## Abstract

Control over the duration of a quantum walk is critical to unlocking its full potential for quantum search and the simulation of many-body physics. Here we report quantum walks of biphoton frequency combs where the duration of the walk, or circuit depth, is tunable over a continuous range without any change to the physical footprint of the system—a feature absent from previous photonic implementations. In our platform, entangled photon pairs hop between discrete frequency modes with the coupling between these modes mediated by electro-optic modulation of the waveguide refractive index. Through control of the phase across different modes, we demonstrate a rich variety of behavior: from walks exhibiting enhanced ballistic transport or strong energy confinement, to subspaces featuring scattering centers or local traps. We also explore the role of entanglement dimensionality in the creation of energy bound states, which illustrates the potential for these walks to quantify high-dimensional entanglement.

## INTRODUCTION

A quantum particle can exist in a superposition of paths, or modes, and interference between the probability amplitudes of these outcomes results in phenomena unique to random walks of quantum systems (*1*–*3*)—enhanced propagation, otherwise called ballistic transport (*4*), or Anderson localization, where the wave function becomes confined in a disordered system (*5*, *6*). Quantum walks of two or more particles can exhibit nonclassical phenomena such as bunching or antibunching for bosons and fermions, respectively (*7*). The complex dynamics observed in these walks cannot be explained by classical models and, therefore, can serve as a probe of entanglement or interactions between particles (*1*, *7*–*11*). Owing to the variety of nonclassical behavior that they can exhibit, quantum walks have the potential to provide a marked speedup in certain computational tasks like physical database searches (*12*) and tests of graph isomorphism (*13*).

Quantum walks come in two flavors—continuous and discrete (*14*). Discrete-time quantum walks can evolve through a sequence of discrete events characterized by a “coin flip” that scatters the walker into adjacent modes. In continuous-time quantum walks, the state evolves through continuous tunneling to neighboring modes. These walks have been observed in atomic systems (*1*), and their depth is determined by the evolution time of the quantum state. Continuous evolution of the quantum state, coupled with the ability to choose an arbitrary walk duration, allows one to explore a wide range of parameter space. Consequently, these walks are especially well suited to simulating Hamiltonian dynamics (*9*) and solving certain black-box problems exponentially faster (*15*). To achieve even comparable performance with discrete quantum walks, additional system complexity is required, primarily through an extra degree of freedom (*14*).

Walks of correlated particles have also been implemented in photonic systems, as they offer a more robust platform in terms of decoherence and room temperature operation (*2*, *3*, *9*, *16*, *17*). However, photonic quantum walks demonstrated to date suffer from the drawback that their circuit depth can only be incremented by physically altering the footprint of the system. Here, we report continuous quantum walks with photon pairs entangled across multiple, discrete frequency modes (*8*, *18*) where the effective duration of the walk is fully tunable without necessitating any physical change to the system. With arbitrary control of the phase across different modes, we demonstrate walks exhibiting enhanced ballistic transport or strong energy confinement, as well as subspaces featuring scattering or trapping of two-photon correlations. We also explore the role of entanglement dimensionality in creating energy bound states—states whose energy remains unaltered over the duration of a walk—and show that their sensitivity to multilevel entanglement hints at the potential for these walks to quantify entanglement in high-dimensional systems.

## RESULTS

### Frequency domain quantum walks with tunable depth

A photon can “walk” across different modes in any one of its many degrees of freedom, whether it be time (*2*, *3*), path (*8*, *16*, *17*, *19*), orbital angular momentum (*20*), or frequency (*21*–*23*). All that is required to observe such behavior is the presence of coupling between different modes in the particular degree of freedom. In the case of a quantum walk in the frequency domain, this coupling is mediated by a periodic (temporal) modulation of the waveguide refractive index. Such coupling, or mode splitting, can be realized in electro-optic phase modulators driven with a single sinusoidal radio frequency (RF) tone. The effect of this perturbation is that the wave function of a photon traversing the waveguide picks up a factor of *e*^{iδcosωmt}. Here, δ corresponds to the strength of the modulating RF field, and ω_{m} denotes the frequency of this RF modulation. Viewed from the perspective of the frequency domain, phase modulation scatters a single frequency into a comb-like spectrum with adjacent frequency modes separated by ω_{m} in frequency (Fig. 1A). The amplitude of a comb line a distance *n*ω_{m} away from the original frequency is given by *n*th-order Bessel function J_{n}(δ). In analogy to quantum walks based on path encoding, the depth of such a frequency domain quantum walk can be incremented simply by cascading one modulator after another. However, a particular strength of the frequency domain approach is that a cascade of *n* identical phase modulators is equivalent to increasing the strength of the modulating RF field in a single modulator by this factor of *n*. In other words, the depth of the walk can be tuned over a continuous range by simply modifying the strength of the modulating RF field. This is in contrast to quantum walks in the spatial domain where evolution of the state is determined by propagation length (*9*, *24*).

Figure 1A shows results from quantum walks of a single photon that starts out in a single-frequency mode. As the strength of the modulating RF field (δ) increases, the extent to which the input mode scatters to outer frequency modes also increases. In Fig. 1A, the output photon distribution is plotted as a function of modulation strength δ, which, in our platform, is equivalent to the effective walk duration. The “rabbit ears” observed in this distribution signify the presence of ballistic energy transport—a signature of random walks with quantum systems. In particular, the standard deviation (SD) of the output photon distribution grows linearly with δ (*25*). For a classical random walk, transport to neighboring modes is not nearly as fast and its SD grows only as *25*).

Quantum walks of entangled particles, particularly those featuring high-dimensional entanglement, exhibit a richer variety of behavior than is possible with just a single photon occupying a single-frequency mode. To explore the effect of entanglement on quantum walks in the frequency domain, we studied the evolution of two-photon quantum frequency combs, commonly referred to as biphoton frequency combs (BFCs) (Fig. 1B), as they traverse the system.

### Quantum walks and multilevel entanglement

BFCs have been generated directly in on-chip optical microresonators (*26*–*29*) or carved from continuous down-conversion spectra (*30*). For results reported here, BFCs were generated by the latter approach as it allows flexibility in the choice of comb linewidth (Δλ) and free spectral range (FSR). Broadband time-energy entangled photons (~5 THz) were generated by type-0 down-conversion in a periodically poled lithium niobate (PPLN) waveguide. In this process, a pump photon from a continuous wave laser (~775 nm) is converted into a pair of daughter photons in the telecommunications band (~1520 to 1580 nm). As energy is conserved in this process, the energies (frequencies) of the daughter photons must add up to that of the pump photon. In other words, the two photons in an entangled pair are anticorrelated in frequency. This two-photon spectrum is carved into a BFC using a Fourier transform pulse shaper. The 3-dB linewidth of each frequency mode is set to 9 GHz—the resolution limit of our pulse shaper. To ensure minimal cross-talk between adjacent modes, the FSR of the BFC is chosen to be 25 GHz.

The pulse shaper is used to manipulate not only the amplitude of the biphoton spectrum but also its phase before any quantum walk. In particular, the spectral phase can be set to vary continuously or to make discrete jumps from one mode to the next. Once the desired state has been prepared, it is sent to an electro-optic phase modulator that implements the mode-mixing operations, which give rise to a quantum walk. Our modulator is driven with a 25 GHz sinusoidal RF waveform, identical to the FSR of the BFC, with the RF power tunable over a continuous range. Downstream of the modulator is a second pulse shaper, which selects a pair of output frequencies and routes each one to a superconducting nanowire single-photon detector (SNSPD). Two-photon events between different frequency modes are identified by correlations in their arrival time, and these data are used to construct a measurement of the joint spectral intensity (JSI) of the BFC—a two-photon correlation map that illustrates the effects of a quantum walk in energy (frequency) space.

*Enhanced ballistic energy transport*. As noted earlier, one hallmark of a quantum walk is the observation of ballistic energy transport of the quantum state across modes of the system. We demonstrate even stronger transport for the case of a BFC entangled across eight dimensions and having the form *31*).

To quantify this energy transfer, we tabulate the total number of events along the antidiagonal and along each line parallel to the antidiagonal, to determine the probability of a biphoton exiting the system with a particular total energy. Energy transfer between the quantum circuit and the two-photon state, expressed in terms of the sum of photon mode indices, is plotted in Fig. 2D as a function of modulation depth δ. A similar walk for a fixed circuit depth was previously demonstrated with high-dimensional, path-entangled photon pairs (*8*).

*Energy bound state*. We break new ground in photonic quantum walks by demonstrating the opposite of ballistic energy transport—strong confinement of the biphoton energy. To achieve this, we modify the spectral phase of the BFC to create a state of the form

*Time-domain perspective*. The evolution of the biphoton, as depicted by two-photon correlation maps (Fig. 2, B and C), can also be understood from a time-domain illustration of the quantum walk that considers the effect of electro-optic phase modulation on the time correlation function of entangled photons (*32*). In Fig. 3, the strength of the modulating RF waveform is shown (in black) as a function of time. The signal photon, which can arrive at the modulator at any time owing to the random nature of the pair generation process, is designated by a blue arrow. In Fig. 3, we only show one possible arrival time. Here, for example, the signal reaches the modulator when the phase of the modulating RF waveform is π/6. While the idler photon also reaches the modulator at a random time, its arrival is highly correlated with that of the signal photon. This correlation is characterized by a distribution of possible values for the delay between signal and idler. The distribution, in delay space, is given by the Fourier transform of the complex biphoton spectrum (*33*). Consequently, for a narrowband biphoton spectrum with entanglement across a limited number of dimensions (“8-dimensional entanglement” in Fig. 3), there is a wide range of possible values for the relative delay between signal and idler. As the entanglement dimensionality of the state increases, i.e., as the biphoton spectrum gets broader, the distribution of possible delays gets narrower (“64-dimensional entanglement” in Fig. 3). The discretization of the biphoton spectrum in frequency space, owing to its comb-like structure, results in a distribution of relative arrival times that repeats at integer multiples of the inverse comb FSR. Since the spacing between comb lines matches the frequency of the RF waveform, this repetition of the distribution in arrival times occurs at integer multiples of the modulation period. The net effect is that both photons “see” nearly the same phase modulation slope (*d*ϕ/*dt*), which means that they experience correlated instantaneous frequency shifts (*34*) (“correlated quantum walk” in Fig. 3).

For the energy bound state, the situation is slightly different. Here, there is a π phase difference between adjacent comb lines, which corresponds to a linear spectral phase ramp or simply a time delay. This time delay corresponds to exactly half the RF modulation period. In other words, the distribution in the relative arrival of signal and idler is now spaced at half-integer multiples of the modulation period. Here, unlike in the case of enhanced ballistic transport described earlier, photons in an entangled pair experience anticorrelated instantaneous frequency shifts (“anticorrelated quantum walk” in Fig. 3), which manifests through confinement of two-photon correlations to the antidiagonal (sum-frequency axis) of a JSI measurement.

*Controllable diffusion in energy subspaces*. The cases of enhanced ballistic scattering (Fig. 2B) and energy confinement (Fig. 2C) are a good illustration of the role that spectral phase plays in the evolution of two-photon correlations. While the diffusion of correlations along, or perpendicular to, the antidiagonal represents the most straightforward forms of transport, they encompass but a subset of possible behavior that one can observe. For example, by preparing BFCs with quadratic spectral phase, we observe remarkable features in two-photon correlations—distinct energy subspaces featuring ballistic scattering or energy confinement. Figure 4 (A and B) shows results from these quantum walks with a 16-dimensional entangled state. Increasing the number frequency modes across which the photons are entangled (16 compared to 8 in preceding experiments) allows us to clearly delineate regions exhibiting correlated energy transfer (scattering) from those exhibiting anticorrelated energy transfer (confinement). In Fig. 4B, we use an additional linear phase pattern, reduce the dispersion, and increase the modulation depth to pin a single site where correlations are “trapped” to the center of the JSI.

While the two-photon correlation maps in Fig. 4 (A and B) use quadratic spectral phase, one can certainly use more complex spectral phase patterns to engineer the features of quantum walks. As a result, the relative delay between the photon pair is controlled by the frequency derivative of the biphoton spectral phase in each frequency subspace. In other words, each input frequency mode pair, or lattice point, exhibits modified scattering amplitudes that depend on the relative phase of the modulating RF waveform. This allows one to simulate certain features of walks with inhomogeneous potentials.

*Effects of high-dimensional entanglement*. The critical role played by spectral phase hints at strong differences between quantum walks featuring coherent superpositions of multiple frequency pairs ∣*m*, − *m*⟩_{SI}, as compared to mixtures of those same frequency pairs. While both states have identical frequency correlations, in the latter, the relative phase between any two basis states (∣*m*, − *m*⟩_{SI} and ∣*m*′, − *m*′ ⟩_{SI} for *m* ≠ *m*′) is completely random. To simulate the effect of this random phase, we construct a JSI measurement of the mixed state by adding together JSI measurements resulting from quantum walks of individual frequency pairs ∣*m*, − *m*⟩_{SI} for *m* = 1, …,8 (see the Supplementary Materials). A clear effect of incoherence is that two-photon correlations are smeared out without any sharp or well-defined features (Fig. 4C).

These results, together with walks presented earlier, suggest that the total energy of the two-photon state after a quantum walk can serve as an indicium of the coherence between frequency mode pairs. One metric to quantify this is the SD of biphoton energy measured at the output. This is presented in terms of the mode index (single-photon case) and the sum of mode indices (two-photon case), as function of walk duration, in Fig. 5A. We see a clear indication of enhanced energy transfer for the two-photon state. In the limit of infinite multilevel entanglement, energy transfer increases at twice the rate for two-photon correlations compared to the single-photon quantum walk since both photons experience exactly the same frequency shift.

To elucidate the effect of the entanglement dimensionality, we present data for biphoton energy transfer in the cases of enhanced ballistic scattering (Fig. 5C) and strong energy confinement (Fig. 5D). Figure 5B shows how the SD of the output biphoton energy changes as the degree of multilevel entanglement increases for a fixed walk depth (δ = 6.1). The clear change in the distribution of biphoton energies as a function of entanglement dimensionality, especially in the case of the energy bound state, points to how these results can potentially be used to certify, or even quantify, high-dimensional frequency-bin entanglement (*28*, *29*).

## DISCUSSION

Here, we have explored the use of electro-optic modulation in a waveguide to realize single-photon and two-photon quantum walks in the frequency domain. Unlike analogous systems in the spatial domain, where the effective walk duration is determined by propagation length, our spectral platform reaches arbitrary walk depths without the need for any physical reconfiguration of the system. All that is needed is a change in the voltage swing of the modulating RF waveform. Furthermore, our experiments were carried out primarily using commercial telecommunications equipment, with little need for any specialized fabrication. Consequently, we were able to delve more deeply into the role of mode-dependent phase on two-particle quantum walks that feature multilevel entanglement.

An avenue ripe for further exploration relates to entanglement certification. The most straightforward way to certify high-dimensional frequency-bin entanglement, under the assumption of a symmetric noise model, is through a measurement of the time correlation function. However, such a measurement becomes challenging as the degree of multilevel entanglement (and biphoton bandwidth) increases because features in the substructure of the time correlation function become finer than the timing jitter of conventional SNSPDs (~60 to 80 ps). Consequently, time-domain techniques are limited in their ability to probe high-dimensional entanglement (*35*). While frequency mixing techniques have been developed to overcome this limitation (*28*–*30*), they become onerous for high-dimensional states. The quantum walks demonstrated here and energy bound states in particular show promise as a tool to probe the joint temporal correlation of broadband, high-dimensional quantum frequency combs. Our results show that the width or timing uncertainty in the time correlation function is mapped to the spread in biphoton energies at the output of a quantum walk. While a rigorous proof is needed to establish the validity of this technique for entanglement certification and quantification (*36*), these walks clearly allow one to probe temporal features in the biphoton that cannot be resolved by direct measurements.

Another key feature of our system is that the uniformity of potentials at each lattice site, i.e., the respective mode coupling coefficients, is not limited by any fabrication tolerances as it is in the case of coupled waveguide arrays. This homogeneity can be a disadvantage from the standpoint of demonstrating phenomena like Bloch oscillations and Anderson localization (*1*, *6*)—effects that stem from site-to-site variations in lattice potentials. However, quantum walks in the spectral domain can accommodate inhomogeneous potentials by using resonant structures (*22*) or loop-based architectures where an element capable of suitable frequency transformations (*37*) is updated on each roundtrip that photons make through the loop. Further complexity can also be introduced by using a second degree of freedom, like the time domain, to provide a high-dimensional coin flip (*38*, *39*). These walks can be used to implement quantum algorithms that certify isomorphism between two high-degree strongly regular graphs (*13*), for example.

In sum, quantum walks in the frequency domain offer an interesting and alternate route for studying quantum walk behaviors. Our spectral platform offers more versatility from the standpoints of choosing an arbitrary walk depth and manipulating the phase across all modes in states featuring a high degree of multilevel entanglement. In addition to applications directed to search (*12*) and simulation (*7*), the phenomena that we demonstrated also show potential for quantifying entanglement in high-dimensional systems.

## MATERIALS AND METHODS

The experimental setup is depicted in Fig. 6. We use a continuous-wave 775-nm laser with about 1-mW power shining on a PPLN crystal to generate broadband time-frequency entangled photons with about 40-nm (5-THz) bandwidth, with a power of about 5 nW. A pulse shaper is then used to carve this spectrum to make a BFC with 25-GHz frequency spacing between the bins and about 9-GHz linewidth. The pulse shaper is also able to manipulate individual frequency bins’ phase. One advantage to the method of state preparation described above is that the degree of multilevel entanglement is limited by the biphoton bandwidth (~5 THz) and the resolution of the pulse shaper (~10 GHz). In other words, with an FSR of 25 GHz, one could prepare a 100-dimensional entangled state.

The duration of the walk is determined by the strength (voltage) of the RF waveform, which, in our experiments, corresponded to a value of up to 6.1 radians. This determines the extent to which distant modes are coupled to one another with a crude metric being the index of the most populated mode in a single-mode and single-particle quantum walk. For the equipment used in our experiments, this index was ~5, which is comparable to that in other platforms (*3*, *17*, *20*, *24*). However, with even small upgrades to our equipment (cabling and amplifiers rated to 25 GHz), one can achieve modulation depths ~10. Furthermore, one can achieve even higher modulation depths by simply connecting more phase modulators in series, as is frequently done in the case of electro-optic comb generation (*40*).

After making the high-dimensional entangled state, it is sent to the quantum walk circuit, which is a phase modulator driven with a 25-GHz RF sinusoidal waveform. After the quantum walk, another pulse shaper picks two frequency bins at a time and sends them to superconducting single-photon detectors (SNSPDs) (Quantum Opus). The relative arrival time of photons on the SNSPD pair is then monitored using an event timer (PicoQuant HydraHarp 400).

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/29/eaba8066/DC1

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**We thank P. Kwiat, R. Ma, C. Reimer, S. Bogdanov, and C. Cortes for valuable discussions.

**Funding:**This work was supported, in part, by the National Science Foundation under award number 1839191-ECCS.

**Author contributions:**P.I. developed the idea and designed the experiment. P.I., N.B.L., and M.S.A. performed the experiments with help from D.E.L. P.I., M.S.A., N.B.L., and A.M.W. carried out the theoretical work, analyzed the data, and prepared the manuscript. A.M.W. supervised the project.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. The data and analysis codes used in this study are available from the corresponding author upon request.

- Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).