Localized quantum walks in quasi-periodic Fibonacci arrays of waveguides

Dan T. Nguyen; Daniel A. Nolan; Nicholas F. Borrelli

doi:10.1364/OE.27.000886

1. Introduction

Quantum walks (QWs) have emerged from fundamental research to one of key processes in quantum computing and quantum communication as the effects are of highly potential use for applications in quantum algorithms [1,2], universal quantum computers [3,4], and quantum simulations [5,6] etc. Those are among the most promising schemes for their unprecedented computing acceleration that can hopefully solve problems that are hard or impossible to solve by classical computers. Photons with their dual wave-particle nature have been demonstrated as excellent “walkers” since they are easily generated and manipulated even in room-temperature conditions. Beam splitter arrays have been used to perform discrete-time QWs (DTQW) [7,8], and evanescently coupled parallel waveguide arrays have been used to perform continuous-time QWs [9–13]. One unique phenomenon in quantum walks is localization in the presence of a disordered medium. This phenomenon, commonly known as Anderson localization, is usually discussed in terms of coherent evolution (e.g. the quantum walks) in the presence of a disordered medium. By breaking the periodicity of the evolution through spatial and/or temporal randomizing the operations with which the dynamics of the system is determined, the effect of a random disordered medium can be resembled and localization can be and has been realized [14–17]. Since then, there have been increased interests and efforts on investigation not only of fundamentals but also applications of LQWs, partially motivated by the possibility of employing localized photonic states for the secure quantum memory [17] and secure transmission of quantum information [18]. Thus, localized QWs show potential for applications in quantum communications.

Anderson published an article in 1958 in which he predicted that the wavefunction of a quantum particle can be localized in the presence of a static disordered potential [19]. Consequently, it is expected that a quantum particle with its dual wave-particle nature and energy transport through a disordered medium should be strongly suppressed. In other worlds, Anderson localization arises from destructive interferences among different scattering paths of a quantum particle propagating in a static disordered medium. Sixty years after its discovery, Anderson localization is still widely studied in many different areas of physics, even recently initiated new research areas [20]. It is well known that deviations from periodicity may result in higher complexity and give rise to a number of surprising effects. One such deviation can be found in the field of optics in the realization of photonic quasi-crystals, a class of structures made from building blocks that are arranged using well-designed patterns but lack translational symmetry. It has been recognized that quasi-periodic systems could also lead to localization in optics [21–23]. A quasi-periodic system is neither a periodic nor a random one so it could be considered as an intermediate between the two. Examples of such systems constructed with Fibonacci sequences include 1D quasi-crystalline Fibonacci dielectric multilayers (FDML) [24] and semiconductor quantum-wells [25], two-dimensional (2D) quasi-crystalline structures [26], and three-dimensional (3D) quasi-crystals [27].

It is worth to note that, disorder-induced localization can be conventionally quantified by averaging over realizations on many systems having the same degree of disorder. Similarly, localized quantum walks (LQWs) in integrated photonics systems have been realized on many randomly disordered arrays of evanescently coupled waveguides. The final results are averaged over all realizations in such arrays of waveguides whose randomness is controlled within a defined range of the disorder. Because of that, realization of LQWs has been proven experimentally to be difficult [14]. Note that, in [15] Lahini et al. proposed a systematically experimental method to determine localized modes in 1D disordered array of waveguides which would greatly benefit investigations of localization effects in randomly disordered systems. Meanwhile, quasi-periodic systems or quasi-crystals provides deterministic disorder deviated from periodicity resulting in localization of light deterministically in those systems [21,22], meaning there is no need to do averaging over many samples or systems. In that spirits, we propose new quasi-periodic AWs constructed with Fibonacci sequences – Fibonacci arrays of waveguides (FAWs) that can be used to realize deterministic LQWs which could be useful for quantum communication applications as proposed recently in [17,18]. The use of FAWs for realizing LQWs have two main benefits (i) the results are deterministic and therefore highly predictable and controllable, (ii) there is no need to do averaging over many realizations in a many experiments and large number of samples as in the case of random disordered systems. Furthermore, our simple construction rules allow us to create symmetrically quasi-periodic FAWs. Consequently, LQWs with symmetrical probability distributions can be realized deterministically in the structures. The proposed FAWs are simple and straightforward to make, but would be potentially useful for many research areas in quantum communication as will be discussed in this paper.

At this point, we would like to note that localization in arrays and lattices of waveguides constructed with Fibonacci sequences in distances between nearest identical waveguides, both 1D and 2D have recently investigated theoretically and experimentally [34,35]. Essentially, the Fibonacci arrays of waveguides considered in [34] are defined as the elements in Fig. 1 in this work; however, A and B were defined as the two fundamental distances 1 (unit) and τ between identical waveguides where τ = 1.618 is golden ratio of Fibonacci sequence. It is important, however, to point out that the arrays of identical waveguides constructed with Fibonacci consequences in distances described in [34,35] have quasi-periodic distribution of coupling coefficients (off-diagonal deterministic disorder) with all waveguides have the same propagation constant. Meanwhile our proposed FAWs are constructed with a Fibonacci sequence with two different waveguides, therefore our FAWs have both on- and off-diagonal deterministic disorders (e.g., both propagation constants and coupling coefficients are quasi-periodic distributions). Localizations have been investigated experimentally in on-diagonal disordered arrays of waveguides [15] and in off-diagonal disordered lattices [16]. Our proposed FWAs would offer systems for investigation of both on- and off-diagonal deterministic disordered structures. Furthermore, the new proposed j^th-order FAW is constructed as an orderly sequence chain of all Fibonacci elements up to j^th-order instead of individual j^th-order Fibonacci element as in [34,35], which are conveniently to make FAWs symmetrically as presented in the next section.

Fig. 1 Elements S₁, S₂ … S₆ of FAWs composed by two waveguides A and B.

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Our paper is organized as follow. Section 1 is the Introduction followed by Section 2 detailing the new concept of FAW. In Section 3, simulation results of QWs in regular arrays of waveguides (RAW) and LQWs due to randomly disorder in RAWs, and unique LQWs in FAWs with symmetrical probability distribution are presented. Section 4 is the discussions and conclusion.

2. Quasi-periodic arrays of waveguides constructed with Fibonacci sequences

Before introducing our new proposed arrays of waveguides constructed with Fibonacci sequences (FAW), let’s review briefly the mathematics of Fibonacci sequences and 1D quasi-periodic Fibonacci dielectric multiple layers (FDML) systems.

First, in mathematics the Fibonacci numbers are the integer numbers following a Fibonacci sequence, characterized by the fact that every number after the first two is the sum of the two preceding ones [28]:

S_{j} = S_{j - 2} + S_{j - 1}, with S_{1} = S_{2} = 1.

From Eq. (1) we can easily write down the first numbers of the Fibonacci sequence as: 1, 1, 2, 3, 5, 8, 13, 21… An important characterization of the Fibonacci sequence is the golden ratio which has been widely used in architectures, arts, and also discovered in nature (many examples can be seen in [28]).

At the beginning, quasi-periodic structures were mainly considered as suitable theoretical models to describe the conceptual transition from randomness to periodic order. Later, it was realized that the structures may offer interesting possibilities for technological applications as well. A particularly interesting 1D quasi-periodic model was proposed in [23] by Kohmoto et al. This model is based on the Fibonacci sequence which is constructed recursively with multiple layers of two materials

S_{j} = S_{j - 2} S_{j - 1}, with S_{1} = A, S_{2} = B,

where A(B) stands for a layer of material with refractive index

n_{A (B)}

and thickness

d_{A (B)}

. From Eq. (2) elements of different orders can be written as in Fig. 1 and so forth.

1D quasi-periodic FDML can be used to create multiple reflection windows mirrors [24]. Note that most of the works on these 1D FDML structures have focused on the properties of the different Fibonacci elements S_j defined in Eq. (2), and the thicknesses of the layers A and B are chosen typically on the order of the wavelength of interest λ (usually λ, λ /2 or λ /4). In [24] an entirely new sequencing rule for the dielectric layers based on the above Fibonacci elements was proposed for multiple reflection windows mirrors as

F_{j} = S_{1} S_{2} S_{3} \dots S_{j} .

In other words, the j^th-order structure F_j is the orderly sequence chain of all Fibonacci elements up to the j^th-order, S_j. Furthermore, the optical thicknesses of the two layers A and B can be on the order of multiples of a quarter-wavelength depending on the number of reflection windows needed. It is worth noting that although the structures are simple and straightforward to make, they have very rich optical spectra. For examples, self-similarity of spectra of chains with different orders, and by changing the thickness ratio between the two material layers, mirrors with different numbers of windows at the same Fibonacci order can be created [24]. More interestingly, even in higher order chains with complicated structures, there are spectral regions where the transmission is ~100% which can be considered as localization of light or transmitted modes in the quasi-crystals.

Let’s now apply the above concept of 1D FDMLs as in [24] for new quasi-periodic AWs. Our proposed AWs are constructed recursively with a Fibonacci sequence e.g., Eq. (2) with two different waveguides A and B instead of two dielectric layers as in FDMLs. As a result, we can easily write down formulae of elements and their corresponding structures. In Fig. 1, we show some first orders elements of FAWs as

Note that, in Fig. 1 above A and B are two different waveguides, specifically we will consider A and B as two single-mode (SM) waveguides placed closely to ensure evanescent coupling between the two waveguides. For elements that have more than two waveguides, all waveguides will be placed in regular positions where the separation distance is constant. In order to increase the complexity of the arrays of waveguides – or to make it less orderly, we define a new j^th-order FAW as in Eq. (3) where S₁, S₂ … S_j again are Fibonacci elements defined in Fig. 1 above. For example, a 6th order FAW (FAW6) can be constructed as shown in Fig. 2 below.

Fig. 2 Diagram of 6th order Fibonacci arrays of waveguides (FAW6) composed by two waveguides A and B (front view). S_j and dotted lines on top of waveguides indicate corresponding Fibonacci elements.

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Although it looks quite simple, the quasi-periodic pattern is clearly shown in FAWs, and therefore we would expect new properties of light propagation in comparison with regular arrays of waveguides. One of those important properties is the localization of light that has been proven theoretically and experimentally in quasi-periodic FDML. It has been shown that localization in elements S_j of FDML are not exponentially localized as in randomly disordered systems but rather quasi-localized (less than exponential decay at large distance) [22,23].

As stated earlier in the Introduction, we would like to explore the quasi-periodic properties of FAW to investigate QWs, especially to realize deterministically LQWs in those structures. It is important to stress here that the disorder of FAWs comes from the Fibonacci pattern of two fundamental elements – two SM waveguides, not from spatially random variation of indexes or positions. It is interesting to note, that the structures are both on- and off-diagonal deterministic disorders (see more details about on- and off diagonal disorder, e.g., in [15,16]). That means we can design and control the disorder of the systems by changing characteristics of FAWs such as the Fibonacci orders, parameters of two fundamental waveguides (NA, size) and core-to-core distance. Because of the deterministic nature of the system, we would not need to have a large number of structures as in the case of randomly disorder. In order to increase the complexity of the AWs – or to make it less orderly, we will consider the j^th-order FAWs F_j instead of the elements S_j. More importantly, to make our AWs symmetrical, a property that is impossible for spatially random disordered systems, we will construct our arrays as in Eq. (4), and as examples the 6th -order of symmetrical FAWs are shown in Fig. 3. We will show later in Section 3 that the symmetrical FAWs would generate a unique distribution of LQWs which is impossible with randomly disordered systems. Since we will mostly focus on symmetrical FAWs in our work, we will call them only as FAWs for short, and we will use symmetrical FAWs to avoid any confusion if needed. Note that, it is easy to construct different quasi-periodic FAWs based on Fibonacci elements defined in Fig. 1 above, however in this paper we will restrict our simulation work in the following two types of symmetrical FWAs that can be describes as in Eqs. (4)a) and (4b):

Fig. 3 Diagrams of two different types of 6th-order symmetrical FAWs $F_{6 A (B)}^{I}$ (or FAW6A(B)-I), and $F_{6}^{I I}$ (or FAW6-II) with different configurations of inputs.

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F_{j}^{I} = S_{j} S_{j - 1} \dots S_{2} S_{1} S_{2} \dots S_{j - 1} S_{j},

F_{j}^{I I} = S_{j} S_{j - 1} \dots S_{2} S_{1} S_{1} S_{2} \dots S_{j - 1} S_{j} .

Note that, both types of the above FAWs are symmetrical, and they can be used with different input configurations as shown in Fig. 3 for 6th order FAWs as examples.

Next, in Section 3, we will first present the method to simulate single photons QWs in AWs using the beam propagation method (BPM). We will then show in detail our simulation of QWs in FAWs in comparison with RAWs, especially LQWs in randomly disordered AWs and in FAWs with symmetrical probability distribution. We will discuss QWs in other quasi-periodic structures and one potential application for the symmetrical LQWs in Section 4.

3. Localized quantum walks in Fibonacci arrays of waveguides

As described in an important review paper on QWs [30], the simplest classical random walks on a line consists of a particle (“the walker”) jumping to either left or right depending on the outcomes of a probability system (“the coin”) with two mutually exclusive results, i.e. the particle moves according to a probability distribution. At each step, an unbiased coin is tossed and depending on the result of the coin up or down the walker makes consecutive left-or-right decisions, respectively. For classical random walks on a line, the resulting distribution is of well-known Gaussian form. The quantum version of the random walk works analogously to its classical counterpart. The main components of discrete-time quantum walks (DTQW) are a walker, a quantum coin, evolution operators for both walker and quantum coin, and a set of observables. QWs can also be continuous-time quantum walks (CTQW) which have been extensively investigated theoretically and experimentally [9–16]. In contrast to DTQWs, CTQWs have no coin operations and the evolutions are defined entirely in position space [10,11,29]. These walks require a well-controlled, continuous coupling between vertices or lattice sites. Integrated photonics systems consist of evanescently coupled waveguides are perfectly suited for CTQWs, and lithographically written waveguides were the first systems used to demonstrate quantum walks on a line with coherent light [9–16]. Those configurations have a common feature: the ‘walking region’ is the evanescent-coupled waveguide arrays. In such structures, spacing between waveguides should be close enough for evanescent coupling to occur, on the order of several micrometers or in strong coupling regimes.

It is important to emphasize that QWs of single photons do not exhibit any different behavior from classical wave propagation, as the intensity distribution corresponds to the probability of detecting the photon at any position. However, when multiple walkers (for example, entangled photons) co-propagate in such systems, truly non-classical correlations appear [29]. In this paper, we will present the results of single photons QWs. The problem of multiphoton QWs is a different subject that will be investigated and published elsewhere.

Let us briefly describe how continuous-time QWs of single photons can be realized using arrays of evanescently coupled waveguides. Refs [10,11,29]. are excellent references of both theoretical and experimental works on QWs in a photonic lattice or array of waveguides. The problem of quantum walks in an AW can be described by a Hamiltonian as

H = ℏ \sum_{n}^{} {β_{n} a_{n}^{†} a_{n} + \sum_{m}^{} κ_{n m} a_{n}^{†} a_{m}},

where

a_{n}^{†}

(

a_{n})

stands for the creation (annihilation) operator of photon in waveguide n, β_n is propagation constants of waveguide n. Note that, κ_nm is coupling coefficient between adjacent waveguides n = m ± 1. The propagation dynamics of a single photon is governed by the Heisenberg equations of motion

\frac{d a_{k}^{†}}{d z} = - \frac{i}{ℏ} [a_{k}^{†}, H] = i (β_{k} a_{k}^{†} + κ_{k, k + 1} a_{k + 1}^{†} + κ_{k, k - 1} a_{k - 1}^{†}) .

As described in detail in Ref [11,29], since the system is conservative and the Hamiltonian is explicitly time independent, we may formally integrate Eq. (6) to obtain the input-output relation for the mode operators as

a_{k}^{†} (z) = \sum_{j = 1}^{N} U_{j, k} a_{j}^{†} (0) .

For the special case of a single photon coupled into site k of a uniform waveguide array ( $κ_{k, k + 1} = κ_{k, k - 1} = κ, a n d β = β_{k})$ one can show that the probability amplitude at site j is analytically described by

U_{k, j} = i^{(k - j)} \exp (i β z) J_{k - j} (2 κ z),

where

J_{k - j} (z)

is a Bessel function of the first kind and order (k-j). When a single photon is coupled to waveguide k, it will evolve to waveguide j with a probability

η_{j} = | U_{k, j} (z) |^{2} = J_{k - j}^{2} (2 κ z)

. The photon spreads across the lattice by coupling from one waveguide to its neighbors in a pattern characterized by two strong “ballistic” lobes [10,11,29].

In general, simulation of QWs in irregular AWs are extremely difficult and it is even impossible to find such analytical solutions e.g. Equation (8) above as in RAWs, and therefore numerical solutions are necessary. Simulation of LQWs in randomly disordered systems is even more challenging even for problems of single-photon QWs. It is important to point out that, single photons QWs do not exhibit any different behavior from classical wave propagation, as the intensity distribution corresponds to the probability of detecting the photon at any position. Therefore, one of powerful methods of simulating light propagation in complicated structures - the beam propagation method (BPM) could be very effective for simulation of single photons QWs in irregular AWs. The BPM can describe very well the evanescent coupling between waveguides, and the method only requires fundamental parameters of the system such as core sizes, indexes and positions of each waveguides. We show in this work BPM is not only capable for simulating QWs in regular AWs, quasi-periodic FAWs but also randomly disordered AWs. Next, we present a brief description of the BPM for simulating single photon QWs in this work.

The paraxial wave equation for the slowly varying electric field envelop of light propagating along the -axis in a waveguide can be written as [31]

\frac{d}{d z} E (x, y, z) = (\hat{D} + \hat{V}) E (x, y, z) .

The operators $\hat{D}$ and $\hat{V}$ are given by

\hat{D} = \frac{i}{2 k} (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}), and \hat{V} = {i k Δ n (x, y) - α (x, y)} .

In Eq. (10) $k = n_{b} k_{0} = \frac{n_{b} ω}{c} = 2 π n_{b} / λ$ where n_b is the background or reference refractive index and λ the free-space wavelength, $Δn = n (x, y, z) - n_{b}$ is the refractive-index profile relative to reference refractive index, and α is the power absorption/loss of the waveguide. A small propagation step is implemented using the following approximation:

E (x, y, z + Δ z) = e^{(\hat{D} + \hat{V}) Δ z} E (x, y, z) \approx e^{\hat{D} ​ \frac{Δ z}{2}} e^{\hat{V} Δ z} e^{\hat{D} ​ \frac{Δ z}{2}} E (x, y, z),

where

e x p (\hat{D} Δ z / 2)

means take a half step of diffraction alone, and

e x p (\hat{V} Δ z)

means take the whole step of linear propagation alone. This calculation is third-order accurate in the step length and requires that the change produced by each step is small compared to unity. Equation (11) can be solved very effectively by a fast Fourier transformation (FFT) [31]. The method has been successfully applied to simulate Yb-doped multicore fiber lasers [32]. We have developed our own Matlab codes, and the simulation results of QWs are presented in the following.

Figures 4 show simulations results of single photons QWs in AWs of 23 identical SM waveguides, Fig. 4(a): RAW without random disorder, Figs. 4(b)-4(e): examples of QWs in AWs with randomly disordered waveguide positions $Δ_{i} = 0.1 d$ . All waveguides have the same core size of 4 μm, index difference between core and cladding Δn = 0.0035, center-to-center separation d = 8 μm, and λ = 1.55μm.

Fig. 4 QWs in AW of 23 identical SM waveguides with input is central core. (a) AW without disorder of waveguide positions; (b, c, d, e) AWs with randomly disorder of positions of 10%. From bottom to top: top-view, front-view and probability distribution of photon (in the same scale). Upper: diagram of array of waveguides.

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The simulation results of QWs for each waveguide structure in Fig. 4 show in the order from bottom to top: top-view, front-view and probability distribution of photon at the end of the walks. Note that, all probability distributions are plotted in the same scale so that the differences can be clearly seen. It is important to note that the results from BPM simulation of QWs in RAW (without disorder) are the same the well-known analytical solutions of Eq. (8). The results show that photons spread across the lattice by coupling from one waveguide to its neighbors in a pattern characterized by two strong “ballistic” lobes [10,11]. Meanwhile, the results in Figs. 4(b)-4(e) show QWs in randomly disordered AWs could be completely different even the structures have the same degree of random disorder. Therefor averaging over a large number of realizations is required for quantifying QWs in randomly disordered systems.

Let’s now consider the cases of Fibonacci AWs. We show in Fig. 5 simulation results of single photons QWs in RAW of 39 waveguides (RAW39) and in two 6th-order FAWs (FAW6A-I1 and I2) in the order from left to right - also consisting of 39 waveguides with input signal in the center waveguide S1 of those structures as described in Fig. 3(a). In RAW39, all 39 SM waveguides are identical with the same core size of 4 μm, and the index difference between the core and cladding is Δn = 0.0035, and the center-to-center separation between the nearest cores is 8 μm. There are two Fibonacci structures: FAW6A-I1 with Δn1 = 0.0035 and Δn2 = 0.0040, and FAW6A-I2 with Δn1 = 0.0035 and Δn2 = 0.0045, core size and separation are the same those of RAW39. The simulation results show clearly LQWs in both Fibonacci systems and stronger localization in FAW6A-I2 with larger index difference. In order to see visually the differences between QWs in RAWs and FAWs, animations are shown in Visualization 1, Visualization 2, and Visualization 5 for QWs in RAW39 and in Visualization 3, Visualization 4, and Visualization 6 for LQWs in FAW6A-I2. Visualization 1 and Visualization 3 are animations of QWs along the waveguide arrays with top view images shown in Figs. 5(a) and Fig. 5(g) for RAW39 and FAW6A-I2, respectively; Visualization 2 and Visualization 4 are animations corresponding to the front-view images shown in Figs. 5(b) and Fig. 5(h), respectively; Visualization 5 and Visualization 6 are animations corresponding to the front-view 3D-distribution images shown in Figs. 6(a) and Fig. 6(b), respectively; The animations are made with 100 frames of pics. As can be seen, visual description of light propagation is one of very useful features of the BPM method. It is worth to note that, this method could also be effective for simulating other single-photon quantum effects. However, as stated earlier the method is not adequate for simulation of multiple photons quantum effects including multiphoton QWs.

Fig. 5 Quantum walks in RAW39 (left), FAW6-I1 (center) and FAW6-I2 (right). From bottom to top: top-view, front-view and probability distribution with the same scale. Input signal is in the center waveguide (red arrows). Visualization 1 (a), Visualization 2 (b), Visualization 3 (g) and Visualization 4 (h).

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Fig. 6 Structural difference of QWs in RAW and FAW: Visualization 5 (a), and Visualization 6 (b) shown 3D photon probability distributions in RAW39 and FAW6A-I2, respectively.

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In order to see more clearly the structural difference of QWs in regular and Fibonacci AWs, we re-plot the probability distributions of RAW39 and FAW6A-I2 (not the same scale) in Fig. 6 below.

Figure 7 shows simulation results of symmetrical LQWs in FAW6B-I and FAW6-II with different input signal schemes: FAW6A-I with input in central S1 (see Fig. 3(a)): FAW6B-I with two near-center S2 waveguides (see Fig. 3(b)), and FAW6-II with input in two center S1 waveguides (see Fig. 3(c)).

Fig. 7 LQWs in FAW6A-I with input in center waveguide S1 (a), FAW6B-I with input in two S2 waveguides near center (b); and FAW6-II (c) with input in two center waveguides S1. From bottom to top: top-view, front-view and probability distribution. Red arrows indicate the input signal.

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As can be seen from Figs. 6 and 7, LQWs in quasi-periodic structures of FAWs are controllable in contrast with the ones in spatially random disordered structures as in Figs. 4(b) - 4(e). Furthermore, LQWs with the symmetrical probability distribution can be conveniently realized in the proposed FAWs. It is worth to note that Chandrashekar and Busch [17] have recently proposed to employ symmetrical LQWs for secure quantum memory application. In order to achieve symmetrical distribution LQWs, they proposed to use temporally disordered operations in spatially ordered systems. However, their approach requires multiple quantum coins for temporally disordered operation which could be extremely difficult in reality. In Section 4, we will discuss about the possibility that LQWS in FAWs, and in other quasi-periodic AWs as well can be used for similar applications.

Wavelength-dependence LWQs in FAWs is another interesting property that are controllable in such systems. We show simulation results of LQWs in FAW6A-I1 for signals of different wavelengths 1.3μm, 1.5μm and 1.7μm in the order from left to right in Fig. 8 below.

Fig. 8 LQWs in FAW6A-I with input in center waveguide S1 with signal of different wavelengths 1.3μm, 1.5μm and 1.7μm in order from left to right. From bottom to top: top-view, front-view and probability distribution (in different scales).

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In Fig. 8, the distance for completing walking (photons reach to the outer waveguides) or walking distance for signal of 1.3μm is 0.58cm, 1.5μm is 0.50cm and for signal of 1.7μm is 0.42cm, or the shorter wavelength, the longer walking distance.

4. Discussions and conclusion

As stated earlier, symmetrical LQWs can be potentially employed for secure quantum memory as recently proposed by Chandrashekhar and Busch [17]. The authors emphasized that because it is impossible to achieve symmetrical LQWs in spatially randomly disordered systems, temporally disordered operations are proposed using multiple quantum coins to realize symmetrical LQWs [17]. Interestingly, the concept is later expanded to quasi-periodic multiple quantum coins with Fibonacci, Thue-Morse, Rudin-Shapiro sequences [33]. It is worthwhile to stress that, experimental implementation of QWs is not quite simple even with only one quantum coin, so it would be extremely difficult with multiple quantum coins operations as proposed by the authors of [17,33]. However, the idea of employing symmetrical LQWs for quantum memory is interesting, and it is worthy of further exploration.

Our simulation results presented in Section 3 show that symmetrical LQWs can be achieved in the proposed Fibonacci arrays of waveguides – FAWs. This is significant in two aspects: (i) LQWs can be symmetrical in FAWs without applying temporal disorder which is extremely difficult due to multiple quantum coins operations. (ii) FAWs are deterministic quasi-periodic systems, meaning that we can design and control the output for every structure without the need of averaging a large number of samples as in the case of randomly disordered systems. We therefore believe the quasi-periodic systems of waveguides in general, in particular the proposed FAWs would have advantages for applications that require symmetrical LQWs, especially in comparison with the cases in which multiple quantum coins are used.

In conclusion, in this work we proposed new quasi-periodic arrays of waveguides constructed with Fibonacci sequences to realize deterministic LQWs. Although the structures of FAWs are straightforward to make, the outcome results of LQWs are predictable and controllable, in contrast with LQWs in randomly disordered systems. Quite importantly, we show that BMP is very effective method for simulation of single-photon QWs in complicated AWs. We have presented, in detail, the main properties of the FAWs. For examples, LQWs with symmetrical probability distributions can be realized in the proposed FAWs. Those are unique features of FAWs that are of potential use for applications requiring symmetrical LQWs. Note that, single photon QWs play important role for understanding the QWs effects and quantum simulations, its applications are limited in comparison with multiphoton QWs. Although our simulation results in this work are about single photon QWs, the results of symmetrical LQWs in FAWs can be used for quantum memory applications where each photons of entangled photon pairs can be stored and retrieved in two different FAWs. It is important to point out that multiphoton quantum states would increase dramatically the information-carrying capacity in quantum communication as compared with single-photon state, and therefore multiphoton states would truly offer a revolution in communication. In [29] Gräfe et al. presents a good review of both theoretical and experimental works on multiphoton QWs. Research on multiphoton QWs in our proposed FWAs are undergoing, and is the subject of other publications of ours in the future. We would, however, like to stress that the advantage of the Fibonacci scheme in this regard relates to the fact that the output photons localize around a particular output waveguide, which can be predictable and controllable with an important wavelength dependence. This greatly simplifies the experimental implementation of multiphoton quantum walks, see for example [36], where observing fermionic statistics is possible with multiphoton QWs and assembly of arrays of waveguides. Multiphoton QWs have been investigated in different systems to explore quantum superiors for applications, for examples the endurance of quantum coherence [37], entangled photons Anderson localization [38], spatial entanglement in Anderson photonic lattices [39], experimental observation of N00N state Bloch oscillations [40], etc. Also regarding quantum communications, again the fact the output localizes around a particular waveguide enables low loss unique components, for example low loss wavelength filters that are not possible with periodic structures but only with quasiperiodic structures.

Finally, we would like to stress here that the symmetrical LQWs could also be realized in other quasi-periodic systems constructed with Thue-Morse [41], Rudin–Shapiro [42] sequences if the construction rules are the same or similar to the rules proposed in this work.

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Name	Description
Visualization 1	Visualization 1 is animation simulation of quatum walks in regular array of waveguides RAW39 corresponding to the top view image shown in Fig. 5a
Visualization 2	Visualization 2 is animation simulation of quatum walks in regular array of waveguides RAW39 corresponding to the front view image shown in Fig. 5b
Visualization 3	Visualization 3 is animation simulation of quatum walks in Fibonacci array of waveguides FAW6A-I2 corresponding to the top view image shown in Fig. 5h
Visualization 4	Visualization 4 is animation simulation of quatum walks in Fibonacci array of waveguides FAW6A-I2 corresponding to the top view image shown in Fig. 5g
Visualization 5	Visualization 5 is animation simulation of quatum walks in regular array of waveguides RAW39 corresponding to the 3D photon propability distribution front view image shown in Fig. 6a
Visualization 6	Visualization 6 is animation simulation of quatum walks in Fibonacci array of waveguides FAW6A-I2 corresponding to the 3D photon propability distribution front view image shown in Fig. 6b

Localized quantum walks in quasi-periodic Fibonacci arrays of waveguides

Abstract

1. Introduction

2. Quasi-periodic arrays of waveguides constructed with Fibonacci sequences

3. Localized quantum walks in Fibonacci arrays of waveguides

4. Discussions and conclusion

References

Supplementary Material (6)

Cited By

Figures (8)

Equations (12)

Optics Express