Quantum walks in synthetic gauge fields with three-dimensional integrated photonics

There is great interest in designing photonic devices capable of disorder-resistant transport and information processing. In this work we propose to exploit three-dimensional integrated photonic circuits in order to realize two-dimensional discrete-time quantum walks in a background synthetic gauge field. The gauge fields are generated by introducing the appropriate phase shifts between waveguides. Polarization-independent phase shifts lead to an Abelian or magnetic field, a case we describe in detail. We find that, in the disordered case, the magnetic field enhances transport due to the presence of topologically protected chiral edge states that do not localize. Polarization-dependent phase shifts lead to effective non-Abelian gauge fields, which could be adopted to realize Rashba-like quantum walks with spin-orbit coupling. Our work introduces a flexible platform for the experimental study of multiparticle quantum walks in the presence of synthetic gauge fields, which paves the way towards topologically robust transport of many-body states of photons.

Figure 1

(a) Accumulated phase acquired by the quantum walker going around two examples of closed trajectories with opposite chirality. This phase depends only on the chirality and the number of elementary cells inside the loop, as in the Aharanov-Bohm effect experienced by a charged particle in a constant magnetic field. (b) In the proposed implementation of the 2D DTQW, the links of the lattice are divided into four groups; each group is depicted with its respective color and number. The motivation is that each site of the lattice corresponds to one waveguide and each link to a beam splitter per time step. Since a beam splitter can only couple two waveguides, the links are divided into four different groups, which together cover the whole lattice. Each group corresponds to a set of commuting beam splitters that implement a unitary $U_i$ from Eq. (5). (c) Diagram of a part of the proposed 3D photonic circuit that realizes the DTQW on a 2D lattice. The $z$ axis represents the direction of time, while the $xy$ plane represents the two spatial dimensions where the QW takes place. One step of the DTQW is composed of four substeps. In each substep, the waveguides are coupled according to the different groups of links depicted in (b). In this scheme, three out of the four different groups of links (beam splitters) from (b) are depicted, with its respective color and number. Each substep implements a set of mutually commuting beam splitters, according to Eq. (5), which can be applied simultaneously. Subsequent substeps could be implemented in the same way along the $z$ direction.

Figure 2

In an ordered lattice, a magnetic field hinders the spreading of the wave function and quantum transport; in a disordered lattice, a magnetic field enhances transport due to the presence of nonlocalized edge states. (a) Variance of the single-photon wave function vs the number of time steps for different values of the magnetic flux $\varphi$. The initial state is localized at the center of the lattice. For nonzero $\varphi$ the QW spreads at a slower rate. (b) Transport efficiency $\eta$ as a function of time in the presence of Gaussian disorder with strength $\delta =0.1$ (defined in Appendix pp2). The quantity $\eta$ measures the accumulated norm at the opposite corner, or the efficiency of quantum transport across the lattice. In the presence of disorder, the applied magnetic field qualitatively enhances transport.

Figure 3

Average distance between photons for the two-photon quantum walk on the IPC, after 20 steps, in a lattice of size $30\times 30$, for different values of the magnetic flux $\varphi$. The initial state is localized at positions $(1,1)$ and $(2,1)$ of the lattice and the photons' polarization states are entangled in a symmetric (antisymmetric) way so that the exchange statistics of the wave function is bosonic (fermionic). We observe, as expected, that for bosonic statistics the photons tend to remain closer than in the fermionic case. Also, the presence of the magnetic field greatly increases the distance between the photons, as each particle overcomes Anderson localization due to the nonzero magnetic field.

Figure 4

Spectrum of the effective Hamiltonian $H_{\mathrm{eff}}=i{log}_{\mathrm{e}}{U}_{\text{step}}$ that generates the Abelian QW proposed here, as a function of the magnetic flux per plaquette $\varphi$. We use units in which the time step $\mathrm{\Delta }T$ and $\hslash$ are 1.

Figure 5

Simulation of the IPC implementation of a DTQW in a 2D lattice in zero magnetic field without disorder, starting at position $(X,Y)=(1,1)$. The probability distribution of a discrete-time QW on a $60\times 60$ ordered lattice is shown after (a) 20, (b) 40, and (c) 60 time steps. The quantum walk propagates quickly from one corner of the lattice to the opposite corner. The white color corresponds to probabilities above 0.01, which can go up to 0.11 in (a), 0.056 in (b), and 0.016 in (c). The total probability for each plot is (a) 1, (b) 1, and (c) 0.77. The coordinates $X$ and $Y$ are merely waveguide labels, so we do not consider units of distance here.

Figure 6

Simulation of the IPC implementation of a DTQW in a 2D lattice in zero magnetic field with disorder, starting at position $(X,Y)=(1,1)$. The probability distribution of a discrete-time QW on a $60\times 60$ disordered lattice is shown after (a) 20, (b) 40, and (c) 60 steps. The strength of disorder is $\delta =0.2$ and the magnetic field is set to zero. The presence of disorder hinders the propagation on the lattice and the photon remains localized close to its starting position, resulting in a low transport efficiency to the opposite corner. The probability amplitude is averaged over 20 disorder realizations. Note that the scales of the plots are different, for better visualization. The total probability for each plot is (a) 1, (b) 1, and (c) 0.997. The coordinates $X$ and $Y$ are merely waveguide labels, so we do not consider units of distance here.

Figure 7

Simulation of the IPC implementation of a DTQW in a 2D lattice in nonzero magnetic field with disorder, starting at position $(X,Y)=(1,1)$. The probability distribution of a discrete-time QW on a $60\times 60$ disordered lattice with magnetic field is shown after (a) 50, (b) 100, and (c) 150 steps. The strength of disorder is $\delta =0.2$ and the magnetic field is $\varphi =\pi /5$. The quantum walk propagates mainly along the edges of the lattice. The presence of topologically protected edge states that do not localize allows for the propagation of the photon to the opposite corner of the lattice. Here we investigate a larger number of steps as compared to Figs. 5 and 6, in order to see the propagation to the opposite corner of the lattice $(X,Y)=(60,60)$. The probability amplitude is averaged over 20 disorder realizations. Note that the scales of the plots are different, for better visualization. The total probability for each plot is (a) 1, (b) 1, and (c) 0.984. The coordinates $X$ and $Y$ are merely waveguide labels, so we do not consider units of distance here.

Figure 8

Probability that the two photons are at the edge of the lattice for the quantum walk on the IPC, after 20 steps, in a lattice of size $30\times 30$, for different values of the magnetic flux $\varphi$. If the photons' polarization states are entangled in a symmetric (antisymmetric) way, their statistics is bosonic (fermionic). The presence of the magnetic field increases significantly the probability that the two photons are at the edge. However, the exchange statistics of the two-photon wave function does not affect this quantity much.