Dynamic-Disorder-Induced Enhancement of Entanglement in Photonic Quantum Walks
Qin-Qin Wang, Xiao-Ye Xu, Wei-Wei Pan, Kai Sun, Jin-Shi Xu, Geng Chen, Yong-Jian Han, Chuan-Feng Li, Guang-Can Guo
DDynamic-Disorder-Induced Enhancement of Entanglement inPhotonic Quantum Walks
Qin-Qin Wang,
1, 2
Xiao-Ye Xu,
1, 2
Wei-Wei Pan,
1, 2
Kai Sun,
1, 2
Jin-Shi Xu,
1, 2
Geng Chen,
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
Entanglement generation in discrete time quantum walks is deemed to be another key propertybeyond the transport behaviors. The latter has been widely used in investigating the localiza-tion or topology in quantum walks. However, there are few experiments involving the former forthe challenges in full reconstruction of the final wave function. Here, we report an experimentdemonstrating the enhancement of the entanglement in quantum walks using dynamic disorder.Through reconstructing the local spinor state for each site, von Neumann entropy can be obtainedand used to quantify the coin-position entanglement. We find that the enhanced entanglement inthe dynamically disordered quantum walks is independent of the initial state, which is differentfrom the entanglement generation in the Hadamard quantum walks. Our results are inspirationalfor achieving quantum computing based on quantum walks. a r X i v : . [ qu a n t - ph ] A ug t ep s Positions -2 0 2
Step 1Step 2Step 3Coins (a) (b) S t ep s Positions -2 0 2
Step 1Step 2Step 3Coins ( c ) ( d )
FIG. 1. Illustrations for (a) ordered QW, where only one kind of coin is employed (red rectangle),and for (b) dynamically disordered QW, where two kinds of coin are employed (red and greenrectangles). The coin-position entanglement as a function of the initial state parameters { θ, φ } for (c) ordered Hadamard QW and for (d) the dynamically disordered QW with the coin tossing S C (define in the text). The dependency of the coin-position entanglement on the initial state issignificantly higher for ordered QW compared to the dynamically disordered one. INTRODUCTION
Entanglement, an intriguing character of quantum systems, plays the critical role inthe quantum information processing [1], such as quantum key distribution [2] and quantumcomputing [3]. However, quantum entanglement is so fragile that it can easily be destroyedby the noises and the environment. Systems become disordered [4] due to the inhomoge-neous environmental conditions and other parameters in the system, which are impossibleto control experimentally. Intuitively, one may expect that such disorder would reduce theentanglement of a given system, which is indeed true for a large variety of systems. In factfor some systems, disorder can enhance the entanglement [5–10]. For example, the genuinemultipartite entanglement of ground state in the quantum spin model [6, 11] can be enhancedby disorder. Another example is the dynamic disorder that can enhance the entanglementgeneration in quantum walks (QW) [8–10].QW on lattices and graphs [12] is a quantum generalization of the classical randomwalks [13], which may play as universal quantum computers [14, 15], quantum simulators [16]and platforms to investigate topological phases [17–19]. The behavior of QW, especially theballistic behavior [20] of the transport properties, is dramatically different from its classi-cal counterpart due to the superposition principle and has been extensively investigated.Besides the probability distribution, the entanglement properties of QW have also been2heoretically studied [21–24]. It is the genuine quantum feature in QW since there is noclassical counterpart. The entanglement (coin-position entanglement) here is different fromits original definition for multiple parties. It is actually entanglement between two modessharing a single particle [25], which has been widely used in some crucial quantum informa-tion protocols [26]. In the ordered QW where the quantum coin is fixed during the wholeevolution process or changes with a deterministic way, the coin-position entanglement [21–23] is highly dependent on the initial state and usually never achieves its maximal value.Entanglement in QW will be affected by dynamic disorder, in which the quantum coin isindependent of the site j and at the same time is randomly chosen for each step. Or it canbe affected by the static disorder, in which the quantum coin is fixed for all the time and atthe same time is randomly chosen for each site j . Entanglement in QW can also be affectedby the combination of both, dynamic and static disorder. Theoretical investigations [8, 9]showed that the entanglement is reduced by the static disorder while the dynamic disorderinduces enhancement of the entanglement independent of the initial states and even withthe appearance of the static disorder.Linear optics is a good platform for implementing QW and thus many technologies havebeen developed: spatial displacers [18], orbital angular momentum (OAM) [27], time mul-tiplexing [28, 29], integrated optical circuits [30] and array of wave guides [31, 32]. Unliketransport behaviors, which have been sufficiently studied just by measuring the final proba-bility distribution, the entanglement properties are still needed to be studied in both orderedand disordered QW. The experimental challenges are twofold: how to reach large-scale QW(the disorder-induced entanglement enhancement can only be demonstrated in the asymp-totic limit) and how to reconstruct the wave function in both the coin and position space [23].Different efforts [33–35] have been made to improve the scalability. For example, all fiber-based QW have reached 62 steps with high fidelity and low loss [36]. Only recently, the finalwave function in a one-dimensional QW of a single cesium (Cs) atom has been obtainedby the local quantum state tomography [37], and complete reconstruction of wave functionwas achieved in OAM [27] and time multiplexing protocol [38] (in Ref. [39] the authors mea-sured the relative phase, 0 or π , between the neighboring sites). In this paper,we reportan experiment for demonstrating the enhancement of entanglement generation in QW bythe dynamic disorder. This experiment is based on our recently developed novel compactplatform for genuine single photon QW in large scale with the ability of full wave function3 BO1L2DM L3BBO2 L4Filter@820nmSPAD PMTL1 MirrorKnife Edge L7 Prism L6 L5 PBS2HWP2FC2RPBS1HWP1QWP1 H W P s ( Q W P s ) & C a l c i t e C r y s t a l s FC1QWP2BBO3
CalciteHWP (QWP) -6 -2 2 6 S t ep FIG. 2. Diagram of the experimental setup (additional details are given in the supplementarymaterial). The system contains four parts: 1. Second harmonic generation in BBO1 to obtainthe ultraviolet pulse; 2. Generation of heralded single photons by adopting the beam-like typeSPDC in BBO2; 3. Time-multiplexing quantum walks realized by birefringent crystals sketched bythe inset at the bottom right corner; 4. Ultra-fast detection of the arrival time of single photonswith an up-conversion single photon detector. 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: fiber collimator, PMT: photomultiplier tubes, SPAD: single-photon avalanche diode. reconstruction. THEORETICAL IDEA
The Hilbert space of a QW is H = H C ⊗ H P , where H C is a two-dimensional Hilbertspace spanned by {|↑(cid:105) , |↓(cid:105)} and H P is an infinite dimensional Hilbert space spanned by aset of orthogonal vectors | j (cid:105) ( j ∈ Z ). A QW is given by a sequence of coin tossing followedby a conditional shift according to the coin state. The time evolution operator for a QWfrom t = 0 to T can be represented by a multi-step unitary operator U ( T ) ≡ (cid:81) Tt =0 U ( t ),where U ( t ) = S · ( C ( t ) ⊗ I P ) with I P is the identity operator in H P and C ( t ) is the cointossing in H C . The shift operator S = (cid:80) j |↑(cid:105)(cid:104)↑| ⊗ | j + 1 (cid:105)(cid:104) j | + |↓(cid:105)(cid:104)↓| ⊗ | j − (cid:105)(cid:104) j | describesthe conditional displacement in lattice, which generates the coin-position entanglement.Generally, the coin tossing C ( t ) in a QW is time and position dependent. In this paper, C ( t ) is assumed to be site independent since we only consider the effect of the dynamicdisorder. In a QW with dynamic disorder, the coin tossing C ( t ) is time dependent: for eachstep, it is randomly chosen from a set C with certain probability distribution (in particularly,homogeneous distribution). According to the literature [8, 9], the type of dynamic disorder(including type of C ) is not important. Without loss of generality, in our experiment, we4ssumed that C consists of Hadamard operator ( H ) and Fourier operator ( F ), where H = 1 √ − , F = 1 √ ii . (1)We also considered an ordered QW, in which the coin tossing is time independent, and wechose Hadamard gate all the time for comparison.The global time evolution operator for a single sample is also unitary in a QW and thefinal state | Ψ( t ) (cid:105) after a t -step walk remains pure if the initial state is pure. The general formof | Ψ( t ) (cid:105) can be written as (cid:80) j [ a ( j, t ) |↑(cid:105) + b ( j, t ) |↓(cid:105) ] ⊗| j (cid:105) , where a ( j, t ) and b ( j, t ) are complexnumbers with the normalization condition (cid:80) j [ | a ( j, t ) | + | b ( j, t ) | ] = 1. With the unitarityfactor, the coin-position entanglement in a QW can be defined by the von Neumann entropy S E ( ρ ( t )) = − Tr[ ρ C ( t ) log ρ C ( t )] , (2)where ρ C ( t ) = Tr P [ ρ ( t )] is the reduced density matrix of the coin with ρ ( t ) = | Ψ( t ) (cid:105)(cid:104) Ψ( t ) | and Tr P is taking the trace over position.For a fixed initial state, with the increase in time, regardless of the ordered or disorderedQW, the coin-position entanglement will be asymptotic to a stable value. Generally, thisasymptotic value in an ordered QW can not reach maximal and is strongly dependent onthe initial state. The entanglement after 20 steps with different initial states is shown inFig. 1.(c). For a dynamically disordered QW, this asymptotical value is found [8, 9] to reachmaximal regardless of the initial states as shown in Fig. 1.(d). EXPERIMENTAL SETUP AND RESULTS
The experimental setup is shown in Fig. 2 and the more detail description is given insupplementary material. Single photons generated from spontaneous parametric down con-version(SPDC) are adopted as the herald walker. These kind of coin states are initializedby sending them through the polarizer PBS1-HWP1-QWP1(see Fig. 2). The state |↑(cid:105) ( |↓(cid:105) )corresponds to a single horizontally polarized photon | H (cid:105) ( | V (cid:105) ), which stands for the hori-zontal (vertical) polarization of the photon (walker). The QW device is composed of wave5 (cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13) (cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:2) (cid:13)(cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13) (cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:8) (cid:2) (cid:13)(cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13) (cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:3) (cid:7) (cid:2) (cid:13)(cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13) (cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:4) (cid:6) (cid:2) (cid:13) Entanglement
N u m b e r o f S t e p s ( b ) (cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13) (cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:2) (cid:13)(cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13) (cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:8) (cid:2) (cid:13)(cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13) (cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:3) (cid:7) (cid:2) (cid:13)
Entanglement ( a )
FIG. 3. (a) Coin-position entanglement measured (dots) for different initial states in orderedHadamard QW. The lines (solid red, dashed blue and dotted yellow) are the theoretical predictionsof the von Neumann entropies. The entanglement generated in ordered QW shows a significant de-pendency on the initial state. Noting that the entanglement involves periodical oscillation aroundan asymptotic value (guided by the dashed black horizontal line) in the limit of infinite steps withan amplitude decay. We have not considered the case { ◦ , ◦ } , which is the same as { ◦ , ◦ } .(b) Coin-position entanglement measured (dots) for different initial states in the dynamically dis-ordered scenario. The lines (solid red, solid blue, solid yellow and solid green) are the theoreticalpredictions. The von Neumann entropies are about 0.98 at 20 steps for all initial state clusters,which significantly differ from the ordered case, where they converge to different asymptotic valuesfor different initial states. The periodical oscillation of entanglement with time degenerates. Onlystatistical errors are considered with total counts 24 thousand in 4 hundred seconds. plates (for realizing coin tossing) and calcite crystals (for implementing conditional shift),and each step contains each one of them. H and F coin tossing was implemented by singleHWP (with its optical axis oriented at π/
8) and QWP (with its optical axis oriented at − π/
4) respectively. The reduced density matrix ρ C in Eq. 2 is equal to (cid:80) j p j ρ j . Equation p j = | a ( j ) | + | b ( j ) | is the probability in site j , ρ j is the local density matrix in site j . Inexperiment, ρ j is obtained through local quantum state tomography (realized by the po-larization analyzer QWP2-HWP2-PBS2 in Fig. 2). Meanwhile p j is directly given by theprojection probabilities in the first two bases, | H (cid:105)(cid:104) H | and | V (cid:105)(cid:104) V | . The lattice is composedof arriving time of signal photons and the time interval is around 5 ps , which is challengingto detect with available commercial detectors. Therefore we constructed the up-conversionsingle photon detectors. 6he initial state in our experiments is located at the original site ( | x = 0 (cid:105) ) and thegeneral state of the coin is | Ψ(0) (cid:105) = [cos( θ/ |↑(cid:105) + e iφ sin( θ/ |↓(cid:105) ] ⊗ | (cid:105) , where θ ∈ [0 , π ]and φ ∈ [0 , π ). In our experiment, the QW step number is limited to 20. First, weexperimentally determined the key characteristics of the coin-position entanglement gen-erated in the standard Hadamard QW (ordered QW). We chose three different initialstates: { θ, φ } = { ◦ , ◦ } , { ◦ , ◦ } and { ◦ , ◦ } . The entanglement dynamics were ex-perimentally obtained for each initial state (Fig. 3(a)). Theoretically, the entanglement fora given initial state will approach an asymptotic value after several oscillations. In addition,the asymptotic value is strongly dependent on the initial state: S E = 0 .
739 for φ = 0 ◦ , S E = 0 .
867 for φ = 90 ◦ and S E = 0 .
977 for φ = 180 ◦ (dashed guided lines in Fig. 3(a)).In our experiment, the entanglement almost approached the theoretical asymptotic vale for φ = 90 ◦ and φ = 180 ◦ . At φ = 0 ◦ , the experimental result was 0 . ± .
027 after a20-step QW and the entanglement was still oscillating. Besides, the ballistic transport be-havior of ordered QW is shown in Fig. 4(a). During the experiment, the fidelities, defined as F ( ρ exp C , ρ th C ) = Tr[ (cid:113)(cid:112) ρ exp C ρ th C (cid:112) ρ exp C ] with ρ exp(th) representing the experimentally measured(theoretically predicted) density matrix, are larger than 0 . ± .
001 for each initial stateand step.We further demonstrated that the dynamic disorder can enhance the coin-position en-tanglement. To achieve this, we first chose the initial coin state as { θ, φ } = { ◦ , ◦ } , wherethe coin-position entanglement after a 20-step Hadamard QW is minimal (see Fig. 1(c)and Fig. 3(a)). We showed that the coin-position entanglement can be dramatically en-hanced to about S E = 0 .
98 by the dynamically disordered coin tossing sequence S C = F F HF HF HHF F F F F HF HHHHH (operated on the coin from left to right). Actually,the sequence S C is one of the optimal sequences to enhance the entanglement for the initialstate { ◦ , ◦ } after 20 steps. S C can also enhance the entanglement for any of the initialstates (the theoretical enhanced entanglement with the sequence S C for any initial statecan be found in Fig. 1(d)). We checked the enhancement with three other initial states: { ◦ , ◦ } , { ◦ , ◦ } and { ◦ , ◦ } . The experimental results are shown in Fig. 3(b),which clearly shows that all the entanglements are improved and approach the asymptoticvalue faster than in the ordered scenario. More importantly, the entanglement approachedthe same value (about 0 . . ± .
003 for each initial state7 ( c )( a )
Second Moment
N u m b e r o f S t e p s ( b )
H a d a m a r d Q W (cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13)(cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:2) (cid:13)(cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13)(cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:8) (cid:2) (cid:13)(cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13)(cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:3) (cid:7) (cid:2) (cid:13) t t D i s o r d e r e d Q W (cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13)(cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:2) (cid:13)(cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13)(cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:8) (cid:2) (cid:13)(cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13)(cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:3) (cid:7) (cid:2) (cid:13)(cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13)(cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:4) (cid:6) (cid:2) (cid:13) - 2 0 - 1 0 0 1 0 2 00 . 00 . 10 . 20 . 3
Probability
P o s i t i o n - 2 0 - 1 0 0 1 0 2 00 . 00 . 10 . 20 . 30 . 4
Probability
P o s i t i o n
FIG. 4. Measured trend (a) of second moment up to 20 steps (dots) for different initial states inordered and dynamically disordered QW. Ballistic transport behavior is observed in ordered QW(solid red line, fitted as 0 . t ). In dynamically disordered QW, a sub-ballistic transport behavior isobserved (solid blue line, fitted as 0 . t . ) and the dotted red line gives the theoretical predictions.The deep yellow line represents the typical diffuse transport behavior in classical random walksstarting at the origin. The error bars are smaller than the point size with only consideration ofstatistical errors. (b) and (c) give the probability distributions after a 18-step QW for initial statewith θ = 51 ◦ and φ = 0 ◦ . It is seen clearly that the spreading velocity in ordered QW (b) issignificantly faster than dynamically disordered one (c). and step in this scenario. Actually, the wave-function transport in a lattice will decelerateand show a sub-ballistic behavior in the presence of disorder [40]. The dynamic disordercan lead to a sub-ballistic transport behavior in a QW, and the transport trend dependson the choice of the two coin operations and the sequences [8, 41]. In Fig. 4(a), we showthe sub-ballistic transport behavior in a dynamically disordered QW. It is obvious that itsspreading velocity is faster than a typical diffuse transport behavior in a classical randomwalk but slower than a ballistic transport behavior in an ordered QW.Theoretically, the enhancement of coin-position entanglement is not dependent on thespecific form of the coin tossing sequence when randomness is introduced, and the numberof the steps is infinite. However, in our experiment, the total amount of step was limited to20. In this case, the enhancement of entanglement is dependent on the sequence S C . Thedependence of the final entanglement after a 20-step disordered QW with the initial state { ◦ , ◦ } on the disordered sequence S C is shown in the Fig. 5. Based on this dependence,the sequences can be divided into several different types (indicated in different colors inFig. 5). We experimentally checked the enhancement of the entanglement after 20-step QWwith different types of sequence (two sequences in each type), and the experimental resultsare shown in Fig. 5. The disordered QW became more powerful than the ordered ones in8 Entanglement
I n d e x o f S e q u e n c e
FIG. 5. Coin-position entanglement measured (opaque bars) for different sequences (forms aregiven in supplementary material) in dynamically disordered QW with the theoretical predictionsshown by transparent bars. The inset shows the rate of sequences generating different S E ∈ { , } at 20 steps with the initial state { ◦ , ◦ } , whose entanglement is lowest (shown with the solidblack line) in ordered Hadamard QW. Note that almost 73% of sequences can generate S E > . (cid:104) S E (cid:105) = 0 .
924 (guided by the black-dash horizontal line) was calculated byaveraging all sequences. Only statistical errors are considered. terms of ability to generate entanglement. If we have no information about the sequence, theentanglement should be obtained by averaging all the sequences. Based on our experimentalresults, the average entanglement (cid:104) S exp E (cid:105) , defined as (cid:104) S exp E (cid:105) = (cid:80) i =1 S iE · P i with P i beingthe rate of entanglement interval, to which S iE belongs, is about 0 . ± . CONCLUSIONS
In conclusion, we reported the first experiment to study the coin-position entanglementgeneration in discrete time quantum walks beyond the usual transport behaviors. We ob-served the initial state dependency of entanglement generation in ordered quantum walks.This entanglement involves periodic oscillations with the amplitude decay around an asymp-totic value, which is dependent on the initial state. More importantly, we found the coin-position entanglement can be significantly improved by the dynamic disorder for any initialstate. Besides, we showed the sub-ballistic transport behavior in dynamically disorderedquantum walks. Based on our experimental results, it seems that the entanglement powerof the coin tossing sequence S C positively correlates with the complexity of the sequence.In the spirit of the Kolmogorov complexity, the complexity of a binary sequence can be9xplicitly defined via Lempel-Ziv complexity C LZ [42, 43]. The complexity of the twelvesequences (from 1 to 12) in the Fig. 5 are 3, 3, 5, 8, 7, 7, 6, 6, 6, 7, 7 and 6, respectively(details given in the supplementary). The complexity measure is qualitative, and our re-sults qualitatively showed that the power of entanglement generation increased with thecomplexity of the sequence. When the sequence length increased, the complexity of randomsequence also increased and the entanglement power of the sequence increased as a resultas well. However, the complexity of periodic sequence will be eventually saturated, and theentanglement power will not increase. When the disordered sequence approached infinity,the complexity will be infinite and the entanglement power of all the disordered sequenceswill be the same and will be a maximal entanglement generator. Our experiment applies away to explore the entanglement in quantum information.Qin-Qin Wang and Xiao-Ye Xu contributed equally to this work. This work was sup-ported by National Key Research and Development Program of China (Nos. 2017YFA0304100,2016YFA0302700), the National Natural Science Foundation of China (Nos. 61327901,11474267, 11774335, 61322506), Key Research Program of Frontier Sciences, CAS (No. QYZDY-SSW-SLH003), the Fundamental Research Funds for the Central Universities (No. WK2470000026),the National Postdoctoral Program for Innovative Talents (No. BX201600146), China Post-doctoral Science Foundation (No. 2017M612073), and Anhui Initiative in Quantum Infor-mation Technologies (Grant No. AHY060300). ∗ [email protected] † cfl[email protected][1] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (CambridgeUniversity Press, Cambridge ; New York, 2000) pp. xxv, 676 p.[2] A. K. Ekert, Phys. Rev. Lett. , 661 (1991).[3] P. W. Shor, in Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposiumon (Ieee, 1994) pp. 124–134.[4] P. W. Anderson, Phys. Rev. , 1492 (1958).[5] A. Niederberger, M. M. Rams, J. Dziarmaga, F. M. Cucchietti, J. Wehr, and M. Lewenstein,Phys. Rev. A , 013630 (2010).
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The details of the experiment setup
Experimental time multiplexing photonic DTQWs —
In this work, the QWs is based onthe time multiplexing protocol. However, for overcoming the problem of the extra loss,birefringence crystal are used to replace the asymmetric Mach-Zehnder interferometer andimplement the spin-orbit coupling. Heralded single photons generated in SPDC are employedas the walker, the polarization is employed as the coin space, such that its polarization statecan be rotated to any of the single qubit states by wave plates, and the arriving time of thephotons, encoded in time bin, acts as the position space. One step quantum walk is realizedby a module composed of a half-wave plate (HWP) or a quarter-wave plate (QWP) and onepiece of birefringence crystal. The coin rotation operator implemented by the former isˆ R HWP ( θ ) = e − i θ ˆ σ y ˆ σ z , ˆ R QWP ( θ ) = e − iθ ˆ σ y e − i ( π/ σ z e iθ ˆ σ y . (3)where θ is the rotation angle of the optical axis of the wave plate, and ˆ σ i denotes thePauli matrices such that the eigenstates of the coin are | H (cid:105) and | V (cid:105) , corresponding tothe horizontal and vertical polarizations respectively, with the condition ˆ σ z | H (cid:105) = | H (cid:105) andˆ σ z | V (cid:105) = −| V (cid:105) . The birefringence of the latter causes the horizontal components to travelfaster inside the crystal than the vertical one. Then, after passing through the crystal, thephotons in the horizontal polarization move a step forward. Considering the dispersion afterpassing through a large number of crystals and the fact that the time bin encoding theposition of the walker in reality is a single pulse with a typical duration of a few hundredfemtoseconds, such a shift in time should be sufficiently large to distinguish the neighborhoodpulses at last. The magnitude of the polarization-dependent time shift by the birefringencecrystal depends on the crystal length and the cut angle. In this experiment, for introducingas weak dispersion as possible with sufficiently large birefringence, calcite crystal is adoptedfor its high birefringence index (0.167 at 800 nm ), and the length is chosen to be 8.98 mm with the optical axis parallel to incident plane, such that the time shift is designed to be5 ps for one-step quantum walk. Heralded single photon adopted as the walker —
The time multiplexing protocol requires13ulse photons, which can be obtained by attenuating a pulse laser or modulating a continuouslaser with an optical chopper. Considering the tradeoff between the operation on the timebins and the final analysis in the time domain, the time duration covers a range fromtens to thousands picoseconds, reaching even a few microseconds. In our experiment, foradopting a genuine single photon as the walker and considering that the length of thecrystals for realizing the time shifter should be as short as possible to reduce dispersion andimprove stability, the pulse duration of the single photon should be as small as possible; itis selected on the level of hundreds of femtoseconds here. Such a short single photon pulsecan be generated from SPDC with an ultrashort femtosecond pulse laser as the pump. Thegenerated photon pairs are time correlated, and as a result, the click of detection on theidler photon can predict the existence of the signal photon. Various architectures exist forgenerating this type of heralded single photons from SPDC. Here, considering the featuresof high brightness and collection efficiency, we adopt the beamlike SPDC. Then, the lengthof the nonlinear crystal can be chosen to be short.
Frequency up conversion single photon detection —
The spectrum of the arriving time ofsingle photons is usually obtained by the technology of time correlated single photon countingbased on commercial single photon detectors. However, in our case, the signals are containedin a single photon pulse train with a pulse duration of approximately 1 ps and a repetitionof 5 ps . Counting and analysis such ultra-fast single photon signals is challenging. Thetime resolution of commercial single photon detectors is limited by the time jitter, which istypically in the range of tens to hundreds picoseconds. As a result, it is unsuitable to directlyuse any commercial single photon detectors in this experiment. The detection of singlephotons with high resolution in time can be realized by transforming the temporal resolutionto a spatial resolution. Based on the optical parameter up conversion, the measurement ofan ultra-fast pulse of single photons can be realized by optical autocorrelation. That is, usingan ultra-fast laser pulse to pump a nonlinear crystal, when the single photon and laser pulsemeet each other inside the crystal, the single photon will be up-converted to have a shortwavelength for the sum frequency process. For the photons with a long wavelength can beconverted to a short one, this technology has been widely used in quantum communication forimproving the detection efficiency in the infrared waveband. Here, we adopt this technologyfor its high resolution in time. Although periodically poled crystals are widely used in thistechnology for their high conversion efficiency, they are useless here for concentrating on the14ime resolution. The thickness of nonlinear crystal should be as thin as possible meanwhiletaking into account the conversion efficiency. There exist two types of structures, collinearand non-collinear sum frequency. We adopt the latter to obtain a better signal to noiseratio (SNR), induced by the spatial divergence between the sum frequency signal and thepump laser. In this work, the crystal used is a 1 mm thick β -BaB O (BBO) crystal, cutfor type-II second harmonic generation in a beam like form. Then, the incidence angles ofthe signal pulse train with single photons and the pump laser are equal to each other, with3 ◦ to the normal direction. For reducing the noise induced by the strong pump laser, adispersion prism in a 4F system is adopted as a spectrum filter. The scattered photons withwavelength longer than 395 nm are blocked by a knife edge. The rising edge in the sidebandof this self-established spectrum filter is less than 1 nm . Similarity (cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13) (cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:2) (cid:13)(cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13) (cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:8) (cid:2) (cid:13)(cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13) (cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:3) (cid:7) (cid:2) (cid:13)
Counts
T i m e D e l a y ( p s ) Counts
T i m e D e l a y ( p s ) × 1 0 Similarity
N u m b e r o f S t e p s (cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13) (cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:2) (cid:13)(cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13) (cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:8) (cid:2) (cid:13)(cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13) (cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:3) (cid:7) (cid:2) (cid:13)(cid:1)(cid:14) (cid:9) (cid:5) (cid:3) (cid:13) (cid:10) (cid:12) (cid:11) (cid:1)(cid:15) (cid:9) (cid:4) (cid:6) (cid:2) (cid:13)
FIG. 6. Plot of the similarity for each step of QW. The similarity is defined as S = (cid:80) x (cid:112) P exp ( x ) P th ( x ), where P exp stands for the experimental probability distribution and P th isthe corresponding theoretical prediction. The results for ordered QW are shown in the top paneland the results for dynamically disordered QW are presented in the bottom panel. The inset ineach panel shows the coincidence counts after a 20-step walk for a initial state with θ = 51 ◦ and φ = 0 ◦ . In the ordered scenarios, the similarity becomes degenerate gradually as the number ofsteps increases, mainly for the decoherence. In the disordered scenarios, there are more occupationsnear the origin (in the middle of time delays), where the interference is more complicated, resultsthe faster degeneration of the similarity compared to the ordered cases. Only statistical errors areconsidered with total counts 24 thousand in 4 hundred seconds. escription of the experimental setup — An ultra fast pulse (140 fs ) train generated bya mode-locked Ti:sapphire laser with a central wavelength at 800 nm and repetition ratio76 MHz is firstly focused by lens L1 to shine on a 2 mm thick β -BaB O crystal (BBO1),cut for type-I second harmonic generation. The frequency-doubled ultraviolet pulse (witha wavelength centered at 400 nm , 100 mW average power and horizontally polarized) andthe residual pump laser are collimated by lens L2, and then separated by a dichroic mirror(DM). The frequency-doubled pulse train is then focused by lens L3 to pump the secondnonlinear crystal (BBO2), cut for type-II nondegenerate beam like SPDC. The signal andidler photons are collimated together with one lens L4 (f=150 mm ). The collimated signalphotons in horizontal polarization with a center wavelength at 780 nm are then guideddirectly in free space to the following quantum walks device. The collimated idler photons invertical polarization with a center wavelength of approximately 821 nm firstly pass througha spectrum filter with a central wavelength 820 nm and bandwidth 12 nm and then arecoupled into a single-mode fibre and sent directly to a single-photon avalanche diode (SPAD)for counting in coincidence with the signal photons. The quantum walks device is composedof HWPs (QWPs) and calcite crystals, and each step contains one piece of HWP (QWP)and one piece of calcite crystal. In the experiment, we have adopted 20 such sets, with only1/4 of them shown in the figure. The initial state is prepared by an apparatus composedof a polarized beam splitter (PBS1), a half wave plate (HWP1) and a quartz wave plate(QWP1). The residual pump in the frequency-double process is adopted as the pump inthe following frequency up conversion single photon detection with the retroreflector R1 fortemporal matching. After the quantum walks is finished, the signal photons are collectedinto a short single mode fibre (10 cm long) by a fibre collimator (FC1) and then guided tothe polarization analyzer composed of QWP2, HWP2 and PBS2 successively. Finally, thearriving time of signal photons is measured by scanning the pump laser and detecting the upconversion signals with a photomultiplier tubes (PMT). For reducing the scattering noise,BBO3 is cut for noncollinear up conversion and a spectrum filter based on a 4F system isconstructed, where a prism is adopted for introducing the dispersion, a knife edge is used toblock the long waves and the signal is reflected to the PMT with a pickup mirror.16 empel-Ziv complexity Lempel-Ziv (LZ) complexity can be introduced to estimate the randomness of finite se-quences, in the spirit of the Kolmogorov complexity. The LZ complexity measure countsthe number of distinct substrings (patterns) in a sequence when scanned from left to rightand then parsed. Note that we only consider binary sources throughout this paper. Thealgorithm is as follows [44]:1. Let S C = s s · · · s n denote a finite 0-1 symbolic sequence; S C ( i, j ) denotes a substringof S C that starts at position i and ends at position j , that is, when i (cid:54) j , S C ( i, j ) = s i s i +1 · · · s j and when i (cid:62) j , S C ( i, j ) = {} (null set); V ( S C ) denotes the vocabulary of asequence S C . It is the set of all substrings S C ( i, j ) of S C , (i.e., S C ( i, j ) for i = 1 , · · · n ; i (cid:54) j ). For example, let S C = 010, we then have V ( S C ) = { , , , , } .2. The parsing procedure needs to scan the sequence S C from left to right. If S C ( i, j )belongs to V ( S C (1 , j − S C ( i, j ) and V ( S C (1 , j − S C ( i, j +1)and V ( S C (1 , j )), respectively. Repeat the previous process until the renewed S C ( i, j )does not belong to the renewed V ( S C (1 , j − S C ( i, j ) to indicate the end of a new sequence. Update S C ( i, j ) and V ( S C (1 , j − S C ( j + 1 , j + 1)(the single symbol in the j + 1 position) and V ( S C (1 , j )), respectively,and the step 2 continues.3. This parsing operation begins with S C (1 ,
1) and continues until j = n , where n is thelength of the symbolic sequence S C .For instance, the sequence which is only composed of units 01 (i.e., 010101 · · · ) is parsedas 0 · · · · · . So the number of distinct substrings c ( n ) is 3. To compute the LZcomplexity, the sequence of H and F should be transformed into a 0-1 symbolic sequence(i.e., map the H and F to 1 and 0). Follow the above method, the twelve randomly selectedsequences in Fig.4 (from 1 to 12) are parsed as:1. 1 · · · → c ( n ) = 32. 1 · · · → c ( n ) = 33. 1 · · · · · → c ( n ) = 517. 1 · · · · · · · · → c ( n ) = 85. 1 · · · · · · · → c ( n ) = 76. 1 · · · · · · · → c ( n ) = 77. 0 · · · · · · → c ( n ) = 68. 1 · · · · · · → c ( n ) = 69. 1 · · · · · · → c ( n ) = 610. 1 · · · · · · · → c ( n ) = 711. 1 · · · · · · · → c ( n ) = 712. 0 · · · · · · → c ( n ) = 6 Detail forms of the sequences in Fig.5 HF HF HF HF HF HF HF HF HF HF HHHHF HHHHF HHHHF HHHHF HHHHHHHF F F F HHHHHF HHH HF HHF HF F F HHHHF F HHF HF HHHF F HHF HF F F HHHF F F HH HF F HHHF HHHF F HF HF F HHF F F HHF F F HF F HF HHF HHF HH HHF F F HHHHHHF F HF HHHF H HHHHHF F HHF F HF HHHF HF F
HHF F F HF HHF F F F HF HF HF H
HHF F F HHHHF F F HF F HF HHH