Dissipatively coupled waveguide networks for coherent diffusive photonics

A photonic circuit is generally described as a structure in which light propagates by unitary exchange and transfers reversibly between channels. In contrast, the term ‘diffusive’ is more akin to a chaotic propagation in scattering media, where light is driven out of coherence towards a thermal mixture. Based on the dynamics of open quantum systems, the combination of these two opposites can result in novel techniques for coherent light control. The crucial feature of these photonic structures is dissipative coupling between modes, via an interaction with a common reservoir. Here, we demonstrate experimentally that such systems can perform optical equalisation to smooth multimode light, or act as a distributor, guiding it into selected channels. Quantum thermodynamically, these systems can act as catalytic coherent reservoirs by performing perfect non-Landauer erasure. For lattice structures, localised stationary states can be supported in the continuum, similar to compacton-like states in conventional flat-band lattices.


Supplementary Note 1: Experimental model
The coherent diffusive photonic circuits, considered in the main text, are described by the following generic quantum master equation: where ρ(t) is the density matrix and A j denote the Lindblad operators for mode j. An interesting feature of the behaviour of light in these devices is the interchangeability between propagation distance and wavelength. The effective time of evolution γt can be altered both by changing the length of the waveguide block, or the wavelength of incident light. As the wavelength is tuned, κ 1 changes almost linearly to keep κ 1 /κ 2 ≈ 0.5, maintaining the correct character of dynamics. Notice that the dependence of diffusion rates, γ j , on time changes neither the diffusive character of the dynamics nor the asymptotic state provided that always γ j (t) > 0.

Supplementary Note 2: Measurement of evanescent coupling
As mentioned in the main text, the control of evanescent coupling is crucial for the experimental realisation of the diffusive equaliser. In Supplementary Fig. 1 we present the measured variation of κ 1,2 as a function of the wavelength of incident light, λ. We fabricated two types of directional couplers (each consisting of two evanescently coupled straight waveguides) which are the building blocks of the photonic circuits shown in Fig. 1 (main text). For the first type, where the two waveguides are at 45 • angle, the coupling constant is κ 1 and that for the second type (consisting of two horizontally separated waveguides) is κ 2 . Measuring the light intensities at the output of these 30-mm-long directional couplers, κ 1,2 (λ) were calculated 2 .
It was observed that for these couplers, the ratio of κ 1,2 remains very close to the desired value of 0.5 with a maximum deviation of ≈ ±0.05. Consider the Glauber P -function for the density matrix, ρ(t), of the state describing the circuit:

Supplementary
where | α = j |α j ; |α j is the coherent state of the j-th mode of the circuit and the amplitude α j represents the j-th elements of the vector α. For the DCC with N + 1 modes and jth Lindblad operator represented as A j = a j − a j+1 , the solution for the P -function is obtained from the following Fokker-Planck equation: Due to the linearity of this equation, the solution can be represented as where dynamics of amplitudes is described by Eq. (2) of the main text. It is instructive to represent the initial state in terms of discrete superposition of coherent state projectors 3 : where amplitudes α jk (t) for the DCC are defined from Eq. (2) of the main text.

As follows from Supplementary Eq. (1), any density matrix which is function of operators
sum , and the vacuum, ρ vac = ∀j |0 0| j , corresponds to a stationary state. These states can be of a quite different nature. The stationary state can be just the pure product of coherent states of individual modes with the same amplitude: However, it can also be quite exotic, for example, it can be a Schrödinger-cat entangled state with where K is the number of different components in our cat-state and w k are scalar weights. The Gibbs state also belongs to the stationary states of the system. This state has maximal entropy for the given sum of the second-order coherences, a † k a l (which is also conserved by the dynamics). As was already mentioned, the stationary state can also be maximally entangled. The α jk . Actually, the DCC drives the initial state to the symmetrical state over all the modes. Note, that the smoothing action of DCC is preserved even for the case of different decay rates, γ j = 0. Stationary states do not depend on them.
Supplementary Note 4: Two-arm distributor structure The prerequisite of the distributing action considered here is the existence of several localised stationary states of the structure described by the master equation, Supplementary Eq. (1). For the sake of simplicity, we consider here pure stationary states. We call the state |χ loc "localised" if exists some subset, {m}, of M < K systems of our dissipatively coupled photonic circuit such that k∈{m} χ loc |a † k a k |χ loc > 0, whereas for systems out of the subset {m} we have These equations describe 1D classical random walk. So, stationary states for arms decoupled from the central node would be vectors with equal elements, α j = α for j = 1 . . . N or j = N +1 . . . 2N and arbitrary α. Also, there is a stationary state localised in two controlling modes, a R and a L , with α R = α L and α j = 0, j = 1 . . . 2N . Obviously, for the whole structure, the equal distribution of amplitudes in both arms α j = α for j = 1 . . . N and j = N + 1 . . . 2N , and equal amplitudes in the controlling modes, α R = α L is also the stationary state. Excitation of just one arm and one of the controlling modes with equal amplitudes (i.e., for example, α j = α for j = 1 . . . N , α R = α and α R = 0, α j = 0 for j = N + 1 . . . 2N ) is also a stationary state. By exciting control modes, a R and a L in certain states, one can make an initial excitation of a particular mode propagate either to the one arm, or to another, or to both arms simultaneously (see Fig. 5 in the main text). Notice, the such a distributing action can be achieved catalytically, since, as it follows from Supplementary Eq. (12), the coherence of two controlling modes are conserved, for any time-moment, t. In Fig. 5b, one can see an illustration of the distribution for the two-arm structure shown in Fig. 5a.

Supplementary Note 5: Double chain and dissipative localisation
For the sake of generalisation, now we consider two parallel dissipatively coupled chains as shown in Supplementary Fig. 2. The chain consists of squares, connected side by sides, so, the Lindblad operator of j-th square is We obtain the following set of equations for the coherent amplitudes: where the matrixÔ has elementsÔ j,k = (−1) j+k , j, k = 1, 2. The vector α j = [α j,+ , α j However, they do affect the de-localised stationary states driving them to the vacuum.
Such localisation phenomena can hold also for infinite perfectly periodic dissipatively coupled photonic lattices. Let us assume Lindblad operators of the following form where {n j } denotes a set of modes coupled to the same dissipative reservoir; x jk are scalar weights describing such a coupling. To avoid trivial localised states, we assume that there are no isolated sets, and for any {n j } there is a set {n l } such that the intersection, {n j } ∩ {n l }, j = l, is not empty, but unequal to any of {n j }. Additionally, for the ideally periodic structures, we assume that any operator, a k , belongs to at least two different sets, and any set transforms to other set by translation along lattice vectors, e i . Obviously, for any localised stationary state we have A j ρ loc = 0 ∀j.
From Supplementary Eq. (15) it follows that any localised state occupies at least two sites of the structure. An example of the honeycomb lattice allowing for dissipative localisation is shown in Supplementary Fig. 3 and briefly discussed in the main text.
To demonstrate basic features of dissipative localisation, let us consider here a simple example of a square lattice (see insets in Supplementary Fig. 4). Denoting the sites in the upper left corner of each square as (j, k), we obtain the following Lindblad operators for such a lattice: The equations for the amplitudes, Supplementary Eqs. (1,16), then read: As can be seen from Supplementary Eqs. (17, 18), the minimal localised states for an infinite square lattice of Supplementary Fig. 4 involve at least four sites (for example, the localised state can be in the set {m} = {(j + 1, k), (j + 2, k), (j + 1, k + 1), (j + 2, k + 1)}). An example of the localised state composed of coherent states is Any closed contour including either 0, 2 or 4 sites of every square can host a localised state. A finite lattice can also support localised edge states with even, as well as odd, number of sites.
where the vector of time-dependent modal amplitudes is a(t) = [α 1,1 (t), α 1,2 (t), α 2,1 (t), Similar heat-like propagation of coherences was found recently in dissipatively coupled 1D spin chains 1 . An illustration of the stationary distribution arising from the initial excitation of just one mode is given in Supplementary Fig. 4a. In Supplementary Fig. 4b,  Naturally, the localised stationary state can be entangled. The simplest example of the entangled states for the minimal localised states of the infinite square lattice of Supplementary Fig. 4a up to the normalization factor is |Ψ loc = |α j+1,k | − α j+2,k |α j+2,k+1 | − α j+1,k+1 + | − α j+1,k |α j+2,k | − α j+2,k+1 |α j+1,k+1 which for |α| > 0 is entangled since an averaging over any mode included in this equation gives a mixed state. Up to the normalization factor, the reduced state of any three modes is given by