Broadband photon-photon interactions mediated by cold atoms in a photonic crystal fiber

We demonstrate theoretically that photon-photon attraction can be engineered in the continuum of scattering states for pairs of photons propagating in a hollow-core photonic crystal fiber filled with cold atoms. The atoms are regularly spaced in an optical lattice configuration and the photons are resonantly tuned to an internal atomic transition. We show that the hard-core repulsion resulting from saturation of the atomic transitions induces bunching in the photonic component of the collective atom-photon modes (polaritons). Bunching is obtained in a frequency range as large as tens of GHz, and can be controlled by the inter-atomic separation. We provide a fully analytical explanation for this phenomenon by proving that correlations result from a mismatch of the quantization volumes for atomic excitations and photons in the continuum. Even stronger correlations can be observed for in-gap two-polariton bound states. Our theoretical results use parameters relevant for current experiments and suggest a simple and feasible way to induce interactions between photons.

SUPPLEMENTARY MATERIAL TO "Broadband photon-photon interactions mediated by cold atoms in a photonic crystal fiber" . Marina Litinskaya, Edoardo Tignone, Guido Pupillo

Kinematic interaction for bare excitons
The Schrödinger equation for two bare excitons (G ≡ 0) on sites n 1 and n 2 interacting via kinematic interaction is with t ij the long-range hopping energy, and the same-site amplitude chosen as C (ex) nn = 0 [1]. Let n be n = |n 1 − n 2 |, and the index µ enumerate the eigenstates of this equation. We rewrite equation (S1) in the nearest neighbour approximation: One can verify that the normalized amplitudes which satisfy equation (S2) are with wave vectors κ µ = 2πµ/(N a) with a half-integer state index µ ∈ [−(N − 1)/2, (N − 1)/2] introduced in equation (10) in the main text. The basis functions g n (µ) form an orthonormal set in both spaces, with the orthonormality conditions reflecting the permutation symmetry and the hard-core condition: The eigenenergies of two-exciton states are also described by the wave vectors κ µ as As the new wave vectors κ µ have a half-integer state index they lie exactly between the positions of the standard wave vectors k ν for non-interacting excitons. As a consequence, the amplitudes C (ex) µ (k ν ) do not have poles, but rather enhanced components k ν ≈ κ µ , as can be seen from the Fourier transform of equation (S3): sin aκ µ cos(ak ν ) + (−1) µ sin ak ν sin(ak ν N/2) cos ak ν − cos aκ µ .
We conclude that the kinematic interaction is a weak, but absolutely non-perturbative effect for excitons. Let us now discuss the effect of the kinematic interaction for polaritons.

Creation of the wave packets
Two-polariton states in the presence of kinematic interaction can be viewed as composed of two subsystems: the non-interacting subsystem (consisting of photon-photon and photon-exciton states) is described by the quantum numbers k ν = 2πν/(N a) with integer ν, whereas the interacting subsystem (consisting of exciton-exciton states) is described by κ µ = 2πµ/(N a) with half-integer µ. In the following discussion we shall stress the role of the two wave vector sets, which will be reflected in the adopted notations. The coupling between these two subsystems is responsible for intermixing the corresponding wave vector sets {k ν } and {κ µ } and eventually leads to the creation of the wave packets in the "original" wave vector set {k ν }. In particular, at the lowest energies the coupling of excitons to photons dominates over exciton-exciton interaction, and the corresponding polaritons are better described by k ν . With the increase of the state number, instead, polaritons enter the exciton-like regime and are better described by κ µ .
We introduce the operators α † n , β † n and γ † n , which describe, respectively, creation of two photons, one photon and one exciton, and two excitons separated by a distance n. The two particle wave function takes the form |Ψ = s [A(s) |α s + B(s) |β s + C(s) |γ s ], and the HamiltonianH eff =H AB +H AB−C is made up of three terms: where the last one describes the coupling between the interacting (C) and non-interacting (AB) subsystems. The resulting Schrödinger equation is identical to the Fourier transform of equations (3) in the main text. The first termH AB describes the subspace "photon-photon photon-exciton", and is diagonalized by the operators (S10) The two-exciton partH (KI) C of the Hamiltonian is instead diagonalized as with energy E (ex) µ defined in equation (S5), and Eventually, the interaction HamiltonianĤ AB−C can be rewritten in terms of ξ-and χ-operators as with coupling coefficients Λ νµ given by The coefficients Λ νµ intermix the wave vector sets k ν and κ µ . The Schrödinger equation for the HamiltonianH eff and the wave function |Ψ = iν p We can exclude the exciton-exciton amplitudes e µ from equations (S15); in the absence of hopping (t ≡ 0) the resulting equation for p iν reduces to with the kernel (S17) The first term in the right-hand side of this equation describes the wave-vector-conserving scattering, while the second describes the formation of wave packets via scattering of non-interacting subsystem through the interacting one. Within these notations, the amplitude for two photons being separated by n lattice sites is N results from a collective effect of p-amplitudes that add up with a vanishing phase; large separation amplitudes are instead averaged out by the oscillating exponentials. The wider is the distribution of p (i) ν , the larger A(0) is expected. Using equations (S15) we get Due to the mismatch between quantum numbers ν and µ the denominator is not a real pole. However, it plays an important role in the establishing of the bunching, which occurs when E = E ρ is resonant with the band of non-interacting states E the two-photon wave function looks unperturbed and exhibits plane-wave-like oscillations. This criterium can be used as a good rule of the thumb when deciding on whether a state with a given energy shows bunching or not. It looks like as if excitons talked to each other via virtual excitations -the eigenstates of the non-interacting subsystem. Indeed, the "real" elementary excitations are one-polariton states, while the energies (S9) do not have an independent physical meaning, except as a virtual scattering channel through which excitons interact.
Using the equality following from |Ψ representation via A, B, C-and p, e-amplitudes, we find that 2e µ = s g s (µ)C(s). For higher ρ showing bunching, we can approximate the polaritonic C ρ -amplitudes by closest in energy (with µ = ρ − 1/2) bare exciton-exciton amplitudes (S3) times a normalization coefficient X (γ) ρ , which accounts for the presence of finite exciton-photon and photon-photon excitation in the total wave function of ρ-th eigenstate. We then obtain, using orthogonality of g-functions, The numerator of the equation (S21) shows that only those state contribute into A(0), which simultaneously have non-vanishing photonic and excitonic amplitudes, which is true only for the strong coupling region. In addition,