# Fluctuating parametric driving of coupled classical oscillators can simulate
dissipative qubits

Lorenzo Bernazzani<EMAIL_ADDRESS>Department of Physics,
University of Konstanz, D-78457 Konstanz, Germany.  Guido Burkard
<EMAIL_ADDRESS>Department of Physics, University of Konstanz,
D-78457 Konstanz, Germany.

###### Abstract

We investigate a system composed of two coupled oscillators subject to
stochastic fluctuations in its internal parameters. In particular, we answer
the question whether the well-known classical analogy of the quantum dynamics
of two-level systems (TLS), i.e. qubits, provided by two coupled oscillators
can be extended to simulate the dynamics of dissipative quantum systems. In
the context of nanomechanics, the analogy in the dissipation free case has
already been tested in multiple experimental setups, e.g., doubly clamped or
cantilever string resonators and optically levitated particles. A well-known
result of this classical analogy is that the relaxation and decoherence times
of the analog quantum system must be equal, i.e. $T_{1}=T_{2}$, in contrast to
the general case of quantum TLS. We show that this fundamentally quantum
feature, i.e. $T_{1}\neq T_{2}$, can be implemented as well in the
aforementioned classical systems by adding stochastic fluctuations in their
internal parameters. Moreover, we show that these stochastic contributions can
be engineered in the control apparatus of those systems.

##  I Introduction

Enquiries on quantum-classical analogies have attracted much interest [1, 2,
3, 4], and it is well established that the dynamics of a quantum N-level
system described by Schrödinger’s equation can be simulated by classical
systems, e.g. coupled classical harmonic oscillators [5, 6, 7, 8, 9, 10, 11,
12]. To date, several papers have outlined that this similarity between the
two dynamics leads to the observation of purely quantum mechanical effects in
classical systems, e.g. Rabi oscillations [5, 13, 14], Landau-Zener
transitions [15], and also Stückelberg interferometry [16, 17, 18, 11]. Quite
surprisingly, some of these effects have been reported also in macroscopic
systems [12], pushing this analogy even out to the macroscopic world. As a
consequence, classical coupled oscillators constitute the simplest platform to
study how a classical system can mimic quantum dynamics.

Mechanics was indeed the first playground of physics. Nowadays, it is, maybe
surprisingly, still investigated in the modern quest for the nanoscale
miniaturization of physical devices [19, 20]. An astounding degree of control
has been achieved in mechanical systems over the past years [21, 22, 23]. This
has fostered the hope [24] to reach mesoscopic quantum superposition of
massive objects and to study quantum effects of gravity in the lab [25, 26,
22], with the aim of uncovering some of the elusive aspects of the quantum-
classical border [27].

This work is meant to give some new insights about the quantum-classical
analogy, drawing from the aforementioned mapping of quantum evolution by means
of classical oscillators. In particular, we address the issue of the trivial
form of the relaxation term [13, 8, 14, 11] in the Schrödinger equation for
the simulated quantum TLS, i.e., the fact that all the components of the Bloch
vector (BV) decay with the same characteristic time. It is well known instead,
that in the quantum case there are two relaxation times, the longitudinal
$T_{1}$ and the transverse $T_{2}$. These are linked to the relaxation of the
populations and of the coherences of the TLS state. Furthermore, they are
related by the equation $T^{-1}_{2}=(2T_{1})^{-1}+T^{-1}_{\phi}$, which also
defines the phase relaxation time $T_{\phi}$ [28]. Our aim is to show how
noise can induce dissipation mechanisms with such quantum features, that were
not grasped by the previous classical model. The addition of stochastic
fluctuations and the solution of the associated Langevin dynamics [29] adds
quantumness to the system, meaning that it provides the phenomenology of this
model with a pure phase relaxation time, dependent on the noise strength.

The quantum phase is the defining concept of quantum mechanics, giving rise to
quantum superpositions, interference phenomena, and many-body entanglement.
The perturbation of microscopic quantum systems leads to the loss of phase
coherence, and, on a macroscopic scale, the emergence of our classical
reality. Decoherence is then a detrimental aspect for quantum information
systems [28]. However, the interaction of quantum systems with the external
degrees of freedom of the environment is unavoidable. The attempt to suppress
the coupling between the system and environment has been complemented with the
modelling of the effects of these interaction on the reduced system dynamics.
While an accurate model can be achieved in simple systems, the computational
complexity scales exponentially with the system size. To cope with this,
simulation of dissipation is a valuable tool [30]. The common approach
consists in adding classical noise to the analog system. In this way the open-
quantum-system, that one is interested in simulating, is mapped onto another
system, more controllable, the dynamics of which is proven to be analog. Our
work also follows this direction by providing a very simple classical analog
system, in which aspects of quantum dissipation can be simulated and even
visualized [12]. The approach to analog simulation of quantum dissipation via
classical noise has a precedent in the context of many-body quantum systems
[31], and the relation between classical and quantum noise in open systems
dynamics has been thoroughly investigated and described [32, 33, 34]. The
novelty of this paper lies in the fact that the system is completely
classical. Therefore it constitutes a furthest simplification of the
aforementioned results that, moreover, can be readily tested in the lab.

##  II The classical two-level atom

Let us start from a system of coupled classical mechanical oscillators
consisting of two bodies with equal mass $m$ connected with springs to each
other and to the adjacent fixed walls, as depicted in Fig. 1. This system is
described by the following linear coupled classical equations of motion for
the positions $x_{1}$ and $x_{2}$ of the two masses,

$\displaystyle m\ddot{x}_{1}+m\gamma\dot{x}_{1}+\biggl{[}k-\frac{\Delta
k(t)}{2}\biggr{]}x_{1}+h\bigl{(}x_{1}-x_{2}\bigr{)}$ $\displaystyle=f(t),$ (1)
$\displaystyle m\ddot{x}_{2}+m\gamma\dot{x}_{2}+\biggl{[}k+\frac{\Delta
k(t)}{2}\biggr{]}x_{2}+h\bigl{(}x_{2}-x_{1}\bigr{)}$ $\displaystyle=0\,,$ (2)

where we assumed that the masses $m$ and the damping coefficient $\gamma$ of
the two oscillators are equal for convenience. The time dependent spring
constants $k_{1,2}(t)=k\mp\Delta k(t)/2$ here account for the parametric
driving mechanism that we are going to discuss extensively later. The (linear)
coupling between the two oscillators is given by $h$. The inhomogeneous term
on the right hand side of the first equation describes the effect of an
external time-dependent deterministic or fluctuating force. This term has the
purpose of injecting energy into the system by displacing the oscillator
$x_{1}$ from its equilibrium position. We discuss later what happens when we
let the system evolve once it has been initialized in a certain state.
Therefore, in the following we consider the time-evolution of the system after
the initial time $t=0$ at which the inhomogeneous force stops. In the rest of
the paper, we set then the inhomogeneus term $f(t)=0$. Nonetheless, since we
have introduced friction through the $\gamma$ coefficient, it would generally
be the case that a fluctuating force of thermal origin will also appear in
$\delta f(t)$. For simplicity, we delegate the treatment of the more general
case $\delta f(t)\neq 0$ to Appendix A. In any case, the suppression of this
thermal noise term can be accomplished with little effort in real systems by
lowering the temperature of the thermal bath or by using feedback mechanisms
[35, 21].

Figure 1: Schematics of the classical-mechanical model under consideration.
Two masses $m$ are connected through a spring with the time-independent spring
constant $h$, and each of them is connected to the neighbouring wall by
springs with time-dependent spring constants $k_{1}(t)$ and $k_{2}(t)$, with
$k_{2}(t)-k_{1}(t)=\Delta k(t)\,$. We also include friction by means of a
damping coefficient $\gamma$ equal for both masses. Moreover, we include an
external driving force $f(t)$. This external driving has the purpose of
initially feeding energy into the system that is then driven parametrically
[36, 23], via the modulation of the spring constants.

Following Ref. 8, let us now divide Eq. (2) by $m$ and introduce the
quantities
$\omega_{0}^{2}\equiv\frac{k+h}{m}\,,\,\Omega^{2}_{c}\equiv\frac{h}{m}\,,\,\Omega^{2}_{d}(t)\equiv\frac{\Delta
k(t)}{2m}\,$. After this relabeling we can rewrite the Eq. (2) in the
following matrix form,

$\Bigl{(}\frac{d^{2}}{dt^{2}}+\gamma\frac{d}{dt}+\omega_{0}^{2}\Bigr{)}\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}+\begin{bmatrix}-\Omega^{2}_{d}&-\Omega^{2}_{c}\\\
-\Omega^{2}_{c}&\Omega^{2}_{d}\end{bmatrix}\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}=\begin{bmatrix}0\\\ 0\end{bmatrix}.$ (3)

Now we are going to make the ansatz
$x_{1,2}(t)=\mathrm{Re}{\bigl{[}\psi_{1,2}(t)\,e^{i\omega_{0}t}\bigr{]}}$.
Basically this consists in factorising the oscillation into an oscillating
component at the carrier frequency $\omega_{0}$ times an amplitude modulation
$\psi_{1,2}(t)$. If we make the further assumption that the envelope
$\psi_{1,2}$ is a slowly varying function of time we can drop the second
derivative in our equations of motion. This has been called the slowly varying
envelope approximation (SVEA) [8]. Furthermore, we assumed
$\gamma\ll\omega_{0}$ [8, 11], which is typically the case in state of the art
micro/nanomechanics since $\gamma=\omega_{0}/Q$, where $Q$ is the quality
factor, which usually varies between $10^{3}$ and $10^{6}$ for most of the
systems of interest [21]. These modifications lead to a matrix equation for
the complex valued amplitudes alone, written in a vectorial form, i.e.,
utilizing the vector $\psi=\bigl{[}\,\psi_{1}\,,\,\psi_{2}\,\bigr{]}^{T}$,

$i\dot{\psi}=\mathrm{H}(t)\psi-i\frac{\gamma}{2}\psi=\,\frac{1}{2}\bigl{[}\Delta\sigma_{x}+\varepsilon(t)\sigma_{z}\bigr{]}\psi-i\frac{\gamma}{2}\psi\,,$
(4)

where $\Delta\equiv\Omega^{2}_{c}/\omega_{0}$ and
$\varepsilon\equiv\Omega^{2}_{d}/\omega_{0}$. In the absence of friction
$\gamma=0$, the two-component-amplitude equation (4) is an analogy of the
Schrödinger equation for a driven TLS (taken with $\hbar=1$) [37].

It is customary now to introduce a harmonic form for the driving term and we
will specifically use $\varepsilon(t)=\varepsilon_{0}+D\cos{\omega t}\,$. It
has already been shown that this parametric driving induces a coherent
dynamics that is quite faithfully resembling a quantum TLS. In fact, we could
introduce either density matrix or Bloch vector (BV) equivalent
representations [38, 39, 40], with the definitions,

$\rho=\begin{bmatrix}|\psi_{1}|^{2}&\psi^{\ast}_{1}\psi_{2}\\\
\psi^{\ast}_{2}\psi_{1}&|\psi_{2}|^{2}\end{bmatrix}\,,\quad\bm{\mathrm{r}}=\begin{bmatrix}\psi^{\ast}_{1}\psi_{2}+\psi^{\ast}_{2}\psi_{1}\\\
i\bigl{(}\psi^{\ast}_{2}\psi_{1}-\psi^{\ast}_{1}\psi_{2}\bigr{)}\\\
|\psi_{2}|^{2}-|\psi_{1}|^{2}.\end{bmatrix}$ (5)

It must be noted however that neither, in the general case, $\psi$ and $\rho$
have modulus or trace equal to $1$ nor the dynamical map of Eq. 4 is trace
preserving. The modulus or trace of the state would be dependent on the energy
fed into the system by the inhomogeneous driving $f(t)$, for $t<0$. From the
point $t=0$ onward anyway, the only source of damping of energy (probability
in the quantum analogy) will be the non-hermitian trivial part of the
Hamiltonian. We are going to get rid of this part in the next section by a
simple coordinate transformation. Notwithstanding, the Hamiltonian with
$\gamma=0$ is conservative in the classical language and Hermitian in the
quantum analogy, therefore that part is energy/trace conserving. Standing
these _caveat_ , we call our $\psi$ and $\rho$ states nonetheless.

Since the Hamiltonian in Eq. (4) can be written in the form
$\mathrm{H}(t)=\bm{\mathrm{B}}(t)\cdot\bm{\sigma}$ this quantum classical
analogy leads to the so-called classical Bloch equations (BE) [8], i.e.,

$\bm{\dot{\mathrm{r}}}=\bm{\mathrm{B}}(t)\times\bm{\mathrm{r}}-\gamma\bm{\mathrm{r}}\,.$
(6)

Here
$\bm{\mathrm{B}}(t)=\bigl{[}\,\Delta\,,\,0\,,\,\varepsilon(t)\,\bigr{]}^{T}$
is the magnetic field vector and its modulus $B$, is the angular velocity with
which the Bloch vector precesses. In a magnetic resonance description this
describes the Larmor precession of a spin with Bloch vector ${\bf r}$ around a
magnetic field along $\bm{\mathrm{B}}$.

The resemblance of quantum dynamics shown by Eq. (6) is striking considered
that they have been derived by purely classical means. However, there is one
evident issue in these equations, which makes the classical simulation less
faithful, i.e., that the relaxation terms are not equal to the general
relaxation present, e.g., in dissipative spin dynamics. More precisely, we
know from the theory of either light-matter interaction or magnetic resonance
that the components of the BV relax with two characteristic times [40, 39,
28]: $T_{2}$ is the relaxation time of the $x$ and $y$ components of the BV,
hence the ones containing the coherences, while $T_{1}$ is the relaxation time
for the $z$ component, related to level populations in the quantum language.
In the following, we will explain how this hallmark quantum feature can be
implemented in the aforementioned analogy.

##  III Fluctuations in the driving

We make a change of coordinates to go into the _diabatic_ basis, i.e.
$\psi^{\prime}=\begin{bmatrix}\psi_{+}\\\
\psi_{-}\end{bmatrix}\equiv\exp\Bigl{(}\frac{i\theta}{2}\sigma_{y}\Bigr{)}\begin{bmatrix}\psi_{1}\\\
\psi_{2}\end{bmatrix}.$ After this rotation by the angle
$\theta=\arctan{[\Delta/\varepsilon_{0}]}$, the Schrödinger equation takes the
form

$i\dot{\psi}^{\prime}=\frac{1}{2}\biggl{[}\Omega\sigma_{z}+\frac{\varepsilon_{0}}{\Omega}\varepsilon^{\prime}(t)\sigma_{z}-\frac{\Delta}{\Omega}\varepsilon^{\prime}(t)\sigma_{x}\biggr{]}\psi^{\prime}-i\frac{\gamma}{2}\psi^{\prime}\,,$
(7)

where $\varepsilon^{\prime}(t)=D\cos(\omega t)$ and
$\Omega=\sqrt{\varepsilon_{0}^{2}+\Delta^{2}}$. It is also useful to
renormalize the state vector in order to get rid of the trivial decay with
rate $\gamma/2$ and thus to ensure a normalized state vector, as costumarily
required by the quantum analogy. Therefore, we rename
$\psi_{\pm}\to\mathcal{N}\psi^{\prime}_{\pm}\,e^{-\gamma t/2}$, where
$\mathcal{N}[E(0)]$ depends on
$E(0)=\int_{-\infty}^{0}f(t)\dot{x}_{1}(t)\,dt$, i.e., the energy fed into the
system before our simulation has started.

Our aim is to add fluctuations to the dynamics of the coupled oscillators
system. We accomplish this by adding a stochastic term to the driving. Along
this route, we write the driving term $\varepsilon(t)$ with an additive
Langevin force (which actually is a multiplicative noise term for the
stochastics problem [41], since it multiplies a random variable),

$\varepsilon^{\prime}(t)=D\cos{(\omega t)}+\Gamma_{d}(t)\,.$ (8)

Here, $\Gamma_{d}$ has the dimension of a frequency. We consider for the noise
term a stationary and Gaussian Ornstein-Uhlenbeck process [42, 29, 41] such
that:

$\langle\Gamma_{d}(t)\rangle=0\,,\qquad\langle\Gamma_{d}(0)\Gamma_{d}(t)\rangle=\,\frac{G}{\tau_{c}}\exp{\biggl{(}-\frac{|t|}{\tau_{c}}\biggr{)}}\,,$

where $G$ is the noise strength.

Therefore, we have to solve the following coupled stochastic differential
equations [43],

$\displaystyle i\dot{\psi}$
$\displaystyle=\Bigl{[}\mathrm{H}_{0}+\mathrm{H}_{1}(t)\Bigr{]}\psi$ (9)
$\displaystyle=\,\frac{1}{2}\biggl{[}\Omega\sigma_{z}+\frac{D\cos{(\omega
t)}+\Gamma_{d}(t)}{\Omega}\bigl{(}\varepsilon_{0}\sigma_{z}-\Delta\sigma_{x}\bigr{)}\biggr{]}\psi\,,$

where $\mathrm{H}_{0}=\frac{1}{2}\Omega\sigma_{z}$ and
$\mathrm{H}_{1}(t)=\mathrm{H}_{d}(t)+\mathrm{H}_{s}(t)$ contains the time-
dependent parts of the Hamiltonian, both deterministic
$\mathrm{H}_{d}(t)\propto\cos(\omega t)$ and stochastic
$\mathrm{H}_{s}(t)\propto\Gamma_{d}(t)$. In order to solve this system of
stochastic equations we rely on the cumulant expansion method. This method was
pioneered by Kubo [44] and then refined by others [45, 46, 47, 48, 49, 50,
51]. These tools have been applied to magnetic resonance in molecular samples
[52]. Particularly useful will be also the extension to coloured noise with
correlations between the longitudinal and transverse component [53, 54].

The aforementioned stochastic formalism allows one to replace the full
stochastic differential equation with a differential equation for the averages
of the stochastic variables, or of their higher moments. Since we are
interested in the motion of the BV we will need the second moments of
$\psi_{\pm}$. This leads us straightforwardly to the stochastic analog of the
BE. Since we need to get to an equation of motion for the avarage of the BV,
this will contain the second moments of the stochastic variables $\psi_{\pm}$.
The effective density matrix, with obvious changes from Eq. 5 is now
$\rho(t)=\sum_{\pm}\psi_{i}^{*}\psi_{j}\,|i\rangle\langle j|$.

### Magnetic resonance analogy

Now we are going to exploit the formal analogy between Eq. 6 and the equations
of magnetic resonance of a single spin in radio-frequency magnetic
spectroscopy. This theory has been outlined by [55, 52, 53, 54, 40, 56]. It
has been lately applied to studies of decoherence problems in qubits [57, 58,
59, 60]. The approach goes as follows. From Eq. 4 we can write a von Neumann
equation for the density matrix defined above, i.e.
$i\dot{\rho}(t)=\bigl{[}\mathrm{H}(t),\rho(t)\bigr{]}=\mathcal{L}(t)\rho(t)$.
Switching to the interaction picture we get rid of the deterministic part of
the Hamiltonian for the time being. Thus we set
$\rho^{\prime}(t)=\mathrm{U}^{\dagger}(t)\rho(t)\mathrm{U}(t)\,$,
$\mathrm{H}^{\prime}(t)=\mathrm{H}_{s}^{\prime}(t)$, where $\mathrm{H}_{s}(t)$
is the stochastic-only part of the starting Hamiltonian (see Eq. 9) and
$\mathrm{U}(t)$ is defined by
$\frac{d\mathrm{U}(t)}{dt}=-i\langle\mathrm{H}(t)\rangle\mathrm{U}(t)$ with
$\mathrm{U}(0)=\openone$ [40]. The previous equation then becomes 111Since
$\frac{d}{dt}\mathrm{U}^{-1}=i\mathrm{U}^{-1}\langle\mathrm{H}\rangle$ (which
follows from the differential equation for $\mathrm{U}(t)$) we have indeed
(plugging the diff. eq. defining $\mathrm{U}$ and $\mathrm{U}^{-1}$ and the
von Neumann equation)
$\dot{\rho}^{\prime}(t)=\dot{\mathrm{U}}^{-1}\rho\mathrm{U}+\mathrm{U}^{-1}\dot{\rho}\mathrm{U}+\mathrm{U}^{-1}\rho\dot{\mathrm{U}}=i\mathrm{U}\langle\mathrm{H}\rangle\rho\mathrm{U}^{-1}-i\mathrm{U}\mathrm{H}\rho\mathrm{U}^{-1}+i\mathrm{U}\rho\mathrm{H}\mathrm{U}^{-1}-i\mathrm{U}\rho\langle\mathrm{H}\rangle\mathrm{U}^{-1}=-i\bigl{[}\mathrm{H}_{s}^{\prime},\rho^{\prime}\bigr{]}\,$.

$i\dot{\rho}^{\prime}(t)=\mathcal{L}^{\prime}(t)\rho^{\prime}(t)=\bigl{[}\mathrm{H}_{s}^{\prime}(t),\rho^{\prime}(t)\bigr{]}\,.$
(10)

Now we seek a solution by iteration [43, 50] we write

$\displaystyle\rho^{\prime}(t)$
$\displaystyle=\rho(0)-i\int_{0}^{t}\mathcal{L}^{\prime}(t_{1})\rho(0)dt_{1}-$
(11)
$\displaystyle-\int_{0}^{t}\int_{0}^{t_{1}}\mathcal{L}^{\prime}(t_{1})\mathcal{L}^{\prime}(t_{2})\rho^{\prime}(t_{2})dt_{1}dt_{2}$
(12)

Going on like this we get to [43, 50]

$\rho^{\prime}(t)=\mathbb{Y}(t|0)\rho(0)\,\implies\,\langle\rho^{\prime}(t)\rangle=\langle\mathbb{Y}(t|0)\rangle\rho(0),$
(13)

since $\rho(0)=\rho^{\prime}(0)$ is not random, and where we have introduced
the non-local kernel
$\mathbb{Y}(t|0)=\openone+\sum_{n=1}^{+\infty}(-i)^{n}\int\dots\int\mathcal{L}^{\prime}(t_{1})\dots\mathcal{L}^{\prime}(t_{n})dt_{1}\dots
dt_{n}$. Differentiating and assuming that $\langle\mathbb{Y}(t|0)\rangle$ is
invertible then

$\langle\dot{\rho}^{\prime}(t)\rangle=\langle\dot{\mathbb{Y}}(t|0)\rangle\rho(0)=\langle\dot{\mathbb{Y}}(t|0)\rangle\langle\mathbb{Y}(t|0)\rangle^{-1}\langle\rho^{\prime}(t)\rangle,$
(14)

where
$\mathbb{K}^{\prime}(t)\equiv\langle\dot{\mathbb{Y}}(t|0)\rangle\langle\mathbb{Y}(t|0)\rangle^{-1}$
is a non-stochastic superoperator by construction, since it connects averaged
quantities. We expand $\mathbb{K}^{\prime}(t)$ in orders of $G$ and truncate
this series at the second order. Utilizing
$\langle\mathcal{L}_{s}(t)\rangle=0$ we see that the cumulants simplify indeed
to the moments of $\mathcal{L}_{s}(t)$,

$\displaystyle\langle\dot{\rho}^{\prime}(t)\rangle=\mathbb{K}^{\rm{II}}(t)\langle\rho^{\prime}(t)\rangle\,,$
(15)
$\displaystyle\mathbb{K}^{\rm{II}}(t)=-\int_{0}^{t}dt^{\prime}\langle\mathcal{L}^{\prime}_{s}(t)\mathcal{L}^{\prime}_{s}(t-t^{\prime})\rangle.$

This is equivalent to the Born approximation in open quantum systems. For that
to be a good approximation it is sufficient that
$|\mathcal{L}_{s}(t)|\tau_{c}\ll 1$, since this is the relative error between
successive orders. The above requirement is equivalent to demanding that the
time scales of the noise and of the deterministic evolution are well
separated, i.e., that $\langle\mathcal{L}(t)\rangle$ varies significantly on a
time scale which is much slower than the noise memory time. Therefore, we will
say that the truncated series is a coarse grained description of the full
dynamics of $\rho(t)$.

Since the time scales are well separated and the noise spectrum exponentially
decays we can extend the limit of integration to $+\infty$, therefore we can
see that our expansion is in fact an expansion in $G\tau_{c}\ll 1$, since
$\tau_{c}$ is the width of the interval where the integrand gives an important
contribution. The extension of the integration interval leads to:

$\displaystyle\langle\dot{\rho}^{\prime}(t)\rangle$
$\displaystyle=\mathbb{K}^{\rm{II}}(+\infty)\langle\rho^{\prime}(t)\rangle$
$\displaystyle=-\int_{0}^{+\infty}dt^{\prime}\langle\mathcal{L}^{\prime}_{s}(t)\mathcal{L}^{\prime}_{s}(t-t^{\prime})\rangle\langle\rho^{\prime}(t)\rangle\,.$
(16)

This is the Markov approximation of our non-Markovian process. We arrive at:

$\displaystyle\langle\dot{\rho}(t)\rangle$
$\displaystyle=-i\bigl{[}\langle\mathrm{H}(t)\rangle,\langle\rho(t)\rangle\bigr{]}-\int_{0}^{+\infty}\langle\bigl{[}\mathrm{H}_{s}(t),$
(17)
$\displaystyle\bigl{[}\mathrm{U}^{\dagger}(t-t^{\prime},t)\mathrm{H}_{s}(t-t^{\prime})\mathrm{U}(t-t^{\prime},t),\langle\rho(t)\rangle\bigr{]}\bigr{]}\rangle
dt^{\prime}\,,$

where
$\mathrm{U}(t-t^{\prime},t)\equiv\mathrm{U}(t-t^{\prime})\mathrm{U}^{\dagger}(t)$.
We approximate $\mathrm{U}(t-t^{\prime},t)$ with
$\exp{\bigl{(}i\mathrm{H}_{0}\,t^{\prime}\bigr{)}}$, which is justified if
$D\tau_{c}\ll 1$ [40]. That means basically that in the relaxation term of the
second order cumulant equation the time evolution operator is
$\mathrm{U}(t)=\exp\bigl{(}-i\mathrm{H}_{0}t\bigr{)}$ (to show this it is
sufficient to plug in $t^{\prime}=t$ in the previous relation and invert).
This procedure is customary in radio frequency magnetic resonance in liquids
and it is usually called the nonviscous-liquid approximation [40]. To compute
the relaxation times we will make use of [40]:

$\displaystyle\langle\dot{\rho}(t)\rangle+i\bigl{[}\langle\mathrm{H}(t)\rangle,\langle\rho(t)\rangle\bigr{]}\approx$
$\displaystyle-\int_{0}^{+\infty}\langle\bigl{[}\mathrm{H}_{s}(t),\bigl{[}e^{-i\mathrm{H}_{0}t^{\prime}}\mathrm{H}_{s}(t-t^{\prime})e^{i\mathrm{H}_{0}t^{\prime}},\langle\rho(t)\rangle\bigr{]}\bigr{]}\rangle
dt^{\prime}$ (18) $\displaystyle=$
$\displaystyle-\,e^{-i\mathrm{H}_{0}t}\int_{0}^{+\infty}dt^{\prime}\langle\bigl{[}\mathrm{H}^{*}_{s}(t),\bigl{[}\mathrm{H}^{*}_{s}(t-t^{\prime}),\langle\rho^{*}(t)\rangle\bigr{]}\bigr{]}\rangle
e^{i\mathrm{H}_{0}t}\approx\mathrm{U}(t)\langle\dot{\rho}^{\prime}(t)\rangle\mathrm{U}^{\dagger}(t)\,.$

This basically means that
$\langle\dot{\rho}^{\prime}(t)\rangle(t)\approx\langle\dot{\rho}^{*}(t)\rangle(t)$
if we remember to sum the correct first order term when we turn back to the
Schrödinger picture. Therefore, our approximation is equivalent to assuming
$\mathbb{K}^{\rm{II}}(+\infty)\approx-\int_{0}^{+\infty}dt^{\prime}\langle\mathcal{L}^{*}_{s}(t)\mathcal{L}^{*}_{s}(t-t^{\prime})\rangle$
(please note that this expression still depends on $t$) in Eq. (15), the
superoperator
$\mathcal{L}^{*}_{s}=\bigl{[}e^{i\mathrm{H}_{0}t}\mathrm{H}_{s}(t)e^{-i\mathrm{H}_{0}t},\,\circ\,\bigr{]}$
being

$\displaystyle\mathcal{L}^{*}_{s}(t)=$
$\displaystyle\frac{\Gamma_{d}(t)}{2\Omega}\Big{\\{}\varepsilon_{0}\bigl{[}\sigma_{z},\,\circ\,\bigr{]}-\Delta
e^{-i\Omega t}\bigl{[}\sigma_{+}\,,\,\circ\,\bigr{]}$ $\displaystyle-\Delta
e^{i\Omega t}\bigl{[}\sigma_{-}\,,\,\circ\,\bigr{]}\Big{\\}}\,.$ (19)

Now we split the density matrix in its spin components
$\\{\sigma_{z},\sigma_{+},\sigma_{-}\\}$, i.e.
$\bm{\mathrm{r}}=\rm{Tr}\bigl{[}\rho^{S}(t)\bm{\sigma}\bigr{]}=\rm{Tr}\bigl{[}\rho^{H}(0)\bm{\sigma}(t)\bigr{]}$.

The equation of motion for $\mathrm{r}_{\alpha}(t)$ with $\alpha=+,-,0$, where
we renamed $\mathrm{r}_{z}$ as $\mathrm{r}_{0}$, are then (averages of these
components are intended, but we do not write them to ease the notation),

$\displaystyle 2\Omega^{2}\,\dot{\mathrm{r}}^{\prime}_{+}(t)=$
$\displaystyle-\bigl{[}2\varepsilon_{0}^{2}k_{0}+\Delta^{2}k_{-}\bigr{]}\mathrm{r}^{\prime}_{+}$
(20) $\displaystyle+\Delta^{2}k_{+}e^{-i2\Omega
t}\mathrm{r}^{\prime}_{-}-2\Delta\varepsilon_{0}k_{+}e^{-i\Omega
t}\mathrm{r}^{\prime}_{0}\,,$ $\displaystyle
2\Omega^{2}\,\dot{\mathrm{r}}^{\prime}_{-}(t)=$
$\displaystyle-\bigl{[}2\varepsilon_{0}^{2}k_{0}+\Delta^{2}k_{+}\bigr{]}\mathrm{r}^{\prime}_{-}$
(21) $\displaystyle+\Delta^{2}k_{-}e^{i2\Omega
t}\mathrm{r}^{\prime}_{+}-2\Delta\varepsilon_{0}k_{-}e^{i\Omega
t}\mathrm{r}^{\prime}_{0}\,,$ $\displaystyle
2\Omega^{2}\,\dot{\mathrm{r}}^{\prime}_{0}(t)=$
$\displaystyle-\Delta^{2}\bigl{[}k_{-}+k_{+}\bigr{]}\mathrm{r}^{\prime}_{0}$
(22) $\displaystyle-\Delta\varepsilon_{0}k_{0}e^{i\Omega
t}\mathrm{r}^{\prime}_{+}-\Delta\varepsilon_{0}k_{0}e^{-i\Omega
t}\mathrm{r}^{\prime}_{-}\,.$

Here we wrote
$k_{\alpha}=\int_{0}^{+\infty}\langle\Gamma_{d}(t)\Gamma_{d}(t-t^{\prime})\rangle
e^{i\alpha\Omega\tau}dt^{\prime}=G\times\frac{1+i\alpha\Omega\tau_{c}}{\alpha^{2}\Omega^{2}\tau_{c}^{2}+1}$
and $\mp$ is a shorthand for $\\{-1\,,\,1\\}$, thus
$\alpha\in\\{-1\,,\,0\,,\,1\\}$. Furthermore, we dropped the subscript of
$\varepsilon_{0}$. We rename these terms using Redfield’s notation [55].

$\langle\dot{\mathrm{r}}^{\prime}_{\alpha}(t)\rangle=-\sum_{\beta}\exp\bigl{[}i(\beta-\alpha)\Omega
t\bigr{]}\mathcal{R}_{\alpha,\beta}\,\langle\mathrm{r}^{\prime}_{\beta}(t)\rangle.$
(23)

Now Eqs. (20)–(23) can be solved perturbatively after Laplace-transforming
them. In this way we finally obtain the relaxation times and the Lamb’s shift
$\delta\Omega$, in closed form. These are found to be:

$\displaystyle T_{1}^{-1}$
$\displaystyle=\mathcal{R}_{0,0}-2\,\mathrm{Re}\biggl{(}\frac{\mathcal{R}_{0,+}\mathcal{R}_{+,0}}{i\Omega+\mathcal{R}_{+,+}}\biggr{)},$
(24) $\displaystyle T_{2}^{-1}$
$\displaystyle=\mathrm{Re}\biggl{(}\mathcal{R}_{+,+}+\frac{\mathcal{R}_{0,+}\mathcal{R}_{+,0}}{i\Omega-\mathcal{R}_{0,0}}\biggr{)},$
(25) $\displaystyle\delta\Omega$
$\displaystyle=\mathrm{Im}\biggl{(}\mathcal{R}_{+,+}+\frac{\mathcal{R}_{0,+}\mathcal{R}_{+,0}}{i\Omega-\mathcal{R}_{0,0}}\biggr{)}-\frac{\bigl{|}\mathcal{R}_{+,-}\bigr{|}^{2}}{2\Omega}.$
(26)

Figure 2: Decay of the polarization $\mathrm{r}^{\prime}_{z}={\rm
Tr}(\sigma_{z}\rho)$ of the Bloch vector (orange), and of the coherences
$\rm{Re}(\mathrm{r}^{\prime}_{+})={\rm Tr}(\sigma_{+}\rho)$ (purple), for
$\varepsilon_{0}=1.5\Delta,G=0.5\Delta$. Solid lines are the plot of the
numerical solution to the Redfield equation (23). Dashed colored lines picture
the exponential decay with times given by Eqs. (24),(25)
($\mathrm{r}^{\prime}_{i}(0)=\mathrm{r}_{i}(0)$). Dotted gray and black lines
represent the corresponding plot neglecting the oscillating terms on the right
hand side of Eq. (20) (secular approximation). (a) Correlated noise,
$G\tau_{c}=0.25$, and (b) white noise limit, $G\tau_{c}=0$.

These results agree with the ones reported by Refs. 53, 54. Here, the
$\mathcal{R}_{i,j}$ constants are given by:

$\displaystyle\mathcal{R}_{0,0}$
$\displaystyle=\frac{G\,\Delta^{2}}{\Omega^{2}(1+\Omega^{2}\tau_{c}^{2})},$
(27) $\displaystyle\mathcal{R}_{0,+}$
$\displaystyle=\frac{G\,\Delta\varepsilon_{0}}{2\Omega^{2}},$ (28)
$\displaystyle\mathcal{R}_{+,0}$
$\displaystyle=\frac{G\,\Delta\varepsilon_{0}(1+i\Omega\tau_{c})}{\Omega^{2}(1+\Omega^{2}\tau_{c}^{2})},$
(29) $\displaystyle\mathcal{R}_{+,+}$
$\displaystyle=\frac{G}{2}\,\biggl{[}\frac{2\varepsilon_{0}^{2}}{\Omega^{2}}+\frac{\Delta^{2}(1-i\Omega\tau_{c})}{\Omega^{2}(1+\Omega^{2}\tau_{c}^{2})}\biggr{]},$
(30) $\displaystyle\mathcal{R}_{+,-}$
$\displaystyle=-\frac{G\,\Delta^{2}(1+i\Omega\tau_{c})}{2\Omega^{2}(1+\Omega^{2}\tau_{c}^{2})}.$
(31)

Now substituting these results into Eqs. (24), (25), and (26), we see that
correlations turn out to be fourth order corrections in those expressions for
the relaxation times and the frequency shift. In Appendix B we show that the
fourth-order term of the cumulant expansion (15) is vanishing due to the
Gaussianity of the noise process and its zero avarage. In Fig. 2 we plot
$\mathrm{Re}[\mathrm{r}^{\prime}_{+}]$ (purple) and $\mathrm{r}^{\prime}_{z}$
(orange) in the interaction picture as a function of time. The solid lines in
the plot are the numerical solution of Eq. (20) while the dashed exponentials
are the result of the perturbative solution given by Eqs. (24) and (25). The
gray and black dotted lines are the solution obtained by discarding the
correlations from the beginning, or by applying the secular approximation to
Eq. (23).

Equations (24), (25), and (23) imply that $T_{2}^{-1}\neq(2T_{1})^{-1}$, the
difference amounting to the pure dephasing rate

$T_{\phi}^{-1}=(T_{2})^{-1}-(2T_{1})^{-1}\approx\frac{G\varepsilon_{0}^{2}}{\,\Omega^{2}}+\frac{G^{2}\Delta^{2}\varepsilon_{0}^{2}}{\,\Omega^{5}(1+\Omega^{2}\tau_{c}^{2})}\,,$
(32)

where we neglected $\mathcal{R}_{0,0}$ and $\mathcal{R}_{+,+}$ in the
denominator of Eqs. (24) and (25). This analytically approximate result and
the graphs of Fig. 2 showing exponential decay with different characteristic
times demonstrate that the addition of noise to the parametric driving of this
coupled classical oscillator system induce a dephasing dynamics, resolving the
problem of equal relaxation times pointed out in the previous models by Refs.
8, 11.

In Fig. 3 we show the BV dynamics on the unit Bloch sphere. These are plotted
using Eq. (20) after transforming to the lab frame (x,y,z) and to the
Schrödinger picture. Both panels represent the time evolution of the BV for
the resonantly driven TLS. The three different initial conditions give rise to
three trajectories on the sphere:
$\bm{\mathrm{r}}(0)=\bigl{[}1/\sqrt{3}\,,\,1/\sqrt{3}\,,\,1/\sqrt{3}\bigr{]}^{T}$
(purple),
$\bm{\mathrm{r}}(0)=\bigl{[}1/\sqrt{2}\,,\,1/\sqrt{2}\,,\,0\bigr{]}^{T}$
(yellow), $\bm{\mathrm{r}}(0)=\bigl{[}0\,,\,0\,,\,-1\bigr{]}^{T}$ (blue). Fig.
3(a) shows the system subject to noise with a memory time $G\tau_{c}=0.15$ and
Fig. 3(b) for memoryless noise, i.e. $G\tau_{c}=0$.

Figure 3: Trajectory of the BV on the unit Bloch sphere for different initial
conditions, i.e.,
$\bm{\mathrm{r}}(0)=\bigl{[}1/\sqrt{3}\,,\,1/\sqrt{3}\,,\,1/\sqrt{3}\bigr{]}^{T}$
(purple),
$\bm{\mathrm{r}}(0)=\bigl{[}1/\sqrt{2}\,,\,1/\sqrt{2}\,,\,0\bigr{]}^{T}$
(yellow), and $\bm{\mathrm{r}}(0)=\bigl{[}0\,,\,0\,,\,-1\bigr{]}^{T}$ (blue).
Both are plotted in the case of resonant driving $\omega=\Delta$ and for (a)
$G=0.3\Delta$, $D=0.5\Delta$ and $\Delta\tau_{c}=0.5$, and (b) $G=0.3\Delta$
and $D=\Delta$, $\tau_{c}=0$. All BV trajectories decay to the infinite
temperature state at the center of the Bloch sphere.

### Why infinite temperature?

It has already been shown by Kubo [62] that the stationary states of systems
governed by stochastic Hamiltonians are the infinite temperature states, i.e.
$\rho\,(t\to+\infty)\propto\openone$. As it is evident from Fig. 3 the BV of
our system always collapses to the center of the sphere, i.e. infinite
temperature state, so the previous result applies here as well. Hereafter we
explain why this is the case for our system, following [33].

We recast the Redfield equation in a Kossakowski-Lindblad form [63]. Since we
have seen in this section that the oscillating terms do not give an important
contribution to the decay of the BV, we ignore those terms here, therefore
applying a secular approximation. the resulting master equation, under the
Born-Markov approximation, from Eqs. (15) and (19) is then,

$\displaystyle\langle\dot{\rho}^{\prime}(t)\rangle$
$\displaystyle=\mathbb{K}^{\rm{II}}(t)\langle\rho^{\prime}(t)\rangle\approx-\frac{G^{2}}{4\Omega^{2}}\Bigl{(}\varepsilon^{2}_{0}k_{0}[\sigma_{z},[\sigma_{z},\langle\rho^{\prime}\rangle]]$
(33)
$\displaystyle+\Delta^{2}k_{+}[\sigma_{+},[\sigma_{-},\langle\rho^{\prime}\rangle]]+\Delta^{2}k_{-}[\sigma_{-},[\sigma_{+},\langle\rho^{\prime}\rangle]]\Bigr{)}.$

Then, we transform back to the Schrödinger picture and apply the secular
approximation.

$\langle\dot{\rho}\rangle\approx-i\bigl{[}\langle\mathrm{H}\rangle,\langle\rho\rangle\bigr{]}-\frac{G^{2}}{4\Omega^{2}}\Bigl{[}2\varepsilon^{2}_{0}k_{0}(\langle\rho\rangle-\sigma_{z}\langle\rho\rangle\sigma_{z})+\Delta^{2}\mathrm{Re}(k_{+})(\langle\rho\rangle-2\sigma_{+}\langle\rho\rangle\sigma_{-}-2\sigma_{-}\langle\rho\rangle\sigma_{+})+i\Delta^{2}\mathrm{Im}(k_{+})[\sigma_{z},\langle\rho\rangle]\Bigr{]}.$
(34)

Here, the first term in the square brackets describes dephasing, while the
second term leads to quantum jumps, and the third term is the Lamb shift. Now
since the rates for upward and downward transitions are the same, i.e.
$\gamma_{\pm}=G^{2}\Delta^{2}\mathrm{Re}(k_{+})/2\Omega^{2}$, we can ascribe
an infinite effective temperature to the system, since we do not have detailed
balance [64, 33]. Clearly this is not a true temperature since we do not
describe a thermal bath, with which the system can thermalize. Nonetheless, it
is a nontrivial consequence of our model and indeed an interesting limit of
the discussed quantum-classical analogy.

##  IV Proposals for experimental tests

We now briefly analyze some experimental setups in which the discussed quantum
analog could be implemented. Among the many possible experimental test-beds we
chose levitated nanoparticles and nanomechanical string resonators, since
these systems are currently attracting much interest thanks to the great
degree of control and isolation from the environment that they guarantee [65,
66, 22]. Nonetheless, a realization with macroscopic setups could also be
within reach [12], favouring the possibility of a macroscopic quantum analogy
of unprecedented depth.

###  IV.1 Levitated nanoparticles

In levitation experiments dieletric particles are trapped in the focus of a
laser beam, using radiation pressure. The harmonic modes of oscillation are
the the center of mass coordinate of the particle, with the equilibrium
position being the focus of the laser. The oscillation along the laser beam
can be frozen so that the motion is effectively restricted to the two modes
belonging to the focal plane. The two modes have different eigenfrequencies
due to the polarization of the laser which creates an elliptically-shaped
potential well. These modes can be coupled by harmonically varying the
polarization direction of the trapping laser [9, 14]. Therefore, periodically
varying the polarization angle according to $\theta=\delta\cos(\omega
t+\varphi)$ realizes a parametric driving of the coupled oscillator system.
When the frequency of this harmonic driving is in resonance with the mode
splitting, the strong coupling regime can be reached and Rabi oscillations can
be implemented in which the energy can be exchanged coherently between the
modes.

Nonetheless, in order to obtain a dephasing dynamics in our system it is
essential to have an off-set parameter such as $\varepsilon_{0}$ in Eq. (9)
that shifts the working point of the system slightly off the center of the
avoided crossing. This makes the driving term appear as
$\theta=\delta\cos(\omega t+\varphi)+\delta_{0}$, with $\delta_{0}\sim\delta$.
Thus, one needs to be able to induce some asymmetry between the mode
eigenfrequencies independently of the coupling/driving mechanism. Otherwise a
shift of the polarization angle of the trapping laser in this system will only
lead to a redefinition of the mode frequencies. However, in these system the
modes eigenfrequency are defined by the polarization of the laser [67].
Therefore, we think that in this setup the effect described here cannot be
observed. We cannot exclude that more complicated configurations exist, in
which this mode frequency asymmetry can be adiabatically tuned, independently
from the driving. In this way, taking the equation (8) from Ref. 14 we can map
it to an equation similar to ours by the coordinate transformation
$\begin{bmatrix}a^{\prime}\\\
b^{\prime}\end{bmatrix}=\exp\bigl{(}i\frac{\theta}{2}\sigma_{y}\bigr{)}\begin{bmatrix}a\\\
b\end{bmatrix}$, where $\theta=\arctan(\delta_{0}/\omega_{\delta})$.

###  IV.2 Nanomechanical string resonators

Considering nanoscale string resonators we propose exploiting the dielectric
protocols for coupling and driving the in-plane and out-of-plane modes of a
nanomechanical doubly clamped beam [68, 69, 15, 70, 13, 16, 18]. On the other
hand, similar physics can be implemented in nanomechanical systems with a
piezoelectrical driving setup [71, 72, 73]. Considering a doubly clamped
string resonator, we have that the two modes (in-plane and out-of-plane) have
different eigenfrequencies, that is $k_{1}\neq k_{2}$, due to the rectangular
cross-section. The oscillations of these two modes will be driven
dielectrically. The parametric driving term can be written as $\Delta
k\approx-\frac{\epsilon_{0}(\epsilon_{d}-1)}{2\pi}\alpha^{2}LF(x)\bigl{[}V_{dc}^{2}+2V_{dc}V_{ac}(t)\bigr{]}=C\bigl{[}V_{dc}^{2}+2V_{dc}A\cos(\omega
t)\bigr{]}$ [69], where $\epsilon_{0}$ and $\epsilon_{d}$ are the dielectric
permittivity of vacuum and of the dielectric composing the beam (e.g. SiN),
$L$ is the total length of the beam, $\alpha$ is the attenuation parameter of
the electric field inside the dielectric and $F(x)$ depends on the geometry of
the device. In this way the system is governed by equations of motion
completely analogous to Eq. (2), where, in the language of Ref. 13, the force
$f(t)$ is the radiofrequency drive that initializes the system, $h$ is the
coupling provided by the cross derivatives of the electric field generated by
the gold electrodes, and $\omega_{1}=\sqrt{2k_{1}/m}$ and
$\omega_{2}=\sqrt{2k_{2}/m}$ are the frequencies of the in-plane and out-of-
plane modes, where $m$ is the total mass of the beam. In our language then
$\varepsilon_{0}\approx C(V_{dc}-V_{0})^{2}$ is proportional to the dc part of
the voltage driving, where $V_{0}$ is the voltage corresponding to the centre
of the level splitting, and $\varepsilon^{\prime}(t)$ is given by the ac part
$CA(V_{dc}-V_{0})V_{ac}\cos(\omega t)$. In this way the coupled modes of these
nanomechanical string resonators are described by the same equations that we
used and we think they can simulate the dephasing dynamics here described.

##  V Conclusions

We showed how the analogy between classical coupled oscillators and quantum
TLS can be extended to include a decoherent dynamics of the TLS. We computed
the relaxation times for BV components and demonstrated that a pure dephasing
$T_{\phi}$ (Eq. (32)) time appears in the TLS dynamics, as a consequence of
the addition of noise to the parametric driving of the classical system. This
analogy can be exploited to effectively simulate or probe quantum decoherence
dynamics and dissipation via classical means, resulting in an effective
simulation of decoherence. However, the simulation in its present form cannot
grasp decay processes such as spontaneous emission. Simulation of open-
quantum-system dynamics by addition of classical noise to analog, more
controllable, physical system has spurred much interest lately [31, 33]. In
particular, Ref. 33 studied how classical noise can mimic quantum dissipation
and derived very clear analogies between the two frameworks, considering for
instance the spin-boson problem. In a more complicated framework Ref. 31
studied how a wide class of master equations for many-body systems can be
simulated in easily controllable systems subject to classical noise.

Our analysis constitutes a simplification of these earlier attempts, since it
does not require any quantum system at all, therefore enabling a purely
classical simulation of quantum dissipation problems, that in principle can be
implemented in systems even beyond the nanoscale. Albeit some other
dissipative phenomena, e.g. spontaneous emission, are still out of reach for
these kinds of simulation due to fundamental issues [32, 33]. Also in the
system discussed here, the equality between the upward and downward transition
rate in Eq. (34) withhold the possibility to simulate spontaneous decay
processes.

Notably, the system investigated here can be valuable also for frequency noise
detection in the aforementioned mechanical systems. Usually, this kind of
noise is hard to investigate because of the interplay of the thermal
fluctuations. However, we see that in the system treated here these
fluctuations are fully decoupled from the frequency noise and do not affect
the relaxation times that we found (see Appendix A). Therefore one could
measure frequency noise features from the expected decay times.

## Acknowledgements

We gratefully acknowledge funding from the Deutsche Forschungsgemeinschaft
Project No. 425217212, SFB 1432.

##  Appendix A Non-homogeneus forcing case

We now analyze what happens considering the full case, i.e. the one with
inhomogeneous forcing. We will be assuming that in this case Eq. 4 can be
easily modified by adding a vector with two noisy complex components, i.e.
$\bm{f}(t)=[\,f_{1}(t)\,f_{2}(t)\,]^{T}$. This is not too far from the
treatment of Ref. 8, even if we should also account for the non-stationarity
of the time factors introduced by the SVEA, that we will nonetheless neglect
for simplicity. Therefore we will consider $f_{1,2}(t)$ as a noise realisation
of a stationary Gaussian process. Let us consider then the larger vectorial
(let us call it a combined Hilbert-Liovillean) space where the following
vector lives:

$\bm{\Psi}=\begin{bmatrix}\psi_{1}\\\ \psi_{2}\\\ \psi_{1}^{*}\psi_{1}\\\
\psi_{1}^{*}\psi_{2}\\\ \psi_{2}^{*}\psi_{1}\\\
\psi_{2}^{*}\psi_{2}\end{bmatrix}$ (35)

So that we can treat everything on the same foot. Assuming that both $D$ and
$G$ are small parameters with respect to
$\Omega=\sqrt{\Delta^{2}+\epsilon^{2}_{0}}\,$, the dynamics of this large
vector can be described in the following way [74]:

$\displaystyle\langle\bm{\dot{\Psi}}(t)\rangle=$
$\displaystyle\,\mathcal{K}(t)\langle\bm{\Psi}(t)\rangle$ (36)
$\displaystyle=$
$\displaystyle\begin{bmatrix}\mathrm{K}(t)&\varnothing_{2\times 4}\\\
\mathbb{F}_{4\times
2}(t)&\mathbb{K}(t)\end{bmatrix}\langle\bm{\Psi}(t)\rangle$
$\displaystyle+\begin{bmatrix}\langle\bm{f}(t)\rangle\\\ \int
dt^{\prime}\langle\tilde{\bm{f}}(t)e^{t^{\prime}\mathrm{H}_{0}}\bm{f}(t-t^{\prime})\rangle\end{bmatrix}\,,$

where the minor $\mathbb{K}(t)$ is an operator that governs the time evolution
of the second moments in the Liouvillean space. This is the analog of Eq. 15.
Let us now focus on the lower left block this is a $4\times 2$ matrix with the
following form:

$\displaystyle\mathbb{F}(t)=\langle\tilde{\bm{f}}(t)\rangle=\langle\bm{f}^{*}(t)\otimes\mathrm{Id}_{2}+\mathrm{Id}_{2}\otimes\bm{f}(t)\rangle=$
$\displaystyle\begin{bmatrix}\langle f_{1}^{*}(t)\rangle\mathrm{Id}_{2}\\\
\langle
f_{2}^{*}(t)\rangle\mathrm{Id}_{2}\end{bmatrix}+\begin{bmatrix}\bm{0}&&\langle\bm{f}(t)\rangle\\\
\langle\bm{f}(t)\rangle&&\bm{0}\end{bmatrix}=\begin{bmatrix}\langle
f_{1}^{*}\rangle&&\langle f_{1}\rangle\\\ 0&&\langle f_{1}^{*}+f_{2}\rangle\\\
\langle f_{2}^{*}+f_{1}\rangle&&0\\\ \langle f_{2}\rangle&&\langle
f_{2}^{*}\rangle\end{bmatrix}$

Note that if $\langle\bm{f}\rangle=0$, i.e. if the fluctuation has zero
average, that is the case considered in the main text, then the super-
superoperator is

$\mathcal{K}(t)=\begin{bmatrix}\mathrm{K}(t)&\varnothing_{2\times 4}\\\
\varnothing_{2\times 4}&\mathbb{K}(t)\end{bmatrix}$ (37)

and notably the dynamics of vector $\bm{\Psi}$ factorizes Eq. (36) into the
dynamics restricted to the two subspaces. In other words, if $\bm{\Psi}\in V$
and $V^{\prime}_{1}\equiv\operatorname{span}\\{\bm{e}_{1},\bm{e}_{2}\\}$ and
$V^{\prime}_{2}\equiv\operatorname{span}\\{\bm{e}_{3},\bm{e}_{4},\bm{e}_{5},\bm{e}_{6}\\}$
are subspaces of $V=V^{\prime}_{1}\oplus V^{\prime}_{2}$ preserved by the
dynamics. It is useful then to decompose the space $V$ as a direct product
$V=V_{1}\otimes V_{2}$, where
$V_{1}\equiv\operatorname{span}\\{\bm{e}_{1},\bm{e}_{2}\\}$ and
$V_{2}\equiv\operatorname{span}\\{\bm{e}_{1},\bm{e}_{2},\bm{e}_{3},\bm{e}_{4}\\}$.
The space $V_{2}$ is the tetradic space. Now the equation restricted to
subspace $V_{1}$ is just the equation for the mean values of the stochastic
variables, i.e. our state vector, which we will write making use of the
cumulant expansion,

$\langle\bm{\dot{\psi}}\rangle=\mathrm{K}(t)\langle\bm{\psi}\rangle,$ (38)

where

$\displaystyle\mathrm{K}(t)=$
$\displaystyle-i\bigr{\langle}\mathrm{H}(t)\bigr{\rangle}$ (39)
$\displaystyle-$
$\displaystyle\int_{0}^{+\infty}\bigr{\langle}\bigr{\langle}\mathrm{H}_{1}(t)\,e^{it^{\prime}\mathrm{H}_{0}}\mathrm{H}_{1}(t-t^{\prime})\bigr{\rangle}\bigr{\rangle}\,e^{-it^{\prime}\mathrm{H}_{0}}dt^{\prime},$

and where $\langle\langle\circ\rangle\rangle$ is the cumulant. Here the
cumulant sign is actually important because the perturbation is
$\mathrm{H}_{1}(t)=\frac{D\cos{(\omega
t)}+G\,\Gamma_{d}(t)}{2\Omega}\bigl{(}\varepsilon_{0}\sigma_{z}-\Delta\sigma_{x}\bigr{)}$
with $\langle\mathrm{H}_{1}(t)\rangle\nequal 0$.

The dynamics in $V_{2}$ space is described by the following dynamic operator

$\displaystyle\mathbb{K}(t)=$
$\displaystyle-i\bigr{\langle}\mathcal{L}(t)\bigr{\rangle}$ (40)
$\displaystyle-$
$\displaystyle\int_{0}^{+\infty}\bigr{\langle}\bigr{\langle}\mathcal{L}_{1}(t)\,e^{it^{\prime}\mathcal{L}_{0}}\mathcal{L}_{1}(t-t^{\prime})\bigr{\rangle}\bigr{\rangle}\,e^{-it^{\prime}\mathcal{L}_{0}}dt^{\prime}$

where $\mathcal{L}_{1}(t)=\mathcal{L}_{d}(t)+\mathcal{L}_{s}(t)$. This is the
analog of Eq. (15) [43]. We can show that in the perturbative regime, this
dynamics actually reduces to the previous result (18) by considering that the
deterministic part of the driving cancels in the cumulants. To see this, let
us expand the cumulant as

$\displaystyle\bigr{\langle}\bigr{\langle}\mathcal{L}_{1}(t)\,e^{it^{\prime}\mathcal{L}_{0}}\mathcal{L}_{1}(t-t^{\prime})\bigr{\rangle}\bigr{\rangle}=$
(41)
$\displaystyle\bigr{\langle}\mathcal{L}_{1}(t)\,e^{it^{\prime}\mathcal{L}_{0}}\mathcal{L}_{1}(t-t^{\prime})\bigr{\rangle}-\bigr{\langle}\mathcal{L}_{1}(t)\bigr{\rangle}\,\bigr{\langle}e^{it^{\prime}\mathcal{L}_{0}}\mathcal{L}_{1}(t-t^{\prime})\bigr{\rangle}$

that follows from the definition of the cumulant and the fact that
$e^{it^{\prime}\mathcal{L}_{0}}$ is non-stochastic. Now we write
$\mathcal{L}_{1}(t)=\mathcal{L}_{d}(t)+\mathcal{L}_{s}(t)$, where
$\langle\mathcal{L}_{s}(t)\rangle=0$, which yields

$\bigl{\langle}\bigl{\langle}\mathcal{L}_{1}(t)\,e^{it^{\prime}\mathcal{L}_{0}}\mathcal{L}_{1}(t-t^{\prime})\bigr{\rangle}\bigr{\rangle}=\bigl{\langle}\mathcal{L}_{s}(t)\,e^{it^{\prime}\mathcal{L}_{0}}\mathcal{L}_{s}(t-t^{\prime})\bigr{\rangle}\,,$
(42)

since the terms containing the deterministic driving part in the RHS cancel
with each other. With this we have for the second order cumulant expansion,

$\mathbb{K}(t)=-i\bigr{\langle}\mathcal{L}(t)\bigr{\rangle}-\frac{1}{4\Omega^{2}}\begin{bmatrix}\Delta^{2}(k_{+}+k_{-})&&-2\Delta\varepsilon_{0}k_{0}&&-2\Delta\varepsilon_{0}k_{0}&&-2\Delta^{2}(k_{+}+k_{-})\\\
-2\Delta\varepsilon_{0}k_{+}&&2\Delta^{2}k_{-}+4\varepsilon_{0}^{2}k_{0}&&2\Delta^{2}k_{+}&&2\Delta\varepsilon_{0}k_{+}\\\
-2\Delta\varepsilon_{0}k_{-}&&-2\Delta\varepsilon_{0}k_{-}&&2\Delta^{2}k_{-}+4\varepsilon_{0}^{2}k_{0}&&2\Delta\varepsilon_{0}k_{-}\\\
-\Delta^{2}(k_{+}+k_{-})&&2\Delta\varepsilon_{0}k_{0}&&2\Delta\varepsilon_{0}k_{0}&&\Delta^{2}(k_{+}+k_{-})\end{bmatrix},$
(43)

with no dependence on the driving in the dissipator.

Note that in this way we cannot be assured that the higher orders vanish
because the perturbation has non-zero stochastic average. However, if we take
their expressions from [46], we can see by inspection that third order and
fourth order terms vanish nonetheless.

Considering now that $\mathcal{L}^{*}_{1}(t)=\frac{D\cos(\omega
t)+\Gamma(t)}{\Omega}\tilde{\mathcal{L}}_{1}^{*}(t)$, we see that both third
order and fourth order vanish.

##  Appendix B Fourth order calculation

To obtain the correction due to the correlations we should then complement our
analysis by a fourth order expansion of Eq. 14. Thus refining our coarse-grain
description by another order. Since the noise is Gaussian and stationary, odd
orders in the perturbation $\mathrm{H}_{s}$ vanish. Using again the non-
viscous liquid approximation we get,

$\displaystyle\mathbb{K}^{\rm{IV}}(+\infty)\approx\iiint_{0}^{+\infty}$
$\displaystyle
dt_{1}dt_{2}dt_{3}\bigl{[}\langle\mathcal{L}^{*}_{s}(t)\mathcal{L}^{*}_{s}(t_{1})\mathcal{L}^{*}_{s}(t_{2})\mathcal{L}^{*}_{s}(t_{3})\rangle-\langle\mathcal{L}^{*}_{s}(t)\mathcal{L}^{*}_{s}(t_{1})\rangle\langle\mathcal{L}^{*}_{s}(t_{2})\mathcal{L}^{*}_{s}(t_{3})\rangle$
$\displaystyle-$
$\displaystyle\langle\mathcal{L}^{*}_{s}(t)\mathcal{L}^{*}_{s}(t_{2})\rangle\langle\mathcal{L}^{*}_{s}(t_{1})\mathcal{L}^{*}_{s}(t_{3})\rangle-\langle\mathcal{L}^{*}_{s}(t)\mathcal{L}^{*}_{s}(t_{3})\rangle\langle\mathcal{L}^{*}_{s}(t_{1})\mathcal{L}^{*}_{s}(t_{2})\rangle\bigr{]}\,.$
(44)

However, the time dependence of
$\mathcal{L}_{s}(t)=\frac{G\Gamma(t)}{\Omega}\bigl{[}\epsilon_{0}\sigma_{z}-\Delta\sigma_{x},\circ\bigr{]}$
is restricted to the noise realization that is the only part relevant for the
integral. Nonetheless, since the process is Gaussian,
$\langle\Gamma(t)\Gamma(t_{1})\Gamma(t_{2})\Gamma(t_{3})\rangle=\langle\Gamma(t)\Gamma(t_{1})\rangle\langle\Gamma(t_{2})\Gamma(t_{3})\rangle+\langle\Gamma(t)\Gamma(t_{2})\rangle\langle\Gamma(t_{1})\Gamma(t_{3})\rangle+\langle\Gamma(t)\Gamma(t_{3})\rangle\langle\Gamma(t_{1})\Gamma(t_{2})\rangle$
and the operator part being all commuting we find that the fourth order
contribution vanishes altogether.

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