# Adiabatic Lindbladian Evolution with Small Dissipators

Alain Joye111 Univ. Grenoble Alpes, CNRS, Institut Fourier, F-38000 Grenoble,
France

Abstract: We consider a time-dependent small quantum system weakly coupled to
an environment, whose effective dynamics we address by means of a Lindblad
equation. We assume the Hamiltonian part of the Lindbladian is slowly varying
in time and the dissipator part has small amplitude. We study the properties
of the evolved state of the small system as the adiabatic parameter and
coupling constant both go to zero, in various asymptotic regimes. In
particular, we analyse the deviations of the transition probabilities of the
small system between the instantaneous eigenspaces of the Hamiltonian with
respect to their values in the purely Hamiltonian adiabatic setup, as a
function of both parameters.

Keywords: Adiabatic approximation, Lindblad generators, Quantum dynamics.

Acknowledgments: This work is partially supported by the ANR grant NONSTOPS
(ANR-17-CE40-0006-01)

##  1 Introduction

The adiabatic approximation of quantum mechanics, designed to address the
time-dependent Schrödinger equation in its Hamiltonian formulation, has been
introduced very soon after the discovery of quantum mechanics [BF, La, Z]. It
has since been developed in order to accommodate more general Hamiltonians and
to improve its accuracy; see [K1, N1, ASY, Te, AE, N2, JKP, JP1, J1, JP2, Sc,
BDF] for examples along these lines. The adiabatic approximation being
instrumental in the analysis of time-dependent phenomena, this mathematical
method was extended and applied to a variety of evolution equations in more
general contexts, like discrete time evolutions, [DKS, HJPR1, HJPR2], non-
linear setups [CFK1, CFK2, GG, S, LLFY, F-KJ], or contracting evolutions in
Banach spaces [Kr, NR, J2, AFGG1] for example.

Specifically, adiabatic approaches were successfully adapted to address the
evolution of open quantum systems consisting of a time-dependent small system
of interest coupled to an environment, be it from a global Hamiltonian
perspective encompassing a modeling of the environment, or from an effective
point of view through a Lindblad evolution equation. In particular, asymptotic
expressions for quantum states solution to an evolution equation driven by
time-dependent Lindblad generators in the adiabatic limit are provided in
[AFGG1, AFGG2, FH], together with detailed analyses of the special case of
dephasing Lindbladians. See [DS, TW, JMS] for results along the same lines,
including a Hamiltonian description of an environment at positive and zero
temperature, while [A-SF, HJPR1, HJPR2, BFJP] focus on entropy production
issues in the adiabatic regime of such systems.

From the point of view of applications to quantum engineering and quantum
control, these adiabatic approaches are suited to describe the evolution of a
small system that can be monitored in a time dependent fashion by external
agents, and which is weakly coupled to its environment, due to imperfect
isolation. In the adiabatic limit, and for a regime of small coupling, one
expects to get a description of the evolution of the small system in terms of
the characteristics of the Hamiltonian, with quantitative information of the
perturbations induced by the effect of the environment. While these questions
have been addressed in the physics literature in various setups, see e.g. the
discussions and references in [AFGG2, FH], the mathematical approaches of
those questions are less numerous and often concern specific cases.

Detailed information about the evolution of the small system is available for
dephasing Lindbladians only [AFGG1, AFGG2, FH, H], in which by definition, the
dissipator is a function of the Hamiltonian. Hence, a time dependence in the
Hamiltonian implies a similar time dependence in the dissipator as well. In
the Hamiltonian model addressed in [JMS], the coupling between the small
system and the (bosonic) environment is assumed to be energy conserving at all
times, which makes it dependent on the Hamiltonian as well. This is arguably a
shortcoming of the approach since the dissipator models the effect of the
coupling of the small system to the environment that, in general, is likely to
be independent of the way the system is monitored and is possibly time-
independent. In the model considered in [DS], the coupling of the small system
to the (fermionic) environment is time independent, however the coupling
constant is determined by the adiabatic parameter.

In case the Lindblad generator is time independent, the dynamics of states can
be inferred from the spectral properties of the Lindbladian. Therefore, a host
of perturbative methods have been designed to identify and approximate the
asymptotic state, as well as to analyse more precise properties of the
dynamics, e.g. unravelling. See for example the recent papers [ABFJ, BCF+,
MGLG, HJ, BBC+], and the references therein for works along these lines, in
various setups.

The present contribution is devoted to the study of the effective dynamics of
a small quantum system weakly coupled to an environment, assuming a
Lindbladian description that consists in a slowly varying time-dependent
Hamiltonian drive and an arbitrary dissipator. We analyse the evolved state of
the small quantum system as both the adiabatic parameter $\varepsilon>0$ and
the coupling constant $g>0$ vanish, in an independent way. Actually we can,
and will, consider time-dependent dissipators, which will allow for
comparisons with some of the results mentioned above.

Under suitable assumptions, we provide leading order approximations of the
density matrix of the small system in the perturbative regime,
$g\ll\varepsilon$, where the coupling constant is much smaller than the
adiabatic parameter, in the slow drive regime, $\varepsilon\ll g$, where the
adiabatic parameter is much smaller than the coupling constant, as well as in
a transition regime, $g\ll\sqrt{\varepsilon}$, bridging the gap between the
two previous regimes. The transition regime is addressed by means of a reduced
dynamics defined on the kernel of the Hamiltonian part of the Lindbladian,
that depends on the single parameter $\varepsilon/g$ that determines the
regime we are in. We show that the reduced dynamics approximates the
asymptotic Lindbladian evolution in the transition regime, that covers the
perturbative regime and, partially, the slow drive regime.

As a consequence, we also derive the asymptotics of the transition
probabilities between instantaneous eigenspaces of the Hamiltonian in these
regimes. In the perturbative regime, the leading order of these transition
probabilities is shown to be given by the familiar expression of order
$\varepsilon^{2}$ depending only on the Hamiltonian if $g\ll\varepsilon^{3}$,
and by an explicit integral expression of order $g/\varepsilon$ that depends
on the dissipator if $\varepsilon^{3}\ll g\ll\varepsilon$. This is in keeping
with [JMS] where a similar transition in the asymptotics of the transition
probability was observed for the Hamiltonian model considered. In case
$g=\varepsilon$, we are in the transition regime and the reduced dynamics is
independent of $\varepsilon$ to leading order. This regime corresponds, in
spirit, to the regime addressed in their Hamiltonian model by [DS], Section 3.
For the slow drive regime, $\varepsilon\ll g$, we get that the transition
probability is independent of $\varepsilon$ and $g$ to leading order, and is
characterised by the kernel of the dissipator.

These features are illustrated for a two-level system with a dissipator
displaying a certain symmetry. This allows for the explicit computation of the
reduced dynamics which interpolates between these regimes.

##  2 Setup and main results

The separable Hilbert space of the small system is denoted by ${\cal H}$ and
$t\mapsto H(t)$ is its time dependent Hamiltonian on ${\cal H}$. The
dissipator of the Lindbladian is constructed by means of a finite sum (for
simplicity) of bounded operators on ${\cal H}$, called jump operators,
$t\mapsto\Gamma_{l}(t)$, $l\in I$, a set of indices. We will assume the
following regularity hypotheses:

Reg
$\bullet$ $H:[0,1]\rightarrow{\cal B}({\cal H})$ is self-adjoint valued and
$C^{\infty}$ in norm (with right and left derivatives at $\\{0,1\\}$).
$\bullet$ $\partial_{t}^{k}H(t)|_{t=0}=0$, for all $k\in{\mathbb{N}}^{*}$.
$\bullet$ For each $j\in I$, $I$ a finite set of indices,
$\Gamma_{j}:[0,1]\rightarrow{\cal B}({\cal H})$ is $C^{\infty}$ in norm (with
right and left derivatives at $\\{0,1\\}$).

Note that while the leading order results stated in the present section do not
require all derivatives of the Hamiltonian at zero to vanish, the arbitrary
high order generalisations of Section 7 require this property. This assumption
ensures that high order adiabatic approximations of the Heisenberg unitary
evolution of the spectral projectors of the Hamiltonian coincide with the
spectral projectors of the Hamiltonian at time zero.

For $g\geq 0$, and each $t\in[0,1]$, the time-dependent Lindblad operator
${\cal L}_{t}^{[g]}(\cdot)\in{\cal B}({\cal B}({\cal H}))$ reads

${\cal L}_{t}^{[g]}(\cdot)={\cal L}_{t}^{0}(\cdot)+g{\cal
L}^{1}_{t}(\cdot)=-{\rm i}[H(t),\cdot]+g\sum_{l\in
I}\Big{(}\Gamma_{l}(t)\cdot\Gamma_{l}^{*}(t)-\frac{1}{2}\big{\\{}\Gamma_{l}^{*}(t)\Gamma_{l}(t),\cdot\big{\\}}\Big{)},$
(2.1)

where the Hamiltonian part ${\cal L}_{t}^{0}(\cdot)$ is time-dependent, while
the dissipator $g{\cal L}^{1}_{t}(\cdot)$ is possibly constant. The
Lindbladian ${\cal L}_{t}^{[g]}$ acts in particular on the Banach space ${\cal
T}({\cal H})$, the set of trace class operators on ${\cal H}$ with norm
denoted by $\|\cdot\|_{1}$. Further specialising, the Lindbladian acts on the
set of density matrices or states, i.e. positive trace class operators of
trace one, in the Schrödinger picture we adopt here.

A special case of interest for which the dissipator depends on time is that of
dephasing Lindbladians characterized by $\Gamma_{l}(t)=F_{l}(H(t))$, where
$F_{l}:{\mathbb{R}}\rightarrow{\mathbb{C}}$ is some smooth function, for each
$l\in I$, see [AFGG1]. Among other things, dephasing Lindbladians enjoy the
following properties for each $t$ fixed:

$\displaystyle{\,\rm Ker\,}{\cal L}_{t}^{[g]}(\cdot)={\,\rm
Ker\,}[H(t),\cdot]\ \ \mbox{in }\ {\cal T}({\cal H}),$
$\displaystyle[\Gamma_{j}(t),P(t)]=0,\ \ \forall\ \mbox{spectral projector}\
P(t)\ \mbox{of}\ H(t).$ (2.2)

We shall work on ${\cal T}({\cal H})$, unless stated otherwise, and the
corresponding operator norm of ${\cal A}\in{\cal B}({\cal T}({\cal H}))$ will
be denoted by $\|{\cal A}\|_{\tau}$. In particular, for any $A\in{\cal
B}({\cal H})$, the maps on ${\cal T}({\cal H})\ni\rho$ given by ${\cal
A}_{l}:\rho\mapsto A\rho$ and ${\cal A}_{r}:\rho\mapsto\rho A$ belong to
${\cal B}({\cal T}({\cal H}))$ and have norms satisfying $\|{\cal
A}_{\\#}\|_{\tau}\leq\|A\|$, $\\#\in\\{l,r\\}$, where $\|\cdot\|$ denotes the
operator norm on ${\cal B}({\cal H})$. For ${\cal A},{\cal B}$ two operators
in ${\cal B}({\cal B}({\cal H}))$, we will denote their composition by ${\cal
A}\circ{\cal B}$, or simply ${\cal A}{\cal B}$ if no risk of confusion arises.

For $\varepsilon>0$, $g\geq 0$, we consider the Lindblad equation

$\left\\{\begin{matrix}\varepsilon\dot{\rho}=({\cal L}_{t}^{0}+g{\cal
L}^{1}_{t})(\rho),&t\in[0,1],\cr\rho(0)=\rho_{0},\hfill&\phantom{x}\rho\in{\cal
T}({\cal H}),\end{matrix}\right.$ (2.3)

in the adiabatic and small coupling regimes, characterised by
$(\varepsilon,g)\rightarrow(0,0)$.

We recall here the main properties of the solutions to (2.3). As is well
known, see [D1, Li] and is recalled in [AFGG1, H] for example, for each fixed
$t\in[0,1]$, the one-parameter family $({\rm e}^{s{\cal L}^{[g]}_{t}})_{s\geq
0}$ considered on ${\cal T}({\cal H})$ forms a norm continuous semigroup of
completely positive and trace preserving (CPTP) applications, which are
contraction operators. Consequently, denoting by $({\cal U}(t,s))_{0\leq s\leq
t\leq 1}$ the two-parameter propagator associated to (2.3),

$\displaystyle\left\\{\begin{matrix}\varepsilon\partial_{t}{\cal
U}(t,s)=({\cal L}_{t}^{0}+g{\cal L}^{1}_{t})({\cal U}(t,s)),\cr{\cal
U}(s,s)={\mathbb{I}},\ \ 0\leq s\leq t\leq 1,\hfill\end{matrix}\right.$ (2.4)

it follows from Thm X.70 in [RS] for example that the propagator is a
contraction:

$\displaystyle\|{\cal U}(t,s)\rho\|_{1}\leq\|\rho\|_{1},\phantom{2}\ \ $
$\displaystyle\forall\rho\in{\cal T}({\cal H}),\ \forall 1\geq t\geq s\geq 0.$
(2.5)

In particular we have $\|{\cal U}(t,s)\|_{\tau}=1$, since ${\cal U}(t,s)$ is
trace preserving.
While we shall stick to the bounded case, note that unbounded Hamiltonians
and/or dissipators could also be accommodated, [D1, J2, FFFS], for example.

We suppose that the spectrum of the Hamiltonian is separated into several
disjoint subsets, which corresponds to the familiar gap assumption of the
adiabatic theory.

Spec
For $2\leq d<\infty$, there exists $G>0$ such that for all $t\in[0,1]$, the
spectrum of $H(t)$, $\sigma(H(t))$, satisfies

$\sigma(H(t))=\cup_{1\leq j\leq d}\,\sigma_{j}(t),\ \ \ \
\inf_{t\in[0,1],1\leq j\neq k\leq d}{\rm
dist}(\sigma_{j}(t),\sigma_{k}(t))\geq G>0.$ (2.6)

Accordingly, we introduce the corresponding self-adjoint spectral projectors
on ${\cal H}$ for $1\leq j\leq d$

$P_{j}(t)=-\frac{1}{2{\rm i}\pi}\oint_{\gamma_{j}}(H(t)-z)^{-1}dz,$ (2.7)

where $\gamma_{j}\in\rho(H(t))$ is a positively oriented simple loop
encircling $\sigma_{j}(t)$ which contains no element of
$\sigma(H(t))\setminus\sigma_{j}(t)$ in its interior, i.e. ${\rm
int}\,\gamma_{j}\cap\sigma(H(t))=\sigma_{j}(t)$.
Moreover, $P_{j}:t\mapsto P_{j}(t)$ is $C^{\infty}$ since $H$ is, and

$\displaystyle P_{j}(t)P_{k}(t)=\delta_{jk}P_{j}(t),\ \ \ \sum_{1\leq j\leq
d}P_{j}(t)={\mathbb{I}}.$ (2.8)

Note that for all $t\in[0,1]$ and $1\leq j\leq d$, $P_{j}(t)$ belongs to
${\,\rm Ker\,}{\cal L}_{t}^{0}$, i.e. ${\cal L}_{t}^{0}(P_{j}(t))\equiv 0$.
Following [K1], we introduce the operator on ${\cal H}$

$K(t)=\sum_{1\leq j\leq d}P^{\prime}_{j}(t)P_{j}(t)=-\sum_{1\leq j\leq
d}P_{j}(t)P^{\prime}_{j}(t),$ (2.9)

and the corresponding parallel transport, or Kato, operator on ${\cal H}$
solution to

$\displaystyle\left\\{\begin{matrix}\partial_{t}W(t,s)=K(t)W(t,s),\hfill\\\
W(s,s)=\mathbb{I},\ \ 0\leq s,t\leq 1.\end{matrix}\right.$ (2.10)

It is unitary and satisfies the well known intertwining relation

$W(t,s)P_{j}(s)=P_{j}(t)W(t,s),$ (2.11)

whose proof is based on the fact that for any smooth projector
$P(t)=P^{2}(t)$, $P(t)P^{\prime}(t)P(t)\equiv 0$, see [K1, K2, Kr]. Note that
the propagator $(W(t,s))_{0\leq s,t\leq 1}$ is actually well defined and
invertible for any set of projectors that satisfy (2.8) in a Banach space
framework, and it satisfies the intertwining relation (2.11) for all $1\leq
j\leq d$.

To keep technicalities to a minimum in this presentation section, we state the
main results of the paper in their leading order formulations, and under
simple assumptions. As will be mentioned along the way, some results are
corollaries of more general statements to be found in later sections. The last
paragraph of the present section indicates the locations in the manuscript
where the proofs of the results and their generalisations are to be found.

###  2.1 Perturbative regime $g\ll\varepsilon\ll 1$

Our first result describes the modification of the adiabatic transition
probabilities between the spectral subspaces $P_{j}(0){\cal H}$ at time zero
and $P_{k}(t){\cal H}$ at time $t$, for $j\neq k$, induced by the presence of
the dissipator $g{\cal L}^{1}_{t}$ in the regime $g\ll\varepsilon$.

Pick an initial state $\rho_{j}\in{\cal T}({\cal H})$ such that
$\rho_{j}=P_{j}(0)\rho_{j}P_{j}(0)$ and denote by ${\cal U}^{0}(t,s)$ the
solution to (2.4) with $g=0$. In absence of dissipator, the transition
probability considered reads ${\,\rm Tr}(P_{k}(t){\cal U}^{0}(t,0)(\rho_{j}))$
and is of order $\varepsilon^{2}$, see Proposition 3.14 and Remark 3.15. In
case both spectral projectors involved are associated to a (potentially
degenerate) eigenvalue, $\sigma_{j}(t)=\\{e_{j}(t)\\}$,
$\sigma_{k}(t)=\\{e_{k}(t)\\}$, one has the explicit expression (3.41) for
$j\neq k$

$\displaystyle{\,\rm Tr}(P_{k}(t){\cal
U}^{0}(t,0)(\rho_{j}))=\varepsilon^{2}{\,\rm
Tr}\Big{\\{}\frac{P_{k}(t)P_{k}^{\prime}(t)\tilde{\rho}_{j}(t)P_{k}^{\prime}(t)P_{k}(t)}{(e_{j}(t)-e_{k}(t))^{2}}\Big{\\}}+{\cal
O}(\varepsilon^{3}),\ \mbox{where}$
$\displaystyle\tilde{\rho}_{j}(t)=W(t,0)\rho_{j}W(0,t).$ (2.12)

Note that due to (2.11)
$\tilde{\rho}_{j}(t)=P_{j}(t)\tilde{\rho}_{j}(t)P_{j}(t)$.

When the dissipator term is turned on we have, in the perturbative regime:

###### Theorem 2.1

Assume Reg and Spec with $\sigma_{j}(t)=\\{e_{j}(t)\\}$ for all $t\in[0,1]$,
and consider a state $\rho_{j}=P_{j}(0)\rho_{j}P_{j}(0)$. Then, the solution
to (2.4) satisfies for $j\neq k$, as $(\varepsilon,g)\rightarrow(0,0)$ with
$g/\varepsilon\rightarrow 0$,

$\displaystyle{\,\rm Tr}(P_{k}(t){\cal U}(t,0)(\rho_{j}))=$
$\displaystyle{\,\rm Tr}(P_{k}(t){\cal U}^{0}(t,0)(\rho_{j}))$
$\displaystyle+\frac{g}{\varepsilon}\sum_{l\in I}\int_{0}^{t}{\,\rm
Tr}(P_{k}(s)\Gamma_{l}(s)\tilde{\rho}_{j}(s)\Gamma_{l}^{*}(s)P_{k}(s))ds+{\cal
O}(g+g^{2}/\varepsilon^{2}),$ (2.13)

with $\tilde{\rho}_{j}(t)=W(t,0)\rho_{j}W(0,t)$.

Further assuming that for $j\neq k$, $\sigma_{j}(t)=\\{e_{j}(t)\\}$ and
$\sigma_{k}(t)=\\{e_{k}(t)\\}$ for all $t\in[0,1]$, we have in the same
regime,

$\displaystyle{\,\rm Tr}(P_{k}(t){\cal U}(t,0)(\rho_{j}))=$ $\displaystyle\
\varepsilon^{2}{\,\rm
Tr}\Big{\\{}\frac{P_{k}(t)P_{k}^{\prime}(t)\tilde{\rho}_{j}(t)P_{k}^{\prime}(t)P_{k}(t)}{(e_{j}(t)-e_{k}(t))^{2}}\Big{\\}}$
$\displaystyle+\frac{g}{\varepsilon}\sum_{l\in I}\int_{0}^{t}{\,\rm
Tr}(P_{k}(s)\Gamma_{l}(s)\tilde{\rho}_{j}(s)\Gamma_{l}^{*}(s)P_{k}(s))ds+{\cal
O}(g+\varepsilon^{3}+g^{2}/\varepsilon^{2}),$ (2.14)

with $\tilde{\rho}_{j}(t)=W(t,0)\rho_{j}W(0,t)$.

The physical interpretation is that in this regime, the dissipator contributes
to the adiabatic transition probabilities of order $\varepsilon^{2}$ by a
history dependent perturbative term of order $g/\varepsilon$; see also Theorem
2.5 below.

###### Remark 2.2

i) The correction to the transition probability due to the dissipator is non
negative.
ii) If we drop the assumption $\sigma_{j}(t)=\\{e_{j}(t)\\}$, formula (2.1)
still holds with $\tilde{\rho}_{j}(s)$ replaced by a state
$\tilde{\rho}_{j}(s,\varepsilon)$ that depends on $\varepsilon$ and also
satisfies
$\tilde{\rho}_{j}(s,\varepsilon)=P_{j}(t)\tilde{\rho}_{j}(s,\varepsilon)P_{j}(t)$,
see (3.34). In any case, if $P_{j}(0)$ is of finite rank and
$\rho_{j}=P_{j}(0)/\dim(P_{j}(0))$, then (2.1) holds with
$\tilde{\rho}_{j}(t)=P_{j}(t)/\dim(P_{j}(0)).$
iii) The transition probability from $P_{j}(0){\cal H}$ to $P_{j}(t){\cal H}$
reads

$\displaystyle{\,\rm Tr}(P_{j}(t){\cal U}(t,0)(\rho_{j}))=$
$\displaystyle{\,\rm Tr}(P_{j}(t){\cal U}^{0}(t,0)(\rho_{j}))$
$\displaystyle-\frac{g}{\varepsilon}\sum_{l\in I}\int_{0}^{t}{\,\rm
Tr}(({\mathbb{I}}-P_{j}(s))\Gamma_{l}(s)\tilde{\rho}_{j}(s)\Gamma_{l}^{*}(s)({\mathbb{I}}-P_{j}(s)))ds+{\cal
O}(g+g^{2}/\varepsilon^{2}).$ (2.15)

iv) In case $g=\varepsilon^{3}$, both contributions in (2.1) are of order
$\varepsilon^{2}$ and the error term is ${\cal O}(\varepsilon^{3})$.
v) If $\varepsilon^{3}\ll g\ll\varepsilon$, the dissipator contribution takes
over, with arbitrary slow decay

$\displaystyle{\,\rm Tr}(P_{k}(t){\cal
U}(t,0)(\rho_{j}))=\frac{g}{\varepsilon}\sum_{l\in I}\int_{0}^{t}{\,\rm
Tr}(P_{k}(s)\Gamma_{l}(s)\tilde{\rho}_{j}(s)\Gamma_{l}^{*}(s)P_{k}(s))ds+{\cal
O}(g+\varepsilon^{2}+g^{2}/\varepsilon^{2}).$ (2.16)

vi) If $g\ll\varepsilon^{3}$, one recovers the adiabatic result to leading
order

$\displaystyle{\,\rm Tr}(P_{k}(t){\cal U}(t,0)(\rho_{j}))=$ $\displaystyle\
\varepsilon^{2}{\,\rm
Tr}\Big{\\{}\frac{P_{k}(t)P_{k}^{\prime}(t)\tilde{\rho}_{j}(t)P_{k}^{\prime}(t)P_{k}(t)}{(e_{j}(t)-e_{k}(t))^{2}}\Big{\\}}+{\cal
O}(\varepsilon^{3}).$ (2.17)

vii) If ${\cal L}_{t}^{[g]}$ is dephasing, the contribution of the dissipator
vanishes, due to (2). This is keeping with Thm 18 of [AFGG1] which yields
transition probabilities of order $\varepsilon g$ with our notations; see also
[AFGG2].
viii) Finally, Theorem 2.1 is a consequence of Theorem 2.5 stated below.

While Theorem 2.1 focuses on transition probabilities, we also provide higher
order approximations of the full propagator ${\cal U}(t,s)$ in Propositions
3.9 and 7.5. The full formulations are too involved for this presentation
section and we limit ourselves here to the leading order expression stated as
Theorem 2.5.

Any state $\rho\in{\cal T}({\cal H})$ can be written as

$\rho=\sum_{1\leq n,m\leq d}P_{n}(t)\rho P_{m}(t),$ (2.18)

with off diagonal elements, or coherences, $P_{n}(t)\rho P_{m}(t)$, for $n\neq
m\in\\{1,\dots,d\\}$, and diagonal elements, or populations, $P_{n}(t)\rho
P_{n}(t)$, for $n\in\\{1,\dots,d\\}$. The extraction of the diagonal part of
$\rho$ is obtained by the action of the projector ${\cal P}_{0}(t)$ on ${\cal
B}({\cal H})$ defined for any $A\in{\cal B}({\cal H})$ by

${\cal P}_{0}(t)(A)=\sum_{1\leq n\leq d}P_{n}(t)AP_{n}(t).$ (2.19)

We take advantage of the fact that the superoperator ${\cal P}_{0}(t)$ acts on
states in ${\cal T}({\cal H})$ in the same way as its dual acts on observables
in ${\cal B}({\cal H})$. When acting on ${\cal T}({\cal H})$, ${\cal
P}_{0}(t)$ is a CPTP map characterised by its Kraus operators. In case
$\sigma_{k}(t)=\\{e_{k}(t)\\}$ for all $1\leq k\leq d$ and all $t\in[0,1]$,
${\cal P}_{0}(t)$ coincides with the spectral projector onto ${\,\rm
Ker\,}{\cal L}_{t}^{0}$.

Then, observe that under Reg, the projector ${\cal P}_{0}(t)$ given by (2.19)
is smooth in trace norm. Consider the parallel transport operator ${\cal
W}_{0}(t,s)$, $0\leq t,s\leq 1$, associated to ${\cal P}_{0}(t)$ via the
equation

$\displaystyle\left\\{\begin{matrix}\partial_{t}{\cal W}_{0}(t,s)=[{\cal
P}_{0}^{\prime}(t),{\cal P}_{0}(t)]{\cal W}_{0}(t,s),\\\ {\cal
W}_{0}(s,s)={\mathbb{I}}\hfill\end{matrix}\right.$ (2.20)

that satisfies, , see [Kr], the intertwining relation

${\cal P}_{0}(t){\cal W}_{0}(t,s)={\cal W}_{0}(t,s){\cal P}_{0}(s)$ (2.21)

and the propagation relation for all $0\leq r,s,t\leq 1$

${\cal W}_{0}(t,s){\cal W}_{0}(s,r)={\cal W}_{0}(t,r).$ (2.22)

The operator ${\cal W}_{0}(t,s)$ enjoys further properties:

###### Lemma 2.3

Assume Reg and Spec Then, for any $0\leq s,t\leq 1$, the operator ${\cal
W}_{0}(t,s){\cal P}_{0}(s)$ is a CPTP map on ${\cal T}({\cal H})$, and ${\cal
W}_{0}(t,s)$ maps ${\,\rm Ran}{\cal P}_{0}(s)$ to ${\,\rm Ran}{\cal P}_{0}(t)$
isometrically in trace norm. Moreover,

${\cal W}_{0}(t,s){\cal P}_{0}(s)(\rho)=W(t,s){\cal P}_{0}(s)(\rho)W(s,t),$
(2.23)

where $W(t,s)$ is the Kato operator defined by (2.10).

###### Remark 2.4

The first statements are shown in [AFGG1], and (2.23) is proven below in lemma
3.5.

We are now ready to give the approximation of ${\cal U}(t,0)$:

###### Theorem 2.5

Assume Reg and Spec with $\sigma_{k}(t)=\\{e_{k}(t)\\}$ for all $1\leq k\leq
d$ and all $t\in[0,1]$. Then the solution to (2.4) satisfies, as
$(\varepsilon,g)\rightarrow(0,0)$ with $g/\varepsilon\rightarrow 0$,

$\displaystyle{\cal U}(t,0){\cal P}_{0}(0)$ $\displaystyle={\cal
U}^{0}(t,0){\cal P}_{0}(0)+\frac{g}{\varepsilon}\int_{0}^{t}{\cal
P}_{0}(t){\cal W}_{0}(t,s){\cal P}_{0}(s){\cal L}_{s}^{1}{\cal
W}_{0}(s,0){\cal P}_{0}(0)ds+{\cal O}(g+(g/\varepsilon)^{2})$ (2.24)
$\displaystyle={\cal P}_{0}(t){\cal W}_{0}(t,0){\cal
P}_{0}(0)+\frac{g}{\varepsilon}\int_{0}^{t}{\cal P}_{0}(t){\cal
W}_{0}(t,s){\cal P}_{0}(s){\cal L}_{s}^{1}{\cal W}_{0}(s,0){\cal
P}_{0}(0)ds+{\cal O}(\varepsilon+(g/\varepsilon)^{2}),$

where, for all $A\in{\cal B}({\cal H})$,

${\cal W}_{0}(t,s){\cal P}_{0}(s)(A)={\cal P}_{0}(t){\cal W}_{0}(t,s){\cal
P}_{0}(s)(A)=\sum_{1\leq n\leq d}W(t,s)P_{n}(s)AP_{n}(s)W(s,t),$ (2.25)

with $W(t,s)$ the Kato operator (2.10).

###### Remark 2.6

i) The first statement compares ${\cal U}(t,0)$ with the Hamiltonian evolution
${\cal U}^{0}(t,0)$, while the second one uses a leading order approximation
of the latter, hence the different error terms.
ii) The projector ${\cal P}_{0}(t)$ on the left of the integral term in (2.24)
shows that the coherences of the correction due to the dissipator vanish to
leading order. Actually, as soon as $\sigma_{j}(t)=\\{e_{j}(t)\\}$, we have
for any $n\neq m$ in the regime $g\ll\varepsilon\ll 1$,

$P_{n}(t){\cal U}(t,0)(\rho_{j})P_{m}(t)=P_{n}(t){\cal
U}^{0}(t,0)(\rho_{j})P_{m}(t)+{\cal O}(g+(g/\varepsilon)^{2}),$ (2.26)

see Lemma 3.12.
iii) Moreover, under the assumptions of Theorem 2.5, Proposition 3.14 shows
that $P_{n}(t){\cal U}^{0}(t,0)(\rho_{j})P_{m}(t)$ is of order
$\varepsilon^{2}$ if $n$ and $m$ are different from $j$, while it is of order
$\varepsilon$ if $n$ or $m$ equals $j$.

###  2.2 Slow drive regime $\varepsilon\ll g\ll 1$

We consider now larger time scales $1/\varepsilon$ for the drive, which
implies the adiabatic dynamics within the instantaneous eigenspaces of the
driving Lindbladian ${\cal L}_{t}^{[g]}$ will dominate. In order to tackle
this regime for $g$ small, we will need more precise spectral information on
${\cal L}_{t}^{[g]}$ that require working in a simpler setup. In particular,
the next result holds under the assumption that ${\cal H}$ is finite
dimensional and that $\sigma(H(t))$ is generic in the following sense:

Gen
$\bullet$ $\dim{\cal H}=d$, $1<d<\infty$.
$\bullet$ $\forall t\in[0,1]$, $\sigma(H(t))=\\{e_{1}(t),\cdots,e_{d}(t)\\}$
is simple and the Bohr frequencies $\\{e_{j}(t)-e_{k}(t)\\}_{1\leq j\neq k\leq
d}$ are distinct.

Note that Spec holds with $\sigma_{j}(t)=\\{e_{j}(t)\\}$ if Gen is satisfied.

Let $\\{\varphi_{j}\\}_{1\leq j\leq d}$ be a fixed orthonormal basis of
eigenvectors of $H(0)$. We consider $\\{\varphi_{j}(t)\\}_{1\leq j\leq d}$,
the orthonormal basis of smooth eigenvectors of $H(t)$ defined by

$\varphi_{j}(t)=W(t,0)\varphi_{j},\ \ \mbox{s.t.} \ \
\langle\varphi_{j}(t)|\varphi_{j}^{\prime}(t)\rangle\equiv 0,$ (2.27)

see (2.10) and (2.11). Therefore

$P_{j}(t)=|\varphi_{j}(t)\rangle\langle\varphi_{j}(t)|\ \ \mbox{and}\ \
H(t)=\sum_{1\leq j\leq d}e_{j}(t)P_{j}(t).$ (2.28)

As a consequence of assumption Gen, ${\cal L}_{t}^{0}$ admits $0$ as a
$d-$fold eigenvalue, with

${\,\rm Ker\,}{\cal L}_{t}^{0}={\,\rm Span}\\{P_{j}(t),1\leq j\leq d\\},$
(2.29)

whereas all its other eigenvalues are purely imaginary and simple. The
spectral projector onto ${\,\rm Ker\,}{\cal L}_{t}^{0}$ is the projector
${\cal P}_{0}(t)$ introduced in (2.19). The splitting of the eigenvalue $0$ of
${\cal L}_{t}^{0}$ by the addition of the dissipator $g{\cal L}_{t}^{1}$ is
thus governed to leading order in $g$ by the operator on ${\cal B}({\cal H})$

$\tilde{\cal L}_{t}^{1}={\cal P}_{0}(t)\circ{\cal L}_{t}^{1}\circ{\cal
P}_{0}(t).$ (2.30)

See (4.15) for the matrix form of $\tilde{\cal L}_{t}^{1}|_{{\,\rm Ker\,}{\cal
L}_{t}^{0}}$ in the ordered basis $\\{P_{1}(t),\dots,P_{d}(t)\\}$.

We assume that the splitting induced by $g{\cal L}_{t}^{1}$ is maximal in the
following sense

Split

$\bullet$ For all $t\in[0,1]$, the spectrum of the restriction of $\tilde{\cal
L}^{1}_{t}$ to ${{\,\rm Ker\,}{\cal L}_{t}^{0}}$ is simple.

Hypothesis Split is generic in the sense that in absence of very specific
symmetry, the dissipator ${\cal L}_{t}^{1}$ fully lifts the degeneracy of the
eigenvalue zero of ${\cal L}_{t}^{0}$, which is equivalent to saying that the
matrix representation of $\tilde{\cal L}_{t}^{1}|_{{\,\rm Ker\,}{\cal
L}_{t}^{0}}$ (4.15) has simple eigenvalues. It implies that for each fixed
$t\in[0,1]$ and $g>0$ small enough, ${\cal L}_{t}^{[g]}$ has one dimensional
kernel, so that the (Cesáro) limit as $s\rightarrow\infty$ of the semigroup
${e^{s{\cal L}_{t}^{[g]}}}$ converges to the projector onto that kernel. This
is true for the example worked out in Section 6, and for the simple quantum
reset models considered in [HJ], section 2.1, in particular.

Consequently, for $g>0$ small enough, the spectrum of the Lindbladian ${\cal
L}_{t}^{[g]}$ is simple, see [K2], with a non trivial kernel. Moreover, thanks
to Remark 4.2 below, the real parts of all the $d-1$ eigenvalues of order $g$
are strictly negative for $g>0$ small enough. The simplicity of the spectrum
of ${\cal L}_{t}^{[g]}$ for small $g>0$ allows us to construct a propagator
$({\cal V}(t,s))_{0\leq s\leq t\leq 1}$, which possesses the intertwining
property with all spectral projectors of ${\cal L}_{t}^{[g]}$, and approches
$({\cal U}(t,s))_{0\leq s\leq t\leq 1}$ under the sole condition
$\varepsilon\ll g\ll 1$: it satisfies for $g>0$ small enough and all $0\leq
s\leq t\leq 1$

$\|{\cal U}(t,s)-{\cal V}(t,s)\|_{\tau}={\cal O}(\varepsilon/g).$ (2.31)

The explicit description of ${\cal V}(t,s)$, which depends on the spectral
data of ${\cal L}_{t}^{[g]}$ and $\varepsilon$, is too involved for this
presentation section, and we refer the reader to Proposition 4.4 for more
details.

We present a statement that holds under the supplementary condition
$\varepsilon\ll g\ll\sqrt{\varepsilon.}$ Our second result describes the
leading order of the density matrix ${\cal U}(t,0)(P_{j}(0))$, which is
characterised by $\tilde{\cal L}^{1}_{t}$, in the slow drive regime.

###### Theorem 2.7

Assume Reg, Gen and Split. Then, for any fixed $0<t\leq 1$, and $j\neq k$, the
solution to (2.4) satisfies for $(\varepsilon,g)\rightarrow(0,0)$ with
$\varepsilon/g\rightarrow 0$ and $g^{2}/\varepsilon\rightarrow 0$,

${\cal U}(t,0)(P_{j}(0))=\tilde{\nu}_{0}(t)+{\cal
O}(g^{2}/\varepsilon+\varepsilon/g),$ (2.32)

where $\tilde{\nu}_{0}(t)={\cal P}_{0}(t)(\tilde{\nu}_{0}(t))$ is determined
by $\tilde{\cal L}_{t}^{1}(\tilde{\nu}_{0}(t))=0$ and ${\,\rm
Tr}(\tilde{\nu}_{0}(t))=1$.

In physical terms, the statements above mean that in the regime considered,
the drive is so slow that the dissipator has time enough to determine the
instantaneous invariant state that the adiabatic dynamics selects, within the
kernel of the Hamiltonian part of the Lindbladian.

###### Remark 2.8

i) The extra constraint $g^{2}\ll\varepsilon$ stems from the fact that we only
retain $\tilde{\cal L}_{t}^{1}$ in the description of ${\cal
U}(t,0)(P_{j}(0))$.
ii) Accordingly, for any $1\leq k\leq d$, the transition probability to
$P_{k}(t)$ starting at $P_{j}(0)$ reads

${\,\rm Tr}(P_{k}(t){\cal U}(t,0)(P_{j}(0)))={\,\rm
Tr}(P_{k}(t)\tilde{\nu}_{0}(t))+{\cal O}(g^{2}/\varepsilon+\varepsilon/g),$
(2.33)

while the coherences all vanish to leading order, since
$\tilde{\nu}_{0}(t)={\cal P}_{0}(t)(\tilde{\nu}_{0}(t))$.
iii) More precise results taking into account the other eigenstates of
$\tilde{\cal L}^{1}_{t}$ can be found in Corollary 5.7.

###  2.3 Transition regime $g\ll\sqrt{\varepsilon}\ll 1$

The asymptotic expressions stated above require either $g\ll\varepsilon$ or
$\varepsilon\ll g$, and thus do not cover the transition regime where
$\varepsilon$ and $g$ are roughly of the same order. Our next result bridges
this gap: for an initial state $\rho_{j}=P_{j}(0)\rho_{j}P_{j}(0)\in{\cal
T}({\cal H})$, it provides an approximation of the evolved state ${\cal
U}(t,0)(\rho_{j})$ which holds as soon as $g\ll\sqrt{\varepsilon}\ll 1$, a
regime that covers partly both the perturbative and slow drive regimes.
Moreover, this result only requires the spectral assumption Spec with
$\sigma_{j}(t)=\\{e_{j}(t)\\}$, for all $1\leq j\leq d$, regardless of the
dimensions of the spectral projectors $P_{j}(t)$ of $H(t)$.

Let $\tilde{\cal L}_{t}^{1}={\cal P}_{0}(t){\cal L}_{t}^{1}{\cal P}_{0}(t)$
and consider $(\tilde{\Psi}_{\delta}(t,s))_{0\leq s\leq t\leq 1}$, defined for
$\delta>0$ by

$\displaystyle\left\\{\begin{matrix}\delta\partial_{t}\tilde{\Psi}_{\delta}(t,s)={{\cal
W}_{0}}(0,t)\tilde{\cal L}_{t}^{1}{\cal
W}_{0}(t,0)\tilde{\Psi}_{\delta}(t,s),\\\
\tilde{\Psi}_{\delta}(s,s)={\mathbb{I}}.\hfill\end{matrix}\right.$ (2.34)

We call $\tilde{\Psi}_{\delta}(t,s)$ the reduced dynamics, since (2.21)
implies $[\tilde{\Psi}_{\delta}(t,s),{\cal P}_{0}(0)]\equiv 0$ .

The following Theorem shows that $\tilde{\Psi}_{\delta}(t,s)$ with
$\delta=\varepsilon/g$ provides an approximation of ${\cal U}(t,0){\cal
P}_{0}(0)$ in the transition regime $g\ll\sqrt{\varepsilon}\ll 1$:

###### Theorem 2.9

Assume Reg and Spec with $\sigma_{j}(t)=\\{e_{j}(t)\\}$ for all $1\leq j\leq
d$ and all $t\in[0,1]$. Then, for all $0\leq t\leq 1$, as
$(\varepsilon,g)\rightarrow 0$ with $g^{2}/\varepsilon\rightarrow 0$, we have
for the solution of (2.4)

${\cal U}(t,0){\cal P}_{0}(0)={\cal
W}_{0}(t,0)\tilde{\Psi}_{\varepsilon/g}(t,0){\cal P}_{0}(0)+{\cal
O}(\varepsilon+g+g^{2}/\varepsilon).$ (2.35)

Consequently, for any state $\rho_{j}=P_{j}(0)\rho_{j}P_{j}(0)\in{\cal
T}({\cal H})$ and any $1\leq j,k\leq d$, $0\leq t\leq 1$, we have in the same
regime,

$\displaystyle{\,\rm Tr}\\{P_{k}(t){\cal U}(t,0)(\rho_{j})\\}$
$\displaystyle={\,\rm
Tr}\\{P_{k}(0)\tilde{\Psi}_{\varepsilon/g}(t,0)(\rho_{j})\\}+{\cal
O}(\varepsilon+g+g^{2}/\varepsilon).$ (2.36)

In this transition regime, the dissipator is not strong enough to ensure
instantaneous relaxation of the dynamics over the adiabatic time scale, while
the coherences are still suppressed by the slow drive.

###### Remark 2.10

i) The coherences vanish to leading order since

${\cal W}_{0}(t,0)\tilde{\Psi}_{\varepsilon/g}(t,0){\cal P}_{0}(0)={\cal
P}_{0}(t){\cal W}_{0}(t,0)\tilde{\Psi}_{\varepsilon/g}(t,0){\cal P}_{0}(0).$
(2.37)

ii) The result holds in particular for $g=\varepsilon$ in which case we have

${\cal U}(t,0)(\rho_{j})={\cal
W}_{0}(t,0)\tilde{\Psi}_{1}(t,0)(\rho_{j})+{\cal O}(\varepsilon),$ (2.38)

where the reduced dynamics is parameter free and thus of order 1.
iii) For any $\delta>0$, the maps $\tilde{\Psi}_{\delta}(t,0){\cal P}_{0}(0)$
and ${\cal W}_{0}(t,0)\tilde{\Psi}_{\delta}(t,0){\cal P}_{0}(0)$ are CPTP, see
Corollary 5.4.
iv) In case $\varepsilon\ll g\ll\sqrt{\varepsilon}$, the reduced dynamic is
itself in an adiabatic regime in the parameter $\delta=\varepsilon/g$, which
allows us to recover the slow drive regime as shown in Corollaries 5.6, 5.7,
further assuming Gen and Split. Whereas for $g\ll\varepsilon\ll 1$ we fall
back on the perturbative regime, see Corollary 5.9, under the present
hypotheses.
v) In Section 6, the reduced dynamics $\tilde{\Psi}_{\varepsilon/g}(t,0)$ is
computed explicitly for a two-level system, under mild symmetry assumptions on
the jump operators $\Gamma_{l}(t)$.
vi) Finally, under Gen, the reduced dynamics can be interpreted as the
transition matrix of an associated classical Markov process, see Lemma 5.5.

The rest of the paper is organised as follows. In the next section, we
consider the perturbative regime, making use of Dyson series. To study this
series, we revisit methods in the adiabatic analysis of evolution equations,
which leads to Proposition 3.9, the main technical result of this section. The
statements of Theorems 2.1 and 2.5 are leading order consequences of this
result, as explained at the very end of that section. Section 4 is devoted to
the slow drive regime, starting with the spectral analysis of the Lindbladian
for $g$ small, to get the approximation (2.31) of the Lindbladian evolution
for $\varepsilon\ll g$ stated as Proposition 4.4, which is proven there. The
transition regime is finally addressed in Section 5, where the reduced
dynamics is introduced and analysed. In particular, various asymptotic values
of $\delta=\varepsilon/g$ allowed by the condition $g\ll\sqrt{\varepsilon}$
are considered as Corollaries 5.4, 5.6, 5.7 and 5.9 of Proposition 5.2, the
main technical result of the section. For instance, Theorem 2.9 corresponds to
Corollary 5.4, while Corollaries 5.6 and 5.7 are shown to yield the statements
of Theorem 2.7, whereas Corollary 5.9 partially recovers results in the
perturbative regime. An application to a two-level system, or Qubit,
illustrating those results is worked out in Section 6, while Section 7 is
devoted to higher order generalisations of the perturbative regime results.
The paper closes with a technical appendix gathering some proofs.

##  3 Perturbative regime $g\ll\varepsilon$

In the regime $g\ll\varepsilon$, the dissipator of the Lindbladian can be
considered a perturbation of the Hamiltonian part, so that a head on approach
using Dyson series in the interaction picture is useful. We work here under
Reg and Spec

Let $({\cal U}^{0}(t,s))_{(t,s)\in{\mathbb{R}}^{2}}$ be the propagator on
${\cal T}({\cal H})$ solution to the equation in ${\cal T}({\cal H})$

$\displaystyle\left\\{\begin{matrix}\varepsilon\partial_{t}{\cal
U}^{0}(t,s)={\cal L}_{t}^{0}({\cal U}^{0}(t,s)),\hfill\cr{\cal
U}^{0}(s,s)={\mathbb{I}},\ \ (t,s)\in[0,1]^{2}.\end{matrix}\right.$ (3.1)

Introducing $(U(t,s))_{(t,s)\in{\mathbb{R}}^{2}}$, the unitary Schrödinger
propagator, solution to the equation in ${\cal H}$

$\left\\{\begin{matrix}{\rm
i}\varepsilon\partial_{t}U(t,s)=H(t)U(t,s),\hfill\cr U(s,s)={\mathbb{I}},\ \
(t,s)\in[0,1]^{2},\end{matrix}\right.$ (3.2)

we check that for any $\rho\in{\cal T}({\cal H})$

${\cal U}^{0}(t,s)(\rho)=U(t,s)\rho U^{*}(t,s),$ (3.3)

showing that ${\cal U}^{0}(t,s)$ is actually unitarily implemented on ${\cal
T}({\cal H})$ and on ${\cal B}({\cal H})$, and is well defined for any $0\leq
s,t\leq 1$.

The integral form of (2.4) and the definition of ${\cal U}^{0}(t,s)$ yield

${\cal U}(t,r)={\cal U}^{0}(t,r)+\frac{g}{\varepsilon}\int_{r}^{t}{\cal
U}^{0}(t,s)\circ{\cal L}^{1}_{s}\circ{\cal U}(s,r)ds,\ \ \forall 0\leq r\leq
t\leq 1.$ (3.4)

By iteration we have for $0\leq s\leq t\leq 1$, $N\geq 1$, and with the
convention $s_{0}=t$,

$\displaystyle{\cal U}(t,s)-{\cal U}^{0}(t,s)$ (3.5)
$\displaystyle=\sum_{n=1}^{N}(g/\varepsilon)^{n}$
$\displaystyle\int_{s}^{t}\int_{s}^{s_{1}}\dots\int_{s}^{s_{n-1}}{\cal
U}^{0}(t,s_{1})\circ{\cal L}^{1}_{s_{1}}\circ{\cal
U}^{0}(s_{1},s_{2})\circ{\cal L}^{1}_{s_{2}}\dots\circ{\cal
L}^{1}_{s_{n}}\circ{\cal U}^{0}(s_{n},s)ds_{n}\dots ds_{2}ds_{1}$
$\displaystyle+(g/\varepsilon)^{N+1}$
$\displaystyle\int_{s}^{t}\int_{s}^{s_{1}}\dots\int_{s}^{s_{N}}{\cal
U}^{0}(t,s_{1})\circ{\cal L}^{1}_{s_{1}}\circ{\cal
U}^{0}(s_{1},s_{2})\circ{\cal L}^{1}_{s_{2}}\dots\circ{\cal
L}^{1}_{s_{N+1}}\circ{\cal U}(s_{N+1},s)ds_{N+1}\dots ds_{2}ds_{1}$
$\displaystyle=\sum_{n=1}^{\infty}(g/\varepsilon)^{n}$
$\displaystyle\int_{s}^{t}\int_{s}^{s_{1}}\dots\int_{s}^{s_{n-1}}{\cal
U}^{0}(t,s_{1})\circ{\cal L}^{1}_{s_{1}}\circ{\cal
U}^{0}(s_{1},s_{2})\circ{\cal L}^{1}_{s_{2}}\dots\circ{\cal
L}^{1}_{s_{n}}\circ{\cal U}^{0}(s_{n},s)ds_{n}\dots ds_{2}ds_{1}.$

The convergence is in the norm operator sense on ${\cal T}({\cal H})$ for the
second expression. In particular, with $\sup_{0\leq s\leq 1}\|{\cal
L}^{1}_{s}\|_{\cal T}=L_{1}$, the norm of the term of order $n$ is bounded
above by $((g/\varepsilon)L_{1}(t-s))^{n}/n!$, since ${\cal U}^{0}(t,s)$ is
isometric.

###### Remark 3.1

The Dyson series converges, irrespectively of the value of the ratio
$g/\varepsilon$.

Thanks to (3.3), the adiabatic approximation of ${\cal U}^{0}(t,s)$ is easily
obtained from that of the Schrödinger propagator $U(t,s)$, under the spectral
hypotheses Spec on the Hamiltonian.

###  3.1 Adiabatic toolbox

The gap hypothesis Spec and the assumed regularity in time of $H(t)$ ensure
the existence of a unitary propagator on ${\cal H}$,
$(V(t,s))_{(t,s)\in[0,1]^{2}}$ defined by

$\left\\{\begin{matrix}{\rm i}\varepsilon\partial_{t}V(t,s)=(H(t)+{\rm
i}\varepsilon K(t))V(t,s),\\\ V(s,s)={\mathbb{I}},\ \
(t,s)\in[0,1]^{2},\hfill\end{matrix}\right.$ (3.6)

where $K(t)$ is given in (2.9). The adiabatic theorem of quantum mechanics
reads, see e.g. [K1, N1, ASY],

###### Lemma 3.2

Under Reg and Spec, there exists $c$ such that for any $0\leq s,t\leq 1$, and
$\varepsilon>0$, the solutions to (3.2) and (3.6) satisfy

$\displaystyle\|U(t,s)-V(t,s)\|\leq c\varepsilon,$ (3.7)

where $V$ possesses the intertwining property

$V(t,s)P_{j}(s)=P_{j}(t)V(t,s),\ \forall 1\leq j\leq d.$ (3.8)

###### Remark 3.3

Thanks to the second point of assumption Reg, we have for $s=0$

$\|U(t,0)-V(t,0)\|\leq ct\varepsilon.$ (3.9)

As a direct consequence, the transition amplitude from the subspace
$P_{j}(s){\cal H}$ at time $s$, to the subspace $P_{k}(t){\cal H}$ at time
$t$, $j\neq k$, is of order $\varepsilon$, as $\varepsilon\rightarrow 0$:

$\|P_{k}(t)U(t,s)P_{j}(s)\|=\|P_{k}(t)V(t,s)P_{j}(s)\|+{\cal
O}(\varepsilon)={\cal O}(\varepsilon).$ (3.10)

Estimate (3.7) in Lemma 3.2 is based on an integration by parts procedure that
will be used below in various situations, and even generalised in Section 7.
Therefore, the argument is presented in Appendix in a general abstract setup
as Lemma 8.2, from which the proof of (3.7) follows. The classical
intertwining property (3.8), obtained by observing that both sides are
solutions to the same differential equation in $t$, with initial condition
$P_{j}(s)$ at $t=s$ can be found in ([K1, K2, Kr]). For later purposes, we
also introduce here a useful decomposition of the operator $V(t,s)$:

Let $W(t,s)$ be the Kato operator defined by (2.10) and
$\Phi_{\varepsilon}(t,s)$ be the dynamical phase operator defined by

$\displaystyle\left\\{\begin{matrix}{\rm
i}\varepsilon\partial_{t}\Phi_{\varepsilon}(t,s)=W^{-1}(t,0)H(t)W(t,0)\Phi_{\varepsilon}(t,s),\\\
\Phi_{\varepsilon}(s,s)=\mathbb{I},\ \ 0\leq s,t\leq
1.\hfill\end{matrix}\right.$ (3.11)

The dynamical phase operator $\Phi_{\varepsilon}(t,s)$ describes the evolution
within the spectral subspaces of $H(t)$, and thus depends on $\varepsilon$. As
the Kato operator, it is well defined in a Banach space framework for bounded
generator, and its key property is that for all $1\leq j\leq d$

$[\Phi_{\varepsilon}(t,s),P_{j}(0)]\equiv 0.$ (3.12)

In case Spec holds with $\sigma_{j}(t)=\\{e_{j}(t)\\}$,
$\Phi_{\varepsilon}(t,s)P_{j}(0)=P_{j}(0){\rm e}^{-\frac{{\rm
i}}{\varepsilon}\int_{s}^{t}e_{j}(r)dr}$, and if this assumption holds for all
$1\leq j\leq d$,

$\Phi_{\varepsilon}(t,s)=\sum_{j=1}^{d}P_{j}(0){\rm e}^{-\frac{{\rm
i}}{\varepsilon}\int_{s}^{t}e_{j}(r)dr},$ (3.13)

which justifies the name of the operator. The link between $V$, $W$ and
$\Phi_{\varepsilon}$ reads, see e.g. [K1, JP2, J2]

###### Lemma 3.4

Under Reg and Spec, one has

$V(t,s)=W(t,0)\Phi_{\varepsilon}(t,s)W^{-1}(s,0),\ \ \forall\ 0\leq t,s\leq
1.$ (3.14)

Under these assumptions, the operators $V(t,s),W(t,s),\Phi_{\varepsilon}(t,s)$
are all unitary.

In turn, the adiabatic approximation (3.7) $V(t,s)$ of $U(t,s)$ on ${\cal H}$
provides an approximation of ${\cal U}^{0}(t,s)$ on ${\cal T}({\cal H})$ up to
${\cal O}(\varepsilon)$. For $\varepsilon>0$, define the isometric operator on
${\cal T}({\cal H})$ (and on ${\cal B}({\cal H})$)

${\cal V}^{0}(t,s)(\rho)=V(t,s)\rho V^{*}(t,s)=V(t,s)\rho V(s,t),\ \
\rho\in{\cal T}({\cal H}).$ (3.15)

Then, for $c$ given in equation (3.7), we get

$\displaystyle\|{\cal U}^{0}(t,s)-{\cal V}^{0}(t,s)\|_{\tau}\leq
2c\varepsilon,$ (3.16)

and the same holds for the operator norm on ${\cal B}({\cal H})$.

To get a better grasp on ${\cal V}^{0}(t,s)$, we proceed by proving here
(2.23) which specifies the action of ${\cal W}_{0}(t,s)$ on the range of
${\cal P}_{0}(s)$:

###### Lemma 3.5

For ${\cal W}_{0}(t,s)$ defined by (2.20) and ${\cal P}_{0}(t)$ by (2.19), we
have for any $\rho\in{\cal B}({\cal H})$,

${\cal W}_{0}(t,s)\circ{\cal P}_{0}(s)(\rho)=W(t,s){\cal
P}_{0}(s)(\rho)W(s,t),$ (3.17)

where $W(t,s)$ is the Kato operator defined by (2.10). Moroever, ${\cal
W}_{0}(t,s)$ is trace preserving.

###### Remark 3.6

i) The particular case $\rho=P_{j}(s)$ which gives ${\cal
W}_{0}(t,s)(P_{j}(s))=P_{j}(t)$ can be found [AFGG1].
ii) If the projectors are all one dimensional, i.e.
$P_{j}(t)=|\varphi_{j}(t)\rangle\langle\varphi_{j}(t)|$, $1\leq j\leq d$, we
have

$\displaystyle{\cal W}_{0}(t,s)\circ{\cal P}_{0}(s)(\rho)=\sum_{1\leq k\leq
d}\langle\varphi_{k}(s)|\rho\varphi_{k}(s)\rangle P_{k}(t).$ (3.18)

Proof: The fact that ${\cal W}_{0}(t,s)$ is trace preserving follows from
${\,\rm Tr}(({\mathbb{I}}-{\cal P}_{0}(t))(A))=0$ for all $A\in{\cal T}({\cal
H})$, and (2.21), together with (3.17) and the fact that ${\cal P}_{0}(s)$ is
trace preserving. The identity is proven by checking that its two sides
satisfy the same differential equation (2.20) with initial condition ${\cal
P}_{0}(s)$ at $t=s$. Considering first the argument $P_{k}(s)\rho
P_{k}(s)={\cal P}_{0}(s)(P_{k}(s)\rho P_{k}(s))$ in place of $\rho$ in (3.17),
the RHS reads $W(t,s)P_{k}(s)\rho P_{k}(s)W(s,t)=P_{k}(t)W(t,s)\rho
W(s,t)P_{k}(t)$ so that with (2.9),

$\displaystyle\partial_{t}\Big{(}W(t,s)P_{k}(s)\rho P_{k}(s)W(s,t)\Big{)}=$
$\displaystyle[K(t),W(t,s)P_{k}(s)\rho P_{k}(s)W(s,t)]$ (3.19)
$\displaystyle=$ $\displaystyle P_{k}^{\prime}(t)P_{k}(t)W(t,s)\rho
W(s,t)P_{k}(t)+P_{k}(t)W(t,s)\rho W(s,t)P_{k}(t)P_{k}^{\prime}(t).$

Then, making use of

$\displaystyle{\cal P}_{0}^{\prime}(t)(\rho)$ $\displaystyle=\sum_{1\leq j\leq
d}P_{j}^{\prime}(t)\rho P_{j}(t)+P_{j}(t)\rho P_{j}^{\prime}(t),$ (3.20)

we get ${\cal P}_{0}(t){\cal P}_{0}^{\prime}(t)(P_{k}(t)W(t,s)\rho
W(s,t)P_{k}(t))\equiv 0$ and

$\displaystyle{\cal P}_{0}^{\prime}(t){\cal P}_{0}(t)(P_{k}(t)W(t,s)\rho$
$\displaystyle W(s,t)P_{k}(t))$ (3.21)
$\displaystyle=P_{k}^{\prime}(t)P_{k}(t)W(t,s)\rho
W(s,t)P_{k}(t)+P_{k}(t)W(t,s)\rho W(s,t)P_{k}(t)P_{k}^{\prime}(t),$

showing the result for the argument $P_{k}(s)\rho P_{k}(s)$. It remains to sum
over $1\leq k\leq d$ to end the proof. $\Box$
As a consequence, we get the following expression for the adiabatic
approximation (3.16) of ${\cal U}^{0}(t,s)(\rho_{j})$ where the state
$\rho_{j}=P_{j}(s)\rho P_{j}(s)$ and $P_{j}(s)$ is associated to a permanently
degenerate eigenvalue:

###### Lemma 3.7

Under Reg and Spec with $\sigma_{j}(t)=\\{e_{j}(t)\\}$ for all $t\in[0,1]$,
for any state $\rho_{j}=P_{j}(s)\rho P_{j}(s)={\cal P}_{0}(s)(\rho_{j})$ it
holds

$\displaystyle{\cal U}^{0}(t,s)(\rho_{j})={\cal V}^{0}(t,s)(\rho_{j})+{\cal
O}(\varepsilon),\ \ \mbox{where}$ $\displaystyle{\cal
V}^{0}(t,s)(\rho_{j})={\cal W}_{0}(t,s)(\rho_{j})=W(t,s)\rho_{j}W(s,t).$
(3.22)

###### Remark 3.8

If $\sigma_{j}(t)=\\{e_{j}(t)\\}$ for all $1\leq j\leq d$ and all $t\in[0,1]$,
then for any $0\leq s\leq t\leq 1$,

${\cal P}_{0}(t){\cal W}_{0}(t,s){\cal P}_{0}(s)={\cal P}_{0}(t){\cal
V}^{0}(t,s)={\cal V}^{0}(t,s){\cal P}_{0}(s).$ (3.23)

Proof: The first estimate is (3.16). Then, Lemma 3.4 yields the expression

$V(t,s)=W(t,0)\Phi_{\varepsilon}(t,s)W(0,s)\ \mbox{ with}\ \
\Phi_{\varepsilon}(t,s)P_{j}(0)=P_{j}(0){\rm e}^{-\frac{{\rm
i}}{\varepsilon}\int_{0}^{t}e_{j}(r)dr},$ (3.24)

since $\sigma_{j}(t)=\\{e_{j}(t)\\}$. Therefore, ${\cal V}^{0}(t,s)$ given by
(3.15) with $\rho_{j}=P_{j}(s)\rho_{j}P_{j}(s)$ and the intertwining relation
(2.11) make the phases ${\rm e}^{\pm\frac{{\rm
i}}{\varepsilon}\int_{s}^{t}e_{j}(r)dr}$ disappear, which justifies the last
two identities. $\Box$
In particular, if $\dim P_{j}(0)<\infty$, then $P_{j}(t)$ belongs to ${\cal
T}({\cal H})$ for all $t\in[0,1]$ so that

$\displaystyle{\cal U}^{0}(t,s)(P_{j}(s))$ $\displaystyle=$ $\displaystyle
P_{j}(t)+{\cal O}(\varepsilon).$ (3.25)

Again, if $P_{j}(0)$ is not trace class, the estimates above hold in operator
norm.

###  3.2 Adiabatic Dyson expansion

We now apply the foregoing to the analysis of the Dyson series.

Equation (3.5) and the above yields the following estimate of the propagator
$({\cal U}(t,s))_{0\leq s\leq t\leq 1}$:

###### Proposition 3.9

Under assumptions Reg and Spec, for any $N\geq 1$, there exists $c<\infty$
(given in Lemma 3.2), such that for all $0\leq s\leq t\leq 1$ (with the
convention $s_{0}=t$ for $N=1$), for all $\varepsilon>0$, all $g\geq 0$, the
propagator ${\cal U}(t,s)\in{\cal B}({\cal T}({\cal H}))$ satisfies

$\displaystyle{\cal U}(t,s)$ $\displaystyle={\cal V}^{0}(t,s)$
$\displaystyle+\sum_{n=1}^{N}(g/\varepsilon)^{n}\int_{s}^{t}\int_{s}^{s_{1}}\dots\int_{s}^{s_{n-1}}{\cal
V}^{0}(t,s_{1})\circ{\cal L}^{1}_{s_{1}}\circ{\cal
V}^{0}(s_{1},s_{2})\circ{\cal L}^{1}_{s_{2}}\dots\circ{\cal
L}^{1}_{s_{n}}\circ{\cal V}^{0}(s_{n},s)ds_{n}\dots ds_{2}ds_{1}$
$\displaystyle+R_{N+1}(t,s,\varepsilon,g)$ (3.26)

where, with $L_{1}=\sup_{0\leq s\leq 1}\|{\cal L}_{s}^{1}\|_{\tau}$,

$\displaystyle\|R_{N+1}(t,s,\varepsilon,g)\|_{\tau}\leq 2c\varepsilon{\rm
e}^{2(t-s)L_{1}(1+2c\varepsilon)g/\varepsilon}+\frac{(L_{1}(t-s))^{N+1}}{(N+1)!}(g/\varepsilon)^{N+1}.$
(3.27)

In particular, if $g/\varepsilon\leq 1$ and $\varepsilon\leq 1/(2c)$,

$\displaystyle\|R_{N+1}(t,s,\varepsilon,g)\|_{\tau}$ $\displaystyle\leq 2{\rm
e}^{4L_{1}}\left(c\varepsilon+((t-s)g/\varepsilon)^{N+1}\right)$
$\displaystyle={\cal O}(\varepsilon+(g/\varepsilon)^{N+1}).$ (3.28)

###### Remark 3.10

i) The isometric operators ${\cal V}^{0}$ depends on $\varepsilon$ and
displays fast oscillations as $\varepsilon\rightarrow 0$.
ii) Keeping ${\cal U}^{0}$ instead of ${\cal V}^{0}$ in the first term of the
RHS of (3.9), the estimate on the remainder reads

$\|R_{N+1}(t,s,\varepsilon,g)\|_{\tau}={\cal O}(g+(g/\varepsilon)^{N+1})).$
(3.29)

iii) In case $s=0$, on can replace $c$ by $tc$, according to Remark 3.3.

Proof: The first estimate follows by replacing ${\cal U}^{0}$ by its
approximation ${\cal V}^{0}$ in each term of the Dyson series, and collecting
the different contributions to the error terms. With

$\Delta=\sup_{0\leq s\leq t\leq 1}\|{\cal U}^{0}(t,s)-{\cal
V}^{0}(t,s)\|_{\tau}\leq 2c\varepsilon,$ (3.30)

the trace norm of the difference of the term of order $n\geq 1$ in (3.5) with
that of order $n$ in (3.9) is bounded above by

$\displaystyle(g/\varepsilon)^{n}\frac{((t-s)L_{1})^{n}}{n!}\sum_{1\leq k\leq
n+1}\begin{pmatrix}n+1\\\
k\end{pmatrix}\Delta^{k}\leq(g/\varepsilon)^{n}\frac{((t-s)L_{1})^{n}}{n!}\sum_{0\leq
j\leq n}\begin{pmatrix}n+1\\\ j+1\end{pmatrix}\Delta^{j+1}$
$\displaystyle\leq(g/\varepsilon)^{n}\frac{((t-s)L_{1})^{n}}{n!}\Delta(n+1)\sum_{0\leq
j\leq n}\begin{pmatrix}n\\\
j\end{pmatrix}\Delta^{j}\leq(g/\varepsilon)^{n}\frac{((t-s)L_{1})^{n}}{n!}\Delta
2^{n}(1+\Delta)^{n}.$ (3.31)

Summing over all $n\in{\mathbb{N}}$ yields the first term in (3.27). The
second term stems from the term of order $N+1$ in (3.5). The second estimate
is a consequence of $g/\varepsilon\leq 1$, $t-s\leq 1$, $1+2c\varepsilon\leq
2$ and $\frac{\alpha^{m}}{m!}\leq e^{\alpha}$, for all $m\geq 1$, $\alpha>0$.
$\Box$

Specialising to the leading order term in $g/\varepsilon$, and taking into
account Remark 3.10 ii) above, we get

###### Corollary 3.11

Under the assumptions of Proposition 3.9, for $\varepsilon\leq 1/(2c)$ and
$g/\varepsilon\leq 1$,

$\displaystyle{\cal U}(t,s)$ $\displaystyle={\cal
U}^{0}(t,s)+\frac{g}{\varepsilon}\int_{s}^{t}{\cal V}^{0}(t,s_{1})\circ{\cal
L}^{1}_{s_{1}}\circ{\cal V}^{0}(s_{1},s)ds_{1}+{\cal
O}(g+g^{2}/\varepsilon^{2}).$ (3.32)

This expression will provide an explicit leading order correction to the
transition probability between spectral subspaces driven by the purely
Hamiltonian dynamics, due to the dissipator.

Consider a state $\rho_{j}=P_{j}(0)\rho_{j}P_{j}(0)\in{\cal T}({\cal H})$ and
recall the definition (2.1) of the dissipator

${\cal L}^{1}_{t}(\cdot)=\sum_{l\in
I}\Gamma_{l}(t)\cdot\Gamma_{l}^{*}(t)-\frac{1}{2}\\{\Gamma_{l}^{*}(t)\Gamma_{l}(t),\cdot\\}.$
(3.33)

The transition probability between $P_{j}(0){\cal H}$ and $P_{k}(t){\cal H}$,
$j\neq k$, induced by the Lindbladian dynamics (2.3) reads ${\,\rm
Tr}(P_{k}(t){\cal U}(t,0)(\rho_{j}))$. Using (3.15) and Lemma 3.2, we have

$\tilde{\rho}_{j}(t,\varepsilon)={\cal
V}^{0}(s,0)(\rho_{j})=V(s,0)\rho_{j}V(s,0)=P_{j}(s)V(s,0)\rho_{j}V(0,s)P_{j}(s),$
(3.34)

so that with $P_{j}(t)P_{k}(t)=0$ and the cyclicity of the trace,

$\displaystyle{\,\rm Tr}(P_{k}(t){\cal U}(t,0)$
$\displaystyle(\rho_{j}))={\,\rm Tr}(P_{k}(t){\cal U}^{0}(t,0)(\rho_{j}))$
(3.35) $\displaystyle+\frac{g}{\varepsilon}\int_{0}^{t}{\,\rm
Tr}(P_{k}(t){\cal V}^{0}(t,s)\circ{\cal L}^{1}_{s}\circ{\cal
V}^{0}(s,0)(\rho_{j}))ds+{\cal O}(g+g^{2}/\varepsilon^{2})$
$\displaystyle\phantom{xxxx}={\,\rm
Tr}(P_{k}(t)U(t,0)P_{j}(0)\rho_{j}P_{j}(0)U(0,t)P_{k}(t))$
$\displaystyle+\frac{g}{\varepsilon}\sum_{l\in I}\int_{0}^{t}{\,\rm
Tr}(P_{k}(s)\Gamma_{l}(s)V(s,0)\rho_{j}V(0,s)\Gamma_{l}^{*}(s)P_{k}(s))ds+{\cal
O}(g+g^{2}/\varepsilon^{2}).$

The first expression on the RHS yields the Hamiltonian adiabatic transition
probability between these subspaces, whereas the non-negative second term of
order $g/\varepsilon$ describes the effect of the environment. Note that in
case $P_{j}(0)$ is finite rank, choosing $\rho_{j}=P_{j}(0)/\dim(P_{j}(0))$
yields the simpler integrands

${\,\rm
Tr}(P_{k}(s)\Gamma_{l}(s)P_{j}(s)\Gamma_{l}^{*}(s)P_{k}(s))/\dim(P_{j}(0)),$
(3.36)

and by Lemma 3.7, $V(s,0)\rho_{j}V(0,s)=\tilde{\rho}_{j}(s)$ is independent of
$\varepsilon$ if $\sigma_{j}(t)=\\{e_{j}(t)\\}$.

Concerning coherences of the integral term in (3.32), we have the following
integration by parts result, whose proof is given in Appendix.

###### Lemma 3.12

Assume Reg, Spec and let $\rho_{j}=P_{j}(0)\rho_{j}P_{j}(0)$ be a state.
Suppose $\sigma_{j}(t)=\\{e_{j}(t)\\}$ for all $t\in[0,1]$. Then, for any
$1\leq n\neq m\leq d$, and all $\varepsilon>0$,

$\frac{g}{\varepsilon}P_{n}(t)\int_{0}^{t}{\cal V}^{0}(t,s)\circ{\cal
L}^{1}_{s}\circ{\cal V}^{0}(s,0)(\rho_{j})ds\,P_{m}(t)={\cal O}(g).$ (3.37)

Actually, a similar result holds for each term in (3.9). For simplicity, we
choose to express it under the supplementary condition
$\sigma_{k}(t)=\\{e_{k}(t)\\}$ for all $1\leq k\leq d$, and postpone its proof
to the Appendix. It will allow us to make contact with the reduced dynamics
later on.

###### Lemma 3.13

Assume Reg and Spec with $\sigma_{k}(t)=\\{e_{k}(t)\\}$ for all $1\leq k\leq
d$ and all $t\in[0,1]$. Then for all $n\geq 1$, there exists
$\varepsilon_{n}>0$ such that for $\varepsilon<\varepsilon_{n}$ and any state
$\rho\in{\cal T}({\cal H})$ (with $s_{0}=t$),

$\displaystyle\int_{0}^{t}\int_{0}^{s_{1}}\dots\int_{0}^{s_{n-1}}{\cal
V}^{0}(t,s_{1}){\cal L}^{1}_{s_{1}}{\cal V}^{0}(s_{1},s_{2}){\cal
L}^{1}_{s_{2}}\dots\cal L^{1}_{s_{n}}{\cal V}^{0}(s_{n},0)(\rho)ds_{n}\dots
ds_{2}ds_{1}$
$\displaystyle=\int_{0}^{t}\int_{0}^{s_{1}}\dots\int_{0}^{s_{n-1}}{\cal
W}^{0}(t,s_{1}){\cal P}_{0}(s_{1}){\cal L}^{1}_{s_{1}}\cdots{\cal
W}^{0}(s_{n-1},s_{n}){\cal P}_{0}(s_{n}){\cal L}^{1}_{s_{n}}{\cal
W}^{0}(s_{n},0){\cal P}_{0}(0)(\rho)ds_{n}\dots ds_{2}ds_{1}$
$\displaystyle\hskip 85.35826pt+{\cal O}_{n}(\varepsilon).$ (3.38)

Here ${\cal O}_{n}(\varepsilon)$ means a quantity bounded by
$C_{n}\varepsilon$, where $C_{n}$ depends on $n$. Consequently, under the
assumptions of Lemma 3.13, for all $N\geq 1$, in the regime
$g\ll\varepsilon\ll 1$, we have the generalisation of Theorem 2.5

$\displaystyle{\cal U}(t,0)={\cal U}^{0}(t,0)$ (3.39)
$\displaystyle+\sum_{n=1}^{N}\Big{(}\frac{g}{\varepsilon}\Big{)}^{n}\int_{0}^{t}\int_{0}^{s_{1}}\dots\int_{0}^{s_{n-1}}{\cal
W}^{0}(t,s_{1}){\cal P}_{0}(s_{1}){\cal L}^{1}_{s_{1}}\cdots{\cal
W}^{0}(s_{n-1},s_{n}){\cal P}_{0}(s_{n}){\cal L}^{1}_{s_{n}}{\cal
W}^{0}(s_{n},0){\cal P}_{0}(0)(\rho)ds_{n}\dots ds_{1}$ $\displaystyle+{\cal
O}_{N}(g+(g/\varepsilon)^{N+1}),$

where all integral terms are independent of $(\varepsilon,g)$. Note that the
error term in (3.39) is negligeable with respect to all explicit terms in the
regime $\varepsilon^{N/(N-1)}\ll g\ll\varepsilon$.

We proceed by recalling the asymptotics of the transition probability between
the spectral subspaces $P_{j}(0){\cal H}$ and $P_{k}(t){\cal H}$ under the
unitarily implemented evolution ${\cal U}^{0}(t,s)$ and of the coherences of
${\cal U}^{0}(t,s)(\rho_{j})$ depending on certain assumptions on the spectral
subsets $\sigma_{k}(t)$, $1\leq k\leq d$.

###### Proposition 3.14

Assume Reg and Spec and consider a state
$\rho_{j}=P_{j}(0)\rho_{j}P_{j}(0)\in{\cal T}({\cal H})$. Then, for
$\sigma_{j}(t)=\\{e_{j}(t)\\}$, the adiabatic transition probability between
$P_{j}(0){\cal H}$ and $P_{k}(t){\cal H}$, $k\neq j$, under the Hamiltonian
evolution is determined by the trace of

$\displaystyle P_{k}(t){\cal U}^{0}(t,0)(\rho_{j})P_{k}(t)$ (3.40)
$\displaystyle=-\frac{\varepsilon^{2}}{(2\pi)^{2}}\Big{\\{}P_{k}(t)\oint_{\gamma_{j}}R(t,z)P_{k}^{\prime}(t)R(t,z)dz\
\tilde{\rho}_{j}(t)\oint_{\gamma_{j}}R(t,z)P_{k}^{\prime}(t)R(t,z)dz\,P_{k}(t)\Big{\\}}+{\cal
O}(\varepsilon^{3}),$

where $\tilde{\rho}_{j}(t)=W(t,0)\rho_{j}W(0,t)$ and $R(t,z)=(H(t)-z)^{-1}$,
for $z\in\rho(H(t))$.

In case $\sigma_{j}(t)=\\{e_{j}(t)\\}$ and $\sigma_{k}(t)=\\{e_{k}(t)\\}$ for
all $t\in[0,1]$, we have

$\displaystyle P_{k}(t){\cal U}^{0}(t,0)(\rho_{j})P_{k}(t)$
$\displaystyle=\varepsilon^{2}\Big{\\{}\frac{P_{k}(t)P_{k}^{\prime}(t)\tilde{\rho}_{j}(t)P_{k}^{\prime}(t)P_{k}(t)}{(e_{j}(t)-e_{k}(t))^{2}}\Big{\\}}+{\cal
O}(\varepsilon^{3}).$ (3.41)

Further assuming $\sigma_{k}(t)=\\{e_{k}(t)\\}$, for all $1\leq k\leq d$, all
$t\in[0,1]$, the coherences read

$\displaystyle P_{n}(t){\cal U}^{0}(t,0)(\rho_{j})P_{m}(t)$
$\displaystyle=\varepsilon^{2}\Big{\\{}\frac{P_{m}(t)P_{m}^{\prime}(t)\tilde{\rho}_{j}(t)P_{n}^{\prime}(t)P_{n}(t)}{(e_{j}(t)-e_{m}(t))(e_{j}(t)-e_{n}(t))}\Big{\\}}+{\cal
O}(\varepsilon^{3}),\phantom{xxxxxx}\mbox{for }m\neq j,n\neq j,$
$\displaystyle P_{j}(t){\cal U}^{0}(t,0)(\rho_{j})P_{m}(t)$
$\displaystyle={\rm
i}\varepsilon\frac{\tilde{\rho}_{j}(t)P_{m}^{\prime}(t)P_{m}(t)}{e_{m}(t)-e_{j}(t)}+{\cal
O}(\varepsilon^{2}),\phantom{xxxxxxxxxxxxxxxxxxxxxxx}\mbox{for }m\neq j.$
(3.42)

###### Remark 3.15

In case the assumption $\sigma_{j}(t)=\\{e_{j}(t)\\}$ is dropped,
$\tilde{\rho}_{j}(t)$ must be replaced by the $\varepsilon-$dependent state

$\tilde{\rho}_{j}(t,\varepsilon)={\cal
V}^{0}(t,0)(\rho_{j})=V(t,0)\rho_{j}V(0,t)=P_{j}(t)\tilde{\rho}_{j}(t,\varepsilon)P_{j}(t)$
(3.43)

in (3.40), see (3.15). Similar integral expressions can be obtained for the
coherences in case condition $\sigma_{k}(t)=\\{e_{k}(t)\\}$ for all $1\leq
k\leq d$ does not hold.

The proof, making use of higher order adiabatic approximations is postponed to
Section 7.

At this point, the proof of Theorem 2.1 follows from (3.35) and Proposition
(3.14): recall that $\sigma_{j}(t)=\\{e_{j}(t)\\}$ implies
$V(t,0)\rho_{j}V(0,t)=W(t,0)\rho_{j}W(0,t)=\tilde{\rho}_{j}(t)$ by (3.15) and
Lemma 3.7, which yields eq. (2.1) from (3.35). Then, (2.1) follows directly
from (3.41).

Similarly, the proof of the first line of (2.24) in Theorem 2.5 is a
consequence of Corollary 3.11 for $s=0$, Lemma 3.12 and Remark 3.8. The second
line follows from Lemma 3.7 and $g\ll\varepsilon$.

##  4 Slow drive regime $g\gg\varepsilon$

As already mentioned in Section 2, the analysis of the regime
$g\gg\varepsilon$, is all the more accurate that we control the spectral
properties of the Lindbladian ${\cal L}^{[g]}_{t}={\cal L}^{0}_{t}+g{\cal
L}^{1}_{t}$. In this section, we provide a fairly explicit approximation of
${\cal U}(t,s)$ based on perturbation theory under the sole condition
$\varepsilon\ll g$, see Proposition 4.4, assuming the Hilbert spaces ${\cal
H}$ is finite dimensional and the Hamiltonian $H(t)$ has simple eigenvalues
that are generic in the sense of Gen.

We recall that for $H(t)=\sum_{1\leq j\leq d}P_{j}(t)e_{j}(t)$, the explicit
description of the approximate evolution operator $V(t,s)$ defined by (3.6)
provided by Lemma 3.4 holds with

$\Phi_{\varepsilon}(t,s)=\sum_{j=1}^{d}P_{j}(0){\rm e}^{-\frac{{\rm
i}}{\varepsilon}\int_{s}^{t}e_{j}(r)dr},$ (4.1)

irrespectively of the dimension of the projectors $P_{j}(t)$, see (3.13).
Under Reg and Gen, we consider $\\{\varphi_{j}(t)\\}_{1\leq j\leq d}$, the
canonical smooth orthonormal basis of eigenvectors of $H(t)$ defined in (2.27)
that satisfies

$\varphi_{j}(t)=W(t,0)\varphi_{j},\ \ \mbox{s.t.} \ \
\langle\varphi_{j}(t)|\varphi_{j}^{\prime}(t)\rangle\equiv 0,$ (4.2)

so that $P_{j}(t)=|\varphi_{j}(t)\rangle\langle\varphi_{j}(t)|$. Introducing
the Hilbert-Schmidt scalar product on ${\cal B}({\cal H})$

$\big{\langle}\big{\langle}A,B\big{\rangle}\big{\rangle}={\,\rm Tr}(A^{*}B),$
(4.3)

the vectors (4.2) yield in turn an instantaneous orthonormal eigenbasis of
${\cal L}_{t}^{0}$ with respect to
$\big{\langle}\big{\langle}\cdot,\cdot\big{\rangle}\big{\rangle}$:

###### Lemma 4.1

For all $t\in[0,1]$, the family of rank one operators on ${\cal H}$,
$\\{|\varphi_{j}(t)\rangle\langle\varphi_{k}(t)|\\}_{1\leq j,k\leq d}$, is a
smooth orthonormal basis of ${\cal B}({\cal H})$ equipped with
$\big{\langle}\big{\langle}\cdot,\cdot\big{\rangle}\big{\rangle}$ such that

$\displaystyle{\cal
L}_{t}^{0}(|\varphi_{j}(t)\rangle\langle\varphi_{k}(t)|)=-{\rm
i}(e_{j}(t)-e_{k}(t))|\varphi_{j}(t)\rangle\langle\varphi_{k}(t)|:=\lambda_{jk}(t)|\varphi_{j}(t)\rangle\langle\varphi_{k}(t)|$
(4.4)

Consequently, the spectrum of ${\cal L}^{0}_{t}$ sits on the imaginary axis,
is simple except for the eigenvalue $0$ which is $d-$fold degenerate, and

${\,\rm Ker\,}{\cal L}^{0}_{t}={\,\rm Span}\\{P_{j}(t),j=1,\dots,d\\}.$ (4.5)

The corresponding spectral decomposition is written as

${\cal L}_{t}^{0}=\sum_{j\neq k}\lambda_{jk}(t){\cal P}_{jk}(t)+0{\cal
P}_{0}(t),$ (4.6)

where

$\displaystyle{\cal P}_{jk}(t)(\rho)$ $\displaystyle=P_{j}(t)\rho
P_{k}(t)=\langle\varphi_{j}(t)|\rho\varphi_{k}(t)\rangle|\varphi_{j}(t)\rangle\langle\varphi_{k}(t)|$
$\displaystyle{\cal P}_{0}(t)(\rho)$ $\displaystyle=\sum_{1\leq j\leq
d}{P_{j}(t)\rho P_{j}(t)}.$ (4.7)

With ${\cal Q}_{0}(t)={\mathbb{I}}-{\cal P}_{0}(t)$, we have ${\cal
L}_{t}^{0}={\cal Q}_{0}(t)\circ{\cal L}_{t}^{0}={\cal L}_{t}^{0}\circ{\cal
Q}_{0}(t)$.

###  4.1 Perturbation theory of ${\cal L}_{t}^{[g]}$

We now address the spectral properties of ${\cal L}_{t}^{[g]}={\cal
L}_{t}^{0}+g{\cal L}^{1}_{t}$ in the perturbative regime $g\rightarrow 0$.
Before doing so, we recall that ${\cal L}_{t}^{[g]}$ being a Lindblad
operator, this imposes the following structural constraints on its spectrum,
for each $t\in[0,1]$:

$0\in\sigma({\cal L}_{t}^{[g]})=\overline{\sigma({\cal
L}_{t}^{[g]})}\subset\\{z\in{\mathbb{C}}\ |\Re z\leq 0\\},\ \ \forall\
g\in{\mathbb{R}}^{+}.$ (4.8)

Indeed, for fixed $t\in[0,1]$, $(e^{s{\cal L}_{t}^{[g]}})_{s\geq 0}$ being a
contraction semigroup on ${\cal T}({\cal H})$, it follows that
$\Re\sigma({\cal L}_{t}^{[g]})\leq 0$ and that all eigenvalues sitting on the
imaginary axis are semisimple, ${i.e.}$ there is no eigennilpotent (Jordan
block) corresponding to those eigenvalues in the spectral decomposition of
${\cal L}_{t}^{[g]}$. Since $e^{s{\cal L}_{t}^{[g]}}$ is a CPTP map for all
$s\geq 0$, it admits $1$ as an eigenvalue, see e.g. [Sch], so that $0$ is an
eigenvalue of the generator ${\cal L}_{t}^{[g]}$. The symmetry ${\cal
L}_{t}^{[g]}(\rho^{*})=({\cal L}_{t}^{[g]}(\rho))^{*}$ which holds for all
$\rho\in{\cal T}({\cal H})$, implies the symmetry of the spectrum, when
applied to eigenvectors of ${\cal L}_{t}^{[g]}$.

Therefore, following Chapter II §2 [K2] and dropping the variable $t$ from the
notation, we denote by ${\cal P}_{jk}^{[g]}$ the spectral projector of ${\cal
L}^{[g]}$ associated with the eigenvalues emanating from the unperturbed
eigenvalue $-{\rm i}(e_{j}-e_{k})$, and by ${\cal P}_{0}^{[g]}$ the spectral
projector on the set of eigenvalues emanating from the $d-$fold degenerate
unperturbed eigenvalue $0$, the so-called $\lambda-$group of eigenvalues, with
$\lambda=0$. Since ${\cal P}_{jk}$ is one dimensional, ${\cal P}_{j,k}^{[g]}$
is one dimensional and analytic in $g\in{\mathbb{C}}$, $|g|$ small enough. On
the other hand, $\dim{\cal P}_{0}=d$ implies that the degenerate eigenvalue
$0$ may split for non zero $g$ and while ${\cal P}_{0}^{[g]}$ is analytic in
$g\in{\mathbb{C}}$, for $|g|$ small enough, it might not be the case for the
projections on the individual eigenvalues emanating from $0$.

Let $\\{\lambda_{j}^{[g]}\\}_{0\leq j\leq m}$, $m\leq d-1$, be the set of
eigenvalues in the $0-$ group for $g\in{\mathbb{C}}\setminus\\{0\\}$ with
$|g|$ small enough. Each $\\{\lambda_{j}^{[g]}\\}_{0\leq j\leq m}$ is an
analytic functions of a (fractional) power of $g$ that tend to zero as
$g\rightarrow 0$ and may be permanently degenerate. For the structural reasons
recalled above, one of these eigenvalues, we denote by $\lambda_{0}^{[g]}$, is
identically equal to zero, $\lambda_{0}^{[g]}\equiv 0$, $\forall
g\in{\mathbb{C}}\setminus\\{0\\}$. In case $\lambda_{0}^{[g]}$ is degenerate,
it is semisimple.

Let us analyse the splitting of the $0$-group of eigenvalues. We have

${\cal P}_{0}^{[g]}=-\frac{1}{2i\pi}\int_{\gamma_{0}}({\cal
L}^{[g]}-z)^{-1}dz={\cal P}_{0}+g{\cal P}_{1}+{\cal O}(g^{2}),$ (4.9)

for $|g|$ small, where $\gamma_{0}$ is a circle of small radius, independent
of $g$, centered at the origin. We mention for completeness, that since $0$ is
a semisimple eigenvalue of ${\cal L}^{0}$,

${\cal P}_{1}=-{\cal P}_{0}{\cal L}^{1}{\cal S}_{0}-{\cal S}_{0}{\cal
L}^{1}{\cal P}_{0}={\cal P}_{0}{\cal P}_{1}{\cal Q}_{0}+{\cal Q}_{0}{\cal
P}_{1}{\cal P}_{0},$ (4.10)

where ${\cal S}_{0}$ is the reduced resolvent of ${\cal L}_{0}$ at $0$,
satisfying ${\cal S}_{0}{\cal P}_{0}={\cal P}_{0}{\cal S}_{0}=0$ and ${\cal
S}_{0}{\cal L}^{0}={\cal L}^{0}{\cal S}_{0}={\cal Q}_{0}$.

The analytic operator that describes the splitting reads

$\displaystyle{\cal P}_{0}^{[g]}{\cal L}^{[g]}{\cal P}_{0}^{[g]}$
$\displaystyle=({\cal P}_{0}+g{\cal P}_{1}+{\cal O}(g^{2}))({\cal
L}^{0}+g{\cal L}^{1})({\cal P}_{0}+g{\cal P}_{1}+{\cal O}(g^{2}))$
$\displaystyle=g{\cal P}_{0}{\cal L}^{1}{\cal P}_{0}+{\cal O}(g^{2}),$ (4.11)

where we used ${\cal L}_{0}{\cal P}_{0}={\cal P}_{0}{\cal L}_{0}=0$.

The restriction to ${{\,\rm Ker\,}{\cal L}^{0}}$ of the leading order term
$\tilde{\cal L}^{1}={\cal P}_{0}{\cal L}^{1}{\cal P}_{0}$ admits a matrix
representation $\tilde{L}$ in the ordered orthonormal basis of rank one
projectors $\\{P_{1},P_{2},\dots,P_{d}\\}$ of ${\cal P}_{0}{\cal B}({\cal H})$
whose elements are determined by the expressions (recall (4) and (3.33))

$P_{k}{\cal L}^{1}(P_{j})P_{k}=\sum_{l\in
I}\left(P_{k}\Gamma_{l}^{*}P_{j}\Gamma_{l}P_{k}-\delta_{jk}P_{k}\Gamma_{l}^{*}\Gamma_{l}P_{k}\right).$
(4.12)

We set $\tilde{L}=\sum_{l\in I}\tilde{L}(l)\in M_{d}({\mathbb{R}})$, where the
matrix elements $\tilde{L}(l)_{jk}$, $1\leq j,k\leq d$, read

$\displaystyle\tilde{L}(l)_{kj}=|\langle\varphi_{k}|\Gamma_{l}\varphi_{j}\rangle|^{2}-\delta_{jk}\|\Gamma_{l}\varphi_{k}\|^{2},$
(4.13)

so that for any $\rho={\cal P}_{0}(\rho)=\sum_{1\leq j\leq d}\rho_{j}P_{j}$,
with $\rho_{j}=\langle\varphi_{j}|\rho\varphi_{j}\rangle$,

$\tilde{\cal L}^{1}(\rho)={\cal P}_{0}{\cal L}^{1}{\cal
P}_{0}(\rho)=\sum_{1\leq j,k\leq d}P_{k}\tilde{L}_{kj}\rho_{j}.$ (4.14)

In other words,

$\displaystyle\tilde{\cal L}^{1}|_{{\,\rm Ker\,}{\cal
L}^{0}}\simeq\tilde{L}=\sum_{l\in
I}\begin{pmatrix}|\langle\varphi_{1}|\Gamma_{l}\varphi_{1}\rangle|^{2}-\|\Gamma_{l}\varphi_{1}\|^{2}&|\langle\varphi_{1}|\Gamma_{l}\varphi_{2}\rangle|^{2}&&|\langle\varphi_{1}|\Gamma_{l}\varphi_{d}\rangle|^{2}\cr|\langle\varphi_{2}|\Gamma_{l}\varphi_{1}\rangle|^{2}&|\langle\varphi_{2}|\Gamma_{l}\varphi_{2}\rangle|^{2}-\|\Gamma_{l}\varphi_{2}\|^{2}&&|\langle\varphi_{2}|\Gamma_{l}\varphi_{d}\rangle|^{2}\cr&&\ddots&\cr|\langle\varphi_{d}|\Gamma_{l}\varphi_{1}\rangle|^{2}&|\langle\varphi_{d}|\Gamma_{l}\varphi_{2}\rangle|^{2}&&|\langle\varphi_{d}|\Gamma_{l}\varphi_{d}\rangle|^{2}-\|\Gamma_{l}\varphi_{d}\|^{2}\end{pmatrix}.$
(4.15)

Note that the real matrices $\tilde{L}(l)$ have non negative off diagonal
elements and satisfy

$\sum_{1\leq j\leq d}\tilde{L}(l)_{jk}=0,$ (4.16)

so that the same properties hold for $\tilde{L}$. This is a reflection of the
fact that ${\cal L}^{1}$ being a lindbladian, we have ${\,\rm Tr}({\cal
L}^{1}(\rho))=0$ for any $\rho\in{\cal B}({\cal H})$, where for $\rho={\cal
P}_{0}(\rho)$,

${\,\rm Tr}({\cal L}^{1}(\rho))={\,\rm Tr}\\{{\cal P}_{0}{\cal L}^{1}{\cal
P}_{0}(\rho)\\}=\sum_{1\leq j,k\leq d}{\,\rm
Tr}(P_{k})\tilde{L}_{kj}\rho_{j}=\sum_{1\leq j,k\leq
d}\tilde{L}_{kj}\rho_{j}.$ (4.17)

In particular $0\in\sigma(\tilde{L})$. Sufficient and necessary conditions for
${\,\rm Ker\,}\tilde{L}$ to be one dimensional are given in [No] or [D2],
Chapter 12..

Note that in the language of classical Markov processes, (4.16) makes the
transpose of $\tilde{L}$ a time-dependent transition rate matrix, or
generator, of a Markov process, see e.g. [YZ]. We further comment on this in
section 5.2.

We suppose that the splitting induced by ${\cal L}^{1}$ is maximal by assuming
Split, i.e. that $\tilde{\cal L}^{1}_{t}|_{{\,\rm Ker\,}{\cal L}_{t}^{0}}$ has
simple spectrum.

###### Remark 4.2

Assumption Split and Gershgorin Theorem imply that for any $t\in[0,1]$ and
$g>0$ small enough, $\Re\sigma({\cal P}_{0}(t)^{[g]}{\cal L}_{t}^{[g]}{\cal
P}_{0}(t)^{[g]}){\setminus\\{0\\}}<0$.

Under this hypothesis on the efficiency of the dissipator, we have the
spectral decomposition

$\tilde{\cal L}^{1}_{t}=0\tilde{\cal
P}_{0}(t)+\sum_{j=1}^{d-1}\tilde{\lambda}_{j}(t)\tilde{\cal P}_{j}(t),$ (4.18)

where the distinct eigenvalues $\tilde{\lambda}_{j}(t)$ and eigenprojectors
$\tilde{\cal P}_{j}(t)$ are smooth in $t\in[0,1]$.

Assumption Split ensures the spectrum of ${\cal L}_{t}^{[g]}$ is simple for
small $g>0$, and its eigenprojectors are all regular as $g\rightarrow 0^{+}$,
despite ${\cal L}_{t}^{[g]}$ is not normal:

###### Proposition 4.3

Assume Reg, Gen, and Split. Then, there exists $g_{0}>0$ such that for all
$t\in[0,1]$, for all $g\in{\mathbb{C}}\setminus\\{0\\}$ with $|g|<g_{0}$,
${\cal L}_{t}^{[g]}$ admits $d$ distinct eigenvalues
$\\{\lambda_{j}^{[g]}(t)\\}_{0\leq j\leq d-1}$, with corresponding one
dimensional eigenprojectors ${\cal P}_{j0}^{[g]}(t)$, that are $C^{\infty}$ in
$t$ and analytic in $g$. Moreover, $\lambda_{0}^{[g]}(t)\equiv 0$, and
$\lim_{g\rightarrow 0}\lambda_{j}^{[g]}(t)/g=\tilde{\lambda}_{j}(t)$ and
$\lim_{g\rightarrow 0}{\cal P}_{j0}^{[g]}(t)=\tilde{\cal P}_{j}(t)$, where
$\\{\tilde{\lambda}_{j}(t)\\}$ and $\tilde{\cal P}_{j}(t)$ are the spectral
data (4.18) of $\tilde{\cal L}_{t}^{1}$ .

Proof: It is a direct consequence of analytic perturbation theory, since for
$t\in[0,1]$ fixed (omitted in the notation), the operator $\frac{1}{g}{\cal
P}_{0}^{[g]}{\cal L}^{[g]}{\cal P}_{0}^{[g]}$ for
$g\in{\mathbb{C}}\setminus\\{0\\}$ with $|g|<g_{0}$ admits an analytic
extension to $\\{g\in{\mathbb{C}},|g|<g_{0}\\}$ with term of order $g^{0}$
given by $\tilde{\cal L}^{1}$ thanks to (4.1), with $\sigma(\tilde{\cal
L}^{1}|_{{\,\rm Ker\,}{\cal L}^{0}})$ simple. $\Box$

###  4.2 Adiabatics and perturbation theory

Under the hypotheses of the previous proposition and for
$g\in{\mathbb{C}}\setminus\\{0\\}$, $|g|$ small enough, $\sigma({\cal
L}_{t}^{[g]})$ is simple and its spectral decomposition reads

${\cal L}_{t}^{[g]}=0{\cal P}_{00}^{[g]}(t)+\sum_{1\leq j\leq
d-1}\lambda_{j}^{[g]}(t){\cal P}_{j0}^{[g]}(t)+\sum_{1\leq j\neq k\leq
d}\lambda_{jk}^{[g]}(t){\cal P}_{jk}^{[g]}(t),$ (4.19)

with analytic data in $g$, where $\lambda_{jk}^{[0]}(t)=-{\rm
i}(e_{j}(t)-e_{k}(t))$, ${\cal P}_{jk}^{[0]}(t)={\cal P}_{jk}(t)$, see (4.6).
For $g>0$ $\sigma({\cal L}_{t}^{[g]})\setminus\\{0\\}\subset\\{z|\Re z<0\\}$.
Moreover

${\cal P}_{0}^{[g]}(t)=\sum_{j=0}^{d-1}{\cal P}_{j0}^{[g]}(t).$ (4.20)

Accordingly, for $0<g<g_{0}$ fixed, we introduce ${\cal V}(t,s)_{0\leq s\leq
t\leq 1}$ by

${\cal V}(t,s)={\cal W}(t,0)\Psi_{\varepsilon}(t,s){\cal W}^{-1}(s,0),$ (4.21)

in keeping with Lemma 3.4, where ${\cal W}(t,s)$ is the solution to

$\displaystyle\partial_{t}{\cal W}(t,s)=\Big{\\{}{\cal
P}_{00}^{[g]}{}^{\prime}(t){\cal P}_{00}^{[g]}(t)+\sum_{1\leq j\leq d-1}{\cal
P}_{j0}^{[g]}{}^{\prime}(t){\cal P}_{j0}^{[g]}(t)+\sum_{1\leq j\neq k\leq
d}{\cal P}_{jk}^{[g]}{}^{\prime}(t){\cal P}_{jk}^{[g]}(t)\Big{\\}}{\cal
W}(t,s),$ $\displaystyle\phantom{xxxxxxx}:={\cal K}^{[g]}_{t}{\cal W}(t,s),$
$\displaystyle{\cal W}(s,s)={\mathbb{I}},\ \ 0\leq s,t\leq 1,$ (4.22)

which satisfies the intertwining property with respect to all spectral
projectors of ${\cal L}_{t}^{[g]}$ by construction, and

$\Psi_{\varepsilon}(t,s)={\cal P}_{00}^{[g]}(0)+\sum_{1\leq j\leq d-1}{\cal
P}_{j0}^{[g]}(0){\rm
e}^{\int_{s}^{t}\lambda_{j}^{[g]}(u)du/\varepsilon}+\sum_{1\leq j\neq k\leq
d}{\cal P}_{jk}^{[g]}(0){\rm
e}^{\int_{s}^{t}\lambda_{jk}^{[g]}(u)du/\varepsilon}.$ (4.23)

Note that ${\cal W}(t,s)$ is independent of $\varepsilon$ and since its
generator is analytic in $g\in{\mathbb{C}}\setminus\\{0\\}$, for $|g|$ small
enough, we have

$\sup_{g_{0}>g>0}\|{\cal W}(t,s)\|\leq C_{W},$ (4.24)

uniformly in $0\leq s\leq t\leq 1$, as revealed by straightforward estimates
of the Dyson series expansion of the solution to (4.2). Also, $\Re\sigma({\cal
L}_{t}^{[g]})\leq 0$ for all $t\in[0,1]$, implies that for all $0\leq s\leq
t\leq 1$,

$\sup_{\varepsilon>0,g_{0}>g>0}\|\Psi_{\varepsilon}(t,s)\|\leq C_{\Psi},$
(4.25)

uniformly in $0\leq s\leq t\leq 1$. One checks that the operator ${\cal
V}(t,s)$ satisfies

$\displaystyle\left\\{\begin{matrix}\varepsilon\partial_{t}{\cal
V}(t,s)=({\cal L}_{t}^{[g]}+\varepsilon{\cal K}^{[g]}_{t}){\cal V}(t,s),\\\
{\cal V}(s,s)={\mathbb{I}},\hfill\end{matrix}\right.$ (4.26)

and,

$\sup_{\varepsilon>0,g>0}\|{\cal V}(t,s)\|_{\tau}\leq C_{V}.$ (4.27)

uniformly in $0\leq s\leq t\leq 1$. Recall also $\|{\cal U}(t,s)\|_{\tau}=1$.

As expected, $({\cal V}(t,s))_{0\leq s\leq t\leq 1}$ approximates the
propagator $({\cal U}(t,s))_{0\leq s\leq t\leq 1}$ solution to (2.4) in the
slow drive regime $\varepsilon\rightarrow 0$, $g\rightarrow 0$,
$\varepsilon\ll g$:

###### Proposition 4.4

Assume Reg, Gen and Split. Then, there exists $g_{0}>0$ and $C<\infty$ such
that for $g<g_{0}$, and all $0\leq s\leq t\leq 1$

$\|{\cal U}(t,s)-{\cal V}(t,s)\|_{\tau}\leq C\varepsilon/g.$ (4.28)

###### Remark 4.5

i) In case some eigenvalues are permanently degenerate in (4.19), the same
result holds, mutatis mutandis. This is the case for dephasing Lindbladians,
as proven in [AFGG1] for $g$ fixed; see also [J2] for results along these
lines in an analytic context.
ii) The condition $\varepsilon\ll g$ to get a useful approximation stems from
the operators ${\cal R}_{j}(B)$ and their derivatives in the integration by
parts formula that contain differences of eigenvalues at the denominators, see
(8.8), and hence have norms of order $1/g$.
iii) The approximation ${\cal V}(t,s)$ depends on both $g$ and $\varepsilon$
and requires the spectral data of ${\cal L}_{t}^{[g]}$.

Proof: This is a direct application of the integration by parts argument
presented in Appendix, Lemma 8.2, keeping track of the dependence in $g>0$ of
the estimates; one makes use of the regularity of the spectral data of ${\cal
L}_{t}^{[g]}$ as $g\rightarrow 0^{+}$ proven in Proposition 4.3, and of the
fact that both ${\cal U}(t,s)$ and ${\cal V}(t,s)$ are uniformly bounded in
$\varepsilon$ and $g$. More precisely, dropping the arguments in the notation,
in the expression provided by Lemma 8.2 for ${\cal U}-{\cal V}$ with ${\cal
X}={\cal U}$, ${\cal Y}={\cal V}$, ${\cal G}={\cal L}^{[g]}$, and ${\cal
K}={\cal K}^{[g]}$, the operators ${\cal X}$ and ${\cal Y}$ are uniformly
bounded in $\varepsilon$ and $g$. The other operators appearing in (8.9) only
depend on $g$ and involve ${\cal L}^{[g]}$, ${\cal K}^{[g]}$ and operators of
the form ${\cal R}_{j}({\cal B})$ and $\partial_{t}{\cal R}_{j}({\cal B})$,
where, see Remark 8.1,

${\cal R}_{j}({\cal B})=\sum_{k\neq j}\frac{{\cal P}_{j}{\cal B}{\cal
P}_{k}+{\cal P}_{k}{\cal B}{\cal P}_{j}}{g_{k}-g_{j}},$ (4.29)

with $\\{g_{j}\\}=\sigma({\cal G})=\sigma({\cal L}^{[g]})$ with corresponding
eigenprojectors ${\cal P}_{j}$, and ${\cal B}=[{\cal K}^{[g]},{\cal P}_{j}]$.
Note also

$\partial_{t}{\cal R}_{j}({\cal B})={\cal R}_{j}(\partial_{t}{\cal
B})+\sum_{k\neq j}(\partial_{t}(g_{j}-g_{k}))\frac{{\cal P}_{j}{\cal B}{\cal
P}_{k}+{\cal P}_{k}{\cal B}{\cal P}_{j}}{(g_{k}-g_{j})^{2}}+\frac{{\cal
P}_{j}^{\prime}{\cal B}{\cal P}_{k}+{\cal P}_{k}^{\prime}{\cal B}{\cal
P}_{j}+{\cal P}_{j}{\cal B}{\cal P}_{k}^{\prime}+{\cal P}_{k}{\cal B}{\cal
P}_{j}^{\prime}}{g_{k}-g_{j}}.$ (4.30)

By Proposition 4.3, ${\cal L}^{[g]}$, ${\cal K}^{[g]}$, ${\cal P}_{j}$ and
their derivatives are uniformly bounded as $g\rightarrow 0$, hence the same
holds for ${\cal B}$. Also, the denominators in (4.29), (4.30) never vanish
for small $g>0$, and there exists $c>0$, such that
$\inf_{t\in[0,1],g>0}|g_{j}-g_{k}|\geq cg$ if both $g_{j}$ and $g_{k}$ stem
from ${\,\rm Ker\,}{\cal L}^{0}$, while $\inf_{t\in[0,1],g>0}|g_{j}-g_{k}|\geq
c$ otherwise. In the former case,
$\sup_{t\in[0,1],g>0}|\partial_{t}(g_{j}-g_{k})/(g_{j}-g_{k})^{2}|\leq c/g$.
Altogether, this shows that $\max(\|{\cal R}_{j}({\cal
B})\|,\|\partial_{t}{\cal R}_{j}({\cal B})\|)={\cal O}(1/g)$, which implies in
turn $\|{\cal U}-{\cal V}\|={\cal O}(\varepsilon/g).$ $\Box$

The next task to get Theorem 2.7 is to make more explicit the dependence in
$(g,\varepsilon)$ of the result above in the regime $g\gg\varepsilon$. This
goal can actually be achieved as a Corollary of another approximation of
${\cal U}(t,s)$ derived in the next section under more general spectral
assumptions, in the transition regime $g\ll\sqrt{\varepsilon}\ll 1$, see
Corollaries 5.7 and 5.6.

##  5 Transition Regime $g\ll\sqrt{\varepsilon}\ll 1$

We work here under the assumptions Reg and Spec with
$\sigma_{j}(t)=\\{e_{j}(t)\\}$ for all $1\leq j\leq d$, all $t\in[0,1]$, so
that $H(t)=\sum_{1\leq k\leq d}e_{k}(t)P_{k}(t)$, where $\dim
P_{k}(t)\leq\infty$.

With $\lambda_{jk}(t)=-{\rm i}(e_{j}(t)-e_{k}(t))$, ${\cal
P}_{jk}(t)(\cdot)=P_{j}(t)\cdot P_{k}(t)$, see (4.4) and (4), and ${\cal
P}_{0}(t)(\cdot)=\sum_{1\leq k\leq d}P_{k}(t)\cdot P_{k}(t)$ given by (2.19),
we denote by $\\{\lambda_{1}(t),\lambda_{2}(t),\dots,\lambda_{r}(t)\\}$ the
non-zero distinct values in $\\{\lambda_{jk}(t)\\}_{1\leq j,k\leq d}$, where
$2\leq r\leq d(d-1)$. Accordingly we define $r$ projectors on ${\cal B}({\cal
H})$ by

$\displaystyle{\cal P}_{n}(t)$ $\displaystyle=\sum_{j\neq k\ {\rm
s.t.}\atop\lambda_{jk}(t)=\lambda_{n}(t)}{\cal P}_{jk}(t),\ \ \ \ 1\leq n\leq
r.$ (5.1)

Then, regardless of the dimension of the projectors $P_{k}(t)$, we have the
smooth spectral decomposition

${\cal L}_{t}^{0}=\sum_{1\leq n\leq r}\lambda_{n}(t){\cal P}_{n}(t)+0{\cal
P}_{0}(t),\ \ \mbox{and}\ \ \sigma({\cal
L}_{t}^{0})=\\{0\\}\cup\\{\lambda_{n}(t)\\}_{1\leq n\leq r}.$ (5.2)

The spectral projectors ${\cal P}_{n}(t)$ have arbitrary dimension, possibly
infinite. In particular,

${\,\rm Ker\,}{\cal L}_{t}^{0}=\Big{\\{}A\in{\cal B}({\cal H})\ {\rm s.t.}\
A=\sum\nolimits_{1\leq j\leq d}P_{j}(t)AP_{j}(t)\Big{\\}},$ (5.3)

the set of diagonal operators with respect to $\\{P_{j}(t)\\}_{1\leq j\leq
d}$.

Since we are interested in the transitions between spectral projectors of the
Hamiltonian, that belong to the kernel of the Lindbladian at zero coupling, we
focus on the evolution restricted to the projector ${\cal P}_{0}^{[g]}(t)$
associated with the piece of spectrum of ${\cal L}_{t}^{[g]}={\cal
L}_{t}^{0}+g{\cal L}_{t}^{1}$ a distance of order $g$ away from zero, see
(4.20), by perturbation theory.

Let ${\cal W}^{[g]}_{0}(t,s)$ be the Kato operator defined by

$\displaystyle\left\\{\begin{matrix}\partial_{t}{\cal W}^{[g]}_{0}(t,s)=[{\cal
P}_{0}^{[g]}{}^{\prime}(t),{\cal P}_{0}^{[g]}(t)]{\cal W}^{[g]}_{0}(t,s),\\\
{\cal W}^{[g]}_{0}(s,s)={\mathbb{I}},\hfill\end{matrix}\right.$ (5.4)

and $\Psi_{\varepsilon}^{[g]}(t,s)$, $0\leq s\leq t\leq 1$, be the dynamical
phase operator solution to

$\displaystyle\left\\{\begin{matrix}\varepsilon\partial_{t}\Psi^{[g]}_{\varepsilon}(t,s)={{\cal
W}^{[g]}_{0}}(0,t){\cal L}_{t}^{[g]}{\cal
W}^{[g]}_{0}(t,0)\Psi^{[g]}_{\varepsilon}(t,s),\\\
\Psi^{[g]}_{\varepsilon}(s,s)={\mathbb{I}},\hfill\end{matrix}\right.$ (5.5)

which commutes with ${\cal P}_{0}^{[g]}(0)$. Similarly to (4.21), we set for
$0\leq s\leq t\leq 1$,

${\cal V}_{0}^{[g]}(t,s)={\cal
W}^{[g]}_{0}(t,0)\Psi^{[g]}_{\varepsilon}(t,s){\cal W}^{[g]}_{0}(0,s).$ (5.6)

The generators in these evolution equations being bounded on ${\cal T}({\cal
H})$, the corresponding propagators have finite operator norms on ${\cal
T}({\cal H})$ as well. We have the estimates

###### Lemma 5.1

Assume Reg and Spec with $\sigma_{j}(t)=\\{e_{j}(t)\\}$ for all $1\leq j\leq
d$. There exist $C<\infty$, $C_{\Psi}<\infty$, $\varepsilon_{0}>0$ and
$g_{0}>0$ such that for all $\varepsilon<\varepsilon_{0}$ and $g<g_{0}$,

$\displaystyle\|{\cal U}(t,s)-{\cal V}^{[g]}_{0}(t,s)\|_{\tau}\leq
C\varepsilon,$ (5.7) $\displaystyle\sup_{0\leq s\leq t\leq
1}\|\Psi^{[g]}_{\varepsilon}(t,s)\|_{\tau}\leq C_{\Psi},$ (5.8)

where

${\cal V}^{[g]}_{0}(t,0){\cal P}_{0}^{[g]}(0)={\cal P}_{0}^{[g]}(t){\cal
V}^{[g]}_{0}(t,0).$ (5.9)

Proof: The integration by parts argument in Appendix, more precisely Corollary
8.3 with ${\cal X}={\cal U}$, ${\cal G}={\cal L}^{[g]}$ and ${\cal Y}={\cal
V}_{0}$, ${\cal K}=[{\cal P}_{0}^{[g]}{}^{\prime},{\cal P}_{0}^{[g]}]$, yield
(5.7) since

$\varepsilon\partial_{t}{\cal V}^{[g]}_{0}(t,s)=({\cal
L}_{t}^{[g]}+\varepsilon[{\cal P}_{0}^{[g]}{}^{\prime}(t),{\cal
P}_{0}^{[g]}(t)]){\cal V}^{[g]}_{0}(t,s),$ (5.10)

and $\|{\cal U}(t,0)\|_{\tau}=1$. The uniformity in $g>0$ of the estimate is
ensured by perturbation theory: the fact that ${\cal P}_{0}^{[g]}(t)$ is
associated with a piece of the spectrum of ${\cal L}_{t}^{[g]}={\cal
L}_{t}^{0}+g{\cal L}_{t}^{1}$, of size of order $g$ and separated by a gap of
order $1$ from the rest of the spectrum, implies that for all $t\in[0,1]$

${\cal P}_{0}^{[g]}(t)={\cal P}_{0}(t)+{\cal O}(g).$ (5.11)

This estimate remains true for derivatives with respect to $t$, so that

$[{\cal P}_{0}^{[g]}{}^{\prime}(t),{\cal P}_{0}^{[g]}(t)]=[{\cal
P}_{0}^{\prime}(t),{\cal P}_{0}(t)]+{\cal O}(g),$ (5.12)

which yields uniformity in $g$ of the estimate so that $\sup_{0\leq s\leq
t\leq 1\atop 0<\varepsilon<\varepsilon_{0},0<g<g_{0}}\|{\cal
V}^{[g]}_{0}(t,s)\|_{\tau}<\infty$. Moreover, uniformly in $s,t\in[0,1]$,

${\cal W}^{[g]}_{0}(t,s)={\cal W}_{0}(t,s)+{\cal O}(g),$ (5.13)

where ${\cal W}_{0}(t,s)$ is defined by (2.20), as a consequence of Duhamel
formula

${\cal W}^{[g]}_{0}(t,s)={\cal W}_{0}(t,s)+\int_{s}^{t}{\cal
W}_{0}(t,u)([{\cal P}_{0}^{[g]}{}^{\prime}(u),{\cal P}_{0}^{[g]}(u)]-[{\cal
P}_{0}^{\prime}(u),{\cal P}_{0}(u)]){\cal W}^{[g]}_{0}(u,s)du.$ (5.14)

The estimate (5.13) and (5.6), together with ${\cal W}_{0}(t,s)^{-1}={\cal
W}_{0}(s,t)$, see (2.22), imply (5.8). $\Box$

###  5.1 Reduced dynamics

We now consider the reduced dynamics within ${\cal P}_{0}(0){\cal B}({\cal
H})$, depending on a time scale $1/\delta$ and driven by the splitting
operator, that will approximate $\Psi^{[g]}_{\varepsilon}(t,0){\cal
P}_{0}^{[g]}(0)$ in certain regimes.

Let $\tilde{\cal L}_{t}^{1}={\cal P}_{0}(t){\cal L}_{t}^{1}{\cal P}_{0}(t)$
and recall, see (2.34), that $\tilde{\Psi}_{\delta}(t,s)$, $0\leq s\leq t\leq
1$ is defined for $\delta>0$ by

$\displaystyle\left\\{\begin{matrix}\delta\partial_{t}\tilde{\Psi}_{\delta}(t,s)={{\cal
W}_{0}}(0,t)\tilde{\cal L}_{t}^{1}{\cal
W}_{0}(t,0)\tilde{\Psi}_{\delta}(t,s),\\\
\tilde{\Psi}_{\delta}(s,s)={\mathbb{I}}.\hfill\end{matrix}\right.$ (5.15)

Note that the Dyson series for $\tilde{\Psi}_{\delta}(t,0)$ has the same
integral terms as those provided in Lemma 3.13. Also by definition, recall
${\cal Q}_{0}(t)={\mathbb{I}}-{\cal P}_{0}(t)$,

$\tilde{\Psi}_{\delta}(t,s)={\cal P}_{0}(0)\tilde{\Psi}_{\delta}(t,s){\cal
P}_{0}(0)+{\cal Q}_{0}(0),$ (5.16)

where $\|{\cal Q}_{0}(0)\|_{\tau}\leq 2$, since ${\cal P}_{0}(0)$ is CPTP.

The next Proposition is the main technical step regarding the approximation of
the evolution ${\cal U}(t,s)$ in the transition regime considered, which holds
regardless of the dimension of the projectors $P_{j}(t)$.

###### Proposition 5.2

Assume Reg and Spec with $\sigma_{j}(t)=\\{e_{j}(t)\\}$, for all $1\leq j\leq
d$. There exist $\tilde{C}$, $\tilde{C}_{\Psi}<\infty$ and $g_{0}$,
$\varepsilon_{0}$, $\alpha_{0}>0$ such that for all $g\leq g_{0}$,
$\varepsilon\leq\varepsilon_{0}$, $g^{2}/\varepsilon\leq\alpha_{0}$, and
$0\leq s\leq t\leq 1$,

$\|\Psi^{[g]}_{\varepsilon}(t,s){\cal
P}_{0}^{[g]}(0)-\tilde{\Psi}_{\varepsilon/g}(t,s){\cal
P}_{0}^{[g]}(0)\|_{\tau}\leq\tilde{C}(t-s)g^{2}/\varepsilon,$ (5.17)

and $\|\tilde{\Psi}_{\varepsilon/g}(t,s)\|_{\tau}\leq\tilde{C}_{\Psi}$.

###### Remark 5.3

The ratio $\varepsilon/g$ which determines the time scale in the reduced
dynamics $\tilde{\Psi}_{\varepsilon/g}(t,s)$ is not required to be small here.

Proof: Recall (4.1) which states that, uniformly in $0\leq t\leq 1$,

$\displaystyle{\cal L}_{t}^{[g]}{\cal P}_{0}^{[g]}(t)={\cal
P}_{0}^{[g]}(t){\cal L}_{t}^{[g]}{\cal P}_{0}^{[g]}(t)=g{\cal P}_{0}(t){\cal
L}_{t}^{1}{\cal P}_{0}(t)+{\cal O}(g^{2}),$ (5.18)

and the intertwining relation ${\cal W}^{[g]}_{0}(t,s){\cal
P}_{0}^{[g]}(s)={\cal P}_{0}^{[g]}(t){\cal W}^{[g]}_{0}(t,s)$, consequence of
the definition (5.4). Composing (5.18) by ${\cal W}^{[g]}_{0}(0,t)$ and ${\cal
W}^{[g]}_{0}(t,0)$ and using (5.11) and (5.13), we get that the generator of
$\Psi^{[g]}_{\varepsilon}(t,s){\cal P}_{0}^{[g]}(0)$, see (5.5), satisfies

${\cal W}^{[g]}_{0}(0,t){\cal L}_{t}^{[g]}{\cal W}^{[g]}_{0}(t,0){\cal
P}_{0}^{[g]}(0)-g{\cal W}_{0}(0,t){\cal P}_{0}(t){\cal L}_{t}^{1}{\cal
P}_{0}(t){\cal W}_{0}(t,0){\cal P}_{0}^{[g]}(0)=g^{2}\Lambda(t,g),$ (5.19)

where $\|\Lambda(t,g)\|_{\tau}\leq C_{\Lambda}$, uniformly in $0\leq t\leq 1$
and $g>0$ small enough. Therefore, making use of
$[\Psi^{[g]}_{\varepsilon}(t,s),{\cal P}_{0}^{[g]}(0)]\equiv 0$, Duhamel
formula yields

$\displaystyle\Psi^{[g]}_{\varepsilon}(t,s){\cal P}_{0}^{[g]}(0)$
$\displaystyle=\tilde{\Psi}_{\varepsilon/g}(t,s){\cal
P}_{0}^{[g]}(0)+\frac{g^{2}}{\varepsilon}\int_{s}^{t}\tilde{\Psi}_{\varepsilon/g}(t,r)\Lambda(r,g)\Psi^{[g]}_{\varepsilon}(r,s){\cal
P}_{0}^{[g]}(0)ds.$ (5.20)

Hence

$\|\Psi^{[g]}_{\varepsilon}(t,s){\cal
P}_{0}^{[g]}(0)-\tilde{\Psi}_{\varepsilon/g}(t,s){\cal
P}_{0}^{[g]}(0)\|_{\tau}\leq
C_{\Lambda}(t-s)\frac{g^{2}}{\varepsilon}\sup_{0\leq r\leq t\leq
1}\|\tilde{\Psi}_{\varepsilon/g}(t,r)\|_{\tau}\sup_{0\leq r\leq s\leq
1}\|\Psi^{[g]}_{\varepsilon}(r,s){\cal P}_{0}^{[g]}(0)\|_{\tau}.$ (5.21)

Now, $\sup_{0\leq s\leq t\leq 1}\|\Psi^{[g]}_{\varepsilon}(t,s){\cal
P}_{0}^{[g]}(0)\|_{\tau}:=C_{\Psi P}$ is uniformly bounded for $\varepsilon>0$
and $g>0$ small enough, see (5.8). Moreover, thanks to (5.16) and (5.11),
there exists $c<\infty$ such that

$\displaystyle\|\tilde{\Psi}_{\varepsilon/g}(t,s)\|_{\tau}\leq\|\tilde{\Psi}_{\varepsilon/g}(t,s){\cal
P}_{0}(0)\|_{\tau}+2,$ (5.22)
$\displaystyle\|\tilde{\Psi}_{\varepsilon/g}(t,s){\cal
P}_{0}^{[g]}(0)-\tilde{\Psi}_{\varepsilon/g}(t,s){\cal P}_{0}(0)\|_{\tau}\leq
cg\|\tilde{\Psi}_{\varepsilon/g}(t,s)\|_{\tau}.$ (5.23)

Consequently, making use of the identity (recall (5.16))

$\displaystyle\tilde{\Psi}_{\varepsilon/g}(t,s)$ $\displaystyle={\cal
Q}_{0}(0)+\Psi_{\varepsilon}^{[g]}(t,s){\cal P}_{0}^{[g]}(0)$
$\displaystyle+(\tilde{\Psi}_{\varepsilon/g}(t,s){\cal
P}_{0}^{[g]}(0)-\Psi^{[g]}_{\varepsilon}(t,s){\cal
P}_{0}^{[g]}(0))+\tilde{\Psi}_{\varepsilon/g}(t,s)({\cal P}_{0}(0)-{\cal
P}_{0}^{[g]}(0)).$ (5.24)

we get with the above and (5.21),

$\sup_{0\leq s\leq t\leq 1}\|\tilde{\Psi}_{\varepsilon/g}(t,s)\|_{\tau}\leq
2+C_{\Psi P}\Big{(}1+C_{\Lambda}\frac{g^{2}}{\varepsilon}\sup_{0\leq r\leq
t\leq 1}\|\tilde{\Psi}_{\varepsilon/g}(t,r)\|_{\tau}\Big{)}+cg\sup_{0\leq
s\leq t\leq 1}\|\tilde{\Psi}_{\varepsilon/g}(t,s)\|_{\tau}.$ (5.25)

Therefore, there exists $0<\tilde{C}_{\Psi}<\infty$ such that if
$\varepsilon>0$, $g>0$ and $g^{2}/\varepsilon$ are small enough,

$\sup_{0\leq s\leq t\leq
1}\|\tilde{\Psi}_{\varepsilon/g}(t,s)\|_{\tau}\leq\frac{2+C_{\Psi
P}}{1-C_{\Psi P}C_{\Lambda}\frac{g^{2}}{\varepsilon}-cg}\leq\tilde{C}_{\Psi},$
(5.26)

irrespectively of the value of $\varepsilon/g$. By inserting this estimate
into (5.21), we get the first statement with
$\tilde{C}=C_{\Lambda}\tilde{C}_{\Psi}C_{\Psi P}$. $\Box$

We are now in a position to approximate the evolution ${\cal U}(t,s)$ and the
transition probabilities between the spectral projectors $P_{j}(t)$ within
${\,\rm Ker\,}{\cal L}_{t}^{0}$, which is the content of Theorem 2.9:

###### Corollary 5.4

Assume Reg and Spec with $\sigma_{j}(t)=\\{e_{j}(t)\\}$, for all $1\leq j\leq
d$. There exists $C_{0}<\infty$, $\varepsilon_{0}>0$, $g_{0}>0$, and
$\alpha_{0}>0$ such that for all $0\leq t\leq 1$,
$\varepsilon<\varepsilon_{0}$, $g\leq g_{0}$, $g^{2}/\varepsilon<\alpha_{0}$,

$\|{\cal U}(t,0){\cal P}_{0}(0)-{\cal
W}_{0}(t,0)\tilde{\Psi}_{\varepsilon/g}(t,0){\cal P}_{0}(0)\|_{\tau}\leq
C_{0}(\varepsilon+g+g^{2}/\varepsilon).$ (5.27)

Consequently, for any state $\rho_{j}=P_{j}(0)\rho_{j}P_{j}(0)\in{\cal
T}({\cal H})$, for any $1\leq j,k\leq d$,

$\displaystyle{\,\rm Tr}\\{P_{k}(t){\cal U}(t,0)(\rho_{j})\\}$
$\displaystyle={\,\rm
Tr}\\{P_{k}(0)\tilde{\Psi}_{\varepsilon/g}(t,0)(\rho_{j})\\}+{\cal
O}(\varepsilon+g+g^{2}/\varepsilon).$ (5.28)

Also, for any $\delta>0$ and any $0\leq t\leq 1$, the map
$\tilde{\Psi}_{\delta}(t,0){\cal P}_{0}(0)$ is CPTP on ${\cal T}({\cal H})$.

Proof: The first statement follows immediately from Lemma 5.1, Proposition 5.2
and estimate (5.11).
To access the transition probabilities ${\,\rm Tr}(P_{k}(t){\cal
U}(t,0)(\rho_{j}))$, we get the action of ${\cal W}_{0}(t,0)$ on the projector
${\cal P}_{0}(0)$ by means of Lemma 3.5. Thus, using
$\tilde{\Psi}_{\varepsilon/g}(t,0){\cal P}_{0}(0)={\cal
P}_{0}(0)\tilde{\Psi}_{\varepsilon/g}(t,0){\cal P}_{0}(0)$, we get the
transition probabilities in terms of the reduced dynamics in the regime
$g^{2}\ll\varepsilon\ll 1$

$\displaystyle{\,\rm Tr}\\{P_{k}(t){\cal U}(t,0)(P_{j}(0))\\}$
$\displaystyle={\,\rm Tr}\\{P_{k}(t){\cal
W}_{0}(t,0)\circ\tilde{\Psi}_{\varepsilon/g}(t,0)(P_{j}(0))\\}+{\cal
O}(\varepsilon+g+g^{2}/\varepsilon)$ $\displaystyle={\,\rm
Tr}\\{P_{k}(0)\tilde{\Psi}_{\varepsilon/g}(t,0)(P_{j}(0))\\}+{\cal
O}(\varepsilon+g+g^{2}/\varepsilon).$ (5.29)

Finally, given $\delta>0$, (5.27) for $\varepsilon=\delta g$ yields ${\cal
W}_{0}(t,0)\tilde{\Psi}_{\delta}(t,0){\cal P}_{0}(0)=\lim_{g\rightarrow
0}{\cal U}(t,0){\cal P}_{0}(0)$ in $\|\cdot\|_{\tau}$-norm , where ${\cal
U}(t,0){\cal P}_{0}(0)$ is CPTP on ${\cal T}({\cal H})$, so the same is true
for ${\cal W}_{0}(t,0)\tilde{\Psi}_{\delta}(t,0){\cal P}_{0}(0)$. Since ${\cal
W}_{0}(0,t){\cal P}_{0}(t)$ is CPTP as well, see Lemma 2.3, and
$\tilde{\Psi}_{\delta}(t,0){\cal P}_{0}(0)={\cal
P}_{0}(0)\tilde{\Psi}_{\delta}(t,0){\cal P}_{0}(0)$,

$\tilde{\Psi}_{\delta}(t,0){\cal P}_{0}(0)={\cal W}_{0}(0,t){\cal
W}_{0}(t,0){\cal P}_{0}(0)\tilde{\Psi}_{\delta}(t,0){\cal P}_{0}(0)={\cal
W}_{0}(0,t){\cal P}_{0}(t){\cal W}_{0}(t,0)\tilde{\Psi}_{\delta}(t,0){\cal
P}_{0}(0)$ (5.30)

is CPTP, as a composition of such maps. $\Box$

###  5.2 Associated Markov Process

Let us proceed with a remark about the generic finite dimensional case. If Reg
and Gen hold (without condition on the Bohr frequencies, actually), the
generator ${\cal W}_{0}(0,t)\tilde{\cal L}_{t}^{1}{\cal W}_{0}(t,0)$ of the
reduced dynamics $\tilde{\Psi}_{\delta}(t,0)$ has a matrix expression in the
fixed basis $\\{P_{1}(0),P_{2}(0),\dots,P_{d}(0)\\}$ given by the time
dependent matrix $\tilde{L}(t)$ (4.15). Indeed, (4.14) and Remark 3.6 i),
yield

${\cal W}_{0}(0,t)\tilde{\cal L}_{t}^{1}{\cal W}_{0}(t,0)(P_{j}(0))={\cal
W}_{0}(0,t){\cal P}_{0}(t){\cal L}_{t}^{1}(P_{j}(t))=\sum_{1\leq k\leq
d}P_{k}(0)\tilde{L}_{kj}(t).$ (5.31)

In other words, ${\cal W}_{0}(0,t)\tilde{\cal L}_{t}^{1}{\cal
W}_{0}(t,0)\simeq\tilde{L}(t)$, where $\sum_{1\leq k\leq
d}\tilde{L}_{kj}(t)=0$ for any $1\leq j\leq d$, recall (4.16). Hence, the
matrix representation of the reduced dynamics in the same basis,
$\tilde{\Psi}_{\delta}(t,0)|_{{\,\rm Span}\\{P_{1}(0),\dots,P_{d}(0)\\}}$, is
such that its transpose is a stochastic matrix, see e.g. [YZ]. Therefore, we
can associate to the reduced dynamics a $d$-state classical continuous-time
Markov process:

###### Lemma 5.5

Under Reg and Gen, the reduced dynamics $\tilde{\Psi}_{\delta}(t,0){\cal
P}_{0}(0)$ is associated to a continuous-time Markov process $(X_{t})_{t\geq
0}$ in the state space $\\{P_{1}(0),\dots,P_{d}(0)\\}:=\\{1,\dots,d\\}$ by the
relation for any $t\geq 0$

${\mathbb{P}}(X_{t}=j|X_{0}=i)={\,\rm
Tr}\big{(}P_{j}(0)\tilde{\Psi}_{\delta}(t,0)(P_{i}(0))\big{)}.$ (5.32)

###  5.3 Back to the slow drive regime

Specialising to the simpler generic framework given by assumptions Gen, and
supposing the dissipator splits ${\,\rm Ker\,}{\cal L}_{t}^{0}$ maximally, we
can further approximate the reduced evolution $\tilde{\Psi}_{\delta}(t,s)$ for
$\delta=\varepsilon/g\ll 1$.

By Assumption Split, Remark 4.2 and (4.18), the generator of
$\tilde{\Psi}_{\delta}(t,0)$ reads

${{\cal W}_{0}}(0,t)\tilde{\cal L}^{1}_{t}{{\cal W}_{0}}(t,0)={\cal
P}_{0}(0)\Big{(}0\tilde{\cal
Q}_{0}(t)+\sum_{j=1}^{d-1}\tilde{\lambda}_{j}(t)\tilde{\cal
Q}_{j}(t)\Big{)}{\cal P}_{0}(0)+0{\cal Q}_{0}(0),$ (5.33)

with rank one spectral projectors $\tilde{\cal Q}_{j}(t)={{\cal
W}_{0}}(0,t)\tilde{\cal P}_{j}(t){{\cal W}_{0}}(t,0)$ and corresponding
eigenvalues $\tilde{\lambda}_{j}(t)$ with negative or zero real parts. Thus
$\tilde{\Psi}_{\delta}(t,s){\cal Q}_{0}(0)\equiv{\cal Q}_{0}(0)$, and
$[\tilde{\Psi}_{\delta}(t,s),{\cal P}_{0}(0)]\equiv 0$. In case the time scale
$1/\delta$ is large, the following adiabatic approximation holds: There exists
$\delta_{0}>0$ and $\tilde{c}<\infty$ such that for all $\delta<\delta_{0}$,
and all $0\leq s\leq t\leq 1$,

$\Big{\|}\tilde{\Psi}_{\delta}(t,s){\cal P}_{0}(0)-\tilde{\cal
W}(t,0)\Big{(}\sum_{j=0}^{d-1}{\rm
e}^{\int_{s}^{t}\tilde{\lambda}_{j}(r)dr/\delta}\tilde{\cal
Q}_{j}(0)\Big{)}\tilde{\cal W}(0,s){\cal
P}_{0}(0)\Big{\|}\leq\tilde{c}\delta,$ (5.34)

where $\tilde{\cal W}(t,s)$ is defined by

$\displaystyle\left\\{\begin{matrix}\partial_{t}\tilde{\cal
W}(t,s)=\Big{(}\sum_{j=0}^{d-1}\tilde{\cal Q}_{j}^{\prime}(t)\tilde{\cal
Q}_{j}(t)\Big{)}\tilde{\cal W}(t,s)\\\ \tilde{\cal
W}(s,s)={\mathbb{I}},\hfill\end{matrix}\right.$ (5.35)

so that $\tilde{\cal W}(t,s){\cal Q}_{0}(0)={\cal Q}_{0}(0)\tilde{\cal
W}(t,s)\equiv{\cal Q}_{0}(0)$ and the following non trivial intertwining
relations hold for $0\leq j\leq d-1$

$\tilde{\cal W}(t,s)\tilde{\cal Q}_{j}(s)=\tilde{\cal Q}_{j}(t)\tilde{\cal
W}(t,s).$ (5.36)

Indeed, the integration by parts argument Lemma 8.2 applies with
$\varepsilon=\delta$ to ${\cal X}(t,s)=\tilde{\Psi}_{\delta}(t,s){\cal
P}_{0}(0)$, ${\cal Y}(t,s)=\tilde{\cal W}(t,0)\Big{(}\sum_{j=0}^{d-1}{\rm
e}^{\int_{s}^{t}\tilde{\lambda}_{j}(r)dr/\delta}\tilde{\cal
Q}_{j}(0)\Big{)}\tilde{\cal W}(0,s){\cal P}_{0}(0)$, ${\cal G}(t)={{\cal
W}_{0}}(0,t)\tilde{\cal L}^{1}_{t}{{\cal W}_{0}}(t,0){\cal P}_{0}(0)$, and
${\cal K}(t)=\sum_{j=0}^{d-1}\tilde{\cal Q}_{j}^{\prime}(t)\tilde{\cal
Q}_{j}(t)$. Since ${\cal Y}(t,s)$ is uniformly bounded in $\delta$, Corollary
8.3 yields estimate (5.34).

Hence, in the restricted slow drive regime $\varepsilon\ll
g\ll\sqrt{\varepsilon}\ll 1$, we can take advantage of (5.34) to express
$\tilde{\Psi}_{\varepsilon/g}(t,0)$ in terms of the spectral data of
$\tilde{\cal L}_{t}^{1}$, in the framework given by assumption Gen to
approximate ${\cal U}(t,0){\cal P}_{0}(0)$. Indeed, making use of (5.27),
(5.34) for $s=0$, and ${\cal P}_{0}(0)\tilde{\cal Q}_{j}(0)=\tilde{\cal
Q}_{j}(0)$, for all $0\leq j\leq d-1$, and taking into account the regime
considered, we immediately get the

###### Corollary 5.6

Assume Reg, Gen and Split. There exist $C_{1}<\infty$, $\varepsilon_{0}>0$,
$g_{0}>0$, $\alpha_{0}>0$ and $\beta_{0}>0$ such that for all $0\leq t\leq 1$,
$\varepsilon\leq\varepsilon_{0}$, $g\leq g_{0}$,
$g^{2}/\varepsilon\leq\alpha_{0}$, $\varepsilon/g\leq\beta_{0}$

$\Big{\|}{\cal U}(t,0){\cal P}_{0}(0)-{\cal W}_{0}(t,0)\tilde{\cal
W}(t,0)\Big{(}\sum_{j=0}^{d-1}{\rm
e}^{\frac{g}{\varepsilon}\int_{0}^{t}\tilde{\lambda}_{j}(r)dr}\tilde{\cal
Q}_{j}(0)\Big{)}\Big{\|}\leq C_{1}(g^{2}/\varepsilon+\varepsilon/g).$ (5.37)

In order to compute the transition probability between the eigenprojectors of
the Hamiltonian, we make explicit $\tilde{\cal Q}_{j}(t)$, $0\leq j\leq d-1$,
the rank one eigenprojectors of ${{\cal W}_{0}}(0,t)\tilde{\cal
L}^{1}_{t}{{\cal W}_{0}}(t,0)$. In keeping with (4.3), for $A,B\in{\cal
B}({\cal H})$, we define a rank one operator on ${\cal B}({\cal H})$ by

$\big{|}A\big{\rangle}\big{\rangle}\big{\langle}\big{\langle}B\big{|}:C\mapsto\big{\langle}\big{\langle}B,C\big{\rangle}\big{\rangle}A={\,\rm
Tr}(B^{*}C)A.$ (5.38)

Hence there exist $\nu_{j}(t),\mu_{j}(t)\in{\cal P}_{0}(0){\cal B}({\cal
H})={\,\rm Span}\\{P_{k}(0),k\in\\{1,\dots,d\\}\\}$ such that

$\displaystyle\tilde{\cal
Q}_{j}(t)=\big{|}\nu_{j}(t)\big{\rangle}\big{\rangle}\big{\langle}\big{\langle}\mu_{j}(t)\big{|},\
\ \mbox{where}\ \ $
$\displaystyle\big{\langle}\big{\langle}\mu_{j}(t),\nu_{j}(t)\big{\rangle}\big{\rangle}\equiv
1,\ \ \
\big{\langle}\big{\langle}\mu_{j}(t),\mu_{j}(t)\big{\rangle}\big{\rangle}\equiv
d.$ (5.39)

where the last identity serves normalisation purposes. Since ${{\cal
W}_{0}}(0,t)\tilde{\cal L}^{1}_{t}{{\cal W}_{0}}(t,0)$ is smooth, these
operators can be chosen smooth as well. Moreover, for $\tilde{\cal W}(t,s)$
defined by (5.35), we have for all $j\in\\{0,1,\dots,d-1\\}$

$\tilde{\cal W}(t,s)(\nu_{j}(s))=\nu_{j}(t){\rm
e}^{-\int_{s}^{t}\small\langle\small\langle\mu_{j}(u),\partial_{u}\nu_{j}(u)\small\rangle\small\rangle
du}.$ (5.40)

This identity follows from (5.36) together with $\tilde{\cal
Q}_{j}(t)\partial_{t}\\{\tilde{\cal W}(t,s)(\nu_{j}(s))\\}\equiv 0$.

In particular, for $j=0$, the eigenprojector ${\cal Q}_{0}(t)$ associated with
$\lambda_{0}(t)\equiv 0$ takes the form

$\tilde{\cal
Q}_{0}(t)=\big{|}\nu_{0}(t)\big{\rangle}\big{\rangle}\big{\langle}\big{\langle}{\mathbb{I}}\big{|},$
(5.41)

where $\nu_{0}(t)={\cal P}_{0}(0)(\nu_{0}(t))\in{\,\rm Ker\,}{{\cal
W}_{0}}(0,t)\tilde{\cal L}^{1}_{t}{{\cal W}_{0}}(t,0)$. Moreover, (5.3)
implies

$\tilde{\cal W}(t,s)(\nu_{0}(s))=\nu_{0}(t),$ (5.42)

and the justification that $\mu_{0}(t)={\mathbb{I}}$ stems from ${\,\rm
Tr}{\cal L}_{t}^{1}(\rho)\equiv 0$, see (4.17). Equivalently, by Lemma 3.5,
$\nu_{0}(t)$ is characterised by

$\tilde{\cal L}^{1}_{t}(\tilde{\nu}_{0}(t))=0\ \ \mbox{where}\ \
\tilde{\nu}_{0}(t)={\cal W}_{0}(t,0)(\nu_{0}(t))=W(t,0)\nu_{0}(t)W(0,t).$
(5.43)

Note that since $\Re\tilde{\lambda}_{j}(t)<0$ for $j\neq 0$, we get that for
all fixed $t>0$, ${\rm
e}^{\frac{g}{\varepsilon}\int_{0}^{t}\tilde{\lambda}_{j}(r)dr}={\cal
O}((\varepsilon/g)^{\infty})$. Hence, from Corollary 5.6 and (5.43), for all
fixed $0<t<1$, and all $P_{j}(0)$, we have

${\cal U}(t,0)(P_{j}(0))=\tilde{\nu}_{0}(t)+{\cal
O}(g^{2}/\varepsilon+\varepsilon/g),$ (5.44)

which is the statement of Theorem 2.7.

Let us turn to the transition probabilities. Writing for any $1\leq l\leq d$

$\nu_{l}(t)=\sum_{1\leq k\leq d}{\,\rm
Tr}(P_{k}(0)\nu_{l}(t))P_{k}(0):=\sum_{1\leq k\leq d}\nu_{l}^{k}(t)P_{k}(0),$
(5.45)

we thus have

${\,\rm Tr}\\{P_{k}(0)\tilde{\cal W}(t,0)\circ\tilde{\cal
Q}_{l}(0)(P_{j}(0))\\}=\bar{\mu}_{l}^{j}(0)\nu_{l}^{k}(t){\rm
e}^{-\int_{0}^{t}\small\langle\small\langle\mu_{l}(s),\nu_{l}^{\prime}(s)\small\rangle\small\rangle
ds}.$ (5.46)

We are in a position to estimate the transition probabilities in the adiabatic
regime for the reduced evolution, to complete Theorem 2.7:

###### Corollary 5.7

Assume Reg, Gen and Split. Then $\forall t\in[0,1]$, $j\neq k$, we have for
$\varepsilon\ll g\ll\sqrt{\varepsilon}\ll 1$,

${\,\rm Tr}(P_{k}(t){\cal U}(t,0)(P_{j}(0)))=\sum_{0\leq l\leq d-1}{\rm
e}^{g/\varepsilon\int_{0}^{t}\tilde{\lambda}_{j}(s)ds}\,\bar{\mu}_{l}^{j}(0)\nu_{l}^{k}(t){\rm
e}^{-\int_{0}^{t}\small\langle\small\langle\mu_{l}(s),\nu_{l}^{\prime}(s)\small\rangle\small\rangle
ds}+{\cal O}(g^{2}/\varepsilon+\varepsilon/g).$ (5.47)

In particular, for any fixed $t>0$, we have in the same regime

${\,\rm Tr}(P_{k}(t){\cal U}(t,0)(P_{j}(0)))={\,\rm
Tr}(P_{k}(t)\tilde{\nu}_{0}(t))+{\cal O}(g^{2}/\varepsilon+\varepsilon/g),$
(5.48)

where $\tilde{\nu}_{0}(t)$ is uniquely defined by $\tilde{\cal
L}_{t}^{1}(\tilde{\nu}_{0}(t))=0$ and ${\,\rm Tr}(\tilde{\nu}_{0}(t))=1$.

###### Remark 5.8

The first statement stems from Corollary 5.6, while the second one takes
advantage of ${\rm
e}^{g/\varepsilon\int_{0}^{t}\tilde{\lambda}_{j}(s)ds}=~{}{\cal
O}((\varepsilon/g)^{\infty})$ for $j>0$ if $t>0$ is independent of
$\varepsilon/g$, since $\Re\tilde{\lambda}_{j}(s)<0$ for such $j$’s. The
reformulation of the leading order is a consequence of the considerations
above and ${\cal W}_{0}(t,s){\cal P}_{0}(s)$ being trace preserving.

###  5.4 Back to the perturbative regime

Finally, we briefly check that Corollary 5.4 reduces to a statement of
Proposition 3.9 in the perturbative regime $g\ll\varepsilon$. We note that the
definition (5.15) allows for an approach of $\tilde{\Psi}_{\delta}$ via Dyson
series which gives for $\delta=\varepsilon/g\gg 1$,

$\tilde{\Psi}_{\delta}(t,0)={\mathbb{I}}+\frac{1}{\delta}\int_{0}^{t}{\cal
W}_{0}(0,s)\tilde{\cal L}_{s}^{1}{\cal W}_{0}(s,0)ds+{\cal O}(1/\delta^{2}),$
(5.49)

since $\|\tilde{\Psi}_{\delta}(t,s)\|_{\tau}$ is uniformly bounded in $0\leq
s\leq t\leq 1$ and $\delta$. Hence, given the definition of $\tilde{\cal
L}_{s}^{1}$,

$\displaystyle{\cal W}_{0}(t,0)\tilde{\Psi}_{\varepsilon/g}(t,0){\cal
P}_{0}(0)$ $\displaystyle={\cal W}_{0}(t,0){\cal P}_{0}(0)+{\cal
W}_{0}(t,0)\frac{g}{\varepsilon}\int_{0}^{t}{\cal W}_{0}(0,s)\tilde{\cal
L}_{s}^{1}{\cal W}_{0}(s,0){\cal P}_{0}(0)ds+{\cal O}(g^{2}/\varepsilon^{2})$
$\displaystyle={\cal W}_{0}(t,0){\cal P}_{0}(0)+\frac{g}{\varepsilon}{\cal
W}_{0}(t,0)\int_{0}^{t}{\cal P}_{0}(0){\cal W}_{0}(0,s){\cal L}_{s}^{1}{\cal
W}_{0}(s,0){\cal P}_{0}(0)ds+{\cal O}(g^{2}/\varepsilon^{2}).$ (5.50)

Since $\sigma_{j}(t)=\\{e_{j}(t)\\}$ for all $1\leq j\leq d$ and all
$t\in[0,1]$, Remark 3.8 applies which, noting the intertwining relation
(2.21), yields for any $0\leq s\leq t\leq 1$,

${\cal W}_{0}(t,s){\cal P}_{0}(s)={\cal V}^{0}(t,s){\cal P}_{0}(s),$ (5.51)

where ${\cal V}^{0}(t,s)$ is defined in (3.15). Thus, further assuming
$g\ll\varepsilon$ in Corollary 5.4, we recover the perturbative regime
estimate of Proposition (3.9) for $N=1$ under the form

###### Corollary 5.9

Assume Reg and Spec with $\sigma_{j}(t)=\\{e_{j}(t)\\}$, for all $1\leq j\leq
d$ and all $t\in[0,1]$. In the regime $(\varepsilon,g)\rightarrow(0,0)$ and
$g/\varepsilon\rightarrow 0$, (5.27) yields

${\cal U}(t,0){\cal P}_{0}(0)={\cal V}^{0}(t,0){\cal
P}_{0}(0)+\frac{g}{\varepsilon}\int_{0}^{t}{\cal P}_{0}(t){\cal
V}^{0}(t,s){\cal L}_{s}^{1}{\cal V}^{0}(s,0){\cal P}_{0}(0)ds+{\cal
O}(\varepsilon+(g/\varepsilon)^{2}),$ (5.52)

where ${\cal V}^{0}(t,s){\cal P}_{0}(s)={\cal P}_{0}(t){\cal V}^{0}(t,s)={\cal
P}_{0}(t){\cal W}_{0}(t,s){\cal P}_{0}(s)$ is independent of $\varepsilon$.

###### Remark 5.10

i) The error term is smaller than the explicit integral term for
$\varepsilon^{2}\ll g\ll\varepsilon$.
ii) We recover this way the second statement of Theorem 2.5.

##  6 Example

We consider here a two-level system or Qubit, for which the reduced dynamics
$\tilde{\Psi}_{\delta}(t,s)$ can be computed explicitly under some symmetry of
the Lindblad operators. Beyond its intrinsic interest, this example allows us
to illustrate the different regimes we encountered in the general case.

We assume Reg and Gen and with the notations introduced so far, for ${\cal
H}={\mathbb{C}}^{2}$, we consider the two-level Hamiltonian

$H(t)=\sum_{1\leq j\leq 2}e_{j}(t)P_{j}(t),\ \ \mbox{with}\ \
P_{j}(t)=|\varphi_{j}(t)\rangle\langle\varphi_{j}(t)|,$ (6.1)

and $\varphi_{j}(t)=W(t,0)\varphi_{j}(0)$, $W(t,0)$ being the unitary Kato
operator. We assume the dissipator

${\cal L}_{t}^{1}(\cdot)=\sum_{l\in
I}\Big{(}\Gamma_{l}(t)\cdot\Gamma_{l}^{*}(t)-\frac{1}{2}\big{\\{}\Gamma_{l}^{*}(t)\Gamma_{l}(t),\cdot\big{\\}}\Big{)},$
(6.2)

has jump operators in ${\cal B}({\mathbb{C}}^{2})$ satisfying the symmetry
condition

$\sum_{l\in
I}|\langle\varphi_{1}(t)|\Gamma_{l}(t)\varphi_{2}(t)\rangle|^{2}=\sum_{l\in
I}|\langle\varphi_{2}(t)|\Gamma_{l}(t)\varphi_{1}(t)\rangle|^{2}.$ (6.3)

This is the case in particular if all jump operators are self-adjoint. Note
that condition (6.3) is independent of the normalised basis of eigenvectors of
$H(t)$ used to express it.

Then we have:

###### Proposition 6.1

Let ${\cal H}={\mathbb{C}}^{2}$, and assume Reg and Gen for $d=2$. Further
suppose the dissipator satisfies the symmetry condition (6.3) and set
$\gamma(t)=\sum_{l\in
I}|\langle\varphi_{1}(t)|\Gamma_{l}(t)\varphi_{2}(t)\rangle|^{2}\in{\mathbb{R}}^{+}$.
Then, for any $\delta>0$, the reduced dynamics takes the explicit form in the
ordered basis $\\{P_{1}(0),P_{2}(0)\\}$ of ${\cal P}_{0}(0)$:

$\displaystyle\tilde{\Psi}_{\delta}(t,s)|_{{\,\rm
Span}\\{P_{1}(0),P_{2}(0)\\}}=\frac{1}{2}\begin{pmatrix}1&1\cr
1&1\cr\end{pmatrix}+{\rm
e}^{-\frac{2}{\delta}\int_{s}^{t}\gamma(u)du}\frac{1}{2}\begin{pmatrix}1&-1\cr-1&1\cr\end{pmatrix}.$
(6.4)

Hence, in the regime $g\ll\sqrt{\varepsilon}\ll 1$, for any initial state
$\rho_{0}=\rho_{1}(0)P_{1}(0)+\rho_{2}(0)P_{2}(0)$, any $t\in[0,1],$

$\displaystyle{\cal
U}(t,0)(\rho_{0})=\tilde{\rho}_{1}(t)P_{1}(t)+\tilde{\rho}_{2}(t)P_{2}(t)+{\cal
O}(\varepsilon+g+g^{2}/\varepsilon),$ (6.5)

where

$\displaystyle\tilde{\rho}_{1}(t)$ $\displaystyle=\frac{1}{2}\Big{(}1+{\rm
e}^{-2\frac{g}{\varepsilon}\int_{0}^{t}\gamma(s)ds}(\rho_{1}(0)-\rho_{2}(0))\Big{)},$
$\displaystyle\tilde{\rho}_{2}(t)$ $\displaystyle=\frac{1}{2}\Big{(}1+{\rm
e}^{-2\frac{g}{\varepsilon}\int_{0}^{t}\gamma(s)ds}(\rho_{2}(0)-\rho_{1}(0))\Big{)}.$
(6.6)

In particular, the transition probabilities read in the same regime

$\displaystyle{\,\rm Tr}(P_{2}(t){\cal U}(t,0)(P_{1}(0)))={\,\rm
Tr}(P_{1}(t){\cal U}(t,0)(P_{2}(0)))=\frac{1}{2}\Big{(}1-{\rm
e}^{-2\frac{g}{\varepsilon}\int_{0}^{t}\gamma(s)ds}\Big{)}+{\cal
O}(\varepsilon+g+g^{2}/\varepsilon).$ (6.7)

###### Remark 6.2

i) For $0<t\leq 1$ fixed such that $\int_{0}^{t}\gamma(s)ds>0$, if
$\varepsilon\ll g\ll\sqrt{\varepsilon}\ll 1$,

${\cal U}(t,0)(\rho_{0})=\frac{1}{2}{\mathbb{I}}+{\cal
O}(g^{2}/\varepsilon+(\varepsilon/g)^{\infty}),$ (6.8)

which corresponds to Theorem 2.7. Note that ${\mathbb{I}}$ spans ${\,\rm
Ker\,}\tilde{\cal L}_{t}^{1}|_{{\cal P}_{0}(t){\cal H}}$.
ii) If $g\ll\varepsilon\ll\sqrt{\varepsilon}\ll 1$,

$\displaystyle{\cal U}(t,0)(\rho_{0})=$
$\displaystyle\rho_{1}(0)P_{1}(t)+\rho_{2}(0)P_{2}(t)$
$\displaystyle-\frac{g}{\varepsilon}\int_{0}^{t}\gamma(s)ds\,(\rho_{1}(0)-\rho_{2}(0))(P_{1}(t)-P_{2}(t))+{\cal
O}(\varepsilon+g^{2}/\varepsilon^{2})$ (6.9)

which corresponds to Corollary 5.9 and Theorem 2.5.
iii) The state ${\cal U}(t,0)(\rho_{0})$ is determined by the asymptotics of
the scalar factor ${\rm e}^{-2\frac{g}{\varepsilon}\int_{0}^{t}\gamma(s)ds}$.
In case $g\simeq\varepsilon$, i.e. $g=\alpha\varepsilon$, for some fixed
$\alpha>0$, the leading order of ${\cal U}(t,0)(\rho_{0})$ takes the form of
any diagonal state with respect to the eigenbasis of $H(t)$, depending on the
value of ${\rm e}^{-2\alpha\int_{0}^{t}\gamma(s)ds}$.
iv) The Markov process interpretation of Lemma 5.5 remains in force here, with
$\tilde{\Psi}_{\delta}(t,0)|_{{\,\rm Span}\\{{\cal P}_{0}(0){\cal
B}({\mathbb{C}}^{2})\\}}$ being bistochastic.

Proof: The arguments leading to Lemma 5.5 show that the generator of the
reduced dynamics ${\cal W}_{0}(0,t)\tilde{\cal L}_{t}^{1}{\cal W}_{0}(t,0)$
has the following matrix form in the basis $\\{P_{1}(0),P_{2}(0)\\}$ (dropping
the variable $t$ from the notation)

$\displaystyle\tilde{L}=$ $\displaystyle\sum_{l\in
I}\begin{pmatrix}|\langle\varphi_{1}|\Gamma_{l}\varphi_{1}\rangle|^{2}-\|\Gamma_{l}\varphi_{1}\|^{2}&|\langle\varphi_{1}|\Gamma_{l}\varphi_{2}\rangle|^{2}\cr|\langle\varphi_{2}|\Gamma_{l}\varphi_{1}\rangle|^{2}&|\langle\varphi_{2}|\Gamma_{l}\varphi_{2}\rangle|^{2}-\|\Gamma_{l}\varphi_{2}\|^{2}\cr\end{pmatrix}=\sum_{l\in
I}\begin{pmatrix}-|\langle\varphi_{2}|\Gamma_{l}\varphi_{1}\rangle|^{2}&|\langle\varphi_{1}|\Gamma_{l}\varphi_{2}\rangle|^{2}\cr|\langle\varphi_{2}|\Gamma_{l}\varphi_{1}\rangle|^{2}&-|\langle\varphi_{1}|\Gamma_{l}\varphi_{2}\rangle|^{2}\cr\end{pmatrix},$
(6.10)

thanks to property (4.16). The assumed symmetry (6.3) allows us to get
(restoring the time variable)

$\displaystyle\tilde{L}(t)=$ $\displaystyle\sum_{l\in
I}|\langle\varphi_{2}(t)|\Gamma_{l}(t)\varphi_{1}(t)\rangle|^{2}\begin{pmatrix}-1&1\cr
1&-1\cr\end{pmatrix}=\gamma(t)\begin{pmatrix}-1&1\cr 1&-1\cr\end{pmatrix}.$
(6.11)

Therefore, in the same basis, the reduced dynamics solution to (5.15) with
$\delta>0$ reads

$\displaystyle\tilde{\Psi}_{\delta}(t,s)|_{{\,\rm
Span}\\{P_{1}(0),P_{2}(0)\\}}=\frac{1}{2}\begin{pmatrix}1&1\cr
1&1\cr\end{pmatrix}+{\rm
e}^{-\frac{2}{\delta}\int_{s}^{t}\gamma(u)du}\frac{1}{2}\begin{pmatrix}1&-1\cr-1&1\cr\end{pmatrix}.$
(6.12)

Then, Corollary 5.4 together with Remark 3.6 i), yield (6.5) and (6.7). $\Box$

##  7 Generalisation

We present here a generalisation of the results concerning the perturbative
regime to arbitrary high order in the adiabatic parameter. This is made
possible by the use of a systematic improvement of the adiabatic
approximations of the Schrödinger propagator $U(t,s)$ (3.2), allowed by our
general setup, see e.g. [ASY, N2, JP2]. We briefly present here the approach
of [JP2] based on a hierarchy labelled by $q\in{\mathbb{N}}$ of smooth
hamiltonians in ${\cal B}({\cal H})$, before spelling out the improvement it
brings to the leading order results of Section 3.

###  7.1 Higher order adiabatics

Set

$\displaystyle H^{0}(t)$ $\displaystyle=$ $\displaystyle H(t)$ (7.1)
$\displaystyle P_{j}^{0}(t)$ $\displaystyle=$ $\displaystyle P_{j}(t),\
\forall 1\leq j\leq d$ (7.2) $\displaystyle K^{0}(t)$ $\displaystyle=$
$\displaystyle K(t).$ (7.3)

and define the self-adjoint operator

$H^{1}(t)=H(t)-{\rm i}\varepsilon K^{0}(t).$ (7.4)

For $\varepsilon$ small enough, the gap hypothesis Spec holds for all
$t\in[0,1]$, and we set for all $j\in\\{1,\dots,d\\}$, $\varepsilon$ small
enough, with ′ denoting the time derivative,

$\displaystyle P_{j}^{1}(t)$ $\displaystyle=$ $\displaystyle-\frac{1}{2\pi
i}\int_{\gamma_{j}}(H^{1}(t)-z)^{-1}\,dz,$ (7.5) $\displaystyle K^{1}(t)$
$\displaystyle=$ $\displaystyle\sum_{1\leq j\leq
d}{P_{j}^{1}}^{\prime}(t)P_{j}^{1}(t).$ (7.6)

Note that $H^{1}$, $P^{1}$, and $K^{1}$ are $\varepsilon$-dependent and smooth
on $[0,1]$. We define inductively, for $\varepsilon$ small enough, the
following hierarchy of operators for $q\geq 1$, and all $j\in\\{1,\dots,d\\}$

$\displaystyle H^{q}(t)$ $\displaystyle=$ $\displaystyle H(t)-{\rm
i}\varepsilon K^{q-1}(t)$ (7.7) $\displaystyle P_{j}^{q}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi i}\int_{\gamma_{j}}(H^{q}(t)-z)^{-1}\,dz,$ (7.8)
$\displaystyle K^{q}(t)$ $\displaystyle=$ $\displaystyle\sum_{1\leq j\leq
d}{P_{j}^{q}}^{\prime}(t)P_{j}^{q}(t).$ (7.9)

It is proven in [JP2], see also [JP3], that in our $C^{\infty}$ framework, the
following holds:

###### Proposition 7.1

For any $q\in{\mathbb{N}}^{*}$, there exists $\varepsilon_{q}>0$ and
$C_{q},\kappa_{q}<\infty$, such that for all $\varepsilon\leq\varepsilon_{q}$,
$t\in[0,1]$, $j\in\\{1,\dots,d\\}$, $H^{n}(t),P_{j}^{n}(t),K^{n}(t)$ are well
defined and smooth for all $0\leq n\leq q$. Moreover, these operators and all
their derivatives admit an asymptotic expansion in powers of $\varepsilon$ and
the following estimates hold

$\displaystyle\|K^{q}(t)-K^{q-1}(t)\|\leq C_{q}\varepsilon^{q}$ (7.10)
$\displaystyle\|K^{q}(t)\|\leq\kappa_{q}.$ (7.11)

###### Remark 7.2

i) The hierarchy above was actually designed to reach exponential accuracy in
the adiabatic approximation, in an analytic framework, in which case it
provides an estimate on the behaviour in $q$ of the constant $C_{q}$.
ii) At $t=0$, the second point of assumption Reg ensures that for all
$q\in{\mathbb{N}}^{*}$, $H^{q}(0)=H(0)$ and $P^{q}(0)=P(0)$.
iii) For any time $t$, $q\in{\mathbb{N}}^{*}$, $1\leq j\leq d$,
$P_{j}^{q}(t)=P_{j}(t)+{\cal O}(\varepsilon)$, by perturbation theory.

Let $\varepsilon<\varepsilon_{q}$ and consider the unitary propagator
$V_{q}(t,s)_{0\leq s,t\leq 1}$, defined as the solution to

$\displaystyle\left\\{\begin{matrix}{\rm
i}\varepsilon\partial_{t}V_{q}(t,s)=(H^{q}(t)+{\rm i}\varepsilon
K^{q}(t))V_{q}(t,s),\\\ V_{q}(s,s)=\mathbb{I},\ \ 0\leq s,t\leq
1.\hfill\end{matrix}\right.$ (7.12)

As is well known, see [K2, Kr], $V_{q}$ also satisfies

$V_{q}(t,s)P_{j}^{q}(s)=P_{j}^{q}(t)V_{q}(t,s),\ \ 0\leq s,t\leq 1.$ (7.13)

Note that since $H^{q}=H-{\rm i}\varepsilon K^{q-1}$, we get that

$H^{q}(t)+{\rm i}\varepsilon K^{q}(t)=H(t)+{\rm
i}\varepsilon(K^{q}(t)-K^{q-1}(t))$ (7.14)

is a smooth perturbation of $H(t)$. Thus, the difference between $U(t,s)$ and
$V_{q}(t,s)$ reads

$U(t,s)-V_{q}(t,s)=-\int_{s}^{t}V_{q}(t,r)(K^{q}(r)-K^{q-1}(r))U(r,s)dr.$
(7.15)

This identity and the previous proposition immediately yield

$\|U(t,s)-V_{q}(t,s)\|\leq C_{q}|t-s|\varepsilon^{q}.$ (7.16)

We improve the error term to ${\cal O}(\varepsilon^{q+1})$ by performing an
integration by parts on (7.15) (see the Appendix), at the cost of slightly
altering the definition of $V_{q}$: Let $\hat{V}_{q}(t,s)$ be the unitary
solution to

$\displaystyle\left\\{\begin{matrix}{\rm
i}\varepsilon\partial_{t}\hat{V}_{q}(t,s)=(H^{q}(t)+{\rm
i}\varepsilon(K^{q}(t))+{\cal D}_{q}K^{q-1}(t))\hat{V}_{q}(t,s),\\\
\hat{V}_{q}(s,s)=\mathbb{I},\ \ 0\leq s,t\leq 1,\hfill\end{matrix}\right.$
(7.17)

where

${\cal D}_{q}K^{q-1}(t)=\sum_{1\leq j\leq
d}P^{q}_{j}(t)K^{q-1}(t)P^{q}_{j}(t),$ (7.18)

and, by convention, $K^{-1}=0$ to recover $\hat{V}_{0}=V$. This allows us to
get the following generalisations of (3.7) and (3.8), see [JP3].

###### Proposition 7.3

Under assumptions Reg and Spec, for all $q\in{\mathbb{N}}$, there exists
$\varepsilon_{q}>0$ and $c_{q}<\infty$ such that for all
$\varepsilon<\varepsilon_{q}$, all $j\in\\{1,\dots,d\\}$, and for all
$(t,s)\in[0,1]^{2}$

$\displaystyle\hat{V}_{q}(t,s)P_{j}^{q}(s)=P_{j}^{q}(t)\hat{V}_{q}(t,s)$
$\displaystyle\|U(t,s)-\hat{V}_{q}(t,s)\|\leq c_{q}\varepsilon^{q+1}$ (7.19)

###### Remark 7.4

i) As a consequence, the quantum evolution follows the instantaneous subspace
$P_{j}^{q}(t){\cal H}$, up to an error of order $\varepsilon^{q+1}$:
$\|P_{k}^{q}(t)U(t,s)P_{j}^{q}(s)\|={\cal O}(\varepsilon^{q+1})$ if $j\neq k$.
ii) The loss of factor $|t-s|$ stems from the integration by parts procedure,
see (7.16). Again, for $s=0$, we have $\|U(t,0)-\hat{V}_{q}(t,0)\|\leq
c_{q}t\varepsilon^{q+1}$.
iii) For $s=0$, $j\neq k$, $\|P_{k}^{q}(t)U(t,0)P_{j}(0)\|={\cal
O}(t\varepsilon^{q+1})$, since $P_{j}^{q}(0)=P_{j}(0)$.

###  7.2 Higher Order Adiabatic Dyson expansion

Making use of the adiabatic approximation $\hat{V}_{q}(t,s)$ of $U(t,s)$ on
${\cal H}$ leads to an approximation of ${\cal U}^{0}(t,s)$ on ${\cal T}({\cal
H})$ to order ${\cal O}(\varepsilon^{q+1})$ and to the improvement of
Proposition 3.9 given in Proposition 7.5.

Let $\varepsilon\leq\varepsilon_{q}$ and define the isometric operator on
${\cal T}({\cal H})$ (and on ${\cal B}({\cal H})$)

${\cal V}^{0}_{q}(t,s)(\rho)=\hat{V}_{q}(t,s)\rho\hat{V}_{q}{}^{*}(t,s),\ \
\rho\in{\cal T}({\cal H}).$ (7.20)

Then, for $c_{q}$ given in Proposition 7.3, we get

$\displaystyle\|{\cal U}^{0}(t,s)-{\cal V}^{0}_{q}(t,s)\|_{\tau}\leq
2c_{q}\varepsilon^{q+1},$ (7.21)

and the same holds for the subordinate operator norm on ${\cal B}({\cal H})$.
If $\dim P_{j}(0)<\infty$, then $P_{j}^{q}(t)$ belongs to ${\cal T}({\cal H})$
for all $t\in[0,1]$ and $q\geq 0$, so that

$\displaystyle{\cal U}^{0}(t,s)(P_{j}^{q}(s))$ $\displaystyle=$ $\displaystyle
P^{q}_{j}(t)+{\cal O}(\varepsilon^{q+1}),$ (7.22) $\displaystyle{\cal
U}^{0}(t,0)(P_{j}(0))$ $\displaystyle=$ $\displaystyle P^{q}_{j}(t)+{\cal
O}(t\varepsilon^{q+1}),$ (7.23)

see Remark ii), 7.4. If $P_{j}(0)$ is not trace class, the estimates above
hold in operator norm. Consequently, the first equality in equation (3.5) and
the above yields the following estimate of the propagator $({\cal
U}(t,s))_{0\leq s\leq t\leq 1}$:

###### Proposition 7.5

Under assumptions Reg and Spec, for any $N\geq 1$, any $q\geq 1$, there exist
$\varepsilon_{q}>0$, $c_{q}<\infty$ (given in Propositions 7.1 and 7.3), such
that for all $0\leq s\leq t\leq 1$, for all $\varepsilon<\varepsilon_{q}$, all
$g\geq 0$, the propagator ${\cal U}(t,s)\in{\cal B}({\cal T}({\cal H}))$
satisfies (with the convention $s_{0}=t$),

$\displaystyle{\cal U}(t,s)$ $\displaystyle={\cal V}^{0}_{q}(t,s)$
$\displaystyle+\sum_{n=1}^{N}(g/\varepsilon)^{n}\int_{s}^{t}\int_{s}^{s_{1}}\dots\int_{s}^{s_{n-1}}{\cal
V}^{0}_{q}(t,s_{1})\circ{\cal L}^{1}_{s_{1}}\circ{\cal
V}^{0}_{q}(s_{1},s_{2})\circ{\cal L}^{1}_{s_{2}}\dots\circ{\cal
L}^{1}_{s_{n}}\circ{\cal V}^{0}_{q}(s_{n},s)ds_{n}\dots ds_{2}ds_{1}$
$\displaystyle+R^{q}_{N+1}(t,s,\varepsilon,g)$ (7.24)

where, with $L_{1}=\sup_{0\leq s\leq 1}\|{\cal L}_{s}^{1}\|_{\tau}$,

$\displaystyle\|R^{q}_{N+1}(t,s,\varepsilon,g)\|_{\tau}\leq
2c_{q}\varepsilon^{q+1}{\rm
e}^{2(t-s)L_{1}(1+2c_{q}\varepsilon^{q+1})g/\varepsilon}+\frac{(L_{1}(t-s))^{N+1}}{(N+1)!}(g/\varepsilon)^{N+1}.$
(7.25)

In particular, if $g/\varepsilon\leq 1$ and $\varepsilon^{q+1}\leq
1/(2c_{q})$,

$\displaystyle\|R^{q}_{N+1}(t,s,\varepsilon,g)\|_{\tau}$ $\displaystyle\leq
2{\rm
e}^{4L_{1}}\left(c_{q}\varepsilon^{q+1}+((t-s)g/\varepsilon)^{N+1}\right)$
$\displaystyle={\cal O}_{q}(\varepsilon^{q+1}+(g/\varepsilon)^{N+1}).$ (7.26)

where the notation stresses the dependence in the sole order $q$ of the
constants involved.

###### Remark 7.6

If one keeps ${\cal U}^{0}$ instead of ${\cal V}_{q}^{0}$ in the first term of
the RHS of (7.5), the estimate on the remainder reads
$\|R^{q}_{N+1}(t,s,\varepsilon,g)\|_{\tau}={\cal
O}_{q}((g/\varepsilon)(\varepsilon^{q+1}+(g/\varepsilon)^{N}))$.

Proof: We replace ${\cal U}^{0}$ by its approximation ${\cal V}^{0}_{q}$ in
each term of the Dyson series, and collect the error terms. With
$\Delta_{q}=\|{\cal U}^{0}(t,s)-{\cal V}_{q}^{0}(t,s)\|_{\tau}$, the trace
norm of the difference of the term of order $n\geq 1$ in (3.5) with that of
order $n$ in (7.5) is bounded above as in (3.2) with $\Delta_{q}$ in place of
$\Delta$. Then, summing over all $n\in{\mathbb{N}}$ yields the first term in
(7.25), while the second one stems from the term of order $N+1$ in (3.5). The
second estimate is a consequence of $g/\varepsilon\leq 1$, $t-s\leq 1$,
$1+2c_{q}\varepsilon^{q+1}\leq 2$ and $\frac{\alpha^{m}}{m!}\leq e^{\alpha}$,
for all $m\geq 1$, $\alpha>0$. $\Box$

Specialising to the leading order term in $g/\varepsilon$, and taking into
account Remark 7.6 above, we get

###### Corollary 7.7

Under the assumptions of Proposition 7.5, for
$\varepsilon\leq\tilde{\varepsilon}_{q}=\min(\varepsilon_{q},1/(2c_{q})^{1/(q+1)})$
and $g/\varepsilon\leq 1$,

$\displaystyle{\cal U}(t,s)$ $\displaystyle={\cal
U}^{0}(t,s)+\frac{g}{\varepsilon}\int_{s}^{t}{\cal
V}^{0}_{q}(t,s_{1})\circ{\cal L}^{1}_{s_{1}}\circ{\cal
V}^{0}_{q}(s_{1},s)ds_{1}+{\cal O}(g\varepsilon^{q}+g^{2}/\varepsilon^{2}).$
(7.27)

Let $0\leq\rho_{j}\in{\cal T}({\cal H})$ be a state such that
$\rho_{j}=P_{j}(0)\rho_{j}P_{j}(0)$, and recall the definition (2.1) of the
dissipator. For any $q\in{\mathbb{N}}$, the transition probability between
$P_{j}^{q}(0){\cal H}=P_{j}(0){\cal H}$ and $P_{k}^{q}(t){\cal H}$, $j\neq k$,
induced by the Lindbladian dynamics (2.3) reads ${\,\rm Tr}(P^{q}_{k}(t){\cal
U}(t,0)(\rho_{j}))$, since at initial time $s=0$, one has
$P_{j}(0)=P^{q}_{j}(0)$. Using (7.20) and Proposition 7.3, we have

${\cal
V}^{0}_{q}(s,0)(\rho_{j})=\hat{V}_{q}(s,0)\rho_{j}\hat{V}_{q}(0,s)=P^{q}_{j}(s)\hat{V}_{q}(s,0)\rho_{j}\hat{V}_{q}(0,s)P^{q}_{j}(s),$
(7.28)

so that with $P_{j}^{q}(t)P^{q}_{k}(t)=0$, we get from (7.27)

$\displaystyle{\,\rm Tr}(P^{q}_{k}(t){\cal U}(t,0)$
$\displaystyle(\rho_{j}))={\,\rm Tr}(P^{q}_{k}(t){\cal U}^{0}(t,0)(\rho_{j}))$
(7.29) $\displaystyle+\frac{g}{\varepsilon}\sum_{l\in I}\int_{0}^{t}{\,\rm
Tr}(P^{q}_{k}(s)\Gamma_{l}(s)\hat{V}_{q}(s,0)\rho_{j}\hat{V}_{q}(0,s)\Gamma_{l}^{*}(s)P^{q}_{k}(s))ds+{\cal
O}(g\varepsilon^{q}+g^{2}/\varepsilon^{2}).$

The Hamiltonian adiabatic transition probability between these subspaces is of
order $\varepsilon^{2(q+1)}$ according to (7.3), whereas the effect of the
environment is of order $g/\varepsilon$: due to (7.21), ${\cal
V}^{0}_{q}(t,0)(\rho_{j})={\cal V}^{0}(t,0)(\rho_{j})+{\cal O}(\varepsilon)$
in trace norm, so that by Lemma 3.7 the dependence in $\varepsilon$ of
$\hat{V}_{q}(s,0)\rho_{j}\hat{V}_{q}(0,s)$ disappears to leading order when
$\sigma_{j}(t)=\\{e_{j}(t)\\}$. In case $P_{j}(0)$ has finite rank, choosing
$\rho_{j}=P_{j}(0)/\dim(P_{j}(0))$ yields the integrand (see iii), Remark 7.2)
:

${\,\rm
Tr}(P^{q}_{k}(s)\Gamma_{l}(s)P^{q}_{j}(s)\Gamma_{l}^{*}(s)P^{q}_{k}(s))={\,\rm
Tr}(P_{k}(s)\Gamma_{l}(s)P_{j}(s)\Gamma_{l}^{*}(s)P_{k}(s))+{\cal
O}(\varepsilon).$ (7.30)

Hence, the correction term prevents the solution from following the
instantaneous subspace $P^{q}_{j}(t)$ up to an error of order
$\varepsilon^{(q+1)}$, unless $g\simeq\varepsilon^{q+2}$.

The coherences with respect to the iterated projectors of the integral term in
(7.27) also vanish to leading order, thanks to (7.21) and Lemma 3.12:

###### Lemma 7.8

Assume Reg, Spec and let $\rho_{j}=P_{j}(0)\rho_{j}P_{j}(0)$ be a state and
suppose $\sigma_{j}(t)=\\{e_{j}(t)\\}$ for all $t\in[0,1]$. For any $1\leq
n\neq m\leq d$, and $q\geq 0$, for $\varepsilon$ small enough

$\frac{g}{\varepsilon}P^{q}_{n}(t)\int_{0}^{t}{\cal V}^{0}_{q}(t,s)\circ{\cal
L}^{1}_{s}\circ{\cal V}^{0}_{q}(s,0)(\rho_{j})ds\,P^{q}_{m}(t)={\cal O}(g).$
(7.31)

Without going into the details, we note that a similar result holds for each
term in (7.5).

Let us close this section by justifying the adiabatic expressions used
throughout the paper for the populations and coherences of ${\cal
U}^{0}(t,0)(\rho_{j})$, making use of the hierarchy (7.7).

Proof of Proposition 3.14:

Thanks to (7.21) and Proposition 7.3 for $q=2$ we have

$\displaystyle P_{k}(t){\cal U}^{0}(t,0)(\rho_{j})P_{k}(t)$
$\displaystyle=P_{k}(t)P_{j}^{2}(t)\hat{V}_{2}(t,0)\rho_{j}\hat{V}_{2}(0,t)P_{j}^{2}(t)P_{k}(t)+{\cal
O}(\varepsilon^{3})$ $\displaystyle=P_{k}(t)P_{j}^{2}(t){\cal
V}_{2}^{0}(t,0)(\rho_{j})P_{j}^{2}(t)P_{k}(t)+{\cal O}(\varepsilon^{3}).$
(7.32)

For $j\neq k$, we have (dropping the variable $t$ in the notation)

$\displaystyle P_{k}P_{j}^{2}$
$\displaystyle=P_{k}(P_{j}^{2}-P_{j})=P_{k}(P_{j}^{2}-P_{j}^{1}+P_{j}^{1}-P_{j}).$
(7.33)

By perturbation theory see e.g. [K2], Proposition 7.1 implies for
$\varepsilon$ small enough,

$P_{j}^{q}-P^{q-1}_{j}={\cal O}(H^{q}-H^{q-1})={\cal
O}(\varepsilon(K^{q-1}-K^{q-2}))={\cal O}(\varepsilon^{q}),$ (7.34)

so that

$\displaystyle P_{k}P_{j}^{2}$ $\displaystyle=P_{k}(P_{j}^{1}-P_{j})+{\cal
O}(\varepsilon^{2}).$ (7.35)

Then, $H^{1}=H-{\rm i}\varepsilon K$ with $K$ given by (2.9) yields,

$P_{j}^{1}-P_{j}=-\frac{\varepsilon}{2\pi}\oint_{\gamma_{j}}R(z)KR(z)dz+{\cal
O}(\varepsilon^{2})$ (7.36)

so that making use of (7.21), to write ${\cal V}_{2}^{0}(t,0)(\rho_{j})={\cal
V}^{0}(t,0)(\rho_{j})+{\cal O}(\varepsilon)$, we have

$\displaystyle P_{k}P_{j}^{2}{\cal V}_{2}^{0}(\rho_{j})P_{j}^{2}P_{k}$
$\displaystyle=-\frac{\varepsilon^{2}}{(2\pi)^{2}}P_{k}\oint_{\gamma_{j}}R(z)P_{k}^{\prime}R(z)dz\,{\cal
V}^{0}(\rho_{j})\oint_{\gamma_{j}}R(z)P_{k}^{\prime}R(z)dzP_{k}+{\cal
O}(\varepsilon^{3}).$ (7.37)

Hence we get (3.40) with ${\cal V}^{0}(t,0)(\rho_{j})$ in place of
$\tilde{\rho}_{j}(t)$, as in Remark 3.15. Finally, assuming
$\sigma_{j}(t)=\\{e_{j}(t)\\}$, Lemma 3.7 yields (3.40). In case
$HP_{k}=e_{k}P_{k}$ and $HP_{j}=e_{j}P_{j}$, so that
$R(z)P_{n}=P_{n}/(e_{n}-z)$ for $n\in\\{j,k\\}$ and $z\in\rho(H)$, a direct
application of Cauchy formula yield (3.41).

The expressions (3.14) for the coherences are proven quite similarly. $\Box$

##  8 Appendix: Integration by Parts

We present here a reformulation of the integration by parts argument used in
[ASY] to prove the adiabatic theorem of quantum mechanics, suited to our
setup.

Let ${\cal Z}$ be a Banach space and assume ${\cal G}:[0,1]\rightarrow{\cal
B}({\cal Z})$, ${\cal K}:[0,1]\rightarrow{\cal B}({\cal Z})$ are bounded
operator valued $C^{\infty}$ functions on $[0,1]$, in the norm sense. Let
$\varepsilon>0$, and consider the two-parameter propagators $({\cal
X}(t,s))_{1\leq s\leq t\leq 1}$ and $({\cal Y}(t,s))_{1\leq s\leq t\leq 1}$,
solution to the equations

$\displaystyle\left\\{\begin{matrix}\varepsilon\partial_{t}{\cal X}(t,s)={\cal
G}(t){\cal X}(t,s),\hfill\cr{\cal X}(s,s)={\mathbb{I}},\ \ 0\leq s\leq t\leq
1,\hfill\end{matrix}\right.$ (8.1)

and

$\displaystyle\left\\{\begin{matrix}\varepsilon\partial_{t}{\cal
Y}(t,s)=({\cal G}(t)+\varepsilon{\cal K}(t)){\cal Y}(t,s),\cr{\cal
Y}(s,s)={\mathbb{I}},\ \ 0\leq s\leq t\leq 1.\hfill\end{matrix}\right.$ (8.2)

The smooth propagators ${\cal X}(t,s)$ and ${\cal Y}(t,s)$ are determined by
the corresponding Dyson series, both depend on $\varepsilon>0$ with norms that
diverge as $\varepsilon\rightarrow 0$, a priori . Moreover, they satisfy the
integral relation

${\cal X}(t,r)={\cal Y}(t,r)-\int_{r}^{t}{\cal Y}(t,s){\cal K}(s){\cal
X}(s,r)ds,\ \ \forall 1\geq t\geq r\geq 0.$ (8.3)

Assume the existence of gaps in the spectrum of ${\cal G}(t)$, uniformly in
$t\in[0,1]$. For $d\in{\mathbb{N}}^{*}$,

$\sigma({\cal G}(t))=\cup_{1\leq j\leq d}\,\sigma_{j}(t)\subset{\mathbb{C}},\
\ \inf_{t\in[0,1],1\leq j\neq k\leq d}{\rm
dist}(\sigma_{j}(t),\sigma_{k}(t))\geq G>0.$ (8.4)

Consider the corresponding spectral projector

${\cal P}_{j}(t)=-\frac{1}{2\pi{\rm i}}\int_{\gamma_{j}}({\cal
G}(s)-z)^{-1}dz,$ (8.5)

where $\gamma_{j}$ is a simple loop in $\rho({\cal G}(t))$, the resolvent set
of ${\cal G}(t)$, encircling $\sigma_{j}(t)$ and such that for all $k\neq j$,
$\mbox{int}\gamma_{j}\cap\sigma_{k}(t)=\emptyset$. For ${\cal
B}:[0,1]\rightarrow{\cal B}({\cal Z})$, a smooth bounded operator valued
function, define for any $t\in[0,1]$

${\cal R}_{j}({\cal B})(t)=-\frac{1}{2\pi{\rm i}}\oint_{\gamma_{j}}({\cal
G}(t)-z)^{-1}{\cal B}(t)({\cal G}(t)-z)^{-1}dz,$ (8.6)

with the same loop $\gamma_{j}$ as in (8.5). This operator is smooth as well,
and satisfies the identity

$[{\cal G}(t),{\cal R}_{j}({\cal B})(t)]=[{\cal B}(t),{\cal P}_{j}(t)].$ (8.7)

###### Remark 8.1

If $\sigma_{k}(t)=\\{g_{k}(t)\\}$ for all $1\leq k\leq d$, $({\cal
G}(t)-z)^{-1}=\sum_{1\leq k\leq d}{\cal P}_{k}(t)/(g_{k}(t)-z)$, and

${\cal R}_{j}({\cal B})(t)=\sum_{1\leq k\leq d\atop k\neq j}\frac{{\cal
P}_{j}(t){\cal B}(t){\cal P}_{k}(t)+{\cal P}_{k}(t){\cal B}(t){\cal
P}_{j}(t)}{g_{k}(t)-g_{j}(t)}.$ (8.8)

###### Lemma 8.2

Suppose ${\cal K}(t)$ is off-diagonal for all $t\in[0,1]$, i.e. s.t. ${\cal
P}_{j}(t){\cal K}(t){\cal P}_{j}(t)\equiv 0$, $\forall 1\leq j\leq d$. Then

$\displaystyle{\cal X}(t,r)$ $\displaystyle-{\cal
Y}(t,r)=\frac{1}{2}\sum_{1\leq j\leq d}\varepsilon\Big{(}{\cal R}_{j}([{\cal
K},{\cal P}_{j}])(t){\cal X}(t,r)-{\cal Y}(t,r){\cal R}_{j}([{\cal K},{\cal
P}_{j}])(r)\Big{)}$ (8.9) $\displaystyle+\frac{1}{2}\sum_{1\leq j\leq
d}\varepsilon\int_{r}^{t}\Big{\\{}{\cal Y}(t,s){\cal K}(s)[{\cal G}(s),{\cal
R}_{j}([{\cal K},{\cal P}_{j}])(s)]{\cal X}(s,r)-{\cal
Y}(t,s)(\partial_{s}{\cal R}_{j}([{\cal K},{\cal P}_{j}])(s)){\cal
X}(s,r)\Big{\\}}ds.$

Proof: The operator ${\cal K}$ being off-diagonal and (8.7) give

${\cal K}(t)=\frac{1}{2}\sum_{1\leq j\leq d}\big{[}[{\cal K}(t),{\cal
P}_{j}(t)\big{]},{\cal P}_{j}(t)]=\frac{1}{2}\sum_{1\leq j\leq d}\big{[}{\cal
G}(t),{\cal R}_{j}([{\cal K},{\cal P}_{j}])(t)\big{]}.$ (8.10)

Hence, using (8.3), (8.1) and (8.2),

$\displaystyle{\cal X}(t,r)-{\cal Y}(t,r)=$
$\displaystyle-\frac{1}{2}\sum_{1\leq j\leq d}\int_{r}^{t}{\cal