Quantum walks present novel tools for redesigning quantum algorithms, universal quantum computations, and quantum simulators. Hitherto, one- and two-dimensional quantum systems (lattices) have been simulated and studied with photonic systems. Here, we report the photonic simulation of cyclic quantum systems, such as hexagonal structures. We experimentally explore the wavefunction dynamics and probability distribution of a quantum particle located on a six-site system, along with three- and four-site systems while under different initial conditions. Various quantum walk systems employing Hadamard, C-NOT, and Pauli- gates are experimentally simulated, where we find configurations capable of simulating particle transport and probability density localization. Our technique can potentially be integrated into small-scale structures using microfabrication, and thus would open a venue towards simulating more complicated quantum systems comprised of cyclic structures.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Within the realm of condensed matter physics, there are two major approaches for tackling quantum phenomena. One approach is based on density functional theory [1,2] and ab initio methods, whereas the second one includes methods based on many-body theory and model Hamiltonians [3,4]. However, there are only a handful of systems in nature whose dynamics can be well understood by means of pure analytical methods. The absolute majority of physical systems require approximate and numerical methods, which are computationally expensive. An alternative solution to these issues can be provided by exploiting different types of simulators. Classical simulators, including classical search algorithms, are helpful when dealing with classical and deterministic phenomena such as Brownian motion and chaos theory. Various numerical techniques such as optimization methods are good examples of such classical simulators. However, given that nature is ultimately governed by quantum mechanics, one is led towards exploiting quantum simulators (QSs) in order to simulate quantum systems. While this idea was first proposed by Feynman  in 1982, it is only in the past couple of years that mature quantum technologies have led to an increased interest in the matter and to the first experimental implementations of QSs [6–8]. Among QSs, quantum walks (QWs) have recently been studied extensively and exploited in various fields, ranging from quantum computations to simulating quantum mechanical systems. For instance, a well-known and direct application of QWs in quantum computations is provided by the development of many quantum search algorithms [9–13]. Within the realm of condensed matter, QWs can be used to study a material’s topological properties, as they provide a powerful tool for understanding a variety of physical properties, e.g., electronic transport, thermal properties, and linear response to external fields [14–17]. The majority of QSs used for these systems are photonic-based QW simulators [18–30]. These simulators are designed to model the simplest solid-state systems such as one- and two-dimensional periodic monoatomic crystals. Trivial and non-trivial topologies emerging from the translational symmetry of these systems have been widely investigated via invoking the orbital angular momentum (OAM) and spin angular momentum (SAM) of photons [18–22]. A number of QSs have been successfully implemented by means of the quantum properties of photons [22–30].
As opposed to a classical random walk, a QW bares a different kind of randomness, as all possible paths in the Hilbert space of the walker are covered simultaneously by exploiting the superposition properties of the quantum walker. In the classical random walk, the probability distribution over many steps has a binomial distribution centered at the walker’s initial position and displays no signs of interference; for the QW on the other hand, the probability distribution is lower at the walker’s original position and tends to increase in a ballistic manner toward the end(s) of the lattice due to interference effects. In general, there are two classes of QWs: the first kind, known as the continuous time QW (CTQW), has been studied comprehensively within recent years [31,32]. In the second kind, known as the discrete time QW (DTQW), the walker is bound to take positions along a path consisting of a sequence of random steps. This path in its simplest form is a line with an infinite number of sites, simulating a one-dimensional lattice . Such QWs have been realized in a number of physical systems such as trapped ions [34,35] or atoms  and nuclear magnetic resonance systems . One-dimensional DTQWs of spin half-particles and their topological properties have been extensively studied in recent years [38,39]. Mathematical models for QWs over finite graphs and cycles  as well as introductory schemes to theoretical and experimental methods have been proposed [30,33,40]. Here, we introduce and study a one-dimensional DTQW in the form of a closed loop, capable of simulating wave-packet dynamics of single particles. This could be helpful for studying cyclic structures, e.g., benzene-like rings. Arranged to produce the ring-shaped structure of such systems, the lattice points are folded on both ends to form a ring of discrete sites. A QW on such a structure will be named hereinafter a cyclic quantum walk (CQW). At each step of the CQW, the walker moves either clockwise or counter-clockwise according to the outcome of a random process, such as the flip of a quantum coin. After a few steps, such a model can simulate complex quantum processes such as energy transport and quantum interference effects in ring-shaped systems. We demonstrate an experimental platform to realize the particle transport in ring-shaped geometries using single photons and linear optical components.
As shown in Fig. 1(a), our model describes the propagation of the spatial distribution of a single quantum particle, e.g., the electron, on a one-dimensional ring composed of identical “atomic sites.” An initially prepared spatial state of the particle is represented by a density cloud distributed over one or any number of the geometric sites. Naturally, atomic potentials govern the subsequent motion of the electron. In our simple model, we are not interested in the specific form of the potential produced by the atoms, but rather in their spatial periodic distribution. We couple the particle’s translational degree of freedom to its intrinsic spin state. As a result, the direction of the particle’s subsequent motion is determined solely by its spin state, which can take one of two configurations, with respect to a certain quantization axis. In a given basis, the spin state of the particle can be represented by and , indicating spin up and spin down, respectively. As the configuration space of the periodic system is in the form of a closed loop (in its simplest form, a circle), the spatial wavefunction of the particle depends on only a single spatial coordinate, namely, the azimuthal coordinate, . As a result, the system is subjected to periodic boundary conditions, which quantizes its energy eigenvalues.
Our simple model can be used to obtain approximate energy levels of a wave-packet propagating along the sites of a ring-shaped geometry. The ring behaves like a circular waveguide, allowing the wave-packet to propagate in both directions (see Supplement 1 for details). One can describe the cyclic DTQW of a particle on a ring of identical geometrical sites by a dimensional Hilbert space, , obtained by the tensor product , where and are the walker and coin subspaces, respectively. The particle, representing the walker, moves on the one-dimensional discrete lattice composed of identical sites arranged in a ring. At each step, the walker has two choices depending on the outcome state of the quantum coin, i.e., the particle’s intrinsic spin state. Thus, the -dimensional walker subspace is spanned by the position eigenstates of the particle, which are represented by the fixed positions of each site on the ring, . The two-dimensional coin subspace is spanned by the walker’s spin eigenstates given in a convenient representation, i.e., the representation in which the component of the nonrelativistic spin half-particle’s Pauli matrix is diagonal, . The general state of a quantum particle in a system composed of identical geometric sites is given by , where and represent the particle’s spatial and spin states, respectively. As our bases form a complete set, any state can be represented by superpositions of spatial and also of spin eigenstates as . Here, , , and are in general complex and time-dependent coefficients, satisfying the relations and . We describe the spatial displacement of the walker at each step by a standard conditional shift operator, , given by , where gives the remainder on the division of by . The shift operator displaces the walker from site to the neighboring site if the coin outcome is , and likewise, it displaces the walker from site to site if the outcome of the coin is , thereby effectively entangling the coin and the walker subspaces. The coin operator is given by a unitary matrix represented in the basis in which the component of the spin half-particle Pauli matrix is diagonal. A convenient choice for this operator is , where is the rotation angle, which can be set to take any value in the interval , and is the component of the Pauli spin-operator for a nonrelativistic spin half-particle . We note that this choice of the coin operator is mainly a matter of convenience; an alternative choice for the coin operator is given in Supplement 1. We have chosen to fix the rotation angle at in order to represent a standard Hadamard gate for the QW (for a more general treatment, see Supplement 1). A single step of the walk can now be described by the step operator, . After steps, the system initially prepared in the state will naturally evolve into a state given by . One can also describe the walker state in terms of its “quasi-momentum” , defined in the first Brillouin zone, [18,20,38,39]. The two representations, real space and reciprocal space , are related to each other via the discrete Fourier transforms given by and . The quasi-momentum representation has the advantage of allowing one to study the system by examining its energy band structure [38,39]. In general, standing states, oscillating states, and unidirectional traveling states are related to different combinations of the allowed quasi-momenta eigenvalues. In what follows, we present the results of the energy and velocity relations of the carrier, i.e., the particle in the studied system. A detailed derivation of the particle’s energy dispersion relation, , is given in Supplement 1.
According to the QW protocol used in this study, and for a large number of identical geometric sites, the dispersion relations of the energy eigenvalues for a Hadamard gate are given by1(b). We find that a nonzero energy gap exists between the two bands, indicating that this choice of the coin operator can potentially be related to a nontopological insulator. However, upon manipulation of the coin operator, one can find particular arrangements in which the band gap closes and the topological insulating phase, including protected edge states, is realized. Another feature that reflects the symmetry of the group velocity profile indicating two-fold degenerate energy states is related to wave-packets traveling in opposite directions but at equal speeds. Upon their interference in the periodic structure, standing waves are eventually realized. The possible eigenstates of a six-site system representing the system are specifically indicated by the quasi-momentum eigenvalues shown in Fig. 1(b). We can identify two states related to and representing stationary states (zero group velocity), whereas the states represent traveling states. The eigenstates indicated by are degenerate in energy with the previous ones; however, they describe particles propagating in opposite directions (see Supplement 1 for more details).
The eigenstates related to the energy eigenvalues are presented by an indicative unit vector , which can be assumed to show polarization directions. These vectors indicate points on the Bloch sphere whose possible orientations are the eigenstates of the studied system. The components of the indicative vector in Cartesian coordinates are given by (see Supplement 1)1(c)]. As spans the entire Brillouin zone, the number of times completes a full circle is a topological property of the system and is called [21,29] the winding number . For the six-site geometry and the CQW protocol implemented in this work, the winding number is 1, and the corresponding eigenvectors are given in Supplement 1 and in Supplement 1, Table I. The quantum state of the particle, as the Brillouin zone is traversed in an adiabatic manner, acquires a Berry phase [41–43]. In the CQW, a one-dimensional Brillouin zone is spanned by the state of the system, and thus a Zak phase is introduced . The Zak phase, , is directly proportional to . In our case, the calculated value for is , which is very close to the value calculated for the standard QW on a one-dimensional lattice (see Supplement 1 and  for details). From a theoretical perspective, one can observe that there is no fundamental difference between the calculated values of Zak phases for linear and cyclic QWs. As the number of sites in the CQW increases, the difference between the calculated Zak phase tends to vanish. This can be attributed to the periodic boundary conditions imposed on the quasi-momenta, which composes the parameter space that the Zak phase is evaluated and is identical to those for a linear QW.
Simulations of the aforementioned wave-packet dynamics in ringed systems are realized experimentally by means of invoking single photons to play the role of single quantum particles. Note that this is an accurate approach only for “single” particles, given that effects arising from the different quantum statistics of bosons and fermions are irrelevant. The particle’s spin states are simulated in our experiment by the single photon’s polarization states as both are spinors living in a 2D Hilbert space, . Here, and represent the photon’s horizontal and vertical polarization states, respectively. Figure 2 shows the sketch of the experiment used for the realization of the photonic CQW. Pairs of photons at 810 nm are generated through spontaneous parametric down-conversion by pumping a periodically pulled potassium titanyl phosphate (ppKTP) crystal with a 405 nm, 100 mW diode laser. One of the photons is sent directly to a single-photon avalanche diode (SPAD), which is used to trigger an intensified CCD (ICCD) camera for detecting its partner photon in coincidence. The other photon is coupled to a single-mode fiber (SMF) and enters the CQW setup.
In the CQW setup, the photon is incident on a spatial light modulator (SLM), where a spatial distribution pattern is generated to simulate the vertices on the cyclic system spanning the walker’s spatial degree of freedom. These vertices are accounted for on the displayed pattern as symmetrically distributed petals that can be added or removed at will with arbitrary phases to produce the initial spatial state of the walker. Thereby, the initial spatial distribution of the walker, , is prepared. After reflection from the SLM, the photon’s initial polarization state is prepared by a half-wave plate (HWP) and a quarter-wave plate (QWP). Now, the initial state of the “quantum particle” is prepared as . The photon enters a 50:50 beam splitter (BS), where half of the time it is reflected and sent to the ICCD camera to register the initial pattern, and the other half of the time it passes through to a HWP, whose action is equivalent to that of the coin operator on the initial state. The polarization is now a superposition of horizontal and vertical states, .
The photon then enters a polarizing BS (PBS), where its superposition in polarization is translated into a superposition of paths in the two arms of a Sagnac interferometer. Within the interferometer, a Dove prism (DP) is fixed at a specific rotation angle, , where is the number of sites, with respect to the beam axis. Beams from the two arms of the interferometer are rotated by an angle , one in clockwise and the other in counter-clockwise directions, i.e., the photon is in a superposition of being translated to its left and right neighboring sites. Thus, the Sagnac interferometer, together with the DP, corresponds to the conditional shift operator . As the photon exits the first interferometer, the first step of the walk is finalized. After the first interferometer, the photon passes through a HWP and enters a second, identical Sagnac interferometer, thereby performing the second step. Afterwards, the photon is guided back to the 50:50 BS, where it is either sent to the ICCD or fed back into the two interferometers to repeat the process, thus simulating the third and fourth steps, and so on. Due to inherent losses in the scheme, circulating the interferometer more than three rounds (six steps) in the current setup is impractical. However, this problem can be circumvented by simply cascading more interferometers instead of feeding back the beam in a loop.
One thing to note here is that a single photon and coherent states would have the same dynamics in a single-particle QW. It is therefore also possible to perform this experiment with a laser. However, due to the looped configuration of this setup, one would need to distinguish between the photons that have traveled through the setup once to those that have traveled through many times. Using photon pairs generated through spontaneous parametric down conversion (SPDC), this is easy to distinguish. After detecting the trigger photon by the SPAD, we simply adjust the electronic delay in the camera to match the travel time of the partner photon in the setup. The experiment can certainly be performed with a pulsed laser, but one would need to be careful of the looped-back pulses overlapping with other subsequent pulses causing contamination of the results. If the setup is not looped, then certainly one could use a laser to perform the walk, but the trade-off would be to build an interferometer for each step of the QW.
The performed experiments involved a series of measurements over specific initial conditions, namely, the special superpositions of the polarization states representing the polarization eigenstates discussed earlier. In Fig. 3, the experimental and theoretical results for a localized initial state with different polarization states, namely, linear, [Fig. 3(a)], and circular, [Fig. 3(b)] are presented and compared. Here, we prepared the initial state of the particle to be localized on a single vertex (node ), namely, a single petal in the SLM pattern of a six-site geometry. We find that in the first two steps the systems both evolve similarly. However, starting with the third step, interference effects result in different time evolutions, leading to biased propagation only for linear polarization. These results show that the evolution of the observed patterns strongly depends on the initial state of the systems. Moreover, the initial states of the walker result only in its distribution among nodes of a given parity. For odd steps, the walker always distributes itself along odd nodes; for even steps, it always distributes itself along the even nodes. This observation reminds one of the two different bonding (single or double) patterns observed in the Kekulé structure of the benzene ring, whereby single bonds and double bonds between neighboring carbon atoms are formed alternatively. For the circular polarization case, a semi-stable probability distribution over different nodal parities of the walker is observed, which can be explained as a result of circular polarization being an eigenstate of the system [see Fig. 1(c)]. In the linear polarization case, although there appears to be an oscillatory behavior of the walker over even and odd nodes, the intensity is not evenly distributed among them. The state vectors for each step are calculated in detail in Supplement 1.
Next, we realize a situation indicative of a stationary state. As for the case of the hexagonal ring, our 12-dimensional Hilbert space allows 12 such stationary states, which are the eigenstates of the single-step unitary time evolution operator of the CQW. Figure 4(a) shows the results for one of such eigenstates. The walker is initially prepared as . As an eigenstate of the system, remains intact as time goes on. Therefore, upon time evolution, “nothing” happens, a result not observed in the linear QW. This can be seen from the experimental results shown in Fig. 4, where both the spatial distribution [Fig. 4(a)] and polarization [Fig. 4(b)] remain unchanged throughout the walk. A more detailed analysis of the evolution of the walker’s quantum state for various initial conditions of a six-site ring is provided in Supplement 1. Theoretical predictions and experimental results for the three- and four-site geometries have been evaluated and are presented in Supplement 1. In particular, and as an illustration, we have examined the nonstationary behavior of the walker over a four-site system. Here, as an interesting case, we experimentally consider an evenly nonlocalized distribution over three of the nodes with a vertical polarization, . After three consecutive steps, the walker redistributes itself evenly over three nodes, two of which are the same as the original ones. However, the polarization on each site remains linear and becomes anti-diagonal. Based on interference effects and symmetry arguments, this resembles a semi-stable dynamical evolution, and the emergent distribution pattern in intensity preserves information of its initial state. The computational details for the first three steps as well as the periodic behavior of an initially localized walker for a four-site system are presented in Supplement 1.
In summary, we have theoretically proposed a platform to simulate wave-packet transport in ring-shaped systems and experimentally implemented a CQW system based on single-photon propagation through linear optical devices. With different combinations of the initial spatial state of the walker, we were able to investigate the propagation of the particle in special points along the band structure of the cyclic systems. This engineering of the initial state of the walker can be used to simulate complex QW dynamics through the combination of suitable delocalized initial conditions in addition to standard QW evolution. The simulation of finite ring-shaped structures and the controlled dynamics of particle transfer may have potential applications in the field of quantum computations, where charge transport through so-called molecular electronic devices plays an important role in high-speed and efficient communication systems. In addition, we have theoretically and experimentally investigated the topological properties of this special type of CQW and considered the properties emerging from the intrinsic symmetries of such systems, as well as the state evolution of a quantum particle in the CQW. With further modifications of the current setup, one can extend this experiment to multiple-particle propagation, as well as interactions with weak external magnetic fields. Also, experimentally implementing the role of intermolecular potentials for multi-particle systems in cyclic structures can lead to simulations of many-body dynamics in aromatic systems such as the benzene molecule. This makes simulating processes, such as valence charge transport, bond formation, and topological properties, feasible. Alternatively, and with further improvement, the current CQW can be experimentally coupled to a linear QW or another cyclic walk, thereby simulating more complex structures and topologies. With our current work, the bedrock of these possible future projects has already been laid and experimentally realized in this new and rapidly developing field combining physics, mathematics, and electronics.
Canada Research Chairs; Canada Foundation for Innovation (CFI); Canada First Excellence Research Fund (CFREF).
See Supplement 1 for supporting content.
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