* Landau and Lifshitz (2013) Lev Davidovich Landau and Evgenii Mikhailovich Lifshitz,  _Fluid Mechanics: Landau and Lifshitz: Course of Theoretical Physics, Volume 6_ , Vol. 6 (Elsevier, 2013).
  * Alder and Wainwright (1970) B. J. Alder and T. E. Wainwright, “Decay of the velocity autocorrelation function,” Phys. Rev. A 1, 18–21 (1970).
  * Jain and Kovtun (2022) Akash Jain and Pavel Kovtun, “Late Time Correlations in Hydrodynamics: Beyond Constitutive Relations,” Phys. Rev. Lett. 128, 071601 (2022), arXiv:2009.01356 [hep-th] .
  * de Boer  _et al._ (2019) Jan de Boer, Michal P. Heller, and Natalia Pinzani-Fokeeva, “Holographic Schwinger-Keldysh effective field theories,” JHEP 05, 188 (2019), arXiv:1812.06093 [hep-th] .
  * Glorioso  _et al._ (2018) Paolo Glorioso, Michael Crossley, and Hong Liu, “A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems,” (2018), arXiv:1812.08785 [hep-th] .
  * Ghosh  _et al._ (2021) Jewel K. Ghosh, R. Loganayagam, Siddharth G. Prabhu, Mukund Rangamani, Akhil Sivakumar, and V. Vishal, “Effective field theory of stochastic diffusion from gravity,” JHEP 05, 130 (2021), arXiv:2012.03999 [hep-th] .
  * Policastro  _et al._ (2002) Giuseppe Policastro, Dam T. Son, and Andrei O. Starinets, “From AdS / CFT correspondence to hydrodynamics,” JHEP 09, 043 (2002), arXiv:hep-th/0205052 .
  * Herzog and Son (2003) C. P. Herzog and D. T. Son, “Schwinger-Keldysh propagators from AdS/CFT correspondence,” JHEP 03, 046 (2003), arXiv:hep-th/0212072 .
  * Skenderis and van Rees (2009) Kostas Skenderis and Balt C. van Rees, “Real-time gauge/gravity duality: Prescription, Renormalization and Examples,” JHEP 05, 085 (2009), arXiv:0812.2909 [hep-th] .
  * Nickel and Son (2011) Dominik Nickel and Dam T. Son, “Deconstructing holographic liquids,” New J. Phys. 13, 075010 (2011), arXiv:1009.3094 [hep-th] .
  * Khemani  _et al._ (2018) Vedika Khemani, Ashvin Vishwanath, and D. A. Huse, “Operator spreading and the emergence of dissipation in unitary dynamics with conservation laws,” Phys. Rev. X 8, 031057 (2018), arXiv:1710.09835 [cond-mat.stat-mech] .
  * Chan  _et al._ (2018) Amos Chan, Andrea De Luca, and J. T. Chalker, “Solution of a minimal model for many-body quantum chaos,” Phys. Rev. X 8, 041019 (2018), arXiv:1712.06836 [cond-mat.stat-mech] .
  * Bauer  _et al._ (2017) Michel Bauer, Denis Bernard, and Tony Jin, “Stochastic dissipative quantum spin chains (i): Quantum fluctuating discrete hydrodynamics,” SciPost Physics 3, 033 (2017).
  * Claeys  _et al._ (2022) Pieter W. Claeys, Austen Lamacraft, and Jonah Herzog-Arbeitman, “Absence of Superdiffusion in Certain Random Spin Models,” Phys. Rev. Lett. 128, 246603 (2022), arXiv:2110.06951 [cond-mat.stat-mech] .
  * Forster  _et al._ (1977) Dieter Forster, David R. Nelson, and Michael J. Stephen, “Large-distance and long-time properties of a randomly stirred fluid,” Phys. Rev. A 16, 732–749 (1977).
  * Jain  _et al._ (2021) Akash Jain, Pavel Kovtun, Adam Ritz, and Ashish Shukla, “Hydrodynamic effective field theory and the analyticity of hydrostatic correlators,” JHEP 02, 200 (2021), arXiv:2011.03691 [hep-th] .
  * Delacretaz (2020) Luca V. Delacretaz, “Heavy Operators and Hydrodynamic Tails,” SciPost Phys. 9, 034 (2020), arXiv:2006.01139 [hep-th] .
  * Chaikin  _et al._ (1995) Paul M Chaikin, Tom C Lubensky, and Thomas A Witten,  _Principles of condensed matter physics_ , Vol. 10 (Cambridge university press Cambridge, 1995).
  * Son and Stephanov (2002) D. T. Son and Misha A. Stephanov, “Real time pion propagation in finite temperature QCD,” Phys. Rev. D 66, 076011 (2002), arXiv:hep-ph/0204226 .
  * Torabian and Yee (2009) Mahdi Torabian and Ho-Ung Yee, “Holographic nonlinear hydrodynamics from AdS/CFT with multiple/non-Abelian symmetries,” JHEP 08, 020 (2009), arXiv:0903.4894 [hep-th] .
  * Fernandez-Melgarejo  _et al._ (2017) Jose J. Fernandez-Melgarejo, Soo-Jong Rey, and Piotr Surówka, “A New Approach to Non-Abelian Hydrodynamics,” JHEP 02, 122 (2017), arXiv:1605.06080 [hep-th] .
  * Spohn (2012) Herbert Spohn,  _Large scale dynamics of interacting particles_ (Springer Science & Business Media, 2012).
  * Katz  _et al._ (1984) Sheldon Katz, Joel L Lebowitz, and Herbert Spohn, “Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors,” Journal of statistical physics 34, 497–537 (1984).
  * Hager  _et al._ (2001) J. S. Hager, J. Krug, V. Popkov, and G. M. Schütz, “Minimal current phase and universal boundary layers in driven diffusive systems,” Phys. Rev. E 63, 056110 (2001).
  * Baek  _et al._ (2017) Yongjoo Baek, Yariv Kafri, and Vivien Lecomte, “Dynamical symmetry breaking and phase transitions in driven diffusive systems,” Phys. Rev. Lett. 118, 030604 (2017), arXiv:1609.06732 [cond-mat.stat-mech] .
  * Gritsev and Polkovnikov (2017) Vladimir Gritsev and Anatoli Polkovnikov, “Integrable Floquet dynamics,” SciPost Phys. 2, 021 (2017).
  * Ljubotina  _et al._ (2019) Marko Ljubotina, Marko Žnidarič, and Toma ž Prosen, “Kardar-parisi-zhang physics in the quantum heisenberg magnet,” Phys. Rev. Lett. 122, 210602 (2019).
  * Ljubotina  _et al._ (2017) Marko Ljubotina, Marko Žnidarič, and Tomaž Prosen, “Spin diffusion from an inhomogeneous quench in an integrable system,” Nature communications 8, 16117 (2017).
  * Verstraete  _et al._ (2004) F. Verstraete, J. J. García-Ripoll, and J. I. Cirac, “Matrix product density operators: Simulation of finite-temperature and dissipative systems,” Phys. Rev. Lett. 93, 207204 (2004).
  * Vidal (2004) Guifré Vidal, “Efficient simulation of one-dimensional quantum many-body systems,” Phys. Rev. Lett. 93, 040502 (2004).
  * Fishman  _et al._ (2022) Matthew Fishman, Steven R. White, and E. Miles Stoudenmire, “The ITensor Software Library for Tensor Network Calculations,” SciPost Phys. Codebases , 4 (2022).
  * Abanin  _et al._ (2017) Dmitry Abanin, Wojciech De Roeck, Wen Wei Ho, and François Huveneers, “A rigorous theory of many-body prethermalization for periodically driven and closed quantum systems,” Communications in Mathematical Physics 354, 809–827 (2017).
  * De Nardis  _et al._ (2020) Jacopo De Nardis, Marko Medenjak, Christoph Karrasch, and Enej Ilievski, “Universality Classes of Spin Transport in One-Dimensional Isotropic Magnets: The Onset of Logarithmic Anomalies,” Phys. Rev. Lett. 124, 210605 (2020), arXiv:2001.06432 [cond-mat.stat-mech] .
  * Ljubotina  _et al._ (2023) Marko Ljubotina, Jean-Yves Desaules, Maksym Serbyn, and Zlatko Papić, “Superdiffusive Energy Transport in Kinetically Constrained Models,” Phys. Rev. X 13, 011033 (2023), arXiv:2210.01146 [cond-mat.stat-mech] .
  * Chen  _et al._ (2023) Chun Chen, Yan Chen, and Xiaoqun Wang, “Superdiffusive to ballistic transports in nonintegrable rydberg chains,” (2023), arXiv:2304.05553 [cond-mat.str-el] .
  * Wienand  _et al._ (2023) Julian F. Wienand, Simon Karch, Alexander Impertro, Christian Schweizer, Ewan McCulloch, Romain Vasseur, Sarang Gopalakrishnan, Monika Aidelsburger, and Immanuel Bloch, “Emergence of fluctuating hydrodynamics in chaotic quantum systems,” (2023), arXiv:2306.11457 [cond-mat.quant-gas] .
  * Richter and Pal (2022) Jonas Richter and Arijeet Pal, “Anomalous hydrodynamics in a class of scarred frustration-free Hamiltonians,” Phys. Rev. Res. 4, L012003 (2022), arXiv:2107.13612 [cond-mat.stat-mech] .
  * Gopalakrishnan  _et al._ (2022) Sarang Gopalakrishnan, Alan Morningstar, Romain Vasseur, and Vedika Khemani, “Distinct universality classes of diffusive transport from full counting statistics,” (2022), arXiv:2203.09526 [cond-mat.stat-mech] .
  * Takeuchi and Sano (2010) Kazumasa A. Takeuchi and Masaki Sano, “Universal fluctuations of growing interfaces: Evidence in turbulent liquid crystals,” Phys. Rev. Lett. 104, 230601 (2010).
  * Sasamoto and Spohn (2010) Tomohiro Sasamoto and Herbert Spohn, “One-dimensional kardar-parisi-zhang equation: An exact solution and its universality,” Phys. Rev. Lett. 104, 230602 (2010).
  * Ferrari and Frings (2011) Patrik L. Ferrari and René Frings, “Finite time corrections in kpz growth models,” Journal of Statistical Physics 144, 1123 (2011).
  * White  _et al._ (2018) Christopher David White, Michael Zaletel, Roger S. K. Mong, and Gil Refael, “Quantum dynamics of thermalizing systems,” Phys. Rev. B 97, 035127 (2018).
  * Parker  _et al._ (2019) Daniel E. Parker, Xiangyu Cao, Alexander Avdoshkin, Thomas Scaffidi, and Ehud Altman, “A universal operator growth hypothesis,” Phys. Rev. X 9, 041017 (2019).
  * Klein Kvorning  _et al._ (2022) Thomas Klein Kvorning, Loïc Herviou, and Jens H Bardarson, “Time-evolution of local information: thermalization dynamics of local observables,” SciPost Physics 13, 080 (2022).
  * Rakovszky  _et al._ (2022) Tibor Rakovszky, CW von Keyserlingk, and Frank Pollmann, “Dissipation-assisted operator evolution method for capturing hydrodynamic transport,” Physical Review B 105, 075131 (2022).
  * Kamenev (2023) Alex Kamenev,  _Field theory of non-equilibrium systems_ (Cambridge University Press, 2023).
  * Akyuz  _et al._ (2023) Can Onur Akyuz, Garrett Goon, and Riccardo Penco, “The Schwinger-Keldysh Coset Construction,” (2023), arXiv:2306.17232 [hep-th] .
  * Winer and Swingle (2022) Michael Winer and Brian Swingle, “Hydrodynamic Theory of the Connected Spectral form Factor,” Phys. Rev. X 12, 021009 (2022), arXiv:2012.01436 [cond-mat.stat-mech] .
  * Sogabe and Yin (2022) Noriyuki Sogabe and Yi Yin, “Off-equilibrium non-Gaussian fluctuations near the QCD critical point: an effective field theory perspective,” JHEP 03, 124 (2022), arXiv:2111.14667 [nucl-th] .
  * Abbasi  _et al._ (2022) Navid Abbasi, Matthias Kaminski, and Omid Tavakol, “Ultraviolet-regulated theory of non-linear diffusion,” (2022), arXiv:2212.11499 [hep-th] .
  * Chao and Schaefer (2023) Jingyi Chao and Thomas Schaefer, “N-particle irreducible actions for stochastic fluids,” (2023), arXiv:2302.00720 [hep-ph] .
  * Kovtun  _et al._ (2011) Pavel Kovtun, Guy D. Moore, and Paul Romatschke, “The stickiness of sound: An absolute lower limit on viscosity and the breakdown of second order relativistic hydrodynamics,” Phys. Rev. D 84, 025006 (2011), arXiv:1104.1586 [hep-ph] .
  * Gao  _et al._ (2020) Ping Gao, Paolo Glorioso, and Hong Liu, “Ghostbusters: Unitarity and Causality of Non-equilibrium Effective Field Theories,” JHEP 03, 040 (2020), arXiv:1803.10778 [hep-th] .
  * Zanardi (2001) Paolo Zanardi, “Entanglement of quantum evolutions,” Phys. Rev. A 63, 040304 (2001).
  * Hartmann  _et al._ (2009) Michael J. Hartmann, Javier Prior, Stephen R. Clark, and Martin B. Plenio, “Density matrix renormalization group in the heisenberg picture,” Phys. Rev. Lett. 102, 057202 (2009).
  * Žnidarič  _et al._ (2008) Marko Žnidarič, Toma ž Prosen, and Iztok Pižorn, “Complexity of thermal states in quantum spin chains,” Phys. Rev. A 78, 022103 (2008).
  * von Keyserlingk  _et al._ (2022) Curt von Keyserlingk, Frank Pollmann, and Tibor Rakovszky, “Operator backflow and the classical simulation of quantum transport,” Phys. Rev. B 105, 245101 (2022).


# Supersymmetric reshaping and higher-dimensional rearrangement of photonic
lattices

Tom A. W. Wolterink<EMAIL_ADDRESS>Institute for Physics,
University of Rostock, Albert-Einstein-Straße 23, 18059 Rostock, Germany
Matthias Heinrich  Institute for Physics, University of Rostock, Albert-
Einstein-Straße 23, 18059 Rostock, Germany  Alexander Szameit  Institute for
Physics, University of Rostock, Albert-Einstein-Straße 23, 18059 Rostock,
Germany

###### Abstract

Integrated $J_{\mathrm{x}}$ photonic lattices, inspired by the quantum
harmonic oscillator and due to their equidistant eigenvalue spectrum, have
been proven extremely useful for various applications, such as perfect imaging
and coherent transfer of quantum states. However, to date their large-scale
implementation remains challenging. We apply concepts from supersymmetry to
construct two-dimensional (2D) systems with spectra identical to that of one-
dimensional (1D) $J_{\mathrm{x}}$ lattices. While exhibiting different
dynamics, these 2D systems retain the key imaging and state transfer
properties of the 1D $J_{\mathrm{x}}$ lattice. Our method extends to other
systems with separable spectra, facilitates experimental fabrication, and may
increase robustness to fabrication imperfections in large-scale photonic
circuits.

Evolution dynamics in wave-mechanical systems are governed by the full set of
its modes and their respective eigenvalues. In particular, the key task of
imaging light between two specific planes can be understood in this framework
in ways that readily generalize from conventional free-space settings
involving mirrors and lenses to waveguide-based integrated optical systems
where crosstalk between the individual channels cannot be globally suppressed:
imaging occurs whenever the overall phases of the individual modes
simultaneously cancel except for integer multiples of $2\pi$. In discrete
settings with symmetric band structures this can be enforced by actively
swapping the excitations of mode pairs whose propagation constants are
arranged symmetrically with respect to the center of the band [1, 2], by
continuously sweeping the wave packet through the Brillouin zone in reciprocal
space by means of a transverse potential gradient (Bloch oscillations [3, 4]),
or, resonantly, via dynamic modulations [5, 6, 7]. In contrast to these active
methods, image recurrence can also occur naturally as a consequence of the
structure of the eigenvalue spectrum: regardless of the specific excitation,
any structure with a commensurable set of eigenvalues $\\{\lambda_{i}\\}$
features periodic propagation dynamics that yield imaging after a distance
corresponding to the least common multiple of the inverse eigenvalues
$\\{\lambda_{i}^{-1}\\}$. This mechanism is particularly effective if the
spectrum is equidistantly spaced similar to that of the harmonic oscillator.
In the context of finite-size discrete systems, the so-called $J_{\mathrm{x}}$
lattice fulfills this condition and allows for various operations of relevance
to classical and quantum photonic information processing, such as coherent
transfer of quantum states [8, 9, 10], discrete fractional Fourier transforms
[11], and the realization of saturable absorbers [12]. However, despite their
inherently finite size, implementing large-scale $J_{\mathrm{x}}$ arrays
remains experimentally challenging, as the underlying parabolic coupling
profile of an $N$-site $J_{\mathrm{x}}$ array necessitates the precise
realization of approximately $N/2$ different nearest-neighbor interaction
strengths across a substantial dynamic range of values. To overcome these
limitations, we leverage the concept of supersymmetric (SUSY) photonics [13]
to design families of compact two-dimensional systems that share the spectral
and imaging characteristics of $J_{\mathrm{x}}$ lattices while requiring
dramatically fewer distinct couplings.

In its original context of quantum field theory, supersymmetry was developed
to facilitate the theoretical description of bosons and fermions in a unified
mathematical framework [14]. Subsequently, these methods have been adapted to
nonrelativistic quantum mechanics [15] to systematically identify Hamiltonians
with identical sets of eigenvalues. Similarly, in photonics, SUSY techniques
enable the synthesis of globally phase-matched partner structures [13, 16] as
well as the parametric design of families of isospectral [17] and scattering-
equivalent refractive index landscapes [18]. Likewise, SUSY transformations of
non-Hermitian systems containing gain and loss [19] have been proposed for
applications in active systems such as laser arrays [20, 21, 22]. While the
factorization techniques typically employed in generating supersymmetric
partner structures tend to be restricted to one dimension (1D), separation-of-
variables methods allow them to be adapted to certain two-dimensional systems
(2D) [22], opening up a second spatial degree of freedom. Here, we present a
method to construct compact 2D systems that faithfully reproduce the signature
spectral and dynamic features of one-dimensional $J_{\mathrm{x}}$ arrays while
dramatically reducing the number of individual structural parameters. We
experimentally realize these systems in evanescently coupled waveguide arrays
and systematically investigate their imaging properties.

In conventional $J_{\mathrm{x}}$ arrays, the matrix elements of the quantum-
mechanical angular momentum operator’s $x$ component are mapped onto the
discrete Hamiltonian a one-dimensional array of $N$ sites [8] with a parabolic
distribution of nearest-neighbor coupling coefficients
$c_{n,n+1}=\frac{1}{2}\sqrt{n\left(N-n\right)}$ between sites $n$ and $n+1$.
The resulting eigenvalues form an equidistant ladder of spacing 1,
illustrating the close connection of this finite system to the continuous
quantum-mechanical harmonic oscillator.

In order to illustrate our approach for constructing two-dimensional
$J_{\mathrm{x}}$ equivalent systems, we consider the example of such a system
with $N=12$ (see Fig. 1). Applying a discrete SUSY transformation to the
Hamiltonian of the $J_{\mathrm{x}}$ array results in its first-order unbroken
superpartner, which retains the spectrum of the original array with the
exception of a single removed eigenvalue. In our example, the lowest
eigenvalue was removed via Cholesky factorization [13]. Note that, in contrast
to the QR factorization method employed to selectively remove states from of
$J_{\mathrm{x}}$ lattices in [23], the Cholesky factorization relates
$J_{\mathrm{x}}$ lattices with different numbers of sites, and, in our case,
the superpartner turns out to be a truncated 11-site $J_{\mathrm{x}}$ array
subject to a global detuning of $+\frac{1}{2}$. A single isolated site with
on-site potential $\beta$ separately hosts the mode corresponding to the
removed eigenvalue $-\frac{11}{2}$. Iteratively repeating this procedure on
the remaining array results in a sequence of higher-order superpartners and a
growing number of isolated detuned sites. Along these lines, the 8th
superpartner consists of a four-site $J_{\mathrm{x}}$ array, globally detuned
by $+4$, and eight individually detuned isolated sites.

Next, we reattach these sites by a series of inverse supersymmetry
transformations in the orthogonal $y$ direction, thereby constructing two-
dimensional arrays. The resulting Hamiltonian can be described as the
Kronecker sum $H^{(\mathrm{x})}\oplus H^{(\mathrm{y})}$ of independent
components in $x$ and $y$ directions, with $H^{(\mathrm{x})}$ in our case
corresponding to the Hamiltonian of the four-site $J_{\mathrm{x}}$ array
detuned by $+4$ in $x$ direction and $H^{(\mathrm{y})}=0$ representing the
single layer in $y$ direction. We apply inverse SUSY transformations to
$H^{(\mathrm{y})}$ while keeping $H^{(\mathrm{x})}$ constant. In a first step,
this reconnects the four sites of highest eigenvalue resulting in a two-layer
structure, with $H^{(\mathrm{y})}$ now representing a two-site
$J_{\mathrm{x}}$ array with coupling scaled by a factor of $4$ and detuned by
$-2$. Repeating this procedure to reconnect the remaining sites yields a
three-site $J_{\mathrm{x}}$ array with its coupling constants scaled by a
factor of $4$ and a detuning of $-4$ for $H^{(\mathrm{y})}$. The final array
of $4\times 3$ sites is a two-dimensional $J_{\mathrm{x}}$ array where
coupling in $y$ direction is increased fourfold compared to the $x$ direction,
while the opposite detuning in $H^{(\mathrm{x})}$ and $H^{(\mathrm{y})}$
cancels out. This system resembles an anisotropic two-dimensional quantum
harmonic oscillator. The newly constructed two-dimensional array shares all
eigenvalues of the original one-dimensional $J_{\mathrm{x}}$ array and can be
interpreted as its isospectral (broken) superpartner.

Figure 1: Transforming a one-dimensional $J_{\mathrm{x}}$ array of twelve
sites into a two-dimensional array through subsequent supersymmetry
transformations. Color/linewidth indicates the relative strength of the
coupling and detuning between sites.

Note that this procedure, although detailed here for the specific case of the
$J_{\mathrm{x}}$ array, holds for all systems with separable eigenvalue
spectra $\Lambda=\Lambda^{(1)}\oplus\Lambda^{(2)}\oplus\cdots$ and may be also
extended to higher dimensions. Moreover, by virtue of its equidistantly spaced
eigenvalues, a $J_{\mathrm{x}}$ array can be transformed into different two-
dimensional arrays, depending on the number $N$ of involved sites. In fact,
for a $J_{\mathrm{x}}$ array of $N=N^{(x)}N^{(y)}$ sites, an entire class of
isospectral two-dimensional arrays exists with Hamiltonians $H^{(x)}\oplus
N^{(x)}H^{(y)}$ (or, equivalently, $N^{(y)}H^{(x)}\oplus H^{(y)}$), each
representing an $N^{(x)}\times N^{(y)}$ two-dimensional $J_{\mathrm{x}}$ array
whose couplings in $y$ direction have been scaled the number of sites in the
$x$ direction.

From an experimental point of view, the challenge in constructing large-scale
$J_{\mathrm{x}}$ arrays lies in the requirement of precisely matching
approximately $N/2$ distinct coupling constants spanning a range of
$c\in\left(\frac{1}{2}\sqrt{N}\ldots\frac{1}{4}N\right)$ in the limit of
$N\rightarrow\infty$. The coupling values needed in the two-dimensional
systems similarly fall between
$c^{(x)}\in\left(\frac{1}{2}\sqrt{N^{(x)}}\ldots\frac{1}{4}N^{(x)}\right)$ in
$x$ direction and
$c^{(y)}\in\left(\frac{1}{2}N^{(x)}\sqrt{N^{(y)}}\ldots\frac{1}{4}N^{(x)}N^{(y)}\right)$
along $y$, respectively. While the highest coupling constant that occurs in
either system is the same, our 2D $J_{\mathrm{x}}$-equivalent systems offer
two advantages for implementation. First, instead of $N/2$, the nearest-
neighbor couplings only take $\left(N^{(x)}+N^{(y)}\right)/2$ distinct values,
a reduction of up to $2/\sqrt{N}$. Secondly, instead of more or less
homogeneously covering the entire range, the couplings of the 2D system occur
in a well-separated bimodal distribution with a set of strong coupling in one
of the spatial directions, and a set of weak coupling in the other. This
convenient grouping allows for a more precise calibration of fabrication
parameters around the respective mean values, thereby further simplifying the
implementation while also allowing for potential inherent anisotropies of the
experimental platform to be taken advantage of.

To illustrate the imaging dynamics in different isospectral systems, we
consider the 2D superpartners of a six-waveguide $J_{\mathrm{x}}$ lattice and
its Hamiltonian $H_{6}$. In line with the prime factor decomposition $6=2\cdot
3$, this structure can either be converted into a $3\times 2$ array described
by $H_{3}^{(\mathrm{x})}\oplus 3H_{2}^{(\mathrm{y})}$, or a $2\times 3$ array
with $H_{2}^{(\mathrm{x})}\oplus 2H_{3}^{(\mathrm{y})}$, respectively. As the
equidistantly spaced eigenvalue spectrum is invariant under our construction,
their individual dynamics can be transformed between them via projections onto
their respective eigenvectors. As a result, the imaging properties of the
$J_{\mathrm{x}}$ array carry over to its superpartners, which both display
perfect state transfer between opposing sites at integer multiples of the
propagation distance $z=\pi$. Figure 2 shows the numerically calculated
intermediate steps of the evolution dynamics resulting from an excitation of
the top-left waveguide of each of the systems. In the 2D superpartners, light
independently evolves in $x$ and $y$ directions. Due to the appropriate
scaling of the coupling, the evolution in $y$ is $N^{(\mathrm{x})}$ times
faster than evolution in $x$. Therefore, the input wave packet is mirrored in
both $x$ and $y$ directions for odd $N^{(\mathrm{x})}$, and for even
$N^{(\mathrm{x})}$ only in the $x$ direction.

Figure 2: Propagation dynamics in reshaped $J_{\mathrm{x}}$ lattices. In the
schematic drawing (left), color/linewidth indicates the relative strength of
the coupling between waveguides. Numerically calculated evolution (right) of
an excitation of the top-left waveguide in a conventional six-waveguide
$J_{\mathrm{x}}$ (top) as well as its 2D superpartners (center, bottom). In
all three systems, perfect state transfer occurs at integer multiples of a
normalized propagation distance of $z=\pi$.

We experimentally probe the imaging dynamics of these systems in arrays of
evanescently coupled waveguides fabricated by femtosecond-laser direct writing
technique [24]. The individual guides are inscribed by focusing 274 fs laser
pulses from an ultrafast fiber laser amplifier (Coherent Monaco) at a
repetition rate of 333 kHz and an average power of 61 mW at a wavelength of
512 nm into fused silica (Corning 7980) through a $50\times$ microscope
objective ($NA=0.6$). The glass sample is translated at a speed of 100 mm/min
by means of a high-precision stage (Aerotech ALS180). Non-bridging oxygen hole
centers formed during the inscription process allow for intensity dynamics of
guided light to be recorded through fluorescence microscopy when excited by a
helium-neon laser (633 nm). Inscribing the arrays in a slightly rotated frame
of reference allows us to simultaneously observe all waveguides even in 2D
lattices. In post-processing, the decrease of signal due to propagation losses
is compensated by normalizing the measured intensity to the total intensity in
the array at any given propagation distance.

In a first set of experiments, we implement the conventional six-waveguide
$J_{\mathrm{x}}$ array ($H_{6}$) and its two 2D superpartners ($H_{3}\oplus
3H_{2}$ and $H_{2}\oplus 2H_{3}$, respectively). The observed light evolution
for two distinct input waveguides in each of these systems is shown in Fig. 3.
In the conventional $J_{\mathrm{x}}$ array (Fig. 3(a)), light follows the
well-known transfer trajectories that coalesce to the respective single target
waveguide on the opposite side of the array at the imaging distance $z=\pi$.
As the evolution continues for larger $z$, light begins to flow in the
opposite direction during the second imaging period. The corresponding
measurements for the two-dimensional $H_{3}\oplus 3H_{2}$ superpartner array
are shown in Fig. 3(b). Here, the input distribution is mirrored in both $x$
and $y$ directions. Note that, as the stronger couplings are assigned to the
$y$ dimension of this structure, light here undergoes three transfers in $y$
to sync up with the single transfer in $x$ at $z=\pi$. In contrast, the
evolution along $x$ in the $2H_{3}\oplus H_{2}$ system (Fig. 3(c)) proceeds
twice as fast as in $y$, such that the input distribution is only mirrored in
the $y$ direction. Crucially, while exhibiting systematically different
dynamics, each of these configurations supports the desired perfect state
transfer enabled by their identical spectra. The measured dynamics are in good
agreement with a theoretical coupled-mode model, illustrating that fabrication
imperfections in both detuning and coupling coefficients are negligible. These
effects as well as the presence of a certain amount of diagonal coupling only
will begin to significantly impact the state transfer performance at larger
propagation distances.

Figure 3: Measured (left) and calculated (right) evolution dynamics of light
in a) a six-waveguide $J_{\mathrm{x}}$ array ($H_{6}$) and its two-dimensional
superpartner configurations b) $H_{3}\oplus 3H_{2}$ and c) $2H_{3}\oplus
H_{2}$. In each panel, the initially excited and imaging target waveguides are
marked by arrows. The imaging distance $z=\pi$ is indicated by semi-
transparent overlay (experiments) and dashed lines (simulations),
respectively.

To demonstrate the versatility of our approach in more extended systems, we
turn to a ten-waveguide $J_{\mathrm{x}}$ array ($H_{10}$) and its 2D
counterpart, a $5\times 2$ array described by $2H_{5}\oplus H_{2}$. As shown
in Fig. 4(a), the experimentally observed pattern deviates quite visibly from
the desired perfect state transfer, with only 53% of the light contained in
the target waveguide at the imaging distance $z=\pi$. This is due to the fact
that the wave packet has to sequentially traverse a substantial number of
links. In contrast, the 2D superpartner is substantially more compact, and
thus less susceptible to such perturbations, as can be readily seen for the
three exemplary excitations in Fig. 4(b), where the target waveguides contain
approximately 64%, 58% and 60% of the light at $z=\pi$.

Figure 4: Measured (left) and calculated (right) evolution of light in a) a
ten-waveguide $J_{\mathrm{x}}$ array ($H_{10}$) and b) its 2D superpartner
$2H_{5}\oplus H_{2}$. In each panel, the initially excited and imaging target
waveguides are marked by arrows. The imaging distance $z=\pi$ is indicated by
semi-transparent overlay (experiments) and dashed lines (simulations),
respectively.

In summary, we have presented a SUSY-based method to transform planar
$J_{\mathrm{x}}$ lattices into isospectral two-dimensional systems. The
viability of this approach was experimentally demonstrated by observing the
imaging dynamics of these structures in laser-written waveguide arrays. The 2D
partner arrays faithfully reproduce the hallmark features of the
$J_{\mathrm{x}}$ array, such as perfect state transfer and mirroring of input
field distributions, on a more compact footprint, while requiring a
systematically lower number of individual structural parameters to be matched
during inscription. As such, our method readily allows for increased
robustness to perturbations and fabrication inaccuracies and provides the
tools to maintain coherence during state transfer in large-scale photonic
circuits for applications in high-dimensional quantum gates [25]. Beyond the
context of $J_{\mathrm{x}}$-type lattices, our method readily translates to
any discrete system with a separable eigenvalue spectrum, and can be extended
to higher dimensions, e.g. by leveraging the polarization degree of freedom in
birefringent waveguides [26], for even higher packaging density.

###### Acknowledgements.

The authors thank M.-A. Miri for valuable discussions. We thank C. Otto for
preparing the high-quality fused-silica samples used in this work. This work
was funded by Deutsche Forschungsgemeinschaft via SFB 1477 “Light-Matter
Interactions at Interfaces”, project no. 441234705. T.A.W.W. is supported by a
European Commission Marie Skłodowska-Curie Actions Individual Fellowship
“Quantum correlations in PT-symmetric photonic integrated circuits”, project
no. 895254.

## References

  * Longhi [2008] S. Longhi, Image reconstruction in segmented waveguide arrays, Opt. Lett. 33, 473 (2008).
  * Szameit  _et al._ [2008] A. Szameit, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, E. Suran, F. Louradour, A. Barthélémy, and S. Longhi, Image reconstruction in segmented femtosecond laser-written waveguide arrays, Appl. Phys. Lett. 93, 181109 (2008).
  * Peschel  _et al._ [1998] U. Peschel, T. Pertsch, and F. Lederer, Optical bloch oscillations in waveguide arrays, Opt. Lett. 23, 1701 (1998).
  * Pertsch  _et al._ [1999] T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Optical bloch oscillations in temperature tuned waveguide arrays, Phys. Rev. Lett. 83, 4752 (1999).
  * Longhi [2005] S. Longhi, Self-imaging and modulational instability in an array of periodically curved waveguides, Opt. Lett. 30, 2137 (2005).
  * Longhi  _et al._ [2006] S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Observation of dynamic localization in periodically curved waveguide arrays, Phys. Rev. Lett. 96, 243901 (2006).
  * Szameit  _et al._ [2009] A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, Polychromatic dynamic localization in curved photonic lattices, Nat. Phys. 5, 271 (2009).
  * Christandl  _et al._ [2004] M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, Perfect state transfer in quantum spin networks, Phys. Rev. Lett. 92, 187902 (2004).
  * Perez-Leija  _et al._ [2013a] A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. M. Rodríguez-Lara, A. Szameit, and D. N. Christodoulides, Coherent quantum transport in photonic lattices, Phys. Rev. A 87, 012309 (2013a).
  * Perez-Leija  _et al._ [2013b] A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, Perfect transfer of path-entangled photons in ${J}_{x}$ photonic lattices, Phys. Rev. A 87, 022303 (2013b).
  * Weimann  _et al._ [2016] S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Gräfe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, Implementation of quantum and classical discrete fractional fourier transforms, Nat. Commun. 7, 11027 (2016).
  * Teimourpour  _et al._ [2017] M. H. Teimourpour, A. Rahman, K. Srinivasan, and R. El-Ganainy, Non-hermitian engineering of synthetic saturable absorbers for applications in photonics, Phys. Rev. Appl. 7, 014015 (2017).
  * Miri  _et al._ [2013a] M.-A. Miri, M. Heinrich, R. El-Ganainy, and D. N. Christodoulides, Supersymmetric optical structures, Phys. Rev. Lett. 110, 233902 (2013a).
  * Ramond [1971] P. Ramond, Dual theory for free fermions, Phys. Rev. D 3, 2415 (1971).
  * Cooper  _et al._ [1995] F. Cooper, A. A. Khare, and U. Sukhatme, Supersymmetry and quantum mechanics, Phys. Rep. 251, 267 (1995).
  * Heinrich  _et al._ [2014a] M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, and D. N. Christodoulides, Supersymmetric mode converters, Nat. Commun. 5, 3698 (2014a).
  * Miri  _et al._ [2014] M.-A. Miri, M. Heinrich, and D. N. Christodoulides, Susy-inspired one-dimensional transformation optics, Optica 1, 89 (2014).
  * Heinrich  _et al._ [2014b] M. Heinrich, M.-A. Miri, S. Stützer, S. Nolte, D. N. Christodoulides, and A. Szameit, Observation of supersymmetric scattering in photonic lattices, Opt. Lett. 39, 6130 (2014b).
  * Miri  _et al._ [2013b] M.-A. Miri, M. Heinrich, and D. N. Christodoulides, Supersymmetry-generated complex optical potentials with real spectra, Phys. Rev. A 87, 043819 (2013b).
  * Teimourpour  _et al._ [2016a] M. H. Teimourpour, L. Ge, D. N. Christodoulides, and R. El-Ganainy, Non-hermitian engineering of single mode two dimensional laser arrays, Sci. Rep. 6, 33253 (2016a).
  * Hokmabadi  _et al._ [2019] M. P. Hokmabadi, N. S. Nye, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, Supersymmetric laser arrays, Science 363, 623 (2019).
  * Qiao  _et al._ [2021] X. Qiao, B. Midya, Z. Gao, Z. Zhang, H. Zhao, T. Wu, J. Yim, R. Agarwal, N. M. Litchinitser, and L. Feng, Higher-dimensional supersymmetric microlaser arrays, Science 372, 403 (2021).
  * Teimourpour  _et al._ [2016b] M. H. Teimourpour, D. N. Christodoulides, and R. El-Ganainy, Optical revivals in nonuniform supersymmetric photonic arrays, Opt. Lett. 41, 372 (2016b).
  * Szameit and Nolte [2010] A. Szameit and S. Nolte, Discrete optics in femtosecond-laserwritten photonic structures, J. Phys. B 43, 163001 (2010).
  * [25] M. Ehrhardt, K. Tschernig, M. Heinrich, D. Guzmán-Silva, D. N. Christodoulides, K. Busch, A. Szameit, and A. Perez-Leija, Integrated photonic swap gates for qudits and qubit registers, Research Square: rs-1593687 .
  * Ehrhardt  _et al._ [2021] M. Ehrhardt, R. Keil, L. J. Maczewsky, C. Dittel, M. Heinrich, and A. Szameit, Exploring complex graphs using three-dimensional quantum walks of correlated photons, Sci. Adv. 7, eabc5266 (2021).