We apply the notion of discrete supersymmetry based on matrix factorization to quantum systems consisting of coupled bosonic oscillators to construct isospectral bosonic quantum networks. By using the algebra that arises due to the indistinguishability of bosonic particles, we write down the Schrödinger equations for these oscillators in the different boson-number sectors. By doing so, we obtain, for every partner quantum network, a system of coupled differential equations that can be emulated by classical light propagation in optical waveguide arrays. This mathematical scheme allows us to build quasi-two-dimensional optical arrays that are either isospectral or share only a subset of their spectrum after deliberately omitting some chosen eigenstates from the spectrum. As an example, we use this technique (which we call bosonic discrete supersymmetry or BD-SUSY) to design two optical, silica-based waveguide arrays consisting of six and three elements, respectively, with overlapping eigenspectrum.
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Supersymmetry (SUSY) was first proposed in the context of high energy physics to relate fermionic and bosonic particles [1–3]. Later, SUSY was introduced in quantum mechanics [4–6]. By using the mathematical analogy between the single particle Schrödinger and the electromagnetic wave equation, the SUSY notion was recently extended to the field of photonics [7,8] with possible applications in optical communications , building new metamaterials , optical circuits , Bragg gratings , and mode selection , to mention a few examples. From a technological perspective, implementing continuous supersymmetry requires precise control of the refractive index profile at a subwavelength scale. Additionally, it may also require refractive index values beyond those obtained by current photonic material systems. To overcome these difficulties, a discrete version of SUSY (D-SUSY) that relies on matrix factorization schemes can be used [9,14]. In this case, a waveguide (or resonator) array described by a matrix Hamiltonian obtained by using coupled mode formalism can be used to construct a supersymmetric partner array that exhibits the same spectrum or a chosen subset of it [9,14]. The advantage of D-SUSY is that the supersymmetric partner array can be implemented by using the standard photonic materials. The tuning parameters here are the width/height of the waveguides as well as their separations, which can be well-controlled during the fabrication processes. This technique, together with non-Hermitian engineering, was recently used to propose [15,16] and later demonstrate  a universal solution for a long-standing problem in laser engineering, namely to build on-chip phase-locked laser arrays [17,18]. In most of these works, however, the D-SUSY construction was one-dimensional. The reason is that the various factorization techniques (i.e. Cholesky or QR ) preserve the structure of the original matrix only if it is tridiagonal. Within the next-neighbor approximation, tridiagonal matrices describe one-dimensional arrays. On the other hand, if the array is two-dimensional, then it is described by a more complex matrix structure. In this case, matrix factorization techniques introduce new long-range couplings which are not easy to implement using planar photonic technology. To alleviate some of these difficulties, recent works proposed a different route to D-SUSY based on Householder algorithm [16,20], which in effect reduces the array dimensionality.
Here we introduce a new approach towards building two-dimensional supersymmetric photonic structures. Essentially, we apply the notion of D-SUSY not directly to the classical array but to a quantum version made of one-dimensional coupled bosonic oscillators. By applying D-SUSY to this quantum array, we obtain a supersymmetric quantum array that shares the same eigenvalues but different eigenvectors. By properly shifting the eigenvalues of the original array, one can also eliminate some chosen eigenvalues from the SUSY array. The next step is to use the Schrödinger equation in order to expand the associated Hamiltonians in the relevant boson-number sectors. By doing so, we obtain a system of coupled differential equations for every partner quantum network. These classical equations can be emulated by light propagation in optical waveguide arrays [21,22]. Interestingly, when an -boson sector is considered in a quantum array having more than two oscillators, then the resultant optical arrays are two-dimensional for . This powerful mathematical tool (which we call bosonic discrete SUSY, or BD-SUSY for short) thus opens the door for engineering a wide range of two-dimensional geometries with well-specified spectral relations — a task that is not possible using conventional D-SUSY method.
2. RESULTS AND DISCUSSION
In order to demonstrate the power of this new technique, we consider a system of one-dimensional three coupled bosonic oscillators (extension to larger systems is mathematically straightforward; though, it can lead to challenges in implementations):1) can be expressed using the Heisenberg equation of motions for the operators , (again, we used the distance instead of time to relate these equations directly to waveguide structures [21,22]):
Let us now assume that the bosonic networks described by are populated by bosons. They can be distributed according to (indicating bosons in site ) where . These states can be arranged in any order and expressed in terms of one index, say , where . The general wavefunction can be thus written as . Similarly, for , we obtain . By substituting and back in and projecting on the states and , respectively, we obtain two different sets of coupled ordinary differential equations (ODEs): one for the coefficients and another for . Each of these coupled ODEs can be emulated by a classical waveguide array.
Particularly, for concreteness, we will consider an example with . Additionally, we will use this example to illustrate how one can build a partner network after eliminating some of the eigenvalues. To do so, we make the choice . As a result, we find that1 shows the corresponding classical networks of these quantum oscillators when populated with two bosons. In this case, the main and partner arrays have eigenvalues that correspond to their optical super modes given by and , respectively. In other words, the partner array shares all the eigenvalues of the main array except . Importantly, by inspecting Eq. (7), we note that the waveguides in the main and partner arrays are identical, which simplify the implementation significantly.
Figure 2 plots a realistic implementation of these networks in silica-glass-based waveguide systems. The design parameters along with the resultant coupling coefficients are all listed in the figure caption.
Figure 3 depicts the eigenmodes and eigenvalues associated with the photonic networks in Fig. 2 as obtained by full-wave finite element simulations. These results clearly illustrate the partial spectral overlap as predicted exactly by the discrete model.
Finally, we also study the evolutionary dynamics of light in these waveguide arrays when a small coupling () is introduced between the rightmost waveguide of the main array and the leftmost channel of the partner array. Figures 4(a)–4(e) plot the optical intensity profiles at different propagation cross sections when only mode ① of the main array is excited (i.e., there is zero input in the partner array). Clearly, at a distance that corresponds to a full period (which in the current design corresponds to in units of meter), the total power is transferred to mode ① of the partner array with an efficiency as can be seen from Fig. 4(f). On the other hand, when mode ② of the main array is excited, we observe almost no power transfer to the partner array, as expected to due to the phase mismatch (see Fig. 3). Importantly, we note that similar designs with scaled parameters can be implemented using shorter structures  or even in resonators where propagation distance is replaced by time.
In conclusion, to the best of our knowledge, we have introduced a new approach for engineering two-dimensional optical arrays that exhibit complete or partial spectral overlap. Our method relies on the bosonic algebra of coupled quantum oscillators, essentially applying the D-SUSY to the corresponding Heisenberg equation of motions followed by expanding the relevant Hamiltonians in their Fock space. The resultant ODEs can be then emulated by classical waveguide arrays. When considering a certain boson sector that has more than one photon, this technique gives rise to optical arrays that have two-dimensional connectivities (i.e., are not represented by tridiagonal matrices). We have demonstrated our technique by designing two different arrays having six and three waveguides, with the spectrum of the latter being a subset of that of the former. We have also presented an implementation for these structures based on waveguide arrays in glass platforms. Our full-wave simulations using the finite element method and coupled mode theory confirm our theoretical predictions. Finally, we emphasize that our technique can in principle be applied to any 1D network structure in order to obtain a quasi 2D SUSY partner. However, implementing more complex arrangements using optical platforms maybe challenging, mainly due to the geometric restrictions imposed by the waveguides (or resonators). In this regard, a platform that can be of potential interest is superconducting circuits which was shown recently to provide more degrees of freedom for implementing hyperbolic lattices . We will investigate how D-SUSY can benefit from this promising platform in future work.
Army Research Office (W911NF-17-1-0481, W911NF-16-1-0013); Air Force Office of Scientific Research (FA9550-14-1-0037); Office of Naval Research (N0001416-1-2640, N00014-18-1-2347, N00014-19-1-2052); National Science Foundation (DMR-1420620, ECCS 1454531, ECCS 1757025, ECCS 1807552); Defense Advanced Research Projects Agency (D18AP00058, HR00111820038, HR00111820042); United States-Israel Binational Science Foundation (2016381).
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