1. Introduction

During the past few decades, localization of light has drawn considerable attention. Anderson localization, which was originally formulated for electrons in disordered lattices [1], now being theoretically and experimentally demonstrated for light in various photonic systems: in disordered two-dimensional photonic lattices [2], in synthetic photonic lattices [3] and even in biological nanostructures of native silk [4].

In this work light localization takes place in synthetic photonic lattice (SPL) - fiber-based optical system, consisting of two loops with different lengths connected via a tunable directional coupler (Fig. 1(a)). Due to lengths difference, a pulse inserted in one of the loops starts to multiply, creating pulse chains in both loops, which are interfering on each roundtrip and thus creating complex evolution picture. Chain pulse evolution in the SPL is equivalent to pulse evolution in mesh lattices (Fig. 1(b)) [5]. Synthetic photonic lattices were employed to demonstrate a number of fundamental effects including discrete quantum walks [6], Bloch oscillations [7], Parity-Time (PT) symmetry breaking [5,8], discrete solitons [9], optical diametric drive acceleration through action-reaction symmetry breaking [10], defect states in PT-symmetric environment [11], and Talbot effect [12].

 

Fig. 1 a) Synthetic photonic lattice b) Schematic representation of pulse evolution in mesh lattice, equivalent to the SPL. Both lattices are equivalent in terms of pulse evolution after applying the appropriate transformations [5].

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SPL could also be considered a rich platform for observing Anderson localization. Indeed, if one compares SPL to traditional tight binding (TB) models where the subject of Anderson localization was historically introduced [1] and observed [13] (see [2]) for the continuous case), one can see that SPL allows one to realize disorder of two different types. First type is a phase disorder which is equivalent to the diagonal disorder of the tight binding model [1]. Another type of the disorder is the static random coupling analogues of the off-diagonal disorder [14,15]. The case of phase disorder in SPL has been considered previously [3,16]. In the present paper we consider the case of the off-diagonal disorder implemented via static random coupling.

While both cases share some similarities there are also significant differences both in the spectral properties and the photon statistics. For example the off-diagonal disorder in TB model leads to slower than exponential decay of the eigenstates in the middle of the spectral band (zero energy) corresponding to the vanishing Lyapunov exponent (see [14]). This anomaly persists also in the discrete time quantum walks (DTQW) [17] which are related to the fiber-coupled SPL considered in the present paper. Another significant difference obtained here for the SPL is that in the case of the strong static coupling disorder the localization length is extremely sensitive to the position inside the quasi-energy band while for the phase modulated model it can be shown to be insensitive to the spectral parameter [16] (in the limit of strong disorder).

It must also be mentioned that while the spectral properties of the disordered photonic systems are generally well understood (e.g. an exponential localization of output state around the input point is expected dynamically after ensemble averaging) the nature of disorder can have a significant impact on the higher order photon statistics. For instance in recent works [18–20] it was shown both theoretically and experimentally that for a TB model (and its continuous counterpart) only off-diagonal disorder can lead to super-thermal field statistics due to the specific symmetries presented by such systems. Having a model for the off-diagonal disorder is therefore of importance for studying the statistics of DTQW and SPL systems. It should also be mentioned that SPLs additionally provide a unique opportunity for studying position periodic evolution of the pulses (ring topologies) which in the TB case have produced some interesting results on topological influence on photon statistics [21] and provides an experimental basis for studying thermalization gap [22] which can also be exploited in the SPL settings.

The localization properties of a coupling-disorder of the DTQW were studied previously by Obuse and Kawakami [17] using a different parametrization of the coin operator and an artificial topological boundary supporting an edge state. Later Rakovsky and Asboth [23] studied Anderson localization for a different implementation of the DTQW (a so-called split-step model parameterized by 2 consecutive coin and shift operations) and another set of recent result was also presented in [24].

In the present paper we obtain a new analytical result for the inverse localization length in the case of weak disorder which is valid everywhere apart from an exponentially small region around the band center where this quantity is known to vanish. We studied the dynamical properties of the wavepacket spreading in such disordered lattices, and demonstrated numerically that despite the delocalization transition occurring at the band centre the participation ratio (i.e. the effective size of the spreading packet) saturates very quickly even for relatively small levels of disorder.

2. The model

One of the uses of the SPLs is that they represent one of the most easily accessible implementations of a 1D discrete time quantum walk (DTQW [25,26]). Indeed these systems possess all the main ingredients of a DTQW: i) a random walker with an internal degree of freedom (pseudospin or coin state taking values ↑ and ↓) traversing a set of discrete sites n, ii) a position shift operator Ŝ that transposes the walkers position to the right or to the left depending on the coin state and, iii) a unitary coin (or rotation) operator R̂(θ_n) that mixes the right- and left-moving components of the walker at position n.

The actual representation of the position and coin operator vary and depend on the specific implementation chosen. In our setup we follow [6] and the two states correspond to pulse components in the upper ↑ and lower ↓ loops, respectively, the position shift is implemented by having the loops of unequal lengths L_long and L_short with the field sampled at $N=\frac{1}{2}\left({\text{L}}_{\text{long}}+{\text{L}}_{\text{short}}\right)/\left({\text{L}}_{\text{long}}-{\text{L}}_{\text{short}}\right)$ discrete points inside each loop.

As in every example of DTQW our photonic implementation provides access only to stroboscopic observations of the amplitudes of the system after an integer number of roundtrips. It means that after m roundtrips the system is in a state $|\mathrm{\Psi }(m)〉={\sum }_n{u}_n^m|n,\uparrow 〉+v_n^m|n,\downarrow 〉$, and its state after the next round trip is obtained by applying a unitary step operator: |Ψ(m + 1)〉 = Û_SPL |Ψ(m)〉 = Ŝ_SPLR̂_SPL |Ψ(m)〉 where in the basis |↑〉, |↓〉 both operator have the form:

In the chosen basis |n, ↑↓〉 this leads to the following explicit map:

The spectrum of the SPL without modulation of the coupling angles (i.e. when θ_n = θ = const) is well known [5,8]. The eigenmodes are a set of N periodic complex exponentials exp(ik_jn) with k_j spanning the first Brillouin zone [−π, π] and the spectrum is given implicitly by the relation cos β = cos θ cos k_j and contains two symmetric Floquet bands separated by a gap (unless the coupling is absent, θ = 0). Such a gap is a salient feature of almost all DTQW systems.

There are two principle ways in which one can randomly perturb the SPL without breaking the unitary symmetry of the DTQW. One is by introducing random static phase disorder. Anderson localization and the spectral properties in this setup were studied experimentally, numerically and analytically in [3, 6, 16]. In particular it was shown that all eigenmodes in this system are localized (even for an arbitrary small level of phase disorder) and as the level of disorder increases the spectral gap between the two branches closes completely.

In the current paper we study another scenario - the one where the disorder is in the coupling angles. We chose the value θ = π/4 corresponding to 50:50 power split in the coupler as the central value of random distribution of θ_n. The coupling angles were then pseudorandomly generated as θ_n = π/4±rand(Δθ_max), where function rand(x) gives random number from 0 to x and Δθ_max is a maximum disorder level that varies from 0 to π/4.

3. Results

3.1. The spectral and localization properties of the disordered SPL

We start by studying the spectrum of the disordered unitary operator Û_SPL(θ⃗). A lot of useful information can be gained by looking into symmetries of this operator as well as the corresponding effective Hamiltonian Û_SPL = exp(iĤ_eff) [17,23,27,28]. Note that the definition of each symmetry operator depends on the specific protocol used and many implementations exist for the DTQW (mostly corresponding to the different gauge for the rotation operator R̂_SPL). In the chosen photonic implementation (Eq. (3)) the system has particle-hole symmetry reflected here in an antiunitary operator P̂ = σ_zK̂, where K̂ is complex conjugation so that P̂Û_SPLP̂⁻¹ = Û_SPL. This symmetry survives in the disordered case and as its consequence if λ = exp(iβ) is an eigenvalue corresponding to the eigenmode (u_n, v_n), then λ^* = exp(−iβ) is also an eigenvalue corresponding to the eigenmode $\left(u_n^{*},-v_n^{*}\right)$ from the opposite band. The corresponding operator P̂ maps the two Floquet-bands on each other hence the particle-hole analogy with the condensed matter physics (see also [27]).

The next symmetry is a chiral one ${\widehat{\mathrm{\Gamma }}}^{-1}{\widehat{\text{U}}}_{\text{SPL}}\widehat{\mathrm{\Gamma }}={\widehat{\text{U}}}_{\text{SPL}}^{†}$ with the unitary operator $\widehat{\mathrm{\Gamma }}=-i{\sum }_n\left(\text{cos}{\theta }_n{\sigma }_y+\text{sin}{\theta }_n{\sigma }_z\right)\otimes |n〉〈n|$. Lets derive the chiral symmetry of the disordered SPL similarly to the disorder-free case as outlined in [27]. For simplicity we begin with the case where all coupling angles are frozen (θ_n = θ). Such an SPL is characterized by an evolutionary operator: Û_SPL(θ) = ŜR̂(θ) where R̂(θ) = exp(iσ_xπ/2). The effective Hamiltonian is:

This prompts seeking a possible candidate for the chiral symmetry as built on a liner combination of the individual rotation operators corresponding to an “isolated Bloch sphere” for each site n. Such an operator will have the following form:

Again due to the different gauge for the SPL rotation operator the explicit form for the chiral generator Γ̂ differs from that reported in [23,27]. This operator maps an eigenmode (u_n, v_n) with λ = exp(iβ) to an eigenmode (−i sin θ_n u_n − cos θ_n v_n, cos θ_n u_n + i sin θ_nv_n) with the eigenvalue λ^*. Note that such a pairing of eigenmodes as imposed by disorder-immune chiral symmetry is known to have interesting consequences in unipatrite photonic lattices where it leads to thermalization gap and has an impact on photon-statistics and its control [18].

The final symmetry is bipatrite one [17, 23]. Since the walker changes the parity of site n in a single act of hopping, flipping the sign of every odd pair of lattice amplitudes (u_n, v_n), while leaving the even amplitudes unchanged, it corresponds to the case with a mirror flipped eigenvalue −λ eigenmodes, which corresponds to the symmetry ${\widehat{\mathrm{\Lambda }}}^{-1}{\widehat{\text{U}}}_{\text{SPL}}\widehat{\mathrm{\Lambda }}=-{\widehat{\text{U}}}_{\text{SPL}}$ with the corresponding flipping operator being $\widehat{\mathrm{\Lambda }}={\sum }_{n\in \mathit{even}}|n〉〈n|-{\sum }_{n\in odd}|n〉〈n|$.

Therefore the spectrum is symmetric respective to the axis reflections and as pointed out in [17] the middle of each band β = ±π/2 is a special point where delocalization occurs.

Now let us concentrate on the localization properties of the eigenmodes. Due to the symmetry of the model, described by Eq. (3), both components of a given eigenmode, u_n and v_n, should have qualitatively similar behaviour. To demonstrate this we follow the recipe from our previous work and eliminate one of the components from the system of coupled equations representing an eigenvalue problem Û_SPL |Ψ〉 = λ |Ψ〉. As the result we obtain two decoupled recursions (neglecting the boundary values since we are mostly interested in the asymptotic limit of N → ∞):

In Fig. 2(a) we perform direct iteration of the Riccati map (Eq. (11)) using pseudorandom initial condition and considering N = 10⁵ iterations (with the first 10% discarded to remove the memory effects) to obtain the Lyapunov exponent as a function of both pseudoenergy and disorder. At the same time in Fig. 2(b) we also provide the results for the normalized density of states $\rho (\beta )=N^{-1}{\sum }_j\delta (\beta -{\beta }_j)$. One can clearly see that in the case of strong disorder (Δθ_max = π/4) the eigenmodes near the center of the band β = π/2 are least localized (since l_loc ∝ 1/γ(β)) and also the most numerous (Fig. 2(b)). This is attributed in [17] to the special nature of the band centers β = ±π/2, where due to the overlapping of chiral and reflection symmetries the Lyapunov exponent vanishes as inverse log: 1/ln |β ∓ π/2| and the localization length in turn diverges logarithmically. Note also that the width of this zero dip is also exponentially small, which seems to be confirmed by Fig. 2(a). This is in stark contrast to the case of strong phase disorder considered in [3,16] where the chiral symmetry is broken and the Lyapunov exponent and the localization length were shown to be flat in within the spectrum.

 

Fig. 2 (a) The Lyapunov exponent obtained by iterating the Riccati map (Eq. (11)) with Δθ_max from 0 to π/4. (b) Density of states (DOS) obtained for Δθ_max from 0 to π/4.

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It is also possible to obtain analytical expression for the Lyapunov exponent in the case of weak disorder Δθ_max ≪ 1, when the pseudo-energy is not too close to the band centre. To do so it is more convenient to work with the (v) recursion (Eq. (9)). Introducing an auxiliary iteration variable x_n = v_n tan θ_n−1/v_n−1 we obtain the following recursion:

Next as a formal small parameter we introduce a standard deviation of the coupling angle $\epsilon =\sqrt{〈\mathrm{\Delta }{\theta }_n^2〉}$ (for the uniform distribution $\epsilon =\mathrm{\Delta }{\theta }_{\mathit{max}}/\sqrt{3}$) and use a standard procedure of expanding recursion in Eq. (12) up to the second order terms in ε (see e.g. [16,30]). We seek our solution of Eq. (12) in the form $x_n=A\text{exp}\left[\epsilon B_n+{\epsilon }^2{C}_n+\dots \right]\approx A+A{B}_n\epsilon +A\left(C_n+B_n^2/2\right){\epsilon }^2+.\hspace{0.17em}.\hspace{0.17em}.$. Substituting this ansatz into both sides of Eq. (12), introducing normalized r.v. V_n having a symmetric distribution with unit variance so that θ_n = π/4 + εV_n and expanding up to the second order in ε we obtain in each order:

Next from the first order terms it follows that in the limit n → ∞, when the stationary distribution is reached, one has

Then, after averaging the equation for the ε² terms, we obtain

 

Fig. 3 The normalized Lyapunov exponent obtained for the DTQW with Δθ_max = 0.1. Comparison of theoretical result (Eq. (20)) with the full numerics.

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3.2. Anderson localization

In the previous sections we have studied the spectral properties of the stroboscopic evolution operator Û_SPL and the exponential localization of its eigenmodes. In this section we turn to the classical formulation of Anderson localization [1], namely the arrest of spreading of initially localized pulse by disorder. To this end we simulated numerically the statistics of propagation of a single pulse, initially inserted into the short SPL at the prescribed temporal location n₀. As discussed in Section 2 in our model random coupling coefficients θ_n do not change between the roundtrips which corresponds to static (frozen) disorder.

Such a controllable modulation may be potentially realized in an experiment using commercially available high-speed switches. As a separation between adjacent temporal slots of a synthetic photonic lattice is typically of hundreds of nanosecond [3], lithium niobate integrated Mach-Zehnder interferometers with switching of much less than 100 ns or even sub-nanosecond switching [32] could suit the purpose. Together with a fact that in typical experiment one can observe up to 300 consecutive round-trips [33], the experimental observation of discussed effects of Anderson localization in SPL could therefore be achieved under reasonable experimental efforts.

The numerical simulation of the pulse evolution was performed by iterating the map of Eq. (3) with N = 1000 time positions (synthetic mesh points) subject to periodic boundary conditions. The results were averaged over an ensemble of N_avg = 10⁴ realizations of disorder and demonstrated the arrest of pulse spreading signalling Anderson localization (Fig. 4). The initial pulse was localized in a single time slot n₀ = 500 of the short loop. The finite size of mesh lattice in the numerical model together with periodic boundary conditions may become important at later stage of pulse evolution. It can influence dynamical properties of DTQW (similar to e.g. the ones considered by recent work [22]), including quantum revivals, pulse overlap, interference effects, etc. However in the present paper we concentrate on the limit of large model where the mesh size remains larger than the typical size of the wavepacket at all stages of the evolution so that the influence of the boundary conditions is minimal.

 

Fig. 4 Anderson localization in the SPL due to the coupling coefficient random distribution with a) Δθ_max = 0.1π, b) Δθ_max = 0.15π and c) Δθ_max = 0.25π.

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While theoretical analysis of the eigenmode structure above relies on the Lyapunov exponent in numerical simulations it is more customary to work with the participation ratio (and its inverse), defined as $P(m)={\left({\sum }_n{\left|v_n^m\right|}^2\right)}^2/{\sum }_n{\left|v_n^m\right|}^4$, to quantify the strength of the localization. For the isolated eigenmode the inverse participation ratio P⁻¹ is often used as a proxy for the Lyapunov exponent γ(β) and vice versa. However it should be noted that in general P⁻¹ ≳ γ which is due to occasional disorder fluctuations that lead to modes of large diameter ∼ P that can be greater than the scale of exponential decay of the tails, given by l = γ(β)⁻¹ [34].

We have simulated the dynamical participation ratio (also averaged over N_avg = 10⁴ realizations of disorder) and the results are depicted in Fig. 5(a), from which one can see that the greater the level of disorder, the stronger the localization effects. The P(m) curves saturate after less than 100 roundtrips, which indicates the presence of the Anderson localization. The participation ratio curve for the SPL without disorder is also provided for reference, where it grows linearly in according to the well-known property of the ballistic spreading of the unperturbed DTQW.

 

Fig. 5 a) Participation ratio dependence on the roundtrip number m for different Δθ_max b) Inverse participation ratio at the fixed roundtrip m = 1000 dependence on Δθ_max with fluctuations, averaged over an ensemble of N_avg = 2000 realizations of the random coupling coefficient distribution. Number of slots N=1000 for both pictures.

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Note that the delocalized eigenmodes at the band centers β = ±π/2 do not prevent saturation of the participation ratio due to their limited overlap with the initial delta-like distribution of the amplitudes.

To provide additional evidence on the Anderson localization we gradually increased the amplitude of disorder Δθ_max, averaged calculated inverse participation ratios (1/P(m)) at fixed roundtrip m = 1000 over the ensemble of N_avg = 2000 realizations of the random participation ratio distributions and calculated its fluctuation δ(1/P(m)). It resulted in a quasilinear increase of the inverse participation ratio and its fluctuation (Fig. 5(b)), both confirming the presence of the Anderson localization.

4. Conclusion

We considered a synthetic photonic lattice with randomized coupling coefficient, i.e. off-diagonal disorder, and build up a new analytical result for the localization length of eigenmodes in the practical case of weak coupling disorder. We studied dynamical properties of the wavepacket spreading in such lattices and demonstrate numerically that one-dimensional Anderson localization occurs. We show that localization arise at any levels of disorder in the system, despite the symmetry-induced delocalization transition occurring at the middle of the band. Further directions could be focused on experimental realization of SPL with off-diagonal disorder, implementation of the proposed techniques on lattices with discrete time symmetry, studies of the effects of disorder-immune symmetries on higher order photonic statistics (thermalization, bunching, etc.), studies of one dimensional lattices with periodic boundary conditions.