Quantum catalysis in cavity quantum electrodynamics

Catalysis plays a key role in many scientific areas, most notably in chemistry and biology. Here we present a catalytic process in a paradigmatic quantum optics setup, namely the Jaynes-Cummings model, where an atom interacts with an optical cavity. The atom plays the role of the catalyst, and allows 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 obtain an evolution with the following properties. First, the state of the cavity has been modified, and now features non-classicality, as witnessed by sub-Poissonian statistics or Wigner negativity. Second, the process is catalytic, in the sense that the atom is deterministically returned to its initial state exactly, and can be re-used multiple times. What is more, we also show that our findings are robust under dissipation and can be applied to scenarios featuring cavity loss and atomic decay. Finally, we investigate the mechanism of this catalytic process, in particular highlighting the key role of correlations and quantum coherence.


I. INTRODUCTION
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.[1]), including biological processes activated by enzymes, the speed-up of chemical reactions, and the synthesis of nanomaterials.
An interesting direction 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 [39][40][41], where a two-level atom interacts with a single-mode optical cavity.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 evo- We consider the evolution U over a well-chosen time interval (from t = 0 to t = τ) such that (i) the final state of the cavity is non-classical, and (ii) the atom is returned to its intial state exactly.Hence, nonclassicality has been generated via catalysis.lution 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 [24].
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 [42] (see also [43,44]) considers two spin qubits that become entangled via an interaction with a cavity that can be considered catalytic [3].Another relevant direction is that of "multiphoton catalysis" (see e.g.[45][46][47][48]) which is a heralded catalytic process, where the catalyst is returned only with some probability.In contrast, our catalytic protocol is deterministic.

II. CATALYSIS IN THE JAYNES-CUMMINGS MODEL
Consider a setup comprising a system (S) initially prepared in a state ρ S and a catalyst (C) in an initial state χ C .The total system SC is assumed to be closed and evolves via an energy-conserving process for some time τ.This evolution is represented by the unitary U = exp(−iH SC τ), where H SC denotes the Hamiltonian.Consequently, the final state of the total system is given by σ 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 which we refer to as the catalytic constraint.Satisfying this constraint typically requires to carefully choose the initial states of the system ρ S and the catalyst χ C , as well as the interaction time τ.
The main goal of a catalytic evolution is to induce an interesting local dynamics on the system S, i.e.
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.
In this work, 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 [39] (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 characterized by the bosonic annihilation operator a with the photon number operator n S := a † a.The atom has energy levels |g⟩ and |e⟩ and its energy is captured by σ z = |e⟩⟨e| − |g⟩⟨g|.We work in the resonant regime, where the atom and cavity have the same frequency ω.The evolution is governed by the JC Hamiltonian, which in the rotating-wave approximation reads where g is the coupling constant and σ + = |e⟩⟨g| and σ − = |g⟩⟨e|.Note that, as we focus on the resonant regime, the evolution specified by Eq. ( 3) is energy-preserving.
Our goal is to catalytically generate non-classical light in the cavity starting from a (classical) coherent state, i.e.
In this work, we will derive a general characterisation of catalytic processes for the JC model.In particular, we FIG. 2. First illustrative example.Catalytic process for generating non-classicality in the cavity, as captured by the second-order auto-correlation function.Its time evolution, g (2) (t) := g (2) (σ S (t)), is shown in the top panel, while the bottom panel shows the modification of the atomic state, via the distance ∆(t) := ∥χ C − σ C (t)∥ 1 with respect to the initial state.The orange stars indicate the final time (τ ≈ 40) for which the evolution is catalytic.The atom returns to its intial state (∆(τ) = 0), while non-classicality has been activated g (2) (τ) ≈ 0.5 < g (2) (0) = 1.The value of the parameters are: 1. Derive a necessary and sufficient condition for the initial state of the atom χ C and an interaction time τ so that the catalytic constraint (1) is satisfied, 2. Give an analytic expression for the final state of the cavity after a catalytic process, 3. Develop an intuitive understanding of the mechanism behind catalysis in terms of the higher moments of observables, and finally 4. Show that catalytic evolutions can generate nonclassicality in the state of the cavity, even in the presence of dissipation.
Before presenting these general results in Secs III, IV and V, it is illustrative to discuss two specific examples of catalytic processes using complementary figures of merit to witness nonclassicality.
For our first example, we quantify non-classicality via the second-order auto correlation function of the final state of the cavity [49] , i.e.
where ⟨n S ⟩ ρ := Tr ρ S n S .Our goal is to obtain g (2) (σ S ) < 1, which certifies non-classicality.In Fig. 2, we show such a catalytic generation of nonclassicality.The upper plot depicts the time evolution of g (2) .Catalysis occurs at time τ = 40 (represented by orange stars).Starting from the initial state of the cavity ρ S = |α⟩⟨α| S (with α = 1/ √ 2), which has g (2) (ρ S ) = 1 as any coherent state, we obtain a final state σ S for which g (2) (σ S ) ≈ 0.5, hence nonclassical.In parallel, we monitor the time evolution of the atom by calculating the trace distance to its initial state, i.e.
At time τ ≈ 40, we find ∆(τ) = 0, indicating that the atom has returned to its initial state as required for catalysis.As a second example, we demonstrate how Wigner negativity can be catalytically generated.We focus on the Wigner logarithmic negativity [50][51][52] (WLN), defined as where W ρ (x, p) is the Wigner function Starting from a coherent state ρ S = |α⟩⟨α| S which has a positive Wigner function, hence W(ρ S ) = 0, we aim to obtain a final state of the cavity σ S with W(σ 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 τ = 5, we obtain a state σ S that has W(σ S ) ≈ 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).Note that the two examples presented here complementary.The first one shows the generation of non-classicality as witnessed by the g (2) function, while the final state σ S has a positive Wigner function.In the second example, the final state σ S has a negative Wigner function, even though g (2) (σ) > 1.

III. MECHANISM OF CATALYSIS
Here we provide an intuitive understanding of quantum catalysis by identifying a mechanism which allows for generating non-classicality.This allows us to characterize the catalytic regime analytically.
In the Jaynes-Cummings model, the energies of the cavity S and the atom C are specified by number operators n S and n C respectively.Hence, the total energy of both systems is proportional to the number of excitations, and described by a joint operator n SC := n S + n C .Since the JC evolution U(t) that takes ρ S ⊗χ C into σ SC conserves the total energy, we have that [U(t), n S + n C ] = 0 for all t.When the evolution is catalytic, all moments of n C must remain unchanged, in particular ⟨n C ⟩ χ = ⟨n C ⟩ σ and consequently ⟨n S ⟩ ρ = ⟨n S ⟩ σ .Importantly, this is not the case for higher moments of n S .Particularly, in Appendix B we show that the second moment satisfies Hence, the second moment in the final state of the cavity, ⟨n 2 S ⟩ σ , can become smaller (or larger) than the second moment in the initial state ⟨n 2 S ⟩ ρ .This means that using a catalyst allows for modifying the distribution of the local observable n 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. ⟨n 2 S ⟩ σ ⟨n 2 S ⟩ ρ only if σ σ S ⊗ σ C [as seen from Eq. ( 9)].Thus, correlations are essential for observing quantum catalysis.Finally, the above analysis also applies beyond the JC model, see Appendix B-1a for a generalisation of Eq. ( 9) to arbitrary observables and moments.This can help to discover and understand new instances of quantum catalysis.
The above analysis will 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 (1).Moreover, we obtain an analytic expression for the auto-correlation function g 2 , which is based on the second moment ⟨n 2 S ⟩ σ .We consider an arbitrary initial state of the atom as well as a general initial state of the cavity ρ S = ∞ n,m p n,m |n⟩⟨m| with p n := p n,n .Combining Eq. ( 9) with the fact that ⟨n S ⟩ σ = ⟨n S ⟩ ρ , we get (11) where with s n := sin gt , and y n := 2 Im rp n+1,n s n c n .For details see Appendix A-3a.
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 inc + q coh , we get with Q := ∞ n=0 (p n + p n+1 )s 2 n .Interestingly, q inc is specified by the occupations of ρ S , while q coh depends on its coherence in the Fock basis.Moreover, the off-diagonal term r satisfies with a i being auxiliary functions defined as For a detailed derivation of Eqs. ( 15) see Appendix A-1.Importantly, Eqs ( 13) and ( 14) are necessary and sufficient for ensuring the catalytic constraint of Eq. ( 1).Combined with Eq. ( 11), they characterize analytically the catalytic generation of non-classicality in our first example of a catalytic process (see Fig. 2).For the second example, we verify the catalytic constraint using our analytical findings and compute Wigner functions numerically.
A relevant question one might have is whether the atom can be reused in a truly catalytic way during the catalytic protocol.That is, whether the catalytic protocol can be used to prepare multiple non-classical states in different cavities, while the catalyst is prepared only once.In Appendix E we discuss this more operational view of catalysis in detail.< 1.We impose a limit on the interaction time τ ≤ 100 and take 10 6 samples.In both panels, parameters are ω = 2π, g = π.

IV. 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.
First, note that it is unclear a priori whether the catalytic constraint of Eq. ( 1) can be satisfied.Nonetheless, the quantum version of the Perron-Frobenius theorem guarantees that every quantum channel has at least one fixed point [53] 1 .Now, for fixed ρ S and τ, the atom evolves according to an effective quantum channel Consequently, there is always exists a state χ C which is left unchanged by this channel, hence providing a solution to Eq. (1).
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 |α⟩, does there always exist a state of the catalyst χ C which allows to generate nonclassicality?To address this question, we investigate the minimum value of g (2) of the final state σ S , as a function of |α| [see Fig. 4(a)].This is done by combining Eqs.(11)(12) and (13)(14), and imposing a bound on the final time, i.e gτ ≤ 100.For α ∈ (0, 2], we observe that g (2) (σ S ) < 1, indicating a sub-Poissonian statistic for the number of photons and, consequently, evidencing non-classicality.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.
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 Appendix 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. ( 9)], which is impossible when the state of the catalyst is pure.
To further explore this question, we 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 gτ ≤ 100.In Fig. 4(b), 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 further discussed in Appendix A-1, in particular we observe its strong dependence on the intial state of the cavity.
Finally, we investigated scenarios where the cavity's initial state is not a coherent one.Interestingly, we find that catalysis can boost non-classicality.Specifically, starting with an initial non-classical state ρ S , one can get a final state σ S that features more non-classicality, i.e. g (2) (σ S ) < g (2) (ρ S ) (see Appendix C-2 for details).Conversely, there also exist initial states of the cavity (e.g.Fock states) for which non-classicality cannot be increased catalytically (see Appendix C-1).This is intuitive, as Fock states are the most non-classical (i.e.minimize g (2) ) among all states with a fixed mean energy.As a catalytic protocol must preserve the mean energy, the value of g (2) cannot be increased.

V. 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 atomcavity system are described by a Lindblad master equation of FIG. 5. Catalysis in realistic scenarios.Wigner logarithmic negativity and auto-correlation function are plotted as functions of the interaction time τ.The blue solid line represents the scenario with dissipation, while the gray solid line represents the scenario without dissipation.For each time τ, the catalytic constraint is satisfied, and the degree of non-classicality activated in the mode is computed.While W ≥ 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 Γ = 0.05 and a cavity dissipation rate of κ = 0.005.The average number of thermal bath excitations is n th = 0.1 and the other parameters are ω = 2π, g = 0.1π and α = 1/ √ 2.
the form [54][55][56][57]: where as before we take ρ SC := |α⟩⟨α| S ⊗ χ C , and κ and Γ are the cavity and atom dissipation rate, respectively.Moreover, n th = (e 1/T − 1) −1 is the average excitation number and is the Lindblad dissipator with {. , .}denoting the anticommutator.In the case of dissipative evolution ( 16) the (generalized) catalytic constraint becomes where E τ [•] is a quantum channel generated by the evolution from Eq. ( 16) acting for some time τ.For a given stopping time τ the solution to Eq.( 16) that simultaneously satisfies the catalytic constraint can be determined numerically.In Appendix E we present an algorithm for constructing the catalytic set for a general dissipative quantum evolution (i.e.any quantum channel), and provide its Python implementation that can be used on a regular laptop.
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 τ for closed (reversible) and open (irreversible) quantum dynamics.At all stopping times τ we ensure that the generalized catalytic constraint from Eq. ( 17) is met.Our results are summarized in in Fig. 5 which shows the amount of non-classicality that can be generated in the mode S for different choice of interaction times τ.Our first observation is that a catalytic generation of non-classicality is possible even in the presence of dissipation.Secondly, the amount of nonclassicality 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.
Finally, since the catalytic states of the atom C are mixed, it must initially be correlated with some external environment.We emphasise that the presence of such correlations does not influence the dynamics of the catalyst C and the system S.

VI. DISCUSSION
We presented a catalytic process for generating nonclassical 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. [58], 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 [59][60][61], e.g. in cavity QED [62,63] or trapped ions [64,65].Beyond proof-ofprinciple 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 [24], 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.

Eigenproblem
The Jaynes-Cummings interaction Hamiltonian H int = g aσ + + a † σ − couples pairs of atom-field states {|n + 1, g⟩ , |n, e⟩}.Consequently, the Hamiltonian H decouples into a direct product of 2 × 2-matrix Hamiltonians, i.e., H = ∞ n=0 H (n) , where The eigenvalue problem for this Hamiltonian yields the eigenfrequencies where µ n = 2g √ n + 1 is the n-photon Rabi frequency.In the resonance regime, the corresponding eigenstates are: The unitary operator given by equation (A4) describes the dynamics of the Jaynes-Cummings model in resonance and under the rotating-wave approximation.
After the interaction, the joint system σ SC (t) = U(t)(ρ S ⊗ χ C )U † becomes correlated.Let the cavity and the atom be prepared in general mixed states, i.e. ρ S = ∞ n,m=0 p n,m |n⟩⟨m| and χ C = q |g⟩⟨g| + r |g⟩⟨e| + r * |e⟩⟨g| + (1 − q) |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.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(σ SC , σ target ) where σ target is a product state σ target = σ S 1 ⊗ σ S 2 ⊗ . . .⊗ σ S N and σ S i = σ S j for 1 ≤ i, j ≤ N. In Fig. 8, we show the resulting fidelity F(σ SC , σ 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.

FIG. 1 .
FIG. 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 = τ) such that (i) the final state of the cavity is non-classical, and (ii) the atom is returned to its intial state exactly.Hence, nonclassicality has been generated via catalysis.

FIG. 3 .
FIG. 3. Second illustrative example.Catalytic generation of Wigner negativity.Panel (a) shows the time evolution of the Wigner logarithmic negativity W, as well as the Wigner function at initial and final time τ.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.In the trajectory, the state always remains inside the Bloch ball and never touches the surface.This is because states on the surface represent pure states, which cannot be used as catalysts.The initial and final state coincide (red dot).The values of the parameters are: α = 1/ √ 2, ω = 2π and g = π.

FIG. 4 . 2 .
FIG. 4. Which states lead to catalysis?Panel (a) shows the minimal value of g 2 obtained as a function of the amplitude |α| of the initial coherent state of the cavity ρ S = |α⟩⟨α| 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 α = 1/ √ 2. The colour represents different values of g (2) < 1.We impose a limit on the interaction time τ ≤ 100 and take 10 6 samples.In both panels, parameters are ω = 2π, g = π.