Photon-photon polaritons in chi(2) microresonators

We consider a high-Q microresonator with $\chi^{(2)}$ nonlinearity under conditions when the coupling rates between the sidebands around the pump and second harmonic exceed the damping rates, implying the strong coupling regime (SC). Using the dressed-resonator approach we demonstrate that this regime leads to the dominance of the Hermitian part of the operator driving the side-band dynamics over its non-Hermitian part responsible for the parametric gain. This has allowed us to introduce and apply the cross-area concept of the polariton quasi-particles and define their effective masses in the context of $\chi^{(2)}$ ring-microresonators. We further use polaritons to predict the modified spectral response of the resonator to a weak probe field, and to reveal splitting of the bare-resonator resonances, avoided crossings, and Rabi dynamics. Polariton basis also allows deriving a discrete sequence of the parametric thresholds for the generation of sidebands of different orders.

A common SC setup involves a mode of the highfinesse resonator interacting with a transition between the material energy levels. If the coupling rate is large relative to the dissipation rates, then the system can sustain multiple oscillations between the light and matter states [13][14][15]. The prime parameter behind properties of these oscillations is Rabi frequency, Ω. SC regime is most suitably described by the dressed eigenstates hybridising the light and matter degrees of freedom [16][17][18][19]. If the energy-momentum relation, ε(k), is introduced for the dressed states, then one can also define quasi-particles or elementary excitations [20]. A list of hybrid light-matter quasi-particles, or polaritons, includes exciton-polaritons [3][4][5][6][7], plasmon-polaritons [21,22], EIT-polaritons (electromagnetically induced transparency) [23], etc. Polaritons are not only a powerful theoretical concept, but are also a striking experimental feature associated with splitting and avoided crossing of the energy levels, and with Bose-Einstein condensation, see, e.g., [3][4][5]22].
While chip-integrated and bulk-cut high-Q ring microresonators continue to push limits of frequency comb research [24,25], some experimental results are also pointing towards potential these devices hold to study SC. E.g., Ref. [26] reported SC between the whispering gallery modes and atoms, Ref. [27] studied the EITlike regimes in two coupled resonators, and Ref. [28] described measurements of the low-contrast resonance splitting effects using non-degenerate χ (3) four-wave mixing * d.v.skryabin@bath.ac.uk (FWM). Refs. [29,30] developed theories for SC between two sidebands in χ (3) FWM without exploring the possibility to introduce the energy-momentum relations and quasi-particles.
Monolithic-and microresonator devices with χ (2) nonlinearity have been a viable and long-existing option complementing the mainstream of χ (3) work, see, e.g, [31][32][33][34][35][36][37]. Ref. [38] considered SC between the pump and singlemode 2nd harmonic photons in a planar semiconductor cavity. χ (2) response generally provides larger nonlinear phase shifts than χ (3) [33], which is a factor increasing the coupling between sidebands. This is also complemented by the recent progress with simultaneous reduction of losses and volumes of χ (2) microresonators [39][40][41][42][43][44]. A combination of these factors favours considering if a multimode χ (2) microresonator could go beyond simply modifying the light-matter interaction by reducing the density of states, i.e., Purcell regime, available for photon transitions and to cross into the SC regime, where new families of states (dressed states, and associated energy levels) are created.
Below, we demonstrate that χ (2) microresonators with Q approaching 10 8 [33][34][35][36][37], can operate in the SC regime between the sidebands centred around the pump (ordinary polarised) and second harmonic (extraordinary) frequencies. The SC regime becomes accessible far from the resonance, well outside the bistability and soliton regimes [43]. We parametrise energies of the dressed, i.e., hybrid ordinary-extraordinary, states with their momenta, and define microresonator quasi-particles -photon-photon polaritons. They are different from the above-mentioned families of the exciton-polaritons in planar [3][4][5][6][7][8][9] and ring [45] semiconductor microresonators, and from other polaritons involving transitions between the real, e.g, molecular [46], levels, because they do not require such absorbing transitions to exist in the matter component of the hybrid states. Instead, the matter side of the photonphoton polaritons belongs to the continuum of virtual, i.e., far-away from resonances, electronic transitions. χ (2) nonlinearity induced by these transitions is broad-band, and practically non-absorbing and non-dispersive.
CW-state power is now expressed via modulus of Ω, and that the 2nd harmonic amplitude is ψ e = Ω 2 /Ω e γ o .
Following the ethos of the dressed-atom theory [16][17][18][19], we apply the dressed-resonator method consisting from (i) expanding the small amplitude perturbations around the cw-state using the modes of the linear resonator, (ii) separating the Hermitian and non-Hermitian parts of the operator driving the evolution of the perturbations, and (iii) identifying the eigenstates of the Hermitian part, i.e., dressed-states or polaritons.
We now look for a solution of Eqs. (1) in the form, Assuming that the sideband amplitudes, |a µs |, are small we find Here |a µ = (a µo , a µe , a −µo , a −µe ) T is the state vector, The diagonal terms in H µ make up the Hamiltonian of the bare, i.e., undressed, resonator. Ω = 0 provides dressing and, importantly, retains the Hermitian structure. The diagonal terms in H µ are ∼ |Ω| 1 , and drive the flopping between the bare ordinary and extraordinary states with the same µ, see the state-vector structure. Therefore, Ω has the meaning of the Rabi frequency. Like in the two-level atom case [18,19], the bare states are separated by the optical frequency, i.e., by the energy ∼eV. While, the level splitting is characterised by the radio-frequency (RF) scale, |Ω| 1µeV [48]. The off-diagonal terms in H µ are the three-wave mixing (TWM) ones, since they describe adding one µ = 0 ordinary photon, defined by the |Ω| 1 -term, Ω ∼ ψ o , to one ordinary µ = 0 photon to generate one extraordinary photon with the same µ. The two-level atom methodology was also previously developed for the resonator-free sum-frequency generation, see, e.g., [49,50].
Interaction between the µ and −µ ordinary photons is governed by the off-diagonal terms in V . These are the four-wave mixing (FWM) terms engaging two µ = 0 ordinary photons, coming from |Ω| 2 , and two side-band ones, which are also ordinary, but have momenta ±µ. For studies into the FWM gain in χ (2) materials see, e.g., Refs. [33,51]. V also includes dissipation, that competes with the FWM gain. Therefore, the sideband dynamics is expected to be dominated by the TWM Rabi dynamics for || H µ || || V ||. This constitutes the SC conditions, which can be expressed as |Ω e | |Ω| Ω * .    |Ω| Ω * ≡ √ 2κ o κ e is achieved by taking a high-Q resonator and the relatively strong pump, while |Ω| |Ω e | implies small conversion efficiency to the 2nd harmonic due to large value of the frequency mismatch parameter, |2ω 0o −ω 0e | ∼ |Ω e |/8. Note, that Eq. (3) can be linearised for |Ω| 2 /|Ω e ||Ω o | 1, and then the Rabi frequency is estimated as |Ω| ≈

IV. RESULTS
Before using SC conditions to our advantage, we set |a µ = |b µ e −iβµt and solve the full dispersion law det( H µ + V − β µ I) = 0 numerically for every µ. Condition Imβ µ = 0 specifies the boundaries between the areas with and without gain. Fig. 1(a) shows the case of |Ω|/κ o 1, and |Ω|/κ e ∼ 1, i.e., when we have SC for the ordinary, but not yet for the extraordinary photons. Under these conditions, the µ-specific instability domains are arranged as a sequence of the 4-beam stars shifted along the δ o , but nearly coinciding along the δ e , direction. As |Ω| is increased further and provides |Ω|/κ o,e 1, i.e., |Ω| Ω * , we step firmly into the SC regime. Now, a spectacular pattern of the narrow resonances appears along the o δ e direction and the ones along δ o sharpen even more, see Figs. 1(b),(c).
Thus, the SC condition channels gain into the narrow mode-number specific regions in the parameter space. Parametric frequency conversion in χ (2) resonators with F ∼ 10-10 2 , operating in the weak-coupling, i.e., Purcell, regime, that has attracted significant recent attention [52][53][54][55], does not have this effect, and the contrast of the star beams in Fig. 1(a) is reduced further for smaller F. Formation of the narrow instability tongues in the high-F regime is similar to the Arnold-tongues reported in the high-finesse Kerr microresonators [56,57]. However, the matrix operator driving the sideband dynamics in Refs. [56,57] is entirely non-Hermitian, as is V here. Therefore, it does not allow to define the energy and momentum for the quasi-particles in the way possible here, if H µ dominates over V , i.e., in the SC regime.
To leading order, the mode dynamics in the SC regime is driven by i∂ t |a µ = H µ |a µ . H µ is block-diagonal and has two orthogonal sub-spaces of the dressed states. The top-block subspace is span by |b is the Rabi detuning, see Eqs. (8). The corresponding eigenfrequencies are β Here, Ω µ is the effective Rabi frequency [18,19]. The bottom-block eigenstates and eigenfrequencies are |b −µ , and |b Each pair of the dressed states makes its own avoided crossing around the momentum values corresponding to ∆ ±µ = 0, Fig. 2(a). Close to the avoided crossings, the state dressing is increasing via Ω ±µ − ∆ ±µ → |Ω|.
The frequency-momentum dependencies, β (j) µ vs µ for j = 1, 2 and j = 3, 4, Fig. 1(d), define the four families of quasi-particles. These quasi-particles hybridise the ω µo and ω µe photons, and can be naturally calledphoton-photon polaritons. H µ is Hermitian and its eigenfrequencies can be used as the quasi-particle energy spectrum. Hence, the photon-photons carry energy ε = β (j) µ (∼ µeV, D 1 ∼ D 1o ), angular momentum = µ, and the linear momentum k = /R, where R is the resonator radius. Predictably, there is a flexibility with defining the photon-photon zero-energy level. This is controlled by the reference frame choice, i.e., by D 1 entering ∆ µo +∆ µe , while ∆ µ and Ω µ are D 1 independent. The effective photon-photon mass is evaluated as . Away from the avoided crossings, the photon-photons behave as photons and have m eff ± /R 2 |D 2s | ∼ 10 −3 m e , where m e is the electron mass. Close to the avoided crossings, SC modifies linear dispersion, and, because of this, m eff can drop by few orders of magnitude, sharply rise to infinity and change its sign, see Fig. 1
Let us now describe how the dressed spectra allow formulating the TWM and FWM energy conservation laws. TWM drives the Rabi flops between |ψ µo (t)| 2 and |ψ µe (t)| 2 , and the quanta involved satisfy -ω p + ω µe . The above condition combines a pump photon with a dressed ordinary photon with momentum µ to produce a dressed extraordinary photon with the same µ. The other process, is the cross-branch FWM that engages µ and −µ quanta and yields the following energy conservation between two pump laser photons and two dressed ordinary polarised quanta, ω p + ω p = ω An important feature of the β (j) µ vs µ dependencies are the resonances between the photon-photons belonging to the 1, 2 (full circles in Fig. 1(d)) and 3, 4 (empty circles) branches. Four eigenfrequencies generally admit six distinct resonance conditions, β µ . However, all six of them can be realised only for Ω = 0. The dressed resonator, Ω = 0, allows for the four resonance conditions (j 1 ; j 2 ) = (1; 3), (1; 4), (2; 3), and (2; 4). A particular case selected for Fig. 1(d) illustrates two simultaneous resonance conditions β Resolving each of the four resonance conditions for Ω gives the same elegant answer, Eq. (9) is structurally similar to the resonant denominators derived for the susceptibilities emerging through the interaction of the multimode fields with the multilevel atoms [16,18,19]. The resonance lines computed from Eq. (9) and mapped onto Fig. 1(c) precisely follow the maximal gain lines in the parameter space, i.e., max Imβ µ > 0 vs δ s , where the complex β µ are computed from det( H µ + V − β µ I) = 0. Taking Fig. 1(c), we find that the first and last brackets in Eq. (9) determine the resonance separation along δ e , which happens in steps |D 1o −D 1e |, 2nd bracket controls resonances along δ o in steps of |D 2o µ|, while the 3rd bracket does not make a noticeable impact along δ e because |D 2e µ| |D 1o −D 1e |. All resonance lines have a special point triggering a parametric instability relative to the ±µ modes. To find the parametric gain threshold when the system is confined to one of the resonances, i.e., when two energy levels are degenerate, β µ , while their eigenvectors remain different, we apply the degenerate state perturbation theory [58] to Eq. (4) by treating the FWM operator V as a perturbation. The matrix elements of . FWM removes the degeneracy between β (j1) µ and β (j2) µ by making different the loss vs gain balance acquired by the two states. One state becomes lossy, while the other generates gain providing V Opening the above for, e.g., the (j 1 ; j 2 ) = (1; 4) case yields Eq. (10) explicitly expresses the balance between the net gain (left) and the net loss (right). The intersection be- photon-photon states as recorded through the series of the numerical scans of the probe frequency, ω b , for a range of δe values. ω b is scanned around the bare resonator resonance at ω3o ω0o + 3D1o. Color density shows the µ = 3 mode power, τ 0 |ψ3o| 2 dt/τ , see color-bar. Parameters are W b = 0.9mW (probe laser power), W 176mW (pump laser power), δo/2π = −50MHz, τ = 0.4µs. Corresponding cw-state power, i.e., the one that directly drives nonlinear processes in the resonator material, is | ψo| 2 = H 2 * |Ω| 2 /Ω 2 * (ηFW/4π) × (κo/δo) 2 60mW, which is below W because the resonator is driven far off-resonance. tween the hypersurfaces defined by Eqs. (9) and (10) provides an excellent approximation for the locations of the instability boundary tips, see Fig. 1(c), thereby proving that the resonance effect lies at the origin of the channelling of the parametric gain into narrow instability domains.
Dressing of the spectrum, i.e., splitting of the resonances of the bare, Ω = 0, resonator, and the underpinning Rabi dynamics have been also demonstrated by us in a series of numerical experiments reproducing how the photon-photon realm could be measured in a lab. To achieve this, we consider a region in Fig. 1(c) that avoids the FWM gain, and where the power enhancement effects provided by the resonator are diminished or even reversed, while the SC induced dressing effects on the resonator spectrum can be seen clearly. We then include a weak probe field, 'b', into Eqs. (1), through the transformation H → H + H b e iµϑ−i(ω b −ωp)t . This probe field has the non-zero projections on the |b (1) µ , |b (2) µ dressed states via their ordinary components. Therefore, if ω b is tuned to the frequency of one of the dressed states, e.g., to ω (1) µo , it then resonantly excites a superposition state, (2) µ , that exhibits Rabi oscillations. These oscillations are expressed as the antiphase beats of |ψ µo (t)| 2 and |ψ µe (t)| 2 with the frequency ω (1) The probe laser power, and what it achieves inside the resonator during the frequency scan, is relatively weak, while the pump laser is kept at the constant frequency far away from the resonance, and therefore they are not expected to cause significant thermal effects, see Fig. 2 for the power and detuning values. Other pump-probe arrangements in Kerr microresonators have been recently demonstrated in Ref. [59]. Further thermal control can be applied with the techniques as the ones developed for cooling of microresonators and photonic chips [60][61][62]. Fig. 2(a) shows how the |ψ 3o | 2 power responds to the applied probe depending on its frequency, ω b , and through a range of the δ e values. One can see pronounced resonant responses for ω b tuned to ω (1) 3o and ω (2) 3o . This fully confirms our analytical results for the dressed spectrum, see Eqs. (8) and the dashed lines in Fig. 2(a). The avoided crossing region in Fig. 2(a) corresponds to the minimum of Ω 3 in δ e , i.e., ∆ 3 = 0 and Ω 3 = |Ω|. Same avoided crossings can be observed for the whole sequence of the resonances in the 'star'-diagram in Fig. 1(c). Measurements like in Fig. 2(a) are often used as signatures of the polariton existence in other optical systems, see, e.g., Refs. [4,21].
Rabi beats can be observed by measuring the RF spectra of either the total field or the individual modes, µ = 3 in this case. Fig. 2(b) shows the spectrum of |ψ 3o (t)| 2 vs δ e with ω b tuned to ω (1) 3o . It makes obvious the spectral signature of the Rabi splitting, which is also known as the Autler-Townes splitting in spectroscopy [16,27,63]. The sideband power in Fig. 2(b) is proportional to the modulation depth of |ψ 3o (t)| 2 , which is reduced if the probe projection on one of the two dressed states is much larger than on the other, e.g., |α 1 | |α 2 |.

V. SUMMARY
We have proposed and formalised a concept of the polariton quasi-particles, photon-photon polaritons, in the context of multimode high-Q χ (2) microring resonators. The quasi-particle regime becomes possible if a resonator is pumped far-off the resonance, which allows the dominance of the Hermitian part of the side-band coupling operator over the dissipation and parametric gain effects. Such a resonator operates in the strong-coupling regime, shows the splitting of the resonances, their avoided cross-ings, and Rabi oscillations between the ordinary and extraordinary modes. Our results reveal a promising connection between the rapidly expanding research area of frequency conversion in high-Q microresonators and the quasiparticle approach widely used in other branches of photonics and condensed matter physics.