Rabi oscillations of two-photon states in nonlinear optical resonators

We demonstrate that four-wave mixing processes in high-quality non-linear resonators can lead to Rabi-like oscillations in photon occupation numbers and second-order correlation functions, being a characteristic feature of the presence of entangled photon pairs in the optical signal. In the case of a system driven by a continuous coherent pump, the oscillations occur in the transient regime. We show that driving the system with pulsed coherent pumping would generate strongly anti-bunched photon states.

Here, we study FWM in high-quality resonators with discrete spectra [9] and strong Kerr non-linearity: "zerodimensional" exciton-polariton microcavities [10][11][12][13][14] and microwave resonators coupled to superconducting qubits [15][16][17][18]. In these systems the FWM between photon pairs is reversible, so that quantum oscillations can occur between the initial and final states. We show that for a photon pair in a resonator, its states oscillate between two entangled two-photon configurations with the frequency determined by the strength of Kerr non-linearity. For a pumped system, such Rabi-type oscillations would occur in transient regime after the pump is switched on, until losses damp the oscillations establishing a steady state. We suggest that such oscillations would be a characteristic feature of the presence of entangled photon pairs in the optical signal. We also demonstrate that applying pulsed driving to the system would generate photon states in the anti-bunching regime.
To model two-photon correlations in a non-linear microcavity, we employ the master equation approach to the density matrix of the photon system, ρ, [19], In (1), the Hamiltonian H ( = c = 1) describes the conservative part of the photon system [20], a i and a † i are annihilation and creation operators of the photons in the mode ω i , u is a photon-photon coupling constant due to Kerr non-linearity, γ is the photon decay rate, which is taken the same for all photon modes, and H pi describes the photon pump. Below, we consider microcavities excited using one of the two possible pumps sketched in Fig.1, where f (t) and ω pi are the amplitude and frequency of the pumping field. The pump (2a) excites photons in the mode ω 0 , while pump (2b) corresponds to coherent excitation of both ω 1 and ω 2 modes. By solving Eq (1a) for the density matrix determined in the Fock space of states with different photon numbers, m i and n i , in the three modes, ρ = ρ(m 1 , m 2 , m 0 , n 1 , n 2 , n 0 )|m 1 , m 2 , m 0 n 1 , n 2 , n 0 |, we evaluate the occupation numbers of photons and zero time-delay pair correlation functions for each mode ω i ,

RABI OSCILLATIONS OF TWO-PHOTON STATES IN CLOSED CAVITIES
To introduce the idea of Rabi oscillations of twophoton states, let us consider an initial state of the sys- The Hamiltonian of this system (1b) can be projected onto the reduced Fock space of the two-photon states, {|2 ω0 , |1 ω1 , 1 ω2 }, with the eigenvalues and eigenstates, 2u is the two-photon Rabi frequency, which characterises time-dependent oscillations of the probabilities P (2 ω0 ) and P (1 ω1 , 1 ω2 ) to find these two photons,

RABI OSCILLATIONS IN OPEN RESONATORS
In the case of a lossy system, with a finite decay rate γ, this result can be generalised into the formulae for probabilities to find two or one photon in the cavity, illustrated in Figs. 2(a) and (b), These oscillations can manifest themselves in the temporal evolution of the occupation numbers, N i , illustrated in Fig. 2(c), (Ω cos(Ωt) + 2γ sin(Ωt)) , (Ω cos(Ωt) + 2γ sin(Ωt)) .
In principle, the quantum-optical Rabi-type oscillations can occur in systems driven by coherent pump-ing. We demonstrate this for the case of resonant pumping, ω pi = ω i , and δ = 0. In the case of lossless sys-tem (γ = 0) and weak continuous pumping switched on at t = 0, f (t) = (const u) × θ(t), where θ(t) is the Heaviside function, Eq. (1a) can be solved analytically by diagonalizing the Hamiltonian (1b) and (2), which in the interaction representation, can be written in the time-independent basis. This procedure leads to the following eigenvalues, {±f, ± √ 2u} for pump (2a), and {0, ± √ 2f, ± √ 2u} for pump (2b), and for f t 1 results in the oscillations in occupation numbers (δ = 0), for pump (2a), and for pump (2b). Here, the sizeable part of the oscillations has frequency Ω/2.
0 , in the continuously driven system sketched in Fig 1 (b). In For a larger pumping amplitude, f = (const ∼ u) × θ(t), and for finite losses γ = 0, we solve Eq (1a) numerically. We truncate the Fock space photon numbers, m i , n i ≤ N max and check that, for f ∼ u, N max = 10 is enough for the convergence (upon the increase of N max ) of the numerical results. We find that photon pairs manifest themselves as Rabi-like oscillations between the two 1 , for the continuously driven system shown in Fig. 1 (a). In (f), the arrow indicates the first minimum of g states |2 ω0 and |1 ω1 , 1 ω2 as we show in Fig.3 for the pumping scheme of Fig. 1(b). Interestingly, the oscillations of the occupation numbers can be well described by the two harmonics with frequencies close to Ω and Ω/2, as in the case discussed above, and can be fitted by the interpolation formula ( Fig.3(a), (c)), where F (t) is a monotonous (non-oscillating) function.
For weak pumping, f u, the amplitude of the Ω/2harmonic, b 1 , is much larger than the amplitude of the Ω-harmonic, b 2 , which is noticeable in Figs. 3(a). As the pumping amplitude f increases, b 2 goes up and exceeds b 1 at f ∼ u. Thus, the oscillations at f ≥ u have frequency Ω (Fig. 3 (b)-(c)). The oscillations with the same frequencies can also be observed in the two-photon correlation function g (2) 0 , Figs. 3(e)-(f). We find a similar behaviour for the pumping scheme illustrated in Fig. 1(a). In Figs. 4(a)-(d), the oscillation frequency changes from Ω/2 for f = (const u) × θ(t) to Ω for f = (const ∼ u) × θ(t). There are also oscillations in the two-photon correlation function in the mode ω 1 , g 1 , with the period matching the one of N 1 . Note that, during its cycle, g for the system of Fig. 1 (a) with γ = 0.1u driven by the pulses (see inset): (green dot-dashed line) with amplitude f0 = u and τ = 2.6/u, corresponding to the minimum of g (2) 1 of a continuously pumped system (Fig.4(f)).
anti-bunching can be promoted, if the system is driven by pulses with a pump of finite duration τ corresponding to the minimum of g (2) 1 in a continuously pumped system. As we show in Fig.5 for the cavity driven by ω 0 -mode pulses, 1 has a maximum corresponding to the bunching regime at the time when N 1 is small, but then, the system switches into anti-bunching regime. In principle, the form of the ringing tones in oscillations of N 1 and g (2) 1 may depend on the extent and shape of the excitation pulse, however this dependence is weak, as we show in Fig.5, where we compare the systems excited by the pulses of various shapes.

DISCUSSION
In this paper, we have demonstrated that in nonlinear resonators with discrete spectrum correlated photon pairs can manifest themselves as quantum Rabi-like oscillations with the period determined by the strength of non-linearity. In coherently driven systems, these oscillations would occur in the transient regime, t < 1/γ, as shown in Figs. 3 and Fig. 4, thus, they could be observed in a system with sufficiently low damping, γ ≤ u. To mention, the proposed theory describes systems with various frequency ranges. It can be applied to the visiblerange polaritonic microcavities based on GaAs, where the non-linearity is due to exciton-exciton interaction in GaAs [21]. In these systems, the strength of nonlinearity, u ∼ 10µeV , is still too weak as compared to damping, γ ∼ 80µeV [13]. However, we suggest that use of two-dimensional transition metal dichalcogenides [22][23][24] in microcavities, may improve the u/γ ratio due to stronger exciton-exciton interaction. Moreover, the proposed theory of two-photon oscillations is also applicable to microwave photons in superconducting resonators, where one can achieve quality factors in the range of 10 6 − 10 9 [25][26][27], hence reach the regime u γ. Large u/γ ratio can also be achieved by resonantly coupling atoms with level structure allowing for electro- for a high-quality toroidal microcavity coupled to atoms in an EIT regime driven by resonant coherent pumping of Fig. 1(a) for an experimentally achievable set of parameters: u = 1.25 × 10 7 s −1 , γ = 2×10 5 s −1 , and f = 1.25×10 7 s −1 [29,30,32,33]. magnetically induced transparency (EIT) [28,29] and microcavities with high quality factors, Q, such as toroidal (Q > 10 8 ) [30] or microrod (Q > 10 9 ) [31] resonators. In these systems, the strength of non-linearity can reach u ∼ 1.25 × 10 7 s −1 , while the losses can be as low as γ ∼ 2 × 10 5 s −1 [29,30,32,33], hence reaching the desirable regime u γ leading, as we show in Fig.6, to Rabi oscillations in photon occupation numbers and twophoton correlation function.
We have also shown, that application of pulsed pumping to the system could lead to the generation of squeezed states of strongly anti-bunched photons.
We thank D. Krizhanovskii, E. Cancellieri and M. Skolnick for useful discussions. This work was supported by EPSRC Programme Grant EP/J007544.