Down-conversion processes in ab-initio non-relativistic quantum electrodynamics

The availability of efficient photon sources with specific properties is important for quantum-technological applications. However, the realization of such photon sources is often challenging and hence alternative perspectives that suggest new means to enhance desired properties while suppressing detrimental processes are valuable. In this work we highlight that ab-initio simulations of coupled light-matter systems can provide such new avenues. We show for a simple model of a quantum ring that by treating light and matter on equal footing we can create and enhance novel pathways for down-conversion processes. By changing the matter subsystem as well as the photonic environment in experimentally feasible ways, we can engineer hybrid light-matter states that enhance at the same time the efficiency of the down-conversion process and the non-classicality of the created photons. Furthermore we show that this also leads to a faster down-conversion, potentially avoiding detrimental decoherence effects.


I. INTRODUCTION
Quantum-light sources exhibiting two-photon emission, especially entangled photon pairs, are crucial building blocks for quantum-information processing protocols [1,2], cryptography [3], or teleportation [4]. As opposed to the characterization of single-photon sources, or "photon guns", which have sub-Poissonian statistics [5], the characterization of two-photon processes shows statistics that vary from non-classical (sub-Poissonian) to even chaotic (super-Poissonian) behavior [6]. Spontaneous parametric down-conversion or parametric down-conversion (PDC) is the coherent generation of a pair of photons with lower frequency (signal photons) by injecting a higher-frequency photonic field (pump photon) into a nonlinear medium [7]. Three-wave mixing has been mostly used for the generation of entangled photon pairs [8][9][10][11]. The need for on-demand efficient two-photon sources expanded PDC from using nonlinear crystals with picoseconds (ps) pulsed lasers [12,13] to atom-like systems coupled to cavities [14,15], photonpair generation using the biexciton-exciton cascade in quantum dots [9,[16][17][18] and the nascent fields of polaritonic chemistry [19] and circuit quantum electrodynamics (QED) [20,21]. The most widely used of these methods suffer from low photon emission rates and limited scalability. For example, an average of one in every 10 12 photons is down-converted when using a bulk material [22]. Also, it was shown that entanglement between emitted photon pairs could be degraded due to the presence of energy-level splitting of the intermediate excitonic states in, e.g., quantum dots [16,23]. Semiconductor quantum rings, where the charge carriers are confined in the radial direction, have been demonstrated to be excellent quantum emitters which generate single-photon states with strong antibunching under certain conditions [24]. Here the quantum confinement of the charge carriers together with modified ring geometries led to the observation of geometry-dependent photon antibunching. This makes quantum rings a suitable medium for photon generation since the electronic properties of the quantum ring can be modified by varying the geometric confinement parameters which in turn changes the optical spectrum [25].
In order to investigate such down-conversion processes theoretically one usually assumes that light and matter can be separated and treated differently. The nonlinear optics approach, for instance, uses non-linear response functions and susceptibilities of matter-only quantum mechanics to characterize such a process and connects them with a classical description of the light field [12,13,26]. The quantum-optics approach, on the other hand, treats the light field quantized and couples the resulting photons to a simplified few-level description of the matter [10,11,27]. In the latter approach one often even gets rid of the matter part altogether by defining effective photon-only Hamiltonians which model the photon-photon interaction due to the matter system [26,28,29]. In both approaches the efficiency and properties of the down-conversion process in question are usually determined by dipole-transition elements which depends on the symmetry the matter subsystem possesses. For instance, since PDC is a three-wave mixing process, the second-order non-linear susceptibility is the dominant contribution. For systems that possess a single symmetry (like the model considered below) this quantity is negligible [30] resulting in an inefficient or even impossible PDC process. For this reason, one would conventionally break the symmetry of the quantum ring by some external classical field to engineer appropriate dipole-allowed transitions [25,31] or use double-ring structures [24]. However, besides this more conventional perspective, we here highlight the possibilities that arise if we do not make the initial assumption to treat light and matter separately. Instead, by perform numerical simulations of non-relativistic QED, where light and matter are treated on equal quantized footing, we show how novel hybrid light-matter states (polaritons) are created that can act as intermediators for a photon down-conversion process. By making use of the full flexibility of the matter and the photon subsystems we can achieve a high-degree of control.
Our ab-initio simulations in the following are based on the Pauli-Fierz Hamiltonian of non-relativistic QED [32][33][34] in the long-wavelength limit [35][36][37][38]. As a simple yet illustrative example system we consider the aforementioned GaAs semiconductor quantum ring featuring a single effective electron coupled to a photonic environment. Varying the anharmonicity of the quantum ring we change the character of the matter subsystem continuously from harmonic to strongly anharmonic without breaking the radial symmetry. We, however, allow changes in the photonic environment, e.g., the geometry of a multi-mode cavity. This provides means to control the basic coupling parameters and can hence interpolate between free-space-like situations (weak coupling with strong dissipation) and ultra-strong coupling situations. Since the coupling between light and matter is fully quantized, also for the quantum ring (non-dipole allowed or virtual-state) down-conversion processes take place. We then investigate how specific properties and the efficiency of such a process can be optimized without relying on (experimentally challenging) specialized input fields but instead making use of the hybrid nature of coupled light-matter systems. For the case of degenerate (both output modes have the same frequency) downconversion we see that by coupling the photons strongly to a virtual transition while optimizing the configuration of the coupled system, we can enhance the efficiency as well as the non-classicality of the created photons. Just increasing the intensity of the input field does not make the down-conversion process more efficient. Furthermore, increasing the coupling between light and matter shifts the down-conversion process to earlier times. This temporal control potentially allows to suppress the influence of dissipation in desired features like non-classicality and entanglement. All these results for a simple system highlight the possibilities that become available for designing novel photon sources if light and matter are treated on equal footing in an ab-initio description.

II. AB-INITIO DESCRIPTION OF DOWN-CONVERSION PROCESS
The starting point of our investigation is very general and we only make the assumption that our bound matter system is small compared to the wavelength of the relevant photon modes. In this case we can consider the Pauli-Fierz Hamiltonian in dipole approximation [38] Here,Ĥ el is the bare Hamiltonian of the matter subsystem in Coulomb gauge andp is the corresponding momentum. The charge and mass are given by e and m, respectively. The operatorÂ = M α=1 λ α e αqα is the vector potential with the dipole coupling strength λ α determined by the value of the respective mode function (of frequency ω α ) at the position of the matter system (center of charge) and e α the polarization vector [38] of M photon modes. Here,q α = 2ωα â α +â † α is the canonical photon coordinate that satisfies the commutation relation [q α ,p α ] = i δ α,α andâ α ,â † α are the usual annihilation and creation operators, respectively. The photon Hamiltonian is given byĤ In the case that we keep many modes (up to a certain physical cutoff) to simulate the continuum, we need to be aware that this might make it necessary to use the bare mass of the charged particles instead of the already renormalized (physical) mass [32,37,39,40]. Here, we make the usual assumption that only a small part of the photon continuum is changed with respect to the free-space case due to a cavity or photonic nanostructure, and we treat only these changes explicitly. The rest of the continuum of modes is subsumed in the usual renormalized (physical) mass of the charged particles [32]. Thus in practice, out of M -modes of the electromagnetic field, we just keep a few effective modes that are relevant. We note that this change of the photon modes also modifies the free-space Coulomb interaction [34,41]. So care has to be taken when assuming thatĤ el is just due to a free-space matter system as described by the standard Schrödinger equation once the interaction with the photon field is taken into account explicitly. These cautionary remarks translate equivalently to the usual few-level models, whereĤ el is replaced typically by a few states from free-space calculations, or when making the mean-field assumption for the coupling between light and matter which leads to the Maxwell-Schrödinger equation of non-linear optics [33].
A. Semiconductor quantum ring in multi-mode photonic environment For investigating the down-conversion process we need to make a specific choice for the matter system and the photonic environment. We choose for the matter subsystem that mediates the down-conversion process a two-dimensional (2D) semiconductor GaAs quan-  1. (a.) Setup for investigating down-conversion processes featuring a 2D GaAs quantum ring at the center of a multi-mode photonic environment. The fields are linearly polarized in the x, y− plane where the effective quantum ring particle is trapped in a binding potential. The fieldsÂα are polarized perpendicular to the propagation vectors kα. (b.) Simplified energy-level scheme indicating the relevant transitions in a non-degenerate down-conversion process. The simulations take the full level structure into account (see App. A for details) tum ring [31,42]. We describe the quantum ring by a single effective electron confined in a 2D mexicanhat potential (see Fig. 23 in App. A). A more realistic description with many interacting electrons and possibly even phononic excitations would be possible if we employ more efficient first-principle methods such as quantum-electrodynamical density-functional theory (QEDFT) [34,[43][44][45] or polaritonic coupled cluster theory [46,47]. Nevertheless, the current level of description already suffices to demonstrate the novel pathways for down-conversion that become accessible with an abinitio light-matter description. Moreover, by changing the anharmonicity of the GaAs quantum ring we obtain control over the electronic level structure. We can use this control, which is also experimentally realizable [25], to optimize the down-conversion process without breaking the symmetry of the matter subsystem. Finally, due to the rotational symmetry [25,48,49], a simple matteronly analysis would indicate that no (efficient) downconversion takes place. Due to the rotational symmetry of the eigenstates, only transitions that change the angular momentum by one (see also a detailed discussion in App. A) are dipole allowed, and hence a process as indicated in Fig. 1 b.) is not dipole allowed. This becomes evident also in the second-order non-linear susceptibility, which is negligible in this case [30]. Yet, since we do not decouple light and matter, we will still find a certain probability that such a process takes place. The efficiency and the details of the ensuing process will then depend crucially on the details of the photonic environment.
For the photonic subsystem we consider a multi-mode environment. To be precise, we assume that we can manipulate three of these modes, i.e.,Â 1 ,Â 2 andÂ 3 , at will, while the rest of the continuum of modes stays largely unaffected. We considerÂ 1 as the input (pump) mode with frequency ω 1 , polarization direction e 1 and coupling strength λ 1 , andÂ 2 andÂ 3 as the output (signal) modes with corresponding frequencies, polarizations and coupling strengths.
We arrange the matter and the photon system as depicted in Fig. 1, such that only the x and y polarization directions are relevant and couple to the quantum ring. We then choose a coordinate system such thatÂ 1 =Â 1 e x ,Â 2 =Â 2 (− sin θ 2 e x + cos θ 2 e y ) and A 3 =Â 3 (sin θ 3 e x + cos θ 3 e y ), where θ 2 and θ 3 are the angles of the respective polarization vectors. With this sketch of a possible setup we can connect the change in coupling strength to a change in the length of the cavity L, i.e., λ = 2/ L where the dielectric permittivity of the GaAs quantum ring is = 12.7 0 . For simplicity we assume in the following that all three modes have the same coupling strength λ. The rest of the continuum of modes is assumed to be subsumed, on the one hand, in the effective mass of the quantum ring particle (in accordance to the case of no nanophotonic environment we take m = 0.067m e , see also App. A) while we treat a finite number of modes that constitute the linewidth of the enhanced modes explicitly in order to account for dissipation and decoherence in the electron-photon coupled system [39]. The Hamiltonian for this description is given in the following form where the internal system Hamiltonian iŝ and the bath and system-bath coupling that constitute the rest of the (M − 3) modes in the full Hamiltonian iŝ wherep =p x e x +p y e y . For our further considerations the last two terms of Eq. (2) constitutes the active photonic bath and its coupling to the system. For the photonic bath, we consider (M − 3) = M 70 = 70 bath modes that are treated in photon Fock number states together with the three relevant modes. The vector potential of the bath modes isÂ α = λ α e α 2ωα â α +â † α where λ α is the coupling of the bath modes.

B. Time-evolution of the coupled system
To investigate the down-conversion process in detail we perform the time evolution of different initial states |Ψ in (0) of the coupled matter-photon system. To do so we explicitly propagate the time-dependent Schrödinger equation i ∂ ∂t |Ψ(t) =Ĥ|Ψ(t) with the Hamiltonian of Eq. (2). The initial states that we consider in the following are factorizable product states of the form |Ψ in (0) = |ϕ 1 0 |φ 1 |0 2 |0 3 . . . |0 M , where |ϕ 1 0 is the ground state of the uncoupled quantum ring and |0 α is the zero-photon state of mode α. The pump mode α = 1 will take different initial states. The simplest choice is that |φ 1 = |1 1 is just a single-photon Fock state. In most cases we will consider |φ 1 = |ξ 1 = e −|ξ1| 2 /2 ∞ n1=0 ξ n1 1 / √ n 1 ! |n 1 where ξ 1 is the amplitude and |n 1 the Fock states of mode 1. This implies that we have on average |ξ 1 | 2 photons at the beginning in the input mode. If we increase |ξ| 2 1 we approach a classical laser field (see App. B for details). We note that the here chosen factorizable initial state is not special, and we find a similar behavior also with different choices. This is discussed in more detail in Sec. V.
We solve the time-dependent Schrödinger equation of the coupled light-matter system with a Lanczos propagation scheme [50]. We represent the matter Hamiltonian on a two-dimensional uniform real-space grid of N x = N y = 127 grid points (implying 127 2 states are taken into account) with spacing ∆x = ∆y = 0.7052 nm while applying an eighth-order finite-difference scheme for the momentum operator and Laplacian. At several points (for comparison or for numerical efficiency) we instead of the real-space grid represent the matter subsystem by its truncated uncoupled eigenstate basis. This amounts for only a few states to the usual few-level approximation (see App. C 1 for further details). The photon modes are represented in a basis of Fock number states as discussed in App. D for the different input fields and descriptions. In the combined electron-photon space, we explicitly construct matrix representations for all operators. The expectation value for observables are computed for a time step of ∆t = 0.029 fs of the time-evolved wave function.
Of importance is the character of the down-converted photons, which can be determined by computing the Mandel Q α parameter [51] defined as The Mandel Q α measures the deviation of the photon statistics from a Poisson distribution and thus is a measure for the non-classicality. For a field with non-classical properties, the range of values lies between −1 ≤ Q α < 0 which corresponds to sub-Poissonian statistics (antibunching behavior). Fields with super-Poissonian statistics (bunching behavior) have Q α > 0 and for a coherent state with Poissonian statistics we have Q α = 0 [52].
In addition, we compute the second-order crosscorrelation function (intensity correlations) of the photon field as a measure of correlation between the modes. The multi-mode intensity correlation function is defined by The correlation function takes values greater than one for correlated modes. For uncorrelated modes, it is equal to one and it takes values smaller than one if the modes are anti-correlated [52][53][54]. Also, we compute the purity γ α with α = 1, 2, 3 as a measure for entanglement. The purity is obtained by tracing over the square of the one-body reduced density matrix of the respective subsystem (see App. E for details). If the purity is equal to one, the system can be expressed as a factorizable state of the subsystem (modes α = 1, 2, 3) and the rest of the coupled matter-photon system. If the purity is smaller than one, we have a nonfactorizable state, which indicates correlation between the subsystems.

III. SINGLE INPUT-PHOTON DOWN-CONVERSION AND TEMPORAL CONTROL
We start by considering the case were the input mode 1 is occupied by just one photon, the coupling between light and matter is weak such that the usual non-linear optics considerations apply, and we have a strongly anharmonic quantum ring such that a few-level approximation is reasonable as well. We therefore choose |φ 1 = |1 1 , fix a small coupling strength of λ = 0.014 and use an anharmonicity of V 0 = 200 meV (see App. A for details on the quantum ring). By selecting the frequency of mode 1 in resonance with the dipole-allowed transition between the ground-and eleventh-excited state (|ϕ 0 1 ↔ |ϕ 1 7 ) of the bare quantum ring, which is ω 1 = 24.65 meV, and by choosing the signal mode 2 with energy ω 2 = 1.36 meV which is resonant with the tenth-and eleventh-excited states ((|ϕ 1 7 ↔ |ϕ 0 6 ) as well as signal mode 3 with energy ω 3 = 23.29 meV resonant with the ground-and tenth-excited state (|ϕ 0 6 ↔ |ϕ 0 1 ), a few-level picture is appropriate (see also Fig. 1 b). We will, however, not a priori restrict the number of electronic states involved. That means we will use all electronic states or else have checked for convergence with respect to the number of states in all considered observables. Only in Sec. VI we make a comparison with an a priori restriction to only a few levels as well as with a semi-classical treatment, which shows how certain observables are not well captured. We further note that the superindex of the matteronly eigenstates refers to their angular momentum. Thus the transition in resonance with mode 3 is not dipole allowed, since only states that differ by exactly one in their angular quantum number have a non-zero dipole transition element (see App. A for details). Due to treating light and matter fully coupled we will, however, still find a down-conversion process in our ab-initio simulation.
Finally, we need to fix the polarization directions of the input and output modes and also define the bath modes. In order to maximize the photon-pair generation, we choose the mixing angles θ 2 = θ 3 = 90 • such that both fields of the signal modes are horizontally polarized asÂ 2 = −Â 2 e x andÂ 3 =Â 3 e x . This choice results in maximization of the interference terms of Eq. (3) since the sines and cosine of the mixing angles become one (see the detailed investigation in Sec. VII A). With these choices the down-conversion process obeys the energy and momentum conservation ω 1 = ω 2 + ω 3 and k 1 = k 2 + k 3 , respectively. The polarizations of the M 70 bath modes are chosen equally aligned as the modes 1,2,3 (since else the bath modes would just couple more weakly). We sample equally-spaced energy ranges around the main modes of ω 1 , ω 2 , and ω 3 . For ω 2 the energy range is ω B2 = [0.113, 4.521] meV, which is sampled with 20 bath modes, and for ω 2 and ω 3 , 50 bath modes are employed to sample for the combined energy range ω B13 = [11.303, 27.128] meV with equal spacing ∆ω = 0.25 meV. The coupling strength of the bath modes is chosen to be λ = 0.007. For the combined system-bath Hamiltonian of Eq. (2), we treat the M 70 and three relevant modes by including three photon Fock states (vacuum, one-photon and two-photon states) in each of these sampled modes (see App. D for details). In the following, we present the time evolution for the combined system and bath (i.e. Eq. (2)) and system only (i.e. Eq. (3)). For the simulation including the bath, we compute observables only for the relevant three modes while the bath modes serves just as dissipation and decoherence channels.

A. Dissipation and coherence time
We first focus on the influence of the bath modes on the down-conversion by performing two different simulations: one with the bath modes (i.e. time-propagate Eq. (2)) and one without the bath modes (i.e. time-propagate only Eq. (3)). In both cases, since the effective coupling strength g α = λ /2ω α of the transition |ϕ 1 7 ↔ |ϕ 0 1 is small when compared to the coupling of the other main transitions (see discussion in Sec. III B for further details), the effective electron preferably relaxes by cascaded emission into modes 2 and 3. This is inferred as the coupling strength is proportional to the square root of the spontaneous decay rate (see App. (C 1)). We therefore expect that mode 2 and 3 will become strongly populated once the initial photon in the pump mode 1 interacts with the matter subsystem. Thus in the ab-initio description of the PDC process the effective coupling strengths g α become decisive and the bare dipole-transition elements are no longer the only important quantity to consider. And indeed we find in Fig. 2, where we show the mode occupations for the input and two signal modes, that this holds true for both situations. Qualitatively both simulations show the same behavior, with the main difference that the down-conversion from mode 1 to the signal modes is less effective when the bath modes are included and the maximum of the down-converted number of photons appears slightly later. If we would simulate for longer times, the differences would become more pronounced and (provided we do not wait too long such that we hit the unphysical revival time due to having only finitely many modes [55]) the photon numbers would relax to the ones of the coupled matter-photon ground state. For the weak-coupling case this number is effectively zero. We would therefore consider the photons emitted. Since we cannot simulate the full emission process in this numerical setup, we instead make the simple assumption that what appears as the maximum amount of photons in each of the signal modes corresponds to what would be detected outside of the system. And since the bath modes do not seem to change these numbers strongly, even for this weakly coupled case where the bath contributes considerably, the bath-free simulation is a good approximation as long as we do not go beyond the coherence time of roughly 40 picoseconds (ps). By coherence time we mean the time interval in which the bath-free (fully coherent) simulation is a good approximation to the simulation including a bath. If we increased the coupling to the bath modes (stronger dissipation) the coherence time would be shorter. On the other hand, if we consider a stronger coupling to the input and signal modes, while keeping the bath modes fixed (essentially increasing the finesse of the cavity), the coherence times would be longer. It is not surprising that simulating 70 modes coupled to the electronic system is numerically very expensive and would not allow for all the different cases of down-conversion processes that we investigate in this work. In order to compare all these different cases as unbiased as possible we therefore consider them all with a bath-free simulation of a coherence time of about 40 ps. This is not a completely arbitrary number, since quantum rings are known to have long coherence times on the order of ps, before other dissipation channels destroy the coherence [25]. Thus this analysis allows us to avoid the use of (numerically expensive) non-unitary master-equation approaches to approximately treat the effect of the bath on the system [56] or to keep the bath explicitly in our following considerations.
Since we will consider also other quantities, let us check whether the coherent simulation faithfully reproduces these observables as well. The most important quantity in the following will be the Mandel Q parameter to determine the statistics of the down converted photons. We find in Fig. 3 that again both simulations agree qualita-tively and remain close over a long period of time. This demonstrates that the bath-free simulations capture at least qualitatively also the more complex properties of the created photon pair. Since we will be focused on an enhanced coupling to the main modes, the influence in these simulations of the bath modes will be even less important. The same holds true if we consider many input photons and classical external pumping.

B. Temporal control of down-converted photons
While we have above shown the mode occupations and we see a drop in the input mode and an increase in the signal modes, this does not imply that we have indeed converted one photon of ω 1 into a photon with ω 2 and one with ω 3 . For this we need to consider the populations of the different Fock number states in each mode. While in mode 1 we know by construction that we have only one photon at t = 0 and zero in the other modes, it could easily be that we also populate 2 and 3 photon states in the down-conversion process. This would imply also higherorder process play a role and we could not identify it in a simple manner as a two-photon down-conversion process. However, in Figs. 4 and 5 and focusing on the pink solid lines, we find that our intuition about the PDC process was correct. Only the one-photon states have significant population throughout the simulation. If we increase the coupling strength and go from the weak to the ultrastrong coupling domain it is not clear a priori that we still only have single-photon processes and no higher-order (multi-level/multi-photon) contributions become important. We vary the coupling λ by varying the effective cavity length L (see also Fig. 1). This also leads to modified effective coupling strengths g α = λ /2ω α (see Tab. I for detail values), while the bare dipole transition elements stay unchanged and are given in scaled effective atomic units in App. (A). From Figs. 4 and 5 we see that also for the strong coupling calculations the PDC remains qualitatively similar to the weak coupling case. Yet if we look for the total amount of photons in each mode we find in Fig. 6 that we can increase the amount of photons in the signal modes considerably. This demonstrate that the effective coupling strength becomes a decisive quantity for the efficiency of the process in an ab-initio description. It indicates how strongly light and matter mix and how strongly the matter-only eigenstates are modified and turn into hybrid light-matter states poten-  tially beneficial for the PDC process. Indeed, as has been shown in several works [37,42], the vacuum field of the modes can break the rotational symmetry and therefore g α can lead to increased dipole transition elements. Although the dipole elements are no longer a good descriptor for the interaction between light and matter in this regime, they nevertheless provide an intuitive picture for the PDC process. And they also highlight that by engineering the vacuum of the electromagnetic field, the properties of the coupled system can be very different from the individual uncoupled systems. For example, changing the photonic environment by decreasing L increases the coupling g α even in the case of a weak dipole moment. Also, the photon-photon couplings g 1 * g 2 , g 1 * g 3 , g 2 * g 3 between the modes increases with increasing couplings. We note that these terms arise due to the induced diamagnetic currents and are an often overlooked yet important contribution in many light-matter phenomena [36,38,40].
In the remainder of this section we focus on a different effect. In the Figs. 4, 5 and 6 we consistently find that the stronger the coupling, the earlier the down-conversion of photons happens. Having the coherence time in mind, we find that we can potentially beat the undesired dissipative processes by strong and ultra-strong coupling. That is, by shifting the creation of the down-converted photons to earlier times, while the coupled system is still behaving coherently, we can hope for attaining also special, coherence-driven features of the PDC process such as non-classicality and entanglement of the photons. Considering the Mandel Q parameter of the different photons (see Fig. 7), we observe that the time of appearance of non-classical features can be controlled by the coupling strength as well. The overall shape of the time-evolution of the different Mandel Q parameters stays relatively rigid. The same holds true for the intensity correlation functions (see Fig. 8). They nicely show how the photon in mode 1 is anti-correlated with respect to the signal modes, while the signal modes are strongly correlated.
While we do not model the emission of the downconverted photons from the photonic environment, we believe that controlling the timing of the creation process of the down-converted photons by varying the coupling strength will have a direct impact on the features of the emitted photons. At least for a simple description of an instantaneous emission, the features of the created signal photons are carried over to the emitted signal photons. It remains, however, unclear whether these features are only there for the special and in practice highly-demanding choice of a single photon in mode 1 and whether also the temporal control of these features is lost with another initial state or when the pump mode is replaced by a classical external pump field, i.e., a laser.

IV. INPUT MODE IN A COHERENT STATE
The generation of a single photon in a specific mode is highly challenging, and due to the usually low efficiency of the PDC process it is also not easy to observe such a process. So in practice one has to increase the number of photons to observe any down-converted photons. That means we will have more than just one photon in mode 1 at the beginning. It is, however, not clear how this will change the main features observed for the case above. Let us therefore here consider how a change in the initial state influences the different observables.
The pump mode is now initially prepared in a coherent state |φ 1 = |ξ 1 and thus, its vector potential has the strength ξ 1 |Â 1 |ξ 1 = λ 2 /ω 1 |ξ 1 | while that of the signal modes are zero at the beginning. We choose ξ 1 such that the mean photon number â † 1â 1 = |ξ 1 | 2 = 4. We have thereby just changed the occupation slightly. However, this is already enough to no longer be able to identify in a simple manner the usual PDC as one photon being down-converted to two photons, since now many photon states mix (see also App. F). We now find that the two-and three-photon states are also relatively strongly occupied. This becomes evident in Fig. 9, where the second-Fock state populations are an order of magnitude larger than the ones of the single-photon Fock states. This also changes the total mode occupations as can be seen in Fig. 10. Qualitatively the mode occupations remain relatively close to the one-photon case and also the feature of faster down-conversion in the strong coupling limit is preserved. Some details (especially for the input mode 1) are changed in comparison with the singlephoton case when considering the non-classicality of the down-converted photons. As can be seen in Fig. 11, in mode 2 the anti-bunching behavior is increased and mode 3 becomes now more non-classical as well. The crosscorrelation features of the different modes also change strongly. While before the input and the different signal modes where anti-correlated all the time, indicating that the input photon is annihilated and the signal photons created instead, we now have a richer behavior. In Fig. 12 (a.) and (b.) we see that there is a switching between anti-correlation and correlation in time. This change is not so surprising because we have now several photons in the initial state and the simple picture of one photon annihilated in mode 1 and one photon created in mode 2 and 3 respectively, is not so straightforward anymore. However, the main feature of correlated downconverted photons persists also for this initial state (see Fig. 12 (c.)). Conservation of energy and momentum in this cascaded process gives rise to the correlations, implying energy-time entanglement [58] which is important for on-demand generation of entangled photon pairs [9,18]. This interpretation is strengthened if we furthermore consider the purity γ α to be the entanglement measure for the present case. As shown in Fig. 13 the different modes become entangled over time. The entanglement and its time profile can be controlled by the coupling strength and pushed to earlier times as well. Treating the pump and signal fields as quantized modes, the three fields are shown to be entangled (see Fig. 13) and have varying correlations (see Fig. 12) which allows for applications in quantum information networks [59].

V. CLASSICAL INPUT FIELDS
While a coherent initial state is already closer to classical input field, we want to investigate whether a description based on a genuine classical external pumping changes the picture fundamentally. Although it is to be expected that a classical description of the input field for the relatively few photons we considered so far is not perfect, it should show qualitatively similar behavior at least for certain observables. We would then expect that these features are relatively unaffected by the details of the input field, i.e., how we populate the input mode or alternatively how we drive the electronic system. This is a very desirable feature since it would indicate that our findings are robust and do not depend on miniscule details of the setup.
At this point we face a further choice. Pumping the system with a classical field is possible in two distinct ways in an ab-initio description. Firstly we can prescribe an external electromagnetic field that couples to the matter system and its induced currents generate photons in the respective output modes. Alternatively we could prescribe an external current that couples to the input mode and thus we pump this mode directly. Physically these options should not be too different (by Maxwell's The generated photon pairs of modes 2 and 3 are time-entangled for the whole evolution. In (c.), we disregard the time interval between t = 0 and t = 0.58 ps due to finite numerical precision [57] equation both can be connected to each other, see also App. B). To find the influence of these two options we in the following consider both cases. First, we consider the more traditional option, that is, we choose some external electromagnetic field that drives the matter subsystem. Such a situation is often considered to avoid the need to have three explicit photon modes [11,60,61]. In practice this means we replace input mode 1 by a classical external field and keep the rest of the system's Hamiltonian the same, i.e., we havê Here, the time-independent HamiltonianĤ S and external driving termĤ ext (t) are given explicitly in App. B. The external pulse used corresponds to the classical field induced by the below defined external source term for the photon field in the case that mode 1 would be uncoupled.
To facilitate a simple comparison we therefore select the weak coupling regime of Tab. (I) in the following. Else we would need to adopt the form of the external pulse ) and (c.), signal mode 2 is most entangled, signal mode 3 is intermediate, and pump mode 1 is least entangled. In addition, by increasing the coupling from weak to ultra-strong not only changes the entanglement profile but also makes the modes more entangled.
to expect a reasonable agreement. We note that as a result of discarding mode 1 the corresponding observables of this mode become inaccessible in a direct manner. This is not the case for the second approach, where we pump mode 1 by an external current, i.e., a source term for the photons. In this case we can still investigate properties of mode 1. The corresponding Hamiltonian becomesĤ The external current is an envelope Gaussian of the form j 1 (t) = j 1 exp −(t − t 0 ) 2 /τ 2 sin(ω 1 t). The parameters of the Gaussian pulse is chosen such that at time t = 0.23 ps, the pump mode is driven to an excited state with on average n 1 (0) = 4 photons. Thus the vector potential of the pump mode is the same as that of the coherent state at the initial time (discussed in Sec. IV).
For both cases, we choose the initial state to be the nonfactorizable ground-state ofĤ S andĤ S , respectively. Comparing now both cases of external driving in Fig. 14, we see that they agree qualitatively for the mode occupations and for the Mandel Q parameters. We find, FIG. 14. Comparison of the two different external drivings for weak matter-photon coupling λ = 0.014. In red we use an external laser pulse, and in blue we use an external current to pump mode 1 directly. We have chosen the pump pulse and the pump current to be connected via Maxwell's equations such that for weak coupling both lead to similar effects.
however, that at later times, e.g., for Q 3 , they can differ strongly. If we then compare to the case of no external driving and having 4 photons in mode 1 at t = 0 instead (discussed in Sec. IV), we find that the chosen external fields qualitatively reproduce this case (see Fig. 15)

VI. COMPARISON TO STANDARD APPROXIMATIONS
We have found so far that while treating the pump mode differently has an influence on certain observables and that it can make the interpretation of the downconversion process more involved, other observables stay relatively unchanged. Yet, can we capture these effects also with standard approaches and avoid a full ab-initio treatment? By this we mean on the one hand quantumoptical approaches based on the few-level approximation and on the other hand semi-classical approaches which are employed in non-linear optics. The few-level approximation takes only a few matter states into account in contrast to the large amount of states that are considered in ab-initio simulations [62]. In ab-initio simulations one increases the amount of states until either the observables of interest or even the full wave function does no longer change. This basis-set-limit procedure is common place to ensure that the obtained results are not pathologically influenced by a cut off in the matter-photon basis. Disregarding the issue of basis-set convergence, we can instead choose to consider only a few "relevant" matter states, which leads to a Hamiltonian of the form given in App. C 1. In this section we choose four levels (as in Fig. (1b) or the highlighted energy levels in Fig. (24a)) which by simple energy arguments are the most relevant states (for details see App. C 1). By simplifying the matter subsystem in this way discards the possibilities of the existence of other hybridized polariton and virtual states that occur in an ab-initio treatment when the quantized field is coupled to the full matter subsystem. In the few-levels approximation considered here, we include the mode-mode interactions that arises from the diamagnetic term as opposed to cases were they are dropped out [27,62]. Ignoring these terms can lead to results that will differ considerably.
In non-linear optics one would usually use a semiclassical approximation and consider, for instance, nonlinear matter-only response functions. However, in the case of the quantum ring these would just be zero at the employed frequencies [30] and thus no further conclusions could be drawn. We therefore go one rung higher and employ an adopted Maxwell-Schrödinger approximation [34], which is a further common approximation used in the field of non-linear optics. It replaces the quantized photon field by its mean-field expression and hence leads to a set of coupled non-linear equations Here, ϕ(r, t) is the wave function of the matter subsystem and j α (t) is the current that self-consistently couples the mode-resolved Maxwell fields to the electronic subsystem (see App. C 2 for details). We note that we include in the Maxwell-Schrödinger approximation also the mean-field mode-mode interactions (and with this the diamagnetic current). Discarding these terms, as commonly done, will make the Maxwell-Schrödinger approximation less accurate. While the PDC process will be well described by this level of theory for the weak-coupling situation, in the strong-coupling case where hybrid light-matter states emerge, it is expected to be less reliable. In Figs. 16 and 17 we display characteristic observables of the down-conversion process in the strong coupling regime (i.e. λ = 0.017). The Maxwell-Schrödinger approximation fails, as expected, in the few-photon strongcoupling limit for all observables. In this case the quantum features of the electromagnetic field are essential and hence the mean-field-type approach of the Maxwell-Schrödinger theory is inadequate. Nevertheless, once we go to many photons, it becomes much more precise such that in the limit of arbitrarily many photons the full ab-initio theory becomes essentially equivalent to the Maxwell-Schrödinger theory (see also App. C 2). Therefore we can use this level of theory to assess how increasing the number of input photons strongly, affects the down-conversion process (see Sec. VII C). Beside the inadequacy of the semi-classical approximation for describing the down-conversion in the few-photon limit, we also find in Figs. 16 and 17 that the standard few-level approximation is not completely reliable as well. The main reason is that since the transition that is in resonance with mode 3 is not dipole allowed, many more levels have a similar contribution. This highlights that such a reduction to merely a few states can become inaccurate once several observables are considered at once. Only upon having converged the full wave function to an appropriate accuracy can we hope to have access to all possible observables.
Consequently, in the case that light and matter couple strongly and hybrid-light matter states mediate the down-conversion process, an ab-initio light-matter description becomes necessary.

VII. OPTIMIZATION OF THE DOWN-CONVERSION: THE DEGENERATE CASE
In this section we want to find the optimal setting for having, on the one hand, an efficient polariton-mediated down-conversion process, and on the other hand, nonclassical and controlled down-converted photons. In our current context we mean by non-classical the minimum Mandel Q parameter, i.e., anti-bunching behavior. To make the space of parameters as small as possible we will here consider the degenerate case, i.e., mode 2 and 3 have the same frequency. As a result of the degeneracy in energy in both signal modes 2 and 3, we decouple the signal mode 3 from Eq. (3) such that the downconverted photons populate only mode 2. We fix the polarization of signal mode 2 asÂ 2 =Â 2 e y by choosing θ 2 = 0 • and change the polarization of the pump field tô A 1 =Â 1 (cos θ 1 e x +sin θ 1 e y ) where θ 1 is the mixing angle between the photonic and electronic subsystems. In this case the Hamiltonian of Eq. (3) reduces to the two-mode electron-photon Hamiltonian As increasing the coupling strength pushes the downconversion process to earlier times, here we consider its maximal efficiency and minimum Mandel-Q parameter in mode 2. We will vary the polarization directions (see Sec. VII A), the anharmonicity of the binding potential of the GaAs quantum ring (see Sec. VII B), and the coupling strength as well as the number of input photons (see Sec. VII C). To judge the efficiency we consider the maximal amount of mode occupation n 2 (except of in Sec. VII C where we take a more general definition) over the range of the first 40 ps. For the non-classicality we determine the minimal amount of Q 2 over the same time interval.
We simulate the time-evolution dynamics of the abinitio system, the 3-level approximation (due to having a degenerate down-conversion process) and the selfconsistent Maxwell-Schrödinger approximation in the strong-coupling regime (λ = 0.017) starting from a coherent state in mode 1 with 4 photons (ξ 1 = 2). For degenerate two-photon generation, the pump field with energy ω 1 = 1.413 meV drives resonantly the transition between the ground-and first-excited state |ϕ 0 1 ↔ |ϕ 1 2 , thereby populating the state |ϕ 1 2 . The signal mode has half the energy ω 2 = 0.706 meV of the |ϕ 0 1 ↔ |ϕ 1 2 transition corresponding to a two-photon process. Since the coupling g 1 = 0.0398 of the pump mode is less than the coupling g 2 = 0.0563 of the signal mode, the effective electron in state |ϕ 1 2 is likely to relax to the groundstate through a two-photon emission channel via a virtual state. From a standard perspective this process should be again very inefficient. Yet within an ab-initio lightmatter treatment tuning the photonic environment to a virtual state will in the strong and ultra-strong coupling regime lead to the creation of a hybrid state that can efficiently mediate the down-conversion of the input photons.
In Figs. (18) and (19), we show the real-time photon occupations and Mandel Q α parameters of the incoming pump photon at the mode-mixing angle θ 1 = 60 • and the degenerate down-converted photons. For the abinitio results, the profile of n 1 (t) shows that at t = 5.61 ps, 0.82 of a photon is annihilated by absorption to promote the electron to the state |ϕ 1 2 . At t = 5.84 ps, the particle subsequently relaxes to the ground state by emitting photons with maximum mode occupation n 2 = 0.041. The apparent weak profile for n 2 (t) is due to emission via virtual states. The photon statistics of the pump field starts out in a coherent state and leads to a field with super-Poissonian statistics while the photon statistics of the generated photons varies between a field with bunching and anti-bunching features at different times, with the minimal value of Q 2 = −0.0199 at t = 4.94 ps. The few-levels approximation (in this case we only take three states into account, see also App. A) in Figs. 18 (a.) and 19(a.) is relatively accurate, while the Maxwell-Schrödinger performs is qualitatively correct only up t = 5 ps and by construction it remains a coherent field for the entire evolution with constant Q 1 = 0. Both approximations deviate from the full solution in Figs. 18 (b.) and 19 (b.). While the threelevel approximation consistently overestimates the downconversion efficiency as well as the quantumness of the photons in mode 2 (because it restricts the matter system too strongly), the Maxwell-Schrödinger approximation under-estimates these quantities. For the Mandel Q parameter this is by design and for the down-conversion efficiency the Maxwell-Schrödinger discards all the correlation between light and matter that becomes beneficial in the strong coupling case.
We note that this down-conversion scheme is indeed an inverse second-harmonic generation made possible by coupling the signal mode to a virtual state at half the energy of the first degenerate excited state |ϕ 1 2 . This will, however, not be the limiting case since the same can be done for the third-, fourth-and N th-photon generation, thereby defining an inverse harmonic generation for realizing an N -photon photon gun. By tuning the system in an experimentally realizable way, the setup can potentially be used an N -photon source with highly nonclassical properties. Such an optimization will be discussed below for the inverse second-harmonic generation process.

A. Optimization of field polarization
Now we consider which relative polarization is most efficient and at the same time minimizes Q 2 . In our setup this means that we consider the mixing angle θ 1 and the contribution of the interference term 2Â 1Â2 sin(θ 1 ). all the mixing angels. The mean photon occupation n 2 (t) increases with increasing angles due to the fact that 2Â 1Â2 sin(θ 1 ) becomes larger. For θ 1 = 90 • we find the highest down-conversion of photons as we obtain n 2 = 0.0527 since sin(θ 1 ) = 1 and the polarization of both modes are parallel with momenta contribution only in the y-axis (i.e.,p y = 0 whilep x = 0). The Mandel Q 2 (t) for θ 1 = 90 • shows the highest non-classical  [34] takes the mixing angle into account, it misses the induced correlations between the two modes, highlighting that an efficient down-conversion is driven in this strong-coupling case by the quantum correlations between the different modes and the matter system. The three-level approximation, on the other hand, overestimate these quantum correlations.

B. Optimization of the matter spectrum
Next, we investigate the influence of the anharmonicity of the electronic subsystem on the down-conversion process. We choose mode 1 in a coherent state with on average 4 photons, assume strong coupling (λ = 0.017) and choose a mixing angle θ 1 = 60 • such that both momentum components of the matter system are non-zero. We vary the parameter V 0 = 0, 50, 100, 150, 200, 250, 300 meV (see also App. A) which changes the transition energies between states from harmonic (all have the same transition energy) to anharmonic (different transitions have different energies). Table. (III) shows the transition energies from the electronic ground state to the first excited state that the pump energy is resonant to and its down-converted energies. In addition, increasing V 0 also increases the transition elements which leads to a stronger effective coupling between light and matter (see App. (C 1)).
We find (Fig. 21) that with increasing V 0 the maxi- mum photon occupation n 2 increases. Simultaneously, the state of the generated photon pairs becomes increasingly anti-bunched as Q 2 becomes more negative. This results from (i.) the increasing dipole moments due to the reduction of the quantum ring width for increasing V 0 and (ii.) the increase in the effective coupling strength due to the reduced scaling of the energies (see Tab. (III)) as the effective coupling is related to the frequency by g α = λ /2ω α (see App. A). We nicely see how the three-level approximation at V 0 = 0 meV fails, because many more than just three levels are important when all transitions have the same energy and we cannot separate a few specific transitions. Yet even when the system becomes strongly anharmonic and hence a separation of states is more reasonable, the few-level approximation overestimates the results strongly. Again, the Maxwell-Schrödinger approximation consistently under-estimates the results.

C. Optimization of the coupling and the input field
Finally, we consider the influence of the coupling as well as the strength of the input field on the efficiency of the down-conversion process. It is clear that by increasing the number of photons in the input mode 1 we will generate more photons in mode 2. We therefore have to think about a better way to judge the efficiency. We choose here the ratio of the photon energy of down-converted photon to the photon energy of the pump V0 (meV) ω1 (meV) ω2 (meV) Exact n max and minimum Q  H 1 (t 0 )). In this way just increasing the input field will not automatically lead to a higher efficiency. We then first scan for a mixing angle of θ 1 = 60 • , a binding potential of V 0 = 200 meV and a coupling strength of λ = 0.017 different initial states for mode 1 by ξ 1 = 1, 2, 3, ..., 10 such that the the input field has n 1 = 1, 4, 9, ..., 100 photons at the initial time. In Fig. 22 (a.) we display that by increasing the strength of the pump field through the amplitude ξ 1 the efficiency of the down-conversion decreases. Since the ab-initio simulation for such large number of photons becomes numer-ically very expensive, we extrapolate by the three-level and the Maxwell-Schrödinger approximation. As is expected, for more photons the Maxwell-Schrödinger theory becomes increasingly accurate and in the limit of very large photon numbers it should become exact. Since the quantum ring can only convert a finite amount of photons in the considered 40 ps, in this limit η (H) 21 goes to zero. On the contrary, if we instead fix the amplitude of the input mode to ξ 1 = 2 with 4 photons, but we increase the coupling strength λ = 0.014, 0.017, 0.019, 0.026, 0.044, we find (see Fig. 22 (b.)) that the efficiency increases. Interestingly, both approximations overestimate the efficiency for large λ. For very large coupling strengths, it is no longer the quantum correlations that provide a higher down-conversion rate but the shear strength of the coupling terms dominates. By manipulating the photonic environment to reach the ultra-strong coupling limit, the inverse second-order harmonic generation is strongly enhanced.

VIII. SUMMARY AND OUTLOOK
In this work we have highlighted how ab-initio non-relativistic quantum-electrodynamics (QED) simulations, that treat light and matter on equal footing, provide novel ways of engineering a parametric downconversion (PDC) process. By coupling strongly to a photonic environment, hybrid light-matter states (polaritons) emerge that can efficiently mediate and control the down-conversion process. This can be done even for processes via non-dipole allowed or virtual states. We have focused on a simple yet instructive model of a GaAs quantum ring coupled to a photonic environment, e.g., an optical cavity. We have shown that for short times (on the order of several tens of picoseconds) the dissipation of a photonic bath can be ignored in our setup and we have focused on coherent simulations. Changing the coupling strength to the input and signal modes, e.g., by increasing the quality of the cavity, then allowed to control the timing of the down-conversion process and shift it to earlier times. Thus strong coupling can help to limit the detrimental effects of dissipation on the PDC process. We then highlighted that while the interpretation of the PDC process as a single-photon process becomes less straightforward in the case where we have a coherent initial state in the input mode or when we replace the input mode by a classical external pump, the main features of the down-conversion remain intact independent of the exact form of the input mode. This suggests that our results are relatively robust with respect to details of the bath, the initial states or the different pumping schemes. We have then optimized the down-conversion efficiency and the non-classicality with respect to the level-structure of the quantum ring, the coupling strength to the photonic environment and the polarization directions of its modes, and with respect to the number of input photons. We find that paral-lel modes to maximize the matter-mediated mode-mode interactions, strong anharmonicity to reduce the number of electronic states involved and ultra-strong coupling between light and matter maximize both targets. We find that common approximations are helpful to estimate limiting cases but often do not capture the qualitative details of the down-conversion process for strong-coupling situations.
Our results demonstrate the possibilities that become accessible with ab-initio light-matter methods in the context of engineering novel photon sources. Since these methods stay applicable from the weak to the ultrastrong coupling regime and connect few-level models to the Maxwell-Schrödinger picture of non-linear optics seamlessly, we can scan a wide range of parameters to find optimal conditions not only for efficiency but also for other objectives, such as the Mandel Q parameter discussed here. While our simulations are still far from exact, since we do not model, for instance, the emission process or the phononic bath, we think our results demonstrate that using the full flexibility of the matter system and the photonic environment allows a detailed control of down-conversion process also in practice. We envision that using methods from polaritonic chemistry and material sciences to enhance the coupling between light and matter even at room temperature might provide new ways to access this full flexibility. Furthermore, since our description is not geared to only a two-photon down-conversion process but includes all processes, we can consider more intricate processes such as 3 or even more photon down-conversions, and can try to develop completely new ways to generate photons with specific properties on demand. With coupling strongly to many virtual states and the emergence of hybrid light-matter states at these energies not only inverse second-harmonic generation can be realized but also inverse high-harmonic generation seems feasible. This will be the subject of future work.
Since we have restricted to just a single active electron in our exploratory study, the total number of photons down-converted is relatively small. However, if we have an ensemble of quantum rings or a general many-electron system, the total number of down-converted photons can be efficiently increased. Such setups can be accessed with, for instance, QEDFT [43,63] or polaritonic coupled cluster [46,47], which can provide not only qualitative results but also a quantitative prediction how hybrid lightmatter states can generate photons on demand. This in certain cases can even include the nanophotonic structure as part of the simulation as well as the full emission process [34]. The results of this work demonstrate that the design of efficient photon sources is a very interesting working avenue for the emerging field of ab-initio lightmatter interactions.

IX. ACKNOWLEDGEMENTS
We would like to thank Frank Schlawin, Nicolas Tancogne-Dejean and Arunangshu Debnath for insightful discussions, and Sebastian Ohlmann for the help with the efficient massive parallel implementation of our code. We acknowledge financial support from the European Research Council (ERC-2015-AdG-694097) and the SFB925 "Light induced dynamics and control of correlated quantum systems". This work was supported by the Excellence Cluster "CUI: Advanced Imaging of Matter" of the Deutsche Forschungsgemeinschaft (DFG), EXC 2056, project ID 390715994. with energies E n = ω 0 (2n + 1) where n = 0, 1, 2, 3, ... and the degeneracy in energy is (n + 1). For V 0 > 0 meV (and we choose V 0 = 200 meV unless otherwise stated) the eigenstates ϕ l j are labeled by the angular momentum l = 0, ±1, ±2, ±3, ... and the index j = |l| + 1 enumerates over the energy levels (see Fig. (24a)). The ground state and excited state with l = 0 are singlets, whereas the excited states with finite angular momentum differing from l = 0 are doubly degenerate as shown in Fig. (24a). Dipole-allowed transitions occur only between states with consecutive angular momenta [31,42]. . Making use of the dielectric constant = 12.7 0 , we work in scaled effective atomic units as in Ref. [31] by defining effective units as Ha * = (m/ 2 )Ha ≈ 11.30 meV, a * B = (m/ )a 0 ≈ 10.03 nm, and u * t = /Ha * ≈ 58.23 fs. The choice of the quantum ring as the down-conversion medium is motivated by two properties. First, since it is a variable atom-like system, the electronic spectrum can be altered which is usually realized experimentally [25]. For example, the parabolic confinement po-tential of a quantum ring can be altered which changes the geometric properties by reducing the ring width (see Fig. (23.b)) thereby changing the electronic spectra as shown in Fig. (24b). Secondly, transitions between states with the same angular momentum (i.e. ϕ −l j to ϕ l j and vice versa) have been shown to be accessible by driving the transitions with a coherent laser field [31] breaking the inversion symmetry of the quantum ring. Here, we take another route by considering the Pauli-Fierz Hamiltonian in which the electronic subsystem is the GaAs quantum ring and show that the down-conversion is possible between non-dipole allowed transitions since the photon modes are quantized and treated inclusively as dynamical contributions of the coupled system. Specifically, photon emission into mode 3 in the cascaded process is possible for coupling resonantly to the non-dipole allowed transition ϕ 0 1 ↔ ϕ 0 6 . The transition dipoles in effective atomic units for the electronic states considered in this work are given in Tab The mode-resolved classical equation of motion of the photon coordinateq 1 is given by ∂ 2 /∂t 2 + ω 2 1 q 1 (t) = −λ 1 j 1 (t), of which the solution of the mode-resolved Maxwell equation is given as With the solution of the classical trajectory q 1 (t), the time-dependent external pump of Eq.(6) is given bŷ where A 1 (t) = λ 1 q 1 (t) and the down-conversion Hamiltonian iŝ In Fig. (15) for strong coupling, we show the comparison for the evolution dynamics of the down-conversion with an external current j 1 (t) (as in Subsec. (V)) and with an initial factorizable state as in Sec. (IV). At t = 0.23 ps, the external pump j 1 (t) injects n 1 = 4 photons and this qualitatively correspond to the case of an initial state which at the initial time have n 1 = 4 photons. The profiles for the photon occupation are qualitatively the same for the evolved time. The apparent slight deviation in the profiles is attributed to the values of the observables at the initial time. For example, at the initial time, n 2 = 0 for the initial factorizable state and n 2 = 0.0018 for the external current with a correlated ground-state. Therefore, the profiles for the evolved observable quantities are independent of the initial state chosen.
Appendix C: Approximations of the electron-photon coupled system

Few level approximation
The few-level approximation of the real-space electron-photon coupled system approximates Eq. (2) tô Â 2 (− p x ij cos θ 2 + p y ij sin θ 2 ) +Â 3 ( p x ij sin θ 3 + p y ij cos θ 3 ) where p x ij = ϕ i |p x |ϕ j , p y ij = ϕ i |p y |ϕ j represent the transition momenta matrix elements between relevant electronic states of the coupling to the photons. This approximation assumes that the few lowest energy levels of the electronic spectrum are separated in energy from the higher lying energy levels. Assuming this approximation holds, we truncate the Hilbert space to the N −lowest energy levels of interest [62]. Due to the degeneracy in the electronic spectrum, for the non-degenerate down-conversion we considered the four electronic states For free space radiation, the electron photon coupling is related to the radiative decay rate as g . Here we used the relation between the radiative decay rate and the transition dipole matrix elements γ ij = ω 3 ij |dij | 2 3π 0c 3 .
In Fig. (25) we show a comparison between the Maxwell-Schrödinger approximation (mean-field) and the real-space result for the non-degenerate case in weak coupling for the generated field q 3 (t) of mode 3 when the pump field have different amplitudes ξ 1 = 2, 3, 4 with respective photon numbers n 1 = |ξ 1 | 2 = 4, 9, 16 at the initial time. Considering the maximum amplitude of q 3 (t) of the evolution, the mean-field is larger than the realspace for increasing amplitude of the pump field. Also, in Fig. (26) we show for the mean-field case that by increasing the amplitude of the pump mode by an order of magnitude, i.e., ξ 1 = 20, 30, 40, the amplitude of q 3 (t) slightly increases. This is in accord with non-linear optics which for intense fields (large number of photons), the down-conversion have a low efficiency [22]. shows that the down-converted photons are of a singlephoton nature.
In the case when the input field is a coherent state, the one-, two-and three-photon Fock states are populated with increasing amplitude from |1 1 , |2 1 and |3 1 . This is true for the input mode Fig. 28 but also for the signal modes Figs. 9 and 29.