Direct Observation of Topology from Single-photon Dynamics on a Photonic Chip

Yao Wang,1, 2 Yong-Heng Lu,1, 2 Feng Mei,3, 4, ∗ Jun Gao,1, 2 Zhan-Ming

Li,1, 2 Hao Tang,1, 2 Shi-Liang Zhu,5, 2 Suotang Jia,3, 4 and Xian-Min Jin1, 2, †

1State Key Laboratory of Advanced Optical Communication Systems and Networks,School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

2Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, China

3State Key Laboratory of Quantum Optics and Quantum Optics Devices,Institute of Laser Spectroscopy, Shanxi University, Taiyuan, Shanxi 030006, China

4Collaborative Innovation Center of Extreme Optics,Shanxi University, Taiyuan, Shanxi 030006, China

5National Laboratory of Solid State Microstructures and School of Physics,Nanjing University, Nanjing 210093, China

(Dated: November 7, 2018)

Topology manifesting in many branches ofphysics deepens our understanding on state ofmatters. Topological photonics has recentlybecome a rapidly growing field since artificialphotonic structures can be well designed andconstructed to support topological states, espe-cially a promising large-scale implementation ofthese states using photonic chips. Meanwhile,due to the inapplicability of Hall conductance tophotons, it is still an elusive problem to directlymeasure the integer topological invariants andtopological phase transitions for photons. Here,we present a direct observation of topologicalwinding numbers by using bulk-state photondynamics on a chip. Furthermore, we for thefirst time experimentally observe the topologicalphase transition points via single-photon dy-namics. The integrated topological structures,direct measurement in the single-photon regimeand strong robustness against disorder add thekey elements into the toolbox of ‘quantum topo-logical photonics’ and may enable topologicallyprotected quantum information processing inlarge scale.

The introduction of topology into condensed-matter andmaterial sciences originates from the connection of inte-ger quantum Hall conductances with topological Cherninvariants [1], which greatly expands our knowledge onstate of matters. With the birth of topological insula-tors, searching topological state of matters in solid statematerials [2, 3] and photonic systems [4, 5] has recentlybecome a leading research field. In contrast to the chal-lenging experimental requirements for realizing topologi-cal states in solid state materials, photonic systems pro-vide a convenient and versatile platform to design varioustopological lattice models and study different topological

∗ meifeng@sxu.edu.cn† xianmin.jin@sjtu.edu.cn

states, including topological insulator states [6–10] andtopological Weyl points [11, 12]. The found topologicalboundary states potentially can be utilized for develop-ing inherently robust and efficient artificial photonic de-vices [13–16].

In the view of fundamental physics, the topologicalinvariant is a crucial parameter to characterize the topo-logical matter state. In fermion systems, the topologicalinvariant can be revealed by conductance measurements,while the concept of Hall conductance is inapplicable inphotonic systems. New methods for directly detectingthe topological invariants in topological photonics remainto be developed. The pioneering proposals in theory [17–19] and experimental observations have been dedicatedin both integrated photonic lattices [20–22] and bulk op-tics [23–26].

Different from the efforts made to detect the topolog-ical invariant based on probing Berry curvature [17, 20],non-Hermitian photon loss [18, 21] or the dynamics ofedge states [19, 22], we propose a new approach to di-rectly detect the topological invariant via the bulk-statephoton dynamics in the real space, which beyonds thephysical picture where topological invariant is defined onthe equilibrium Bloch state in the momentum space.

To extend promised topological protection into thequantum regime, we have to find an appropriate systemthat is physically scalable and has inherently low losswhen scaling up. Integrated photonics can meet the firstrequirement elegantly by constructing topological struc-tures on a photonic chip in a physically scalable fashion,with which topological states can be generated, manipu-lated and detected in a very high complexity beyond thatconventional bulk optics can do [23–26]. Meanwhile, re-alizing topological states in Hermitian systems can wellmeet the second requirement since the intrinsic loss innon-Hermitian systems [21, 24, 25] will induce an evolu-tion of exponential decay for single photons and multi-plicative inefficiency for multi-photons.

Here, we integrate topological waveguide latticeson a photonic chip and experimentally demonstrate adirect observation of the topological invariants in the

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FIG. 1. Schematic of the integrated topological pho-tonic lattice, model and simulation. a, Sketch of Su-Schrieffer-Heeger model. The unit cell label x is marked start-ing from the edge of the system with 1. Every unit cell consistsof two sites and every site is implemented by a laser-writtenwaveguide. b-c, Simulated results. The values of PPDC oscil-late around 0 and 0.5 for w = 0.1 and w = 0.9 correspondingto the systems in topological trivial and nontrivial phase, re-spectively. The numbers on the right side of the figure presentthe averaged values of PPDC.

constructed Hermitian system using bulk-state photondynamics. Through initially injecting photons into themiddle waveguide to excite the bulk state, the valuesof topological winding numbers can be extracted fromthe chiral photonic density centers associated withthe final output distribution. We further extend thetopological system and measurement into quantumregime by observing the topological phase transitionpoint via single-photon dynamics. With the bulk stateexcited by heralded single photons, we can successfullyidentify the topological phase transition point separatingthe topological trivial and nontrivial phases, even withartificially introduced disorder.

Topological photonic lattice. As is shown inFig.1(a), we fabricate waveguide lattices in borosilicateglass by using the femtosecond laser direct writing tech-nique [27–30] (see Methods for details). The constructedlattices are based on the Su-Schrieffer-Heeger model,which describes a one-dimensional lattice with alternat-ing strong and weak couplings. The Hamiltonian of this

model could be written as [31]

H =∑x

(J1a+x bx + J2b

+x ax+1) + h.c., (1)

where each unit cell in the chain consists of two sites la-beled as a and b, the terms a+x (ax) and b+x (bx) are thecreation (annihilation) operators for the two sites in thex unit cell, and the coefficients J1 and J2 represent theintra-cell and inter-cell coupling strengths, respectively.To study the topological feature, we rewrite Eq. (1) in

momentum space as H =∑kxh(kx), where h(kx) =

dxτx + dy τy, dx = J1 + J2 cos(kx), dy = J2 sin(kx), andτx and τy are the Pauli spin operators defined in the mo-mentum space. The energy bands of the Hamiltonian arecharacterized by the topological winding number

ν =1

2π

∫dkxn× ∂kxn (2)

where n = (nx, ny) = (dx, dy)/√d2x + d2y. We manip-

ulate the coupling coefficients as J1 = g + gt cos(wπ)and J2 = g − gt cos(wπ), where g > 0, 0 < w < 1, and0 ≤ t ≤ 1. The system is in the topological nontrivialphase with the winding number ν = 1 when J1 < J2, i.e.w ∈ (0.5, 1). Otherwise, it is in the topological trivialphase with ν = 0 when J1 > J2, i.e. w ∈ (0, 0.5). Thetopological phase transition point appears when J1 = J2.

Dynamical detection of topological windingnumber. To detect the winding number of the latticeon the topological photonic chip, we introduce a photonpopulation difference center (PPDC) Pd =

∑x x(a+x ax−

b+x bx), where the unit cell index x is shown in Fig.1(a).We inject photons into the middle waveguide to excitethe bulk state. With the evolution of the photons overa distance z in the lattice, the corresponding PPDC canbe denoted as Pd(z) (see Methods). We find that thetopological winding number ν can be measured via theevolution-distance-averaged PPDC Pd(z), which can beexpressed as (see Methods)

ν = 2 limZ→∞

1

Z

∫ Z

0

dtPd(z), (3)

where z is the evolution distance. In Fig.1(b-c), we calcu-late the Pd for different coupling coefficients and latticesizes. The results show that the values of PPDC Pd keeposcillating centered at 0 and 0.5 when the lattice is inthe topological trivial and nontrivial phases, respectively.The topological winding numbers derived as ν = 0 andν = 1 can be directly measured from the output densitydistribution.

In the experiment, we implement a topological pho-tonic lattice consisting of 10 waveguides with t = 1.0.To perform the evolution-distance average, we integrate40 such photonic lattices on a single chip with differentevolution distances varying from 20 mm to 30 mm with a

3

FIG. 2. Experimental results of PPDC. The measuredvalues of PPDC for 10-sited (a, b) and 18-sited (c, d) lattices,which are found oscillating around 0 (a, c) and 0.5 (b, d)for the systems in topological trivial and nontrivial phase.The averaged values of PPDC Pd are 0.045 ± 0.090 (a) and0.540 ± 0.070 (b) for the case of t = 1.0, and are 0.095 ± 0.16(c) and 0.526± 0.014 (d) for the case of t = 0.5, respectively.

step size of 0.2mm. We excite one waveguide in the cen-tral unit cell (x = 3) with a narrowband coherent light at852 nm, and measure the output density from each pho-tonic lattice. The evolution-distance-dependent PPDC isextracted and shown in Fig.2(a-b). The result in Fig.2(a)shows that, when the system is in the topological trivialphase, the values of PPDC Pd keep oscillating centered at0.045±0.090. While the system is in the topological non-trivial phase, Pd keeps oscillating around 0.540 ± 0.070as shown in Fig.2(b). According to Eq. (3), we obtainthe topological winding numbers ν = 0.09 ± 0.18 andν = 1.08 ± 0.14 for the two phases respectively. We cansee that the oscillation of the measured Pd values is moreirregular than that of the simulated result, but the wind-ing number can still be clearly extracted.

To further experimentally demonstrate the reliabilityand universality of our approach, we fabricate anotherset of photonic lattices consisting of 18 waveguideswith t = 0.5. The evolution distance varies from 7 mmto 16 mm with a step size of 0.2 mm. As is shownin Fig.2(c-d), when the lattices are prepared in thetopological trivial and nontrivial phases, the measuredvalues of Pd are oscillating around 0.095 ± 0.16 and

FIG. 3. Theoretical results of TPTS. a, The Su-Schrieffer-Heeger model with the labels marked starting frommiddle of the system for a concise expression of the photonpopulation center. b, The simulated results of TPTS. The dy-namical TPTS value increases and then decreases for a contin-uous transitive system (red point), and the transition pointwill be more distinct in the strong interaction region (bluepoint).

0.526 ± 0.014, which lead to the topological windingnumbers of ν = 0.19± 0.32 and ν = 1.052± 0.28, respec-tively. The results are well consist with the simulatedresults shown in Fig.1(c) and suggest that our proposeddynamical approach of measuring topological invariantsis insensitive to the detailed lattice configurations.

Dynamical detection of topological phase tran-sition. We further extend the topological system andmeasurement into quantum regime by observing thetopological phase transition point via single-photon dy-namics. The topological phase transition in our pho-tonic lattices can also be directly measured from theoutput density distribution. The transition point canbe revealed by the generalized photon population centerPc =

∑x x

2(a+x ax + b+x bx), where the label x is markedas shown in Fig.3(a) for a concise expression. With thebulk state excited from the central unit cell by single pho-tons, the value of generalized photon population centercan be derived as Pc(z) for an evolution distance z (seeMethods). We can further obtain the topological phasetransition signal (TPTS) St = Pc(z)/z

2, and we find that(see Methods)

St =

{J21

2 , J1 < J2J22

2 . J1 > J2(4)

The simulated results are illustrated in Fig.3(b). Fora continuously transitive system from topological non-

4

FIG. 4. Experimental results of TPTS. a, The relationbetween the coupling strength and the separation betweenadjacent waveguides. The blue dash lines mark the exper-imentally accessible range of d1 and d2. b, The couplingstrength of J1 and J2 of the 11 lattices used in experiment(red squares). c, The evolution probability distribution of sin-gle photons in topological nontrivial phase, transition pointand trivial phase. The blank arrows mark the excited sitesin experiment. d, The measured results of TPTS. The topo-logical transition point appears when the system undergoesthe phase transition from the topological nontrivial to trivialphase (red circles), even with artificially introduced disorder(black and blue circles).

trivial to trivial, as is sketched with the red points, theTPTS value increases firstly and then decreases, and themaximum value arises when the system undergoes thetopological phase transition point. Unlike the statisticalmeasurement of PPDC, this approach requires only sin-gle measurement on generalized photon population centerfor a certain structure, the topological transition pointtherefore can be more conveniently observed in experi-ment. When the dynamical TPTS value varies with theincreasing of the value of J1 and J2, the transition points

as sketched with blue points will be more distinct to beobserved in the strong interaction region.

To experimentally observe the TPTS in a continuoustransitive system, we fabricate 11 set of photonic latticeswith lattice constant d=20 µm and different intra-cell(d1) and inter-cell (d2) space. The dimerization ∆d =(d2 − d1)/2 varies from -2 µm to 2 µm in a step of 0.5µm for 9 lattices, and two more lattices are designed nearthe transition point with the dimerization values of -0.2µm and 0.2 µm. The coupling strength is modulated bythe separation between adjacent waveguides, which is notlinear according to coupling mode theory (see Fig.4(a)).The corresponding coupling strength of J1 and J2 definedby the dimerization values in 11 lattices are marked withred squares in Fig.4(b). All the lattices consist of 42sites (21 untie cells) and have an evolution distance of18 mm to ensure that the photons will not evolve to theedge. We prepare heralded single photons at 810 nm viaspontaneous parametric down conversion (see Methods)and excite the bulk state from one of two sites in thecentral unit cell (x = 0).

We show the experimental results in Fig.4(c-d). Di-rectly from the evolution probability distribution of sin-gle photons, we can not find distinct criteria to distin-guish when the lattice is in topological nontrivial phase,transition point and trivial phase (see Fig.4(c)). InFig.4(d), we plot the experimental (red dots) and the-oretical (black dash line) results of TPTS St varyingwith the coupling strength ratio J1/J2 (and dimeriza-tion ∆d). The values of St increase with the square ofJ1 when J1/J2 < 1, and decrease with the square of J2when J1/J2 > 1. As a result, the signal of the topologicaltransition point appears very clearly when the J1/J2 = 1,corresponding to the ∆d = 0.

Besides the advantage of directly observing topologicaltransition point, it would be also interesting to test therobustness of this approach. We manage to introduce thedisorder into the system by adding random fluctuation of±0.1 µm to d1 and d2 in the laser writing process. Wefabricate 11 set of such disorder-embedded lattices andrepeat the experiment twice. As is shown in Fig.4(c),the experimental results retrieved from the 22 latticesindicate that while the measured values of St randomlydeviate from the theoretical curve assumed for the idealcase, the topological transition point still can be clearlyidentified around J1/J2 = 1.

In summary, we experimentally demonstrate a directobservation of the topological invariants and the phasetransition from the photon dynamics in the bulk state.Our approach provides a new route to direct measure-ment of topology via single-particle dynamics in thereal space, which complements the approach in ultracoldatomic systems using Bloch state dynamics in the mo-mentum space [32, 33]. Our approach is also availablefor further generalization and application to other topo-logical systems and higher-dimensional cases.

The demonstrated key elements, including integratedtopological structures, direct measurement in single-

5

photon regime and strong robustness against disorder,can enrich the emerging field of ‘quantum topologicalphotonics’. With the primary attempt to combine topol-ogy with quantum integrated photonics, it is promisingto explore scalable topologically protected quantum in-formation processing on topological photonic chips be-yond classical topological photonics. The prompt ques-tions, but remain open, will be whether we can directlyobserve topology with multi-photon dynamics and howqubit and entanglement behave.

Acknowledgments

The authors thank Jian-Wei Pan for helpful dis-cussions. This research is supported by the Na-tional Key Research and Development Programof China (2016YFA0301803, 2017YFA0303700,2017YFA0304203), National Natural Science Foun-dation of China (NSFC) (Grant No. 61734005,11761141014, 11690033, 11604392), Science and Tech-nology Commission of Shanghai Municipality (STCSM)(15QA1402200, 16JC1400405, 17JC1400403), ShanghaiMunicipal Education Commission (SMEC)(16SG09,2017-01-07-00-02-E00049), PCSIRT (IRT 17R70) andthe Shanxi 1331KSC and 111 Project (D18001), X.-M.J.acknowledges support from the National Young 1000Talents Plan.

Methods

Fabrication and measurement of the integratedtopological photonic lattices: According to the char-acterized coupling coefficients modulated by the separa-tion between two adjacent waveguides, the lattices aredesigned and written in borosilicate glass (refractive in-dex n0 = 1.514) with femtosecond laser with repetitionrate 1MHz, pulse duration 290fs and working wavelength513nm. Before the laser writing beam is focused insidethe borosilicate substrate with a 50X objective lens (nu-merical aperture of 0.55), we control the shape and size ofthe focal volume of the beam with a beam-shaping cylin-drical lens. A high-precision three-axis motion stage isused to move the photonic chip during fabrication witha constant velocity of 5 mm/s.

Experiments are performed by injecting the photons(herald single photon) into the lattices using a 20X ob-jective lens. The evolution output is observed using a 10Xmicroscope objective lens and the CCD (ICCD) cameraafter a total evolution distance within the lattice.

The relationship between topological windingnumber and PPDC: Suppose one waveguide in themiddle unit cell is initially excited, then the initial stateof the photonic chip can be written as |ψ(0)〉. To find the

relationship between the photon dynamics in the waveg-uide lattice and the winding number, we introduce pop-ulation difference in each unit cell and define a photonpopulation difference center (PPDC), i.e.,

Pd =

N∑x=1

x(P eax − Pebx), (5)

where P em = |e〉m〈e| (m = ax, bx) is the photon popula-tion probability. The PPDC associated with the photonicevolution in the waveguide lattice can be described as

Pd(z) = 〈ψ(0)|eiHzPde−iHz|ψ(0)〉. (6)

For the photonic SSH model, we can connect the abovedynamical center to the winding number. We rewrite thePPDC in the momentum space as

Pd(z) =1

2π

∫ π

−πdkx〈χ(0)|eih(kx)zi∂kxτze−ih(kx)z|χ(0)〉.

(7)By substituting h(kx) = dxσx + dyσy into (7), we findthat Pd(Z) can be connected with the topological wind-ing number ν defined, i.e.,

Pd(z) =ν

2− 1

4π

∫dkx cos(2Ez)n× ∂kxn, (8)

where n = (nx, ny) = (dx, dy)/E and E =√J21 + J2

2 + 2J1J2 cos(kx). In the long evolution dis-tance limit, the second term in the above equation willvanish. Then we can obtain a relationship betweenthe winding number and the evolution-distance-averagedPPDC, i.e.,

ν = 2Pd, (9)

where the evolution-distance-averaged PPDC is

Pd = limZ→∞

1

Z

∫ Z

0

dz Pd(z). (10)

where Z is the total evolution distance for the photons inthe waveguide lattice. Note that Pd is just the oscillationcenter of Pd(z) varying with z. The topological windingnumber is twice this oscillation center.

The relationship between TPTS and photonpopulation center: The initial state of the photonicchip is the same as the one in the winding number detec-tion. To find the relation between the photon dynamicsin the waveguide lattice and the winding number, we in-troduce a generalize photon population center operator,i.e.,

Pc =

N∑x=1

x2(P eax + P ebx), (11)

6

where P em = |e〉m〈e| (m = ax, bx) is the photon popula-tion operator. Then the generalized photon populationcenter associated with the evolution of photons in thewaveguide lattice can be described as

Pc(z) = 〈ψ(0)|eiHzPce−iHz|ψ(0)〉. (12)

Based on the above equation, we define a new quan-tity called as topological phase transition signal (TPTS),which is expressed as

St = Pc(z)/z2. (13)

By transferring the above equation into the momentumspace, we can further get

St =1

2π

∫ π

−πdkx〈ψ(0)|eih(kx)z(i∂kx)2e−ih(kx)z|ψ(0)〉/z2.

(14)In the long evolution distance limit, the terms propor-tional to 1/z can be omitted and the above identity canbe simplified into

St =1

2π

∫ π

−πdkx(∂kxE)2

=1

2π

∫ π

−πdkx

J21J

22 sin2(kx)

J21 + J2

2 + 2J21J

22 cos(kx)

. (15)

Based on residue theorem, we can analytically solve the

above integral and get

St =

{J21

2 , J1 < J2J22

2 . J1 > J2(16)

This equation shows that the topological phase transitionin the photonic Su-Schrieffer-Heeger model can be di-rectly observed from the single photon dynamics in bulkstate.

The generation and imaging of the heralded sin-gle photons: We obtain the single-photon source withthe wavelength of 810 nm generated from periodically-poled KTP (PPKTP) crystal via spontaneous paramet-ric down conversion (SPDC). After a long-pass filter anda polarized beam splitter (PBS), the photon pairs areseparated to two components, horizontal and vertical po-larization. The measured evolution patterns would comefrom the thermal-state light rather than single-photonsif we inject only one polarized photon into the latticeswithout external trigger. Therefore, we inject the hori-zontally polarized photon into the lattices, while the ver-tically polarized photon acts the trigger for heralding thehorizontally polarized photons out from the lattices witha time slot of 10ns. We capture each evolution resultusing ICCD camera after accumulating in the ‘external’triggering mode for 2000s.

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