Abstract

A synthetic photonic lattice (SPL) is a re-configurable test-bed for studying the dynamics of one-dimensional mesh lattices including the photonic implementations of discrete time quantum walks. Unlike other realizations of photonic lattices, SPL possesses easy and fast control of lattice parameters. Here we consider disordered SPL where the coupling ratio between the two fiber loops realizing the lattice is random but does not change between the round trips. We obtain a new analytical result for the localization length (inverse Lyapunov exponent) for a practical case of weak coupling disorder. We also numerically study the dynamics of the pulse train circulating within the loops and observe that despite delocalization transition at the band center the pulse spreading is arrested even at small values of the disorder.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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].

 figure: Fig. 1

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(θn) 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 Llong and Lshort with the field sampled at N=12(Llong+Lshort)/(LlongLshort) 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 |Ψ(m)=nunm|n,+vnm|n,, and its state after the next round trip is obtained by applying a unitary step operator: |Ψ(m + 1)〉 = ÛSPL |Ψ(m)〉 = ŜSPLSPL |Ψ(m)〉 where in the basis |↑〉, |↓〉 both operator have the form:

S^SPL=n{|n1,n,|+|n+1,n,|}
R^SPL=neiθnσx|nn|
with the Pauli matrix σx = |↑〉 〈↓| + |↓〉 〈↑|.

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

{unm+1=[cos(θn+1)un+1m+isin(θn+1)vn+1m]vnm+1=[cos(θn1)vn1m+isin(θn1)un1m]
The vector of coupling angles at different position θ⃗ = (θ1, . . ., θN) is assumed to be static but position dependent (which in the fibre loops can be implemented in a controllable way by using fast Mach-Zender modulator). Assuming periodic boundary condition the dynamical evolution of a quantum walker is completely determined by the set of Floquet-Bloch eigenmodes of the step operator ÛSPL. Since this operator is unitary its eigenvalues have the form λj = exp(j) where βj are quasienergies (or propagation constants) defined in the interval (−π, π] and each corresponding eignemode |Ψj=nun(j)|n,+vn(j)|n, evolves in discrete time simply as |Ψj(m)〉 =exp(imβj) |Ψj(0)〉.

The spectrum of the SPL without modulation of the coupling angles (i.e. when θn = θ = const) is well known [5,8]. The eigenmodes are a set of N periodic complex exponentials exp(ikjn) with kj spanning the first Brillouin zone [−π, π] and the spectrum is given implicitly by the relation cos β = cos θ cos kj 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 θn. The coupling angles were then pseudorandomly generated as θn = π/4±rand(Δθmax), where function rand(x) gives random number from 0 to x and Δθmax 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 ÛSPL(θ⃗). A lot of useful information can be gained by looking into symmetries of this operator as well as the corresponding effective Hamiltonian ÛSPL = exp(iĤeff) [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 R̂SPL). In the chosen photonic implementation (Eq. (3)) the system has particle-hole symmetry reflected here in an antiunitary operator P̂ = σzK̂, where K̂ is complex conjugation so that P̂ÛSPL−1 = ÛSPL. This symmetry survives in the disordered case and as its consequence if λ = exp() is an eigenvalue corresponding to the eigenmode (un, vn), then λ* = exp(−) is also an eigenvalue corresponding to the eigenmode (un*,vn*) 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 Γ^1U^SPLΓ^=U^SPL with the unitary operator Γ^=in(cosθnσy+sinθnσz)|nn|. 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 (θn = θ). Such an SPL is characterized by an evolutionary operator: ÛSPL(θ) = ŜR̂(θ) where R̂(θ) = exp(xπ/2). The effective Hamiltonian is:

H^SPL=ππdkβ(k)n(k)σ|kk|
where the quasi-energy β(k) obeys the dispersion law cos β = cos θ cos k, and we have assumed the limit of large system N → ∞ so that the spectrum is continuous with the eigenmodes of the shift operator:
|k=12πneikn|n
In addition we have defined a unit vector n⃗(k) which is projected on the Pauli vector σ⃗ = (σx, σy, σz) and having the components:
n(k)=[cosθcosk,sinθsink,cosθsink]sinβ(k)
As the wave vector k traces the first Brillouin zone [−π, π], the vector n⃗(k) traces a great circle on the Bloch sphere, winding exactly once around the origin corresponding to the winding number (topological charge) equal to 1 [27]. The plane of motion is perpendicular to vector A⃗(θ) = (0, cos θ, sin θ), the direction of which defines the chiral symmetry of the Hamiltonian and the DTWQ: Γ^(θ)1U^SPLΓ^θ=USPL with the unitary operator Γ̂(θ) = exp(−iπA⃗(θ) · σ⃗/2) = −iA⃗(θ) · σ⃗.

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:

Γ=ei(π/2)nAnσ|nn|=nAnσ|nn|,An=An(θn).
While the effective Hamiltonian in the disordered case is not readily available, it is easy to demonstrate by direct calculation that the companion identity Γ^1U^SPLΓ^=U^SPL still holds, which proves the existence of the chiral symmetry.

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 (un, vn) with λ = exp() to an eigenmode (−i sin θn un − cos θn vn, cos θn un + i sin θnvn) 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 (un, vn), while leaving the even amplitudes unchanged, it corresponds to the case with a mirror flipped eigenvalue −λ eigenmodes, which corresponds to the symmetry Λ^1U^SPLΛ^=U^SPL with the corresponding flipping operator being Λ^=neven|nn|nodd|nn|.

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, un and vn, 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 ÛSPL |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 → ∞):

un+11cos(θn+1)[λ+λ1sinθn+1sinθn]un+tanθn+1tanθnun1=0
vn+11cos(θn)[sinθnsinθn1λ+λ1]vn+tanθntanθn1vn1=0
One can immediately see that the recursion for vn is the same recursion for un only in the reverse direction n → −n. Therefore both components of the QW should have the same localization properties as expected from the symmetry considerations. A standard measure of the inverse localization length l−1 is given by the Lyapunov exponent - γ(β), defined via the exponential divergence of the solution of either recursion in Eqs. (8,9) evaluated inside the spectral support λ = exp() starting from some arbitrary non-zero initial values. Using the u-component as an example we get:
γ(β)=1Nlog|uN|=1Nn=1log|rnU|=E[log|rnU|]
where we have introduced the so-called Riccati variable rnU=un+1/un obeying its own recursion:
rn+1U=1cosθn+1[eiβ+eiβsinθn+1sinθn]tanθn+1tanθn1rnU
and the last line in the Eq. (10) implies that asymptotically, when N → ∞, the distribution of log |rn| becomes stationary independent from the initial conditions used for the recursion [29]. It is also possible to work with 2x2 transfer matrices (which is the path chosen by most authors) but we have found that working with a 1D complex Riccati map simplifies both numerical and theoretical analysis.

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 ρ(β)=N1jδ(ββj). One can clearly see that in the case of strong disorder (Δθmax = π/4) the eigenmodes near the center of the band β = π/2 are least localized (since lloc ∝ 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 |βπ/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.

 figure: Fig. 2

Fig. 2 (a) The Lyapunov exponent obtained by iterating the Riccati map (Eq. (11)) with Δθmax from 0 to π/4. (b) Density of states (DOS) obtained for Δθmax 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 Δθmax ≪ 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 xn = vn tan θn−1/vn−1 we obtain the following recursion:

xn+1=[λtan2θnsinθn1+tanθnλcosθn]tan2θnxn,λ=eiβ.
An important property of the variable xn is that it is “causal”, i.e. correlated only with the values of θk with k < n. One has
γ(β)=limNE[ln|vNvN1|]=limNE[ln|xN|]
where the last equality follows from the fact that ln(tan(π/4 + θ)) is an odd function of θ and hence its average, with respect to any symmetric distribution of the coupling angles, vanishes.

Next as a formal small parameter we introduce a standard deviation of the coupling angle ε=Δθn2 (for the uniform distribution ε=Δθmax/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 xn=Aexp[εBn+ε2Cn+]A+ABnε+A(Cn+Bn2/2)ε2+.... Substituting this ansatz into both sides of Eq. (12), introducing normalized r.v. Vn having a symmetric distribution with unit variance so that θn = π/4 + εVn and expanding up to the second order in ε we obtain in each order:

ε0:A=22cosβ1A,
ε1:ABn+1=1ABn+2eiβ(4VnVn1)+32eiβVn4VnA
ε2:A[Cn+1+12Bn+12]=1A[Cn+12Bn2]+4BnAVn++2(8Vn2+32Vn124VnVn1)eiβ+112Vn2eiβ8Vn2A
In the zeroth order one obtains a constant solution of the quadratic equation which gives a Lyapunov exponent of the unperturbed system: γ(0)(β) = ln |A|. Since we are interested in the case where β belongs to the unperturbed spectrum so that cos2 β < 1/2 these two solutions are A±=2cosβ±i12cos2β so that |A±| = 1 and in the zeroth order Lyapunov exponent vanishes inside the spectrum as all the eigenstates are extended.

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

Bn+1=Bn=0,
Bn+12=Bn2=A±2A±41[82A3eiβ+A2(416e2iβ)82A(7cos(β)+isin(β))+16i(sin2β)+52cos(2β)+48]
where we have used the fact that Bn and Vn are statistically independent and additionally multiplied both sides of the equation by Bn to obtain a stationary average E[Bn+1Vn] = E[BnVn−1].

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

Cn+1=Cn=A2+12(A21)Bn2+1A21[19A2eiβ+11A2eiβ8]
Thus for the Lyapunov exponent defined as λ(ξ) = ln |A| + ε Re〈Bn〉 + ε2 Re〈Cn〉 after some tedious albeit straightforward calculation one obtains the final result:
γ(β)=ε2sin2β12cos2β,0<|cosβ|<1/2.
The comparison of this result and full numerical simulation of the Ricatti map (Eq. (11)) is given in Fig. 3. One can see an excellent agreement between theory and the numerics apart from the band centre, where the genuine Lyapunov exponent vanishes (see the discussion above). The failure of the perturbative approach at the band center is the result of the band center anomaly well known for a traditional Anderson model [30, 31]. However, as we shall see below, the delocalization transition at the band centre does not affect significantly the saturation of the participation ratio in a propagating wave packet.

 figure: Fig. 3

Fig. 3 The normalized Lyapunov exponent obtained for the DTQW with Δθmax = 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 ÛSPL 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 θn 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 Navg = 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.

 figure: Fig. 4

Fig. 4 Anderson localization in the SPL due to the coupling coefficient random distribution with a) Δθmax = 0.1π, b) Δθmax = 0.15π and c) Δθmax = 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)=(n|vnm|2)2/n|vnm|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−1γ 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 Navg = 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.

 figure: Fig. 5

Fig. 5 a) Participation ratio dependence on the roundtrip number m for different Δθmax b) Inverse participation ratio at the fixed roundtrip m = 1000 dependence on Δθmax with fluctuations, averaged over an ensemble of Navg = 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 Δθmax, averaged calculated inverse participation ratios (1/P(m)) at fixed roundtrip m = 1000 over the ensemble of Navg = 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.

Funding

Russian Science Foundation (16-12-10402); I.D. Vatnik is supported by Ministry of Education and Science of the Russian Federation (3.7672.2017/8.9);

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20. H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Interferometric control of the photon-number distribution,” APL Photonics 2, 071301 (2017). [CrossRef]  

21. H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Lattice topology dictates photon statistics,” Sci. Rep. 7, 8948 (2017). [CrossRef]   [PubMed]  

22. Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

23. T. Rakovszky and J. K. Asboth, “Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk,” Phys. Rev. A 92, 052311 (2015). [CrossRef]  

24. I. Vakulchyk, M. Fistul, P. Qin, and S. Flach, “Anderson localization in generalized discrete-time quantum walks,” Phys. Rev. B 96, 144204 (2017). [CrossRef]  

25. Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687 (1993). [CrossRef]   [PubMed]  

26. J. Kempe, “Quantum random walks: an introductory overview,” Contemp. Phys. 44, 307–327 (2003). [CrossRef]  

27. T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010). [CrossRef]  

28. J. M. Edge and J. K. Asboth, “Localization, delocalization, and topological transitions in disordered two-dimensional quantum walks,” Phys. Rev. B 91, 104202 (2015). [CrossRef]  

29. L. Tessieri and F. Izrailev, “Anderson localization as a parametric instability of the linear kicked oscillator,” Phys. Rev. E 62, 3090 (2000). [CrossRef]  

30. B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional anderson model: weak disorder expansions,” Eur. Phys. J. 45, 1283–1295 (1984).

31. M. Kappus and F. Wegner, “Anomaly in the band centre of the one-dimensional anderson model,” Z. Phys. B Con. Mat. 45, 15–21 (1981). [CrossRef]  

32. EOSpace, “Optical fiber high-speed switches,” https://www.eospace.com/high-speed-switch/.

33. M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015). [CrossRef]   [PubMed]  

34. B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep.Prog.Phys. 56, 1469 (1993).

References

  • View by:

  1. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958).
    [Crossref]
  2. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52 (2007).
    [Crossref] [PubMed]
  3. I. D. Vatnik, A. Tikan, G. Onishchukov, D. V. Churkin, and A. A. Sukhorukov, “Anderson localization in synthetic photonic lattices,” Sci. Rep. 7, 4301 (2017).
    [Crossref] [PubMed]
  4. S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
    [Crossref] [PubMed]
  5. M.-A. Miri, A. Regensburger, U. Peschel, and D. N. Christodoulides, “Optical mesh lattices with PT symmetry,” Phys. Rev. A 86, 023807 (2012).
    [Crossref]
  6. A. Schreiber, K. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
    [Crossref] [PubMed]
  7. A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
    [Crossref] [PubMed]
  8. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
    [Crossref] [PubMed]
  9. M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
    [Crossref] [PubMed]
  10. M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
    [Crossref]
  11. A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
    [Crossref]
  12. S. Wang, C. Qin, B. Wang, and P. Lu, “Discrete temporal Talbot effect in synthetic mesh lattices,” Opt. Express 26, 19235 (2018).
    [Crossref] [PubMed]
  13. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
    [Crossref] [PubMed]
  14. C. Soukoulis and E. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698 (1981).
    [Crossref]
  15. L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, and D. N. Christodoulides, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19, 13636–13646 (2011).
    [Crossref] [PubMed]
  16. S. Derevyanko, “Anderson localization of a one-dimensional quantum walker,” Sci. Rep. 8, 1795 (2018).
    [Crossref] [PubMed]
  17. H. Obuse and N. Kawakami, “Topological phases and delocalization of quantum walks in random environments,” Phys. Rev. B 84, 195139 (2011).
    [Crossref]
  18. H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “A photonic thermalization gap in disordered lattices,” Nat. Phys. 11, 930 (2015).
    [Crossref]
  19. H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Sub-thermal to super-thermal light statistics from a disordered lattice via deterministic control of excitation symmetry,” Optica 3, 477–482 (2016).
    [Crossref]
  20. H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Interferometric control of the photon-number distribution,” APL Photonics 2, 071301 (2017).
    [Crossref]
  21. H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Lattice topology dictates photon statistics,” Sci. Rep. 7, 8948 (2017).
    [Crossref] [PubMed]
  22. Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).
  23. T. Rakovszky and J. K. Asboth, “Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk,” Phys. Rev. A 92, 052311 (2015).
    [Crossref]
  24. I. Vakulchyk, M. Fistul, P. Qin, and S. Flach, “Anderson localization in generalized discrete-time quantum walks,” Phys. Rev. B 96, 144204 (2017).
    [Crossref]
  25. Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687 (1993).
    [Crossref] [PubMed]
  26. J. Kempe, “Quantum random walks: an introductory overview,” Contemp. Phys. 44, 307–327 (2003).
    [Crossref]
  27. T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010).
    [Crossref]
  28. J. M. Edge and J. K. Asboth, “Localization, delocalization, and topological transitions in disordered two-dimensional quantum walks,” Phys. Rev. B 91, 104202 (2015).
    [Crossref]
  29. L. Tessieri and F. Izrailev, “Anderson localization as a parametric instability of the linear kicked oscillator,” Phys. Rev. E 62, 3090 (2000).
    [Crossref]
  30. B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional anderson model: weak disorder expansions,” Eur. Phys. J. 45, 1283–1295 (1984).
  31. M. Kappus and F. Wegner, “Anomaly in the band centre of the one-dimensional anderson model,” Z. Phys. B Con. Mat. 45, 15–21 (1981).
    [Crossref]
  32. EOSpace, “Optical fiber high-speed switches,” https://www.eospace.com/high-speed-switch/ .
  33. M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
    [Crossref] [PubMed]
  34. B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep.Prog.Phys. 56, 1469 (1993).

2018 (3)

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

S. Wang, C. Qin, B. Wang, and P. Lu, “Discrete temporal Talbot effect in synthetic mesh lattices,” Opt. Express 26, 19235 (2018).
[Crossref] [PubMed]

S. Derevyanko, “Anderson localization of a one-dimensional quantum walker,” Sci. Rep. 8, 1795 (2018).
[Crossref] [PubMed]

2017 (4)

I. D. Vatnik, A. Tikan, G. Onishchukov, D. V. Churkin, and A. A. Sukhorukov, “Anderson localization in synthetic photonic lattices,” Sci. Rep. 7, 4301 (2017).
[Crossref] [PubMed]

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Interferometric control of the photon-number distribution,” APL Photonics 2, 071301 (2017).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Lattice topology dictates photon statistics,” Sci. Rep. 7, 8948 (2017).
[Crossref] [PubMed]

I. Vakulchyk, M. Fistul, P. Qin, and S. Flach, “Anderson localization in generalized discrete-time quantum walks,” Phys. Rev. B 96, 144204 (2017).
[Crossref]

2016 (1)

2015 (5)

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[Crossref] [PubMed]

T. Rakovszky and J. K. Asboth, “Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk,” Phys. Rev. A 92, 052311 (2015).
[Crossref]

J. M. Edge and J. K. Asboth, “Localization, delocalization, and topological transitions in disordered two-dimensional quantum walks,” Phys. Rev. B 91, 104202 (2015).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “A photonic thermalization gap in disordered lattices,” Nat. Phys. 11, 930 (2015).
[Crossref]

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

2013 (2)

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

2012 (2)

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

M.-A. Miri, A. Regensburger, U. Peschel, and D. N. Christodoulides, “Optical mesh lattices with PT symmetry,” Phys. Rev. A 86, 023807 (2012).
[Crossref]

2011 (3)

A. Schreiber, K. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref] [PubMed]

L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, and D. N. Christodoulides, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19, 13636–13646 (2011).
[Crossref] [PubMed]

H. Obuse and N. Kawakami, “Topological phases and delocalization of quantum walks in random environments,” Phys. Rev. B 84, 195139 (2011).
[Crossref]

2010 (2)

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010).
[Crossref]

2008 (1)

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

2007 (1)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52 (2007).
[Crossref] [PubMed]

2003 (1)

J. Kempe, “Quantum random walks: an introductory overview,” Contemp. Phys. 44, 307–327 (2003).
[Crossref]

2000 (1)

L. Tessieri and F. Izrailev, “Anderson localization as a parametric instability of the linear kicked oscillator,” Phys. Rev. E 62, 3090 (2000).
[Crossref]

1993 (2)

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687 (1993).
[Crossref] [PubMed]

B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep.Prog.Phys. 56, 1469 (1993).

1984 (1)

B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional anderson model: weak disorder expansions,” Eur. Phys. J. 45, 1283–1295 (1984).

1981 (2)

M. Kappus and F. Wegner, “Anomaly in the band centre of the one-dimensional anderson model,” Z. Phys. B Con. Mat. 45, 15–21 (1981).
[Crossref]

C. Soukoulis and E. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698 (1981).
[Crossref]

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958).
[Crossref]

Abouraddy, A. F.

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Interferometric control of the photon-number distribution,” APL Photonics 2, 071301 (2017).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Lattice topology dictates photon statistics,” Sci. Rep. 7, 8948 (2017).
[Crossref] [PubMed]

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Sub-thermal to super-thermal light statistics from a disordered lattice via deterministic control of excitation symmetry,” Optica 3, 477–482 (2016).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “A photonic thermalization gap in disordered lattices,” Nat. Phys. 11, 930 (2015).
[Crossref]

L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, and D. N. Christodoulides, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19, 13636–13646 (2011).
[Crossref] [PubMed]

Aharonov, Y.

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687 (1993).
[Crossref] [PubMed]

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958).
[Crossref]

Andersson, E.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Asboth, J. K.

T. Rakovszky and J. K. Asboth, “Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk,” Phys. Rev. A 92, 052311 (2015).
[Crossref]

J. M. Edge and J. K. Asboth, “Localization, delocalization, and topological transitions in disordered two-dimensional quantum walks,” Phys. Rev. B 91, 104202 (2015).
[Crossref]

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52 (2007).
[Crossref] [PubMed]

Batz, S.

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

Berg, E.

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010).
[Crossref]

Bersch, C.

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

Cassemiro, K.

A. Schreiber, K. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref] [PubMed]

Cassemiro, K. N.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Chen, Y.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Choi, K.-H.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Choi, S. H.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Choi, W.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Christodoulides, D.

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[Crossref] [PubMed]

Christodoulides, D. N.

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Interferometric control of the photon-number distribution,” APL Photonics 2, 071301 (2017).
[Crossref]

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Sub-thermal to super-thermal light statistics from a disordered lattice via deterministic control of excitation symmetry,” Optica 3, 477–482 (2016).
[Crossref]

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

M.-A. Miri, A. Regensburger, U. Peschel, and D. N. Christodoulides, “Optical mesh lattices with PT symmetry,” Phys. Rev. A 86, 023807 (2012).
[Crossref]

L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, and D. N. Christodoulides, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19, 13636–13646 (2011).
[Crossref] [PubMed]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Churkin, D. V.

I. D. Vatnik, A. Tikan, G. Onishchukov, D. V. Churkin, and A. A. Sukhorukov, “Anderson localization in synthetic photonic lattices,” Sci. Rep. 7, 4301 (2017).
[Crossref] [PubMed]

Davidovich, L.

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687 (1993).
[Crossref] [PubMed]

Demler, E.

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010).
[Crossref]

Derevyanko, S.

S. Derevyanko, “Anderson localization of a one-dimensional quantum walker,” Sci. Rep. 8, 1795 (2018).
[Crossref] [PubMed]

Derrida, B.

B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional anderson model: weak disorder expansions,” Eur. Phys. J. 45, 1283–1295 (1984).

Di Giuseppe, G.

Dou, J.-P.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Dreisow, F.

Economou, E.

C. Soukoulis and E. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698 (1981).
[Crossref]

Edge, J. M.

J. M. Edge and J. K. Asboth, “Localization, delocalization, and topological transitions in disordered two-dimensional quantum walks,” Phys. Rev. B 91, 104202 (2015).
[Crossref]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52 (2007).
[Crossref] [PubMed]

Fistul, M.

I. Vakulchyk, M. Fistul, P. Qin, and S. Flach, “Anderson localization in generalized discrete-time quantum walks,” Phys. Rev. B 96, 144204 (2017).
[Crossref]

Flach, S.

I. Vakulchyk, M. Fistul, P. Qin, and S. Flach, “Anderson localization in generalized discrete-time quantum walks,” Phys. Rev. B 96, 144204 (2017).
[Crossref]

Gábris, A.

A. Schreiber, K. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref] [PubMed]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Gao, J.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Gao, Z.-W.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Gardner, E.

B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional anderson model: weak disorder expansions,” Eur. Phys. J. 45, 1283–1295 (1984).

Goo, T.-W.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Heinrich, M.

Izrailev, F.

L. Tessieri and F. Izrailev, “Anderson localization as a parametric instability of the linear kicked oscillator,” Phys. Rev. E 62, 3090 (2000).
[Crossref]

Jex, I.

A. Schreiber, K. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref] [PubMed]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Jiao, Z.-Q.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Kappus, M.

M. Kappus and F. Wegner, “Anomaly in the band centre of the one-dimensional anderson model,” Z. Phys. B Con. Mat. 45, 15–21 (1981).
[Crossref]

Kawakami, N.

H. Obuse and N. Kawakami, “Topological phases and delocalization of quantum walks in random environments,” Phys. Rev. B 84, 195139 (2011).
[Crossref]

Keil, R.

Kempe, J.

J. Kempe, “Quantum random walks: an introductory overview,” Contemp. Phys. 44, 307–327 (2003).
[Crossref]

Kim, S.-R.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Kim, S.-W.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Kitagawa, T.

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010).
[Crossref]

Ko, H.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Kondakci, H. E.

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Lattice topology dictates photon statistics,” Sci. Rep. 7, 8948 (2017).
[Crossref] [PubMed]

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Interferometric control of the photon-number distribution,” APL Photonics 2, 071301 (2017).
[Crossref]

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Sub-thermal to super-thermal light statistics from a disordered lattice via deterministic control of excitation symmetry,” Optica 3, 477–482 (2016).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “A photonic thermalization gap in disordered lattices,” Nat. Phys. 11, 930 (2015).
[Crossref]

Kramer, B.

B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep.Prog.Phys. 56, 1469 (1993).

Ku, Z.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Lahini, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Lu, P.

MacKinnon, A.

B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep.Prog.Phys. 56, 1469 (1993).

Martin, L.

Miri, M.-A.

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

M.-A. Miri, A. Regensburger, U. Peschel, and D. N. Christodoulides, “Optical mesh lattices with PT symmetry,” Phys. Rev. A 86, 023807 (2012).
[Crossref]

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Mosley, P. J.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Näger, J.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

Nolte, S.

Obuse, H.

H. Obuse and N. Kawakami, “Topological phases and delocalization of quantum walks in random environments,” Phys. Rev. B 84, 195139 (2011).
[Crossref]

Onishchukov, G.

I. D. Vatnik, A. Tikan, G. Onishchukov, D. V. Churkin, and A. A. Sukhorukov, “Anderson localization in synthetic photonic lattices,” Sci. Rep. 7, 4301 (2017).
[Crossref] [PubMed]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

Pang, X.-L.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Perez-Leija, A.

Peschel, U.

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

M.-A. Miri, A. Regensburger, U. Peschel, and D. N. Christodoulides, “Optical mesh lattices with PT symmetry,” Phys. Rev. A 86, 023807 (2012).
[Crossref]

Potocek, V.

A. Schreiber, K. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref] [PubMed]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Qiao, L.-F.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Qin, C.

Qin, P.

I. Vakulchyk, M. Fistul, P. Qin, and S. Flach, “Anderson localization in generalized discrete-time quantum walks,” Phys. Rev. B 96, 144204 (2017).
[Crossref]

Rakovszky, T.

T. Rakovszky and J. K. Asboth, “Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk,” Phys. Rev. A 92, 052311 (2015).
[Crossref]

Regensburger, A.

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

M.-A. Miri, A. Regensburger, U. Peschel, and D. N. Christodoulides, “Optical mesh lattices with PT symmetry,” Phys. Rev. A 86, 023807 (2012).
[Crossref]

Rudner, M. S.

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010).
[Crossref]

Saleh, B. E.

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Lattice topology dictates photon statistics,” Sci. Rep. 7, 8948 (2017).
[Crossref] [PubMed]

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Interferometric control of the photon-number distribution,” APL Photonics 2, 071301 (2017).
[Crossref]

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Sub-thermal to super-thermal light statistics from a disordered lattice via deterministic control of excitation symmetry,” Optica 3, 477–482 (2016).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “A photonic thermalization gap in disordered lattices,” Nat. Phys. 11, 930 (2015).
[Crossref]

Schreiber, A.

A. Schreiber, K. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref] [PubMed]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52 (2007).
[Crossref] [PubMed]

Segev, M.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52 (2007).
[Crossref] [PubMed]

Silberberg, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Silberhorn, C.

A. Schreiber, K. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref] [PubMed]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Soukoulis, C.

C. Soukoulis and E. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698 (1981).
[Crossref]

Sukhorukov, A. A.

I. D. Vatnik, A. Tikan, G. Onishchukov, D. V. Churkin, and A. A. Sukhorukov, “Anderson localization in synthetic photonic lattices,” Sci. Rep. 7, 4301 (2017).
[Crossref] [PubMed]

Szameit, A.

Tang, H.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Tessieri, L.

L. Tessieri and F. Izrailev, “Anderson localization as a parametric instability of the linear kicked oscillator,” Phys. Rev. E 62, 3090 (2000).
[Crossref]

Tikan, A.

I. D. Vatnik, A. Tikan, G. Onishchukov, D. V. Churkin, and A. A. Sukhorukov, “Anderson localization in synthetic photonic lattices,” Sci. Rep. 7, 4301 (2017).
[Crossref] [PubMed]

Urbas, A. M.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Vakulchyk, I.

I. Vakulchyk, M. Fistul, P. Qin, and S. Flach, “Anderson localization in generalized discrete-time quantum walks,” Phys. Rev. B 96, 144204 (2017).
[Crossref]

Vatnik, I. D.

I. D. Vatnik, A. Tikan, G. Onishchukov, D. V. Churkin, and A. A. Sukhorukov, “Anderson localization in synthetic photonic lattices,” Sci. Rep. 7, 4301 (2017).
[Crossref] [PubMed]

Visbal-Onufrak, M. A.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Wang, B.

Wang, S.

Wang, Y.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Wegner, F.

M. Kappus and F. Wegner, “Anomaly in the band centre of the one-dimensional anderson model,” Z. Phys. B Con. Mat. 45, 15–21 (1981).
[Crossref]

Wimmer, M.

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

Yang, A.-L.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Zagury, N.

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687 (1993).
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APL Photonics (1)

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Interferometric control of the photon-number distribution,” APL Photonics 2, 071301 (2017).
[Crossref]

Contemp. Phys. (1)

J. Kempe, “Quantum random walks: an introductory overview,” Contemp. Phys. 44, 307–327 (2003).
[Crossref]

Eur. Phys. J. (1)

B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional anderson model: weak disorder expansions,” Eur. Phys. J. 45, 1283–1295 (1984).

Nat. Commun. (2)

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

Nat. Phys. (2)

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “A photonic thermalization gap in disordered lattices,” Nat. Phys. 11, 930 (2015).
[Crossref]

Nature (2)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52 (2007).
[Crossref] [PubMed]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

Opt. Express (2)

Optica (1)

Phys. Rev. (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958).
[Crossref]

Phys. Rev. A (4)

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Figures (5)

Fig. 1
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].
Fig. 2
Fig. 2 (a) The Lyapunov exponent obtained by iterating the Riccati map (Eq. (11)) with Δθmax from 0 to π/4. (b) Density of states (DOS) obtained for Δθmax from 0 to π/4.
Fig. 3
Fig. 3 The normalized Lyapunov exponent obtained for the DTQW with Δθmax = 0.1. Comparison of theoretical result (Eq. (20)) with the full numerics.
Fig. 4
Fig. 4 Anderson localization in the SPL due to the coupling coefficient random distribution with a) Δθmax = 0.1π, b) Δθmax = 0.15π and c) Δθmax = 0.25π.
Fig. 5
Fig. 5 a) Participation ratio dependence on the roundtrip number m for different Δθmax b) Inverse participation ratio at the fixed roundtrip m = 1000 dependence on Δθmax with fluctuations, averaged over an ensemble of Navg = 2000 realizations of the random coupling coefficient distribution. Number of slots N=1000 for both pictures.

Equations (20)

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S ^ SPL = n { | n 1 , n , | + | n + 1 , n , | }
R ^ SPL = n e i θ n σ x | n n |
{ u n m + 1 = [ cos ( θ n + 1 ) u n + 1 m + i sin ( θ n + 1 ) v n + 1 m ] v n m + 1 = [ cos ( θ n 1 ) v n 1 m + i sin ( θ n 1 ) u n 1 m ]
H ^ SPL = π π d k β ( k ) n ( k ) σ | k k |
| k = 1 2 π n e i k n | n
n ( k ) = [ cos θ cos k , sin θ sin k , cos θ sin k ] sin β ( k )
Γ = e i ( π / 2 ) n A n σ | n n | = n A n σ | n n | , A n = A n ( θ n ) .
u n + 1 1 cos ( θ n + 1 ) [ λ + λ 1 sin θ n + 1 sin θ n ] u n + tan θ n + 1 tan θ n u n 1 = 0
v n + 1 1 cos ( θ n ) [ sin θ n sin θ n 1 λ + λ 1 ] v n + tan θ n tan θ n 1 v n 1 = 0
γ ( β ) = 1 N log | u N | = 1 N n = 1 log | r n U | = E [ log | r n U | ]
r n + 1 U = 1 cos θ n + 1 [ e i β + e i β sin θ n + 1 sin θ n ] tan θ n + 1 tan θ n 1 r n U
x n + 1 = [ λ tan 2 θ n sin θ n 1 + tan θ n λ cos θ n ] tan 2 θ n x n , λ = e i β .
γ ( β ) = lim N E [ ln | v N v N 1 | ] = lim N E [ ln | x N | ]
ε 0 : A = 2 2 cos β 1 A ,
ε 1 : A B n + 1 = 1 A B n + 2 e i β ( 4 V n V n 1 ) + 3 2 e i β V n 4 V n A
ε 2 : A [ C n + 1 + 1 2 B n + 1 2 ] = 1 A [ C n + 1 2 B n 2 ] + 4 B n A V n + + 2 ( 8 V n 2 + 3 2 V n 1 2 4 V n V n 1 ) e i β + 11 2 V n 2 e i β 8 V n 2 A
B n + 1 = B n = 0 ,
B n + 1 2 = B n 2 = A ± 2 A ± 4 1 [ 8 2 A 3 e i β + A 2 ( 4 16 e 2 i β ) 8 2 A ( 7 cos ( β ) + i sin ( β ) ) + 16 i ( sin 2 β ) + 52 cos ( 2 β ) + 48 ]
C n + 1 = C n = A 2 + 1 2 ( A 2 1 ) B n 2 + 1 A 2 1 [ 19 A 2 e i β + 11 A 2 e i β 8 ]
γ ( β ) = ε 2 sin 2 β 1 2 cos 2 β , 0 < | cos β | < 1 / 2 .

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