# Dynamical signatures of non-Markovianity in a dissipative–driven qubit

Graziano Amati  Albert Ludwigs Universität Freiburg, Hermann Herder Str. 3,
79104 Freiburg, Germany

###### Abstract

We investigate signatures of non-Markovianity in the dynamics of a
periodically-driven qubit coupled to a dissipative bosonic environment. We
propagate the dynamics of the reduced density matrix of the qubit by
integrating the numerically exact hierarchical equations of motion. Non-
Markovian features are quantified by comparing the prediction from diverse and
complementary approaches to quantum dissipation. In particular, we analyze the
distinguishability of quantum states, the decay of the volume accessible to
the qubit on the Hilbert space, the negativity of the canonical rates in the
generalized Lindblad equation and the relaxation of the memory kernels in the
Nakajima–Zwanzig generalized quantum master equation. We discuss the out-of-
equilbrium stationary limit reached by the dissipative–driven dynamics and a
numerical approach to identify resonant driving frequencies leading to maximal
information revival. We discuss that the phenomenon of eternal non-
Markovianity can be observed for sufficiently small system–bath coupling and
enhanced by modulating the frequency of the external drive. The present work
provides a broad theoretical analysis on quantum control within the framework
of open quantum dynamics and quantum information.

##  I Introduction

To date uncontrolled dissipation is a crucial issue limiting the scalability
of quantum-computing technologies. Babu, Tuorila, and Ala-Nissila (2021); Ray
_et al._ (2023); Verney  _et al._ (2019); Harrington, Mueller, and Murch
(2022) Improving the coherence lifetime of quantum devices is a goal of the
utmost importance in order to tackle relevant and challenging numerical
analyses, e.g. in the framework of drug discovery, Outeiral  _et al._ (2021)
artificial intelligence Ahmed and Mähönen (2021) and the simulation of many-
body quantum systems. Smith  _et al._ (2019); Franke  _et al._ (2019)

From a theoretical standpoint, it is in the first place controversial how to
systematically define and measure dissipation in quantum mechanics. It is in
particular debated how this ubiquitous physical process relates to the
occurrence of dynamical memory effects, otherwise called signatures of _non-
Markovianity_ (in contrast to Markovian memory-less processes). Breuer (2012);
Rivas, Huelga, and Plenio (2014)
Memory effects in classical dynamics can be witnessed by measuring the
probability of a physical system of being in a given state in the present,
conditioned on its past history. Breuer  _et al._ (2016) This probabilistic
approach is applicable provided a series of measurements is performed over
time. While measurement processes do not alter classical states, they do
however affect coherence in quantum systems. In particular, projective
measurements lead to the irreversible collapse of entangled quantum states.
Schlosshauer (2005) Hence, to minimize external perturbations induced by
measurement processes, it is necessary to devise alternative methods to
quantify dissipation and non-Markovianity in quantum mechanics. This issue has
been tackled from multiple perspectives.

The _trace-distance approach_ to quantum non-Markovianity measures dissipation
from the loss of distinguishability over time between two maximally
distinguishable densities, due to the interaction with an external
environment.Breuer, Laine, and Piilo (2009); Breuer  _et al._ (2016); Amato,
Breuer, and Vacchini (2018) In particular, one can prepare two copies of a
quantum subsystem in two densities maximizing an operator metric known as the
trace distance. Unitary memory-less dynamics conserve this distance. In the
opposite limit of strongly dissipative dynamics, the trace distance decreases
monotonically over time. It can however occur that, even in the limit of an
infinitely large dissipative bath, the trace distance increases over time,
witnessing a dynamical increase of distinguishability due to _information
backflow_ from the environment. The external bath acts in this case as a
“dynamical hard disk”, partially restoring to the subsystem information on the
initial state.

The _volume of states_ accessible by the qubit provides another measure of
dissipation induced by the environment. Lorenzo, Plastina, and Paternostro
(2013a) It has been discussed in literature that a potential limitation of
this approach is that it can fail to capture dissipation occurring
anisotropically along different channels on the Hilbert space of the
subsystem. This issue is however resolved by analyzing the _canonical rates_ ,
time-dependent coefficients appearing in an exact time-local master equation
for open quantum dynamics in a generalized Lindblad form. Hall  _et al._
(2014); Erika Andersson and Hall (2007); Maldonado-Mundo  _et al._ (2012);
Hayden and Sorce (2022) The latter approach is capable of quantifying memory
effects occurring along each quantum channel. In particular, a generalized
Lindblad equation, uniquely written in canonical form, involves a set of
canonical dissipation rates, which are non-negative functions of time in the
Markovian limit. The negativity of each rate can be utilized as a measure of
non-Markovianity on the respective channel. _Eternal non-Markovianity_ occurs
whenever some rates assume negative values at all times. Shrikant, Srikanth,
and Banerjee (2022); Gulácsi and Burkard (2023)

The _Nakajima–Zwanzig formalism_ is another rigorous, first-principled
approach to quantum dissipation. The method is based on the construction of an
exact integro–differential equation of motion describing the dynamics of an
open quantum system. Dissipation from the environmental bath is captured by an
integral term involving a memory kernel, Nakajima (1958); Zwanzig (1960); Mori
(1965) enclosing complete information on the interaction between the subsystem
and the environment. The kernel vanishes in the limit of unitary dynamics,
while it collapses to a Dirac delta function in the opposite Markovian limit.
The timescale to relaxation of the kernel can therefore be utilized as a
measure of dissipation.

Among other insightful approaches to quantum dissipation we mention the study
of entropy production, Şenyaşa  _et al._ (2022) or the entanglement between
the dissipative system and a non-dissipative ancilla. Rivas, Huelga, and
Plenio (2010)

All the aforementioned approaches provide an interpretation of dissipation
from different perspectives, not necessarily in agreement with each other.
Chruściński, Kossakowski, and Rivas (2011a); Chruściński, Macchiavello, and
Maniscalco (2017); Bhattacharya  _et al._ (2017); Chanda and Bhattacharya
(2016) Hence, one could ask whether a unified approach to quantum dissipation
could be designed in the first place, or whether different methods should
instead be regarded as complementary to each other.

The effects of dissipation on the long-time limits of open dynamical systems
can be inspected from the perspective of ergodic theory. From a classical
standpoint, dissipation due to an external environment leads to the ergodic
relaxation of the initial distribution to its invariant thermal limit,
fulfilling detailed balance.Hawkins (2021) For quantum mechanical systems it
has been conjectured, within the _eigenstate thermalization hypothesis_ , that
thermalization can ultimately occur if the asymptotic limit of quantum
operators represented via the energy eigenstates is consistent with the
microcanonical ensemble. Deutsch (2018); Luca D’Alessio and Rigol (2016)
Predicting the occurrence of chaos and thermalization in quantum dynamics
remains however a hard and unresolved theoretical problem. Analyzing the long-
time behavior of open quantum dynamics becomes even more challenging in
presence of an external time-dependent field. Mori (2023); Mangaud  _et al._
(2018)

The analysis of memory retention in dissipative–driven quantum dynamics is
especially relevant in the framework of quantum information. In particular,
loss of information in a system of qubits occurs naturally due to the
uncontrolled coupling to an external environment. To counterbalance the
effects of environmental noise, a variety of innovative quantum error
correction (QEC) techniques has been developed in the last decades. Lidar and
Brun (2013); Vozhakov  _et al._ (2022); Terhal (2015); Babu, Tuorila, and Ala-
Nissila (2021) Among those, a successful family of approaches is based on the
_dynamical decoupling_ of the interaction between the qubits and the
environment by means of controlled external pulses.Bastrakova, Klenov, and
Satanin (2019); Jirari and Pötz (2006, 2007); Rebentrost and Wilhelm (2009)

Inspired by dynamical decoupling strategies, in the present work we analyze
how the interplay between thermal dissipation and time-dependent drive affects
memory retention in a spin–boson model externally coupled to a periodic
monochromatic field. Engelhardt, Platero, and Cao (2019); Magazzù  _et al._
(2018) We propagate the open quantum dynamics of the qubit by integrating the
hierarchical equations of motion (HEOM). Tanimura (2014, 2020) This allows us
to obtain a numerically exact solution for the reduced density matrix (RDM) of
the qubit. We quantify dissipation by calculating a measure of information
backflow within the trace-distance formalism. Breuer, Laine, and Piilo (2009)
From our numerical analysis it emerges that non-Markovian features can be
measured even in the limit of a classical thermal bath, and that memory
retention can be maximized provided the system is driven at resonant values of
the external drive. Shuang  _et al._ (2000); Mori (2023) By analyzing the
rates of the canonical form of a generalized Lindblad equation we identify the
occurrence of eternal non-Markovianity for sufficiently small values of the
system–bath coupling. We finally study a timescale to relaxation of memory
effects extracted from the Nakajima–Zwanzig generalized quantum master
equation (GQME). Nakajima (1958); Zwanzig (1960)

The present work provides an extensive analysis of quantum dissipation, and
paves the way towards the extended study of dissipation in large composite
quantum-information systems, with the ultimate aim of maximizing coherence
lifetime in quantum-computing devices.

##  II Dissipative–driven dynamics

In the present work we study the dynamics of a fermionic two-level subsystem
($\mathrm{s}$) coupled to a bosonic environmental bath ($\mathrm{b}$) and to
an external time-dependent drive. The total Hilbert space $\mathcal{H}$ can be
factorized in terms of the tensor product of the subsystem and bath Hilbert
spaces, $\mathcal{H}=\mathcal{H}_{\mathrm{s}}\otimes\mathcal{H}_{\mathrm{b}}$
, and the total Hamiltonian is decomposed onto three contributions, according
to

$\hat{H}(t)=\hat{H}_{\mathrm{s}}(t)\otimes\hat{\mathcal{I}}_{\mathrm{b}}+\hat{\mathcal{I}}_{\mathrm{s}}\otimes\hat{H}_{\mathrm{b}}+\hat{H}_{\mathrm{sb}}.$
(II.1)

Here, we denote by $\hat{H}_{\mathrm{s}}(t)$ the Hamiltonian of the qubit
subsystem, coupled to the bosonic bath $\hat{H}_{\mathrm{b}}$ via the cross
term $\hat{H}_{\mathrm{sb}}$. Here $\hat{\mathcal{I}}_{\mathrm{s}}$ and
$\hat{\mathcal{I}}_{\mathrm{b}}$ denote the identity operators of the
subsystem and the bath, respectively. The time evolution of the total density
matrix $\hat{\rho}_{0}$ is expressed via the time-ordered exponential

$\displaystyle\hat{\rho}(t,\tau)$
$\displaystyle=\exp_{+}\left\\{-i\int_{\tau}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\right\\}\hat{\rho}_{0}$
(II.2a)
$\displaystyle=\hat{U}(t,\tau)\hat{\rho}_{0}\hat{U}^{\dagger}(t,\tau),$
(II.2b)

where $\mathcal{L}_{t}\;\cdot=[\hat{H}(t),\cdot]$ denotes the Liouville
superoperator, the subscript $+$ indicates positive time ordering in the
exponential superoperator and

$\hat{U}(t,\tau)=\exp_{+}\left\\{-i\int_{\tau}^{t}\mathrm{d}t^{\prime}\;\hat{H}(t^{\prime})\right\\}$
(II.3)

is the time evolution operator. Here and in the following we fix $\hbar=1$.

The open dynamics of the subsystem are fully described by the reduced density
matrix
$\hat{\rho}_{\mathrm{s}}(t,\tau)=\operatorname{tr}_{\mathrm{b}}\\{\hat{\rho}(t,\tau)\\}$,
where $\operatorname{tr}_{\mathrm{b}}\\{\cdot\\}$ denotes the partial trace
over the Hilbert space of the bath. The total trace $\operatorname{tr}[\cdot]$
is obtained by composing
${\operatorname{tr}[\cdot]=\operatorname{tr}_{\mathrm{s}}[\operatorname{tr}_{\mathrm{b}}\\{\cdot\\}]=\operatorname{tr}_{\mathrm{b}}\\{\operatorname{tr}_{\mathrm{s}}[\cdot]\\}}$,
where $\operatorname{tr}_{\mathrm{s}}[\cdot]$ is the partial trace with
respect to the subsystem. We define $\hat{\rho}(t)=\hat{\rho}(t,0)$ and
similarly for the time evolution of other operators and superoperators.

In the present work we consider an initially factorized total density of
Feynman–Vernon type Feynman and Vernon Jr (1963)

$\hat{\rho}(0)=\hat{\rho}_{0}=\frac{1}{2}\left(v_{i}\hat{\sigma}_{i}+\hat{\mathcal{I}}_{\mathrm{s}}\right)\otimes\hat{\rho}_{\mathrm{b}}=\frac{1}{2}v_{\mu}\hat{\sigma}_{\mu}\otimes\hat{\rho}_{\mathrm{b}},$
(II.4)

where

$\hat{\rho}_{\mathrm{b}}=\frac{1}{Z_{\mathrm{b}}}\,\mathrm{e}^{-\beta\hat{H}_{\mathrm{b}}},\hskip
28.45274ptZ_{\mathrm{b}}=\operatorname{tr}_{\mathrm{b}}\\{\,\mathrm{e}^{-\beta\hat{H}_{\mathrm{b}}}\\},$
(II.5)

denotes the thermal distribution for the uncoupled bath. Throughout the paper
we make use of Einstein’s summation convention, reserving Latin letters
$i,j,k,\cdots\in\\{x,y,z\\}$ for sums over the indices of the three Pauli
matrices $\hat{\sigma}_{i}$’s. We instead utilize Greek letters
$\mu,\nu,\lambda,\cdots\in\\{0,x,y,z\\}$ to include in the sum the identity
operator of the qubit subsystem
$\hat{\sigma}_{0}=\hat{\mathcal{I}}_{\mathrm{s}}$. This quadruplet forms a
complete set of operators for the Hilbert space of the qubit. The coefficients
$v_{\mu}$’s of the Bloch vector can be chosen to be real without loss of
generality. The normalization condition
$\operatorname{tr}\\{\hat{\rho}(t)\\}=1$ holds by fixing $v_{0}=1$.

We discuss in the following different formulations of the open dynamics of the
qubit, and the mutual relations between those approaches.

The dynamics of the RDM can be written under broad conditions in terms of the
Nakajima–Zwanzig generalized quantum master equation (GQME) Nakajima (1958);
Zwanzig (1960)

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\hat{\rho}_{\mathrm{s}}(t)$
$\displaystyle=-i[\hat{H}_{\mathrm{s}}(t),\hat{\rho}_{\mathrm{s}}(t)]-\int_{0}^{t}\mathrm{d}\tau\;k_{t,\tau}\hat{\rho}_{\mathrm{s}}(\tau)$
(II.6a) $\displaystyle=\Lambda_{t}\hat{\rho}_{\mathrm{s}}(t).$ (II.6b)

Here, $k_{t,\tau}$ denotes the kernel superoperator,

$\Lambda_{t}\cdot=-i[\hat{H}_{\mathrm{s}}(t),\cdot]-\int_{0}^{t}\mathrm{d}\tau\;k_{t,\tau}\circ\Phi_{\tau}\Phi_{t}^{-1}\cdot$
(II.7)

is the generator of the open dynamics of $\hat{\rho}_{\mathrm{s}}(t)$ and
$\Phi_{t}$ the corresponding propagator

$\Phi_{t}\;\cdot=\operatorname{tr}_{\mathrm{b}}\left\\{\exp_{+}\left\\{-i\int_{0}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\right\\}\hat{\rho}_{0}\;\cdot\right\\}.$
(II.8)

Equation II.6 is solved by

$\hat{\rho}_{\mathrm{s}}(t)=\exp_{+}\left\\{\int_{0}^{t}\mathrm{d}t^{\prime}\;\Lambda_{t^{\prime}}\right\\}\hat{\rho}_{\mathrm{s}}(0)=\Phi_{t}\hat{\rho}_{\mathrm{s}}(0).$
(II.9)

An extended derivation of Eq. II.6 is provided in Appendix B. In the
Born–Markov approximation, the memory kernel is replaced by a Dirac delta
function [i.e. $k_{t,\tau}\propto\delta(t-\tau)$]. We will discuss in Section
IV.2.4 how deviations from this limit can be utilized as a measure of non-
Markovian effects.
From Eqs. II.7 and II.6b it is evident that a time-local representation of the
dynamics is possible provided the propagator $\Phi_{t}$ is invertible, while
the time-nonlocal expression Eq. II.6b is not affected by this restriction. If
$\Phi_{t}^{-1}$ exists at a given time $t$, it possible to construct a time-
local master equation in the so-called _canonical form_ Hall  _et al._ (2014);
Erika Andersson and Hall (2007); Maldonado-Mundo  _et al._ (2012); Hayden and
Sorce (2022)

$\displaystyle\Lambda_{t}{\hat{\rho}}_{\mathrm{s}}(t)$
$\displaystyle=-i[\hat{H}_{\mathrm{c}}(t),\hat{\rho}_{\mathrm{s}}(t)]$
$\displaystyle\quad+\gamma_{i}(t)\left[\hat{L}_{i}(t)\hat{\rho}_{\mathrm{s}}(t)\hat{L}^{\dagger}_{i}(t)-\tfrac{1}{2}\\{\hat{L}_{i}^{\dagger}(t)\hat{L}_{i}(t),\hat{\rho}_{\mathrm{s}}(t)\\}\right].$
(II.10)

Equation II.10 is expressed in terms of a uniquely defined set of canonical
rates $\gamma_{i}(t)$’s and an orthogonal basis of traceless operators
$\hat{L}_{i}(t)$’s, fulfilling

$\operatorname{tr}_{\mathrm{s}}[\hat{L}_{i}(t)]=0,\hskip
28.45274pt\operatorname{tr}_{\mathrm{s}}[\hat{L}_{i}(t)\hat{L}_{j}(t)]=2\delta_{ij}.$
(II.11)

$\hat{H}_{\mathrm{c}}(t)$ is the canonical ($\mathrm{c}$) Hamiltonian
generating the Hermitian part of the dynamics. A derivation of Eq. II.10 is
provided in Appendix C. According to the _Gorini–Kossakowski–Sudarshan
theorem_ , the non-negativity of the rates $\gamma_{i}(t)$’s is a necessary
and sufficient condition for the complete positivity of the dynamics. Gorini,
Kossakowski, and Sudarshan (1976); Breuer  _et al._ (2016) In Section IV.2.3
we will exploit this feature to measure non-Markovianity in open quantum
dynamics with the only, albeit strong restriction to invertible dynamical
maps.

The spin time-correlation functions (STCF)

$\displaystyle\mathcal{C}_{\mu\nu}(t)$
$\displaystyle=\frac{1}{2}\operatorname{tr}\left[(\hat{\sigma}_{\mu}\otimes\hat{\rho}_{\mathrm{b}})\exp_{-}\left\\{i\int_{0}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\right\\}\hat{\sigma}_{\nu}\right].$
(II.12)

allow to naturally decompose the contributions of populations and coherences
in the dynamics, and to conveniently measure the invertibility of the
dynamical map. The time evolution of Eq. II.12 provides a description of the
dynamics equivalent to the RDM. In fact, we can expand

$\displaystyle\hat{\rho}_{\mathrm{s}}(t)$
$\displaystyle=\frac{1}{2}\operatorname{tr}[\hat{\rho}(t)\hat{\sigma}_{\nu}]\hat{\sigma}_{\nu}=\frac{1}{4}\operatorname{tr}_{\mathrm{s}}[(v_{\mu}\hat{\sigma}_{\mu}\otimes\hat{\rho}_{\mathrm{b}})\hat{\sigma}_{\nu}(t)]\hat{\sigma}_{\nu}$
$\displaystyle=\frac{1}{2}v_{\mu}\mathcal{C}_{\mu\nu}(t)\hat{\sigma}_{\nu}.$
(II.13)

An important property of the matrix of the STCF [with components Eq. II.12] is
that its transposed corresponds to the propagator of the Bloch vector. This
can be shown by expanding

$\hat{\rho}_{\mathrm{s}}(t)=\frac{1}{2}v_{\mu}(t)\hat{\sigma}_{\mu},$ (II.14)

and by making use of Eq. II.8, to obtain

$\displaystyle v_{\mu}(t)$
$\displaystyle=\frac{1}{2}\operatorname{tr}_{\mathrm{s}}[\hat{\sigma}_{\mu}\Phi_{t}\hat{\sigma}_{\nu}]v_{\nu}$
$\displaystyle=\frac{1}{2}\operatorname{tr}_{\mathrm{s}}\left[\hat{\sigma}_{\mu}\operatorname{tr}_{\mathrm{b}}\left\\{\exp_{+}\left\\{-i\int_{0}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\right\\}(\hat{\rho}_{\mathrm{b}}\otimes\hat{\sigma}_{\nu})\right\\}\right]v_{\nu}$
$\displaystyle=\mathcal{C}^{T}_{\mu\nu}(t)v_{\nu}.$ (II.15)

The norm of the Bloch vector is conserved in the dynamics of an isolated
subsystem, while it decreases monotonically for a strongly dissipative bath.
Increasing trends of the trace distance over time signal the occurrence of
revivals of quantum information. Breuer, Laine, and Piilo (2009); Breuer  _et
al._ (2016)

As a concluding remark of this initial overview of open quantum dynamics, we
note that the present formalism can be straightforwardly generalized to a
subsystem consisting of an arbitrary number $N$ of quantum levels. In
particular, one can replace the Pauli matrices with a set of $N^{2}-1$
operators $\\{\hat{\Sigma}_{i}\\}_{i=1}^{N^{2}-1}$ generating the
$\mathfrak{su}(N)$ algebra of the subsystem and obeying the trace relations
Runeson and Richardson (2020); Lorenzo, Plastina, and Paternostro (2013b)

$\operatorname{tr}_{\mathrm{s}}[\hat{\Sigma}_{i}\hat{\Sigma}_{j}]=N\delta_{ij},\hskip
28.45274pt\operatorname{tr}_{\mathrm{s}}[\hat{\Sigma}_{i}]=0.$ (II.16)

The $\Sigma_{i}$’s, together with the $N\times N$ identity operator,
constitute a basis of the Hilbert space of the subsystem.

In Section III we present in detail the model system studied in this work,
while in Section IV we develop an extensive numerical analysis of non-
Markovian effects in dissipative–driven quantum dynamics.

##  III Model system

A qubit bilinearly coupled to a bosonic field and to a time-dependent pulse is
modeled by a periodically driven spin–boson model. Grifoni, Paladino, and
Weiss (1999); Magazzù  _et al._ (2018); Magazzù, Denisov, and Hänggi (2018) We
consider here an external drive exclusively coupled to the Hamiltonian of the
subsystem, leading to periodic oscillations in the population difference.
Specifically, the different terms of Eq. II.1 are

$\displaystyle\hat{H}_{\mathrm{s}}(t)$
$\displaystyle=\Delta\hat{\sigma}_{x}+[\varepsilon_{0}+\varepsilon_{d}\cos(\Omega
t)]\hat{\sigma}_{z},$ (III.1a) $\displaystyle\hat{H}_{\mathrm{b}}$
$\displaystyle=\sum_{\alpha=1}^{F}\omega_{\alpha}\hat{a}^{\dagger}_{\alpha}\hat{a}_{\alpha},$
(III.1b) $\displaystyle\hat{H}_{\mathrm{sb}}$
$\displaystyle=\hat{\sigma}_{z}\otimes\sum_{\alpha=1}^{F}c_{\alpha}(\hat{a}^{\dagger}_{\alpha}+\hat{a}_{\alpha}).$
(III.1c)

The diagonal part of $\hat{H}_{\mathrm{s}}(t)$ includes a static bias
$\varepsilon_{0}$ and a time-dependent term involving periodic oscillations at
frequency $\Omega$ and amplitude $\varepsilon_{\mathrm{d}}$. The two quantum
levels interact through the non-diagonal coupling term with strength $\Delta$.
$\hat{a}_{\alpha}^{\dagger}$’s ($\hat{a}_{\alpha}$’s) are the creation
(destruction) operators of the $F$ bath modes, oscillating at frequencies
$\omega_{\alpha}$’s and coupled to the population difference via the
interaction defined by the coefficients $c_{\alpha}$’s. In this work we
consider an Ohmic spectral density with a Drude cutoff Weiss (2012)

$J(\omega)=\frac{\eta}{\pi}\frac{\omega_{\mathrm{c}}\omega}{\omega_{\mathrm{c}}^{2}+\omega^{2}}.$
(III.2)

$\eta$ and $\omega_{\mathrm{c}}$ in Eq. III.2 denote respectively the
system–bath coupling constant and the cutoff frequency of the bath modes.
Equation III.2 corresponds to the continuous limit of its discrete counterpart

$J_{F}(\omega)=\sum_{\alpha=1}^{F}\frac{c_{\alpha}^{2}}{2m_{\alpha}\omega_{\alpha}}\delta(\omega-\omega_{\alpha}).$
(III.3)

The degrees of freedom of the bath can be rewritten in terms of canonical
configurations $\hat{q}_{\alpha}$’s and momenta $\hat{p}_{\alpha}$’s via the
change of coordinates

$\hat{a}_{\alpha}=\frac{1}{\sqrt{2\omega_{\alpha}}}\left(\sqrt{m_{\alpha}}\omega_{\alpha}\hat{q}_{\alpha}+i\frac{\hat{p}_{\alpha}}{\sqrt{m_{\alpha}}}\right),$
(III.4)

such that

$\displaystyle\hat{H}_{\mathrm{b}}$
$\displaystyle=\frac{1}{2}\sum_{\alpha=1}^{F}\left(m_{\alpha}\omega_{\alpha}^{2}\hat{q}_{\alpha}^{2}+\hat{p}_{\alpha}^{2}\right),$
(III.5a) $\displaystyle\hat{H}_{\mathrm{sb}}$
$\displaystyle=\hat{\sigma}_{z}\otimes\sum_{\alpha=1}^{F}\tilde{c}_{\alpha}\hat{q}_{\alpha},$
(III.5b)

where we made use of the canonical commutation relations
$[\hat{q}_{\alpha},\hat{p}_{\alpha^{\prime}}]=i\delta_{\alpha\alpha^{\prime}}$.
We fix in the following $m_{\alpha}=1$ for all $\alpha$.

##  IV Numerical analysis

###  IV.1 Dynamics and stationary state

In this section we study the time evolution of the STCF defined in Eq. II.12,
and their asymptotic behavior at long times.

Given that the identity $\hat{\sigma}_{0}$ is invariant under the action of
the total propagator as defined in Eq. II.2, $\mathcal{C}_{\mu
0}(t)=\operatorname{tr}[\hat{\rho}_{0}\hat{\sigma}_{\mu}\hat{\sigma}_{0}(t)]=\delta_{\mu
0}$. The time evolution of the other nontrivial correlations is shown in Fig.
1 for increasing values of the frequency of the external drive from $\Omega=0$
(dark blue) to $\Omega=20$ (dark red); the other system parameters are set to
$\beta=0.3$, $\Delta=1$, $\varepsilon_{0}=0$, $\varepsilon_{\mathrm{d}}=1$,
$\eta=1$ and $\omega_{\mathrm{c}}=1$.

Figure 1: Dynamics of the spin time-correlation functions
$\\{\mathcal{C}_{\mu\nu(t)}\\}_{\mu\nu}$ [Eq. II.12], shown for increasing
values of the frequency of the external drive from $\Omega=0$ (dark blue) to
$\Omega=20$ (dark red). The other parameters of the system are set to
$\beta=0.3$, $\Delta=1$, $\varepsilon_{0}=0$, $\varepsilon_{\mathrm{d}}=1$,
$\eta=1$ and $\omega_{\mathrm{c}}=1$. The dashed lines correspond to the
dynamics of the time-independent case ($\varepsilon_{\mathrm{d}}=0$, see
discussion in Section IV.1). The black stars at the final time of each panel
are the long-time benchmarks for the undriven system in the assumption of
ergodic dynamics.

We calculate the dynamics STCF’s by integrating the numerically exact
hierarchical equations of motion (HEOM). These are a set of coupled
differential equations for the reduced density matrix (RDM) of the subsystem
and additional auxiliary densities. Ishizaki and Tanimura (2005); Tanimura
(2020) Although this hierarchy is in principle infinite, it can be safely
truncated to a finite tier for all the systems considered in this work, to
obtain a numerically exact solution of the RDM of the qubit. The integration
of the HEOM requires to fix two numerical parameters, i.e., the number of
Matsubara modes $K$ utilized in the high-temperature expansion of the bath
correlation function, and the highest tier $L$ considered in the hierarchy.
All the results presented in the paper exhibit a satisfactory convergence with
$K\leq 2$ and $L\leq 20$.

We observe that for all considered values of $\Omega$ the only correlation
functions that do not relax to a zero plateau at long times are the ones
involving the identity as a static operator (that is, the panels in the first
row of Fig. 1, with $\mu=0$). This behavior is consistent with a
generalization of the mixing long-time limit for a time-dependent Hamiltonian:
Amati, Runeson, and Richardson (2023); Amati, Mannouch, and Richardson (2023);
Hawkins (2021); Berkovitz, Frigg, and Kronz (2006)

$\mathcal{C}_{\mu\nu}(t)\sim\langle\hat{\sigma}_{\mu}\rangle_{0}\langle\hat{\sigma}_{\nu}\rangle_{\mathrm{st},t},\hskip
28.45274ptt\gg 1.$ (IV.1)

In Eq. IV.1 we denoted by
$\langle\hat{\sigma}_{\mu}\rangle_{0}=\operatorname{tr}[\hat{\rho}(0)\hat{\sigma}_{\mu}]=\delta_{\mu
0}$ the average of the static operator over the initial distribution Eq. II.4,
while

$\langle\hat{\sigma}_{\nu}\rangle_{\mathrm{st},t}=\operatorname{tr}[\hat{\rho}_{\mathrm{st}}(t)\hat{\sigma}_{\nu}]$
(IV.2)

is the average over the stationary out-of-equilibrium density
$\hat{\rho}_{\mathrm{st}}(t)$. According to the Floquet theorem, this density
obeys the same periodicity as the system Hamiltonian. Sato  _et al._ (2020);
Ikeda and Sato (2020) It is in general not straightforward to determine a
closed expression for $\hat{\rho}_{\mathrm{s}t}(t)$. This can be approximately
via perturbative expansions in certain limits, e.g. strong but non-vanishing
drive or small system–bath coupling. Shirai  _et al._ (2016); Engelhardt,
Platero, and Cao (2019) In the limit case of a time-independent Hamiltonian
[$\Omega\to 0$ or $\varepsilon_{\mathrm{d}}\to 0$ in Eq. III.1a] and for
ergodic dynamics, Evans, Searles, and Williams (2009)
$\hat{\rho}_{\mathrm{st}}(t)$ converges to the canonical density
${\hat{\rho}_{\beta}=\,\mathrm{e}^{-\beta\hat{H}}/\operatorname{tr}[\,\mathrm{e}^{-\beta\hat{H}}]}$.
This is confirmed by the good overlap between the long-time limits of the
system with $\varepsilon_{\mathrm{d}}=0$ (black dashed line in Fig. 1) and the
averages
$\langle\hat{\sigma}_{\mu}\rangle_{0}\langle\hat{\sigma}_{\nu}\rangle_{\beta}$
(black stars). The second factor, denoting the thermal average with respect to
$\hat{\rho}_{\beta}$, is calculated via the reaction-coordinate representation
of the spin–boson Hamiltonian, an approach valid provided the bosonic bath can
be approximated by its classical limit. Thoss, Wang, and Miller (2001); Amati,
Runeson, and Richardson (2023); Runeson  _et al._ (2022) The good agreement
between long-time dynamics and statistics supports both the validity of the
mixing assumption and of the classical limit for the nuclear bath (as expected
here, given that $\beta\omega_{\mathrm{c}}<1$Shi  _et al._ (2009); Zhu, Liu,
and Shi (2013)).

By inspecting Fig. 1, we note that, as we increase the frequency $\Omega$
(from blue to red), the driven dynamics converge to the time-independent
limit, with $\varepsilon_{\mathrm{d}}=0$ (black dashed lines). This ‘washout’
effect can be understood by noticing that the dynamics generated by Eqs. II.1
and III.1 depends on the drive exclusively via the factor
$\varepsilon_{\mathrm{d}}\sin(\Omega t)/\Omega$ (see Appendix A for a proof).
This implies that $\Omega\to+\infty$ and $\varepsilon_{\mathrm{d}}\to 0$ are
equivalent limits from a dynamical standpoint.

In the spirit of QEC protocols, one could ask whether it could possible to
identify optimal values for the driving frequency $\Omega$ minimizing
environmental dissipation. We tackle this question in the next section,
focusing in particular on the analysis of the trace distance, an insightful
measure of non-Markovianity and dissipation inferred from the measurement of
quantum distinguishability.

###  IV.2 Non-Markovianity

####  IV.2.1 Trace distance

The trace-distance approach to quantum non-Markovianity relies on the
interpretation of dissipation as loss of distinguishability between quantum
states over time. Vacchini (2012); Breuer  _et al._ (2016) In the present
section we summarize the key features of this approach, and utilize the method
to quantify the competing roles of drive and dissipation on the information
stored in the qubit. Extended discussions and reviews on the method can be
found, e.g., in Refs. Laine, Piilo, and Breuer, 2010; Breuer  _et al._ , 2016.

Let us consider two copies of the qubit prepared in two well-defined initial
states, with densities $\hat{\rho}_{\mathrm{s}}^{(1)}(0)$ and
$\hat{\rho}_{\mathrm{s}}^{(2)}(0)$ and corresponding Bloch vectors
$\bm{v}^{(1)}(0)$ and $\bm{v}^{(2)}(0)$. A measure of distinguishability
between these densities is given by the _trace distance_

$\mathcal{D}^{(1,2)}(t)=\frac{1}{2}\left\lVert\hat{\rho}_{\mathrm{s}}^{(1)}(t)-\hat{\rho}_{\mathrm{s}}^{(2)}(t)\right\rVert,$
(IV.3)

where $\left\lVert\cdot\right\rVert$ is the trace norm on
$\mathcal{H}_{\mathrm{s}}$. Watrous (2018) Not that the trace distance of a
Hermitian operator $\hat{A}\in\mathcal{H}_{\mathrm{s}}$ can be expressed in
terms of the absolute sum of its eigenvalues $d_{i}$’s, that is
$\left\lVert\hat{A}\right\rVert=\sum_{i=1,2,3}|d_{i}|$. Breuer  _et al._
(2016) This, in particular, implies that for a two-level subsystem the trace
distance of two densities Eq. IV.3 coincides with half of the Euclidean
distance between the respective Bloch vectors:

$\mathcal{D}^{(1,2)}(t)=\frac{1}{2}|\bm{v}^{(1)}(t)-\bm{v}^{(2)}(t)|.$ (IV.4)

Equivalently, by making use of Eq. II.15,

$\displaystyle\mathcal{D}^{(1,2)}(t)$
$\displaystyle=\frac{1}{2}\left\\{\sum_{j=x,y,z}\left[\sum_{i=x,y,z}\left(v^{(1)}_{i}-v^{(2)}_{i}\right)\mathcal{C}_{ij}(t)\right]^{2}\right\\}^{1/2}.$
(IV.5)

From Eqs. IV.5 and IV.1 we infer that, for our model system, information on
the initial state is fully dissipated to the environment at long times. This
is guaranteed by the fact that the correlations $\mathcal{C}_{ij}(t)$ (with
$i,j\geq 1$) involved in the calculation of the trace distance ultimately
relax to zero at long times. A consequent caveat is that the trace distance
cannot capture non-Markovian effects due to correlation functions
$\mathcal{C}_{0i}(t)$ generating the non-unital part of the dynamics, Utagi,
Banerjee, and Srikanth (2023) and leading to nonzero stationary limits (as
seen in Fig. 1).

Divisible time-dependent dynamical processes act as contractions of the trace
distance, that is $\dot{\mathcal{D}}^{(1,2)}(t)\leq 0$ for all $\bm{v}^{(1)}$
and $\bm{v}^{(2)}$. Ruskai (1994); Breuer  _et al._ (2016) Non-divisibility
can instead be interpreted as memory effects, given that time propagation on a
finite time interval cannot be arbitrarily decomposed at intermediate times.
The present observations led Breuer, Laine and Piilo (BLP) to introduce as a
witness of non-Markovianity the integral measure

$\displaystyle\mathcal{N}_{\mathrm{BLP}}$
$\displaystyle=\max_{\hat{\rho}_{\mathrm{s}}^{(1)},\hat{\rho}_{\mathrm{s}}^{(2)}}\int_{\dot{\mathcal{D}}^{(1,2)}(t)>0}\mathrm{d}t\;\dot{\mathcal{D}}^{(1,2)}(t).$
(IV.6)

The maximization in Eq. IV.6 guarantees that $\mathcal{N}_{\mathrm{BLP}}$ is
exclusively a function of the dynamical map $\Phi_{t}$ describing the open
quantum dynamics, rather than of any specific couple of initial densities. It
has been proven in Ref. Wißmann  _et al._ , 2012 that the densities maximizing
Eq. IV.6 are mutually orthogonal and lie on the boundary of the Hilbert space
of the subsystem $\mathcal{H}_{\mathrm{s}}$. For the specific case of two-
level systems, those densities are represented by antipodal vectors of unit
norm on the surface of the Bloch sphere. We can hence rewrite Eq. IV.6 as

$\mathcal{N}_{\mathrm{BLP}}=\max_{|\bm{v}|=1}\int_{\dot{\mathcal{D}}^{\bm{v}}(t)>0}\mathrm{d}t\;\dot{\mathcal{D}}^{\bm{v}}(t),$
(IV.7)

where

${\mathcal{D}}^{\bm{v}}(t)=\left\\{\sum_{j=x,y,z}\left[\sum_{i=x,y,z}v_{i}\mathcal{C}_{ij}(t)\right]^{2}\right\\}^{1/2}.$
(IV.8)

The time evolution of the trace distance $\mathcal{D}_{\mathrm{max}}(t)$
maximizing Eq. IV.7 is shown in Figs. 3 and 3. Each figure corresponds to a
fixed value of the system–bath coupling ($\eta=0.1$ and $\eta=1$). In each
figure, results are given for two values of the inverse temperature,
$\beta=0.3$ and $\beta=1.6$ in the upper and lower panels, respectively. As in
Fig. 1, the color scheme denotes increasing values of the frequency of the
external drive $\Omega$ from dark blue for $\Omega=0$ up to dark red for
$\Omega=20$, with increasing warmth. The other system parameters are set to
$\Delta=1$, $\varepsilon_{0}=0$, $\varepsilon_{\mathrm{d}}=1$,
$\omega_{\mathrm{c}}=1$.

Figure 2: Main panels: time evolution of the maximal trace distance
$\mathcal{D}_{\mathrm{max}}(t)$ for the dissipative–driven qubit. Results are
shown for two values of the inverse temperature, $\beta=0.3$ and $\beta=1.6$
and for increasing values of the frequency of the drive from $\Omega=0$ (dark
blue) to $\Omega=20$ (dark red). The other parameters are set to $\Delta=1$,
$\varepsilon_{0}=0$, $\varepsilon_{\mathrm{d}}=1$, $\eta=0.1$ and
$\omega_{\mathrm{c}}=1$. Insets: time evolution of the volume of accessible
states, with the same color code and for the same systems shown in the main
panels (see the discussion in Section IV.2.2.)

Figure 3:  Similar to Fig. 3 but with larger system–bath coupling constant
$\eta=1$.

By increasing the inverse temperature we observe an overall slower decay of
the trace distance over time, as expected due to the decrease of thermal
fluctuations leading to loss of coherence. Furthermore, the time evolution of
$\mathcal{D}_{\mathrm{max}}(t)$ appears to be highly sensitive to modulations
of the driving frequency $\Omega$. To quantify the dependence of the
information flow on the driving frequency, we show in Figs. 5 and 5 phase
diagrams of $\mathcal{N}_{\mathrm{BLP}}$ as a function of $\Omega$ for the
same parameters of Figs. 3 and 3 and two additional values of $\beta$. The
maximization in Eq. IV.7 is achieved by uniform Monte Carlo (MC) sampling of
antipodal Bloch vectors on the surface of the Bloch sphere. Clos and Breuer
(2012) Satisfactory convergence is achieved with number of
$N_{\mathrm{MC}}=10^{6}$ samples for Fig. 5 and $N_{\mathrm{MC}}=10^{5}$
samples for Fig. 5. Interestingly, revivals of information are observed for
all considered systems. This holds true also in the high-temperature regime at
$\beta=0.3$, where the quantum environmental bath can be approximated by its
classical counterpart (see Section IV.1). Our analysis is consistent with
results from other studies, witnessing quantum non-Markovianity in presence of
a classical environment. Bordone, Buscemi, and Benedetti (2012); Xu  _et al._
(2013) In our results we observe maxima of $\mathcal{N}_{\mathrm{BLP}}$ for
specific values of the drive associated to resonant revivals of information.
Shuang  _et al._ (2000) This analysis indicates that, compared to the non-
driven dynamics at $\Omega=0$, the inclusion of a periodic external field can
effectively mitigate dissipation of information to the environment. The
present study further supports the well established role of trace-distance
approaches as insightful tools in the framework of quantum control. Liu  _et
al._ (2011); Chen  _et al._ (2021)

In the above discussion on $\mathcal{N}_{\mathrm{BLP}}$ we pointed out that,
in the case of a qubit, states corresponding to maxima of information backflow
must lay on the surface on the Bloch sphere. Those states also maximize
another scalar quantity corresponding to the volume underneath the surface of
the sphere. We discuss this approach in the next section.

Figure 4: Integral measure of non-Makrovianity $\mathcal{N}_{\mathrm{BLP}}$
[Eq. IV.6], shown for four values of the inverse temperature $\beta$ and for
increasing values of the frequency of the drive $\Omega$. The other parameters
are set to $\Delta=1$, $\varepsilon_{0}=0$, $\varepsilon_{\mathrm{d}}=1$,
$\eta=0.1$ and $\omega_{\mathrm{c}}=1$. The maximization in Eq. IV.7 has been
performed with $N_{\mathrm{MC}}=10^{6}$ Monte Carlo samples.

Figure 5: Similar to Fig. 5 but with $\eta=1$ and $N_{\mathrm{MC}}=10^{5}$.

####  IV.2.2 Volume of accessible states

From the propagation of the Bloch vector in Eq. II.15 we can identify the
volume on the Bloch sphere

$\mathcal{V}(t)=\lvert\det\mathcal{C}^{T}(t)\rvert$ (IV.9)

accessible to the qubit at a given time $t$. Lorenzo, Plastina, and
Paternostro (2013b); Chruściński, Macchiavello, and Maniscalco (2017);
Puthumpally-Joseph  _et al._ (2018); Mangaud  _et al._ (2018) Another
insightful expression for $\mathcal{V}(t)$ follows from the first-order
expansion

$\displaystyle\mathcal{V}(t+\delta t)$
$\displaystyle=\left\lvert\det\left[\mathcal{C}^{T}(t)+\delta
t\dot{\mathcal{C}}^{T}(t)\right]\right\rvert+\mathcal{O}(\delta t^{2})$
$\displaystyle=\left\lvert\det\left[\mathcal{I}_{4}+\delta
t\Xi(t)\right]\det\left[\mathcal{C}^{T}(t)\right]\right\rvert+\mathcal{O}(\delta
t^{2})$ $\displaystyle=\left\lvert\left\\{\mathcal{I}_{4}+\delta
t\operatorname{tr}_{\mathrm{s}}[\Xi(t)]\right\\}\mathcal{V}(t)\right\rvert+\mathcal{O}(\delta
t^{2}),$ (IV.10)

where we introduced the _damping matrix_ Hall  _et al._ (2014)

$\Xi(t)=\dot{\mathcal{C}}^{T}(t)[\mathcal{C}^{T}(t)]^{-1}=\frac{\mathrm{d}}{\mathrm{d}t}\ln[\mathcal{C}^{T}(t)].$
(IV.11)

Equation IV.10 leads to the differential equation

$\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{V}(t)=\operatorname{tr}_{\mathrm{s}}[\Xi(t)]\mathcal{V}(t),$
(IV.12)

solved by

$\mathcal{V}(t)=\mathcal{V}(0)\exp\left\\{\int_{0}^{t}\mathrm{d}t^{\prime}\;\operatorname{tr}_{\mathrm{s}}[\Xi(t^{\prime})]\right\\}.$
(IV.13)

$\Xi(t)$ is also the generator of the dynamics in the Bloch representation. In
particular, from Eq. II.15,

$\displaystyle\dot{v}_{\mu}(t)$ $\displaystyle=\Xi_{\mu\nu}(t)v_{\nu}(t).$
(IV.14)

Equivalently, given that

$\displaystyle\dot{v}_{\mu}(t)$
$\displaystyle=\operatorname{tr}_{\mathrm{s}}\left[\Lambda_{t}[\hat{\rho}_{\mathrm{s}}(t)]\hat{\sigma}_{\mu}\right]$
$\displaystyle=\frac{1}{2}\operatorname{tr}_{\mathrm{s}}\left[\Lambda_{t}[\hat{\sigma}_{\nu}]\hat{\sigma}_{\mu}\right]v_{\nu}(t),$
(IV.15)

we identify

$\Xi_{\mu\nu}(t)=\frac{1}{2}\operatorname{tr}_{\mathrm{s}}\left[\Lambda_{t}[\hat{\sigma}_{\nu}]\hat{\sigma}_{\mu}\right].$
(IV.16)

It can be proven that for completely positive open quantum dynamics the volume
$\mathcal{V}(t)$ decreases monotonically in time. Wolf and Cirac (2008); Raja
_et al._ (2018) This applies in particular to the BM limit. Therefore,
similarly to Eq. IV.6, it has been proposed in Ref. Lorenzo, Plastina, and
Paternostro, 2013a as a witness of non-Markovianity the integral measure

$\mathcal{N}_{\mathcal{V}}=\frac{1}{\mathcal{V}(0)}\int_{\dot{\mathcal{V}}(t)>0}\mathrm{d}t\;\dot{\mathcal{V}}(t).$
(IV.17)

The time evolution of $\mathcal{V}(t)$ is shown in the insets of Figs. 3 and
3. We note that the timescale to relaxation of $\mathcal{V}(t)$ increases for
larger inverse temperature $\beta$ and for smaller system–bath coupling
$\eta$. This suggests that the relaxation of the volume provides a simple and
effective measure of decoherence, sensitive to the strength of dissipation
induced by the bath. However we note that, for all considered system
parameters, $\mathcal{V}(t)$ decreases monotonically in time, hence
$\mathcal{N}_{\mathcal{V}}=0$. This result appears in contrast with the
analysis of the trace distance, which, as discussed in Section IV.2.1, does
witness non-Markovian effects in all the considered systems. A well-known
issue of the volume approach is that $\mathcal{V}(t)$ accounts exclusively for
the average dissipation over all quantum channels, hence it can fail to
capture non-Markovian effects whenever those localize on a subset of channels
on the Hilbert space. Hall  _et al._ (2014); Chruściński, Macchiavello, and
Maniscalco (2017); Mangaud, Meier, and Desouter-Lecomte (2017) We will
extensively discuss this issue in the next Section IV.2.3, devoted to the
decomposition of dissipative effects separately on each quantum channel,
within the framework of the time-local master equation Eq. II.10. Despite
limitations in witnessing non-Markovianity, the analysis $\mathcal{V}(t)$ can
still provide valuable insight on the invertibility of the open quantum
dynamics, a necessary condition to construct the time-local canonical master
equation Eq. II.10. In particular, from the insets of Fig. 3, we observe that,
in the small coupling limit, the monochromatic drive can extend invertibility
to longer times compared to the non-driven system with $\Omega=0$.

####  IV.2.3 Canonical rates

A rigorous approach to measure non-Markovianity in open quantum dynamics
involves the analysis of the canonical rates in the time-local master equation
Eq. II.10. In particular, it can be shown that, in the BM limit of Eq. II.6,
the operators $\hat{L}_{i}$’s become time-independent, and the rates
$\gamma_{i}$’s converge to non-negative constants. Lindblad (1976); de Vega
and Alonso (2017) Beyond the BM approximation, the rates remain non-negative
functions of time in case of completely positive divisible dynamics, as proven
by the _Gorini–Kossakowski–Sudarshan theorem_. Gorini, Kossakowski, and
Sudarshan (1976) In light of these observations, it has been proposed to
measure non-Markovian effects on each channel by studying the negativity of
the respective rate. Hall  _et al._ (2014) This analysis provides in general
deeper insight compared to the study of the volume of accessible states [Eq.
IV.9]. Hall  _et al._ (2014); Chruściński and Maniscalco (2014) In fact, the
exponent of Eq. IV.13 can be rewritten as Hall  _et al._ (2014)

$\operatorname{tr}_{\mathrm{s}}[\Xi(t)]=-2\sum_{i=1,2,3}\gamma_{i}(t).$
(IV.18)

In conjunction with Eq. IV.13, Eq. IV.18 tells us that $\mathcal{V}(t)$
provides information only on the average dissipation occurring on all quantum
channels. The analysis of all rates avoids instead possible compensation
effects, which can in particular impair the detection of weak non-Markovian
effects.

The knowledge of a complete set of STCF for a given system suffices to
calculate the related canonical rates. In fact, the $\gamma_{i}(t)$’s are the
eigenvalues of the _decoherence matrix_ $\xi(t)$, with components

$\displaystyle\xi_{ij}(t)$
$\displaystyle=\frac{1}{4}\operatorname{tr}_{\mathrm{s}}\left[\hat{\sigma}_{\lambda}\hat{\sigma}_{i}\Lambda_{t}[\hat{\sigma}_{\lambda}]\hat{\sigma}_{j}\right]$
$\displaystyle=\frac{1}{4}\operatorname{tr}_{\mathrm{s}}\left[\hat{\sigma}_{\lambda}\hat{\sigma}_{i}\hat{\sigma}_{\rho}\hat{\sigma}_{j}\right]\Xi_{\rho\lambda}(t),$
(IV.19)

and related to $\mathcal{C}(t)$ via the identity Eq. IV.11. A proof of Eq.
IV.19 is shown in Appendix D.

Figure 6:  Time evolution of the canonical rates. Each column corresponds to a
given value of the inverse temperature $\beta$. Results are shown for
increasing values of the driving frequency $\Omega$, with the same color
scheme utilized in the previous figures. The other system parameters are fixed
to $\Delta=1$, $\varepsilon_{0}=1$, $\varepsilon_{\mathrm{d}}=1$, $\eta=0.1$,
$\omega_{\mathrm{c}}=1$.

An important caveat of this approach is that a solution of the rates at a
given time $t$ is possible only provided the matrix $\mathcal{C}(t)$ is
invertible. However, for the systems considered in this work, the determinant
Eq. IV.9 vanishes at long times. This can be inferred from the fact
$\mathcal{V}(t)$ depends only on the vanishing terms
$\\{\mathcal{C}_{ij}(t)\\}_{ij}$, for $i,j\geq 1$ [see Eq. IV.1]. To account
for this divergence, we calculated the solution of the rates up to the lowest
threshold time $\tau_{\mathrm{th}}$ defined such that
$\mathcal{V}(\tau_{\mathrm{th}})=10^{-3}$. The time evolution of the three
canonical rates for the dissipative–driven qubit is given in Fig. 6, for
increasing values of the driving frequency from $\Omega=0$ (dark blue) to
$\Omega=20$ (dark red). Each column corresponds to a fixed value of the
inverse temperature ($\beta=0.3$ and $\beta=1.6$, at left and right,
respectively). The other system parameters are fixed to $\Delta=1$,
$\varepsilon_{0}=1$, $\varepsilon_{\mathrm{d}}=1$, $\eta=0.1$,
$\omega_{\mathrm{c}}=1$. We observe that at both temperatures one of the three
rates is negative at all times, and it relaxes to a nonzero long-time plateau.
The negativity of at least a quantum rate for all times has been named in
literature as _eternal non-Markovianity_. Hall  _et al._ (2014); Megier  _et
al._ (2017); Vaishy, Mitra, and Bhattacharya (2022) The relaxation of the
$\gamma_{i}(t)$’s to non-zero final limits is consistent with the fact that
these rates are functions of the limits of Eq. IV.11, in general non-
vanishing. Remarkably, we observe for $\gamma_{1}(t)$ at $\beta=1.6$ (upper-
right panel) that driving can increase the negativity of the rates at long
times compared to the undriven case (shown as the dark blue curve for
$\Omega=0$). Consistently with our analysis of the trace distance in Figs. 5
and 5, this implies that suitable controlled driving can be proficiently
utilized as a strategy to counterbalance environmental dissipation and to
strengthen non-Markovian effects. Let us stress that the positivity of
${\sum_{i=1,2,3}\gamma_{i}(t)}$ is the reason why non-Markovianity cannot be
detected from the analysis of the accessible volume in Section IV.2.2.

As pointed out above, the canonical rates cannot be calculated for times at
which the dynamics is not invertible. Such limitation does not apply for the
time-nonlocal GQME, a general first-principled approach to quantum dissipation
discussed in the next section.

####  IV.2.4 Generalized quantum master equation

Equation II.6a can be rewritten in terms of an integro–differential equation
of motion for STCF

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{C}(t,\tau)=\mathcal{C}(t,\tau)\mathcal{X}(t)-\int_{\tau}^{t}\mathrm{d}\tau^{\prime}\;\mathcal{C}(\tau^{\prime},\tau)\mathcal{K}(t,\tau^{\prime}).$
(IV.20)

The time-dependent drift $\mathcal{X}(t)$ and the memory kernel matrix
$\mathcal{K}(t,\tau)$ in Eq. IV.20 are defined in Eqs. B.7 and B.6 of Appendix
B, respectively. In the same appendix we provide a derivation of Eq. IV.20,
together with details on how to calculate numerically the memory kernel from
projection-free input time-correlation functions.

The memory kernel encapsulates complete information on the dynamical effects
of the environment on the subsystem. Hence, one could ask at which extent the
kernel relates to the other measures of non-Markovianity discussed in the
present work. To tackle this question, we start by considering the Born–Markov
(BM) approximation of Eq. IV.20, Breuer and Petruccione (2002) corresponding
to the assumption of memory-less dynamics
$\mathcal{K}(t,\tau)\propto\delta(t-\tau)$. This leads to the time-local
master equation

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}{\mathcal{C}}_{\mathrm{BM}}(t,\tau)$
$\displaystyle=\mathcal{C}_{\mathrm{BM}}(t,\tau)\mathcal{M}_{\mathrm{BM}}(t,\tau),$
(IV.21a) $\displaystyle\mathcal{M}_{\mathrm{BM}}(t,\tau)$
$\displaystyle=\mathcal{X}(t)-\int_{\tau}^{t}\mathrm{d}\tau^{\prime}\;\mathcal{K}(t,\tau^{\prime}),$
(IV.21b)

which is solved by

$\displaystyle\mathcal{C}_{\mathrm{BM}}(t,\tau)$
$\displaystyle=[\Phi_{\mathrm{BM}}(t,\tau)\mathcal{I}_{4}]^{T},$ (IV.22a)
$\displaystyle\Phi_{\mathrm{BM}}(t,\tau)$
$\displaystyle=\exp_{+}\left\\{\int_{\tau}^{t}\mathrm{d}t^{\prime}\;\mathcal{M}_{\mathrm{BM}}^{T}(t^{\prime},\tau)\right\\},$
(IV.22b)

where we noticed that the initial value $\mathcal{C}_{\mathrm{BM}}(0,0)$ is
equal to the $4\times 4$ identity matrix $\mathcal{I}_{4}$. Note that the BM
propagator $\Phi_{\mathrm{BM}}(t,\tau)$, being a time-ordered exponential, is
divisible, J Evans and P Morriss (2007) i.e.,

$\Phi_{\mathrm{BM}}(t,0)=\Phi_{\mathrm{BM}}(t,\tau)\Phi_{\mathrm{BM}}(\tau,0),\hskip
8.53581pt0\leq\tau\leq t.$ (IV.23)

This indicates that the BM approximation of the GQME is consistent with the
correspondence between Markovianity and divisibility of the dynamical map,
laying at the foundation of all the approaches to quantum dissipation
discussed in the present work. Breuer  _et al._ (2016); Rivas, Huelga, and
Plenio (2010); Chruściński, Kossakowski, and Rivas (2011b); Chruściński and
Maniscalco (2014)

For a non-driven time-independent Hamiltonian, Eq. IV.20 simplifies to

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{C}(t)=\mathcal{C}(t)\mathcal{X}-\int_{0}^{t}\mathrm{d}\tau^{\prime}\;\mathcal{C}(\tau^{\prime})\mathcal{K}(t-\tau^{\prime}),$
(IV.24)

with corresponding BM approximation

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}{\mathcal{C}}_{\mathrm{BM}}(t)$
$\displaystyle=\mathcal{C}_{\mathrm{BM}}(t)\mathcal{M}_{\mathrm{BM}},$
(IV.25a) $\displaystyle\mathcal{M}_{\mathrm{BM}}$
$\displaystyle=\mathcal{X}-\int_{0}^{+\infty}\mathrm{d}\tau^{\prime}\;\mathcal{K}(\tau^{\prime}),$
(IV.25b)

generating divisible dynamics. One can prove that Eq. IV.25 preserves the same
long-time limits of the exact dynamics Eq. IV.24 (see Appendix B of Ref. Amati
_et al._ , 2022 for further details). This supports the relevance and physical
significance of the present Markovian approximation of the GQME.

In the following we introduce a dynamical approach to measure non-Markovian
effects from the analysis of the GQME. We consider here for simplicity the
case of nondriven dynamics by fixing $\varepsilon_{\mathrm{d}}=0$ in Eq.
III.1a. Panel (a) of Fig. 7, shows the time evolution of the $\mu=\nu=x$
component of the memory kernel, normalized by its initial value,
$\overline{\mathcal{K}}_{xx}(t)={\mathcal{K}}_{xx}(t)/{\mathcal{K}}_{xx}(0)$.
Results are given for increasing values of the system–bath coupling from
purple ($\eta=0.1$) to yellow ($\eta=2)$. The other parameters are fixed to
$\beta=0.3$, $\Delta=1$, $\varepsilon_{0}=1$, $\omega_{\mathrm{c}}=1$. We
observe that the normalized kernel relaxes to zero faster for increasing
values of $\eta$, converging towards the BM limit. To quantify the deviation
between the exact dynamics and the Markovian approximation of the GQME, we
measure a timescale to relaxation of the kernel, defined by

$\tau_{\mathcal{K}}(\delta)=\max_{\mu\nu}\left\\{\tau:\;\;\tfrac{\int_{0}^{\tau}\mathrm{d}t^{\prime}\;|\mathcal{K}_{\mu\nu}(0)|}{\int_{0}^{+\infty}\mathrm{d}t^{\prime}\;|\mathcal{K}_{\mu\nu}(0)|}=\delta\right\\},\hskip
14.22636pt\delta\in(0,1).$ (IV.26)

In our numerical results, we fix $\delta=0.9$, and redefine
$\tau_{\mathcal{K}}=\tau_{\mathcal{K}}(0.9)$. As shown in the inset of panel
(a), $\tau_{\mathcal{K}}$ decreases monotonically for increasing values of
$\eta$. By interpolating the decay of this curve for larger values of the
coupling, it is possible to assess a threshold value for the system–bath
coupling above which the BM equation Eq. IV.25 provides a satisfactory
description of the dynamics.

In panel (b) of Fig. 7 we show the time evolution of the maximal trace
distance, $\mathcal{D}_{\mathrm{max}}(t)$, while the integral measure of non-
Markovianity, $\mathcal{N}_{\mathrm{BLP}}$, [Eq. IV.6] is displayed as a
function of $\eta$ in the respective inset. A steady increase in information
revival occurs for intermediate system–bath coupling $0.5\lesssim\eta\lesssim
1$, at values corresponding to the transition from coherent to incoherent
regime for the spin boson model. By further increasing the coupling
($\eta\gtrsim 1$) we observe a monotonic decrease of information backflow due
to the strong dissipative effects exerted by the environment. Similarly to the
study carried out in Section IV.2.1, the analysis of the trace distance
provides here a well-defined criterion to optimize coherence control in open
quantum dynamics. The complementary GQME approach can be proficiently utilized
to construct an effective Markovian description in the strong-coupling regime,
where revivals of information become negligible.

Figure 7: Panel (a): diagonal components of the memory kernel from Eq. IV.24
and respective timescale to relaxation $\tau_{\mathcal{K}}$ [Eq. IV.26]. Panel
(b): time evolution of the maximal trace distance
$\mathcal{D}_{\mathrm{max}}(t)$ and respective integral measure of non-
Markovianity $\mathcal{N}_{\mathrm{BLP}}$ [Eq. IV.6]. Results are shown for
increasing values of the system–bath coupling from purple ($\eta=0.1$) to
yellow ($\eta=2)$. The other parameters of the system are fixed to
$\beta=0.3$, $\Delta=1$, $\varepsilon_{0}=1$, $\varepsilon_{\mathrm{d}}=0$,
$\omega_{\mathrm{c}}=1$.

##  V Conclusions

In this paper we analyzed the open quantum dynamics of a dissipative.-driven
qubit, from multiple perspective and with several approaches. We integrated
the HEOM of the model to propagate the RDM of the qubit over time, for a large
set of parameters. By decomposing the solution of the RDM onto a complete set
of STCF, we observed that the non-unital part of the dynamics is the only
contributor to the formation of non-equilibrium stationary states, involving
the same periodicity of the total Hamiltonian.

We calculated a witness of non-Markovianity by measuring the time evolution of
the trace distance between two maximally distinguishable RDM’s. This approach
provides a dynamical measure of information exchange between the qubit and the
external environment. By tuning the frequency of an external monochromatic
field, we observed nontrivial competing effects between dissipation and drive,
leading to maximal resonant peaks in information backflow. This study
highlights an insightful strategy to design optimal control protocols aimed at
mitigating the loss of coherence in the dynamics of quantum-information
systems subjected to external sources of dissipation. da Costa Morazotti  _et
al._ (2023); Jones, Ladd, and Fong (2012)

We noticed that for the model systems studied in this work the analysis of the
volume of accessible states fails to witness weak signatures of non-
Markovianity occurring on a finite subset of quantum channels. This issue is
resolved by studying dissipation separately on each channel. In particular, we
split the non-Hermitian part of the dissipative dynamics along the direction
of a time-dependent basis of the Hilbert space. We then studied non-Markovian
effects by measuring canonical rates of dissipation on each channel. Our
numerical analysis allowed us to identify, for sufficiently small values of
the system–bath coupling, stable signatures of _eternal non-
Markovianity_.Gulácsi and Burkard (2023); Hall  _et al._ (2014); Megier  _et
al._ (2017) Furthermore, we showed that sufficiently strong drive can enhance
non-Markovian effects compared to the non-driven dynamics, an important
observation which could be proficiently exploited to enhance quantum control.

The time-nonlocal GQME is an exact equation of motion for the RDM of an open
quantum system. It provides an interpretation of dissipation in terms of
dynamical memory effects, captured by fast-decaying memory kernels. We
discussed that, in the BM approximation, the GQME leads to divisible dynamics,
in agreement with the notion of Markovianity introduced by the other
approaches discussed in this work. For a nondriven Hamiltonian we introduced a
timescale to relaxation of the kernel as a measure of non-Markovian effects.
We showed that, for our model system and parameter domain, dynamical memory
effects in the GQME decrease monotonically for increasing values of the
system–bath coupling. Hence, the GQME formalism can be proficiently utilized
as a numerical tool to assess the validity of an effective Markovian
description of open quantum dynamics. The trace-distance approach allows
instead to identify peaks of information revival in the parameter space of the
system. The present joint analysis strengthens the complementarity of the two
approaches in a multifaceted analysis of quantum dissipation.

Several questions tackled in the present work are worth further investigation
in future research, from both theoretical and technological perspectives.

It has been recently shown how features of eternal non-Markovianity emerge in
the dynamics of superconducting qubits, a leading technology in the framework
of quantum computing. Gulácsi and Burkard (2023) It could be highly insightful
to apply and extend the analysis of the present work to realistic systems of
quantum processors, especially by tackling the challenging problem of
improving coherence lifetime in multi-qubit gates.

In this work we focused on the dissipative effects induced by the environment
on a small quantum subsystem. A complementary theoretical approach worth
investigating involves a somehow reversed perspective, i.e. the study of how a
quantum subsystem affects the dynamics of the environment in a thermal state.
This approach could lead to meaningful insight in particular in QEC protocols
relying on the design of engineered noise as a control resource.Kapit (2017)

The analysis of dissipation from the canonical master equation is based on the
construction of a time-dependent orthonormal set of operators, identifying
canonical channels consisting of a linear combination of the standard Pauli
channels. It could be instructive to implement a analysis of dissipation based
on the topological structure of the canonical channels in the Hilbert space.

Optimal-control theory is a rigorous approach exploiting a variational
principle to design quantum systems subjected to minimal dissipation and
stable coherence over long time scales. Jirari and Pötz (2006); Jirari (2020);
Palao, Kosloff, and Koch (2008) Further perspectives for future research
involve the utilization of tools from control theory to identify decoherence-
free subspaces, Fortunato  _et al._ (2002) in conjunction with all the
analytical methods utilized in this work.

## Data availability

The data that support the findings of this study are available from the author
upon reasonable request.

## Acknowledgments

The author would like to thank Tanja Schilling, Anja Seegebrecht, Fabian Koch,
Heinz-Peter Breuer, Michael Thoss and Andreas Buchleitner for fruitful
discussions and helpful comments. This project has been supported by the
_Deutsche Forschungsgemeinschaft_ (DFG) via the Research Unit FOR5099
“Reducing complexity of nonequilibrium” (project no. 431945604).

##  Appendix A High-frequency limit

In this appendix we determine a factorization of the propagator in Eq. II.3
which allows us to straightforwardly calculate the $\Omega\to+\infty$ limit of
the dynamics.

Firstly let us remark that, due to the non-diagonal term
$\Delta\hat{\sigma}_{x}$ in the total Hamiltonian $\hat{H}(t)$, it is not
possible to remove the time ordering in the integration in Eq. II.3. We can,
however, factorize $\hat{U}(t)$ in two terms whose $\Omega\to\infty$ limit can
be straightforwardly calculated. This is accomplished by introducing the
rotating frame (R), defined by the unitary transformation

$\displaystyle\hat{U}_{\mathrm{R}}(t)$
$\displaystyle=\exp_{+}\left\\{-i\hat{\sigma}_{z}\varepsilon_{\mathrm{d}}\int_{0}^{t}\mathrm{d}t^{\prime}\;\cos(\Omega
t^{\prime})\right\\}$
$\displaystyle=\hat{\mathcal{I}}_{\mathrm{s}}\cos\left[\frac{\varepsilon_{\mathrm{d}}\sin(\Omega
t)}{\Omega}\right]-i\hat{\sigma}_{z}\sin\left[\frac{\varepsilon_{\mathrm{d}}\sin(\Omega
t)}{\Omega}\right],$ (A.1)

rotating a quantum state $\ket{\psi(t)}$ in phase with an external drive. The
rotated state
$\ket{\psi_{\mathrm{R}}(t)}=\hat{U}_{\mathrm{R}}^{\dagger}(t)\ket{\psi(t)}$
obeys the Schrödinger equation Shirai, Mori, and Miyashita (2018)

$\displaystyle i\frac{\partial}{\partial t}\ket{\psi_{\mathrm{R}}(t)}$
$\displaystyle=\left[\hat{H}_{\mathrm{R}}(t)-(\hat{\sigma}_{z}\otimes\hat{\mathcal{I}}_{\mathrm{b}})\varepsilon_{\mathrm{d}}\cos(\Omega
t)\right]\ket{\psi_{\mathrm{R}}(t)},$ (A.2a)

where

$\displaystyle\hat{H}_{\mathrm{R}}(t)$
$\displaystyle=\hat{U}_{\mathrm{R}}^{\dagger}(t)\hat{H}(t)\hat{U}_{\mathrm{R}}(t)$
$\displaystyle=[\varepsilon_{0}+\varepsilon_{\mathrm{d}}\cos(\Omega
t)]\hat{\sigma}_{z}\otimes\hat{\mathcal{I}}_{\mathrm{b}}+\hat{\mathcal{I}}_{\mathrm{s}}\otimes\hat{H}_{\mathrm{b}}+\hat{H}_{\mathrm{sb}}$
$\displaystyle\quad+\Delta\Bigg{[}\cos\left(\frac{2\varepsilon_{\mathrm{d}}\sin(\Omega
t)}{\Omega}\right)\hat{\sigma}_{x}$
$\displaystyle\quad-\sin\left(\frac{2\varepsilon_{\mathrm{d}}\sin(\Omega
t)}{\Omega}\right)\hat{\sigma}_{y}\Bigg{]}\otimes\hat{\mathcal{I}}_{\mathrm{b}}.$
(A.3)

Let us introduce now the propagator

$\hat{V}_{\mathrm{R}}(t)=\exp_{+}\left\\{-i\int_{0}^{t}\mathrm{d}t^{\prime}\;\left[\hat{H}_{\mathrm{R}}(t)-(\hat{\sigma}_{z}\otimes\hat{\mathcal{I}}_{\mathrm{b}})\varepsilon_{\mathrm{d}}\cos(\Omega
t)\right]\right\\},$ (A.4)

solving the Schrödinger equation Eq. A.2a. Given that, for all
$\ket{\psi(0)}$,

$\displaystyle\ket{\psi(t)}$
$\displaystyle=\hat{U}(t)\ket{\psi(0)}=\hat{U}_{\mathrm{R}}(t)\ket{\psi_{\mathrm{R}}(t)}=\hat{U}_{\mathrm{R}}(t)\hat{V}_{\mathrm{R}}(t)\ket{\psi(0)},$
(A.5)

it follows that

$\hat{U}(t)=\hat{U}_{\mathrm{R}}(t)\hat{V}_{\mathrm{R}}(t).$ (A.6)

Equation A.6 allows us to calculate the strong-frequency limit of the driven
dynamics

$\displaystyle\lim_{\Omega\to+\infty}\hat{U}(t)$
$\displaystyle=\lim_{\Omega\to+\infty}\hat{V}_{\mathrm{R}}(t)=\lim_{\varepsilon_{\mathrm{d}}\to
0}\hat{U}(t).$ (A.7)

Eq. A.7 justifies the $\Omega\to+\infty$ limits discussed throughout the paper
(see e.g. Fig. 1).

##  Appendix B GQME for dissipative–driven dynamics

The Nakajima–Zwanzig GQME has affirmed itself as a powerful theoretical tool
within the framework of open quantum systems and nonadiabatic dynamics.
Nakajima (1958); Zwanzig (1960); Brian and Sun (2021); Montoya-Castillo and
Reichman (2016); Kelly, Brackbill, and Markland (2015) Thus far, the approach
has been extensively utilized to analyze and interpret non-Markovian effects
in model systems described by time-independent Hamiltonians. The formalism can
be straightforwardly extended to the case of dissipative–driven driven
dynamics, where the Hamiltonian involves an explicit dependence on time. In
the present appendix we provide a derivation of the approach in this general
case. The following derivation applies specifically to a two-level subsystem,
although it can be extended with no limitations to an arbitrary number $N$ of
levels [see the concluding remark of Section II]. We follow here the
derivation originally presented in Ref. Haake, 1973, with the main difference
that in our case we define projection operators in the spin basis. This
approach is particularly well suited to tackle the analysis of the spin–spin
correlation functions in a variety of models in quantum nonadiabatic dynamics.

Let us consider for now a generic time-independent projection superoperator,
$\mathcal{P}=\mathcal{P}^{2}$, acting on the total Hilbert space of system and
environment. Its complementary is defined by
$\mathcal{Q}=\mathbb{1}-\mathcal{P}$, where $\mathbb{1}$ is the identity
superoperator.

We aim to derive a non-Markovian equation of motion for the open quantum
dynamics of correlation functions of the subsystem. To do so, we start by
decomposing the equation of motion of the two propagator onto the two
directions of projection:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\exp_{-}\left\\{i\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\right\\}=i\exp_{-}\left\\{i\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\right\\}\mathcal{P}\mathcal{L}_{t}$
$\displaystyle\quad+i\exp_{-}\left\\{i\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\right\\}\mathcal{Q}\mathcal{L}_{t}.$
(B.1)

We can now make use of the Dyson identity Montoya-Castillo and Reichman (2016)

$\displaystyle\exp_{-}\left\\{\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\;\mathcal{B}_{t^{\prime}}\right\\}$
$\displaystyle=\exp_{-}\left\\{-\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\;\mathcal{A}_{t^{\prime}}\right\\}-\int_{t_{0}}^{t}\mathrm{d}\tau\;\exp_{-}\left\\{-\int_{t_{0}}^{\tau}\mathrm{d}t^{\prime}\;\mathcal{A}_{t^{\prime}}\right\\}$
$\displaystyle\quad\times(\mathcal{A}_{\tau}+\mathcal{B}_{\tau})\exp_{-}\left\\{\int_{\tau}^{t}\mathrm{d}t^{\prime}\;\mathcal{B}_{t^{\prime}}\right\\},$
(B.2)

defined for two generic superoperators $\mathcal{A}_{t}$ and
$\mathcal{B}_{t}$. By replacing $\mathcal{A}_{t}=-i\mathcal{L}_{t}$ and
$\mathcal{B}_{t}=i\mathcal{Q}\mathcal{L}_{t}$, we obtain

$\displaystyle\exp_{-}\left\\{i\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\right\\}$
$\displaystyle=\exp_{-}\left\\{i\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\;\mathcal{Q}\mathcal{L}_{t^{\prime}}\right\\}+i\int_{t_{0}}^{t}\mathrm{d}\tau\;\exp_{-}\left\\{i\int_{t_{0}}^{\tau}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\right\\}$
$\displaystyle\quad\times\mathcal{P}\mathcal{L}_{\tau}\exp_{-}\left\\{i\int_{\tau}^{t}\mathrm{d}t^{\prime}\;\mathcal{Q}\mathcal{L}_{t^{\prime}}\right\\},$
(B.3)

which can be replaced onto the right-hand side of Eq. B.1 to find Kelly  _et
al._ (2016)

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\exp_{-}\left\\{i\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\right\\}=i\exp_{-}\left\\{i\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\right\\}\mathcal{P}\mathcal{L}_{t}$
$\displaystyle\quad+i\exp_{-}\left\\{i\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\;\mathcal{Q}\mathcal{L}_{t^{\prime}}\right\\}\mathcal{Q}\mathcal{L}_{t}$
$\displaystyle\quad-\int_{t_{0}}^{t}\mathrm{d}\tau\;\exp_{-}\left\\{i\int_{t_{0}}^{\tau}\mathrm{d}t\;\mathcal{L}_{t^{\prime}}\right\\}$
$\displaystyle\quad\times\mathcal{P}\mathcal{L}_{\tau}\exp_{-}\left\\{i\int_{\tau}^{t}\mathrm{d}t^{\prime}\;\mathcal{Q}\mathcal{L}_{t^{\prime}}\right\\}\mathcal{Q}\mathcal{L}_{t}.$
(B.4)

To construct an equation of motion for the STCF Eq. II.12, we consider a
Redfield-type projection superoperator Kelly  _et al._ (2016) spanning the
full Hilbert space of the subsystem. This is written in Liouville space
Mukamel (1999) as

$\mathcal{P}=|\hat{\sigma}_{\lambda}\rrangle\llangle\hat{\rho}_{0}\hat{\sigma}_{\lambda}|.$
(B.5)

After multiplying Eq. B.4 from left and right by
$\llangle\hat{\rho}_{0}\hat{\sigma}_{\mu}|$ and $|\hat{\sigma}_{\nu}\rrangle$
respectively, we obtain the GQME Eq. IV.20, involving the memory kernel

$\displaystyle\mathcal{K}_{\mu\nu}(t,\tau)=\llangle\hat{\rho}_{0}\hat{\sigma}_{\mu}|\mathcal{L}_{\tau}\exp_{-}\left\\{i\int_{\tau}^{t}\mathrm{d}t^{\prime}\;\mathcal{Q}\mathcal{L}_{t^{\prime}}\right\\}\mathcal{Q}\mathcal{L}_{t}|\hat{\sigma}_{\nu}\rrangle$
$\displaystyle=\llangle\hat{\rho}_{0}\hat{\sigma}_{\mu}|\mathcal{L}_{\tau}\mathcal{Q}\exp_{-}\left\\{i\int_{\tau}^{t}\mathrm{d}t^{\prime}\;\mathcal{Q}\mathcal{L}_{t^{\prime}}\right\\}\mathcal{Q}\mathcal{L}_{t}|\hat{\sigma}_{\nu}\rrangle$
(B.6)

and drift matrix

$\mathcal{X}_{\mu\nu}(t)=i\llangle\hat{\rho}_{0}\hat{\sigma}_{\mu}|\mathcal{L}_{t}|\hat{\sigma}_{\nu}\rrangle.$
(B.7)

The present choice of projection operators allows for a convenient
simplification in the structure of the open quantum dynamics. In fact, the
matrix elements of the complicated projected dynamics in the second term of
the right-hand side of Eq. B.4 vanish identically:

$\displaystyle
i\llangle\hat{\rho}_{0}\hat{\sigma}_{\mu}|\exp_{-}\left\\{i\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\;\mathcal{Q}\mathcal{L}_{t^{\prime}}\right\\}\mathcal{Q}\mathcal{L}_{t}|\hat{\sigma}_{\nu}\rrangle$
$\displaystyle=i\llangle\hat{\rho}_{0}\hat{\sigma}_{\mu}|\mathcal{Q}\exp_{-}\left\\{i\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\mathcal{Q}\right\\}\mathcal{Q}\mathcal{L}_{t}|\hat{\sigma}_{\nu}\rrangle$
$\displaystyle=i\llangle\hat{\rho}_{0}\hat{\sigma}_{\mu}|\exp_{-}\left\\{i\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\mathcal{Q}\right\\}\mathcal{Q}\mathcal{L}_{t}|\hat{\sigma}_{\nu}\rrangle$
$\displaystyle\quad-i\llangle\hat{\rho}_{0}\hat{\sigma}_{\mu}|\hat{\sigma}_{\lambda}\rrangle\llangle\hat{\rho}_{0}\hat{\sigma}_{\lambda}|\exp_{-}\left\\{i\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\mathcal{Q}\right\\}\mathcal{Q}\mathcal{L}_{t}|\hat{\sigma}_{\nu}\rrangle=0,$
(B.8)

where we made use of the orthogonality of the scalar product
$\llangle\hat{\rho}_{0}\hat{\sigma}_{\mu}|\hat{\sigma}_{\nu}\rrangle=\operatorname{tr}[\hat{\rho}_{0}\hat{\sigma}_{\mu}\hat{\sigma}_{\nu}]=\delta_{\mu\nu}$.

The memory kernel in Eq. B.6 depends on the projected dynamics, which is hard
to simulate numerically, both exactly and with approximated quasiclassical
approaches. Amati  _et al._ (2022) To circumvent this issue, we follow an
established strategy originally introduced in Ref. Shi and Geva, 2003. In
particular, we construct a recursive relation for the memory kernel by
inserting again the Dyson identity Eq. B.3 onto Eq. B.6:

$\displaystyle\mathcal{K}_{\mu\nu}(t,\tau)=\llangle\hat{\rho}_{0}\hat{\sigma}_{\mu}|\mathcal{L}_{\tau}\mathcal{Q}\exp_{-}\left\\{i\int_{\tau}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\right\\}\mathcal{Q}\mathcal{L}_{t}|\hat{\sigma}_{\nu}\rrangle$
$\displaystyle\quad-i\int_{\tau}^{t}\mathrm{d}\tau^{\prime}\;\llangle\hat{\rho}_{0}\hat{\sigma}_{\mu}|\mathcal{L}_{\tau}\mathcal{Q}\exp_{-}\left\\{i\int_{\tau}^{\tau^{\prime}}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\right\\}|\hat{\sigma}_{\lambda}\rrangle$
$\displaystyle\quad\times\llangle\hat{\rho}_{0}\hat{\sigma}_{\lambda}|\mathcal{L}_{\tau^{\prime}}\exp_{-}\left\\{i\int_{\tau^{\prime}}^{t}\mathrm{d}t^{\prime}\;\mathcal{Q}\mathcal{L}_{t^{\prime}}\right\\}\mathcal{Q}\mathcal{L}_{t}|\hat{\sigma}_{\nu}\rrangle,$
(B.9)

written in matrix form as

$\mathcal{K}(t,\tau)=\mathcal{K}^{(1)}(t,\tau)+\int_{\tau}^{t}\mathrm{d}\tau^{\prime}\;\mathcal{K}^{(3)}(\tau^{\prime},\tau)\mathcal{K}(t,\tau^{\prime}),$
(B.10)

where we introduced the two auxiliary kernels

$\displaystyle\mathcal{K}_{\mu\nu}^{(1)}(t,\tau)$
$\displaystyle=\llangle\hat{\rho}_{0}\hat{\sigma}_{\mu}|\mathcal{L}_{\tau}\mathcal{Q}\exp_{-}\left\\{i\int_{\tau}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\right\\}\mathcal{Q}\mathcal{L}_{t}|\hat{\sigma}_{\nu}\rrangle,$
(B.11a) $\displaystyle\mathcal{K}_{\mu\nu}^{(3)}(t,\tau)$
$\displaystyle=-i\llangle\hat{\rho}_{0}\hat{\sigma}_{\mu}|\mathcal{L}_{\tau}\mathcal{Q}\exp_{-}\left\\{i\int_{\tau}^{t}\mathrm{d}t^{\prime}\;\mathcal{L}_{t^{\prime}}\right\\}|\hat{\sigma}_{\nu}\rrangle.$
(B.11b) Equations B.10 and B.11 allow us to conveniently calculate the
solution of kernel Eq. B.11 via projection-free input correlation functions.

In particular, by expanding the projectors in Eq. B.11 we obtain the relations

$\displaystyle\mathcal{K}^{(1)}(t,\tau)$
$\displaystyle=\frac{\mathrm{d}^{2}}{\mathrm{d}t\mathrm{d}\tau}\mathcal{C}(t,\tau)+\mathcal{X}(\tau)\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{C}(t,\tau)$
$\displaystyle\quad-\frac{\mathrm{d}}{\mathrm{d}\tau}\mathcal{C}(t,\tau)\mathcal{X}(t)-\mathcal{X}(\tau)\mathcal{C}(t,\tau)\mathcal{X}(t),$
(B.12a) $\displaystyle\mathcal{K}^{(3)}(t,\tau)$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}\tau}\mathcal{C}(t,\tau)+\mathcal{X}(\tau)\mathcal{C}(t,\tau).$
(B.12b)

By comparing Eqs. B.12a and B.12b, we find a relation between the two
auxiliary kernels

$\mathcal{K}^{(1)}(t,\tau)=\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{K}^{(3)}(t,\tau)-\mathcal{K}^{(3)}(t,\tau)\mathcal{X}(t).$
(B.13)

Therefore, to obtain with projection-free input the full kernel Eq. B.6 it
suffices to calculate directly only $\mathcal{K}^{(3)}(t,\tau)$ via Eq. II.12,
and then to make use of the identities Eqs. B.13 and B.10.

This general derivation of the GQME applies to any arbitrary two-level quantum
subsystem and environment. In the specific case of the driven spin–boson model
considered in this work [Eq. III.1] it is possible to obtain an analytic
expression for the drift matrix in Eq. B.7:

$\displaystyle\mathcal{X}_{\mu\nu}(t)$
$\displaystyle=i\llangle\hat{\rho}_{0}\hat{\sigma}_{\mu}|[\hat{H}_{\mathrm{s}}(t),\hat{\sigma}_{\nu}]\rrangle$
$\displaystyle=-2\Big{[}\Delta(\delta_{\mu z}\delta_{\nu y}-\delta_{\mu
y}\delta_{\nu z})+\epsilon(t)(\delta_{\mu y}\delta_{\nu x}-\delta_{\mu
x}\delta_{\nu y})\Big{]},$ (B.14)

or, in matrix form,

$\displaystyle\mathcal{X}(t)$ $\displaystyle=2\begin{pmatrix}0&0&0&0\\\
0&0&\varepsilon(t)&0\\\ 0&-\varepsilon(t)&0&\Delta\\\
0&0&-\Delta&0\end{pmatrix}.$ (B.15)

##  Appendix C Canonical form of the time-local master equation

Studying the dynamics of open quantum systems via the time-local master
equation Eq. II.10 provides a rigorous and well defined strategy to witness
the occurrence of non-Markovian effects. In this appendix we present a
detailed derivation of this master equation, following Refs. Mangaud, Meier,
and Desouter-Lecomte, 2017; Hall  _et al._ , 2014; Erika Andersson and Hall,
2007.

First of all, we observe that the linearity of the generator $\Lambda_{t}$
[Eq. II.6] implies that, if the propagator $\Phi_{t}$ is invertible, two sets
of operators $\\{\hat{A}_{\mu}(t)\\}_{\mu}$ and $\\{\hat{B}_{\mu}(t)\\}_{\mu}$
exist such that

$\displaystyle\Lambda_{t}\hat{\rho}_{\mathrm{s}}(t)=\hat{A}_{\mu}(t)\hat{\rho}_{\mathrm{s}}(t)\hat{B}_{\mu}^{\dagger}(t).$
(C.1)

By expanding

$\hat{A}_{\mu}(t)=\hat{\sigma}_{\nu}a_{\nu\mu}(t),\hskip
28.45274pt\hat{B}_{\mu}(t)=\hat{\sigma}_{\nu}b_{\nu\mu}(t),$ (C.2)

we rewrite Eq. C.1 as

$\displaystyle\Lambda_{t}\hat{\rho}_{\mathrm{s}}(t)$
$\displaystyle=a_{\nu\mu}(t)\hat{\sigma}_{\nu}\hat{\rho}_{\mathrm{s}}(t)b^{*}_{\lambda\mu}(t)\hat{\sigma}_{\lambda}=\xi_{\nu\lambda}(t)\hat{\sigma}_{\nu}\hat{\rho}_{\mathrm{s}}(t)\hat{\sigma}_{\lambda},$
(C.3)

where we introduced the _decoherence matrix_ $\xi(t)$, with components
$\xi_{\nu\lambda}(t)=a_{\nu\mu}(t)b^{*}_{\lambda\mu}(t)$ [see also Eq. IV.19
for an alternative expression]. Given that $\hat{\rho}_{\mathrm{s}}(t)$ and
$\Lambda_{t}\hat{\rho}_{\mathrm{s}}(t)$ are Hermitian operators, $\xi(t)$ is a
Hermitian matrix at all times. In fact,

$\displaystyle[\Lambda_{t}\hat{\rho}_{\mathrm{s}}(t)]^{\dagger}$
$\displaystyle=\xi_{\nu\lambda}^{*}(t)\hat{\sigma}_{\lambda}\hat{\rho}_{\mathrm{s}}(t)\hat{\sigma}_{\nu}=\xi_{\lambda\nu}^{*}(t)\hat{\sigma}_{\nu}\hat{\rho}_{\mathrm{s}}(t)\hat{\sigma}_{\lambda},$
(C.4)

and, by comparing Eqs. C.3 and C.4, it follows that
$\xi_{\nu\lambda}(t)=\xi_{\lambda\nu}^{*}(t)$. To identify in Eq. C.3 the
Hermitian part of the open dynamics, we expand

$\displaystyle\Lambda_{t}\hat{\rho}_{\mathrm{s}}(t)$
$\displaystyle=\xi_{00}(t)\hat{\rho}_{\mathrm{s}}(t)+\xi_{i0}(t)\hat{\sigma}_{i}\hat{\rho}_{\mathrm{s}}(t)+\xi_{0j}(t)\hat{\rho}_{\mathrm{s}}(t)\hat{\sigma}_{j}$
$\displaystyle\quad+\xi_{ij}(t)\hat{\sigma}_{i}\hat{\rho}_{\mathrm{s}}(t)\hat{\sigma}_{j},$
(C.5)

and, by defining

$\hat{h}_{\mathrm{c}}(t)=\tfrac{1}{2}\xi_{00}(t)\hat{\mathcal{I}}_{\mathrm{s}}+\xi_{i0}(t)\hat{\sigma}_{i},$
(C.6)

we recast Eq. C.5 as

$\displaystyle\Lambda_{t}\hat{\rho}_{\mathrm{s}}(t)$
$\displaystyle=\hat{h}_{\mathrm{c}}(t)\hat{\rho}_{\mathrm{s}}(t)+\hat{\rho}_{\mathrm{s}}(t)\hat{h}_{\mathrm{c}}^{\dagger}(t)+\xi_{ij}(t)\hat{\sigma}_{i}\hat{\rho}_{\mathrm{s}}(t)\hat{\sigma}_{j}.$
(C.7)

By taking the trace of Eq. C.7 and making use of the trace-preservation of the
density matrix
($\operatorname{tr}_{\mathrm{s}}[\Lambda_{t}\hat{\rho}_{\mathrm{s}}(t)]=0$),
we obtain

$\displaystyle\hat{h}_{\mathrm{c}}(t)+\hat{h}_{\mathrm{c}}^{\dagger}(t)=-\xi_{ij}(t)\hat{\sigma}_{j}\hat{\sigma}_{i}.$
(C.8)

The Hermitian operator

$\hat{H}_{\mathrm{c}}(t)=\frac{i}{2}\left[\hat{h}_{\mathrm{c}}(t)-\hat{h}_{\mathrm{c}}^{\dagger}(t)\right],$
(C.9)

allows us to discern the Hermitian and the dissipative contributions in Eq.
C.7:

$\displaystyle\Lambda_{t}\hat{\rho}_{\mathrm{s}}(t)$
$\displaystyle=-i[\hat{H}_{\mathrm{c}}(t),\hat{\rho}_{\mathrm{s}}(t)]$
$\displaystyle\quad+\xi_{ij}(t)\left[\hat{\sigma}_{i}\hat{\rho}_{\mathrm{s}}(t)\hat{\sigma}_{j}-\tfrac{1}{2}\left\\{\hat{\sigma}_{j}\hat{\sigma}_{i},\hat{\rho}_{\mathrm{s}}(t)\right\\}\right].$
(C.10)

The negativity of the eigenvalues of $\xi(t)$ provides a measure of the
discrepancy from the Markovian limit of the dynamics. In particular, we can
decompose $\xi(t)$ in terms of its real eigenvalues $\gamma_{i}(t)$’s and
corresponding orthonormal eigenvectors $\bm{u}^{(i)}(t)$’s according to

$\displaystyle\xi_{ij}(t)$
$\displaystyle=u_{i}^{(k)}(t)\gamma_{k}(t)\left[u^{(k)}_{j}(t)\right]^{*}=\mathcal{U}_{ik}(t)\gamma_{k}(t)\mathcal{U}_{ik}^{\dagger}(t),$
(C.11)

where $\mathcal{U}_{ij}(t)=u_{i}^{(j)}(t)$. The time-dependent operators

$\displaystyle\hat{L}_{i}(t)$
$\displaystyle=\mathcal{U}_{ij}(t)\hat{\sigma}_{j},$ (C.12)

allow us to rewrite Eq. C.10 as

$\displaystyle\Lambda_{t}\hat{\rho}_{\mathrm{s}}(t)$
$\displaystyle=-i[\hat{H}_{\mathrm{c}}(t),\hat{\rho}_{\mathrm{s}}(t)]$
$\displaystyle\quad+\left(\mathcal{U}^{\dagger}_{ki}(t)\xi_{ij}(t)\mathcal{U}_{jl}(t)\right)\hat{L}_{k}(t)\hat{\rho}_{\mathrm{s}}(t)\hat{L}^{\dagger}_{l}(t)$
$\displaystyle\quad-\frac{1}{2}\left(\mathcal{U}^{\dagger}_{li}(t)\xi_{ij}(t)\mathcal{U}_{jk}(t)\right)\\{\hat{L}^{\dagger}_{k}(t)\hat{L}_{l}(t),\hat{\rho}_{\mathrm{s}}(t)\\},$
(C.13)

which corresponds to Eq. II.10, given that
$\left[\mathcal{U}^{\dagger}(t)\xi(t)\mathcal{U}(t)\right]_{ij}=\delta_{ij}\gamma_{j}(t)$.

##  Appendix D Decomposition of superoperators of the subsystem

In the present appendix we include the derivation of Eq. IV.19, an identity
relating the canonical rates to the damping matrix Eq. IV.11. The following
proof corresponds to Lemma 2.2 Ref. Gorini, Kossakowski, and Sudarshan, 1976.

Lemma: Let us denote by $\\{\hat{\pi}_{\mu}\\}_{\mu}$ a complete orthonormal
set (COS) of operators on $\mathcal{H}_{\mathrm{s}}$. For
$\hat{O}:\mathcal{H}_{\mathrm{s}}\to\mathcal{H}_{\mathrm{s}}$, the family of
superoperators $\\{\mathcal{G}_{\mu\nu}\\}_{\mu\nu}$ defined by

$\hat{O}\mapsto\mathcal{G}_{\mu\nu}\hat{O}=\hat{\pi}_{\mu}\hat{O}\hat{\pi}_{\nu}^{\dagger},$
(D.1)

is a COS in $L(\mathcal{H}_{\mathrm{s}})$, the linear space of superoperators
acting on $\mathcal{H}_{\mathrm{s}}$.

Proof: First of all, we notice that

$\hat{\pi}_{\mu}^{\dagger}\hat{O}\hat{\pi}_{\mu}=\hat{\mathcal{I}}_{\mathrm{s}}\operatorname{tr}_{\mathrm{s}}[\hat{O}].$
(D.2)

In fact, being the left-hand side of Eq. D.2 invariant under a change of COS,
we can replace $\\{\hat{\pi}_{\mu}\\}_{\mu}$ with
$\\{\hat{E}_{ab}=\ket{a}\bra{b}\\}_{a,b=0,1}$, and expand

$\hat{E}_{ab}^{\dagger}\hat{O}\hat{E}_{ab}=\ket{b}\braket{a}{\hat{O}}{a}\bra{b}=\hat{\mathcal{I}}_{\mathrm{s}}\operatorname{tr}[\hat{O}].$
(D.3)

$L(\mathcal{H}_{\mathrm{s}})$ is a unitary space with inner product

$\displaystyle\langle\mathcal{A},\mathcal{B}\rangle=\operatorname{tr}_{\mathrm{s}}[(\mathcal{A}\hat{\tau}_{\mu})^{\dagger}(\mathcal{B}\hat{\tau}_{\mu})],\hskip
14.22636pt\mathcal{A},\mathcal{B}\in L(\mathcal{H}_{\mathrm{s}}),$ (D.4)

also independent on any specific choice of COS $\\{\hat{\tau}_{\mu}\\}_{\mu}$.
We can now prove that $\\{\mathcal{G}_{\mu\nu}\\}_{\mu\nu}$ is a COS in
$L(\mathcal{H}_{\mathrm{s}})$ . In fact, via Eq. D.2,

$\displaystyle\langle{\mathcal{G}}_{\alpha\beta},{\mathcal{G}}_{\mu\nu}\rangle$
$\displaystyle=\operatorname{tr}_{\mathrm{s}}\left[({\mathcal{G}}_{\alpha\beta}\hat{\tau}_{\lambda})^{\dagger}({\mathcal{G}}_{\mu\nu}\hat{\tau}_{\lambda})\right]$
$\displaystyle=\operatorname{tr}_{\mathrm{s}}\left[\left(\hat{\pi}_{\alpha}\hat{\tau}_{\lambda}\hat{\pi}_{\beta}^{\dagger}\right)^{\dagger}\left(\hat{\pi}_{\mu}\hat{\tau}_{\lambda}\hat{\pi}_{\nu}^{\dagger}\right)\right]$
$\displaystyle=\operatorname{tr}_{\mathrm{s}}\left[\hat{\pi}_{\beta}(\hat{\tau}^{\dagger}_{\lambda}\hat{\pi}_{\alpha}^{\dagger}\hat{\pi}_{\mu}\hat{\tau}_{\lambda})\hat{\pi}_{\nu}^{\dagger}\right]$
$\displaystyle=\operatorname{tr}_{\mathrm{s}}[\hat{\pi}_{\alpha}^{\dagger}\hat{\pi}_{\mu}]\operatorname{tr}_{\mathrm{s}}[\hat{\pi}_{\beta}\hat{\pi}_{\nu}^{\dagger}]=\delta_{\alpha\mu}\delta_{\beta\nu}.\;\;\blacksquare$

The above result implies that the generator $\Lambda_{t}$ can be decomposed on
the basis $\\{\mathcal{G}_{\mu\nu}\\}_{\mu\nu}$ according to

$\Lambda_{t}\;\cdot=\langle\Lambda_{t},{\mathcal{G}}_{\nu\lambda}\rangle{\mathcal{G}}_{\nu\lambda}\;\cdot\;\;.$
(D.5)

Equation IV.19 follows by applying Eq. D.5 to $\hat{\rho}_{\mathrm{s}}(t)$,
expressing both the scalar product and the superoperators
$\mathcal{G}_{\mu\nu}$ in terms of the COS
$\tfrac{1}{\sqrt{2}}\\{\hat{\sigma}_{\mu}\\}_{\mu}$, and by comparing with Eq.
C.3.

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