Jacobi photonic lattices and their SUSY partners

We present a classical analog of quantum optical deformed oscillators in arrays of waveguides. The normal modes of these one-dimensional photonic crystals are given in terms of Jacobi polynomials. We show that it is possible to attack the problem via factorization by exploiting the quantum optical analogy. This allows us to provide an unbroken supersymmetric partner of the proposed Jacobi lattices. Thanks to the underlying SU(1,1) group symmetry of the lattices, we present the analytic propagators and impulse functions for these one-dimensional photonic crystals.


Introduction
Supersymmetry (SUSY) was first used as a way to unify bosonic and fermionic sectors in string models and, along the time, it has been used to unify space-time and internal symmetries in high energy physics, to generalize gravity in relativistic physics, to find and categorize analytically solvable potentials in quantum mechanics, just to mention a few examples [1][2][3]. In optics, planar waveguides with some particular refractive index profiles have been shown to accept isospectral partners in the paraxial regime [4] and SUSY has provided a method to generate a family of isospectral potentials to optimize quantum cascade lasers [5,6]. In quantum optics, isospectral partners for ion-trap Hamiltonians have been used to propose entanglement generation by adiabatic ground-state transition [7], the simulation of the Wess-Zumino SUSY model in 2+1 dimensions with cold atom-molecule mixtures in the presence of a two-dimensional optical lattice [8] and that of the electric dipole moment of neutral relativistic particles related to SUSY models in ion-traps setups [9].
Optical analogies of quantum systems realized in waveguide arrays have recently impacted the field of integrated optical structures [10]. In particular, SUSY photonic lattices can be used to provide phase matching conditions between large number of modes allowing the pairing of isospectral crystals [11,12]. In the following, we provide a class of photonic lattices showing discrete-SUSY and find their analytic spectrum, propagator and impulse function. Our onedimensional crystals are also the classical analog to a class of generalized deformed oscillators [13,14] and can simulate non-classical squeezed light with the propagation of classical light [15]. The normal modes coefficients of these waveguide arrays are given in terms of Jacobi polynomials and this the origin of their denomination. We follow a factorization method [16] to study our Jacobi lattices and to propose a feasible SUSY parter for them. The analogy between the quantum and the classical optics systems allows us to present a closed form for the propagator and impulse function of our photonic crystals.

Jacobi lattices
Let us consider the Schödinger-like equation for a generalized deformed oscillator [13], where the shorthand notation ∂ z stands for derivation with respect to z, the creation (ahhihilation) and number operators are given byâ † (â) andn, respectively, and the parameter α is a real number that characterizes the deformed oscillator such that α = ±1. The action of the operators over a Fock state, also known as number state | j , is given byâ † | j = √ j + 1| j + 1 , a| j = √ j| j − 1 andn| j = j| j . This allows us to write any given state of the system as the superposition |ψ = ∑ ∞ j=0 E j | j leading to the differential equation set This differential set can be related to an array of photonic waveguides as shown in Fig.  1 [17,18], such that we have a classical analog for our quantum optics system. The experimental realization of this class of photonic lattices is feasible; cf. a discussion on a similar type of refractive index scaling and coupling given in [19][20][21] and the fact that our lattices are a particular subclass of those given in [15] with parameters N = 1, β i = 1 + α 2 and β s = 0. In order to study our waveguide array, we will take advantage of the quantum optics model and follow an algebraic approach [13]. For this reason, we can rewrite the Schrödinger-like equation in (1) as −i∂ z |ψ =Ĥ|ψ with the Hamiltonian, in terms of the elements of the SU(1, 1) group,K 0 =n + 1, where we have used tanh ξ = 2α/(1 + α 2 ). Thus, the spectrum of our photonic lattice is given by The corresponding normal modes, |α k =R(−ξ )|k , are given by a rotation over the basis in the diagonal representation, given by the number states |k , and can be reduced to the form in terms of the Jacobi polynomials P γ,β n (x) [22]. It is for this reason that we christen our waveguide arrays as Jacobi lattices. Figure 2 shows the squared amplitudes of the ground and tenth normal mode for different values of the deformation parameter.
As we have shown in the past for a photonic lattice with SU(2) symmetry [23], we can follow an equivalent algebraic approach with the SU(1, 1) group and calculate the propagator function by using the quantum optics analogy: Fig. 1. The effective refractive index, ω j (α), and coupling, g j (α), functions for the Jacobi lattices.  where |ψ(z) = ∑ ∞ j=0 E j (z)| j is a vector containing the information of the propagating classical field amplitude at the jth waveguide, E j (z). Thus, the classical field amplitude at the jth waveguide for an initial field impinging just the kth waveguide, also known as impulse function, is given in terms of Jacobi polynomials and the auxiliary function Note that as (3) is a compact operator we will expect coherent oscillations for a single waveguide input as shown in Fig. 3(a) for an initial classical field impinging at the tenth waveguide, j = 9, of a Jacobi array with parameter α = 0.5; although (3) is composed by compact,K 0 , and non-compact, (K + +K − ), members of SU(1, 1) the relation 1 + α 2 ≥ |α| makes it a compact operator [24], The numerical and theoretical results are in good agreement, Fig. 3(b), with differences of the order of 10 −15 . We used a lattice size of 200 waveguides in the numerical simulations, which is in the experimental range, and the light intensity at the last waveguide was of the order of 10 −17 for the example shown in Fig. 3.

SUSY partner for Jacobi lattices
By defining deformed, nonlinear, parametrized creation and anihilation operators, it is possible to factorize (1) into Now, it is straightforward to propose a SUSY partner [1] to our Jacobi lattice in the form: This leads to an array of waveguides described by the differential set and shown in Fig. 4. Here, we have to use an alternative representation for the elements of the SU(1, 1) group:K 0 =n + 1/2,K + = √nâ † andK − =â √n such that the equivalent partner Hamiltonian for the quantum optics analog is given bŷ Otherwise, the procedure to find the spectrum and normal modes is identical to that in the past section; we define a rotationR = e −ξ (K+−K−)/2 such that diagonalizes the quantum analog Hamiltonian,Ĥ Fig. 4. The effective refractive index, ω j (α), and coupling, g j (α), functions for the SUSY partner of our Jacobi lattices. where we have used tanh ξ = 2α/(1 + α 2 ) again. In this case the spectrum is identical, up to an extra first term, to that in (7), Ω (p) k (α) = 1 − α 2 k, k = 0, 1, 2, . . . .