, C. Volk, W. I. L. Lawrie, D. Brousse, L. Tryputen, B. P. Wuetz, A. Sammak, M. Veldhorst, G. Scappucci, and L. M. K. Vandersypen, Nature 609, 919 (2022).
  * Borsoi  _et al._ [2022] F. Borsoi, N. W. Hendrickx, V. John, S. Motz, F. van Riggelen, A. Sammak, S. L. de Snoo, G. Scappucci, and M. Veldhorst, Shared control of a 16 semiconductor quantum dot crossbar array (2022).
  * Xue  _et al._ [2022] X. Xue, M. Russ, N. Samkharadze, B. Undseth, A. Sammak, G. Scappucci, and L. M. Vandersypen, Nature 601, 343 (2022).
  * Noiri  _et al._ [2022] A. Noiri, K. Takeda, T. Nakajima, T. Kobayashi, A. Sammak, G. Scappucci, and S. Tarucha, Nature 601, 338 (2022).
  * Mills  _et al._ [2022] A. R. Mills, C. R. Guinn, M. J. Gullans, A. J. Sigillito, M. M. Feldman, E. Nielsen, and J. R. Petta, Science Advances 8, eabn5130 (2022).
  * Maurand  _et al._ [2016] R. Maurand, X. Jehl, D. Kotekar-Patil, A. Corna, H. Bohuslavskyi, R. Laviéville, L. Hutin, S. Barraud, M. Vinet, M. Sanquer, and S. De Franceschi, Nat. Commun. 7, 13575 (2016).
  * Zwerver  _et al._ [2022] A. Zwerver, T. Krähenmann, T. Watson, L. Lampert, H. C. George, R. Pillarisetty, S. Bojarski, P. Amin, S. Amitonov, J. Boter, _et al._ , Nature Electronics 5, 184 (2022).
  * Xue  _et al._ [2021] X. Xue, B. Patra, J. P. van Dijk, N. Samkharadze, S. Subramanian, A. Corna, B. Paquelet Wuetz, C. Jeon, F. Sheikh, E. Juarez-Hernandez, _et al._ , Nature 593, 205 (2021).
  * Ruffino  _et al._ [2022] A. Ruffino, T.-Y. Yang, J. Michniewicz, Y. Peng, E. Charbon, and M. F. Gonzalez-Zalba, Nature Electronics 5, 53 (2022).
  * Petersson  _et al._ [2010] K. D. Petersson, C. G. Smith, D. Anderson, P. Atkinson, G. A. Jones, and D. A. Ritchie, Nano Lett. 10, 2789 (2010).
  * Vigneau  _et al._ [2023] F. Vigneau, F. Fedele, A. Chatterjee, D. Reilly, F. Kuemmeth, M. F. Gonzalez-Zalba, E. Laird, and N. Ares, Applied Physics Reviews 10, 10.1063/5.0088229 (2023).
  * House  _et al._ [2016] M. G. House, I. Bartlett, P. Pakkiam, M. Koch, E. Peretz, J. van der Heijden, T. Kobayashi, S. Rogge, and M. Y. Simmons, Physical Review Applied 6, 044016 (2016).
  * Oakes  _et al._ [2023] G. A. Oakes, V. N. Ciriano-Tejel, D. F. Wise, M. A. Fogarty, T. Lundberg, C. Lainé, S. Schaal, F. Martins, D. J. Ibberson, L. Hutin, B. Bertrand, N. Stelmashenko, J. W. A. Robinson, L. Ibberson, A. Hashim, I. Siddiqi, A. Lee, M. Vinet, C. G. Smith, J. J. L. Morton, and M. F. Gonzalez-Zalba, Phys. Rev. X 13, 011023 (2023).
  * van der Heijden  _et al._ [2018] J. van der Heijden, T. Kobayashi, M. G. House, J. Salfi, S. Barraud, R. Laviéville, M. Y. Simmons, and S. Rogge, Science Advances 4, eaat9199 (2018).
  * Ahmed  _et al._ [2018a] I. Ahmed, A. Chatterjee, S. Barraud, J. J. Morton, J. A. Haigh, and M. F. Gonzalez-Zalba, Commun. Phys. 1, 1 (2018a).
  * Chawner  _et al._ [2021] J. M. A. Chawner, S. Barraud, M. F. Gonzalez-Zalba, S. Holt, E. A. Laird, Y. A. Pashkin, and J. R. Prance, Phys. Rev. Appl. 15, 034044 (2021).
  * Oakes  _et al._ [2022] G. A. Oakes, L. Peri, L. Cochrane, F. Martins, L. Hutin, B. Bertrand, M. Vinet, A. G. Saiz, C. J. B. Ford, C. G. Smith, and M. F. Gonzalez-Zalba, A quantum dot-based frequency multiplier (2022), arXiv:2211.14127 [cond-mat.mes-hall] .
  * Cochrane  _et al._ [2022a] L. Cochrane, T. Lundberg, D. J. Ibberson, L. A. Ibberson, L. Hutin, B. Bertrand, N. Stelmashenko, J. W. A. Robinson, M. Vinet, A. A. Seshia, and M. F. Gonzalez-Zalba, Phys. Rev. Lett. 128, 197701 (2022a).
  * Mi  _et al._ [2017] X. Mi, J. V. Cady, D. M. Zajac, P. W. Deelman, and J. R. Petta, Science 355, 156 (2017).
  * Samkharadze  _et al._ [2018] N. Samkharadze, G. Zheng, N. Kalhor, D. Brousse, A. Sammak, U. C. Mendes, A. Blais, G. Scappucci, and L. M. K. Vandersypen, Science (2018).
  * Hansen  _et al._ [2022] I. Hansen, A. E. Seedhouse, K. W. Chan, F. E. Hudson, K. M. Itoh, A. Laucht, A. Saraiva, C. H. Yang, and A. S. Dzurak, Applied Physics Reviews 9, 031409 (2022).
  * Seedhouse  _et al._ [2021] A. E. Seedhouse, I. Hansen, A. Laucht, C. H. Yang, A. S. Dzurak, and A. Saraiva, Physical Review B 104, 235411 (2021).
  * Mizuta  _et al._ [2017] R. Mizuta, R. M. Otxoa, A. C. Betz, and M. F. Gonzalez-Zalba, Phys. Rev. B 95, 045414 (2017).
  * Esterli  _et al._ [2019] M. Esterli, R. Otxoa, and M. Gonzalez-Zalba, Applied Physics Letters 114, 10.1063/1.5098889 (2019).
  * Cottet  _et al._ [2011] A. Cottet, C. Mora, and T. Kontos, Phys. Rev. B 83, 121311(R) (2011).
  * Crepieux and Lavagna [2022] A. Crepieux and M. Lavagna, Physical Review B 106, 115439 (2022).
  * Gambetta  _et al._ [2008] J. Gambetta, A. Blais, M. Boissonneault, A. A. Houck, D. I. Schuster, and S. M. Girvin, Physical Review A 77, 012112 (2008).
  * [29] J. Gambetta, A. Blais, D. I. Schuster, A. Wallraff, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Physical Review A 74, 042318, number: 4, cond-mat/0602322 .
  * Slichter  _et al._ [2012] D. H. Slichter, R. Vijay, S. J. Weber, S. Boutin, M. Boissonneault, J. M. Gambetta, A. Blais, and I. Siddiqi, Physical Review Letters 109, 153601 (2012).
  * Derakhshan Maman  _et al._ [2020] V. Derakhshan Maman, M. F. Gonzalez-Zalba, and A. Palyi, Physical Review Applied 14, 064024 (2020).
  * Yamaguchi  _et al._ [2017] M. Yamaguchi, T. Yuge, and T. Ogawa, Physical Review E 95, 012136 (2017).
  * Manzano [2020] D. Manzano, AIP Advances 10, 025106 (2020), publisher: American Institute of Physics.
  * Mori [2023] T. Mori, Annual Review of Condensed Matter Physics 14, 35 (2023).
  * Gardiner and Zoller [2004] C. W. Gardiner and P. Zoller,  _Quantum noise: a handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics_ , 3rd ed., Springer series in synergetics (Springer, 2004).
  * Sowa  _et al._ [2018] J. K. Sowa, J. A. Mol, G. A. D. Briggs, and E. M. Gauger, The Journal of Chemical Physics 149, 154112 (2018), arXiv:1807.08502 [cond-mat, physics:physics].
  * Gardiner and Gardiner [2009] C. W. Gardiner and C. W. Gardiner,  _Stochastic methods: a handbook for the natural and social sciences_ , 4th ed., Springer series in synergetics (Springer, Berlin, 2009).
  * Am-Shallem  _et al._ [2015] M. Am-Shallem, A. Levy, I. Schaefer, and R. Kosloff, Three approaches for representing lindblad dynamics by a matrix-vector notation (2015), arXiv:1510.08634 [quant-ph] .
  * Yan [1998] Y. Yan, Physical Review A 58, 2721 (1998).
  * Kohler  _et al._ [2005] S. Kohler, J. Lehmann, and P. Hänggi, Physics Reports 406, 379 (2005), cond-mat/0409251 .
  * Cochrane  _et al._ [2022b] L. Cochrane, A. A. Seshia, and M. F. G. Zalba, Intrinsic noise of the single electron box (2022b), arXiv:2209.15086 [cond-mat.mes-hall] .
  * Sowa  _et al._ [2020] J. K. Sowa, N. Lambert, T. Seideman, and E. M. Gauger, The Journal of Chemical Physics 152, 064103 (2020), arXiv:1912.11003 [cond-mat, physics:physics].
  * Kohler  _et al._ [1997] S. Kohler, T. Dittrich, and P. Hänggi, Physical Review E 55, 300 (1997), publisher: American Physical Society.
  * Kohler [2018] S. Kohler, Physical Review A 98, 023849 (2018), publisher: American Physical Society.
  * Childs  _et al._ [2001] A. M. Childs, E. Farhi, and J. Preskill, Physical Review A 65, 012322 (2001).
  * Brasil  _et al._ [2013] C. A. Brasil, F. F. Fanchini, and R. d. J. Napolitano, Revista Brasileira de Ensino de Física 35, 01 (2013), arXiv:1110.2122 [quant-ph].
  * Dann  _et al._ [2018] R. Dann, A. Levy, and R. Kosloff, Physical Review A 98, 052129 (2018), arXiv:1805.10689 [quant-ph].
  * Gonzalez-Zalba  _et al._ [2015] M. F. Gonzalez-Zalba, S. Barraud, A. J. Ferguson, and A. C. Betz, Nat Commun 6, 6084 (2015).
  * Jang [2022] S. J. Jang, The Journal of Chemical Physics 157, 104107 (2022), arXiv:2203.02812 [cond-mat, physics:physics, physics:quant-ph].
  * Xu and Cao [2016] D. Xu and J. Cao, Frontiers of Physics 11, 110308 (2016).
  * Wilner  _et al._ [2015] E. Y. Wilner, H. Wang, M. Thoss, and E. Rabani, Physical Review B 92, 195143 (2015), arXiv:1508.07793 [cond-mat].
  * Lee  _et al._ [2009] W. Lee, N. Jean, and S. Sanvito, Physical Review B 79, 085120 (2009).
  * Esposito and Galperin [2010] M. Esposito and M. Galperin, The Journal of Physical Chemistry C 114, 20362 (2010).
  * Hou  _et al._ [2015] D. Hou, S. Wang, R. Wang, L. Ye, R. Xu, X. Zheng, and Y. Yan, The Journal of Chemical Physics 142, 104112 (2015).
  * Ikeda  _et al._ [2021] T. Ikeda, K. Chinzei, and M. Sato, SciPost Physics Core 4, 033 (2021).
  * Benito  _et al._ [2017] M. Benito, X. Mi, J. M. Taylor, J. R. Petta, and G. Burkard, Physical Review B 96, 235434 (2017).
  * Koski  _et al._ [2018] J. V. Koski, A. J. Landig, A. Palyi, P. Scarlino, C. Reichl, W. Wegscheider, G. Burkard, A. Wallraff, K. Ensslin, and T. Ihn, Physical Review Letters 121, 043603 (2018).
  * Hartung  _et al._ [2022] T. Hartung, K. Jansen, and C. Sarti, Zeta-regularized lattice field theory with lorentzian background metrics (2022), arXiv:2208.08223 [hep-lat] .
  * Tien and Gordon [1963] P. K. Tien and J. P. Gordon, Physical Review 129, 10.1103/PhysRev.129.647 (1963).
  * Platero and Aguado [2004] G. Platero and R. Aguado, Physics Reports 395, 1 (2004).
  * Rudner and Lindner [2020] M. S. Rudner and N. H. Lindner, The floquet engineer’s handbook (2020), arXiv:2003.08252 [cond-mat, physics:quant-ph].
  * Albert  _et al._ [2016] V. V. Albert, B. Bradlyn, M. Fraas, and L. Jiang, Physical Review X 6, 041031 (2016), arXiv:1512.08079 [cond-mat, physics:math-ph, physics:quant-ph].
  * House  _et al._ [2015] M. G. House, T. Kobayashi, B. Weber, S. J. Hile, T. F. Watson, J. van der Heijden, S. Rogge, and M. Y. Simmons, Nature Communications 6, 8848 (2015).
  * Chatterjee  _et al._ [2018] A. Chatterjee, S. N. Shevchenko, S. Barraud, R. M. Otxoa, F. Nori, J. J. L. Morton, and M. F. Gonzalez-Zalba, Physical Review B 97, 045405 (2018).
  * Ivakhnenko  _et al._ [2023] O. V. Ivakhnenko, S. N. Shevchenko, and F. Nori, Physics Reports Nonadiabatic Landau-Zener-Stückelberg-Majorana transitions, dynamics, and interference, 995, 1 (2023).
  * Shevchenko  _et al._ [2010] S. Shevchenko, S. Ashhab, and F. Nori, Phy Rep 492, 1 (2010).
  * Blais  _et al._ [2004a] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 69, 062320 (2004a).
  * McTague and Foley [2022] J. McTague and J. J. Foley, The Journal of Chemical Physics 156, 154103 (2022), https://doi.org/10.1063/5.0091953 .
  * Dutra [1997] S. M. Dutra, European Journal of Physics 18, 194 (1997).
  * Roccati  _et al._ [2022] F. Roccati, S. Lorenzo, G. Calajò , G. M. Palma, A. Carollo, and F. Ciccarello, Optica 9, 565 (2022).
  * Yan  _et al._ [2016] F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. Sears, D. Hover, T. J. Gudmundsen, D. Rosenberg, G. Samach, S. Weber, J. L. Yoder, T. P. Orlando, J. Clarke, A. J. Kerman, and W. D. Oliver, Nature Communications 7, 12964 (2016).
  * Blais  _et al._ [2004b] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Physical Review A 69, 062320 (2004b).
  * Sears  _et al._ [2012] A. P. Sears, A. Petrenko, G. Catelani, L. Sun, H. Paik, G. Kirchmair, L. Frunzio, L. I. Glazman, S. M. Girvin, and R. J. Schoelkopf, Physical Review B 86, 180504(R) (2012).
  * [74] M. Malekakhlagh, E. Magesan, and L. C. G. Govia, Time-dependent schrieffer-wolff-lindblad perturbation theory: measurement-induced dephasing and second-order stark shift in dispersive readout, issue: arXiv:2206.12467, 2206.12467 [cond-mat, physics:quant-ph] .
  * von Horstig  _et al._ [2023] F.-E. von Horstig, D. J. Ibberson, G. A. Oakes, L. Cochrane, N. Stelmashenko, S. Barraud, J. A. W. Robinson, F. Martins, and M. F. Gonzalez-Zalba, Multi-module microwave assembly for fast read-out and charge noise characterization of silicon quantum dots (2023), arXiv:2304.13442 [cond-mat.mes-hall] .
  * Ahmed  _et al._ [2018b] I. Ahmed, A. Chatterjee, S. Barraud, J. J. L. Morton, J. A. Haigh, and M. F. Gonzalez-Zalba, Communications Physics 1, 1 (2018b).
  * Scarlino  _et al._ [2022] P. Scarlino, J. H. Ungerer, D. J. van Woerkom, M. Mancini, P. Stano, C. Muller, A. J. Landig, J. V. Koski, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, Physical Review X 12, 031004 (2022).
  * Whittaker and Watson [1996] E. T. Whittaker and G. N. Watson,  _A Course of Modern Analysis_, 4th ed., Cambridge Mathematical Library (Cambridge University Press, 1996).
  * Cahill and Glauber [1999] K. E. Cahill and R. J. Glauber, Physical Review A 59, 1538 (1999), arXiv:physics/9808029.