# Eternally non-Markovian dynamics of a qubit interacting with a single-photon
wavepacket

Anita Da̧browska  Institute of Theoretical Physics and Astrophysics,
University of Gdańsk, ul. Wita Stwosza 57, 80-308 Gdańsk, Poland  Dariusz
Chruściński  Institute of Physics, Faculty of Physics, Astronomy and
Informatics, Nicolaus Copernicus University, Grudziadzka 5/7, 87-100 Toruń,
Poland  Sagnik Chakraborty  Institute of Physics, Faculty of Physics,
Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5/7,
87-100 Toruń, Poland  Gniewomir Sarbicki  Institute of Physics, Faculty of
Physics, Astronomy and Informatics, Nicolaus Copernicus University,
Grudziadzka 5/7, 87-100 Toruń, Poland

###### Abstract

An evolution of a two-level system (qubit) interacting with a single-photon
wave packet is analyzed. It is shown that a hierarchy of master equations
gives rise to phase covariant qubit evolution. The temporal correlations in
the input field induce nontrivial memory effects for the evolution of a qubit.
It is shown that in the resonant case whenever time-local generator is regular
(does not display singularities) the qubit evolution never displays
information backflow. However, in general the generator might be highly
singular leading to intricate non-Markovian effects. A detailed analysis of
the exponential profile is provided which allows to illustrate all
characteristic feature of the qubit evolution.

single-photon state, open quantum systems, non-Markovianity, master equation

##  I Introduction

Any real quantum system is never perfectly isolated and hence has to be
treated as an open system BP02 . Therefore, its evolution is no longer unitary
and gives rise to well known processes of dissipation, decay, and decoherence
induced by the nontrivial system–environment interaction Alicki . When the
interaction between system and environment is sufficiently weak and the
characteristic time scales are well separated one uses well known Markovian
approximation governed by the celebrated Markovian master equation
$\dot{\rho}=\mathcal{L}\rho$, with $\mathcal{L}$ being the Gorini-Kossakowski-
Lindblad-Sudarshan (GKLS) generator GKS ; L

$\mathcal{L}(\rho)=-i[H_{S},\rho]+\sum_{k}\gamma_{k}\Big{(}L_{k}\rho
L_{k}^{\dagger}-\frac{1}{2}\\{L_{k}^{\dagger}L_{k},\rho\\}\Big{)},$ (1)

where $H_{S}$ stands for the effective system’s Hamiltonian (including Lamb
shift correction), $L_{k}$ are jump (Lindblad) operators, and $\gamma_{k}\geq
0$ (in what follows we keep $\hbar=1$). In non-Markovian regime due to
correlations between the system and environment the reduced evolution of the
system is no longer governed by (1). One observes characteristic memory
effects such as information bacflow or recoherence. Laboratory techniques of
quantum engineering enable nowadays to test experimentally theoretical
descriptions and concepts of non-Markovian quantum dynamics. Scope of
applications of these theoretical tools is constantly growing and the field of
non-Markovian quantum processes attracts a lot of interest (cf. recent reviews
NM1 ; NM2 ; NM3 ; NM4  together with Piilo1 ; Piilo2 ; Modi2 ; Modi3  and the
recent tutorial Modi4 ).

In this paper we study non-Markovian dynamics of the two-level system (qubit)
interacting with a single-photon wave packet and the vacuum field. So we
consider a scattering of a single-photon field on the two-level system.
Propagating wave packets of light of definite numbers of photons play a key
role in photonics. The continuous-mode single-photon state L00 ; M08  is an
example of non-classical states of light having applications in quantum
computation KLM01 ; Ralph03 , metrology D08 ; Munro02 , communication
Scarani09 , and simulation Aaronon11 . Constantly developing techniques of
producing, storing and detecting of single-photon states of light can serve in
a wide range of practical applications CWSS13 ; PSZ13 ; RR15 ; Leong2016 ;
Zoller2017 .

We describe the evolution of qubit which is driven by the single-photon field
and at the same time it undergoes damping process. Note that the single-photon
field not only drives the system but it is also the source of stimulated
emission. The key ingredient of our model are temporal correlations present in
the input state of the field which is characterised by the time dependent
profile $\xi(t)$ (for $t\geq 0$). The temporal correlations are responsible
for all non-Markovian memory effects of the qubit dynamics. The reduced
evolution of the qubit, obtained within the input-output formalism GarCol85 ,
is represented then by a hierarchy of coupled equations GEPZ98 ; WMSS11 ;
BCBC12 ; Gough12a ; Baragiola17 ; DSC17 ; DSC19 ; D20 . We provide the
analytical solution to this hierarchy of equations for any initial state of
the system and an arbitrary profile $\xi(t)$ and use it to show that the set
of these equations is equivalent to a single time-local master equation. The
price one pays for this reduction is highly nontrivial structure of time-
dependent rates in the time-local master equation. The corresponding time-
local generator is phase covariant and its general properties, recently
studied, providing us with the necessary tools to analyze non-Markovanity
effects SKHD16 ; Sabrina-2016 ; Sabrina-NJP ; Francesco ; HSH19 ; Sergey-2020
. The time dependent rates governing damping (cooling), heating and
decoherence processes are fully characterized by the wave packet profile. Such
formulae were never presented before. An immediate consequence of our analysis
is the observation that in general the dynamical map governing the qubit
evolution is not invertible which implies the singularity of rates in the
corresponding time-local master equation. Similar observation for phase
covariant dynamics was recently reported in Francesco . Non-invertible maps
were considered in quantum optics scenario, for instance, in Cresser1 ;
Cresser2 ; Hou ; SKHD16  (cf. also PRL-2018 ; CC19  and recent paper
Jyrki-2021 ).

The analyzed system provides excellent platform to test indicators of non-
Markovianity. It turns out that in the resonant case, which is of particular
importance in photonic experiments, the evolution is never CP-divisible but it
also never displays information backflow. We provide a detailed analysis of
the qubit evolution interacting with exponential profile. In this case we are
able to derive both analytical formulae for time-local rates and characterize
the regime when the dynamics does not allow for information backflow.

The paper is organized as follows. Section II introduces basic properties of
dynamical maps and discusses recent concepts of Markovianity. In Section III
we introduce the hierarchy equations describing the reduced evolution of the
two-level system interacting with the single-photon field and the vacuum, and
eventually provide the solution for the qubit evolution. Sections IV and V
contain the general properties of qubit evolution and the corresponding time-
local generator, respectively. In Section VI the detailed analysis of the
exponential profile is carried out: we discuss both the resonant and off
resonant case. The key observation is that for an exponential profile the
qubit evolution is eternally non-Markovian Erika , that is, the corresponding
time-local generator always contains a strictly negative transition rate for
any time $t>0$. Actually, we conjecture that this characteristic feature holds
for an arbitrary photon profile. Final conclusions are presented in section
VII. Technical details are presented in the appendices.

##  II Preliminaries

Evolution of a quantum system is represented by a dynamical map
$\\{\Lambda_{t}\\}_{t\geq 0}$ which maps an initial state $\rho$ at $t=0$ into
a state at a current time $t$, i.e. $\rho(t)=\Lambda_{t}(\rho)$. Usually, the
map is realized as a reduced evolution of the system + environment

$\rho(t)=\mathrm{Tr}_{E}\left(U(t)\rho\otimes\rho_{E}U^{\dagger}(t)\right),$
(2)

where $U(t)$ is the unitary evolution operator corresponding to the total
Hamiltonian $H=H_{S}+H_{E}+H_{int}$, and ${\rm Tr}_{E}$ denotes the partial
trace over the environmental degrees of freedom BP02 . As a result (2) gives
rise to a map $\rho\to\rho(t)$ which is completely positive and trace
preserving (CPTP). Recall that any such map allows for the Kraus
representation

$\Lambda_{t}(\rho)=\sum_{\alpha}K_{\alpha}(t)\rho K_{\alpha}^{\dagger}(t),$
(3)

where the time-dependent Kraus operator $K_{\alpha}(t)$ satisfies the
additional normalization condition
$\,\sum_{\alpha}K_{\alpha}^{\dagger}(t)K_{\alpha}(t)=\mathbbm{1}\,$ which
guaranties that the map $\Lambda_{t}$ is trace-preserving, i.e. ${\rm
Tr}\,\Lambda_{t}(\rho)={\rm Tr}\,\rho$ for any $\rho$. Reduced dynamics
defined via (2) provides a generalization of Markovian dynamical semigroups
governed by GKLS generator (1). For differentiable quantum dynamical map the
system’s density operator $\rho(t)$ satisfies time-local master equation

$\dot{\rho}(t)=\mathcal{L}_{t}\rho(t),$ (4)

with time-dependent generator $\mathcal{L}_{t}$. Note, that given
$\\{\Lambda_{t}\\}_{t\geq 0}$ the time-local generator is formally defined by
$\,\mathcal{L}_{t}=\dot{\Lambda}_{t}\Lambda_{t}^{-1}$ provided the map
$\Lambda_{t}$ is invertible. If the map is not invertible at $t=\tau$, then
$\mathcal{L}_{t}$ displays singularities at $\tau$. This is not only a
mathematical curiosity. Non-invertible maps were already considered in quantum
optical systems Francesco ; Cresser1 ; Cresser2 ; Hou ; SKHD16  (cf. also
PRL-2018 ; CC19  and recent papers Piilo1 ; Piilo2 ). The system we study in
this paper gives rise to a perfectly regular map $\\{\Lambda_{t}\\}_{t\geq 0}$
which, however, needs not be invertible and hence the corresponding master
equation is governed by a singular generator. Interestingly, singularities of
$\mathcal{L}_{t}$ provide important physical insight into the dynamical
properties of the system.

Any time-local generator $\mathcal{L}_{t}$ has essentially the same structure
as (1), where now the effective Hamiltonian $H_{S}$, the noise operators
$L_{k}$, and the transition rates $\gamma_{k}$ are time-dependent. Given
$\mathcal{L}_{t}$ the formal solution for the dynamical map reads as follows

$\Lambda_{t}=\mathcal{T}\exp\left(\int_{0}^{t}\mathcal{L}_{\tau}d\tau\right),$
(5)

where $\mathcal{T}$ stands for the chronological product. It should be
stressed that in the time-dependent case transition rates $\gamma_{k}(t)$ are
not necessarily non-negative. This makes the characterization of admissible
generators extremely hard Erika ; Angel . Actually, temporal negativity of
$\gamma_{k}(t)$ is usually interpreted as manifestation of non-Markovianity. A
quantum dynamical map $\\{\Lambda_{t}\\}_{t\geq 0}$ is called divisible if for
any $t\geq s$ one has $\,\Lambda_{t}=V_{t,s}\Lambda_{s}$. Note, that any
invertible map is necessarily divisible since
$V_{t,s}=\Lambda_{t}\Lambda_{s}^{-1}$. However, for not invertible maps the
issue of divisibility is much more subtle BOGNA ; datta ; Acin-cor ; Johansson
; PRL-2018 ; CC19 ; Jyrki-2021 . In a recent paper Ujan  the construction of
propagators for non-invertible maps in terms of generalized inverse was
proposed. One calls the map $\\{\Lambda_{t}\\}_{t\geq 0}$ P-divisible if
$V_{t,s}$ is positive and trace-preserving, and CP-divisible if $V_{t,s}$ is
completely positive and trace-preserving. Following RHP  we call the evolution
represented by $\\{\Lambda_{t}\\}_{t\geq 0}$ Markovian if it is CP-divisible.
An invertible map is CP-divisible if and only if all rates satisfy
$\gamma_{k}(t)\geq 0$. This is a direct generalization of Markovian semigroup.
For non-invertible maps the full characterization is still missing PRL-2018 .
One of the most interesting implication of non-Markovianity is an information
backflow defined as follows BLP : if the evolution is Markovian, then for any
pair of initial states $\rho_{1}$ and $\rho_{2}$

$\frac{d}{dt}\|\Lambda_{t}(\rho_{1}-\rho_{2})\|_{1}\leq 0,$ (6)

where $\|X\|_{1}={\rm Tr}|X|$ denotes the trace norm of $X$. Note, that the
quantity $\|\rho_{1}-\rho_{2}\|_{1}$ describes the distinguishability of
$\rho_{1}$ and $\rho_{2}$. Now, the violation of (6) is interpreted as an
information backflow (from the environment to the system). In what follows we
call (6) a BLP condition. An interesting example of the qubit evolution which
satisfies BLP condition but is not CP-divisible was provided in Erika : the
corresponding time-local generator reads as follows

$\mathcal{L}_{t}(\rho)=\frac{1}{2}\sum_{k=1}^{3}\gamma_{k}(t)(\sigma_{k}\rho\sigma_{k}-\rho),$
(7)

with $\gamma_{1}=\gamma_{2}=1$ and $\gamma_{3}(t)=-\tanh t<0$. Interestingly,
one of the rate is always negative (for $t>0$) but still the corresponding
dynamical map is completely positive. The authors of Erika  call such
evolution eternally non-Markovian. This analysis was further generalized in
Nina  and Kasia  for qudit systems. In this paper we face another example of
qubit evolution for which for any $t>0$ some rate $\gamma_{k}(t)$ is negative.
Nevertheless, the evolution is represented by completely positive dynamical
map. This might be, therefore, considered as an another example of eternally
non-Markovian evolution. This example, however, corresponds to well motivated
physical system representing a qubit interacting with an incoming external
photon.

It should be stressed that dealing with evolution of open quantum systems the
very notion of Markovian evolution is often used in different contexts and
actually several concept of (non)Markovian evolution are available in the
current literature (cf. recent reviews NM1 ; NM2 ; NM3 ; NM4  and Piilo1 ;
Piilo2 ). Authors of NM4  shown an intricate hierarchy of various notions of
quantum Markovianity and clearly indicates that this concept is highly
context-dependent. In the present paper we are characterizing non-Markovian
process using only the properties of the corresponding dynamical map. An
interesting approach to quantum non-Markovianity beyond a dynamical map based
on so called quantum process tensor was recently analyzed in Modi2 ; Modi3
(cf. also recent tutorial Modi4 ). This approach was further applied in
Andrea1 ; Andrea2 ; Filippov-Modi  in the study of open quantum systems
dynamics.

A simple indicator of non-Markovianity was proposed in LPP13  (see also
further discussion in CMM17 ): for any Markovian evolution

$\frac{d}{dt}|{\rm det}\Lambda_{t}|\,\leq\,0.$ (8)

The volume of the accessible states at time $t$ is given by
$\mathrm{Vol}(t)=|{\rm det}\Lambda_{t}|\rm{Vol}(0)$ and hence (8) implies
monotonic decrees of $\rm{Vol}(t)$. Any P-divisible (and hence also CP-
divisible) satisfies (8). In what follows we analyze a qubit interacting with
an incoming photon field and show how the photon’s profile $\xi(t)$ decides
about the validity of BLP condition (6), geometric condition (8), and
eventually (non)Markovianity of the evolution.

##  III Hierarchy of master equations

Figure 1: The system S interacts from the left with a single-photon field
$|1_{\xi}\rangle$ and from the right with a vacuum field $|vac\rangle$. We
introduced $\Gamma=\Gamma_{1}+\Gamma_{2}$ and the parameter $\kappa\in[0,1]$
such that $\Gamma_{1}=\kappa\Gamma$, and $\Gamma_{2}=(1-\kappa)\Gamma$.

We consider a two-level system interacting by two channels with a continuous-
mode electromagnetic field L00  schematically depicted in Fig. 1. The system
interacts with the single-photon field incoming from the left and the vacuum
field incoming from the right. We assume that, in general, the coupling of the
system with the single-photon field and the vacuum could be different. In the
model some standard assumptions in quantum optics are made (rotating wave-
approximation, a flat coupling constant, and an extension of the lower limit
of integration over frequency to minus infinity). The bandwidth of the
spectrum is assumed to be much smaller that the central frequency of the pulse
$\Delta\omega\ll\omega_{c}$ and the central frequency $\omega_{c}$ is taken to
be close the transition frequency of the two-level system $\omega_{0}$. A
detailed discussion on the assumptions for the model one can find, for
instance, in FTVRS17 . We assume that the composed system is initially in the
state $\rho(0)\otimes|1_{\xi}\rangle\langle 1_{\xi}|\otimes|vac\rangle\langle
vac|$, where

$|1_{\xi}\rangle=\int_{-\infty}^{+\infty}d\omega\hat{\xi}(\omega)a^{\dagger}_{1}(\omega)|vac\rangle$
(9)

and $|vac\rangle$ is the continuous-mode vacuum state,
$a^{\dagger}_{i}(\omega)$, $a_{i}(\omega)$ $(i=1,2)$ are annihilation and
creation operators of $\omega$-mode satisfying the standard canonical
commutation relations

$[a_{i}(\omega),a^{\dagger}_{j}(\omega^{\prime})]=\delta_{ij}\delta(\omega-\omega^{\prime}).$
(10)

The profile of the photon in the time domain is defined as follows

$\xi(t)=\int_{-\infty}^{+\infty}d\omega\hat{\xi}(\omega)e^{-i(\omega-\omega_{c})t}$
(11)

with the normalization condition

$\int_{0}^{+\infty}dt|\xi(t)|^{2}=\int_{-\infty}^{+\infty}d\omega|\hat{\xi}(\omega)|^{2}=1.$
(12)

Note that the profile, $\xi(t)$, defines a slow varying envelope of the time
amplitude of the pulse. Since we are close to the resonance
$(\omega_{c}\approx\omega_{0}$) it is convenient to pass to the interaction
picture defined by the central frequency $\omega_{c}$, that is, a qubit state
$\rho(t)$ ‘rotates’ via $e^{-i\omega_{c}\sigma_{z}/2}$. The interaction
Hamiltonian in the interaction-picture has the form

$H_{int}(t)=i\sqrt{\Gamma_{1}}\left(\sigma_{-}a^{\dagger}_{1}(t)-\sigma_{+}a_{1}(t)\right)+i\sqrt{\Gamma_{2}}\left(\sigma_{-}a_{2}^{\dagger}(t)-\sigma_{+}a_{2}(t)\right),$
(13)

where $a_{i}(t)$ and $a^{\dagger}_{i}(t)$ are quantum white-noise operators
GZ10  obeying the communication relation

$[a_{i}(t),a_{j}^{\dagger}(t^{\prime})]=\delta_{ij}\delta(t-t^{\prime}).$ (14)

Using the ‘rotating frame’ one derives the following hierarchy of coupled
equations for the family of qubit operators ${\varrho}^{kl}$ with $k,l=0,1$

$\dot{\varrho}^{11}(t)=\mathcal{L}{\varrho}^{11}(t)+\sqrt{\Gamma_{1}}\;\xi^{\ast}(t)[\sigma_{-},{\varrho}^{10}(t)]-\sqrt{\Gamma_{1}}\;\xi(t)[\sigma_{+},{\varrho}^{01}(t)],$
(15)
$\dot{{\varrho}}^{10}(t)=\mathcal{L}{\varrho}^{10}(t)-\sqrt{\Gamma_{1}}\;\xi(t)\left[\sigma_{+},{\varrho}^{00}(t)\right],$
(16)
$\dot{{\varrho}}^{01}(t)=\mathcal{L}{\varrho}^{01}(t)+\sqrt{\Gamma_{1}}\;\xi^{\ast}(t)\left[\sigma_{-},{\varrho}^{00}(t)\right],$
(17) $\dot{{\varrho}}^{00}(t)=\mathcal{L}{\varrho}^{00}(t)$ (18)

with the super-operator

$\mathcal{L}({\varrho})=-\frac{i\Delta_{0}}{2}[{\varrho},\sigma_{z}]-\frac{\Gamma}{2}\left\\{\sigma_{+}\sigma_{-},{\varrho}\right\\}+\Gamma\sigma_{-}{\varrho}\sigma_{+},$
(19)

where $\Delta_{0}=\omega_{c}-\omega_{0}$, $\sigma_{z}=|e\rangle\langle
e|-|g\rangle\langle g|$, $\Gamma=\Gamma_{1}+\Gamma_{2}$, and
$\Gamma_{1},\Gamma_{2}\geq 0$. By $|e\rangle$ and $|g\rangle$ the excited and
ground states of the two-level system are respectively denoted. One defines
the density operator of the system $\rho(t):=\varrho^{11}(t)$, and initially

$\varrho^{11}(0)=\varrho^{00}(0)=\rho(0)\ ,\ \
\varrho^{01}(0)=\varrho^{10}(0)=0.$ (20)

Note, that $\varrho^{10}(t)=\left(\varrho^{01}(t)\right)^{\dagger}$. The
system is initially in an arbitrary state, $\rho(0)$. The set of equations
defines the qubit dynamics during the scattering process. The derivations of
the set of equations describing the evolution of a quantum system interacting
with the single-photon field was given, for instance, in GEPZ98 ; BCBC12 ;
WMSS11 ; Gough12a ; Baragiola17 . The details of its determination in a
collision model Ciccarello2021  one can find in DSC17 . Let us notice that the
above set of equations can be written in the form

$\frac{d}{dt}\begin{pmatrix}\varrho^{11}(t)\\\ \varrho^{10}(t)\\\
\varrho^{01}(t)\\\
\varrho^{00}(t)\end{pmatrix}=\begin{pmatrix}\mathcal{L}&\mathcal{L}_{\xi}&\mathcal{L}_{\xi}^{\ddagger}&0\\\
0&\mathcal{L}&0&\mathcal{L}_{\xi}^{\ddagger}\\\
0&0&\mathcal{L}&\mathcal{L}_{\xi}\\\
0&0&0&\mathcal{L}\end{pmatrix}\begin{pmatrix}\varrho^{11}(t)\\\
\varrho^{10}(t)\\\ \varrho^{01}(t)\\\ \varrho^{00}(t)\end{pmatrix},$ (21)

where we introduced

$\mathcal{L}_{\xi}(X)=\sqrt{\Gamma_{1}}\;\xi^{*}(t)[\sigma_{-},X],$ (22)

and its dual

$\mathcal{L}^{\ddagger}_{\xi}(X)=-\sqrt{\Gamma_{1}}\;\xi(t)[\sigma_{+},X],$
(23)

where ${\rm Tr}(\mathcal{L}^{\ddagger}_{\xi}(X)Y)={\rm
Tr}(X\mathcal{L}_{\xi}(Y))$ for any pair $X,Y$.

Note, that operators $\varrho^{01}(t)$ and $\varrho^{10}(t)$ are not
Hermitian. However, introducing a pair of Hermitian operators

$\varrho^{2}(t):=\varrho^{01}(t)+\varrho^{10}(t)\ ,\ \ \
\varrho^{3}(t):=-i(\varrho^{01}(t)-\varrho^{10}(t))\ ,$ (24)

together with $\varrho^{1}(t):=\varrho^{11}(t)$ and
$\varrho^{4}(t):=\varrho^{00}(t)$, the systems of equations (21) may be
rewritten as follows

$\displaystyle\dot{\varrho}^{1}(t)$ $\displaystyle=$
$\displaystyle\mathcal{L}\varrho^{1}(t)-\frac{i}{2}\sqrt{\Gamma_{1}}\,\Big{(}[\sigma^{\xi}_{y},\varrho^{2}(t)]+[\sigma^{\xi}_{x},\varrho^{3}(t)]\Big{)},$
$\displaystyle\dot{\varrho}^{2}(t)$ $\displaystyle=$
$\displaystyle\mathcal{L}\varrho^{2}(t)-i\sqrt{\Gamma_{1}}\,[\sigma^{\xi}_{y},\varrho^{4}(t)],$
$\displaystyle\dot{\varrho}^{3}(t)$ $\displaystyle=$
$\displaystyle\mathcal{L}\varrho^{3}(t)-i\sqrt{\Gamma_{1}}\,[\sigma^{\xi}_{x},\varrho^{4}(t)],$
(25) $\displaystyle\dot{\varrho}^{4}(t)$ $\displaystyle=$
$\displaystyle\mathcal{L}\varrho^{4}(t),$

where

$\sigma^{\xi}_{x}:=\xi^{*}\sigma_{-}+\xi\sigma_{+}=\left(\begin{array}[]{cc}0&\xi^{*}\\\
\xi&0\end{array}\right),\ \ \ \
\sigma^{\xi}_{y}:=i(\xi^{*}\sigma_{-}-\xi\sigma_{+})=\left(\begin{array}[]{cc}0&i\xi^{*}\\\
-i\xi&0\end{array}\right).$ (26)

The set of equations (III) has a structure of hierarchical master equations of
motion (HEOM) Florian ; Tanimura

$\dot{\varrho}^{i}(t)=\sum_{j=1}^{n}\mathcal{L}_{ij}\varrho^{j}(t)\ ,\ \
i,j=1,\ldots,n,$ (27)

for some $n\geq 1$. In Eq. (27) only $\varrho^{1}$ is considered as a system’s
state and the remaining operators $\varrho^{i}$ are auxiliary objects. For the
system (III) one has $n=2^{2}$. Interestingly, in a more general case, when
the field is prepared in $N$-photon state, the evolution of the system is
governed by the set of $(N+1)^{2}$ coupled equations BCBC12 ; DSC19 .

Note, that if $\xi(t)=0$, i.e. there is no photon, then (III) reduces to

$\dot{\varrho}^{k}(t)=\mathcal{L}\varrho^{k}(t)\ ,\ \ \ k=1,2,3,4,$ (28)

and the auxiliary operators $\\{\varrho^{2},\varrho^{3},\varrho^{4}\\}$ are
totally decoupled from the evolution of the system itself. The photon profile
$\xi(t)$ makes the systems (III) nontrivial and eventually influences the
evolution of $\rho(t):=\varrho^{1}(t)$. As we shall see such evolution is no
longer Markovian.

##  IV General properties of the qubit dynamics

The hierarchy of equations (21) may be solved (cf. Appendix A) and eventually
one finds the following dynamical map $\Lambda_{t}$ representing qubit
evolution D20

$\rho(0)\ \to\
{\rho}(t)=\Lambda_{t}\rho(0)=\left(\begin{array}[]{cc}1-P_{e}(t)&C(t)\rho_{ge}(0)\\\
C^{\ast}(t)\rho_{eg}(0)&P_{e}(t)\\\ \end{array}\right),$ (29)

where the population of the excited state $P_{e}(t)=\rho_{ee}(t)$ reads as
follows

$P_{e}(t)=A(t)+B(t)P_{e}(0),$ (30)

together with

$A(t)=\kappa\Gamma e^{-\Gamma
t}\left|\int_{0}^{t}ds\xi(s)e^{\left(-i\Delta_{0}+\frac{\Gamma}{2}\right)s}\right|^{2},$
(31) $B(t)=e^{-\Gamma
t}\left(1-4\kappa\Gamma\mathrm{Re}\int_{0}^{t}ds\xi^{\ast}(s)e^{\left(i\Delta_{0}+\frac{\Gamma}{2}\right)s}\int_{0}^{s}d\tau\xi(\tau)e^{\left(-i\Delta_{0}-\frac{\Gamma}{2}\right)\tau}\right),$
(32)

and

$C(t)=e^{\left(-i\Delta_{0}-\frac{\Gamma}{2}\right)t}\left(1-2\kappa\Gamma\int_{0}^{t}ds\xi^{\ast}(s)e^{\left(i\Delta_{0}-\frac{\Gamma}{2}\right)s}\int_{0}^{s}d\tau\xi(\tau)e^{\left(-i\Delta_{0}+\frac{\Gamma}{2}\right)\tau}\right).$
(33)

We introduce here the parameter $\kappa=\Gamma_{1}/\Gamma$, hence
$\kappa\in[0,1]$. Thus, if $\kappa=0$, we deal with the system interacting
only with the vacuum part, and if $\kappa=1$, we observe only an interaction
with the single-photon field. The above formulae provide a complete
description of the qubit evolution for an arbitrary photon profile $\xi(t)$.
Note, that $B(t)$ together with $\\{C(t),C^{*}(t)\\}$ define time dependent
eigenvalues of the dynamical map $\Lambda_{t}$:

$\Lambda_{t}(|e\rangle\langle e|-|g\rangle\langle g|)=B(t)(|e\rangle\langle
e|-|g\rangle\langle g|)\ ,\ \ \Lambda_{t}|g\rangle\langle
e|=C(t)|g\rangle\langle e|\ ,\ \ \Lambda_{t}|e\rangle\langle
g|=C^{*}(t)|e\rangle\langle g|.$ (34)

Clearly, the evolution highly depends upon the photon profile and the
dependence of $\\{A(t),B(t),C(t)\\}$ upon $\xi(t)$ is quite nontrivial.
However, the asymptotic state is universal: the atom eventually relaxes to the
ground state $|g\rangle$ irrespective of $\xi(t)$. It follows from the
asymptotic analysis

$\lim_{t\to+\infty}A(t)=0,\;\;\lim_{t\to+\infty}B(t)=0,\;\;\lim_{t\to+\infty}C(t)=0.$
(35)

Hence, we obtain $\displaystyle{\lim_{t\to+\infty}}P_{e}(t)=0$ and
$\displaystyle{\lim_{t\to+\infty}}\rho(t)=|g\rangle\langle g|$. It should be
stressed, however, that $|g\rangle\langle g|$ is not an invariant state of the
evolution. This is essential difference between Markovian semigroup and non-
Markovian evolution PRA-Saverio . For a semigroup an asymptotic state always
defines an invariant state. However, for the evolution (29) if one starts with
$\rho(0)=|g\rangle\langle g|$, then

$\rho(t)=A(t)|e\rangle\langle e|+[1-A(t)]|g\rangle\langle g|,$ (36)

that is, the evolution does not create coherence but it does create population
controlled by $A(t)$ which asymptotically vanishes. The problem of an optimal
profile, i.e. a profile maximizing $A(t)=P_{e}(t)$ for some finite $t$, was
analyzed by several authors SAL10a ; Rephali2010 ; WMSS11 ; RB17 . Our
analysis allows to formulate the following

###### Proposition 1

If $\rho(0)=|g\rangle\langle g|$, then the maximal population of the excited
state $P_{e}(t)$ at time $t>0$ reads as follows

$P^{\rm max}_{e}(t):=\max_{\xi(t)}P_{e}(t)=\kappa\left(1-e^{-\Gamma
t}\right),$ (37)

and is realized only at the resonance (i.e. $\Delta_{0}=0$) by the following
pulse

$\xi(\tau)=\sqrt{\frac{\Gamma}{e^{\Gamma t}-1}}\,e^{\frac{\Gamma}{2}\tau}$
(38)

for $\tau\in[0,t]$, and $\xi(\tau)=0$ for $\tau>t$.

For the proof cf. Appendix B. Let us note that for any given profile,
$\xi(t)$, one may discuss the problem of choosing its parameters which
provides the maximal temporal excitation of the system WMSS11 ; RB17 .

We conclude this section with the following statements. There are two
essentially different scenario for the qubit evolution:

  1. 1.

The quantities $B(t)>0$ and $|C(t)|>0$ for all $t\geq 0$. In this case the
dynamical map $\Lambda_{t}$ is invertible (eigenvalues
$\\{B(t),C(t),C^{*}(t)\\}$ do not vanish), that is, knowing a qubit state
$\rho(t)$ at time $t$ one may reconstruct the initial state $\rho(0)$.

  2. 2.

The dynamical map $\Lambda_{t}$ is no longer invertible, i.e. there exists
$t<\infty$ such that either $B(t)=0$ or $C(t)=0$. Note, that whenever
$B(t)=0$, the density operator $\rho(t)$ reads as

${\rho}(t)=\left(\begin{array}[]{cc}1-A(t)&C(t)\rho_{ge}(0)\\\
C^{\ast}(t)\rho_{eg}(0)&A(t)\\\ \end{array}\right),$ (39)

and hence it does not depend upon initial population $P_{e}(0)$. Actually,
diagonal elements display the same structure as in (36). So if initially
$\rho_{ge}(0)=(\rho_{eg}(0))^{\dagger}=0$, then all trajectories of different
$P_{e}(0)$ cross at some point. Whenever $C(t)=0$, then

${\rho}(t)=\left(\begin{array}[]{cc}1-P_{e}(t)&0\\\ 0&P_{e}(t)\\\
\end{array}\right),$ (40)

and for this moment the information about qubit coherence is lost.

##  V Time-local generator and non-Markovianity conditions

To discuss non-Markovianity of qubit evolution it is convenient to introduce
the corresponding time-local master equation for the density matrix. The
formula (29) for time evolution of the qubit density operator $\rho(t)$
defines a dynamical map $\rho(0)\to\Lambda_{t}\rho(0)$. By differentiating
(29) with respect to time, one easily finds that $\Lambda_{t}$ satisfies time-
local master equation $\dot{\Lambda}_{t}=\mathcal{L}_{t}\Lambda_{t}$, with the
following time-local generator

$\mathcal{L}_{t}(\rho)=-i\frac{\omega(t)}{2}[\sigma_{z},\rho]+\frac{\gamma_{+}(t)}{2}\left(\sigma_{+}\rho\sigma_{-}-\frac{1}{2}\\{\sigma_{-}\sigma_{+},\rho\\}\right)+\frac{\gamma_{-}(t)}{2}\left(\sigma_{-}\rho\sigma_{+}-\frac{1}{2}\\{\sigma_{+}\sigma_{-},\rho\\}\right)+\frac{\gamma_{z}(t)}{2}(\sigma_{z}\rho\sigma_{z}-\rho),$
(41)

where $\sigma_{\pm}=\frac{1}{2}(\sigma_{x}\pm i\sigma_{y})$, and $\sigma_{x}$,
$\sigma_{y}$, $\sigma_{z}$ are the Pauli operators. The real valued time-
dependent rates $\gamma_{+}(t)$, $\gamma_{-}(t)$, and $\gamma_{z}(t)$,
describing, respectively, pumping, damping and pure dephasing, are defined as
follows

$\gamma_{+}(t)=2\frac{\dot{A}(t)B(t)-A(t)\dot{B}(t)}{B(t)}\ ,\ \ \
\gamma_{-}(t)=-2\frac{\dot{B}(t)}{B(t)}-\gamma_{+}(t),$ (42)

$\gamma_{z}(t)=\frac{1}{2}\frac{\dot{B}(t)}{B(t)}-\frac{\frac{d}{dt}{|C(t)|}}{|C(t)|}=\frac{1}{2}\frac{\dot{B}(t)}{B(t)}-{\rm
Re}\,\frac{\dot{C}(t)}{C(t)},$ (43)

and

$\omega(t)=\frac{\dot{C}(t)|C(t)|-C(t)\frac{d}{dt}|C(t)|}{iC(t)|C(t)|}={\rm
Im}\,\frac{\dot{C}(t)}{C(t)}.$ (44)

It should be emphasized that time-local generator $\mathcal{L}_{t}$ becomes
singular whenever $B(t)=0$ or $C(t)=0$. In this case $\gamma_{\pm}(t)$ and
$\gamma(t)$ diverge. Nevertheless, the dynamical map
$\Lambda_{t}=\mathcal{T}\exp(\int_{0}^{t}\mathcal{L}_{u}du)$ is always well
defined for any $t\geq 0$. It shows that perfectly regular hierarchy of the
equations (15)-(16) may give rise to singular master equation for a system’s
density operator $\rho(t)$. One can easily check that if $\xi(t)=0$ for any
$t\geq 0$, then we get $A(t)=0,\,B(t)=e^{-\Gamma
t},\,C(t)=e^{-\left(i\Delta_{0}+\Gamma/2\right)t}$, and
$\gamma_{+}(t)=0,\,\gamma_{-}(t)=2\Gamma,\,\gamma_{z}(t)=0,\,\omega(t)=\Delta_{0}$,
that is, $\mathcal{L}_{t}=\mathcal{L}$.

Interestingly, as shown in SKHD16 , (41) provides the most general time-local
generator which satisfies the following covariance property

$U_{\varphi}\mathcal{L}_{t}(\rho)U^{\dagger}_{\varphi}=\mathcal{L}_{t}(U_{\varphi}\rho
U^{\dagger}_{\varphi}),$ (45)

for any diagonal unitary operator

$U_{\varphi}=e^{i\varphi}|g\rangle\langle g|+e^{-i\varphi}|e\rangle\langle
e|,$ (46)

which means that the map $\Lambda_{t}$ commutes with the free evolution of the
system.

Conditions for Markovianity, as well as positivity and complete positivity of
phase covariant qubit dynamical map have been extensively studied recently
Sabrina-2016 ; Sabrina-NJP ; Sergey-2020 . Clearly, if $\gamma_{\pm}(t)\geq 0$
and $\gamma_{z}(t)\geq 0$ for all $t\geq 0$, then the evolution is CP-
divisible. However, $\gamma_{\pm}(t)$ and $\gamma_{z}(t)$ can be temporally
negative without violating CP condition for $\Lambda_{t}$. Of course, by
construction the evolution defined by (15)-(18) is always CP. For the time-
local generator (41) one can provide the following characterization Sabrina-
NJP ; Sergey-2020 .

###### Theorem 1

The qubit evolution generated by (41)

  * •

is CP-divisible iff $\gamma_{\pm}(t)\geq 0$ and $\gamma_{z}(t)\geq 0$,

  * •

is P-divisible iff

$\gamma_{\pm}(t)\geq 0,\ \ \
\sqrt{\gamma_{+}(t)\gamma_{-}(t)}+2\gamma_{z}(t)\geq 0,$ (47)

  * •

satisfies BLP condition iff

$\gamma_{+}(t)+\gamma_{-}(t)\geq 0,\ \ \
\gamma_{+}(t)+\gamma_{-}(t)+4\gamma_{z}(t)\geq 0,$ (48)

  * •

satisfies a geometric criterion (8) iff

$\gamma_{+}(t)+\gamma_{-}(t)+2\gamma_{z}(t)\geq 0,$ (49)

for all $t\geq 0$.

Note that the BLP as well as the geometric condition are fully controlled by
$B(t)$ and $C(t)$, only. One can easily check that

$\displaystyle\gamma_{+}(t)+\gamma_{-}(t)=-2\frac{\dot{B}(t)}{{B(t)}}\ ,\ \ \
\gamma_{+}(t)+\gamma_{-}(t)+4\gamma_{z}(t)=-4{\rm
Re}\,\frac{\dot{C}(t)}{C(t)}.$ (50)
$\displaystyle\gamma_{+}(t)+\gamma_{-}(t)+2\gamma_{z}(t)=-\frac{\dot{B}(t)}{{B(t)}}-2{\rm
Re}\,\frac{\dot{C}(t)}{C(t)}.$ (51)

It clearly shows the difference between BLP condition and P-divisibility which
is more demanding and depends also upon $A(t)$. Detailed properties of qubit
evolution depend upon the profile $\xi(t)$. Assuming resonance we can provide
the detailed analysis for a large class of profiles. For the profile
$\xi(t)\in\mathbb{R}_{+}$ the evolution is never CP-divisible. However, for
invertible dynamical map the weaker condition for no information backflow is
always satisfied.

###### Proposition 2

Let $\Delta_{0}=0$ and $\xi(t)\in\mathbb{R}_{+}$. If $B(t)>0$ for all $t>0$,
then there is no information backflow (BLP condition holds). One has $C(t)>0$
for all $t>0$ and the coherence, $C(t)|\rho_{ge}(0)|$, decreases monotonically
in the course of time. Moreover,

$\gamma_{+}(t)\geq 0\ ,\ \ \gamma_{-}(t)\geq 2\Gamma\ ,\ \ \gamma_{z}(t)\leq
0,$ (52)

and hence the evolution is eternally non-Markovian.

For the proof see Appendix C. In this case one has the following
representation: introducing local relaxation rates (longitudinal and
transversal)

$\gamma_{\rm L}(t):=\gamma_{+}(t)+\gamma_{-}(t)\ ,\ \ \ \gamma_{\rm
T}(t):=\frac{\gamma_{+}(t)+\gamma_{-}(t)}{2}+2\gamma_{z}(t),$ (53)

and

$\Gamma_{\rm L}(t)=\int_{0}^{t}\gamma_{\rm L}(\tau)d\tau\ ,\ \ \ \Gamma_{\rm
T}(t)=\int_{0}^{t}\gamma_{\rm T}(\tau)d\tau\ ,$ (54)

one finds Sabrina-NJP

$B(t)=\exp(-\Gamma_{\rm L}(t)/2)\ ,\ \ \ C(t)=\exp(-\Gamma_{\rm T}(t)).$ (55)

Actually, for any qubit dynamical semigroup a set of three relaxation rates
$\\{G_{1},G_{2},G_{3}\\}$ satisfy the following relation Gen3

$G_{k}\leq\frac{1}{2}(G_{1}+G_{2}+G_{3})=\frac{1}{2}G_{\rm total}.$ (56)

Recall, that $G_{k}=-{\rm Re}\,\ell_{k}$ and $\ell_{k}$ are (in general
complex) eigenvalues of the qubit generator, i.e.
$\mathcal{L}(X_{k})=\ell_{k}X_{k}$. Interestingly, in a recent paper Gen  (cf.
also Gen2 ) it was conjectured that the above relation can be generalized for
any dynamical semigroup of $d$-level quantum system as follows

$G_{k}\leq\frac{1}{d}\sum_{m=1}^{d^{2}-1}G_{m}=\frac{1}{d}G_{\rm total}.$ (57)

This relation is satisfied for several important classes of Markovian
semigroups Gen . Moreover, it is strongly supported by numerical analysis.
Now, if the generator (41) is time-dependent and all three transition rates
$\gamma_{\pm}(t)\geq 0$ and $\gamma_{z}(t)\geq 0$, then the local time-
dependent relaxation rates

$G_{1}(t)=G_{2}(t)=\gamma_{\rm T}(t)\ ,\ \ \ G_{3}(t)=\gamma_{\rm L}(t)$ (58)

satisfy (56) for any $t\geq 0$, that is,

$\gamma_{\rm T}(t)\leq\frac{1}{2}\gamma_{\rm total}(t)\ ,\ \ \ \gamma_{\rm
L}(t)\leq\frac{1}{2}\gamma_{\rm total}(t),$ (59)

where the total rate reads

$\gamma_{\rm total}(t)=2(\gamma_{+}(t)+\gamma_{-}(t)+2\gamma_{z}(t)).$ (60)

Conditions (59) are necessary for CP-divisibility of he corresponding
dynamical map. Therefore, for the non-Markovian evolution the validity of (59)
is not guaranteed. Note, that conditions (59) imply

$\gamma_{\rm L}(t)=\gamma_{+}(t)+\gamma_{-}(t)\geq 0\ ,\ \ \ \gamma_{z}(t)\geq
0,$ (61)

and hence (59) guarantee that the evolution satisfies BLP condition (48).
Interestingly, $\gamma_{\rm total}(t)\geq 0$ is equivalent to the geometric
criterion (49). However, as we shall see in general conditions (59) are
violated (cf. Section VI).

##  VI Exponential profile: a case study

We illustrate the behaviour of the system and the problem of non-Markovianity
of its evolution for the exponential profile

$\xi({t})=\begin{cases}0&\mathrm{for}\;\;t<0\\\
\sqrt{\Gamma_{p}}e^{-\Gamma_{p}t/2}&\mathrm{for}\;\;t\geq 0.\end{cases}$ (62)

The wave-packet of such shape can be experimentally produce by controlled
emission Forn17 . We perform the analysis introducing the positive parameter
$\alpha$ and writing $\Gamma_{p}$ as a fraction of the total coupling constant
$\Gamma$ i.e. $\Gamma_{p}=\alpha\Gamma$. From the general formulae (31)-(33),
we obtain then

$A(t)=\frac{4\kappa\alpha\Gamma^{2}}{(1-\alpha)^{2}\Gamma^{2}+4\Delta_{0}^{2}}\left(e^{-\Gamma
t}-2e^{-\frac{(1+\alpha)\Gamma t}{2}}\cos\Delta_{0}t+e^{-\alpha\Gamma
t}\right)$ (63) $\displaystyle B(t)$ $\displaystyle=$ $\displaystyle
e^{-\Gamma
t}\left\\{1-\frac{8\kappa\Gamma}{(1+\alpha)^{2}\Gamma^{2}+4\Delta_{0}^{2}}\Bigg{[}(1+\alpha)\Gamma\left(e^{-\alpha\Gamma
t}-1\right)\right.$ (64)
$\displaystyle\left.+\frac{2\alpha\Gamma}{(1-\alpha)^{2}\Gamma^{2}+4\Delta_{0}^{2}}\bigg{[}\left((1-\alpha^{2})\Gamma^{2}-4\Delta_{0}^{2}\right)\left(e^{\frac{(1-\alpha)\Gamma
t}{2}}\cos\Delta_{0}t-1\right)+4\Delta_{0}\Gamma e^{\frac{(1-\alpha)\Gamma
t}{2}}\sin\Delta_{0}t\bigg{]}\Bigg{]}\right\\}$
$C(t)=e^{-(i\Delta_{0}+\frac{\Gamma}{2})t}\left\\{1-\frac{4\kappa\Gamma}{(\Gamma-2i\Delta_{0})^{2}-\alpha^{2}\Gamma^{2}}\left[(-2i\Delta_{0}+(1+\alpha)\Gamma)(1-e^{-\alpha\Gamma
t})+2\alpha\Gamma\left(e^{\left(i\Delta_{0}-\frac{(1+\alpha)\Gamma}{2}\right)t}-1\right)\right]\right\\}.$
(65)

Let us note that the formula for $A(t)$ allows one to compute the maximal
excitation probability for an exponential profile.

###### Proposition 3

The maximal excitation probability for an exponential profile reads

$P_{e}^{\rm max}=\frac{4\kappa}{e^{2}}.$ (66)

and is realized for $\alpha=1$ and $\Delta_{0}=0$ at $t=2/\Gamma$.

For the proof cf. Appendix D. Recently, this result for $\kappa=0.5$ was
derived via different methods in Branczyk  (it was found numerically in WMSS11
). Clearly, the fewer $\kappa$ one takes, the smaller excitation one obtains.
We mention it here because it indicates the values of the parameters for which
we observe the maximum influence of the single-photon field on the evolution
of the atom. From the physical point of view the cases when $\alpha\ll 1$ or
$\alpha\gg 1$, or when the central frequency of the field is far from the
resonance they are rather not interesting. Briefly, if $\alpha\ll 1$, then the
interaction between the atom and the single-photon field is too weak to drive
the system. If $\alpha\gg 1$, then the mean time of interaction, given by
$\Gamma_{p}^{-1}=\alpha^{-1}\Gamma^{-1}$, is too short to change the state of
the system. The evolution goes then to the semigroup. One can notice it
considering the respective limits of formulae (63)-(65).

###  VI.1 Resonant case $\Delta_{0}=0$

First, we carefully analyse the resonant case. Then, if $\alpha\neq 1$ we get

$A(t)=\frac{4\kappa\alpha}{(1-\alpha)^{2}}\left(e^{-\frac{\Gamma
t}{2}}-e^{-\frac{\alpha\Gamma t}{2}}\right)^{2}$ (67)

$\displaystyle B(t)$ $\displaystyle=$ $\displaystyle\frac{e^{-(1+\alpha)\Gamma
t}}{\alpha^{2}-1}\left[16\kappa\alpha e^{\frac{(1+\alpha)\Gamma
t}{2}}+[\alpha^{2}-8\kappa(1+\alpha)-1]e^{\alpha\Gamma
t}-8\kappa(\alpha-1)\right]$ (68)

$C(t)=\frac{e^{-(1+\alpha)\Gamma
t}}{\alpha^{2}-1}\left[(1-\alpha)(4\kappa-\alpha-1)e^{\left(\alpha+\frac{1}{2}\right)\Gamma
t}-4\kappa(1+\alpha)e^{\frac{\Gamma t}{2}}+8\kappa\alpha e^{\frac{\alpha\Gamma
t}{2}}\right].$ (69)

And for $\alpha=1$, we have the formulae

$A(t)=\kappa\Gamma^{2}t^{2}e^{-\Gamma t},$ (70)

$B(t)=e^{-\Gamma t}\left\\{1-4\kappa\Gamma\left[t+\frac{1}{\Gamma}(e^{-\Gamma
t}-1)\right]\right\\},$ (71)

$C(t)=e^{-\frac{\Gamma t}{2}}\left\\{1-2\kappa\Gamma\left[-te^{-\Gamma
t}-\frac{1}{\Gamma}\left(e^{-\Gamma t}-1\right)\right]\right\\}.$ (72)

Now, using (42) and (43) one computes the expressions for $\gamma_{+}$,
$\gamma_{-}$, and $\gamma_{z}$ (see Appendix E). Let us notice that even for a
relatively simple exponential pulse in resonance the formulae are quite
involved. However, they do allow to provide the analysis of memory effects of
the corresponding evolution. For a resonant case one can easily discriminate
between invertible and non-invertible dynamical maps.

###### Proposition 4

The evolution is invertible iff $\alpha\geq 8\kappa+1$.

For the proof cf. Appendix F. Thus, for a given $\kappa$, we can always
indicate the range of $\alpha$ producing an invertible evolution.

###### Corollary 1

In the regular case, i.e. $\alpha\geq 8\kappa+1$, the evolution satisfies BLP
condition, i.e. there is no information backflow.

Numerical analysis also shows that in the resonant case P-divisibility and BLP
condition coincide. However, we were not able to provide analytical proof. The
two scenarios: for invertible and non-invertible maps for $\kappa=1$ are
illustrated in Fig. 2. For the regular case, defined then by $\alpha\geq 9$,
we observe the monotonic decreasing functions $B(t)$ and $C(t)$ which remain
positive and approach zero values for large time. Note that for non-invertible
maps, whenever $B(t_{*})=0$, then $B(t)$ stays negative for all $t>t_{*}$ and
asymptotically approaches zero. The same we observe for $C(t)$. Thus in the
second scenario, first we see a decrease of coherence, then we deal with
increase of coherence, and finally we observe the effect of decoherence. The
case of non-invertible map is illustrated by $\alpha=1$ giving the maximal
excitation of the atom. In Fig. 2 the probability of excitation, $P_{e}(t)$,
for a chosen initial state and the function $A(t)$ are also depicted. Of
course, $P_{e}(t)$ as well as $A(t)$ are non-monotonic functions of time in
the both, regular and singular, cases. Note that for the regular case we deal
with the eternally non-Markovian evolution i.e $\gamma_{z}(t)<0$ for all $t>0$
as depicted in Fig. 3. A singularity of the generator $\mathcal{L}_{t}$,
displaying for $\kappa=1$ whenever $\alpha<9$, manifests in the singularities
of all rates which are shown for $\alpha=1$ in Fig. 3. It is interesting (see
Fig. 3) that $\gamma_{+}(t)$ and $\gamma_{z}(t)$ have always opposite signs:

$\forall t\geq 0\;\;\;\;\gamma_{+}(t)\gamma_{z}(t)\leq 0.$ (73)

In general, one sees that at least one of the three rates is negative at a
given time.

Figure 2: The plots of $A(t)$, $B(t)$, $C(t)$, and $P_{e}(t)$ for $\kappa=1$,
$\Delta_{0}=0$ and $P_{e}(0)=0.5$. Left: regular case $\alpha=9.5$. Right:
singular case $\alpha=1$.

Figure 3: The plots of $\gamma_{+}(t)$, $\gamma_{-}(t)$, and $\gamma_{z}(t)$
for $\Gamma=1$, $\kappa=1$, $\Delta_{0}=0$. Left: regular case $\alpha=9.5$.
Right: singular case $\alpha=1$.

###### Proposition 5

One has the following asymptotic behaviour for $\gamma_{\pm}(t)$ and
$\gamma_{z}(t)$:

  * •

for $\alpha\geq 1$

$\lim_{t\to\infty}\gamma_{+}(t)=\lim_{t\to\infty}\gamma_{z}(t)=0\ ,\ \ \
\lim_{t\to\infty}\gamma_{-}(t)=2\Gamma$ (74)

  * •

for $\alpha\in(0,1)$

$\lim_{t\to\infty}\gamma_{+}(t)=0\ ,\ \ \
\lim_{t\to\infty}\gamma_{z}(t)=\frac{1}{4}\Gamma(1-\alpha)\ ,\ \ \
\lim_{t\to\infty}\gamma_{-}(t)=\Gamma(1+\alpha).$ (75)

It implies that asymptotically one has

  * •

for $\alpha\geq 1$

$\mathcal{L}_{t}(\rho)\ \to\
{\Gamma}\left(\sigma_{-}\rho\sigma_{+}-\frac{1}{2}\\{\sigma_{+}\sigma_{-},\rho\\}\right)$
(76)

  * •

for $\alpha\in(0,1)$

$\mathcal{L}_{t}(\rho)\ \to\
\frac{\Gamma(1+\alpha)}{2}\left(\sigma_{-}\rho\sigma_{+}-\frac{1}{2}\\{\sigma_{+}\sigma_{-},\rho\\}\right)+\frac{\Gamma(1-\alpha)}{8}(\sigma_{z}\rho\sigma_{z}-\rho).$
(77)

Let us note that the asymptotic evolution of coherence governed by (76) and
(77) is the same. For details cf. Appendix E.

###  VI.2 Off-resonant case $\Delta_{0}\neq 0$

Now, the formulae for $A(t)$, $B(t)$, and $C(t)$ are much more involved and we
were not able to find analytical condition for
$\\{\Delta_{0},\Gamma,\Gamma_{p}\\}$ which guaranties invertible dynamical map
$\Lambda_{t}$. We observe that contrary to the resonant case $B(t)$ may vanish
at several moments of time which makes the generator even more singular than
in the resonant case. Figure 4 shows that increasing $\Delta_{0}$ from ‘0’
(resonant case) to $\Delta_{0}=1.5$ one creates additional singular points for
$\kappa=1$ and $\alpha=1.5$. However, if $\Delta_{0}=2.5$ all singularities
are removed. Second interesting observation is related to indicators of non-
Markovianity. In the resonant case if the generator is free from singularities
(i.e. $\alpha\geq 8\kappa+1$) the evolution satisfies BLP condition and hence
does not display information backflow. It is no longer true if $\Delta_{0}\neq
0$. Figure 5 shows that even if the generator is free from singularities the
validity of BLP condition still depends for a given $\kappa$ on the parameters
$\\{\Delta_{0},\alpha\\}$. Since $\gamma_{+}(t)$ might be negative, it is
clear, that even if the dynamics satisfies BLP condition it needs not be
P-divisible (cf. Theorem 1). Therefore, in the off-resonant case these two
indicators, that is, P-divisibility and BLP condition do not coincide. This
observation is quite interesting since in majority of studied examples these
two notions coincide. It simply means that for off-resonant evolution the
monotonicity condition

$\frac{d}{dt}\|\Lambda_{t}(X)\|_{1}\leq 0,$ (78)

is violated even if it is satisfied for all traceless Hermitian operators
(i.e. $X\sim\rho_{1}-\rho_{2}$). This analysis shows that no information
backflow does not guarantee the existence of positive trace-preserving
propagators $V_{t,s}$. Actually, one may violate not only BLP condition but
much weaker condition based on the geometric indicator LPP13  which measures
the volume of accessible states. One finds that if the evolution is Markovian
then

$\frac{d}{dt}|{\rm
det}\Lambda_{t}|=\frac{d}{dt}|\lambda_{1}(t)\lambda_{2}(t)\lambda_{3}(t)|\,\leq\,0,$
(79)

where $\lambda_{k}(t)$ denote eigenvalues of the dynamical map ($k=1,2,3$).
This condition is equivalent to (49) Sabrina-NJP , which is much weaker than
BLP condition. The geometric and the BLP criteria of non-Markovianity are
shown in Fig. 6. One sees that for parameters indicated in the left panel of
Figure 6 this condition is violated proving that evolutions display strong
memory effects. Finally, let us note that Fig. 5 shows that for the chosen
values of parameters the rates oscillate with decreasing amplitudes around
their asymptotic values, namely $\gamma_{+}(t)$ and $\gamma_{z}(t)$ oscillate
around zero, and $\gamma_{-}(t)$ around $2\Gamma$. Interestingly, whenever
$\gamma_{+}(t)$ is zero $\gamma_{z}(t)$ is also equal to zero, and
$\gamma_{-}(t)$ is equal to $2\Gamma$, so for these moments
$\mathcal{L}_{t}=\mathcal{L}$.

Figure 4: The plots of $A(t)$, $B(t)$, and $|C(t)|$ for $\kappa=1$,
$\Gamma=1$, and $\alpha=1.5$. Left: $\Delta_{0}=0$. Center: $\Delta_{0}=1.5$.
Right: $\Delta_{0}=2.5$.

Figure 5: The plots of $\gamma_{+}(t)$, $\gamma_{-}(t)-2\Gamma$, and
$\gamma_{z}(t)$ for $\kappa=1$, $\Gamma=1$, and $\alpha=1.5$. Left:
$\Delta_{0}=3.0$ — BLP condition is violated. Right: $\Delta_{0}=6.5$ — BLP
condition is satisfied.

Figure 6: The plots of $\gamma_{+}(t)+\gamma_{-}(t)$,
$\gamma_{+}(t)+\gamma_{-}(t)+2\gamma_{z}(t)$, and
$\gamma_{+}(t)+\gamma_{-}(t)+4\gamma_{z}(t)$ for $\kappa=1$, $\Gamma=1$,
$\alpha=1.5$. Left: $\Delta_{0}=3.0$. Right: $\Delta_{0}=6.5$.

##  VII Conclusion

We provided a detailed discussion of the evolution of a qubit interacting with
single-photon wave packet. Clearly, this problem was already analyzed by
several researchers (see e.g. WMSS11 ; SAL10a ; Francesco ; Branczyk ). In
this paper it is shown that a set of hierarchical equations, already derived
in GEPZ98 ; BCBC12 ; DSC17 ; DSC19 ; D20 , is equivalent to a single time-
local master equation for the qubit density operator. To show this equivalence
we make use of the general form of solution to this set defining a dynamical
map $\\{\Lambda_{t}\\}_{t\geq 0}$ for an arbitrary photon profile $\xi(t)$.
The price one pays for this reduction is highly nontrivial structure of time-
dependent rates in the time-local master equation. Interestingly, the
corresponding time-local generator has a structure of phase covariant
generator which has been already extensively studied SKHD16 ; Sabrina-2016 ;
Sabrina-NJP ; Francesco ; HSH19 ; Sergey-2020 . We provided the formulae for
time dependent rates governing damping (cooling), heating and decoherence
processes. These rates are fully controlled by the photon profile $\xi(t)$
which is a key ingredient of the entire analysis. It turn out that temporal
correlations encoded into $\xi(t)$ give rise to all characteristic non-
Markovian memory effects of the corresponding qubit evolution. Due to the
interaction with a non-locally (in time) entangled field the system undergoes
such phenomena as recoherence. But the recoherence is not universal, for some
parameters the photon becomes only the source of decoherence.

Note that a dynamical map $\\{\Lambda_{t}\\}_{t\geq 0}$ representing the qubit
evolution might be non-invertible for some time moments which implies the
singularity of transition rates in the corresponding time-local master
equation. The phenomenon of non-invertible maps were already considered in
quantum optics for example in Cresser1 ; Cresser2 ; SKHD16 . Similar
observation for phase covariant dynamics was also recently reported in
Francesco . It should be stressed that being singular the time-local generator
$\mathcal{L}_{t}$ gives rise to perfectly regular evolution D20  (represented
by $\\{\Lambda_{t}\\}_{t\geq 0}$). Non-invertible dynamical maps were recently
analyzed in PRL-2018  and CC19  in connection to divisibility (cf. also recent
paper Jyrki-2021 ). Divisibility which is equivalent to the existence of a
propagator $V_{t,s}$ is always guaranteed for invertible maps. For maps which
are not invertible divisibility holds only under additional conditions
PRL-2018 .

We provided the detailed analysis of the exponential profile. In particular we
derived the formula for the maximal excitation probability. Such formula was
also derived in a recent paper Branczyk  via different methods. For
exponential profile we derived explicit formulae for all time dependent rates
of the time-local generator. The key observation is that for the exponential
profile qubit evolution is never Markovian (i.e. CP-divisible). Interestingly,
in the resonant case the dynamical map $\\{\Lambda_{t}\\}_{t\geq 0}$ is either
invertible or looses invertibility in at most two moments of time. However,
whenever it is invertible the evolution satisfies BLP condition which
guarantees no information backflow. Off-resonant scenario is much more
intricate since the number of singular points for the time-local generator
highly depends upon the detuning parameter $\Delta_{0}$. Still, one can
observe some universal properties of qubit dynamics. In particular
$\gamma_{+}(t)\gamma_{z}(t)\leq 0$ for all $t\geq 0$. It shows that for any
$t$ one of the transition rate is strictly negative and hence the dynamics is
eternally non-Markovian Erika . Similar analysis was performed for a square
pulse $\xi(t)$. The conclusions are very similar and they are not included in
the manuscript.

Beyond exponential profile it is still possible to observe universal
properties. We shown that in the resonant case assuming
$\xi(t)\in\mathbb{R}_{+}$ the evolution satisfies BLP condition whenever it is
invertible. Still the evolution is eternally non-Markovian ($\gamma_{z}(t)\leq
0$) but invertibility protects the system against the information backflow.
This observation clearly shows that being invertible is not only a
mathematical curiosity but implies direct physical consequences.

The problem of non-Markovian dynamics of the qubit interacting with a single-
photon field was studied in Francesco  by making use an infinite waveguide in
the real-space approach. Authors of Francesco  analyzed an exponential profile
in resonance and $\kappa=\frac{1}{2}$. We consider off resonant case as well
and allow for $\kappa\in[0,1]$. Moreover, we derive general properties for an
arbitrary profile satisfying $\xi(t)\in\mathbb{R}_{+}$. In particular it is
shown that in the resonant case whenever the dynamical map is invertible the
evolution satisfies BLP condition meaning there is no information backflow.
Exponential profile is just an example of this general property.

The system we consider provides a perfect platform to test and compare various
aspects of non-Markovianity. Detailed analysis of the exponential profiles
shows that the evolution of the qubit is never Markovian (CP-divisible).
Moreover, it is eternally non-Markovian Erika , that is, one of the transition
rate is always negative. We claim that the same holds for an arbitrary profile
meaning that the interaction with a single-photon field always results in the
eternally non-Markovian dynamics. In our opinion this is the first proper
physical model displaying such non-trivial behavior. The analysis of non-
Markovian dynamics of the system interacting with single-photon state is of
particular importance for effective control of quantum systems and the storage
and retrieving of quantum information. It would be interesting using similar
theoretical methods to analyze the memory effects of a qubit interacting with
$N$-photon field (the corresponding hierarchical equation were already derived
in DSC19 ).

###### Acknowledgements.

This research was supported by the National Science Centre project
2018/30/A/ST2/00837.

##  Appendix A Solution of hierchical equations

The solution to (16)-(18) reads as

$\displaystyle{\varrho}^{01}(t)=\left(\begin{array}[]{ccc}E(t)\rho_{eg}(0)&&F(t)P_{e}(0)+G(t)\\\
0&&-E(t)\varrho_{eg}(0)\end{array}\right)$ (82)
$\displaystyle{\varrho}^{00}(t)$
$\displaystyle=\left(\begin{array}[]{ccc}1-P_{e}(0)e^{-\Gamma
t}&&\varrho_{ge}(0)e^{-\left(i\Delta_{0}+\frac{\Gamma}{2}\right)t}\\\
\varrho_{eg}(0)e^{\left(i\Delta_{0}-\frac{\Gamma}{2}\right)t}&&P_{e}(0)e^{-\Gamma
t}\end{array}\right)$ (85) $E(t)=\sqrt{\kappa\Gamma}e^{-\Gamma
t}\int_{0}^{t}ds\xi^{\ast}(s)e^{\left(i\Delta_{0}+\frac{\Gamma}{2}\right)s},$
(86)
$F(t)=2\sqrt{\kappa\Gamma}e^{-\left(i\Delta_{0}+\frac{\Gamma}{2}\right)t}\int_{0}^{t}ds\xi^{\ast}(s)e^{\left(i\Delta_{0}-\frac{\Gamma}{2}\right)s},$
(87)
$G(t)=-\sqrt{\kappa\Gamma}\int_{0}^{t}ds\xi^{\ast}(s)e^{\left(i\Delta_{0}+\frac{\Gamma}{2}\right)(s-t)},$
(88)

and $\varrho^{10}(t)=\left(\varrho^{01}(t)\right)^{\dagger}$.

##  Appendix B Proof of Proposition 1

The proof is very simple: let
$\tilde{\xi}(t)=e^{-i\omega_{c}t}\xi(t)\in\mathbb{C}$. Assuming
$\rho(0)=|g\rangle\langle g|$ one obtains

$P_{e}(t)=\kappa\Gamma e^{-\Gamma
t}\left|\int_{0}^{t}ds\tilde{\xi}(s)e^{\left(i\omega_{0}+\frac{\Gamma}{2}\right)s}\right|^{2}.$
(89)

To maximize this expression one has to maximize

$\left|\int_{0}^{t}ds\tilde{\xi}(s)e^{\left(i\omega_{0}+\frac{\Gamma}{2}\right)s}\right|^{2}=|\langle\tilde{\xi}|f\rangle_{t}|^{2}$
(90)

where the function $f(s)=e^{\left(-i\omega_{0}+\frac{\Gamma}{2}\right)s}$, and
we introduce an inner product

$\langle\tilde{\xi}|f\rangle_{t}:=\int_{0}^{t}\tilde{\xi}(s)f^{\ast}(s)ds.$
(91)

It is evident that it is maximized for the profile $\tilde{\xi}(s)$ which is
parallel to $f(s)$, that is, $\tilde{\xi}(s)=c(t)f(s)$, where $c(t)$ is a
constant (depending on a fixed time $t$), and

$\langle\tilde{\xi}|\tilde{\xi}\rangle_{t}=1.$ (92)

This normalization condition allows to calculate $c(t)$ leading to

$\tilde{\xi}(\tau)=\sqrt{\frac{\Gamma}{e^{\Gamma
t}-1}}\,e^{\left(-i\omega_{0}+\frac{\Gamma}{2}\right)\tau}$ (93)

for $\tau\in[0,t]$, and $\tilde{\xi}(\tau)=0$ for $\tau>t$ and then to (37).
Clearly, the maximum excitation is realized at resonance
$\omega_{c}=\omega_{0}$, i.e. $\Delta_{0}=0$. $\Box$

##  Appendix C  Proof to Proposition 2

Let us assume that

$\Delta_{0}=0,\ \ B(t)>0\ \ \mathrm{and}\ \ \xi(t)\in\mathbb{R}_{+}\ \
\mathrm{for\ \ all}\ \ t\geq 0.$ (94)

One can check that then

$\dot{A}(t)B(t)-\dot{B}(t)A(t)=2\kappa\Gamma\xi(t)e^{-\frac{\Gamma
t}{2}}\left[B(t)\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma
s}{2}}+2A(t)\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma s}{2}}\right]\geq 0,$ (95)

therefore $\gamma_{+}(t)\geq 0$ for all $t\geq 0$. In the next step one can
find that

$\gamma_{-}(t)=2\Gamma-4\kappa\Gamma\xi(t)e^{-\frac{\Gamma
t}{2}}\frac{B(t)\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma
s}{2}}+2\left(A(t)-1\right)\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma s}{2}}}{B(t)}$
(96)

and show that

$B(t)\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma
s}{2}}+2\left(A(t)-1\right)\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma s}{2}}\leq 0,$
(97)

which can be written in the form

$\displaystyle\left[1-4\kappa\Gamma\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma
s}{2}}\int_{0}^{s}d\tau\xi(\tau)e^{-\frac{\Gamma\tau}{2}}+2\kappa\Gamma\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma
s}{2}}\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma s}{2}}\right]$ $\displaystyle\times
e^{-\Gamma t}\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma
s}{2}}-2\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma s}{2}}\leq 0.$ (98)

This inequality can be proved by using

$e^{-\Gamma t}\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma
s}{2}}\leq\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma s}{2}},$ (99)

which follows from

$e^{-\Gamma t}\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma s}{2}}\leq e^{-\frac{\Gamma
t}{2}}\int_{0}^{t}ds\xi(s)\leq\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma s}{2}}$
(100)

and by applying

$2\kappa\Gamma\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma
s}{2}}\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma
s}{2}}<1+4\kappa\Gamma\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma
s}{2}}\int_{0}^{s}d\tau\xi(\tau)e^{-\frac{\Gamma\tau}{2}}.$ (101)

Let us notice that on both sides of the last inequality we deal with non-
decreasing functions of $t$ and one can prove it by comparing values of these
function at $t=0$ and their derivatives. Hence we obtain that
$\gamma_{-}(t)\geq 2\Gamma$ for all $t\geq 0$.

Now we show that if (94) holds, then $C(t)>0$. Let us notice that from

$1-4\kappa\Gamma\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma
s}{2}}\int_{0}^{s}d\tau\xi(\tau)e^{-\frac{\Gamma\tau}{2}}>0,$ (102)

which follows from $B(t)>0$, and

$\displaystyle\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma
s}{2}}\int_{0}^{s}d\tau\xi(\tau)e^{-\frac{\Gamma\tau}{2}}=-\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma
s}{2}}\int_{0}^{s}d\tau\xi(\tau)e^{\frac{\Gamma\tau}{2}}+\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma
s}{2}}\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma s}{2}}$ (103)

we obtain

$\displaystyle 1-2\kappa\Gamma\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma
s}{2}}\int_{0}^{s}ds\xi(\tau)e^{\frac{\Gamma\tau}{2}}>\;\;\;\;$ $\displaystyle
2\kappa\Gamma\left(2\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma
s}{2}}\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma
s}{2}}-3\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma
s}{2}}\int_{0}^{s}d\tau\xi(\tau)e^{\frac{\Gamma\tau}{2}}\right).$ (104)

To check that

$\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma
s}{2}}\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma
s}{2}}\geq\frac{3}{2}\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma
s}{2}}\int_{0}^{s}d\tau\xi(\tau)e^{\frac{\Gamma\tau}{2}}$ (105)

we can use the fact that on both sides of the above inequality we deal with
non-decreasing functions of $t$ having the same values for $t=0$ whose
derivatives satisfies the inequality

$\xi(t)e^{-\frac{\Gamma t}{2}}\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma
s}{2}}+\xi(t)e^{\frac{\Gamma t}{2}}\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma
s}{2}}\geq\frac{3}{2}\xi(t)e^{-\frac{\Gamma
t}{2}}\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma s}{2}}.$ (106)

Hence $C(t)>0$ for all $t\geq 0$. Using this results one can easily show that

$\dot{C}(t)=-\frac{\Gamma}{2}C(t)-2\kappa\Gamma e^{-\Gamma
t}\xi(t)\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma s}{2}}<0$ (107)

so we see that $\gamma_{+}(t)+\gamma_{-}(t)+4\gamma_{z}(t)>0$ is satisfied for
all $t\geq 0$.

Finally let us derive

$\dot{B}(t)C(t)-2B(t)\dot{C}(t)=4\kappa\Gamma\xi(t)e^{-\frac{\Gamma
t}{2}}\left[B(t)e^{-\frac{\Gamma t}{2}}\int_{0}^{t}ds\xi(s)e^{\frac{\Gamma
s}{2}}-C(t)\int_{0}^{t}ds\xi(s)e^{-\frac{\Gamma s}{2}}\right].$ (108)

To prove that this expression is less or equal to zero one can use the
properties (103), (105), and (100). This property leads to $\gamma_{z}(t)\leq
0$ for all $t\geq 0$ and ends the proof. $\Box$

##  Appendix D  Proof to Proposition 4

Assuming $\rho(0)=|g\rangle\langle g|$ one has

$P_{e}(\alpha,\Delta_{0},t)=\frac{4\kappa\alpha\Gamma^{2}}{(1-\alpha)^{2}\Gamma^{2}+4\Delta_{0}^{2}}\left(e^{-\Gamma
t}-2e^{-\frac{(1+\alpha)\Gamma t}{2}}\cos\Delta_{0}t+e^{-\alpha\Gamma
t}\right).$ (109)

Introducing $\chi$ such that $\Gamma t/2=\chi$ one finds that the function

$P_{e}=f(\alpha,\Delta_{0},\chi)=\frac{4\kappa\alpha\Gamma^{2}}{(1-\alpha)^{2}\Gamma^{2}+4\Delta_{0}^{2}}\left(e^{-2\chi}-2e^{-(1+\alpha)\chi}\cos\Delta_{0}t+e^{-\alpha\chi}\right).$
(110)

attains its local maximum for $\alpha=1$, $\Delta_{0}=0$, and $\chi=1$ which
corresponds to $t=2/\Gamma$. Finally,

$\lim_{\alpha\to 1}f(\alpha,0,1)=\frac{4\kappa}{e^{2}},$ (111)

which ends the proof. $\Box$

##  Appendix E formulae for $\gamma_{+}$, $\gamma_{-}$, and $\gamma_{z}$ for
$\Delta_{0}=0$

If $\alpha\neq 1$, then for (67)-(69) we obtain

$\displaystyle\gamma_{+}(t)$ $\displaystyle=16\alpha\Gamma\kappa
e^{-\frac{1}{4}(\alpha+1)\Gamma t}\sinh\left[\frac{1}{4}(\alpha-1)\Gamma
t\right]\frac{\left(\alpha^{2}-16\kappa-1\right)\sinh\left[\frac{\alpha\Gamma
t}{2}\right]+\left(\alpha^{2}-1\right)\cosh\left[\frac{\alpha\Gamma
t}{2}\right]+16\alpha\kappa\sinh\left[\frac{\Gamma
t}{2}\right]}{(\alpha-1)\left\\{8\kappa(1-\alpha)+16\alpha\kappa
e^{\frac{1}{2}(\alpha+1)\Gamma t}+(\alpha+1)(\alpha-8\kappa-1)e^{\alpha\Gamma
t}\right\\}},$ (112) $\displaystyle\gamma_{-}(t)$
$\displaystyle=-\gamma_{+}(t)+\frac{2(\alpha+1)\Gamma\left\\{8\kappa(1-\alpha)+8\alpha\kappa
e^{\frac{1}{2}(\alpha+1)\Gamma t}+(\alpha-8\kappa-1)e^{\alpha\Gamma
t}\right\\}}{8\kappa(1-\alpha)+16\alpha\kappa e^{\frac{1}{2}(\alpha+1)\Gamma
t}+(\alpha+1)(\alpha-8\kappa-1)e^{\alpha\Gamma t}},$ (113)
$\displaystyle\gamma_{z}(t)$
$\displaystyle=-\frac{1}{4}(\gamma_{+}(t)+\gamma_{-}(t))-\frac{\Gamma\left\\{-8\alpha(\alpha+2)\kappa
e^{\frac{\alpha\Gamma t}{2}}+4(\alpha+1)(2\alpha+1)\kappa e^{\frac{\Gamma
t}{2}}-(\alpha-1)(\alpha-4\kappa+1)e^{\left(\alpha+\frac{1}{2}\right)\Gamma
t}\right\\}}{16\alpha\kappa e^{\frac{\alpha\Gamma t}{2}}-8(\alpha+1)\kappa
e^{\frac{\Gamma
t}{2}}+2(\alpha-1)(\alpha-4\kappa+1)e^{\left(\alpha+\frac{1}{2}\right)\Gamma
t}}.$ (114)

For $\alpha=1$, from (70)-(72) we get

$\displaystyle\gamma_{+}(t)$ $\displaystyle=\frac{4\Gamma^{2}\kappa
t\Big{\\{}2\kappa(\Gamma t+2)+e^{\Gamma t}\big{[}2\kappa(\Gamma
t-2)-1\big{]}\Big{\\}}}{4\kappa e^{\Gamma t}+e^{2\Gamma t}(4\kappa(\Gamma
t-1)-1)}$ (115) $\displaystyle\gamma_{-}(t)$
$\displaystyle=-\gamma_{+}(t)+\frac{2\Gamma\Big{\\{}8\kappa+e^{\Gamma
t}\big{[}4\kappa(\Gamma t-2)-1\big{]}\Big{\\}}}{4\kappa+e^{\Gamma
t}(4\kappa(\Gamma t-1)-1)}$ (116) $\displaystyle\gamma_{z}(t)$
$\displaystyle=-\frac{1}{4}(\gamma_{+}(t)+\gamma_{-}(t))+\frac{\Gamma}{2}\left[\frac{4\Gamma\kappa
t}{2(\kappa+\Gamma\kappa t)+(1-2\kappa)e^{\Gamma t}}+1\right]$ (117)

Hence one can find the limits (76) and (77).

##  Appendix F  Proof to Proposition 4

One can rewrite (68) as

$B(t)=\frac{16\kappa\alpha}{\alpha^{2}-1}e^{-\frac{1+\alpha}{2}\Gamma
t}+\left(1-\frac{8\kappa}{\alpha-1}\right)e^{-\Gamma
t}-\frac{8\kappa}{\alpha+1}e^{-(1+\alpha)\Gamma t}$ (118)

If $\alpha<1$, then we rewrite the above as:

$B(t)=e^{-\frac{1+\alpha}{2}\Gamma
t}\left(\frac{16\kappa\alpha}{\alpha^{2}-1}+\left(1-\frac{8\kappa}{\alpha-1}\right)e^{-\frac{1-\alpha}{2}\Gamma
t}-\frac{8\kappa}{\alpha+1}e^{-\frac{1+\alpha}{2}\Gamma t}\right)$ (119)

The limit of the second factor is $16\kappa\alpha/(\alpha^{2}-1)<0$, hence
$B(t)$ is negative for $t$ large enough. Because $B(0)=1$, there exists $t$
such that $B(t)=0$.

If $\alpha\in(1,8\kappa+1)$, then rewriting (118) as:

$B(t)=e^{-\Gamma
t}\left(\frac{16\kappa\alpha}{\alpha^{2}-1}e^{-\frac{\alpha-1}{2}\Gamma
t}+\left(1-\frac{8\kappa}{\alpha-1}\right)-\frac{8\kappa}{\alpha+1}e^{-\alpha\Gamma
t}\right)$ (120)

one can notice that the limit of the second factor is $1-8\kappa/(\alpha-1)$
which now is negative due to assumption about the range of $\alpha$ and again
there exists $t$ such that $B(t)=0$.

If $\alpha=1$, then the second factor in (71) is equal 1 for $t=0$ and it is
unbounded from below, hence there exists $t$ such that $B(t)=0$ . Hence for
the whole range $\alpha<8\kappa+1$ the dynamics is not invertible.

If $\alpha\geq 8\kappa+1$, then $1-8\kappa/(\alpha-1)\geq 0$ and estimating
(118) as follows:

$\displaystyle B(t)$
$\displaystyle=\frac{16\kappa\alpha}{\alpha^{2}-1}e^{-\frac{1+\alpha}{2}\Gamma
t}+\left(1-\frac{8\kappa}{\alpha-1}\right)e^{-\Gamma
t}-\frac{8\kappa}{\alpha+1}e^{-(1+\alpha)\Gamma t}$
$\displaystyle\geq\frac{16\kappa\alpha}{\alpha^{2}-1}e^{-\frac{1+\alpha}{2}\Gamma
t}+\left(1-\frac{8\kappa}{\alpha-1}\right)e^{-\frac{1+\alpha}{2}\Gamma
t}-\frac{8\kappa}{\alpha+1}e^{-(1+\alpha)\Gamma
t}=\left(1+\frac{8\kappa}{\alpha+1}\right)x-\frac{8\kappa}{\alpha+1}x^{2}$
(121)

where $x=\exp(-(1+\alpha)\Gamma t/2)\in(0,1]$, one can observe that the above
expression is positive for $x\in(0,1]$, hence $B(t)$ is positive for all $t$.

Finally, using the result from Appendix C i.e whenever $B(t)>0$ then $C(t)>0$,
we see that for the range $\alpha\geq 8\kappa+1$, we get $C(t)>0$, and we
conclude that the dynamics is invertible.

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