Optical diametric drive acceleration through action reaction symmetry breaking

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1 Optical diametric drive acceleration through action reaction symmetry breaking Martin Wimmer 1,4, Alois Regensburger 1, Christoph Bersch 1, Mohammad-Ali Miri 2, Sascha Batz 1, Georgy Onishchukov 1,3, Demetrios N. Christodoulides 2, and Ulf Peschel 1 1 Institute of Optics, Information and Photonics, University of Erlangen-Nuernberg, Staudtstraße 7/B2, Erlangen, Germany 2 CREOL, College of Optics and Photonics, University of Central Florida, Orlando, Florida , USA 3 Max Planck Institute for the Science of Light, Guenther-Scharowsky-Str. 1, Bau 24, Erlangen, Germany 4 School of Advances Optical Technologies (SAOT) Table of contents 1. Supplementary Methods Experimental setup Time multiplexing Effective nonlinearity χχ Photonic band structure of the mesh lattice Analogy to a relativistic particle with constant acceleration Generation of pulse sequences with Gaussian envelope Supplementary Figures Scattering from positive and negative barriers (Fig. 2 in main paper) Nonlinear case: Soliton and diametric drive (Figs. 3 and 4 in main paper) References NATURE PHYSICS 1

2 1. Supplementary Methods The following sections provide the necessary background to understand our experimental arrangement and the dynamics of light pulses observed in it. 1.1 Experimental setup Supplementary Figure S1 Detailed scheme of the nonlinear fiber loops. Pulses circulate in two loops of optical fiber which are connected by a central 50/50 coupler. All components are fiber-coupled. PD: Photodiode; AOM: Acousto-optic modulator; PM: Phase modulator; WDM: Wavelength-division multiplexing coupler; EDFA: Erbium-doped fiber amplifier; DCF: Dispersion-compensating fiber; χχ: total nonlinearity coefficient; ISO: Optical isolator. Linear versions of the fiber-loop setup depicted in Fig. S1 have been reported in Refs. [S1-S3] These three publications are also accompanied by Supplementary Materials which contain detailed explanations on experimental methods as well as the time-multiplexing approach. Time multiplexing is implemented in experiment by using two fiber loops as depicted in Fig. S1 with average length, which are linked by a central 50/50 coupler and differ in length by. 2 NATURE PHYSICS

3 Initially, a 50 ns long signal laser pulse is produced by a signal generation block at a wavelength of. The signal generator consists of a CW distributed feedback laser which is followed by a Mach-Zehnder amplitude modulator to create the pulses. The pulse train is amplified by an erbiumdoped fiber amplifier (EDFA) booster to reach high peak powers, and subsequently filtered with a bandpass filter to remove excess noise from the EDFA. Finally, an acousto-optic modulator (AOM) operated in first diffraction order selects one specific pulse from the pulse train and controls the loop operation. This initial pulse is now inserted into the long loop with a 50/50 coupler. Each loop contains an EDFA which compensates all round-trip losses, thereby maintaining the pulse power. To reduce transients in the optical amplification, each EDFA is preceded by a wavelength-division multiplexing coupler, through which a CW pilot signal of a few milliwatt at a wavelength of is inserted. This pilot signal is also used to tune the gain in each loop. Behind the EDFA, the pilot signal is removed by a bandpass filter which also removes excess noise caused by amplified spontaneous emission. The high-power pulses are now injected into a dispersion-compensating fiber (DCF, type: OFS-HSDK, [S4]). Each fiber has length of ca. 4 km (effective length ) and a nonlinear parameter of γ = 7/(W km), so that a peak power of about 300 mw is required to induce a nonlinear phase shift of 2π. To ensure unidirectional operation, isolators are installed in both loops. The net gain and loss per round trip can be controlled in each loop with the pilot signal power and an AOM. These are operated in zeroth order, so they do not affect the signal wavelength. The short loop contains an electro-optic phase modulator, which allows applying an arbitrary phase shift to each pulse. The pulses in the right, short loop are detected by a photodiode which is attached to a 90/10 monitor coupler. In the left, long loop, the open port of the 50/50 input coupler is connected to a second photodiode to monitor the intensities of all circulating light pulses. After a single shot of the measurement has been finished, the AOMs in both loops are switched to their highly attenuating state to clear the two loops of all circulating optical pulses. During this time span, the signal generation block injects a pulse train into the loop in order to keep the average power inside the EDFAs approximately at the same level, thereby reducing transients between consecutive measurements. At the beginning of a measurement, the two AOMs inside the loops are switched back to their transparent state until a single pulse (or two pulses, depending on the experiment) has passed. This signal circulates in the loops and evolves into a sequence of optical pulses. In the first round trips a special modulation scheme is applied to the phase modulator and the AOMs to generate Gaussian pulse sequences with either negative or positive effective mass (see section 1.6). 1.2 Time multiplexing The idea of using a recursive loop scheme as a resource-efficient way for studying discrete-time optical lattices has been first introduced in the frequency domain by Bouwmeester et al [S5] and was later transferred to time steps by Schreiber et al. [S6-S8]. Figure S2 visualizes the underlying principle: The dynamics expected from a spatial mesh lattice [S9] are fully reproduced in the two-loop fiber setup (Fig. S1). NATURE PHYSICS 3

4 At each round trip, a sequence of pulses arrives at the detectors attached to the monitor ports and their temporal shape is recorded by an oscilloscope. As the arrival time of each pulse is precisely known as a function of position and round-trip number, the peak intensities can be easily extracted from the data (see Supplementary Material of Ref. [S2]). (a) (c) (b) (d) position time Supplementary Figure S2 Time multiplexing a, A spatial mesh lattice consists of waveguides or fibers which are periodically coupled to their next neighbors (all coupling ratios are 50/50). b, Similar to a random walk, pulses in the spatial lattice step to the left and right on a discrete 1D grid of positions n. c, In the coupled fiber loops with length difference ΔL, a sequence of pulses is formed which obey the same dynamics as in the spatial mesh. d, Here, pulses in the short (long) loop are advanced (delayed) in time, thus moving to the left or right on a 1D grid of time slots. The resulting interference patterns are identical to those encountered in the spatial domain. Figure is adapted from Ref. [S9]. Although the loops are quite long, with uncontrolled temporal drifts and weak protection against external vibrations, no countermeasures like active interferometric stabilization [S10,S11] have to be taken. Still stable interference is observed after 300 round trips, i.e. after the light pulses have travelled more than 1200 km inside the optical fiber. This rather surprising robustness is typical for timemultiplexing and gives an understanding of the prospects and limitations of this experimental approach. The same succession of optical components is cycled again and again by the light pulses in each loop. In the time-emulated mesh lattice, several different paths end at the same temporal position n after a certain number of round trips m. But all these paths through the network only differ in the sequential order of passages of the short and the long loop corresponding to left-right steps. The total number of round trips in the long loop (steps to the right) is always, whereas the pulse circulates in the short loop times (steps to the left). All pulses that arrive simultaneously and interfere at the central 50/50 coupler have therefore passed through all fibers and optical devices in the loops the same number of times. If these components have not changed their properties considerably within the short 4 NATURE PHYSICS

5 time span of a few milliseconds required for 100 steps guaranteed., stable interference between all light pulses is This also explains why long-term phase drifts of the setup do not become apparent in the final intensity measurements, as long as the system starts with a single laser pulse. Each of the realizations only takes place in a brief time interval, shorter than the typical time scales of external disturbances like temperature drifts or vibrations. Afterwards, averaging is performed over many such single-shot measurements, which further decreases the noise level. In contrast, the setup is extremely sensitive to all changes of the optical components that occur on time scales smaller than the duration of a single measurement shot. If the optical path length or losses of a loop change between the round trips or positions, the interference patterns change. In experiment, the fast phase modulator in the short loop allows to take advantage of this possibility to coherently manipulate pulse spreading. Moreover, the AOMs can be used to quickly control the net gain or loss of both loops. 1.3 Effective nonlinearity χχ In each round trip, light pulses propagate through an approximate length of 4 km of dispersioncompensating single-mode fiber (DCF, type: OFS-HSDK, [S4]). Because of this long interaction length and the high field concentration in the fiber s core, it is possible to achieve significant nonlinear phase shifts despite the low material nonlinearity of silica [S12]. As silica is inversion-symmetric, its optical nonlinearity is dominated by the third-order susceptibility χχ. The experimental configuration and parameters were chosen such that we can exclude influence of any further nonlinear material response like Raman and Brillouin scattering or acoustic resonance effects on the observed propagation patterns. Moreover, gain transients in the EDFAs are suppressed using pilot signals and keeping the average power of optical signal in the EDFAs nearly constant during and between the measurements. This has the consequence that in each round trip every light pulse acquires an additional nonlinear, power-dependent phase shift due to self-phase modulation. Both DCF spools have a length of, which is much longer than the total length of the remaining fiber pigtails and the standard single-mode fiber used to realize the length difference =45m. So it is fair to assume that the nonlinear phase shifts are acquired in the DCFs only. Taking into account fiber losses, this phase shift can be expressed as χχ (S1), where or are the pulse amplitudes in the short and long loop, respectively. The absorption coefficient and the nonlinearity parameter are fiber-specific parameters, the values for our fibers were taken from Ref. [S4]. Here, is the effective mode area of the fiber and is the nonlinear index coefficient, while is a constant factor arising from our choice of NATURE PHYSICS 5

6 units. The nonlinearity will thus enter the iteration equations of our previous publication on linear light transport (Eqs. (1) in the main paper and in Ref. [S1]) in the following way: χχχ χχχ χχχ χχχ (S2) Note that due to experimental imperfections (splice losses, fiber parameters), the actual peak power of pulses in the fibers can differ between the two spools of DCF fiber and the nonlinear phase shifts acquired in the short and the long loops might be slightly different. However, our numerical simulations have shown that the final pulse dynamics are rather insensitive to a difference between the nonlinearities χχ and χχ in the short and long loop. In Eqs. (2) of the main paper and in all numerical simulations we, therefore, make the approximation that the effective nonlinearities are equal in both loops (χχ χχχ ). 1.4 Photonic band structure of the mesh lattice To derive the band structure of the time-multiplexed mesh lattice in the linear optical regime (χχ χ χχ, a plane-wave expansion of Eqs. (1) in the main paper is necessary [S2, S9,S13,S14]: ( ) ( ) (S3) Here, is the wave number in position coordinate and is the propagation constant. After each double step, the plane wave is rotated by a phase of. Each mode has an eigenvector ( ) which determines the relative phase and amplitude of the plane wave in the short and long loop. The unit cell of the mesh lattice with no phase potential has a size of 2 positions by 2 steps. Inserting this ansatz into a two-step version of Eqs. (1) in the main paper and solving the determinant yields the dispersion relation [S2,S9]: The eigenvectors of these harmonic modes can also be calculated analytically as a function of the wave number : (S4) ( ) ( ) in the upper, focusing band with [ ] (S5) ( ) ( ) in the lower, defocusing band with [ ]. 6 NATURE PHYSICS

7 Supplementary Figure S3 Band structure of an empty mesh lattice. Here, no phase potential is present, i.e.. The parameter is determined by. Figure S3 shows that there are two connected bands with a band gap in between. As the mesh lattice is periodic in both and, the band structure is also periodic in both of its coordinates and no higher-order bands exist (see Figure S4). From the shape of these bands, we can infer the main features of the spatial evolution patterns in the empty mesh lattice: The slope or first derivative of a band determines the speed of a wave packet. In addition, the second derivative gives a measure for the diffractive broadening [S9] and is also proportional to the inverse of the effective mass. Supplementary Figure S4 Twofold periodicity of bands. The band structure is periodic in both its coordinates because of the high symmetry of the system. Its Brillouin zone (marked area, shown in Fig. 1c of the main paper) consists of two bands with opposite curvature. Red: positive effective mass; blue: negative effective mass. NATURE PHYSICS 7

8 1.5 Analogy to a relativistic particle with constant acceleration The diametric drive allows for a continuous acceleration of both masses and thus raises the question if there are mechanisms which will asymptotically stop the propulsion. Surprisingly, an analogy between objects which are constantly accelerated towards relativistic velocities and our system of two attracting Gaussian beams can be drawn. Special relativity imposes the speed of light as the maximum velocity. Since the acceleration of the two mechanical particles constituting the diametric drive is time independent, they can be treated as a composite object which is constantly accelerated in its moving reference frame. As calculated by Born in 1909 [S15], the solution of the relativistic equation of motion in a stationary reference frame results in a hyperbolic trajectory, (S6) where is the initial position and the start time. This relativistic trajectory is now compared to the motion of the optical diametric drive in our timemultiplexed system: In the photonic lattice, the first derivative of the photonic band structure defines the group velocity of the two beams and limits its absolute value to a maximum velocity positions per round trip. The effective mass on the other hand is proportional to the inverse of the second derivative and thus diverges at the edge of the Brillouin zone, where the speed reaches the maximum. Although the dynamics in the periodic potential are not related to special relativity, the limited velocity and the increasing mass with higher velocities coincide qualitatively with special relativity. To visualize the similarity between special relativity and the motion in the synthetic photonic lattice the hyperbolic solution is overlaid to Fig. 4 of the main paper and compared with the experimental and simulated data. The speed of light in Eq. (S6) is set to the maximum achievable velocity of. The acceleration parameter is chosen manually and equals for the experiment in Fig. 4a and for the simulation in Fig. 4b. 1.6 Generation of pulse sequences with Gaussian envelope A direct way to excite in experiments an arbitrary section of the band structure either with negative or positive effective mass is to start with a broad distribution of pulses in the loops. The Gaussian envelope ( ) has a width parameter and a phase tilt. Provided that the excitation matches the eigenvector ( either one of the bands centered at the wave number. ), it becomes possible to selectively excite narrow regions in In section 1.2 of this Supplementary Material, we mentioned that to guarantee a stable phase relationship between light pulses, the system must always start with a single laser pulse as input. Therefore, to study the spreading of a broad pulse distribution with Gaussian envelope, this sequence has to be generated inside the system itself after starting from a single pulse. 8 NATURE PHYSICS

9 A well-suited method to generate Gaussian distributions has already been introduced in Ref. [S1]. There, a single-loop version of our fiber setup with a special role of optical losses was presented. Starting from a single laser pulse, the pulses quickly approach the desired Gaussian envelope when circulating in the one-loop setup. In the two-loop setup of Fig. S1, the same dynamics can be achieved when all pulses circulating in the short loop are completely absorbed at every second round trip. We therefore set the AOM in the short loop to a highly absorptive state ( ) for every second round trip. This way, a single initial pulse indeed evolves into a broad Gaussian pulse sequence. In Fig. 2 of the main paper, the Gaussian envelopes were prepared during 80 round trips, before the actual measurement started and in Fig. 3 and 4 of the main paper, 55 round trips were used to generate of the input state. At the last round trip during which the AOM is set to maximum absorption, the step number is set to. At AOM:, we find a Gaussian pulse sequence in the long loop while the short loop is cleared by the ( ) (S7) In absence of phase modulation, this Gaussian pulse sequence excites modes at the edge of the Brillouin zone (, i.e. ). To shift the phase tilt to any arbitrary wave number along the band structure, the phase modulation needs to be set to for all steps, i.e. during the generation of the Gaussian pulse sequence (see Figs. S5 and S6). Setting the phase modulation to afterwards for shifts the phase tilt of the whole pulse sequence, finally producing a Gaussian sequence with in the long loop. Now, a superposition of modes from both bands is excited around the desired value of. To select a single band, it is necessary to match the respective eigenvector ( ). To this end, we apply a phase potential during the round trip. For the upper (focusing) band,, whereas the lower band is excited when setting. Moreover, one loop AOM can be driven to provide a damping or amplification factor of with during the round trip. This was not necessary for the relatively small initial tilt of needed for the measurements of Fig. 2, as in this case, but it becomes important when exciting modes with higher spatial frequency in a single band only (not shown). Performing one iteration of Eqs. (1) to the pulse distribution of Eq. (S7) yields the desired initial pulse distribution: ( ( )) ( ) ( ) (S8) NATURE PHYSICS 9

10 For sufficiently broad Gaussian sequences, this reduces to ( ) ( ) ( ) ( ). (S9) Depending on the choice of, both eigenvectors of Eq. (S5) can therefore be matched. Starting from, the AOMs are set to have no net gain or loss. From here on, the propagation is fully governed by Eqs. (1) or Eqs. (2) in the nonlinear case of χχχχ. In the measurement of Fig. 4, two incoherent Gaussians beams with positive and negative mass are generated to form an accelerating bound state. To achieve this, we start with two temporally separated initial laser beams which are incoherent to each other because we modulate the current of the signal laser source in between them to induce a slight frequency drift which results in the loss of temporal coherence. We then generate a Gaussian beam from each of them using the protocol described above. Afterwards, the phase shifts imprinted on the two Gaussian pulse sequences are programmed such that one excites the upper band at and the other one the corresponding mode in the lower band at the same spatial frequency. This way, both Gaussians are incoherent to each other while all pulses within the same sequence are fully coherent. Fig. S6 displays the same measurement of Fig. 3a in the main paper but also shows the round trips from to during which the Gaussian sequence is prepared by blocking the short loop during every second round trip. Supplementary Figure S5 Tailoring the Gaussian pulse sequences to excite specific modes of the band structure. a, Modulation scheme of the phase modulator. In all round trips, a phase shift of is applied 1. At, the shift becomes and afterwards, Three cases are shown: Dashed black line: excites the two bands at the edge of the Brillouin zone ( in part b with unbalanced weight. Magenta line: excites the two bands in the center ( with equal weight. Dashed green line: shifts the excitation to the 1 Because of experimental imperfections, a slight additional offset to the actual modulation value of is necessary. 10 NATURE PHYSICS

11 center as in the previous case, and an additional shift of only (green cross in part b). leads to an excitation of the upper band Supplementary Figure S6 Modulation scheme for generating a Gaussian beam. a, In the first 56 rounds the phase modulator induces a piecewise phase shift from over to 0 to ensure that only the focusing band at the center of the Brillouin zone is excited. Same modulation as the dashed green line in Fig. S5a. b,d, Every second row the AOM in the short loop (b) amplifies the pulses while the AOM in the long loop (d) eliminates them. This provides a high output power level of the pulse chain. c,e, The resulting signal in the short (c) and in the long loop (e) shows the preparation and propagation of the excited Gaussian beam. NATURE PHYSICS 11

12 2. Supplementary Figures 2.1 Scattering from positive and negative barriers (Fig. 2 in main paper) The linear phase potential applied in the measurements shown in Figure 2 of the main paper is ( ) ( ) (S10) hill valley where can be either 0 or 1 and discriminate between the four different cases a-d (see Fig. S7). The maximum amplitude of the potential is. The initial velocity or phase tilt of the two beams with Gaussian envelope is set to in experiment. The amplitude of the Gaussian envelopes is set to a low level in order to avoid nonlinear effects. Although some minor degree of self-focusing or - defocusing is visible in the experiment (Fig. S7 e-h) the completely linear simulation (χχ χ χχ Fig. S7 k-n) is in excellent agreement with the measurements. In simulations 2, the initial velocity of the left, focusing beam is set to while the velocity of the defocusing beam is assumed to be. In addition, the pulse dynamics in the photonic lattice is compared to the problem of a mechanical particle at position with speed and mass M. It is subject to a continuous version of the potential in Eq. (S10): ( ) ( ) The resulting trajectory of the particle can only be expressed by an integral expression ( ) due to the non-trivial form of the potential. Therefore, there is no closed-form expression for the trajectory of the particle. A numerically calculated trajectory of this problem is overlaid on Figs. S7e-n to provide an additional comparison between mechanical motion and the experimental and simulated results of our photonic lattice. The dimensionless masses and, the positions and as well as the initial velocities and were manually chosen for a best fit to the simulations in Figs. S7k-n. 2 Due to experimental imperfections and a slight degree of nonlinearity, the actual effective mass and initial velocity of the positive- and negative mass beams were a bit different. This was accounted for by adjusting the simulation parameters. 12 NATURE PHYSICS

13 Supplementary Figure S7 Scattering of a focusing and a defocusing beam on different potentials. (a) The external potential is switched off (A,B = 0), so that the pulse propagation (e,k) is completely undisturbed. The slight asymmetry between the two beams in (e) is caused by a very small degree of nonlinearity present in the experiment. (b) In case of a positive smooth barrier ( ) the focusing beam is totally reflected like a marble rolling upwards with insufficient speed. (c,g,m) The opposite scenario ( ) shows the acceleration of the focusing beam and the retardation of the defocusing beam until it finally reverses its movement. (d,h,n) A superposition of both potentials ( ) leads to a reflection of both beams where the focusing beam is initially accelerated in the valley. All simulations (k-n) were performed for. The data shows the pulse distribution of the short loop. Red dashed lines: Comparison to trajectories of classical particles (see text above). NATURE PHYSICS 13

14 2.2 Nonlinear case: Soliton and diametric drive (Figs. 3 and 4 in main paper) Supplementary Figure S8 Generation of a focusing and defocusing Gaussian beam and a combination of both. All measurements where performed in the linear low-power limit. a, A linear defocusing and b, a focusing Gaussian beam. The defocusing beam is slightly tilted to the left while the focusing beam tends to the right. The superposition of both beams is shown in (c). All measurements are in a good agreement with simulations for an effective nonlinearity of and a beam width of. In case of (f) the beams were initially separated by a distance of 20. The first row of every measurement or simulation is normed to a total power of ½ in each of the two loops. Only data for the short loop are shown. Supplementary Figure S9 Same as in Figure S8 for the nonlinear case. a,d, Shows in the experiment as well as in the simulation the additional defocusing due to nonlinearity. In the focusing case (b,e) a soliton is formed and propagates stable for 150 coupling lengths. The superposition of both (c,f) forms a 14 NATURE PHYSICS

15 diametric drive. Simulations where done assuming an effective nonlinearity of 1.5 for two beams (f) and half of this value in case of a single beam (d,e) because of power normalization. Additionally, an initial beam width of was assumed. The initial distance of both beams in (f) is assumed to be 20. The first row of every figure is normed to a total power of 1 in the two-loop system. Measurement and simulation show the data of the short loop. In measurements where two Gaussian pulse sequences are inserted (Figs. S8c,f and S9c,f), these two effective beams are not coherent to each other. This is achieved by generating the Gaussian sequences from two separate initial laser pulses which are slightly detuned in frequency (see section S1.6). Therefore, the Gaussians do not linearly interfere with each other. In the nonlinear regime, they interact via cross-phase modulation. In numerical simulations, this is modeled by assigning the focusing Gaussian pulse sequence with positive mass the amplitudes and for the short and long loop, while the other Gaussian has pulse amplitudes and. Now there are two sets of iteration equations coupled only via the nonlinear phase shifts: and ( ( χχχ χ )) ( χχχ χ )) ) ( ( χχχ χ )) ( χχχχ χ )) ) ( ( χχχ χ )) ( χχχ χ )) ) ( ( χχχ χ )) ( χχχχ χ )) ) A diametric drive does not form in any case. As the excitation displayed in Fig. S9f was not appropriate there is mainly a scattering of the focusing beam at the defocusing one. Therefore power is lost and acceleration stops before the maximum possible velocity is reached. To this end, another set of parameters was experimentally investigated and compared to simulations. For the case displayed in Fig. 4 and in Fig. S10 more power was injected into the negative mass beam to compensate for reflective losses which occur in the cause of the formation of the diametric drive. Indeed, a diametric drive is formed and constant acceleration (in the sense of a relativistic approximation) is observed. For comparison, the hyperbolic trajectories for a relativistic particle subject to a constant acceleration and are overlaid to both parts of the Figure (see section 1.5 of this document). The acceleration is assumed to start at row since there is no contact between the beams before. Above, a by vertical dashed line for a resting particle is drawn for steps to. NATURE PHYSICS 15

16 Supplementary Figure S10 Measurement and Simulation of the optical diametric drive. a,b, Show the experimental and the simulated results for the short loop as in Fig. 4 of the main paper. Most of the power of the defocusing beam does not contribute to the bound state while the power in the focusing band enters much more efficiently into the accelerated soliton. Therefore, the initial ratio between the power of the defocusing beam and the power of the focusing is set in simulations to 4. In experiment the ratios of the powers equals about 2 and is chosen to get best results. The assumed distance of the beams in the simulation equals 15 and the width positions. The effective nonlinearity is set to 1.2, which is slightly lower than in Fig. S9 due to the higher power of the defocusing beam. The first row of both figures is normalized to a total power of 1 in the two-loop system. The measured and simulated cross sections are shown for in the according insets (c) and (d). 16 NATURE PHYSICS

17 3. References S1. Regensburger, A. et al. Photon Propagation in a Discrete Fiber Network: An Interplay of Coherence and Losses. Phys. Rev. Lett. 107, (2011). S2. Regensburger, A. et al. Parity time synthetic photonic lattices. Nature 488, (2012). S3. Regensburger, A. et al. Observation of Defect States in PT-Symmetric Optical Lattices. Phys. Rev. Lett. 110, (2013). S4. Gruner-Nielsen, L. et al. Dispersion-compensating fibers. Journal of Lightwave Technology 23, (2005). S5. Bouwmeester, D., Marzoli, I., Karman, G., Schleich, W. & Woerdman, J. Optical Galton board. Phys. Rev. A 61, (1999). S6. Schreiber, A. et al. Photons Walking the Line: A Quantum Walk with Adjustable Coin Operations. Phys. Rev. Lett. 104, (2010). S7. Schreiber, A. et al. Decoherence and Disorder in Quantum Walks: From Ballistic Spread to Localization. Phys. Rev. Lett. 106, (2011). S8. Schreiber, A. et al. A 2D quantum walk simulation of two-particle dynamics. Science 336, 55 8 (2012). S9. Miri, M.-A., Regensburger, A., Peschel, U. & Christodoulides, D. Optical mesh lattices with PT symmetry. Phys. Rev. A 86, (2012). S10. Xavier, G. B. & Von der Weid, J. P. Stable single-photon interference in a 1 km fiber-optic Mach Zehnder interferometer with continuous phase adjustment. Optics Letters 36, 1764 (2011). S11. Cho, S.-B. & Noh, T.-G. Stabilization of a long-armed fiber-optic single-photon interferometer. Optics Express 17, (2009). S12. Agrawal, G. P. Nonlinear Fiber Optics. Nonlinear Science at the Dawn of the 21st Century 15, 30 (Academic Press: 2001). S13. Knight, P., Roldan, E. & Sipe, J. E. Propagating quantum walks: The origin of interference structures. Journal of Modern Optics 51, (2004). S14. Nayak, A. & Vishwanath, A. Quantum Walk on the Line. arxiv:quant-ph/ (2000). S15. Born, M. Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzips. Annalen der Physik 335, 1 56 (1909). NATURE PHYSICS 17