dissipation. For energy-incoherent initial and final states one can infer from
Ref. [228] that appropriately tuned fluctuations can significantly reduce
dissipation, however nothing is known for states with coherence.
Unfortunately, since thermal operations are time-translation covariant, such
that coherence and athermality form independent resources [234, 234, 239], it
seems unlikely that the current approach can be easily generalised. Thirdly,
one could try to extend our results on pure states to allow for non-identical
systems and to derive a bound working for all $N$, not only for $N\to\infty$
(i.e., replace the proving technique based on central limit theorem by the one
based on a version of Berry-Esseen theorem).

In this chapter we have also provided a number of physical applications of our
fluctuation-dissipation theorems by considering several scenarios and
explaining how our results can be useful to describe fundamental and well-
known thermodynamic and information-theoretic processes. We derived the
optimal value of extractable work in a thermodynamic distillation process as a
function of the transformation error associated to the work quality. This,
together with the knowledge of the actual final free energy of the battery
system provided by Theorem 3.2, could potentially be used to clarify the
notion of imperfect work [208, 240, 241], and to construct a comparison
platform allowing one to continuously distinguish between work-like and heat-
like forms of energy. We have also shown how our results yield the optimal
trade-off between the work invested in erasing $N$ independent bits prepared
in arbitrary states, and the erasure quality measured by the infidelity
distance between the final state and the fully erased state. This can of
course be straightforwardly extended to higher-dimensional systems and
arbitrary final erased state (not necessarily the ground state). Finally, we
have investigated the optimal encoding rate into a collection of non-
interacting subsystems consisting of energy-incoherent or pure states using
thermal operations. We derived the optimal rate (up to second-order
asymptotics) of encoding information with a given average decoding error and
without spending thermodynamic resources. This provides an operational
interpretation of the resourcefulness of athermal quantum states for
communication scenarios under the restriction of using thermal operations.

We would also like to point out to some possible technical extensions of our
results. Firstly, we used infidelity as our quantifier of transformation
error, but we expect that similar results could be derived using other
quantifiers, e.g., the trace distance. Secondly, our investigations were
performed in the spirit of small-deviation analysis (where we look for
constant transformation error and total free energy dissipation of the order
$O(\sqrt{N})$), but possibly other interesting fluctuation-dissipation
relations could be derived within the the moderate and large deviation
regimes. Thirdly, our result for pure states is limited to Hamiltonians with
incommensurable spectrum, but we believe this is just a technical nuisance
that one should be able to get rid of. Lastly, within the framework of general
resource theories, it might be possible to derive analogous fluctuation-
dissipation relations, but with free energy replaced by a resource quantifier
relevant for a given resource theory.

##  Chapter 7 Quantum catalysis in cavity quantum electrodynamics

The effect of catalysis involves using an auxiliary system (a catalyst) to
enable a process that would either not occur spontaneously or would occur very
slowly. Catalysis manifests across a variety of fields (see e.g., [242]),
including biological processes activated by enzymes, the speed-up of chemical
reactions, and the synthesis of nanomaterials.

More recently, the phenomenon of catalysis has also become relevant in the
context of quantum information; see recent reviews [243, 106]. First examples
focused on entanglement manipulation [89, 244, 245, 246, 247, 248, 249, 250,
251], and then spread to quantum thermodynamics [57, 255, 256, 257, 258, 259,
260, 261, 262, 252, 201, 253, 254, 63], coherence theory [263, 264, 265, 266,
267, 268] and other areas [269, 270, 271, 272, 273, 274]. These results are
typically formulated within the framework of quantum resource theories [61].
This abstract approach is particularly useful for characterising the
fundamental limits of manipulating quantum resources, including scenarios
involving catalytic systems of arbitrary complexity.

An interesting question is whether quantum catalysis is also relevant and
useful in a more practical context, potentially even in experiments. Here we
investigate quantum catalysis in a paradigmatic setup of quantum optics,
namely the Jaynes-Cummings model [275, 276, 277], where a two-level atom
interacts with a single-mode optical cavity. In this chapter, we uncover a
catalytic process enabling the generation of a non-classical state of light in
the cavity, using the atom as a catalyst. Specifically, we consider the cavity
to be initially prepared in a “classical” coherent state, and uncorrelated to
the atom. By carefully setting the initial state of the atom and the
interaction time, we obtain a final state such that _(i)_ the atom is back in
its initial state exactly, and _(ii)_ the state of the cavity is now non-
classical, i.e., featuring Wigner negativity or sub-Poisonian statistics.
Hence, non-classicalilty of the cavity has been generated without perturbing
the state of the atom (see Fig. 1). The process is catalytic and the atom
could be re-used, for example by coupling it to another cavity.

We investigate the mechanism of this catalytic process, and identify two
crucial ingredients. First, the final state of the atom and cavity must
feature correlations. Second, the evolution of the state of the catalyst must
involve quantum coherence (i.e., superpositions of the energy basis states).
The latter is an interesting aspect, as typical instances of quantum catalysis
in resource theories involve only diagonal states (i.e., without coherence),
so that they can be understood as a stochastic process involving the
probability distributions of the system and catalysis. Instead here, the
system and catalyst experience a genuinely quantum evolution. This is a novel
instance of the effect of _coherent quantum catalysis_ , recently investigated
in quantum thermodynamics [254].

Before proceeding, it is worth discussing previous works in quantum optics
that relate to the concept of catalysis. Notably, the pioneering proposal for
quantum computing in ion traps [278] (see also [279, 280]) considers two spin
qubits that become entangled via an interaction with a cavity that can be
considered catalytic [106]. Another relevant direction is that of “multi-
photon catalysis” (see e.g., [281, 282, 283, 284]) which is a heralded
catalytic process, where the catalyst is returned only with some probability.
In contrast, our catalytic protocol is deterministic.

Figure 1: _Quantum catalysis in the Jaynes-Cummings model._ (a) An atom (the
catalyst C) interacts with a single-mode optical cavity (the system S),
initially prepared in a “classical” coherent state. (b) We consider the
evolution $U$ over a well-chosen time interval (from $t=0$ to $t=\tau$) such
that (i) the final state of the cavity is non-classical, and (ii) the atom is
returned to its intial state exactly. Hence, non-classicality has been
generated via catalysis.

###  Section 1 Setting the scene

Consider a setup comprising a system $(\textsf{S})$ initially prepared in a
state $\rho_{\textsf{S}}$ and a catalyst $(\textsf{C})$ in an initial state
$\chi_{\textsf{C}}$. The total system SC is assumed to be closed and evolves
via an energy-conserving process for some time $\tau$. This evolution is
represented by the unitary $U=\exp({-iH_{\textsf{SC}}\tau})$, where
$H_{\textsf{SC}}$ denotes the joint Hamiltonian. Consequently, the final state
of the total system is given by
$\sigma_{\textsf{SC}}:=U(\rho_{\textsf{S}}\otimes\chi_{\textsf{C}})U^{\dagger}$.

The evolution is said to be _catalytic_ when the catalyst is returned in
exactly the same state as it was initially prepared. Formally, we demand that

$\sigma_{\textsf{C}}:=\Tr_{\textsf{S}}[U(\rho_{\textsf{S}}\otimes\chi_{\textsf{C}})U^{\dagger}]=\chi_{\textsf{C}},$
(1)

which we refer to as the _catalytic constraint_. Satisfying this constraint
typically requires to carefully choose the initial states of the system
$\rho_{\textsf{S}}$ and the catalyst $\chi_{\textsf{C}}$, as well as the
interaction time $\tau$.

The main goal of a catalytic evolution is to induce an interesting local
dynamics on the system S, i.e.,

$\displaystyle\rho_{\textsf{S}}\rightarrow\sigma_{\textsf{S}}:=\Tr_{\textsf{C}}[U(\rho_{\textsf{S}}\otimes\chi_{\textsf{C}})U^{\dagger}],$
(2)

while leaving the state of the catalyst unchanged. Notably, it is possible to
induce an evolution on S [as in Eq. (2)] that would not be possible without
the presence of the catalyst.

####  Subsection 1 Jaynes-Cummings model

In this Section, we discuss the phenomenon of catalysis in the Jaynes-Cummings
(JC) model, describing the interaction between a single-mode optical cavity
and a two-level atom [275] (see Fig. 1a).

We choose the cavity to represent the system S, while the atom will play the
role of the catalyst C. The cavity is characterised by the bosonic
annihilation operator $a$ with the photon number operator
$n_{\textsf{S}}:=a^{\dagger}a$. The atom has energy levels $\ket{g}$ and
$\ket{e}$ and its energy is captured by
$\sigma_{z}=\outerproduct{e}{e}-\outerproduct{g}{g}$. We work in the resonant
regime, where the atom and cavity have the same frequency $\omega$. The
evolution is governed by the JC Hamiltonian, which in the rotating-wave
approximation reads

$\displaystyle H_{\textsf{SC}}=\omega
a^{\dagger}a+\frac{\omega}{2}\sigma_{z}+g\left(\sigma_{+}a+\sigma_{-}a^{\dagger}\right),$
(3)

where $g$ is the coupling constant and $\sigma_{+}=\outerproduct{e}{g}$ and
$\sigma_{-}=\outerproduct{g}{e}$ are the raising and lowering operators. Note
that, as we focus on the resonant regime, the evolution specified by Eq. (3)
is energy-preserving.

The Jaynes-Cummings interaction Hamiltonian
$H_{\text{int}}=g\left(a\sigma_{+}+a^{\dagger}\sigma_{-}\right)$ couples pairs
of atom-field states $\\{\ket{n+1,g},\ket{n,e}\\}$. Consequently, the
Hamiltonian $H_{\textsf{SC}}$ decouples into a direct product of $2\times
2$-matrix Hamiltonians, i.e.,
$H_{\textsf{SC}}=\bigoplus_{n=0}^{\infty}H_{\textsf{SC}}^{(n)}$, where

$H_{\textsf{SC}}^{(n)}\begin{bmatrix}\ket{n+1,g}\\\
\ket{n,e}\end{bmatrix}=\begin{bmatrix}(n+1/2)\omega&g\sqrt{n+1}\,\\\ \\\
g\sqrt{n+1}&(n+1/2)\omega\end{bmatrix}\begin{bmatrix}\ket{n+1,g}\\\
\ket{n,e}\end{bmatrix}.$ (4)

The eigenvalue problem for this Hamiltonian yields the eigenfrequencies

$\omega^{(n)}_{\pm}=\quantity(n+\frac{1}{2})\omega\pm\frac{1}{2}\mu_{n},$ (5)

where $\mu_{n}=2g\sqrt{n+1}$ is the $n$-photon Rabi frequency. In the
resonance regime, the corresponding eigenstates are:

$\displaystyle\ket{n,+}=\frac{1}{\sqrt{2}}(\ket{n+1,g}+\ket{n,e})\>,\>\ket{n,-}=\frac{1}{\sqrt{2}}(\ket{n+1,g}-\ket{n,e}).$
(6)

The eigenstates $\ket{n,\pm}$ are referred to as dressed atom states. The
unperturbed atomic eigenstates, denoted as $\ket{g}$ and $\ket{e}$, are
modified (or ’dressed’) due to their interaction with the cavity field,
resulting in a shift of their eigenfrequencies by an amount that is determined
by the strength of the coupling, as illustrated in Fig. 1. [-2.6cm]  _Jaynes-
Cummings ladder_. Resonant atom-light interaction lifts the degeneracy of the
unperturbed states $\\{\ket{n,e},\ket{n+1,g}\\}$. The level splitting
$\mu_{n}=2g\sqrt{n+1}$ depends on the number of photons. Given that we know
the eigenvalues and eigenstates of the Jaynes–Cummings Hamiltonian, the
unitary time evolution operator $U(t)=e^{-iHt}$ takes on the form of

$\displaystyle U(t)=e^{i\omega t}\outerproduct{0,g}{0,g}$ $\displaystyle\hskip
17.07182pt+\sum_{n=0}^{\infty}e^{-i\quantity(n+\frac{1}{2})\omega
t}\Bigl{\\{}\cos\frac{\mu_{n}t}{2}\Bigl{(}\outerproduct{n\\!+\\!1,g}{n\\!+\\!1,g}+\outerproduct{n,e}{n,e}\Bigl{)}$
$\displaystyle\hskip
76.82234pt-i\sin\frac{\mu_{n}t}{2}\Bigl{(}\outerproduct{n\\!+\\!1,g}{n,e}+\outerproduct{n,e}{n\\!+\\!1,g}\Bigl{)}\Bigl{\\}}.$
(7)

The unitary operator (1) describes the full dynamics of the Jaynes-Cummings
model. In particular, we derive the final reduced states of the cavity S and
the atom C, and explicitly determine the set of atomic states that satisfy the
catalytic constraint [Eq. (1)]. Here, we will omit the technical details of
the derivation and instead focus on the discussion of our results (see Sec. 1
for the derivation of the reduced states of the system and cavity, and Sec. 2
for the solution of the catalytic constraint.).

Our objective is to catalytically generate non-classical light in the cavity.
To investigate the possibility of doing so, we will use two complementary
figures of merit as non-classicality witnesses. [1.2cm]  _Hanbury Brown and
Twiss experiment_. (a) A light source $S$ emits photons that are transmitted
through or reflected by a mirror M. Two detectors, namely $D_{1}$ and $D_{2}$,
count the number of photons, and the correlation between them is determined.
Photon detections are recorded as a function of time for (b) bunching (chaotic
light), (c) randomness (e.g., a coherent state, laser beam), and (d)
antibunching (e.g., light emitted from a single atom).

####  Subsection 2 Figures of merit

_Second-order coherence_

As a first figure of merit, we quantify non-classicality via the second-order
auto correlation function (or simply second-order coherence) of the final
state of the cavity [285], i.e.,

$\displaystyle g^{(2)}(\sigma_{\textsf{S}})=\frac{\langle a^{\dagger
2}a^{2}\rangle_{\sigma}}{\langle
a^{\dagger}a\rangle^{2}_{\sigma}}=\frac{\langle
n_{\textsf{S}}^{2}\rangle_{\sigma}-\langle
n_{\textsf{S}}\rangle_{\sigma}}{\langle n_{\textsf{S}}\rangle_{\sigma}^{2}},$
(8)

where $\langle
n_{\textsf{S}}\rangle_{\rho}:=\Tr[\rho_{\textsf{S}}n_{\textsf{S}}]$.

Physically, the concept of second-order coherence can be interpreted as a
response to the following question [286]: “ _If I have detected a photon or
multiple photons at time $t$, what is the probability or correlation of
detecting another photon or a similar number of photons at a later time
$t+\tau$?_”. Depending on the type of light source, this question yields
different answers.

To delve deeper into the physics behind the answer to such a question, let us
discuss the Hanbury Brown and Twiss experiment [287]. Firstly, let us remember
that quantum theory views a beam of light as a stream of photons, with the
measurement of beam intensity interpreted as the counting of photon arrivals
at a phototube. In the Hanbury Brown and Twiss experiment [see Fig. 1a for a
schematic representation], two detectors, $D_{1}$ and $D_{2}$, count the
number of photons transmitted through or reflected from the mirror $M$.
Consider a series of measurements where the numbers of photons, $n_{1}$ and
$n_{2}$, counted by the two detectors during a constant time interval, are
recorded. If the series of identical experiments is sufficiently long, each
detector registers the same average number of counts, $\langle
n_{1}\rangle=\langle n_{2}\rangle$. However, the number of counts in each
detector in a single run is not necessarily the same. In the quantum realm,
each incident photon either passes through the half-silvered mirror or is
reflected from it. Consequently, the two split beams are not exact replicas of
each other; it is only their average properties over an extended series of
experiments that are identical.

When a beam of light from an ordinary source is directed onto a
photomultiplier tube, which records the arrival of photons, an intriguing
phenomenon known as _photon bunching_ can be observed. In this phenomenon, the
photons detected by the phototube do not scatter randomly in time; rather,
they are detected in the form of clusters or bunches [see Fig. 1b]. Computing
the degree of second-order coherence of thermal (chaotic) light yields a value
of 2. This number indicates that a thermal light field has _super-Poissonian
statistics_. However, if the light source is now prepared as coherent light,
it exhibits no classical intensity fluctuations or photon bunching. As a
result, the counts in the two detectors are uncorrelated, and only random
coincidences occur, leading to a correlation function equal to $g^{(2)}=1$
[see Fig.1d]. Coherent light exhibits Poissonian statistics, yielding random
photon spacing. Interestingly, within such an experimental setup, we can
observe a phenomenon known as _antibunching_ , a counterpart to photon
bunching. This unique type of light is generated through nonlinear
interactions of laser light with matter and is solely compatible with the
quantum theory. Unlike coherent light, where the degree of second-order
coherence equals one, antibunched light exhibits a degree of second-order
coherence below unity, i.e., $g^{(2)}<1$ [see Fig. 1b]. Consequently, it is
considered to be anticorrelated or antibunched, highlighting its quantum
nature. In this case, the light field follows a sub-Poissonian photon
statistics, which means that the photon number distribution has a variance
that is less than the mean. Such a discussion can be summarised as follows:

$g^{(2)}=\begin{cases}2\quad\>\>\Longrightarrow\>\text{Super-Poissonian
statistics.}\\\ 1\quad\>\>\Longrightarrow\>\text{Poissonian statistics}\\\
<1\>\>\Longrightarrow\>\text{Sub-Poissonian photon statistics}\end{cases}$ (9)

_Wigner logarithmic negativity_

Our second figure of merit is the Wigner logarithmic negativity (WLN) [288,
114, 289], defined as

$\displaystyle\mathsf{W}\left(\rho\right):=\log\left(\int\\!\mathrm{d}x\,\mathrm{d}p\,\left|W_{\rho}\left(x,p\right)\right|\right),$
(10)

where $W_{\rho}\left(x,p\right)$ is the Wigner function

$\displaystyle W_{\rho}(x,p)=\frac{1}{\pi}\int e^{2ipx^{\prime}}\langle
x-x^{\prime}|\rho|x+x^{\prime}\rangle\,\text{d}x^{\prime}.$ (11)

Negativity of the Wigner function has long been recognised as an important
quantum feature, especially the volume of the negative part, which has been
introduced as a nonclassicality quantifier [290]. The crucial aspect of the
Wigner function’s logarithmic negativity (WLN) is its property as an additive
monotoneA resource theory for continuous-variable systems has been proposed
[114, 289], which quantifies both quantum non-Gaussianity (defined as the
convex hull of Gaussian states being the set of free states) and Wigner
negativity (with states having positive Wigner functions considered as free
states). In both cases, the Wigner logarithmic negativity is a resource
monotone., as the Wigner function of separable states can be factorised.
Additionally, it is computable through numerical integration.

Since the Wigner function is normalised, it is greater than zero
($\mathsf{W}>0$) whenever $W_{\rho}(x,p)$ is negative for some region of the
phase space.

###  Section 2 Generating non-classical states of light

The task we would like to achieve catalytically is the generation of non-
classical light in the cavity. We consider an initial state of the cavity that
is classical, namely a coherent state

$\ket{\alpha}=e^{-|\alpha|^{2}/2}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}\ket{n}.$
(12)

This state represents the closest approximation to an ideal, perfectly stable,
classical field with a well-defined amplitude and phase. Coherent states
exhibit Poissonian statistics with the mean number of photons $\langle
n_{\mathrm{S}}\rangle_{\ket{\alpha}}=|\alpha|^{2}$. The second-order coherence
for a coherent state is equal to unity, i.e.,
$g^{(2)}(\outerproduct{\alpha}{\alpha})=1$, and, since the Wigner function of
a coherent state is Gaussian, it implies that it is everywhere non-negative,
resulting in $\mathsf{W}=0$.

Figure 2: _First illustrative example._ Catalytic process for generating non-
classicality in the cavity, as captured by the second-order coherence. The
time evolution, $g^{(2)}(t):=g^{(2)}(\sigma_{\mathrm{S}}(t))$, is shown in the
top panel, while the bottom panel displays the change of the atomic state,
measured via the distance
$\Delta(t):=|\chi_{\mathrm{C}}-\sigma_{\mathrm{C}}(t)|_{1}$ concerning the
initial state. The orange stars indicate the final time ($\tau\approx 40$) for
which the evolution is catalytic. The atom returns to its initial state
($\Delta(\tau)=0$), while non-classicality has been activated,
$g^{(2)}(\tau)\approx 0.5<g^{(2)}(0)=1$. The values of the parameters are:
$\alpha=1/\sqrt{2}$, $\omega=2\pi$, $g=\pi$.

Our goal is to find an initial state of the atom $\chi_{\textsf{C}}$ and an
interaction time $\tau$ such that the evolution is catalytic [i.e., satisfying
Eq. (1)], while also leading to a final state of the cavity
$\sigma_{\textsf{S}}$ that exhibits non-classicality. Below, we present two
illustrative examples of such processes, using two complementary figures of
merit to witness non-classicality. A detailed analysis of these examples will
be discussed in the next section.

In Fig. 2, we present the catalytic activation of non-classicality. The upper
plot depicts the time evolution of $g^{(2)}$. Catalysis occurs at time
$\tau\approx 40$, indicated by the orange stars. Starting from the initial
state of the cavity
$\rho_{\mathrm{S}}=\outerproduct{\alpha}{\alpha}_{\mathrm{S}}$ (with
$\alpha=1/\sqrt{2}$), which has $g^{(2)}(\rho_{\mathrm{S}})=1$ as any coherent
state, we obtain a final state $\sigma_{\mathrm{S}}$ for which
$g^{(2)}(\sigma_{\mathrm{S}})\approx 0.5$, indicating non-classicality.
Simultaneously, we monitor the time evolution of the atomic state by
calculating the trace distance to its initial state, i.e.,

$\displaystyle\Delta(t):=\norm{{\chi_{\textsf{C}}-\sigma_{\textsf{C}}(t)}}_{1}.$
(13)

We observe that at the final time $\tau\approx 40$, we find $\Delta(\tau)=0$,
indicating that the atom has returned to its initial state

$\chi_{\textsf{C}}\approx\begin{pmatrix}0.16&0.36i\\\ 0.36i&0.84\\\
\end{pmatrix}$ (14)

as required for catalysis.

As a second example, we demonstrate how Wigner negativity can be catalytically
generated. Starting from a coherent state
$\rho_{\textsf{S}}=\outerproduct{\alpha}{\alpha}_{\textsf{S}}$ which has a
positive Wigner function, hence $\mathsf{W}(\rho_{\textsf{S}})=0$, we aim to
obtain a final state of the cavity $\sigma_{\textsf{S}}$ with
$\mathsf{W}(\sigma_{\textsf{S}})>0$, therefore certifying its non-
classicality. In Fig. 3, we present an example of such an evolution. First, we
plot WLN as a function of time $t$. At the final time $\tau\approx 5$, we
obtain a state $\sigma_{\textsf{S}}$ that has
$\mathsf{W}(\sigma_{\textsf{S}})\approx 0.1$, and we plot its Wigner function.
To verify the catalytic nature of the evolution, we display the evolution of
the atomic state via its trajectory in the Bloch sphere. Crucially, the
trajectory is closed, as the initial and final state of the atom exactly
coincide (red dot). In this example, the atom was prepared in the following
state:

$\chi_{\textsf{C}}\approx\begin{pmatrix}0.154&6.84.10^{-4}i\\\
-6.84.10^{-4}i&0.846\\\ \end{pmatrix}.$ (15)

Figure 3: _Second illustrative example._ Catalytic generation of Wigner
negativity. Panel $(a)$ shows the time evolution of the Wigner logarithmic
negativity $\mathsf{W}$, as well as the Wigner function at initial and final
time $\tau$. Non-classicality is clearly generated, while the process is
catalytic. Panel $(b)$ shows the trajectory of the atomic state, i.e., the
catalyst C, on the Bloch sphere. The initial and final state coincide (red
dot). The values of the parameters are: $\alpha=1/\sqrt{2}$, $\omega=2\pi$ and
$g=\pi$.

Note that the two examples we analysed are complementary. The first one
demonstrates the generation of non-classicality as witnessed by the $g^{(2)}$
function, while the final state $\sigma_{\mathrm{S}}$ has a positive Wigner
function. In the second example, the final state $\sigma_{\mathrm{S}}$ has a
negative Wigner function, despite $g^{(2)}(\sigma)>1$.

###  Section 3 Mechanism of catalysis

Here, we present a more intuitive understanding of quantum catalysis by
identifying a mechanism that allows for the activation of non-classicality. We
start by stating the following lemma:

######  Lemma 3.1 (Higher order moments).

Let
$O_{\textsf{SC}}=O_{\textsf{S}}\otimes\mathbbm{1}_{\textsf{C}}+\mathbbm{1}_{\textsf{S}}\otimes
O_{\textsf{C}}$ be an additive and conserved observable on the joint system
SC. In a catalytic protocol governed by a unitary operator $U$, such that
$\sigma_{\textsf{SC}}=U(\rho_{\textsf{S}}\otimes\chi_{\textsf{C}})U^{\dagger}$,
where $\sigma_{\textsf{C}}=\chi_{\textsf{C}}$, and $[U,O_{\textsf{SC}}]=0$,
the $k$-th moment of the observable $O$ on the system S satisfies:

$\langle O^{k}_{\textsf{S}}\rangle_{\sigma}=\langle
O_{\textsf{S}}^{k}\rangle_{\rho}+\Tr[\Delta_{k}(\rho_{\textsf{SC}}-\sigma_{\textsf{SC}})],$
(16)

where

$\Delta_{k}=\sum_{i=1}^{k-1}\binom{k}{i}O^{k-i}_{\textsf{S}}\otimes
O^{i}_{\textsf{C}}.$ (17)

###### Proof.

Let us use the short notation
$O_{\textsf{S}}=O_{\textsf{S}}\otimes\mathbbm{1}_{\textsf{C}}$ and
$O_{\textsf{C}}=\mathbbm{1}_{\textsf{S}}\otimes O_{\textsf{C}}$. Then, to
calculate $\langle O^{k}_{\textsf{S}}\rangle_{\sigma}$, we write
$O_{\textsf{SC}}$ as
$O^{k}_{\textsf{S}\textsf{C}}=O^{k}_{\textsf{S}}+O^{k}_{\textsf{C}}+\Delta_{k}$,
where
$\Delta_{k}=O^{k}_{\textsf{S}\textsf{C}}-O^{k}_{\textsf{S}}-O^{k}_{\textsf{C}}$
as given by Eq. (17). Thus, the $k$th moment of the observable
$O^{k}_{\textsf{S}\textsf{C}}$ in the state $\mathcal{A}$ is given by

$\langle
O^{k}_{\textsf{S}\textsf{C}}\rangle_{A}=\tr(O^{k}_{\textsf{S}}\mathcal{A})+\tr(O^{k}_{\textsf{C}}\mathcal{A})+\tr(\Delta_{k}\mathcal{A}).$
(18)

Next, re-writing $O^{k}_{\textsf{S}}$ as a function of
$O_{\textsf{S}\textsf{C}}$, $O_{\textsf{C}}$, and $\Delta_{k}$ allows one to
express $\langle O^{k}_{\textsf{S}}\rangle_{\sigma}$ as

$\displaystyle\langle O^{k}_{\textsf{S}}\rangle_{\sigma}$
$\displaystyle=\tr(O^{k}_{\textsf{S}\textsf{C}}\sigma)\\!-\\!\tr(O^{k}_{\textsf{C}}\sigma)\\!-\\!\tr(\Delta_{k}\sigma)=\tr(O^{k}_{\textsf{S}\textsf{C}}\rho)-\tr(O^{k}_{\textsf{C}}\rho)-\tr(\Delta_{k}\sigma)$
$\displaystyle=\tr(O^{k}_{\textsf{S}}\rho)+\tr(O^{k}_{\textsf{C}}\rho)+\tr(\Delta_{k}\rho)-\tr(\Delta_{k}\sigma)-\tr(O^{k}_{\textsf{C}}\rho).$
(19)

In the first line, we first used the fact that
$[U,O_{\textsf{S}\textsf{C}}]=0$ and that all moments of C are preserved. This
allows one to replace $\sigma$’s to $\rho$’s. Second, we use Eq. (16) to re-
write $\tr(O^{k}_{\textsf{S}\textsf{C}}\rho)$. Finally, simplifying Eq. (3)
gives

$\langle O^{k}_{\textsf{S}}\rangle_{\sigma}=\langle
O_{\textsf{S}}^{k}\rangle_{\rho}+\Tr[\Delta_{k}(\rho_{\textsf{SC}}-\sigma_{\textsf{SC}})].$
(20)

∎

In the Jaynes-Cummings model, the energies of the cavity S and the atom C are
specified by local number operators $n_{\textsf{S}}$ and $n_{\textsf{C}}$,
respectively. Hence, the total energy of both systems is proportional to the
number of excitations, and described by a joint operator
$n_{\textsf{SC}}:=n_{\textsf{S}}+n_{\textsf{C}}$. Since the JC evolution
$U(t)$ that takes $\rho_{\textsf{S}}\otimes\chi_{\textsf{C}}$ into
$\sigma_{\textsf{SC}}$ conserves the total energy, we have that
$[U(t),n_{\textsf{S}}+n_{\textsf{C}}]=0$ for all $t$. When the evolution is
catalytic, all moments of $n_{\textsf{C}}$ must remain unchanged, in
particular $\langle n_{\textsf{C}}\rangle_{\chi}$ = $\langle
n_{\textsf{C}}\rangle_{\sigma}$. Consequently, the first moment of
$n_{\textsf{S}}$ is also preserved, that is $\langle
n_{\textsf{S}}\rangle_{\rho}=\langle n_{\textsf{S}}\rangle_{\sigma}$.
Importantly, this is not the case for higher moments of $n_{\textsf{S}}$.
Particularly, from Lemma. 3.1, the second moment satisfies

$\displaystyle\langle n^{2}_{\textsf{S}}\rangle_{\sigma}=\langle
n^{2}_{\textsf{S}}\rangle_{\rho}+2\Bigg{(}\langle
n_{\textsf{S}}\rangle_{\sigma}\langle n_{\textsf{C}}\rangle_{\sigma}-\langle
n_{\textsf{S}}\otimes n_{\textsf{C}}\rangle_{\sigma}\Bigg{)}.$ (21)

Hence, the second moment in the final state of the cavity, $\langle
n^{2}_{\textsf{S}}\rangle_{\sigma}$, can become smaller (or larger) than the
second moment in the initial state $\langle n^{2}_{\textsf{S}}\rangle_{\rho}$.
This means that using a catalyst allows for modifying the distribution of the
local observable $n_{\textsf{S}}$ of the system S: while its average must
remain the same, the higher moments can change. Importantly, this can only
happen if the system becomes correlated with the catalyst, i.e., $\langle
n^{2}_{\textsf{S}}\rangle_{\sigma}\neq\langle
n^{2}_{\textsf{S}}\rangle_{\rho}$ only if
$\sigma\neq\sigma_{\textsf{S}}\otimes\sigma_{\textsf{C}}$ [as seen from Eq.
(21)]. Thus, these correlations are essential for observing quantum catalysis.
Note that the above analysis also applies beyond the JC model to arbitrary
observables and moments. The only requirement is the conservation of local
observables and the catalytic constraint.

The above analysis will also serve as a basis for the characterisation of the
parameter regime leading to catalysis. In particular, we derive a necessary
and sufficient condition for satisfying the catalytic constraint of Eq. (1).
This, in turn, allows one to obtain an analytic expression for the second-
order coherence $g^{2}$, which is based on the second moment $\langle
n^{2}_{\textsf{S}}\rangle_{\sigma}$. To do so, let us consider an arbitrary
initial state of the atom:

$\displaystyle\chi_{\textsf{C}}=q\outerproduct{g}{g}+r\outerproduct{g}{e}+r^{*}\outerproduct{e}{g}+[1-q]\outerproduct{e}{e},$
(22)

as well as a general initial state of the cavity
$\rho_{\textsf{S}}=\sum_{n,m}^{\infty}p_{n,m}\outerproduct{n}{m}$ with
$p_{n}:=p_{n,n}$. Combining Eq. (21) with the fact that mean energy of the
cavity is conserved, i.e., $\langle n_{\textsf{S}}\rangle_{\sigma}=\langle
n_{\textsf{S}}\rangle_{\rho}$, we arrive at

$\displaystyle\\!g^{(2)}(\sigma_{\textsf{S}})=g^{(2)}(\rho_{\textsf{S}})-\frac{2}{\langle
n_{\textsf{S}}\rangle_{\rho}^{2}}\\!\left[\langle n_{\textsf{S}}\otimes
n_{\textsf{C}}\rangle_{\sigma}\\!-(1-q)\langle
n_{\textsf{S}}\rangle_{\rho}\right]\\!,\\!$ (23)

where we have

$\displaystyle\\!\\!\\!\langle n_{\textsf{S}}\otimes
n_{\textsf{C}}\rangle_{\sigma}\\!=\\!\sum_{n=0}^{\infty}n\left[(1-q)p_{n}c_{n}^{2}+y_{n}+qp_{n+1}s_{n}^{2}\right],$
(24)

with $s_{n}\\!:=\\!\sin(gt\sqrt{n+1}),c_{n}\\!:=\\!\cos(gt\sqrt{n+1})$ and
$y_{n}\\!:=\\!2\operatorname{Im}(rp_{n+1,n})s_{n}c_{n}$.

In order to satisfy the catalytic constraint, we obtain a set of equations for
the components of the atomic state. Decomposing the diagonal term as
$q=q_{\text{inc}}+q_{\text{coh}}$, we get

$\displaystyle
q_{\text{inc}}=\frac{1}{Q}\sum_{n=0}^{\infty}p_{n}s_{n}^{2},\qquad
q_{\text{coh}}=\frac{1}{Q}\sum_{n=0}^{\infty}y_{n},$ (25)

with $Q:=\sum_{n=0}^{\infty}(p_{n}+p_{n+1})s_{n}^{2}$. Interestingly,
$q_{\text{inc}}$ is specified by the occupations of $\rho_{\textsf{S}}$, while
$q_{\text{coh}}$ depends on its coherence in the Fock basis. Moreover, the
off-diagonal term $r$ satisfies

$r=\frac{i(a_{3}a^{*}_{4}+a^{*}_{1}a_{4})}{|a_{1}|^{2}-|a_{3}|^{2}}-\frac{i(a_{3}a^{*}_{2}+a^{*}_{1}a_{2})}{|a_{1}|^{2}-|a_{3}|^{2}}q,$
(26)

with $a_{i}$ being auxiliary functions defined as

$\displaystyle\begin{split}a_{1}&=\sum_{n=0}^{\infty}p_{n,n}c_{n-1}c_{n}-e^{-i\omega\tau},\qquad
a_{3}=\sum_{n=0}^{\infty}p_{n,n+2}s_{n}s_{n+1},\\\
a_{2}&=\sum_{n=0}^{\infty}p_{n,n+1}s_{n}\quantity[c_{n-1}+c_{n+1}],\hskip
10.0pta_{4}=\sum_{n=0}^{\infty}p_{n,n+1}s_{n}c_{n+1}.\end{split}$ (27)

For a detailed derivation of Eqs. (27) see Sections 1 and 2.

Importantly, Eqs (25) and (26) are necessary and sufficient for ensuring the
catalytic constraint of Eq. (1). In combination with Eq. (23), we can now
characterise analytically the effect of non-classicality activation for the
$g^{2}$ function in the catalytic regime. This is done for finding the
parameters (initial states and interaction time) for the first illustrative
example of a catalytic process (see Fig. 2). For the second example, we use
again such analytic results to ensure the validity of the catalytic
constraint, while the Wigner functions are computed numerically.

###  Section 4 How general is catalysis?

An interesting problem is to understand how typical the effect of catalysis
is. Here we discuss different aspects of this question.

To begin with, note that it is not obvious a priori whether the catalytic
constraint of Eq. (1) can be satisfied. However, due to the quantum version of
the Perron-Frobenius theorem, every quantum channel has at least one positive
semi-definite fixed point [291]. Now, for a fixed input state
$\rho_{\textsf{S}}$ and a fixed interaction time $\tau$, the state of the atom
C evolves according to an effective quantum channel
$\chi_{\textsf{C}}\rightarrow\Tr_{\textsf{S}}[U(\tau)(\rho_{\textsf{S}}\otimes\chi_{\textsf{C}})U^{\dagger}(\tau)]$.
Consequently, there always exists an initial state $\chi_{\textsf{C}}$ which
is left unchanged by this channel, hence providing at least one solution to
Eq. (1).

Next, one might wonder how often a catalytic evolution leads to a non-
classical state of the cavity. In particular, when preparing the cavity in a
coherent state $\ket{\alpha}$, does there always exist a state of the catalyst
$\chi_{\textsf{C}}$ which allows to generate non-classicality? To address this
question, we investigate the minimum value of $g^{(2)}$ of the final state
$\sigma_{\textsf{S}}$, as a function of $|\alpha|$ [see Fig. 4a]. This is done
by combining Eqs. (23-24) and (25-26), and imposing a bound on the final time,
i.e $g\tau\leq 100$. For $\alpha\in(0,2]$, we observe that
$g^{(2)}(\sigma_{\textsf{S}})<1$, indicating that non-classicality generation
via catalysis is generic here. Note that when the initial coherent state has
low energy, the final state of the cavity is close to the intial one, but with
a slightly reduced variance, leading to a value of $g^{(2)}$ approaching zero.

Figure 4: _Which states lead to catalysis?_ Panel (a) shows the minimal value
of $g^{2}$ obtained as a function of the amplitude $|\alpha|$ of the initial
coherent state of the cavity
$\rho_{\textsf{S}}=\outerproduct{\alpha}{\alpha}_{\textsf{S}}$. The orange
star corresponds to our first illustrative example. Panel (b) displays the
atomic states (in the $y-z$ plane of the Bloch sphere) that satisfy the
catalytic constraint and generate non-classical states, for an intial coherent
state $\alpha=1/\sqrt{2}$. The colour represents different values of
$g^{(2)}<1$. We impose a limit on the interaction time $\tau\leq 100$ and take
$10^{6}$ samples. In both panels, parameters are $\omega=2\pi$, $g=\pi$.

Although we have focused on the generation of states with Wigner negativity or
sub-Poissonian distribution, catalysis also allows the generation of specific
non-classical states. In this regard, we show in Sec. C-2 that one can also
use our catalytic protocol to generate squeezed states of light. We further
determine the possible final states of the cavity that can be obtained under a
catalytic protocol.

Let us now ask the converse question, i.e., whether every atomic state can
lead to a catalytic evolution. A first observation is that pure states cannot
act as useful catalysts in general, and in particular cannot catalytically
generate non-classicality. Indeed, a key ingredient for catalysis is the fact
that the system and catalyst become correlated [see Eq. (21)], which is
impossible when the state of the catalyst is pure.

To further explore this question, we also investigated which states of the
atom can catalytically generate sub-Poissonian statistics (i.e., $g^{(2)}<1$),
given an initial coherent state of the cavity and a limited interaction time
$g\tau\leq 100$. In Fig. 4b, we display an example of such a set of catalytic
states. Interestingly, this set appears to contain states that are almost
pure. The structure of the catalytic set is highly non-trivial, and, in
particular, we observe a strong dependence on the initial state of the cavity.
This leads to the following question:

_Which states of the atom allow to generate non-classicality?_

We refer to the set of all atomic states that satisfy Eq. (1) as the _set of
catalytic states_. Formally, one can define it as follows:

$\mathcal{C}:=\big{\\{}\omega_{\textsf{C}}\,\big{|}\,\omega_{\textsf{C}}=\Tr_{\textsf{S}}[U(\tau)(\rho_{\textsf{S}}\otimes\omega_{\textsf{C}})U^{\dagger}(\tau)],\,\tau\geq
0\big{\\}}.$ (28)

To explore the geometry of this set, we characterise the catalytic set for
three initial coherent states of the cavity corresponding to
$\alpha\in\\{0.2,1/\sqrt{2},25\\}$. The results are presented in Fig. 5, where
the catalytic set is shown in grey. Additionally, we determine which states in
the catalytic set can generate non-classicality, focusing on the $g^{(2)}$
function. Similar to Fig. 4b, we represent these states in color, with the
latter indicating the level of non-classicality being generated.
Interestingly, these sets vary significantly with $|\alpha|$. Moreover, when
$|\alpha|$ is large, catalytic states are distributed close to the equatorial
plane of the Bloch sphere, and we could find no instance where non-
classicality is generated. This can be understood by noticing that the atom is
“too small” to significantly perturb the field. Furthermore, when the initial
state of the cavity is prepared as an incoherent mixture of Fock states, the
set of catalytic states is diagonal and lies along the $z$ axis.

Figure 5: _Set of catalytic state for different initial state preparation_.
Atomic states (projection of the $y-z$ plane of the Bloch ball) that satisfy
the catalytic constraint (gray) for (a) $\alpha=0.2$, (b) $\alpha=1/\sqrt{2}$,
and (c) $\alpha=25$ are highlighted by a uniform gradient if they produce
$g^{2}(\sigma_{\textsf{S}})<1$ All panels were generated by imposing a limit
on interaction time with $\tau\leq 100$ and taking $10^{6}$ samples. The
parameters are $\omega=2\pi$ and $g=\pi$.

Finally, one can ask what happens when the initial state of the cavity is not
a coherent one. Let us start by examining Eqs. (51) and (52) under the
assumption that the initial state of the cavity is an incoherent mixture of
Fock states, i.e., $\rho_{\textsf{S}}=\sum_{n,m}p_{n,m}\outerproduct{n}{m}$
with $p_{n,m}=0$ if $n\neq m$. In this case, from Eqs. (25) and (26), we can
infer that the only feasible states of the catalyst are those with
$q_{\text{coh}}=0$ and $r=0$. Consequently, the atomic state is incoherent in
the energy basis and its ground state occupation takes the form of

$q=\frac{\sum_{n=0}^{\infty}p_{n}\sin^{2}(gt\sqrt{n+1})}{\sum_{n=0}^{\infty}(p_{n}+p_{n+1})\sin^{2}(gt\sqrt{n+1})}.$
(29)

This leads to the following theorem:

######  Theorem 4.1 (Fock states under catalytic transformations).

The non-classicality of a pure Fock state is non-decreasing under catalytic
evolution

###### Proof.

Proving that the second-order second-order coherence of a Fock state
$\rho_{\textsf{S}}=\outerproduct{k}{k}_{\textsf{S}}$, cannot decrease under a
catalytic evolution is equivalent to show the quantity

$\displaystyle g^{(2)}(\sigma_{\textsf{S}})-g^{(2)}(\rho_{\textsf{S}})$
$\displaystyle=\frac{\Tr[(a^{\dagger
2}a^{2})\sigma_{\textsf{S}}]}{\Tr[(a^{\dagger}a)\sigma_{\textsf{S}}]}-\frac{\Tr[(a^{\dagger
2}a^{2})\rho_{\textsf{S}}]}{\Tr[(a^{\dagger}a)\rho_{\textsf{S}}]}$
$\displaystyle=\frac{\Tr[(a^{\dagger
2}a^{2})(\sigma_{\textsf{S}}-\rho_{\textsf{S}})]}{\Tr[(a^{\dagger}a)\rho_{\textsf{S}}]}$
(30)

is nonnegative. This will be accomplished by showing that the following
inequality holds:

$\Tr[(a^{\dagger 2}a^{2})(\sigma_{\textsf{S}}-\rho_{\textsf{S}})]\geq 0.$ (31)

In what follows, we omit the index S as well as any explicit reference to the
variables’ dependence on time. When the initial state of the cavity is a Fock
state, then Eq. (1) implies that the state of the cavity after the catalytic
evolution is given by

$\displaystyle\sigma=$
$\displaystyle\quantity[(1-q)\cos^{2}\quantity(gt\sqrt{k+1})+q\cos^{2}\quantity(gt\sqrt{k})]\outerproduct{k}{k}$
$\displaystyle+q\sin^{2}\quantity(gt\sqrt{k})\outerproduct{k\\!-\\!1}{k\\!-\\!1}+(1-q)\sin^{2}\quantity(gt\sqrt{k+1})\outerproduct{k\\!+\\!1}{k\\!+\\!1},$
(32)

where $q$ is determined by Eq. (29), which for this particular case takes the
form:

$\displaystyle
q=\frac{\sin^{2}\quantity(gt\sqrt{k+1})}{\sin^{2}\quantity(gt\sqrt{k+1})+\sin^{2}\quantity(gt\sqrt{k})}$
(33)

By substituting Eq.(33) into Eq.(1) and introducing the notation
$\psi=\frac{1}{2}(\outerproduct{n-1}{n-1}+\outerproduct{n+1}{n+1})$, we
obtain:

$\displaystyle\sigma=$
$\displaystyle\frac{\sin^{2}\quantity(gt\sqrt{k})\cos^{2}\quantity(gt\sqrt{k+1})+\sin^{2}\quantity(gt\sqrt{k+1})\cos^{2}\quantity(gt\sqrt{k})}{\sin^{2}\quantity(gt\sqrt{k+1})+\sin^{2}\quantity(gt\sqrt{k})}\rho$
$\displaystyle\hskip
99.58464pt+\frac{2\sin^{2}\quantity(gt\sqrt{k})\sin^{2}\quantity(gt\sqrt{k+1})}{\sin^{2}\quantity(gt\sqrt{k+1})+\sin^{2}\quantity(gt\sqrt{k})}\psi.$
(34)

Alternatively, we can express the above equation as $\sigma=t\rho+(1-t)\psi$,
where

$t=\frac{\sin^{2}\quantity(gt\sqrt{k})\cos^{2}\quantity(gt\sqrt{k+1})+\sin^{2}\quantity(gt\sqrt{k+1})\cos^{2}\quantity(gt\sqrt{k})}{\sin^{2}\quantity(gt\sqrt{k+1})+\sin^{2}\quantity(gt\sqrt{k})}.$
(35)

With these results at hand, we can manipulate Eq. (31) to obtain

$\displaystyle\Tr[(a^{\dagger 2}a^{2})(\sigma-\rho)]$
$\displaystyle=(1-t)\Tr[(a^{\dagger 2}a^{2})(\psi-\rho)]$
$\displaystyle=\frac{(1-t)}{2}\Tr[(a^{\dagger
2}a^{2})(\outerproduct{k+1}{k+1}+\outerproduct{k-1}{k-1})]$
$\displaystyle\hskip 113.81102pt-(1-t)\Tr[(a^{\dagger
2}a^{2})\outerproduct{k}{k}]$
$\displaystyle=\frac{(1-t)}{2}\quantity[(k+1)k+(k-1)(k-2)-2k(k-1)]$
$\displaystyle=(1-t)\geq 0.$ (36)

Thus, we conclude that second-order coherence $g^{(2)}$ can only increase
during a catalytic processes involving a pure Fock state, i.e.,

$g^{(2)}(\sigma)-g^{(2)}(\rho)\geq 0.$ (37)

∎

However, note that the above result is only valid for pure Fock states. For an
incoherent mixture of Fock states, we observed that catalysis can further
boost non-classicality (see Section C-2 for details).

####  Subsection 1 Generating squeezed states

The state of the cavity after the catalytic process is described by Eq. (1),
with $q$ and $p$ given by Eqs. (52) and (51). In the main text, we have
discussed that both states with Wigner negativity and sub-Poissonian states
can be produced catalytically, but the specific form of the states of light
was not directly addressed. Here, we focus on this question and show that it
is possible to prepare squeezed states in a catalytic way.

Defining the field quadratures by $X_{1}=(a^{\dagger}+a)/\sqrt{2}$ and
$X_{2}=i(a^{\dagger}-a)/\sqrt{2}$, which satisfy the commutation relation
$[X_{1},X_{2}]=i$. The Heisenberg uncertainty principle is given by $\Delta
X_{1}\Delta X_{2}\geq 1/2$, where $(\Delta X_{1})^{2}=\langle
X_{1}^{2}\rangle-\langle X_{1}\rangle^{2}$ and similarly for $(\Delta
X_{2})^{2}$. For the coherent field state, the uncertainties are equal to
$\Delta X^{(\alpha)}_{1}=\Delta X^{(\alpha)}_{2}=1/\sqrt{2}$. A field state is
called squeezed if the uncertainty of one of the quadratures is below the
vacuum level, i.e., $\Delta X_{1}<1/\sqrt{2}$ or $\Delta X_{2}<1/\sqrt{2}$.
Here, we consider the squeezing parameter

$\xi=\frac{\Delta X_{1}}{\Delta X^{(\alpha)}_{1}}=\sqrt{2}\Delta X_{1}$ (38)

and search for conditions where $\xi<1$, which manifests field squeezing.

The catalytic generation of squeezed states can be observed in Fig 6. The left
plot depicts the minimal value of $\xi$ obtained as a function of the
amplitude $|\alpha|$ of the initial coherent state of the cavity. We impose a
limit on the interaction time $\tau\leq 100$ and observe that $\xi$ goes below
the shot noise level, implying the presence of squeezing. In the right plot,
we depict the evolution of $\xi$. Catalysis occurs at time $\tau=18$
(represented by an orange star). Starting from the initial state of the cavity
$\rho_{\textsf{S}}=\outerproduct{\alpha}{\alpha}_{\textsf{S}}$, which has
$\xi=1$, we obtain a final state $\sigma_{\textsf{S}}$ for which $\xi\approx
0.79$.

Figure 6: _Generating squeezed states via quantum catalysis._ The left panel
shows the minimal value of $\xi$ obtained as a function of the amplitude
$|\alpha|$ of the initial coherent state of the cavity within the interval of
interaction time $\tau\leq[0,100]$. Taking the minimal value of $|\alpha|$,
the right panel displays its time evolution, defined as
$\xi(t):=\xi(\sigma_{\textsf{S}}(t))$. The orange star indicates the final
time ($\tau\approx 18$) at which the evolution becomes catalytic. At this
point, the atom returns to its initial state, while a squeezed state with
$\xi(\tau)\approx 0.79<\xi(0)=1$ has been produced. The values of the
parameters are: $\alpha=1/2$, $\omega=\pi$, and $g=2\pi$.

###  Section 5 Catalysis in realistic scenarios

####  Subsection 1 Catalysis in the presence of dissipation

So far we have discussed an idealized (noise-free) scenario in quantum optics
to understand the physics behind catalysis in quantum systems. However, any
experiment implementing the Jaynes-Cummings model will necessarily feature
cavity loss and atomic decay. We will now illustrate that the catalytic effect
is robust even after incorporating these two effects in the model. To
understand why our results remain valid in the presence of dissipation, it is
worth noting that the Perron-Frobenius theorem, which assures that the
catalytic constraint can be satisied, is valid for all quantum channels (hence
also dissipative). Consequently, even in the presence of losses and noise, the
catalytic constraint can still be satisfied. The interesting problem is
whether non-classicality can still be generated in such a dissipative
protocol.

In order to understand the role of dissipation in our model let us assume that
the cavity S is coupled to an environment characterized by temperature $T$
which accounts for the cavity losses. We also consider the possibility of atom
decay due to photon emission. Consequently, the dynamics of the atom-cavity
system are described by a Lindblad master equation of the form [292, 293, 294,
295]:

$\dot{\rho}_{\textsf{SC}}\\!=\\!-i\left[H_{\textsf{JC}},\rho_{\textsf{SC}}\right]+\kappa(n_{\textsf{th}}+1)\mathcal{L}[a]+\kappa
n_{\textsf{th}}\mathcal{L}[a^{\dagger}]+\Gamma\mathcal{L}[\sigma_{-}],$ (39)

where as before we take
$\rho_{\textsf{SC}}:=\outerproduct{\alpha}{\alpha}_{\textsf{S}}\otimes\chi_{\textsf{C}}$,
and $\kappa$ and $\Gamma$ are the cavity and atom dissipation rate,
respectively. Moreover, $n_{\textsf{th}}=(e^{1/T}-1)^{-1}$ is the average
excitation number and
$\mathcal{L}[L]=L\rho_{\textsf{SC}}L^{\dagger}-\frac{1}{2}\\{L^{\dagger}L,\rho_{\textsf{SC}}\\}$
is the Lindblad dissipator with $\\{.\,,\,.\\}$ denoting the anticommutator.
In the case of dissipative evolution (39) the (generalized) catalytic
constraint becomes

$\sigma_{\textsf{C}}:=\Tr_{\textsf{S}}[\mathcal{E}_{\tau}(\rho_{\textsf{SC}})]=\chi_{\textsf{C}},$
(40)

where $\mathcal{E}_{\tau}(\cdot)$ is a quantum channel generated by the
evolution from Eq. (39) acting for some time $\tau$.

For a given stopping time $\tau$ the solution to Eq.(39) that simultaneously
satisfies the catalytic constraint can be determined numerically.

In order to study the robustness of our results to dissipation we will now
analyse two figures of merit, WLN and the auto-correlation function $g^{(2)}$,
as a function of the stopping time $\tau$ for closed (reversible) and open
(irreversible) quantum dynamics. At all stopping times $\tau$ we ensure that
the generalized catalytic constraint from Eq. (40) is met. Our results are
summarized in in Fig. 1 which shows the amount of non-classicality that can be
generated in the mode S for different choice of interaction times $\tau$. Our
first observation is that a catalytic generation of non-classicality is
possible even in the presence of dissipation. Secondly, the amount of non-
classicality that can be generated in this way generally diminishes with
longer interaction times. This is intuitive, as the longer we wait, the higher
is the probability of losing photons from the cavity, especially since we are
considering a constant dissipation rate for both the cavity and atom. Very
interestingly, even for short evolution times, we can achieve relatively high
levels of non-classicality as compared to the unitary case. [-8cm]  _Catalysis
in realistic scenarios._ Wigner logarithmic negativity and auto-correlation
function are plotted as functions of the interaction time $\tau$. The blue
solid line represents the scenario with dissipation, while the gray solid line
represents the scenario without dissipation. For each time $\tau$, the
catalytic constraint is satisfied, and the degree of non-classicality
activated in the mode is computed. While $\mathsf{W}\geq 0$ indicates non-
classical behaviour, $g^{(2)}<1$ also witnesses non-classicality (reddish area
in the right panel). Both plots use constant dissipation rates: an atom
dissipation rate of $\Gamma=0.05$ and a cavity dissipation rate of
$\kappa=0.005$. The average number of thermal bath excitations is
$n_{\text{th}}=0.1$ and the other parameters are $\omega=2\pi$, $g=0.1\pi$ and
$\alpha=1/\sqrt{2}$.

####  Subsection 2 Catalytic protocol in the presence of multiple cavities

A relevant question one might have when investigating the catalytic protocol
is whether the atom can be reused in a _truly_ catalytic way. That is, whether
the catalytic protocol can be performed multiple times, while the catalyst is
prepared only once. To see whether our protocol fulfills this requirement we
consider a single atom C interacting sequentially with $N$ cavities
$\textsf{S}:=\textsf{S}_{1}\textsf{S}_{2}\ldots\textsf{S}_{N}$ with the same
matching mode according to a catalytic process satisfying Eq. (1). More
specifically, we imagine the following protocol:

  1. 1.

For a given choice of parameters $\omega$, $g$, and $\tau$, determine the
state of the atom $\chi_{\textsf{C}}$ as discussed in Sec. 3.

  2. 2.

Let the atom interact sequentially with $N$ cavities with the same frequency
$\omega$ and interaction strenght $g$ and for the same amount of time $\tau$.
This leads to the following unitary transformation of the composite system

$\sigma_{\textsf{SC}}=\bigotimes_{i=1}^{N}[U_{i}\,(\rho_{\textsf{S}_{i}}\otimes\chi_{\textsf{C}})\,U^{\dagger}_{i}],$
(41)

where for all $i\in\\{1,...,N\\}$, we have
$\rho_{{\textsf{S}_{i}}}=\outerproduct{\alpha}{\alpha}_{\textsf{S}_{i}}$ and
$U_{i}$ is the unitary from Eq. (1) applied to $\textsf{S}_{i}$ and C.

  3. 3.

After interacting with all the cavities the atom is removed from the last
cavity ($\textsf{S}_{N}$) and allowed to dissipate its energy to the
environment. This removes the correlations built between the atom and the
cavities.

After discarding the atom C the joint state of the cavities is given by
$\sigma_{\textsf{S}}=\Tr_{\textsf{C}}[\sigma_{\textsf{SC}}]=\sigma_{\textsf{S}_{1}\textsf{S}_{2}\ldots\textsf{S}_{n}}$.
Importantly, each of the states $\sigma_{\textsf{S}_{i}}$ produced in this way
is the same and non-classical, and can be further used in some information
processing task.

In certain applications one might be also interested in the amount of
correlations developed between the cavities themselves. Such correlations
could, in principle, limit the protocol’s applicability to certain tasks which
require producing multiple uncorrelated non-classical states. Luckily, the
amount correlations developed between the cavities after discarding the atom
is relatively low. To see this we will use the quantum fidelity
$F(\sigma_{\textsf{SC}},\sigma_{\text{target}})$ where
$\sigma_{\text{target}}$ is a product state
$\sigma_{\text{target}}=\sigma_{\textsf{S}_{1}}\otimes\sigma_{\textsf{S}_{2}}\otimes\ldots\otimes\sigma_{\textsf{S}_{N}}$
and $\sigma_{\textsf{S}_{i}}=\sigma_{\textsf{S}_{j}}$ for $1\leq i,j\leq N$.
In Fig. 2, we show the resulting fidelity
$F(\sigma_{\textsf{SC}},\sigma_{\text{target}})$ for up to $N=5$ cavities.
Notably, we observe that even for $N=5$ the fidelity is relatively high,
indicating tha the cavities are approximately uncorrelated after the catalytic
protocol, and hence can be treated as independent. [-5.2cm]  _Fidelity as a
function of the number of cavities_. In the regime $g\tau\approx\pi$ (red
dot), a high fidelity is observed, indicating the feasibility of employing the
catalytic protocol to activate the non-classicality across five modes.

###  Section 6 Derivation of the results

The first part of this Section provides the eigenproblem solution for the
Jaynes-Cummings model (for further information, see references [296, 297]). In
the following, we derive the final reduced states of the cavity S and the atom
C, and explicitly determine the set of atomic states that satisfy the
catalytic constraint [Eq. (1)].

####  Subsection 1 Reduced states of subsystems

The following unitary operator describes the dynamics of the Jaynes-Cummings
model in resonance and under the rotating-wave approximation:

$\displaystyle U(t)=$ $\displaystyle e^{i\omega t}\outerproduct{0,g}{0,g}$
$\displaystyle\\!\\!+\sum_{n=0}^{\infty}e^{-i\quantity(n+\frac{1}{2})\omega
t}\Bigl{\\{}\cos\frac{\mu_{n}t}{2}\Bigl{(}\outerproduct{n+1,g}{n+1,g}+\outerproduct{n,e}{n,e}\Bigl{)}$
$\displaystyle\\!\\!\hskip
59.75095pt-i\sin\frac{\mu_{n}t}{2}\Bigl{(}\outerproduct{n+1,g}{n,e}+\outerproduct{n,e}{n+1,g}\Bigl{)}\Bigl{\\}}.$
(42)

After the interaction, the joint system
$\sigma_{\textsf{S}C}(t)=U(t)(\rho_{\textsf{S}}\otimes\chi_{\textsf{C}})U^{\dagger}$
becomes correlated. Let the cavity and the atom be prepared in general mixed
states, i.e.,
$\rho_{\textsf{S}}=\sum^{\infty}_{n,m=0}p_{n,m}\outerproduct{n}{m}$ and
$\chi_{\textsf{C}}=q\outerproduct{g}{g}+r\outerproduct{g}{e}+r^{*}\outerproduct{e}{g}+(1-q)\outerproduct{e}{e}$.
Then, the reduced state of the cavity at time $t$ is obtained by taking the
partial trace over the atom’s degrees of freedom, i.e.,
$\sigma_{\textsf{S}}(t):=\text{tr}_{\textsf{C}}[{\sigma_{\textsf{S}C}}(t)]$.
More specifically,

$\displaystyle\sigma_{\textsf{S}}(t)=$ $\displaystyle
qp_{0,0}\outerproduct{0}{0}+\Bigl{\\{}\sum_{n=0}^{\infty}e^{i(n+1)\omega
t}\quantity[qp_{0,n+1}c_{n}+irp_{0,n}s_{n}]\outerproduct{0}{n+1}+\text{h.c}\Bigl{\\}}$
$\displaystyle+\sum_{n,m=0}^{\infty}e^{-i(n-m)\omega
t}qp_{n+1,m+1}\quantity[s_{n}s_{m}\outerproduct{n}{m}+c_{n}c_{m}\outerproduct{n\\!+\\!1}{m\\!+\\!1}]$
$\displaystyle+\sum_{n,m=0}^{\infty}e^{-i(n-m)\omega
t}(1-q)p_{n,m}\quantity[c_{n}c_{m}\outerproduct{n}{m}+s_{n}s_{m}\outerproduct{n\\!+\\!1}{m\\!+\\!1}]$
$\displaystyle+\Bigl{\\{}\sum_{n,m=0}^{\infty}ie^{-i(n-m)\omega
t}c_{n}s_{m}\quantity[p_{n+1,m}r\outerproduct{n\\!+\\!1}{m\\!+\\!1}+p_{n,m+1}r^{*}\outerproduct{n}{m}]$
$\displaystyle\hskip 207.7052pt+\text{h.c}\Bigl{\\}}.$ (43)

The atomic state is obtained by marginalising over the photonic degrees of
freedom, i.e., $\chi(t):=\text{tr}_{\textsf{S}}[{\sigma_{\textsf{S}C}}(t)]$,
which leads to

$\displaystyle\chi_{\textsf{C}}(t)=q(t)\outerproduct{g}{g}+r(t)\outerproduct{g}{e}+r^{*}(t)\outerproduct{e}{g}+[1-q(t)]\outerproduct{e}{e},$
(44)

Note that we identify $r:=r(0)$ and $q:=q(0)$ in what follows. The
coefficients $q(t)$ and $r(t)$ are given by

$\displaystyle q(t)$
$\displaystyle=q\sum_{n=0}^{\infty}p_{n}c^{2}_{n-1}+(1-q)p_{n}s^{2}_{n}+2\textrm{Re}[irp_{n+1,n}]s_{n}c_{n},$
(45) $\displaystyle r(t)$ $\displaystyle=-ie^{i\omega
t}\sum_{n=0}^{\infty}p_{n,n+1}s_{n}c_{n}+re^{i\omega
t}\sum_{n=0}^{\infty}p_{n}c_{n-1}c_{n}+r^{*}\sum_{n=0}^{\infty}p_{n,n+2}s_{n}s_{n+1})$
$\displaystyle\hskip 113.81102pt+ie^{i\omega
t}q\sum_{n=0}^{\infty}p_{n,n+1}s_{n}\quantity[c_{n-1}+c_{n}].$ (46)

####  Subsection 2 Catalytic constraint

Here we determine the set of atomic states that evolve catalytically by
explicitly solving the catalytic constraint from Eq. (1). More specifically,
we are looking for the solution to the following operator equation:

$\displaystyle\chi_{\textsf{C}}(\tau)=\Tr_{\textsf{S}}\\{U(\tau)[\rho_{\textsf{S}}\otimes\chi_{\textsf{C}}(\tau)]U(\tau)^{\dagger}\\},$
(47)

for a fixed time $\tau$. From this point forward, we will abbreviate the
diagonal elements of the state $\rho_{\textsf{S}}$ as $p_{n}:=p_{n,n}$. To
obtain the set of states that satisfy Eq.(47), we first define the auxiliary
functions:

$\displaystyle\tilde{a}_{1}(t)$ $\displaystyle=e^{i\omega
t}\sum_{n=0}^{\infty}p_{n}\cos(gt\sqrt{n})\cos(gt\sqrt{n+1})-1,$
$\displaystyle\tilde{a}_{2}(t)$ $\displaystyle=ie^{i\omega
t}\sum_{n=0}^{\infty}p_{n,n+1}\sin(gt\sqrt{n+1})\quantity[\cos(gt\sqrt{n})+\cos(gt\sqrt{n+2})],$
$\displaystyle\tilde{a}_{3}(t)$ $\displaystyle=e^{i\omega
t}\sum_{n=0}^{\infty}p_{n,n+2}\sin(gt\sqrt{n+1})\sin(gt\sqrt{n+2}),$
$\displaystyle\tilde{a}_{4}(t)$ $\displaystyle=-ie^{i\omega
t}\sum_{n=0}^{\infty}p_{n,n+1}\sin(gt\sqrt{n+1})\cos(gt\sqrt{n+2}).$ (48)

Note that $\tilde{a}_{i}=e^{i\omega t}a_{i}$. Next, we observe that Eq. (47)
gives rise to a set of two equations with two variables. By considering the
ground state occupation $q(t)$, we find that the states satisfying Eq. (47)
are given by:

$q(t)=\frac{\sum_{n=0}^{\infty}p_{n}\sin^{2}(gt\sqrt{n+1})+\operatorname{Re}\quantity[ir(t)p_{n+1,n}]\sin(2gt\sqrt{n+1})}{\sum_{n=0}^{\infty}(p_{n}+p_{n+1})\sin^{2}(gt\sqrt{n+1})},$
(49)

whereas the coherence $r(t)$ obeys the equation

$r(t)\tilde{a}_{1}(t)+q(t)\tilde{a}_{2}(t)+r^{*}(t)\tilde{a}_{3}(t)+\tilde{a}_{4}(t)=0,$
(50)

whose solution is given by

$r(t)=\frac{\tilde{a}_{3}(t)\tilde{a}^{*}_{4}(t)-\tilde{a}^{*}_{1}(t)\tilde{a}_{4}(t)}{|\tilde{a}_{1}(t)|^{2}-|\tilde{a}_{3}(t)|^{2}}+\frac{\tilde{a}_{3}(t)\tilde{a}^{*}_{2}(t)-\tilde{a}^{*}_{1}(t)\tilde{a}_{2}(t)}{|\tilde{a}_{1}(t)|^{2}-|\tilde{a}_{3}(t)|^{2}}q(t).$
(51)

Substituting Eq. (51) into Eq. (49), we find that

$q(t)=\frac{\sum_{n=0}^{\infty}p_{n}s^{2}_{n}+2\operatorname{Re}\quantity[i\quantity(\frac{\tilde{a}_{3}(t)\tilde{a}^{*}_{4}(t)-\tilde{a}^{*}_{1}(t)\tilde{a}_{4}(t)}{|\tilde{a}_{1}(t)|^{2}-|\tilde{a}_{3}(t)|^{2}})p_{n+1,n}]s_{n}c_{n}}{\sum_{n=0}^{\infty}(p_{n+1}+p_{n})s^{2}_{n}-2\operatorname{Re}\quantity[i\quantity(\frac{\tilde{a}_{3}(t)\tilde{a}^{*}_{2}(t)-\tilde{a}^{*}_{1}(t)\tilde{a}_{2}(t)}{|\tilde{a}_{1}(t)|^{2}-|\tilde{a}_{3}(t)|^{2}})p_{n+1,n}]s_{n}c_{n}}.$
(52)

Therefore, for a given value of $g$ and time $\tau$, Eqs. (51) and (52)
uniquely determine a state of the catalyst.

####  Subsection 3 Second moment of photon statistics in the catalytic Jaynes-
Cummings evolution

Here, we use Lemma. (3.1) to obtain an explicit expression for the second
moment in a catalytic evolution as specified by the Jaynes-Cummings
Hamiltonian. We start by considering the second moment of photon statistics

$\displaystyle\langle n^{2}_{\textsf{S}}\rangle_{\sigma}=\langle
n^{2}_{\textsf{S}}\rangle_{\rho}+2\left[(1-q)\,\langle
n_{\textsf{S}}\rangle_{\rho}-\langle
n_{\textsf{S}}\otimes\outerproduct{e}{e}_{\textsf{C}}\rangle_{\sigma}\right],$
(53)

where $q:=\langle e|\chi_{\textsf{C}}|e\rangle$ is the excited-state
occupation of the catalyst,
$\sigma=U(\rho_{\textsf{S}}\otimes\chi_{\textsf{C}})U^{\dagger}$ and
$\Tr_{\textsf{S}}[\sigma]=\chi_{\textsf{C}}$. Let us focus on the following
term:

$\displaystyle\langle
n_{\textsf{S}}\otimes\outerproduct{e}{e}_{\textsf{C}}\rangle_{\sigma}$
$\displaystyle=\Tr\left[U^{\dagger}(n_{\textsf{S}}\otimes\outerproduct{e}{e}_{\textsf{C}})U(\rho_{\textsf{S}}\otimes\chi_{\textsf{C}})\right]$
$\displaystyle=\sum_{k=0}^{\infty}k\Tr[U^{\dagger}\outerproduct{k,e}{k,e}U(\rho_{\textsf{S}}\otimes\chi)],$
(54)

Using Eq. (1), we can write

$\displaystyle U^{\dagger}\ket{k,e}$ $\displaystyle=e^{i(k+\frac{1}{2})\chi
t}\left(c_{k}\ket{k,e}-is_{k}\ket{k+1,g}\right),$ (55)

Substituting the above result into Eq. (3) leads to

$\displaystyle\langle
n_{\textsf{S}}\otimes\outerproduct{e}{e}_{\textsf{C}}\rangle_{\sigma}=\sum_{k=0}^{\infty}k\bigl{[}(1-q)c_{k}^{2}p_{k,k}$
$\displaystyle+2s_{k}c_{k}\text{Im}(p_{k+1,k}r)$
$\displaystyle+(1-q)s_{k}^{2}p_{k+1,k+1}\bigl{]}.$ (56)

####  Subsection 4 Boosting non-classicality via a catalytic process for
incoherent mixtures of Fock states

For an incoherent mixture of Fock states
$\rho=\sum_{n}p_{n}\outerproduct{n}{n}$, the second-order coherence is given
by

$g^{(2)}(\rho)=\frac{\sum_{n=0}^{\infty}n(n-1)p_{n}}{(\sum_{n=0}^{\infty}np_{n})^{2}}.$
(57)

Assuming that the initial state of the cavity is prepared in a state
$\rho_{\textsf{S}}=\frac{1}{4}\outerproduct{0}{0}+\frac{3}{4}\outerproduct{2}{2})$,
then its second-order coherence is $g^{(2)}(\rho_{\textsf{S}})=2/3$. According
to Eq. (1), the state of the cavity after the catalytic protocol (at time
$t=\tau$) takes the form of

$\displaystyle\sigma_{\textsf{S}}=$
$\displaystyle\frac{1}{4}\quantity[q\\!+\\!(1\\!-\\!q)\cos^{2}gt]\\!\outerproduct{0}{0}\\!+\\!\frac{3q}{4}\quantity[\sin^{2}\quantity(g\tau\sqrt{2})\\!+\\!\frac{(1-q)}{4}\sin^{2}gt]\\!\outerproduct{1}{1}$
$\displaystyle+\frac{3}{4}\quantity[q\cos^{2}\quantity(g\tau\sqrt{2})+(1-q)\cos^{2}\quantity(g\tau\sqrt{3})]\outerproduct{2}{2}$
$\displaystyle+\frac{3(1-q)}{4}\sin^{2}\quantity(g\tau\sqrt{3})\outerproduct{3}{3}$
(58)

where $q$ is determined by Eq. (29):

$q=\frac{\sin^{2}\quantity(g\tau)+3\sin^{2}\quantity(g\tau\sqrt{3})}{\sin^{2}\quantity(g\tau)+3\quantity[\sin^{2}\quantity(g\tau\sqrt{3})+\sin^{2}\quantity(g\tau\sqrt{2})]}.$
(59)

Using Eq. (57), the second-order coherence for the final state is given by

$g^{(2)}(\sigma_{\textsf{S}})=\frac{2}{3}\quantity{q\cos^{2}\quantity(g\tau\sqrt{2})+(1-q)\quantity[1+2\sin^{2}\quantity(g\tau\sqrt{3})]}.$
(60)

Therefore, for $g\tau=7.5\pi$, we obtain $g^{(2)}\approx 0.505$, indicating
that non-classicality in the mode was catalytically increased.

###  Section 7 Concluding remarks

We presented a catalytic process for generating non-classical states of light
in an optical cavity via interaction with an atom. Our results are valid for
any coupling regime in which the Jaynes-Cummings model is applicable and also
for realistic scenarios where the presence of noise and loss cannot be
neglected. This fundamentally stems from the fact that a catalytic state of
the atom exists for any coupling strength and even under the presence of
dissipation. This work shows that quantum catalysis, a concept so far explored
in the abstract framework of resource theories, is relevant in a practical
context (see also Ref. [298], which discuss the need to turn resource-theory
notions into real-world applications). Furthermore, our protocol could
potentially be implemented in state-of-the-art experimental setups [299, 300,
301], e.g. in cavity QED [302, 303] or trapped ions [304, 305]. Beyond proof-
of-principle experiments, it would also be interesting to investigate whether
such a catalytic protocol offers a practical advantage. Indeed, the key point
of catalysis is that the catalyst is returned exactly in the same state as it
was initially prepared. Hence the same atom could in principle be used
repeatedly for activating non-classicality in different cavities, or in the
same cavity, but at different times. It would be interesting to uncover
further instances of quantum catalysis in realistic setups, e.g. exploring
other platforms and models in quantum optics, as well as for applications in
quantum information and metrology.

Another relevant aspect of our work is that the catalyst (atom) is, in
general, a coherent quantum system. Specifically, to generate non-
classicality, the atom’s state must exhibit coherence in the energy basis over
the course of its evolution. This constitutes an example of coherent quantum
catalysis [254], which contrasts with most previous examples of quantum
catalysis. A deeper understanding of the role of coherence in catalysis is an
exciting future direction.

Finally, our analysis also shed light on the key role of correlations in
quantum catalysis. This represents a distinctive feature of quantum catalysis.
It would be interesting to see if this effect can occur in different physical
models.

##  Chapter 8 Conclusions

Quantum thermodynamics arose from a desire to formalise and generalise the
concepts of macroscopic thermodynamics to the quantum realm. Its emergence
evolved naturally with our increasing ability to manipulate and control
systems at finer scales. This led theoreticians to probe the intricate
dynamics of small-scale systems, asking how quantum phenomena, such as
entanglement and coherence, might affect classical thermodynamic formulations.
However, theoretical inquiries were not the sole catalyst for the advent of
this field. As we have witnessed the race towards so-called ”quantum
technologies”, understanding the path to optimality brings us closer to the
idea that one day we might develop devices that outperform current ones by
using purely quantum effects. Since then, joint theoretical efforts from
quantum optics, statistical mechanics, and quantum information theory are
bundled to pave the way for a new era of quantum engineering ahead.

This thesis asks simple questions that naturally arise when we attempt to
probe the quantum realm thermodynamically. By relying on a model-independent
approach and using minimal assumptions, our main goal consisted of
characterising thermodynamic transformations across different regimes and
identifying optimal protocols. To ask and answer these questions, we lay out a
powerful theoretical toolkit that provides a robust approach to studying the
thermodynamics of small systems: the resource-theoretic approach. The main
mathematical tools that consistently appear within this framework were
presented in Chapter 2, while the framework itself was detailed in Chapter 3.

Chapter 4 analysed the structure of thermal cones. Specifically, for a
$d$-dimensional classical state of a system interacting with a heat bath, we
found explicit construction of the past thermal cone and the incomparable
region. Moreover, we provided a detailed analysis of their behaviour based on
thermodynamic monotones given by the volumes of thermal cones. Then, we
discussed the applicability of these results to other majorisation-based
resource theories (such as that of entanglement and coherence), since the
partial ordering describing allowed state transformations is then the opposite
of the thermodynamic order in the infinite temperature limit. Finally, we also
generalised the construction of thermal cones to account for probabilistic
transformations and for a coherent qubit.

In Chapter 5, we developed a resource-theoretic framework that allowed one to
bridge the gap between two approaches to quantum thermodynamics based on
Markovian thermal processes. Our approach was built on the notion of memory-
assisted Markovian thermal processes, where memoryless thermodynamic processes
were promoted to non-Markovianity by explicitly modelling ancillary memory
systems initialised in thermal equilibrium states. Within this setting, we
proposed a family of protocols composed of sequences of elementary two-level
thermalisations that approximated all transitions between energy-incoherent
states accessible via thermal operations. We proved that, as the size of the
memory increased, these approximations became arbitrarily good for all
transitions in the infinite temperature limit, and for a subset of transitions
in the finite temperature regime. Furthermore, we presented solid numerical
evidence for the convergence of our protocol to any transition at finite
temperatures. We also explained how our framework could be used to quantify
the role played by memory effects in thermodynamic protocols such as work
extraction. Finally, our results showed that elementary control over two
energy levels at a given time was sufficient to generate all energy-incoherent
transitions accessible via thermal operations if one allowed for ancillary
thermal systems.

In Chapter 6, we derived a fluctuation-dissipation theorem version within a
resource-theoretic framework, where one investigates optimal quantum state
transitions under thermodynamic constraints. More precisely, we first
characterised optimal thermodynamic distillation processes, and then proved a
relation between the amount of free energy dissipated in such processes and
the free energy fluctuations of the initial state of the system. Our results
applied to initial states given by either asymptotically many identical pure
systems or an arbitrary number of independent energy-incoherent systems, and
allowed not only for a state transformation, but also for the change of
Hamiltonian. The fluctuation-dissipation relations we derived enabled us to
find the optimal performance of thermodynamic protocols such as work
extraction, information erasure, and thermodynamically-free communication, up
to second-order asymptotics in the number $N$ of processed systems. We thus
provided a first rigorous analysis of these thermodynamic protocols for
quantum states with coherence between different energy eigenstates in the
intermediate regime of large but finite $N$.

Finally, in the last chapter of this thesis, we went beyond thermodynamics and
presented a catalytic process in a paradigmatic quantum optics setup, namely
the Jaynes-Cummings model, where an atom interacted with an optical cavity.
The atom played the role of the catalyst, and allowed for the deterministic
generation of non-classical light in the cavity. Considering a cavity prepared
in a ”classical” coherent state, and choosing appropriately the atomic state
and the interaction time, we obtained an evolution with the following
properties. First, the state of the cavity had been modified, and now featured
non-classicality, as witnessed by sub-Poissonian statistics or Wigner
negativity. Second, the process was catalytic, in the sense that the atom was
deterministically returned to its initial state exactly, and could then in
principle be re-used multiple times. We investigated the mechanism of this
catalytic process, in particular highlighting the key role of correlations and
quantum coherence.

plain

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