2. Periodic and quasiperiodic photonics lattices

In this section, we will present the general construction of photonics lattices (PLs) that will be used for our numerical investigation of single-photon quantum walks (QWs). Figure 1 shows diagrams of several PLs that are the subject of our investigation. Figure 1(a) and (b) shows diagrams of two different types of PPLs: type-I consists of identical waveguide PPLs (IPPL), and type-II are lattices of two (or more) different waveguide PPLs (DPPL). Figure 1(c) and (d) shows two different QPLs, one with the Fibonacci sequence or Fibonacci QPLs (FQPL) and the other with the Thue-Morse sequence or Thue-Morse QPLs (TMQPL).

In Figure 1, the structures are arrays of single-mode (SM) waveguides having core diameter a, center-to-center distance between cores d, and index difference between core and clad Δn. In the structures composed of two different waveguides as in the cases of DPPL, FQPL, and TMQPL, each waveguide is characterized by V-number. For example, waveguides A and B are characterized by V<sup>A</sup> = πaANAA/λ and V<sup>B</sup> = πaBNAB/λ, respectively, where aA(B) stands for core diameter and NAA(B) is the numerical aperture of waveguide A(B). Note that the numerical aperture NA can be determined by the index difference between core and clad Δn, and we will use Δn to characterize waveguides in our calculations. For example, the IPPL shown

#### Figure 1.

Diagrams of different photonics lattices of single-mode waveguides having diameter a and center-to-center distance d. (a) IPPL with all cores have the same index difference Δn. (b) DPPL is composed of two different cores A (blue) and B (yellow) having ΔnA and ΔnB, respectively. (c) Sixth-order FQPL of 39 cores composed of A and B waveguides. (d) Fourth-order TMQPL of 29 waveguides composed of A and B waveguides. The red arrows indicate the position of the input signal. The construction rules for FQPL and TMQPL are explained in the text.

in Figure 1(a) is a regular array of 39 identical SM waveguides with a = 4 μm, d = 8 μm, and Δn = 0.0035. The DPPL of 39 cores in Figure 1(b) is a periodic array of two different waveguides A (blue) and B (yellow) with Δn<sup>A</sup> = 0.0045 and Δn<sup>B</sup> = 0.0035, respectively. Both A and B have the same a = 4 μm and d = 8 μm. Those parameters are used in all numerical simulations in this work. Figure 1 also shows a Fibonacci QPL of 39 waveguides of sixth order and a Thue-Morse QPL of 29 waveguides of fourth order. Both FQPL and TMQPL are constructed with two different waveguides A and B whose properties are defined above.

Below we will describe the construction rules for our quasiperiodic photonics lattices based on Fibonacci and Thue-Morse sequences. It is worth mentioning that these rules have been first proposed to construct one-dimensional Fibonacci quasiperiodic multiple dielectric layers [22–25] and Fibonacci quasiperiodic arrays of waveguides [29, 30]. In general Fibonacci QPLs and Thue-Morse QPLs (TMQPL) are constructed recursively with Fibonacci and Thue-Morse sequences, respectively, starting with two different single-mode waveguides A and B. We can easily write down the formulae of the elements and their corresponding structures as in Figure 2 for the first-order elements of FQPLs and TQPLs in the upper and lower panels, respectively.

We now define a new j th-order quasiperiodic photonics lattice as [29, 30]

$$F\_j = \mathbb{S}\_j \mathbb{S}\_{j-1} \cdots \mathbb{S}\_2 \mathbb{S}\_1 \mathbb{S}\_2 \cdots \mathbb{S}\_{j-1} \mathbb{S}\_j,\tag{1}$$

Rudin-Shapiro sequence, etc. Details of the new construction rules and the meaning of the deterministic disordered nature of such QPL can be seen in [29, 30]. Next, in Section 3, we will first present the method to simulate single-photon QWs in those PLs using the beam propagation method (BPM). We will then show in detail our simulations of QWs in IPPLs and DPPLs in comparison with FQPLs and TMQPLs. Especially, we will show unique localized quantum walks in FQPLs and TMQPLs

3. Localized quantum walks in quasiperiodic photonics lattices

The most well-known example of classical random walks on a line consist of a walker (e.g., a particle) walking to either the left or right depending on the outcomes of an unbiased toasting coin (probability system) with two mutually exclusive results, i.e., the walker moves according to a probability distribution. At each step of the walking process, an unbiased coin is tossed, and the walker makes consecutive left-or-right decisions depending on the result of the coin (up or down), respectively. For classical random walks, it is well known that both 1D and 2D distributions are Gaussian distributions [31]. For the quantum version of the random walk—the quantum walks (QWs)—the main components of discrete-time quantum walks (DTQWs) are a quantum particle, "the walker," a quantum coin, evolution operators for both walker and quantum coin, and a set of observables. QWs can also be in another form that has no classical counterpart, such as the CTQWs which have been extensively investigated in photonics lattices [10–17]. In contrast to DTQWs, CTQWs (QWs for short in the following) have no coin operations, and the walking evolutions are defined entirely in a position space where continuous coupling between vertices or lattice sites is required. Integrated photonics lattices consisting of evanescently coupled 1D and 2D arrays of waveguides are perfectly suited for the investigation of QWs. In such structures, spacing between waveguides typically is on the order of several micrometers or is in strong

In this chapter, we will restrict ourselves to the single-photon QWs in both periodic and quasiperiodic PLs. First, we would like to emphasize that singlephoton QWs do not behave any differently from classically coherent wave propagation and the distribution of light intensity corresponds to the probability distribution of photons that can be detected [32]. It is important to stress, however, that in multiple-photon QWs (indistinguishable or entangled photons), truly nonclassical features will appear. Although single-photon QWs have limited features, the effects are still very important not only for understanding but also for

Let us briefly describe how single-photon QWs can be modeled mathematically and realized experimentally in PLs. Below, we follow the description from Refs. [11, 12, 32] which are excellent references on QWs in PLs. In general, the

Hamiltonian description of the problem of quantum walks in a PL can be written as

nan þ ∑ m

<sup>n</sup>(anÞ is the creation (annihilation) operator of a photon in the nth-waveguide, β<sup>n</sup> is the propagation constant of the nth-waveguide, and κnm is the coupling coefficient between nearest neighbor sites n = m � 1. The propagation of a

κnma† nam (2)

βna†

single photon can be described by the Heisenberg equation as below:

with symmetrical probability distribution.

Quantum Walks in Quasi-Periodic Photonics Lattices DOI: http://dx.doi.org/10.5772/intechopen.87758

coupling regimes for evanescent couplings to occur.

H ¼ ℏ ∑ n

quantum applications.

where a†

63

where S1, S<sup>2</sup> … Sj are Fibonacci and Thue-Morse elements defined in Figure 2. As an example, Figure 1(c) is a diagram of the sixth-order Fibonacci QPL with 39 cores. It is clear from Figures 1 and 2 that the photonics lattices constructed with Eq. (1), with Sj being the elements of Fibonacci and Thue-Morse sequences, are quasiperiodic and symmetrical.

In general, we can apply the same rules, e.g., Eq. (1), to construct symmetrical quasiperiodic photonics lattices based on other quasiperiodic sequences, such as the

Figure 2.

First-order elements of FQPLs (upper) and TMQPLs (lower) composed by two waveguides A and B.

Quantum Walks in Quasi-Periodic Photonics Lattices DOI: http://dx.doi.org/10.5772/intechopen.87758

in Figure 1(a) is a regular array of 39 identical SM waveguides with a = 4 μm, d = 8 μm, and Δn = 0.0035. The DPPL of 39 cores in Figure 1(b) is a periodic array of two different waveguides A (blue) and B (yellow) with Δn<sup>A</sup> = 0.0045 and Δn<sup>B</sup> = 0.0035, respectively. Both A and B have the same a = 4 μm and d = 8 μm. Those parameters are used in all numerical simulations in this work. Figure 1 also shows a Fibonacci QPL of 39 waveguides of sixth order and a Thue-Morse QPL of 29 waveguides of fourth order. Both FQPL and TMQPL are constructed with two

Below we will describe the construction rules for our quasiperiodic photonics lattices based on Fibonacci and Thue-Morse sequences. It is worth mentioning that these rules have been first proposed to construct one-dimensional Fibonacci quasiperiodic multiple dielectric layers [22–25] and Fibonacci quasiperiodic arrays of waveguides [29, 30]. In general Fibonacci QPLs and Thue-Morse QPLs (TMQPL) are constructed recursively with Fibonacci and Thue-Morse sequences, respectively, starting with two different single-mode waveguides A and B. We can easily write down the formulae of the elements and their corresponding structures as in Figure 2 for the first-order elements of FQPLs and TQPLs in the upper and lower

where S1, S<sup>2</sup> … Sj are Fibonacci and Thue-Morse elements defined in Figure 2. As an example, Figure 1(c) is a diagram of the sixth-order Fibonacci QPL with 39 cores. It is clear from Figures 1 and 2 that the photonics lattices constructed with Eq. (1), with Sj being the elements of Fibonacci and Thue-Morse sequences, are

In general, we can apply the same rules, e.g., Eq. (1), to construct symmetrical quasiperiodic photonics lattices based on other quasiperiodic sequences, such as the

First-order elements of FQPLs (upper) and TMQPLs (lower) composed by two waveguides A and B.

th-order quasiperiodic photonics lattice as [29, 30]

Fj ¼ SjSj�<sup>1</sup>⋯S2S1S2⋯Sj�<sup>1</sup>Sj, (1)

different waveguides A and B whose properties are defined above.

Advances in Quantum Communication and Information

panels, respectively.

Figure 2.

62

We now define a new j

quasiperiodic and symmetrical.

Rudin-Shapiro sequence, etc. Details of the new construction rules and the meaning of the deterministic disordered nature of such QPL can be seen in [29, 30]. Next, in Section 3, we will first present the method to simulate single-photon QWs in those PLs using the beam propagation method (BPM). We will then show in detail our simulations of QWs in IPPLs and DPPLs in comparison with FQPLs and TMQPLs. Especially, we will show unique localized quantum walks in FQPLs and TMQPLs with symmetrical probability distribution.
