1. Introduction

Quantum walks (QWs) have been used to construct exponential speedup quantum algorithms [1–3] and quantum simulations [4, 5], to implement universal gates for quantum computers [6, 7], etc. For the last few decades, scientists have made tremendous progress on research and development of those areas which are the most promising for solving problems that are intractable by classical computers. Among different schemes (or approaches) quantum photonics has the advantage of highly advanced technology that can easily generate and manipulate almost any desired photons—the walkers, even in room-temperature conditions. Discrete-time QWs (DTQW) have been demonstrated using beam splitter arrays [8, 9], and continuous-time quantum walks (CTQWs) have been investigated both theoretically and experimentally in evanescently coupled parallel waveguide arrays [10–17]. Integrated photonics lattices consisting of evanescently coupled waveguides are perfectly suited for investigation of CTQWs, and in fact, laser-written waveguides were the first systems used to demonstrate quantum walks on a line with coherent light [10–13]. In those photonics lattices, the walking process occurs in the region of evanescent-coupled waveguides. As a result, spacing between waveguides in those lattices is close enough, typically on the order of several micrometers to ensure evanescent coupling to occur. It is well established both theoretically and experimentally that in a uniform and/or periodic array of coupled waveguides (photonics lattices), the probability distribution of single-photon QWs spreads across the waveguide lattice by coupling from one waveguide to its neighbors in a pattern characterized by two strong "ballistic" lobes [11, 12].

In contrast to the normal QWs in periodic photonics lattices (PLs), quantum walks can be localized in disordered ones. This phenomenon belongs to a more general concept commonly known as Anderson localization (AL). Moreover, AL is usually discussed in terms of coherent evolution in the presence of a randomly disordered medium. By breaking the order of structures and/or the periodicity of the QWs' evolution through spatially and temporally randomizing operations, localized QWs (LQWs) have been realized theoretically and experimentally [15–17]. For the last decade, there have been increased interests and efforts on exploration of LQWs for applications. The research works have been inspired by proposals of employing LQWs for quantum communications, such as secure transmission of quantum information [18], and quantum memory [19]. Besides, LQWs in photonics lattices are also a very effective approach to simulate the AL using quantum optics.

realizations. Furthermore, our simple construction rules allow us to create symmetric QPLs. Consequently, LQWs with symmetric probability distributions can be realized deterministically in these structures. The proposed QPLs are simple and straightforward to make but would be potentially useful for many research areas in

The chapter is organized as follows. Section 1 is the introduction. The new concept of QPLs based on Fibonacci and Thue-Morse sequences is described in Section 2. In Section 3, simulation results of QWs in periodic photonics lattices (PPLs) and LQWs in the new class of photonics lattices—the quasiperiodic photonics

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

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

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

lattices (QPLs)—are presented. Section 4 is the discussion and conclusion.

other with the Thue-Morse sequence or Thue-Morse QPLs (TMQPL).

quantum communication as will be discussed in this chapter.

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

2. Periodic and quasiperiodic photonics lattices

Figure 1.

the text.

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Anderson predicted that the wave function of a quantum particle can be localized in the presence of a static disordered potential in his famous paper [20]. As a result, evolution of a quantum particle with its dual wave-particle nature through a disordered medium can be strongly suppressed, depending on the degree of the disorder. More generally, in a static disordered medium, destructive interferences among different propagating paths of a quantum particle could generate AL. Although AL has been widely studied in quantum solid-state physics, localization of light has recently been explored with potentially important applications [21]. One unique phenomenon is that deviations from periodicity may result in higher complexity and many surprising effects would arise. For decades, such deviations have been investigated in optics in the realization of photonic quasicrystals: a class of structures made from optical elements that are arranged in different patterns but lack translational symmetry. A quasiperiodic structure is neither a periodic nor a random system, so it could be considered as an intermediate between the order and disorder systems. Furthermore, it has been demonstrated that quasiperiodic crystals can also lead to the localization of light [22–24]. Examples of such quasiperiodic structures constructed with the Fibonacci sequence include one-dimensional (1D) Fibonacci quasiperiodic dielectric multilayers [25], semiconductor quantum wells [26], 2D quasicrystals [27], and 3D quasicrystals [28].

It is important to stress here that AL has been conventionally realized in randomly disordered systems, and the effect of disorder-induced localization is quantified by averaging over a large number of realizations on many systems having the same degree of disorder. Similarly, LQWs in integrated photonics lattices have been realized on many randomly disordered waveguide lattices. The results of LQWs are averaged over all realizations in those waveguide lattices that are well controlled in a defined range of randomness of the disorder. Experimentally, it is not simple as is shown in [15]. Note that an experimental method has been proposed to determine localized modes systematically in 1D-disordered waveguide lattices [16], and the method is useful for investigating LQWs in randomly disordered lattices. As mentioned earlier, optically quasiperiodic structures or quasicrystals can provide deterministic disorders deviated from periodicity. Localization of light has been realized deterministically in those quasicrystals [23–25], and there is no need to average over a large number of structures. In that spirit, we propose a new class of quasiperiodic photonics lattices (QPL) to realize LQWs deterministically. As presented in this chapter, the new class of QPLs could be useful for quantum communication applications as proposed recently in [18, 19]. The new quasiperiodic structures are constructed symmetrically with Fibonacci, Thue-Morse, and other quasiperiodic sequences. Benefits of using the new class of QPL for realizing LQWs are twofold: (i) LQWs can be realized deterministically and therefore are highly programmable and optimizable, and (ii) it is much simpler for realization of LQWs in comparison with random disordered systems as there is no need to do averaging over many

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

In contrast to the normal QWs in periodic photonics lattices (PLs), quantum walks can be localized in disordered ones. This phenomenon belongs to a more general concept commonly known as Anderson localization (AL). Moreover, AL is usually discussed in terms of coherent evolution in the presence of a randomly disordered medium. By breaking the order of structures and/or the periodicity of the QWs' evolution through spatially and temporally randomizing operations, localized QWs (LQWs) have been realized theoretically and experimentally [15–17]. For the last decade, there have been increased interests and efforts on exploration of LQWs for applications. The research works have been inspired by proposals of employing LQWs for quantum communications, such as secure transmission of quantum information [18], and quantum memory [19]. Besides, LQWs in photonics lattices are also a very effective approach to simulate the AL using quantum optics. Anderson predicted that the wave function of a quantum particle can be localized in the presence of a static disordered potential in his famous paper [20]. As a result, evolution of a quantum particle with its dual wave-particle nature through a disordered medium can be strongly suppressed, depending on the degree of the disorder. More generally, in a static disordered medium, destructive interferences among different propagating paths of a quantum particle could generate AL. Although AL has been widely studied in quantum solid-state physics, localization of light has recently been explored with potentially important applications [21]. One unique phenomenon is that deviations from periodicity may result in higher complexity and many surprising effects would arise. For decades, such deviations have been investigated in optics in the realization of photonic quasicrystals: a class of structures made from optical elements that are arranged in different patterns but lack translational symmetry. A quasiperiodic structure is neither a periodic nor a random system, so it could be considered as an intermediate between the order and disorder systems. Furthermore, it has been demonstrated that quasiperiodic crystals can also lead to the localization of light [22–24]. Examples of such quasiperiodic structures constructed with the Fibonacci sequence include one-dimensional (1D) Fibonacci quasiperiodic dielectric multilayers [25], semiconductor quantum wells

Advances in Quantum Communication and Information

[26], 2D quasicrystals [27], and 3D quasicrystals [28].

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It is important to stress here that AL has been conventionally realized in randomly disordered systems, and the effect of disorder-induced localization is quantified by averaging over a large number of realizations on many systems having the same degree of disorder. Similarly, LQWs in integrated photonics lattices have been realized on many randomly disordered waveguide lattices. The results of LQWs are averaged over all realizations in those waveguide lattices that are well controlled in a defined range of randomness of the disorder. Experimentally, it is not simple as is shown in [15]. Note that an experimental method has been proposed to determine localized modes systematically in 1D-disordered waveguide lattices [16], and the method is useful for investigating LQWs in randomly disordered lattices. As mentioned earlier, optically quasiperiodic structures or quasicrystals can provide deterministic disorders deviated from periodicity. Localization of light has been realized deterministically in those quasicrystals [23–25], and there is no need to average over a large number of structures. In that spirit, we propose a new class of quasiperiodic photonics lattices (QPL) to realize LQWs deterministically. As presented in this chapter, the new class of QPLs could be useful for quantum communication applications as proposed recently in [18, 19]. The new quasiperiodic structures are constructed symmetrically with Fibonacci, Thue-Morse, and other quasiperiodic sequences. Benefits of using the new class of QPL for realizing LQWs are twofold: (i) LQWs can be realized deterministically and therefore are highly programmable and optimizable, and (ii) it is much simpler for realization of LQWs in comparison with random disordered systems as there is no need to do averaging over many

realizations. Furthermore, our simple construction rules allow us to create symmetric QPLs. Consequently, LQWs with symmetric probability distributions can be realized deterministically in these structures. The proposed QPLs are simple and straightforward to make but would be potentially useful for many research areas in quantum communication as will be discussed in this chapter.

The chapter is organized as follows. Section 1 is the introduction. The new concept of QPLs based on Fibonacci and Thue-Morse sequences is described in Section 2. In Section 3, simulation results of QWs in periodic photonics lattices (PPLs) and LQWs in the new class of photonics lattices—the quasiperiodic photonics lattices (QPLs)—are presented. Section 4 is the discussion and conclusion.
