Measuring the winding number in a large-scale chiral quantum walk
Xiao-Ye Xu, Qin-Qin Wang, Wei-Wei Pan, Kai Sun, Jin-Shi Xu, Geng Chen, Jian-Shun Tang, Ming Gong, Yong-Jian Han, Chuan-Feng Li, Guang-Can Guo
MMeasuring the winding number in a large-scale chiral quantumwalk
Xiao-Ye Xu,
1, 2
Qin-Qin Wang,
1, 2
Wei-Wei Pan,
1, 2
Kai Sun,
1, 2
Jin-Shi Xu,
1, 2
Geng Chen,
1, 2
Jian-Shun Tang,
1, 2
Ming Gong,
1, 2
Yong-Jian Han,
1, 2, ∗ Chuan-Feng Li,
1, 2, † and Guang-Can Guo
1, 21
CAS Key Laboratory of Quantum Information,University of Science and Technology of China,Hefei 230026, People’s Republic of China Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China,Hefei 230026, People’s Republic of China (Dated: August 29, 2018)
Abstract
We report the experimental measurement of the winding number in an unitary chiral quantumwalk. Fundamentally, the spin-orbit coupling in discrete time quantum walks is implemented viabirefringent crystal collinearly cut based on time-multiplexing scheme. Our protocol is compactand avoids extra loss, making it suitable for realizing genuine single-photon quantum walks at alarge-scale. By adopting heralded single-photon as the walker and with a high time resolutiontechnology in single-photon detection, we carry out a 50-step Hadamard discrete-time quantumwalk with high fidelity up to 0.948 ± a r X i v : . [ qu a n t - ph ] A ug opological phases show distinctive characteristics that are beyond the Landau-Ginzburg-Wilson paradigm of symmetry breaking [1–3]. In topological phases, there is no local orderparameter exhibited in the conventional phases due to symmetry breaking, and these phasesare distinguished by some non-local topological invariants [4] of their ground state wave-functions which are quantized and robust. The classification of the topological phases isone of the main tasks in this field. Significant progresses have been made toward theircomplete classification in recent years [5]. For a non-interaction system with some symme-tries, such as particle-hole, time-reversal or chiral symmetry, a “period table” of topologicalphases has been given [6, 7]. Understanding the topological phases not only is a fundamentalproblem but also has potential important applications in quantum information for their ro-bustness. Such topological features have been explored in condensed matter systems [8–10],high-energy physics [11], photonic systems [12, 13] and atomic physics [14–16]. Nevertheless,direct measurement of the topological invariants in these systems remains a difficult task.Quantum walks (QWs) [17], which naturally couple the spin and movement of particles,provide a unique platform to investigate the topological phases in non-interaction systemswith certain symmetries [18]. It can reveal all topological phases occurring in one- and two-dimensional systems of non-interaction particles. Topologically protected bound states [19]between two different bulk topological phases have been observed in QWs. The statisticalmoments of the final distribution in the lattice are introduced for monitoring phase transi-tions between trivial and non-trivial topological phases [20]. However, exploring topologicalphases by the topological invariants of the bulk observable is still a current challenge [21–23]and only few examples in QWs have been demonstrated. Probing the topological invariantsthrough the accumulated Berry phase in a Bloch oscillating type QW has been proposedin [24] and realized subsequently in [25]. The topology in the non-Hermitian QWs hasalso been investigated recently [26–30]. It is proposed to understand the topology in QWsusing scattering theory [21], which has been demonstrated recently in [31]. As a periodi-cally driven one-dimensional(1D) system, the complemented determination of its topologyhas been well studied theoretically [32–37] with an experimental demonstration presentedrecently through measuring the mean chiral displacement (proportional to the Zak phase)in an unitary QW [38].In this work, we report a novel platform for QWs and experimentally show that it is apowerful tool to investigate the topological characters in 1D discrete time quantum walks2 a) ~13ns~5ps Step Step Step CalciteHWP …… (b) State Preparation TomographyL1BBO1BBO2L2DML3Filer@820nm L5PBS HWP QWP Calcite FC PMTQuantum Walks ModuleStep Step N BBO3 K n i f e E dge Prism R R m S M F N o r m a li z ed P r obab ili t y Position . FIG. 1. (a) diagram of the experimental setup with (b) presenting the compact time-multiplexingprotocol by using birefringent crystals. The system contains four parts: 1. Second harmonic gener-ation in BBO1 for obtaining the ultraviolet pulse; 2. Preparation of the heralded single photons byadopting the beam-like SPDC in BBO2; 3. Time-multiplexing QWs realized by birefringent crys-tals; 4. Ultra-fast detection of the arriving time of the single photons via up-conversion in BBO3.BBO: β -BaB O , L: lens, DM: dichroic mirror, PBS: polarization dependent beam splitter, HWP:half-wave plate, QWP: quarter-wave plate, R: reflector, FC: fibre collimator, SMF: single modefibre, PMT: photomultiplier tubes, SPAD: single-photon avalanche diode. Detail descriptions areprovided in supplementary. (DTQWs), such as, directly obtaining the topological invariants. There have been manyapproaches to realize photonic QWs (reviewed in [39]). Our scheme is based on the frame-work of time multiplexing, which is free of mode matching. Although there have been manyreports [40–43], one obstacle limiting their developments in scale and rejecting the employ-ment of genuine single photons as the walker is the extra loss induced by the loop structure.For overcoming this obstacle, we propose using birefringent crystal to realize spin-orbit cou-pling. With its features of compactness and free of extra loss, our method can be used torealize large-scale QWs of single photons. The photon’s polarization is adopted as the coinspace and the coin tossing in each step can be varied arbitrarily by wave plates. Experi-mentally, for detecting and analyzing the single-photon signals with time intervals of a fewpicoseconds, we adopt the frequency up conversion single-photon detector for tomographyof the spinor state in each lattice site [44].Particularly, the complete wave-function of the system can be reconstructed in real spaceby local tomography of the spinor state in each site [45] and interference measurements be-tween the nearest neighbor sites [46], then the wave-function in quasi-momentum space can3
25 -20 -15 -10 -5 0 5 10 15 20 250.000.020.040.060.080.100.12 N o r m a li z ed P r obab ili t y Position (a) (b)
FIG. 2. (a) histogram of the experimentally measured final probability distribution after a 50-stepHadamard QW starting from the origin with the spinor initialized in | L (cid:105) = √ ( | H (cid:105) + i | V (cid:105) ) (purpleline guides the theoretical expectation). (b) shows the reconstructed spinor states (purple points)for each quasi-momentum k (increasing in the first Brillouin zone indexed by the number) aftera 20-step split-step QW with θ = π/ , θ = π/
18. The theoretical vectors are shown with thearrows. be obtained via Fourier transform. Concretely, suppose the system after t -step walks isin state | Ψ t (cid:105) = (cid:80) x p t ( x ) e − iφ t ( x ) | ψ t ( x ) (cid:105) ⊗ | x (cid:105) . For each site x there is a normalized localspinor state | ψ t ( x ) (cid:105) with a complex amplitude p t ( x ) e − iφ t ( x ) ( p t ( x ) is a real valued quantity),where | ψ t ( x ) (cid:105) = cos[ θ t ( x ) / |↑(cid:105) + e iδ t ( x ) sin[ θ t ( x ) / |↓(cid:105) with θ t ( x ) ∈ [0 , π ] and δ t ( x ) ∈ [0 , π ).Experimentally, we obtain these parameters through three steps (details are given in supple-mentary): firstly, we perform local tomography on the spin for each site and get a count set S ; then we shift all of the spin-up components a step backward and perform local tomogra-phy again to obtain an additional count set ˜ S (local interference measurements); finally, wecarry out a numerical global optimization program based on simulated annealing algorithmto find the optimal pure state | Ψ t (cid:105) from the two count sets S + ˜ S . Actually, our reconstruc-tion method is an interferometric approach [31, 47]. In addition, it can be systematicallyimproved by increasing the rank of the target density matrices [48, 49] (current pure statesituation corresponds to rank 1).The layout of the apparatus is shown in Fig. 1. The signal photon of the photon-pair frombeam-like spontaneous parametric down conversion (SPDC) [50] is adopted as the walker,whose polarization can be initialized to any state by a typical polarizer. The birefringentcrystals in collinear cut are used to realize the spin-orbit couplings and HWPs are insertedbetween them for coin tossing. The walker’s final spinor states for all sites (time bins withequal interval 5 ps ) are analyzed with a polarizer and the corresponding amplitudes aremeasured with an up-conversion single-photon detector [51]. Our method in realizing time-4 a) (b) (c) (d) (e) FIG. 3. Topological characterizations of a split-step DTQW in the standard time-frame. (a)-(c)depict the energy bands (upper) and the corresponding eigenvectors (bottom) in three differenttopological phases, whose diagram formed by the various combinations of the two rotation anglesis presented in (d). The guiding arrows in the energy band show the different winding featurewith their exact forms n ( k ) presented in the bottom Bloch spheres. (e) sketches the method forreconstructing n ( k ) through final states from three different steps ( S - S ). multiplexing QWs can avoid the extra loss in previous schemes and its collinear feature, asa result, without mode matching, can guarantee the visibility and stability in the interfer-ence. Here, we report, for the first time in a photonic system, a conventional HadamardDTQW [17] of heralded single photons on the scale of 50-step with high fidelity. The finalprobability distribution is presented in Fig. 2(a). We compare the experimental distribution P exp with its theoretical prediction P th by the classical indicator similarity [42], defined as S = [ (cid:80) x (cid:112) P th ( x ) P exp ( x )] , which gives 0.948 ± k after a 20-stepQW starting from the origin. The fidelity defined as |(cid:104) Ψ th | Ψ exp (cid:105)| is 0 . ± . U ( θ , θ ) = T − R ( θ ) T + R ( θ ). T ± := (cid:80) x ( | x ± (cid:105) (cid:104) x | ⊗ |±(cid:105) (cid:104)±| + | x (cid:105) (cid:104) x |⊗|∓(cid:105) (cid:104)∓| ) is the shift operator with | + (cid:105) = |↑(cid:105) , |−(cid:105) = |↓(cid:105) and R ( θ ) is the coin tossing(here we can take it as a spinor rotation along σ y without loss of generality). Completely clas-sifying the topological phases in such a periodically driven system, we apply two nonequiva-lent shifted time-frames as [33, 35]: U (cid:48) ( θ , θ ) = R ( θ / T − R ( θ ) T + R ( θ /
2) & U (cid:48)(cid:48) ( θ , θ ) = R ( θ / T + R ( θ ) T − R ( θ / σ x ), which is in-dependent of the system’s parameters, can then be defined. Therefore, we can define thecorresponding winding numbers ν (cid:48) and ν (cid:48)(cid:48) through the Berry phases accumulated by theeigenvectors n ( k ) as the quasi-momentum k runs from − π to π in the first Brillouin zone.Two invariants ν = ( ν (cid:48) + ν (cid:48)(cid:48) ) / ν π = ( ν (cid:48) − ν (cid:48)(cid:48) ) / A ( θ ) in the standard time-frame)for different topological phases are shown in Fig. 3.As a periodically driven system, the time evolution operator U can be represented bythe evolution of an effective Hamiltonian as: U ( θ , θ ) = e − iH eff ( θ ,θ ) with H eff ( θ , θ ) = (cid:82) π − π dk [ E ( k ) n ( k ) · ˆ σ ] ⊗ | k (cid:105)(cid:104) k | . Physically, n ( k ) defines an axis of rotation for each k , aroundwhich the spinor states are revolved, as shown in Fig. 3(e). As a result, the spinor states foreach given k with different steps will be constrained to lie on a plane that is perpendicularto n ( k ) (the normal vector of the plane). Conversely, for fixed θ , the eigenvectors n ( k )for each k can be determined with the plane formed by at least three different spinor states(which can be reconstructed in our platform). The sign for n ( k ) (‘plus’ or ‘minus’ corre-sponding to two eigenvectors) remains uncertain. Resorting to the continuation of n ( k ) in k space and assuming the direction of n ( k ) (where k can be arbitrarily chosen) is fixed,the entire eigenvectors can then be uniquely determined. With the obtained eigenvectorsfor every k , we can directly read out the winding number. Obviously, the method proposedhere can be directly extended to the high winding number ( ≥
2) situations. We need tonote here that our method can be taken as a kind of dynamical measurement of system’s6 b) (a) (c) (d) FIG. 4. Theoretical predictions (upper rows) and experimental results (lower rows) for directlymeasuring the topological invariants with (a)&(b) in a standard time-frame and (c)&(d) in ashifted time-frame. The total size of the lattice is 21 after a 20-step QW. Experimentally measuredeigenvectors are given by the colored points on the Bloch sphere. The eigenvectors start windingfrom n (1) to n (21) with the direction presented by the arrow. In trivial phase (a), winding of n ( k )forms an arc corresponding to a winding number W = 0. In non-trivial phase (b)-(d), winding of n ( k ) forms an circle corresponding to a winding number W = ±
1, where the sign can be definedby the winding direction. The parameters ( θ , θ ) for each scenario are given at the bottom. Eachexperimentally measured n ( k ) is an average over ten different measurements and the translucentbars in each n ( k ) give the associated standard errors, which indicate the total noise. eigenstates, which has been demonstrated in atoms system [23].In our experiment, we use three spinor states in the k -space for each k , i.e., the initialstate (a single lattice site state), the relevant states reconstructed from the 1- and 20-step walks starting from the initial state, to form a plane and obtain its normal vector (thespinor eigenvector n ( k )). The resolution of the momentum space is determined by the largeststeps–20. Concretely, we chose the parameters ( θ , θ ) to be (22 . ◦ , ◦ ) in non-trivial phaseand (22 . ◦ , ◦ ) in trivial phase in the standard time-frame as examples. Resorting to themethod introduced in the previous section, we obtained the entire n ( k ) and depicted theseeigenvectors on the Bloch sphere in Fig. 4(a)&(b). Due to the imperfections in experiment,the reconstructed eigenvectors will extend to a certain range (depicted with the color bandwith ± π/
10 divergence) on the surface of the Bloch sphere instead of constrained on a plane(predicted in theory as shown in the upper row). However, in this practical situation, thewinding of n ( k ) around the axis A ( θ ) (in the standard time-frame) is also well defined. Itcan be calculated by projecting these eigenvectors to the plane which is perpendicular to A ( θ ). Our experimental results clearly show the different winding features in the topologicaltrivial and non-trivial phases.To complete the classification of the non-trivial phases in the QWs, two winding numbers, ν (cid:48) & ν (cid:48)(cid:48) , defined in two nonequivalent shifted time-frames [33, 35] are necessary. In the split-7tep protocol, the time evolution operators U (cid:48) and U (cid:48)(cid:48) are identical only by switching θ and θ . In consequence, the critical requirement is to catch the feature that in non-trivial phasesthe winding number ν (cid:48) is +1 for θ ∈ { , π } and − θ ∈ {− π, } ( ν (cid:48)(cid:48) is trivial in thisscenario) [35]. We further perform QWs in the shifted time-frame ( U (cid:48) ) and reconstruct n ( k )(shown in Fig.4(c)&(d)). Our results show that the winding numbers of the parameters(22 . ◦ , 10 ◦ ) and ( − . ◦ , 10 ◦ ) are different. With the winding numbers ν (cid:48) and ν (cid:48)(cid:48) obtainedin the two nonequivalent time-frames, we can obtain the invariants ( ν , ν π ) and completethe classification of the topological phases in QWs.One of the critical characters of topological phases is its robustness. We consider thewinding of the eigenvectors n ( k ) in the presence of dynamic disorder, that is, the parame-ters θ randomly change its value at each step within a given range ∆ θ . Without loss ofgenerality, we consider the QW with a winding number W = − U (cid:48) . The chiral symmetry is not destroyed in this dynamic disorder as its definition is in-dependent of system parameters. We numerically investigate the influences of the disorderstrength ∆ θ = 3 ◦ , ◦ , ◦ and different steps of walks on the winding of eigenvectors(the results are presented in supplementary). Our simulations show that although the n ( k )will diverge from the chiral plane defined by σ x and the divergence will increase with theincrease of disorder strength and the steps of the walk, the robust of the winding numberholds provided that the disorder strength is smaller than the gap size and the evolutiontime has not extended beyond the coherence time. Large disorder strength or long timeevolution (beyond the coherent time) will significantly mix the system, which results in thefailure of reading out the eigenvectors. Therefore, the robust of the winding number will bedestroyed [53, 54].In summary, we develop a photonic platform for realizing single-photon DTQWs with theability of reconstructing the full final wave-function. Our scheme is robust in overcomingthe visibility and stability problems experienced in previous trails. Additionally, based onthe technology in extracting the full information of the system, we perform an experimentto obtain the system’s spinor eigenvectors and directly read out the winding number of thebulk observable. The method proposed here is general and can be extended to determinehigh winding number topological phases. In prospect, our approach in directly measuringthe topology based on reconstructing the spinor states in quasi-momentum space may beapplicable to the investigation and observation of other classes of phase transitions in more8omplex quantum systems [55–57], thus providing new perspectives for investigating thetopology in physics.This work was supported by National Key Research and Development Program ofChina (Nos. 2017YFA0304100, 2016YFA0302700), the National Natural Science Founda-tion of China (Nos. 11474267, 61327901, 61322506, 11774335, 61725504), Key ResearchProgram of Frontier Sciences, CAS (No. QYZDY-SSW-SLH003), the Fundamental Re-search Funds for the Central Universities (No. WK2470000026), the National PostdoctoralProgram for Innovative Talents (No. BX201600146), China Postdoctoral Science Foun-dation (No. 2017M612073) and Anhui Initiative in Quantum Information Technologies(No. AHY020100). ∗ [email protected] † cfl[email protected][1] L. D. Landau, E. M. Lifsh > its, and L. P. Pitaevski˘ı, Statistical Physics , 3rd ed. (PergamonPress, New York, 1980).[2] S. Sachdev,
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Experimental time multiplexing photonic DTQWsDTQWs in time domain
In this work, the implementation of DTQW is based on the time multiplexing proto-col [39]. However, for overcoming the problem of the extra loss, birefringent crystals areused to implement the spin-orbit coupling instead of the asymmetric Mach-Zehnder inter-ferometers. Heralded single photon generated from SPDC is employed as the walker. Thepolarization degree of the photon is employed as the coin space, such that its coin state canbe optionally rotated via wave plates. The arriving time of the photon, encoded in timebin, acts as the position space. One step QW is realized by a module composed of a halfwave plate (HWP) and one piece of birefringent crystal. The coin rotation operator can bewritten as ˆ R HWP ( θ ) = e − i θ ˆ σ y ˆ σ z , (1)where θ is the rotation angle of the optical axis of the HWP and ˆ σ i ( i ∈ { x, y, z } ) denote thePauli matrices. The eigenstates of the coin are | H (cid:105) and | V (cid:105) , corresponding to the horizontaland vertical polarization respectively, with the condition ˆ σ z | H (cid:105) = | H (cid:105) and ˆ σ z | V (cid:105) = −| V (cid:105) .The birefringence causes the horizontal components to travel faster inside the crystal thanthe vertical one. As a result, after passing through the crystal the photon in state | H (cid:105) movesa step forward. Considering the dispersion after passing through a large number of crystalsand the fact that the time bin encoding the position of the walker in reality is a singlepulse with a typical duration of a few hundred femtoseconds, such a shift in time shouldbe sufficiently large to distinguish the neighborhood pulses at last. The magnitude of thepolarization-dependent time shift by the birefringent crystal depends on the crystal lengthand the cut angle. In our experiment, for introducing as weak dispersion as possible withsufficiently large birefringence, calcite crystal is adopted for its high birefringence index(0.167 at 800 nm ). The length is chosen to be 8.98 mm with its optical axis parallel toincident plane, such that the time shift is designed to be 5 ps for one-step.13 eralded single photon adopted as the walker The time multiplexing protocol requires pulse photons, which can be obtained by atten-uating a pulse laser or modulating a continuous laser with an optical chopper. Consideringthe tradeoff between the operation on the time bins and the final analysis in time domain,the pulse duration covers a range from tens to thousands of picoseconds, reaching even a fewmicroseconds. In our experiment, for adopting a genuine single photon as the walker andconsidering that the length of the crystal should be as short as possible to reduce dispersionand improve stability, the duration of the single photon pulse should be as small as possible.It is selected on the level of hundreds of femtoseconds. Such a short single photon pulse canbe generated via SPDC with an ultra-short femtosecond pulse laser as the pumper. Thegenerated photon pairs are time correlated, the click of detection on the idler photon canpredict the existence of the signal photon. Various architectures exist for generating thistype of heralded single photon from SPDC. Here, considering the features of high brightnessand collection efficiency, we adopt the beam-like SPDC [50].
Frequency up conversion single photon detection
The spectrum of the arriving time of single photon is usually measured with the technol-ogy of time correlated single photon counting and commercial single photon detectors [58].However, in our case, the signals are contained in a single photon pulse train with a pulseduration of approximately 1 ps and a repetition of 5 ps . Counting and analyzing such ultra-fast single photon signals are challenging. The time resolution of commercial single photondetectors is limited by the time jitter, which is typically in the range of tens to hundredspicoseconds [59]. That is to say, it is unsuitable to directly use any commercial single photondetectors in our experiment. The detection of single photon with high resolution in time canbe realized by transforming the temporal resolution to a spatial resolution. The measure-ment of an ultra-fast pulse of single photons can be realized via optical auto-correlation [60],a technology developed from the optical parameter up conversion. That is, using an ultra-fast laser pulse to pump a nonlinear crystal, when the single photon and pumper pulse meeteach other inside the crystal, the single photon will be up-converted to a shorter wavelengthvia the sum frequency process. For the photon with a long wavelength can be converted to14 short one, this technology has been widely used in quantum communication for improvingthe detection efficiency in the infrared waveband [61]. Here, we adopt this technology forits high resolution in time. Although periodically poled crystals are widely used in thistechnology for their high conversion efficiency, they are useless in our experiment for con-centrating on the time resolution. The thickness of nonlinear crystal should be as thin aspossible meanwhile taking into account the conversion efficiency. There exist two typesof structures, collinear and non-collinear sum frequency. We adopt the latter to obtain abetter signal to noise ratio (SNR), which is induced by the spatial divergence between thesum frequency signal and the pump laser. The crystal used in our experiment is a 1 mm thick β -BaB O (BBO) crystal, cut for type-II second harmonic generation in a beam-likeform. Then, the incidence angle of the signal pulse train and the pump laser are equal toeach other, with 3 ◦ to the normal direction. For reducing the noise induced by the strongpump laser, a dispersion prism in a 4F system is adopted as a spectrum filter. The scatteredphotons with wavelength longer than 395 nm are blocked by a knife edge. The rising edgein the sideband of this self-established spectrum filter is less than 1 nm . Detail description of the experimental setup
An ultra fast pulse (140 fs ) train generated by a mode-locked Ti:sapphire laser with acentral wavelength at 800 nm and repetition ratio 76 MHz is firstly focused by lens L1 toshine on a 2 mm thick β -BaB O crystal (BBO1), cut for type-I second harmonic generation.The frequency-doubled ultraviolet pulse (with a wavelength centered at 400 nm , 100 mW average power and horizontally polarized) and the residual pump laser are collimated bylens L2, and then separated by a dichroic mirror (DM). The frequency-doubled pulse trainis then focused by lens L3 to pump the second nonlinear crystal (BBO2), cut for type-II non-degenerate beam-like SPDC. The signal and idler photons are collimated togetherwith one lens (f=150 mm ). The collimated signal photons in horizontal polarization witha center wavelength at 780 nm are then guided directly in free space to the following QWdevice. The collimated idler photons in vertical polarization with a center wavelength ofapproximately 821 nm firstly pass through a spectrum filter with a central wavelength 820 nm and bandwidth 12 nm and then are coupled into a single-mode fibre and sent directly to asingle-photon avalanche diode for counting in coincidence with the signal photons. The15uantum walks device is composed of HWPs and calcite crystals, and each step containsone piece of HWP and one piece of calcite crystal. In the experiment, we have adopted50 such sets. The initial state is prepared by an apparatus composed of a PBS, a HWPand a QWP orderly. A reference laser beam for calibration is coupled into the QW devicewith this PBS. The residual pump in the frequency-double process is split by a PBS (notshown) into two beams, one acting as the reference laser and the other with most of theresidual power adopted as the pump in the following frequency up conversion single photondetection. The partition of their power is realized by a HWP (not shown) and both of themare delayed with retroreflectors for matching the arriving time of the signal photons. Afterthe quantum walks is finished, the signal photons are collected into a short single mode fibre(10 cm long) by a fibre collimator and then guided to the polarization analyzer composedof QWP, HWP and PBS orderly. Finally, the arriving time of signal photons is measuredby scanning the pump laser and detecting the up conversion signals with a photomultipliertubes. For reducing the scattering noise, BBO3 is cut for non-collinear up conversion and aspectrum filter based on a 4F system is constructed, where a prism is adopted for introducingthe dispersion, a knife edge is used to block the long waves and the signal is reflected tothe PMT with a pickup mirror. The inset in the bottom right corner gives the diagram ofthe time-multiplexing split-step quantum walks using birefringence crystals. The sites aredefined as the arriving times of the photons. For each site, the photons are located withina pulse with a duration of a few hundred femtoseconds. For each split step, the horizontalphotons will travel approximately 5 ps faster than the vertical ones for the birefringence inthe calcite crystals, equivalently, the walker jumps to the right neighbouring site when thecoin state is in horizontal. For a complete step, the original point is redefined, which resultsin the jump to the left neighbouring site when the coin state is vertical. The repetition rateof our laser is 76 MHz , which corresponds to a time interval ∼ ns , significantly large thanthe total length of the lattice of approximately 0.25 ns .16 x y A- Without Extra Crystal -10 -5 0 5 100.00.10.20.30.4 (d) P r obab ili t y Position -10 -5 0 5 100.00.10.20.30.4 (c) P r obab ili t y Position -10 -5 0 5 100.00.10.20.30.4 (b) P r obab ili t y Position -10 -5 0 5 100.00.10.20.30.4 P r obab ili t y Position (a) z x y B- With Extra Crystal -10 -5 0 5 100.00.10.20.30.4 (d) P r obab ili t y Position -10 -5 0 5 100.00.10.20.30.4 (c) P r obab ili t y Position -10 -5 0 5 100.00.10.20.30.4 (b) P r obab ili t y Position -10 -5 0 5 100.00.10.20.30.4 P r obab ili t y Position (a)
FIG. 5. Distributions after projecting to the bases given in Eq.4 for full wave-function reconstruc-tion. A, before inserting the extra crystal. B, after inserting the extra crystal. In both of A andB, the spin is projected to four bases, (a) for s H , (b) for s V , (c) for s R and (d) for s D , whichcorresponds to the eigenstates of the Pauli matrices, σ z = ± , σ y = − σ x = +1. Experimen-tal results are presented with histograms and the red lines gives the ideally expected values. Theerrors are given with considering the statistical noise. Full reconstruction of the final wave-functionReconstruction of wave-function in site space
We consider to experimentally reconstruct the full wave-function which was deemed toa challenge in usual interferometer based QWs [62]. Suppose the walker starts at the originand consider the walker state at a certain step t , | Ψ t (cid:105) = N (cid:88) x = − N p t ( x ) e − iφ t ( x ) | ψ t ( x ) (cid:105) ⊗ | x (cid:105) , (2)where the position index x ∈ [ − N, N ] (integer) and the lattice size is 2 N + 1. For each posi-tion x there is a local normalized spinor state | ψ t ( x ) (cid:105) with a complex amplitude p t ( x ) e − iφ t ( x ) .For convenience, we write local spinor state in | ψ t ( x ) (cid:105) = cos θ t ( x )2 | H (cid:105) + e iδ t ( x ) sin θ t ( x )2 | V (cid:105) , (3)with θ t ( x ) ∈ [0 , π ] and δ t ( x ) ∈ [0 , π ). In experiment, we perform three steps to obtainthe parameters, p t ( x ) , φ t ( x ) , θ t ( x ) and δ t ( x ). Noting that the first lattice site phase φ t ( − N )is meaningless and the normalization condition (cid:80) N − N | p t ( x ) | ≡ N + 1) − t -step walks starting from the original position ( N = t ).Firstly, for each site x , we perform local projection measurement on the conventionalbases s H = | H (cid:105) (cid:104) H | ,s V = | V (cid:105) (cid:104) V | ,s R = | R (cid:105) (cid:104) R | ,s D = | D (cid:105) (cid:104) D | , (4)where | R (cid:105) = √ ( | H (cid:105) − i | V (cid:105) ) , | D (cid:105) = √ ( | H (cid:105) + | V (cid:105) ). We note the corresponding expectedcounts n H ( x ), n V ( x ), n R ( x ), and n D ( x ) ( x ∈ [ − N, N ]). We can obtain a set of 4(2 N + 1)counts which is labeled by S .Secondly, an extra crystal is inserted and a spin echo is performed to shift all the horizontalbins a step backward. As a result, the spinor state at site x is changed to be | ψ (cid:48) t ( x ) (cid:105) = N (cid:18) p t ( x + 1) e − iφ t ( x +1) cos θ t ( x +1)2 p t ( x ) e i [ − φ t ( x )+ δ t ( x )] sin θ t ( x )2 (cid:19) , (5)where N is a renormalization coefficient. Then again we perform local projection measure-ment on the same bases and note the corresponding expected counts ˜ n H ( x ), ˜ n V ( x ), ˜ n R ( x ),and ˜ n D ( x ) ( x ∈ [ − N, N ]). At this stage, we can obtain a set of 4(2 N −
1) counts which islabeled by ˜ S .At last, we carry out a numerical global optimization program based on simulated an-nealing algorithm to find the optimal pure state | Ψ t (cid:105) which can give the data set S + ˜ S .Following the method frequently used in measurement of qubits, we optimize the pure state | Ψ t (cid:105) by finding the minimum of the following “likelihood” function [45], L = N (cid:88) x = − N (cid:88) i ∈{ H,V,R,D } [ N n i ( x ) − n i exp ( x )] N n i ( x ) + N − (cid:88) x = − N (cid:88) i ∈{ H,V,R,D } [ N ˜ n i ( x ) − ˜ n i exp ( x )] N ˜ n i ( x ) , (6)where n i exp ( x ) (˜ n i exp ( x )) stands for the experimentally measured counts and N is a normal-ization coefficient. The total number of bases we have measured is 4(2 N + 1) + 4(2 N − IG. 6. Diagram of the spinor state rotation in k − space (left sphere) and the reconstruction ofeigenvectors n ( k ) from the spinor states (right sphere). The initial state | φ ( k ) (cid:105) will be rotatedalone the vector n ( k ) after a certain step of quantum walk. As a result, the states for different steps1 , , , ..., t will be constrained to lie on a plane determined by n ( k ). For the spinor states aftervarious steps of quantum walks are constrained to lie on a plane determined by n ( k ), generally,three individual points (black points, S , S , S ) are enough for obtaining the normal vector of thatplane. Obtaining the wave-function in quasi-momentum space
The wave-function in quasi-momentum space | Φ t (cid:105) corresponds to the Fourier transfor-mation of the wave-function in site space. To obtain | Φ t (cid:105) , we perform a discrete Fouriertransform to the reconstructed wave-function | Ψ t (cid:105) . In details, | Ψ t (cid:105) contains two components,the complex amplitude p t ( x ) e iϕ t ( x ) cos θ t ( x )2 for horizontal polarization and the complex am-plitude p t ( x ) e i ( − ϕ t ( x )+ δ t ( x )) sin θ t ( x )2 for vertical polarization. Two individual discrete Fouriertransforms are performed to the two components. Then with a normalization for each quasi-momentum k , we can get the normalized spinor state φ t ( k ) for each k . Theoretically, thetime evolution operator U ( θ , θ ) is diagonalized in Fourier basis, the final sate of spin foreach quasi-momentum k , i.e., | φ t ( k ) (cid:105) can then be obtained directly by performing a unitaryoperation (actually a rotation) on the initial state according to the time evolution operator. Reconstruction of eigenvectors
Theoretically, the spinor states | φ t ( k ) (cid:105) ( t = 0 , , , ... ) for a fixed quasi-momentum k after a t -step quantum walk, can be obtained by rotating the initial state | φ ( k ) (cid:105) withthe angle t.E ( k ) around the axis n ( k ), as sketched in the left panel of Fig. 6. In otherwords, the spinor states | φ t ( k ) (cid:105) will be constrained to lie on a plane that is perpendicularto n ( k ) on the Block sphere. As a result, there exists correspondence between the plane19etermined by the spinor states | φ t ( k ) (cid:105) and spinor eigenvectors n ( k ). To determine thespinor eigenvectors n ( k ), generally, three different steps are enough to determine a plane(for the special cases that the vectors are linear dependent, the steps need more), sketchedin right panel of Fig.6. Using this method, the sign of n ( k ) (‘plus’ or ‘minus’, correspondingto two spin eigenvectors) remains uncertain. Resorting to the continuation of n ( k ) in k space and assuming the direction of the first eigenvector n ( k ) (where k can be chosen tobe − π or 0) is fixed; the entire n ( k ) can then be uniquely determined. It should be notedthat there exists a null set where the energy band is flat ( θ = ± π or θ = ± π ). For thecase θ = ± π , located in the trivial phase, the walker will stand at the origin all the timeand no changes can be observed. For the case θ = ± π , there only exist one different finalstates compared to the initial state, therefore it will be failure to determine the plane and n ( k ) with only two points. While for other values of the parameters θ and θ , our methodis valid. Reading the topological phase from the reconstructed spinor eigenvectorsStandard time frame
The topology in split-step DTQWs is firstly investigated in a standard time frame [18],that is, U ( θ , θ ) = T − R ( θ ) T + R ( θ ) . (7)In this scenario, the eigenvectors n ( k ) are constrained to lie on a plane defined by the vector A ( θ ), and as a result, the chiral symmetry can be defined with A ( θ ). Although the windingnumbers in spit-step DTQWs can be understood by the effective n ( k ) winding around A ( θ )on the Bloch sphere, directly getting these eigenstates is full of challenge. Alternativeshave been developed to detecting the topological phase indirectly. The edge states havebeen observed [19], and the phase transitions between them are identified by the statisticalmoments of the final distributions [20]. Recently, they improved their method, the Zakphase connected to the winding number of the bulk of this system has been experimentallyexploited by measuring the so called mean chiral displacement [38]. In our experiment, withthe ability of full reconstruction of the final spinor states in k -space, we can read the windingnumber of n ( k ). 20 hifted time frame Further works show that the topological phase in periodically driven system is much morecomplicated and its complete classification should be modified with two bulk invariants [32–37]. It is suggested to introduce nonequivalent shifted time frames, both of which maintainthe chiral symmetry, to fully determine the topological phase[33, 35]. The time evolutionoperators are given by U (cid:48) ( θ , θ ) = R ( θ / T − R ( θ ) T + R ( θ / , (8) U (cid:48)(cid:48) ( θ , θ ) = R ( θ / T + R ( θ ) T − R ( θ /
2) (9)The topological phases are then determined by the combined invariants ( ν , ν π ), where ν = ( ν (cid:48) + ν (cid:48)(cid:48) ) / ν (cid:48) − ν (cid:48)(cid:48) ) / ν (cid:48) and ν (cid:48)(cid:48) are conventional winding numbers definedthrough the Berry phase for U (cid:48) and U (cid:48)(cid:48) respectively. For U (cid:48) and U (cid:48)(cid:48) are identical only byswitching θ and θ , what we measure in experiment is ν (cid:48) , which is theoretically given bywhen sin ( θ ) − sin ( θ ) > , ν (cid:48) = < θ < π ) − − π < θ <
0) (10)else, ν (cid:48) = 0 (11) Figures for discussion of robustnessStatistical moments
The analytic forms of the statistical moments can be given in our scenario, which are M = tan ( θ / − max ( | sin( θ / | , | sin( θ / | )] (12)for the second order moment and M = tan ( θ / − max ( | sin( θ / | , | sin( θ / | )][ (cid:104) ψ | ( σ x + tan ( θ / σ z ) | ψ (cid:105) ] (13)for the first order moment (which is dependent on the initial state | ψ (cid:105) ). In Fig.8, we showthe measured statistical moments (1st and 2nd order) for 50-step and 20-step in different21 teps = 20 steps = 30 steps = 40steps = 20 (b)(c) steps = 40 (a) FIG. 7. Numerical simulations of the winding of eigenvectors in the presence of dynamic disorder.Without loss of generality, the disorder is introduced via fluctuations of the first rotation angle θ over evolution time, the second rotation angle is θ = 10 ◦ and choose a mean value of the firstrotation angle as ¯ θ = − . ◦ . With the given parameters ( − . ◦ , ◦ ), the system is expectedto yield a topological phase with its winding number W = −
1. For comparison, (a) shows thetheoretical eigenvectors reconstructed from the 0-, 1- and 40-step quantum walks in the absence ofdisorder. In (b), the eigenvectors reconstructed from the 0-, 1- and 20-(30-, 40-, from left to right)step quantum walks in the presence of the dynamic disorder with disorder strength | ∆ θ | = 3 ◦ . Itcan be observed that in this scenario, the winding of the eigenvector is robust when the number ofthe step is small, no matter the divergence of the eigenvectors to their ideal expectations (dashedcircle). When the time gets larger, the decoherence induced by the dynamic disorder will divergethe eigenvector to reach the x − z plane, which results in the failure of reading out the windingnumber. In (c), the eigenvectors reconstructed from the 0-, 1- and 20-step quantum walks withthree different disorder strength ∆ θ = 3 ◦ , 4 ◦ , 10 ◦ . We can see that the winding of the eigenvectoris robust provided that the strength of the disorder is small (the energy gap is larger than noisespectral bandwidth). With increasing the disorder strength, reading out the eigenvector will befailed for the divergence of the eigenvector reaches the x − z plane, as shown in the scenario∆ θ = 10 ◦ . The number of samples is 100 in (b) and (c). topological phases. 22 FIG. 8. Points show the experimentally measured normalized statistical moments after a 50-step QW (black triangles) and 20-step (red circles): first order (upper) and second order (lower)with the blue dashed lines representing the theoretical expectations. θ is fixed at 22 . ◦ with θ = 5 ◦ , ◦ , ◦ in non-trivial phase and θ = 30 ◦ , ◦ , ◦ in trivial phase for the 50-stepscenario. In such large scale, the system is more sensitive and hard to measure. The decoherencewill degenerate the QW’s quantum feature, that is, experimentally measured M is lower than itstheoretical predictions. In a smaller scale (20-step), the results match the theory very well (redcircle). The region with shadow represents the trivial phase (non-trivial phase without shadow).We consider the statistical error and present the error bar with standard variance via numericalsimulation. Experimental imperfections, decoherence and noise.
One of the main concerns of this work is to realize a large-scale DTQW with genuinesingle photons adopted as the walker. As a result, there are many challenges which usuallycan be neglected in DTQWs with small scale or attenuated laser as the walker.
Blocking the scattering photons
First of all, the walker is in the level of single photon, although the stray photons inenvironment are in extremely low level, the detection of signal adopts a high power laser(approximately 300 mW ) as the pump, whose direction of propagation is the same as thewalker, thus it is straightforward for the scattering photons from the strong pump to ulti-mately enter the single photon detectors. Here, we adopt three technologies to overcomethis noise: Firstly, the SPDC for generating the heralded single photons is chosen to be23on-degenerate in wavelength, then by inserting a spectrum filter before detecting the idlerphoton the weak reflection of the strong pump laser is blocked, and with the help of coin-cidence counting the noise can also be reduced. Secondly, the nonlinear crystal adopted forthe parametric up conversion detector is cut in type-II and with a 3 ◦ incidence angle for thesignal and pump, the scattering photons from the pump are then filtered in the dimensionsof both polarization and momentum. Thirdly, a spectrum filter based on 4F system with adispersion prism inside is built up for blocking the scattering photons with a similar wave-length as the signal, which are generated from the weak nonlinear parametric process of thehigh power pump inside the crystal. Loss and detection efficiency
Our scheme can overcome the extra loss that exists in the previous time multiplexingDTQWs, while the losses induced by the imperfect anti-reflection coating and the intrinsicabsorption of the crystals should be considered in the case of a large-scale. Although thetransmittance of a single crystal can be as high as 0.995, the total transmittance is reducedto 50% after passing through two hundred surfaces and crystals with total length as longas half a meter. Another imperfection is the low detection efficiency in our parametricup conversion single photon detector. This imperfection arises from two reasons: the lowdetection efficiency (typical 20%) for the photomultiplier tube and the low up conversionefficiency induced by the small nonlinear coefficient of the crystal and the pulse broadeninginduced by dispersion inside the crystal. The total measured detection efficiency can be ashigh as 0.5% at last. Considering all of these disadvantages, the total coincidence counts is6 pairs/(s · mW) with a pump power of 300 mW in the up conversion. Noise analysis
Tolerance of optical axis-
Firstly we consider the noise from the poor orientation of thewave plates for realizing the tossing operation. The experimentally accessible orientationprecision for each wave plate is typical 0 . ◦ . We have supposed that the optical angles ofthe wave plates are oriented randomly in a range defined by a Possian distribution and haveadopted the Monte Carlo simulations to evaluate the error. The numerical results show that24his type of error is on the order of 10 − for considering the fidelity. Statistical noise-
Another important noise is the shot noise, which is also estimated bythe Monte Carlo simulations. In our experiment, the total counts is around 2 × , whichwill introduce an error ( ∼ − ) to the measured probability distribution and its fidelity,and the fidelity in full wave function reconstruction. Amplitude damping-
Note that amplitude damping exists in our scheme for the loss men-tioned above is weakly polarization dependent. Because the damping is systematic, we canovercome it through compensation. We have also checked the degeneration of the extinctionratio and interference visibility as the system increases in size, and they are consistent withthe exponent rule, which indicates that based on the high visibility for single Mach Zehnderinterferometer, these type of degeneration can be neglected.
Phase drifting-
The collinear structure in the interferometer induces the extreme stabilityfor single step. However, as the system size increases, the stability degenerates in exper-iment. The vibration is typically responsible for this stability degeneration in other bulkinterferometers. In our experiment, the vibration amplitude of the rotation stages (fixingthe crystals) is only in the typical level of µ rad, which will not responsible for the system’sstability degeneration. The main governing factor is the temperature of crystals. We havemonitored the drifting of the environment temperature and the stability of the DTQWs with50 steps. The two patterns match each other very well in term of the period, approximateone thousand seconds, primarily because of the thermal expansion (typically on the orderof 10 − / ◦ C). Although the refraction index also changes with the temperature at the samelevel, the birefringence is much more lower. To eliminate the influence of the temperature, wefirstly reduce the temperature fluctuation to less than 0 . ◦ C, and monitor the temperatureas the data is collected.
Mode matching-
Mode matching is unnecessary for the collinear structure in our scheme,while the imperfections of optical elements will introduce mode mismatching. We mainlyconsider the influences of two imperfections which will cause the decoherence: the first one isthe length tolerance of birefringent crystals. Similar to the spatial shift interferometer, thelength of the crystal determines the degree of the temporal shift. The tolerance of the lengthis ± .
01 mm, implying that the corresponding phase tolerance is slightly greater than fourwaves. We fixed the crystals sequentially by carefully tilting the crystal around its opticalaxis that is perpendicular to the table to find the maximal interference visibility with a25roadband laser. The same method is also used for locating the destructive interferencein experiment. Another imperfection is that of the cut angle of the optical axis, which issupposed to be parallel to the surface for introducing a pure time shift. An imperfection inthe cut angle of the optical axis will introduce a walk-off effect for the extraordinary ray.Considering a typical cut angle tolerance of 0 . ◦ , the two spots for the two perpendicularpolarizations will be displaced by approximately 70 µm , which is large relative to the spotsize 1 mm . To overcome this decoherence, we carefully tilted the pitch angle of crystals tofind the maximal visibility and then used a short single mode fibre (0.1 mm