A synthetic gauge field for two-dimensional time-multiplexed quantum random walks

TL;DR

This paper demonstrates a 2D topological quantum random walk using synthetic gauge fields, leading to multiple bandgaps, spatial confinement, and topological edge states, expanding the simulation capabilities of photonic systems.

Contribution

It introduces a novel 2D quantum random walk with synthetic gauge fields, showcasing topological features not previously realized in time-multiplexed photonic systems.

Findings

Synthetic gauge fields induce multiple bandgaps.

Spatial confinement observed in the random walk distribution.

Topological edge states appear at domain interfaces.

Abstract

Temporal multiplexing provides an efficient and scalable approach to realize a quantum random walk with photons that can exhibit topological properties. But two dimensional time-multiplexed topological quantum walks studied so far have relied on generalizations of the Su-Shreiffer-Heeger (SSH) model with no synthetic gauge field. In this work, we demonstrate a 2D topological quantum random walk where the non-trivial topology is due to the presence of a synthetic gauge field. We show that the synthetic gauge field leads to the appearance of multiple bandgaps and consequently, a spatial confinement of the random walk distribution. Moreover, we demonstrate topological edge states at an interface between domains with opposite synthetic fields. Our results expand the range of Hamiltonians that can be simulated using photonic random walks.

Equations

$$E_{\pm }=\pm \arccos \left(\sin \left(k_x\right)\sin \left(k_y\right)\right),$$

$$E_{n,\pm }=\frac{n\pi }{2}\pm \frac{1}{4}\arccos \left(1-\frac{1}{2}{\sin }^2\left(2k_x\right){\sin }^2\left(2k_y\right)\right)$$

Full text

Photonics provides a compelling platform to study quantum random walks Aharonov et al. (1993). Photons can propagate over long distances without losing coherence, enabling complex random walks that can implement various quantum computing algorithms Kempe (2003); Shenvi et al. (2003); Childs et al. (2003), and also simulate a broad range of quantum Hamiltonians Childs et al. (2013). Photonic random walks in both one and two dimensions can be implemented in spatial degrees of freedom using beam splitters De Nicola et al. (2014); Sansoni et al. (2012); Broome et al. (2010) or coupled waveguide arrays Poulios et al. (2014); Tang et al. (2018a, b). But such approaches are difficult to scale to large number of steps, particularly when going to higher dimensions.

Synthetic spaces provide an alternative approach to scale the state-space of the walker without requiring complex photonic circuits. Examples of synthetic spaces include frequency Navarrete-Benlloch et al. (2007); Bouwmeester et al. (1999); Lin et al. (2016, 2018); Yuan et al. (2018), orbital angular momentum Goyal et al. (2013); Cardano et al. (2015, 2016, 2017), and transverse spatial modes as recently experimentally realized Lustig et al. (2018). Time-multiplexing is another synthetic space that is particularly easy to work with Schreiber et al. (2010, 2011, 2012); Nitsche et al. (2016); Barkhofen et al. (2017); Chen et al. (2018). Time-multiplexed random walks have the advantage that they can span an extremely large state-space with only a few optical elements and can efficiently scale to higher number of walker dimensions.

Recently, time-multiplexed quantum walks have been used to explore topological physics and the associated edge states in both one and two dimensional systems Barkhofen et al. (2017); Chen et al. (2018). Most realizations of such topological quantum walks are based on the split-step quantum walk protocol Kitagawa et al. (2010a, b, 2012); Kitagawa (2012). Similar to the Su-Shreiffer-Heeger (SSH) model, here the non-trivial topology is a result of the direction-dependent hopping strength between the lattice sites. However, many of the most interesting topological Hamiltonians, such as the integer quantum Hall effect Klitzing et al. (1980), the Haldane model Haldane (1988), and the quantum-spin Hall effect Konig et al. (2007), require gauge fields that generate direction-dependent hopping phases. So far, time-multiplexed random walks with synthetic gauge fields have only been implemented in 1D, which severely restricts the number of topological Hamiltonians that can be explored.

Here, we demonstrate a two-dimensional topological synthetic gauge field in a time-multiplexed quantum walk. We show that in our discrete-time quantum walk, the pseudo-energy band structure exhibits multiple bandgaps depending on the magnitude of the synthetic gauge field. These bandgaps result in the confinement of the random walker, as opposed to ballistic diffusion that would otherwise occur Yalcnkaya and Gedik (2015). Moreover, we demonstrate the presence of multiple topological edge bands at an interface between two domains with opposite magnetic fields. Because of the presence of two topological edge bands, our system supports two sets of non-degenerate topological edge states that travel in forward and backward directions along the interface.

To implement a gauge field, pulses must accumulate a net phase shift when walking around a closed trajectory. Figure 1a shows how we implement this condition. We apply a phase shift of $y\phi$ when the walker moves to the right, and $-y\phi$ when the walker steps to the left, where $y$ is the vertical coordinate of the walker. This phase convention realizes a uniform magnetic field in the Landau gauge. Consequently, this phase modulation scheme can be harnessed as an artificial gauge field affecting the evolution of the random walk.

In our time-multiplexed photonic random walk, optical delays map the walker state-space into time-delays of optical pulses. Similar to earlier studies Schreiber et al. (2012); Chen et al. (2018) of two dimensional quantum random walks, we implement these delays using a pair of nested fiber delay loops. Figure 1b shows the schematic of the experimental setup, and the full details are explained in the Supplementary Material sup . The experimental setup essentially consists of two beam-splitters with their ports connected to fibers of different lengths such that they map the $\pm x$ and $\pm y$ directions to different time delays. One complete propagation of an optical pulse around the loop is then equivalent to hopping of the walker to one of the four possible corners in the synthetic space (Fig. 1a). Two semiconductor optical amplifiers (SOAs) are employed to partially compensate for the losses that the optical pulses experience in each round trip. In this setup, we study the random walk distribution at each time step via two photodetectors analyzing two channels that we refer to as the up and down channels as labeled in Fig. 1b. We initialize the random walk through a single incident laser pulse that is injected into the up channel starting the evolution of random walk distribution from the origin in synthetic space. Here, we have analyzed the evolution of the random walk based on the pulses detected in the down channel. The electro-optic modulators which are driven by programmable voltage waveforms are used for producing the desired phase shifts to generate the synthetic gauge field.

Figure 2 compares the evolution of the random walk distribution with and without an applied gauge field. Figure 2a shows the experimental results for the evolution under no applied gauge field. In this figure, the distribution of the random walker is shown at three different time steps of 1, 5, and 9. In the absence of a gauge field, the random walk exhibits rapid diffusion. These results are consistent with the theoretical predictions shown in Fig. 2b. Figure 2c shows the experimental results for the evolution of the random walk distribution in the presence of a gauge field with $\phi=\pi/2$. The gauge field leads to suppressed diffusion and confinement of the random walk distribution. The experimental results shown in Fig. 2c for the case of a gauge field with $\phi=\pi/2$ are consistent with the theoretical predictions shown in Fig. 2d.

We calculate the fidelity of the measured distributions relative to the theoretical distributions ($P_{th}$) based on $F\left(n\right)=\sum_{x,y}\sqrt{P_{th}\left(x,y;n\right)P_{exp}\left(x,y;n\right)}$. For the case of no applied gauge field, we obtained fidelities of 0.999, 0.996, and 0.985 for time steps of 1, 5, and 9, respectively. Similarly, for the random walk under the gauge field we determined fidelities of 0.999, 0.993, and 0.972 for time steps of 1, 5, and 9, respectively.

To provide a more quantitative analysis of the effect of the gauge field on the particle confinement, we calculate the variation of the spatial quadratic means of the random walk distribution as a function of time step. Figures 3a and 3b plot the quadratic means in the $x$ and $y$ directions with gauge fields of $\phi=0$ and $\phi=\pi/2$. With no applied gauge field, the quadratic means show nearly linear variation with the time step, consistent with ballistic diffusion (See Supplementary Material sup for analytical explanation). But under the application of the gauge field with $\phi=\pi/2$, the quadratic means show reduced diffusion. The decrease of the quadratic means in both directions is due to the confinement of the random walk’s distribution under a constant pseudo magnetic field. Figures 3a and 3b confirm the agreement of the experimental results with the theoretical predictions, both with and without the effect of the gauge field. (See also Fig. S3 for the numerical study of the variation of quadratic means over a larger number of steps)

In order to better understand the confinement of the random walker in the presence of a gauge field, we first calculate the band structure of the random walk. The full evolution of the walker is determined by the single-step propagation matrix $U$, which advances the random walker distribution by one time-step. According to Floquet band theory, the single-step propagation matrix determines the effective Hamiltonian from $U=e^{-iH_{eff}}$, which gives the band structure of the walker. With no synthetic gauge field ($\phi=0$), we can analytically solve for the dispersion relation of the walker (See theoretical analysis section in the Supplementary Material sup ). The quasi-energy bands of the system in this case are

[TABLE]

where $k_{x}$ and $k_{y}$ are the momentum wave vectors in inverse synthetic space. Figure 3c shows the corresponding band structure of the system. Because of the discrete nature of the random walk, the quasi-energy spectrum wraps every $2\pi$, and therefore we restrict the quasi-energies to the range of $-\pi$ and $\pi$. As Fig. 3c shows, the system is gapless, and there are four Dirac points, two at $E=0$ and two at $E=\pm\pi$.

We next consider the effect of the synthetic gauge field on the band structure. Figure 3d shows the band diagram for the case of $\phi=\frac{\pi}{2}$. An analytical solution for this case also exists (see Supplementary Material sup ), with a quasi-energy band structure given by

[TABLE]

for $n\in\mathbb{Z}$. Similar to the case of integer quantum Hall effect, the introduction of a gauge field produces a series of topological bands. For $\phi=\frac{\pi}{2}$, we observe four doubly degenerate bands. However, because of the wrapping of pseudo-energy, one set of bands is split and appears close to energies $\pm\pi$. In contrast to the zero gauge field, the band structure in the presence of a synthetic gauge field exhibits bandgaps that lead to the confinement of the random walker. We have also obtained the corresponding band diagrams for several other choices of $\phi$. We have presented these results (See Fig. S2) along with their derivation in the Supplementary Material sup .

One consequence of a gauge field is the presence of edge states at the boundaries. In this synthetic space, we can make a boundary by applying two different gauge fields to two neighboring regions. Here, using a phase modulation pattern of $\phi_{y}=y\phi$ for $y>0$ and $\phi_{y}=-y\phi$ for $y<0$, we realize two domains with opposite magnetic fields ($y>0$ and $y<0$), as illustrated in Fig. 4a. Figure 4b shows the band structure for such a phase pattern with $\phi=\frac{\pi}{2}$. The band diagram contains multiple bandgaps hosting unidirectional edge states that propagate at the boundary in opposite directions. The corresponding band diagrams for several other choices of $\phi$ are also presented in the Supplementary Material sup (See Fig. S4).

Figure 4c shows experimentally measured results for the phase modulation pattern shown in Fig. 4a. We start the random walker at the interface between the two magnetic domains, precisely where edge states should be present. In this case the random walker predominantly walks along the edge, remaining confined to the boundary between the two regions. These results are consistent with the numerical simulations demonstrating how the edge states cause the random walk distribution to move mainly along the boundary (Fig. 4d).

Typical topological quantum walks result in unidirectional edge state propagation. Here, however, we do not see unidirectional movements because we are initializing the walker at a position eigenstate, which is a superposition of all energy eigenstates of the band structure. As can be seen from Fig. 4b, different energy bands support topological edge states propagating in either the left or right direction. We could excite specific edge modes by engineering the initial distribution of the random walk to be confined in corresponding energies. Additionally, more complicated phase modulation patterns can be harnessed to produce sharp edges in the synthetic space.

In conclusion, we have implemented time-multiplexed two-dimensional quantum random walks with a synthetic gauge field. This gauge field leads to the confinement of the walker evolution. Through application of an in-homogeneous gauge field on this random walk, we observed the creation of topological edge states that are confined at the boundary of two distinct gauge fields. These results demonstrate a versatile approach to create various types of band structures with tunable number of bandgaps. In order to increase the number of steps, we used optical amplifiers to compensate for round-trip losses without damaging the phase coherence of the optical pulses. These losses can be reduced by decreasing coupler losses through fiber splicing and the use of modulators with lower insertion loss. Eliminating these losses opens up a path towards quantum random walks that can be implemented at the single photon level, or in higher dimensions. Addition of optical nonlinearities and integration of this platform with single photon emitters could provide another interesting opportunity to study topological band structures with optical interactions Chalabi and Waks (2018); Pichler and Zoller (2016). Ultimately, our results expand the toolbox of quantum photonic simulation and provide a scalable architecture to study photonic random walks with non-trivial topologies.

This work was supported by the Air Force Office of Scientific Research-Multidisciplinary University Research Initiative (Grant FA9550-16-1-0323), the Physics Frontier Center at the Joint Quantum Institute, the National Science Foundation (Grant No. PHYS. 1415458), and the Center for Distributed Quantum Information. The authors would also like to acknowledge support from the U.S. Department of Defense.