Robustness of topologically protected edge states in quantum walk experiments with neutral atoms
Thorsten Groh, Stefan Brakhane, Wolfgang Alt, Dieter Meschede, Janos Asbóth, Andrea Alberti
RRobustness of topologically protected edge states in quantum walk experimentswith neutral atoms
Thorsten Groh, Stefan Brakhane, Wolfgang Alt, Dieter Meschede, Janos K. Asbóth, and Andrea Alberti ∗ Institut für Angewandte Physik, Universität Bonn, Wegelerstr. 8, D-53115 Bonn, Germany Institute for Solid State Physics and Optics, Wigner Research Centre for Physics,Hungarian Academy of Sciences, H-1525 Budapest P.O. Box 49, Hungary (Dated: June 7, 2016)Discrete-time quantum walks allow Floquet topological insulator materials to be explored usingcontrollable systems such as ultracold atoms in optical lattices. By numerical simulations, we studythe robustness of topologically protected edge states in the presence of decoherence in one- andtwo-dimensional discrete-time quantum walks. We also develop a simple analytical model quan-tifying the robustness of these edge states against either spin or spatial dephasing, predicting anexponential decay of the population of topologically protected edge states. Moreover, we present anexperimental proposal based on neutral atoms in spin-dependent optical lattices to realize spatialboundaries between distinct topological phases. Our proposal relies on a new scheme to implementspin-dependent discrete shift operations in a two-dimensional optical lattice. We analyze under real-istic decoherence conditions the experimental feasibility of observing unidirectional, dissipationlesstransport of matter waves along boundaries separating distinct topological domains.
PACS numbers: 67.85.-d, 03.65.VfKeywords: Floquet topological phases; discrete-time quantum walks; decoherence; neutral atoms in opticallattices
I. INTRODUCTION
Topological insulators are quantum materials behavinglike an ordinary insulator in the bulk, and yet allowing,in two dimensions and above, matter waves to propagatealong their boundaries through a discrete number of edgemodes [1, 2]. The distinguishing property of these mate-rials is the existence of so-called topologically protected(TP) edge modes, which are robust against continuousdeformations of the material’s parameters including spa-tial disorder, providing the bulk remains insulating (i.e.,no gap closing). In one dimension (1D), a discrete num-ber of TP edge states can exist in the presence of spe-cial symmetries (e.g., particle-hole symmetry in super-conducting quantum wires), with their energy being ex-actly pinned to the midpoint of the energy gap. In twodimensions (2D), the most notable example of a topo-logical insulator is a two-dimensional electron gas in ahigh magnetic field, where the transverse conductance isfound to be quantized in multiples of e /h (integer quan-tum Hall effect, IQHE) [3]. Over the years, this effect hasbeen verified by experiments to one part in despiteimpurities and other imperfections, which unavoidablyoccur in actual physical samples [4]. Its robustness is to-day well understood in terms of the topological structureof the Landau levels, which form well-separated energybands [5].In general, the robustness of edge states in these insu-lating materials results from energy bands with nontrivialtopological character. Topologically nontrivial bands areoften related to an obstruction to define the Bloch wave ∗ [email protected] functions over the whole Brillouin zone using a singlephase convention [6]. This obstruction to a global choiceof the gauge can be understood as resulting from a twistof the Bloch wave functions, much as the twist in theMöbius strip represents an obstruction to define an ori-ented surface. The twists of the energy bands are quan-tified by topological invariants, which are integer quan-tum numbers assigned to each isolated band of the bulk.These can be, for instance, winding numbers Z (e.g., forthe Su-Schrieffer-Heeger model), or just Z numbers withtwo possible values denoting trivial and nontrivial topo-logical phases (e.g., for particle-hole-symmetric quantumwires). The characteristic of such invariants is that theyare unchanged under a continuous modification of thesystem parameters, provided that the energy gap and therelevant symmetries are preserved. In particular, two in-sulators are said to belong to different topological phasesif the sum of the topological invariants of occupied bandsare different [7, 8].A topological argument with far-reaching physical im-plications, known as the bulk-boundary correspondenceprinciple , establishes a relation between the topologicalinvariants and the number of TP edge modes at theboundary between two topological phases [9]. Simplystated, it predicts that any spatial crossover region sepa-rating two bulks hosts a minimum number of edge modesgiven by the difference of the bulk invariants. Thesemodes are topologically protected as they cannot disap-pear by a continuous deformation of the system parame-ters, including a deformation of the boundary’s shape. Inthe IQHE, for instance, the number of current-carryingTP edge modes is equal to the sum of the Chern numbersof the Landau levels below the Fermi energy [10].TP edge modes at the boundary of a 2D topological a r X i v : . [ qu a n t - ph ] J un insulator are immune to Anderson localization. Even ifwe allow for local disorder (of any amount in the regionadjacent to the boundary), including shape irregulari-ties, topological arguments predict that TP edge statesmaintain their metallic-like character notwithstandingthe disorder their wave functions being fully delocal-ized around the whole length of the insulator [9]. Asa consequence, any wave packet formed by a superposi-tion of TP edge states propagates coherently along theboundary, instead of being confined within some regionby the disorder. Moreover, transport along the bound-ary is virtually immune to backscattering too [11], for thewave packet would need to tunnel to the opposite edgeof the insulator material in order to couple to a counter-propagating edge mode a process that is exponentiallysuppressed with the size of the sample.Besides being interesting per se , topological insulatorshave stimulated great interest for the possibility to ex-ploit TP edge states for engineering ballistic electronictransport in dissipationless solid-state devices and forenabling topological protection of quantum information[12]. In recent years, IQHE devices have attained anexquisite level of control, which enabled the demonstra-tion of quantum devices such as an electronic Mach-Zehnder interferometer [13] and a two-electron Hong-Ou-Mandel-like interferometer [14]. However, these systemsstill require high magnetic fields on the order of Tin order to make the energy gap between Landau lev-els (i.e., the cyclotron frequency) larger than cryogenictemperatures below K. Larger gaps are obtained withhigh-mobility graphene IQHE devices, holding promiseto operate at room temperature, though still requiringhigh magnetic fields [15]. In a different approach, thequantum anomalous Hall effect avoids external magneticfields by exploiting a ferromagnetic topological-insulatorstate induced by spontaneous magnetization, though de-manding, in return, cryogenic temperatures well belowboth the Curie point and the magnetically induced en-ergy gap [16, 17]. The discovery of quantum spin Halleffect in HgTe/CdTe quantum wells started the quest fortopological insulators with large gap, and yet not relyingon magnetic fields [18]. However, the gap size of thesenovel materials still imposes, at least so far, cryogenictemperatures < K to function [19].Topological insulator materials are challenging to syn-thesize, and only a few topological phases have hith-erto been accessible with solid-state materials [20]. Thishas motivated the search for topological phases in non-electronic systems, which also allow implementing thesame wave-mechanical principles underlying topologicalinsulators. Because of their high degree of control andflexibility, ultracold atoms trapped in an optical latticeare ideal systems to shed new light on the origin and dy-namics of topological insulators. In particular, these sys-tems have enabled the direct measurement of the Berry-Zak phase [21] and Wilson lines [22], the realization of theHaldane model [23], the measurements of the anomaloustransverse velocity [24], demonstration of the Thouless pump mechanism [25, 26], the realization of compactedartificial dimensions [27, 28], and the measurement of theBerry flux [29] as well as Berry curvature [30]. Besidesultracold atom systems, TP edge modes have also beenobserved in microwave photonic crystals [31], photonicquasicrystals [32, 33], and even mechanical spring sys-tems [34, 35].Discrete-time quantum walks (DTQWs) with trappedultracold atoms [36] offer a versatile and highly controlledplatform for the experimental investigation of topologicalinsulators. We note that even a single atom coherentlydelocalized on a periodic potential is sufficient to sim-ulate topology-induced transport phenomena, providedthat the energy bands have a nontrivial topological struc-ture. In DTQW experiments, an ultracold atom trappedin an optical lattice undergoes a periodic sequence of in-ternal rotations and spin-dependent translations. Thisapproach can be understood to fall under the more gen-eral class of Floquet topological insulators systems thatare periodically driven in time with a period T . Afteran integer number of periods (i.e. steps), their quantumevolution is reproduced by an effective (Floquet) Hamil-tonian that is topologically nontrivial [37]. Varying theprotocol for the DTQW is a mean to engineer the ef-fective Hamiltonian. In this way, effective Hamiltoniansfrom all universality classes of topological insulators [7, 8]can be realized by quantum walks [38].Floquet topological insulators are especially attractivefor the possibility to control their topological propertiesvia an external periodic drive [39], yet avoiding any ex-ternal magnetic field. An optical analogue of Floquettopological insulators was demonstrated using an array ofevanescently coupled waveguides on a honeycomb lattice[40], with the external periodic drive being effectively im-plemented by a helicoidal deformation of the waveguides.DTQWs are well suited for creating TP edge modes, onthe fly, by locally controlling the parameters of the exter-nal drive. Furthermore, beyond simulating static topo-logical insulators, DTQWs allow us to explore the richertopological structure inherent to Floquet systems, whichis not entirely represented in the effective Hamiltonian,but instead rooted in the details of the quantum walk se-quence. For example, a one-dimensional quantum walkcan host TP edge states between domains having thesame effective Hamiltonian [41]. Experimental evidenceof this phenomenon was shown in a photonic DTQWsetup, though using only a small number of steps [42].In our laboratory we choose a single massive Cs atomwith two long-lived hyperfine states as the quantumwalker, which we coherently delocalized in optical lat-tices over ten and more lattice sites [43]. However, quan-tum superposition states in such a large Hilbert spaceare always highly fragile because they are subject to de-coherence and dephasing mechanisms arising from theopenness of the quantum system. In DTQWs decoher-ence leads to a quantum-to-classical transition of thewalk evolution dominated by the dephasing process af-fecting the coherences in the coin degree of freedom, aswe have shown previously [43]. It is generally acceptedthat disturbances with frequencies beyond the energy gaplead to the destruction of the TP edge states. However,in most condensed matter systems, these effects are of-ten suppressed by operating at cryogenic temperatures[44]. In DTQWs, disturbances on the coin operation, aswell as spin dephasing, effectively act with infinitely widespectrum and therefore extend over the whole band gap,so that we expect the loss of protection in the long timelimit. In the 1D split-step walk, Obuse et al. [45] hasshown, that while topological protection is preserved un-der weak spatial disorder, temporal fluctuations of thecoin angles destroy it. However, a quantitative model-ing of decoherence effects, which is essential for futureexperiments, is still missing.In this paper, we study how environment-induceddephasing affects TP edge states in one- and two-dimensional quantum walk setups and how diffusivespreading has an impact on the existence and form ofTP edge states in general. Moreover, we formulate anexperimental proposal under realistic conditions on howto observe ballistic transport of quantum walks using ul-tracold atoms in optical lattices.The paper is structured as follows: In Sec. II, we in-troduce DTQW protocols in one and two dimensions andprovide a short overview of their topological structureand corresponding TP edge states. We discuss the aris-ing edge phenomena and analyze their robustness underspatial deformations of the topological phase boundary.In Sec. III, we investigate how the shape and evolutionof the edge states is affected under decoherence. Fur-thermore, we give insight into the limits concerning themodel of stroboscopic decoherence, which was employedin Ref. 43. The numerical simulations in this analysisare carried out using realistic experimental parameters,which are chosen based on the experimental proposal dis-cussed in Sec. IV. In Sec. IV, we present a new experimen-tal scheme to realize a two-dimensional spin-dependentoptical lattice, and discuss the experimental requirementsto create spatial boundaries between topological phasesas well as to observe TP edge states under realistic deco-herence conditions. II. TOPOLOGICAL PHASES INDISCRETE-TIME QUANTUM WALKSA. The system
We consider a particle with two internal spin states,labeled s ∈ {↑ , ↓} , that is positioned on a cubic latticewith lattice constant a . We will specifically address thecases of N = 1 and N = 2 dimensions, which can beimplemented in present experimental apparatuses, as ex-plained in detail in Sec. IV. We label the nodes of the N -dimensional cubic lattice with x = ( x, y, . . . ) ∈ Z N .Thus, in the absence of decoherence, the quantum stateof the walker after n steps is a pure state | ψ n (cid:105) , which comprises a superposition of the basis states | x , s (cid:105) .The dynamics of the DTQW is defined by a sequence ofunitary operations ( protocol ), which can be of two types:the coin toss operation and spin-dependent shift opera-tions. The coin toss is realized by a unitary rotation ofthe spin state into superpositions of |↑(cid:105) and |↓(cid:105) , C ( θ ) = (cid:88) x | x (cid:105)(cid:104) x | ⊗ e − i σ θ/ , (1)where σ i is the i -th Pauli matrix. The coin angle θ de-termines the amount of rotation of the spin state and is,in general, a function of the lattice position x , θ = θ ( x ) .The rotation axis does not depend on the position, in-stead, and is chosen to be along the y -direction of theBloch sphere. Note that different choices of the rota-tion axis in the x - y plane are equivalent up to a unitarytransformation of the spin basis vectors {|↑(cid:105) , |↓(cid:105)} .Different choices of the rotation axis are equivalentto a unitary transformation of the spin basis vectors {|↑(cid:105) , |↓(cid:105)} .The spin-dependent shift operation S sd ( s ∈ {↑ , ↓} , d ∈{ x, y } ) is defined as S ↑ d = (cid:88) x | x + e d (cid:105)(cid:104) x | ⊗ |↑(cid:105)(cid:104)↑| + | x (cid:105)(cid:104) x | ⊗ |↓(cid:105)(cid:104)↓| , (2) S ↓ d = (cid:88) x | x − e d (cid:105)(cid:104) x | ⊗ |↓(cid:105)(cid:104)↓| + | x (cid:105)(cid:104) x | ⊗ |↑(cid:105)(cid:104)↑| , (3)where e d denotes the unit lattice vector in the d -direction. S ↑ d ( S ↓ d ) shifts the walker’s spin up (down)component in the positive (negative) e d -direction by onelattice site, while the other spin component is unchanged.The evolution of a pure state | ψ n (cid:105) in time is describedby a unitary walk operator W applied periodically atdiscrete time steps t = n T , n ∈ N : | ψ n (cid:105) = W n | ψ (cid:105) . (4)Note that the quantum evolution of the walker is period-ically driven in time with a Floquet period T , which isthe duration of a single step.In this work we focus on two DTQW protocols, whichallow us to study the most relevant physical properties oftopological phases of discrete-time quantum walks in one-and two-dimensions. In a one-dimensional (1D) lattice,we consider the so-called split-step walk protocol definedin Ref. 38 as W = S ↓ x C ( θ ) S ↑ x C ( θ ) , (5)consisting of two spin rotations separated by spin-dependent shifts in x -direction. In a two-dimensional(2D) lattice, we study the quantum walk defined by W = S ↓ y S ↑ y C ( θ ) S ↓ x S ↑ x C ( θ ) , (6)where after each coin operation both spin states areshifted in opposite directions [46]. Note that the shiftoperators commute, [ S ↑ d , S ↓ d ] = 0 . a z y x W ' W '' q u a s i e n e r g y ε b c π - π - π /a π /aquasimomentum k Figure 1. Topological twist in the 1D split-step quantum walk with ( θ , θ ) = ( π/ , (Hadamard walk). ( a ) Quasienergyspectrum with two energy gaps occurring at energy (cid:15) = 0 and (cid:15) = π . ( b , c ) The corresponding quasienergy eigenstates of theupper band in the two time frames, Eqs. (7), (8), displayed on the Bloch sphere. Chiral symmetry constrains the eigenspinors tolie in a plane, x = 0 , while the quasimomentum is varied across the Brillouin zone performing a closed loop. The color gradientindicates the winding direction around the Brillouin zone. The (signed) winding number associated with transformation differsin the two time frames, ν (cid:48) = 1 in ( b ) and ν (cid:48)(cid:48) = 0 in ( c ). The topological invariants of the bulk are given by the sum anddifference of the two winding numbers, ( ν , ν π ) = ( ν (cid:48) + ν (cid:48)(cid:48) , ν (cid:48) − ν (cid:48)(cid:48) ) / / . See also Fig. 2(a) for the related phase diagram. B. Topological phases and symmetries
In the context of Floquet theory, the evolution ofthe quantum state can be expressed by the action ofa time-independent effective Hamiltonian H , defined by W = e − i H [47, 48]. Due to the discrete spatial transla-tional invariance implied by the lattice, the correspond-ing eigenstates are Bloch waves characterized by a quasi-momentum k , which takes values within the Brillouinzone ( − π/a, π/a ] N . Likewise, the discreteness of thetime evolution implies that the eigenvalues of the theeffective Hamiltonian H are quasienergies, denoted by (cid:15) ,which in our notation take dimensionless values in theinterval ( − π, π ] . Note that physical energy units canbe restored trough multiplication by the quantity (cid:126) /T .In DTQWs, the quasienergy spectrum reveals a bandstructure with two bands resulting from the two internalstates, as can be seen in Fig. 1(a), where we provide thequasienergy spectrum for the 1D split-step protocol with ( θ , θ ) = ( π/ , ( Hadamard walk ). For a generic choiceof the coin parameters, these two bands are gapped. Thegapped spectrum relates quantum walks to static sys-tems like insulator materials. However, unlike in staticsystems, the Floquet quasienergy spectrum can also havea gap at (cid:15) = π , since quasienergies (cid:15) = − π and (cid:15) = π are identified. In addition, artificial electric [49, 50] andmagnetic fields [51, 52] can lead to a higher number ofbands, which can possess nontrivial topological proper-ties as well.Adapting methods developed for static topological in-sulators to the effective Hamiltonian H , Demler et al. [38]have shown that DTQWs can reproduce all ten classes ofnontrivial topological phases in one- and two-dimensionsfor non-interacting particles [7, 8]. Topological phasescan be assigned to different realizations of the effectiveHamiltonian and the corresponding topological invari- ants occur in the form of winding numbers of the Blochenergy eigenstates [1].However, a closer inspection of DTQWs reveals thattheir so-called Floquet topological phases exhibit an evenricher structure, which can only be accessed by analyzingthe full time evolution of the walk. This holds for both1D and 2D DTQWs [41, 53, 54]. For instance, the topo-logical phases of the 1D split-step protocol originate froma special symmetry of the walk protocol, which is called chiral symmetry . A walk operator W exhibits chiral sym-metry if a unitary operator Γ exists, which transforms itas follows: Γ W Γ † = W † ⇔ Γ H Γ † = − H . Although thesplit-step walk operator W defined in Eq. (5) does nothave chiral symmetry, one can show that the two walkoperators W (cid:48) = C ( θ / S ↓ x C ( θ ) S ↑ x C ( θ / , (7) W (cid:48)(cid:48) = C ( θ / S ↑ x C ( θ ) S ↓ x C ( θ / , (8)obtained through a cyclic permutation of the single walkoperations, do exhibit chiral symmetry, with the symme-try operator being Γ = σ [55]. The cyclic permutationhas split the coin operations into two parts, C ( θ i ) = C ( θ i / C ( θ i / , i = 1 , . Since the walk operationsrepeat themselves periodically, a cyclic permutation ofthese operations corresponds to a change of basis preserv-ing the underlying topological structure. Likewise, cyclicpermutations allowed identifying time-reversal symme-try in Floquet topological insulators [56]. Hence, the twowalk operators in Eqs. (7), (8) are chiral-symmetric rep-resentations of the same walk, but expressed in two dif-ferent time frames. It results from chiral symmetry thateach eigenstate at quasienergy (cid:15) has a chiral-symmetricpartner eigenstate at quasienergy − (cid:15) . In particular, ifeigenstates exist with quasienergy either (cid:15) = 0 or (cid:15) = π ,these states can be their own symmetry partners, i.e., beeigenstates of the symmetry operator Γ . This character- )0,1( )1,0(0, )0( )1,1( θ θ θ θ -1 1+ a b Figure 2. Topological invariants assigned to the coin anglesof the 1D split-step walk ( a ) and the 2D protocol ( b ). Due tothe form of the coin operator, C ( θ ) , the walk possesses a π -periodicity in the coin angles. At the phase boundaries, thegap closes at quasienergy (cid:15) = 0 (dotted), (cid:15) = π (dashed), orboth at (cid:15) = 0 and (cid:15) = π (dash-dotted). The coin angle pairschosen in the numerical examples in this work, and the corre-sponding phase transitions defined in Eqs. (9), (10) are alsodisplayed ( — ). The 1D Hadamard walk ( θ , θ ) = ( π/ , ,which is discussed in Fig. 1, is also shown ( ). istic ensures the robustness of TP edge states in the 1Dsplit-step walk (see Sec. II C).We obtain a geometrical representation of the topolog-ical twist of the 1D split-step walk by displaying on theBloch sphere the eigenspinors of the two chiral-symmetricwalk operators defined in Eqs. (7), (8). The eigen-spinors ± n ( k ) with quasimomentum k are determinedby the translational invariant effective Hamiltonian, H = (cid:80) k (cid:15) ( k ) | k (cid:105)(cid:104) k | ⊗ n ( k ) · σ . It directly follows fromchiral symmetry that the eigenspinors with quasienergy (cid:15) (cid:54) = 0 , π lie in the plane x = 0 . This holds true, inparticular, for the bulk eigenstates, whose quasienergieslie outside of the gaps, as shown in Fig. 1(a). Hence,if we vary the quasimomentum k across the the wholeBrillouin zone, the eigenspinor rotates in the plane per-forming a closed trajectory, winding a (signed) numberof times around the origin, as shown in Fig. 1(b,c). Thedifference and sum of the signed winding numbers asso-ciated with the two time frames yield a pair Z × Z oftopological invariants [41, 57, 58]. For the derivation ofthe winding numbers, the reader is referred to Ref. 55.These invariants classify the topological phases of thesplit-step walk, and depend only on the coin angles ( θ , θ ) , as shown by the phase diagram in Fig. 2(a).In essence, the pair of topological invariants, ( ν , ν π ) ,count the minimal number of times the band gap closesat quasienergy (cid:15) = 0 and (cid:15) = π , respectively, as the walkis continuously transformed into the topological phasecharacterized by (0 , . Note, however, that the topolog-ical protection of these states holds only for perturbationsthat can be continuously contracted to unity. For non-continuous perturbations, instead, the topological phasediagram relies on a single signed winding number, as re-cently demonstrated in Ref. [59].In two dimensions, a Floquet topological invariant Z , the so-called Rudner winding number [54], identifies thetopological phases of the 2D DTQW protocol [60]. Thetopological phase diagram is shown in Fig. 2(b) as a func-tion of the coin angles. Remarkably, due to the Floquetcharacter of the DTQW protocol, nontrivial topologicalphases exist even if the topological invariants assigned tothe effective Hamiltonian (i.e., the Chern numbers) arezero. Moreover, we note that, unlike in one dimension,the 2D DTQW protocol possesses nontrivial topologicalphases without need for specific symmetries. C. Topologically protected edge states
We consider a spatially inhomogeneous DTQW inwhich the coin angles depend on the position. The coinangles are allowed to assume any value inside a spatiallyconfined region at the interface between bulk regions,where the coin angles are kept constant, instead. Whenthese bulk regions are associated with different topolog-ical invariants, TP edge states occur at energies lyingin the gaps of the bulk insulators. More precisely, thebulk-boundary correspondence principle states that theminimum number of edge states is equal to the algebraicdifference (in absolute value) between the topological in-variants of the individual bulk phases.For the investigation of TP edge states in the 1D pro-tocol, we choose ( θ , θ ) = (cid:40) ( − π/ , π/ x (cid:28) − π/ , π/ x (cid:29) (9)realizing two spatially adjacent topological phases withinvariants ( ν , ν π ) = (0 , for x (cid:28) and (1 , for x (cid:29) ,as delineated in 2(a). We thus expect a TP edge statewith quasienergy (cid:15) = 0 to be localized at the boundaryaround the site x = 0 . To account for realistic experi-mental conditions, we considered a regular variation ofthe coin angles over (cid:39) lattice sites, as displayed inFig. 3(a), without abrupt changes. The width of thetransition is related to the optical resolution of our ex-periment, introduced in Sec. IV. Under these conditions,we studied the time evolution of a walker initially pre-pared in the single-site state | ψ (cid:105) = | , ↓(cid:105) . The resultsfor the ideal situation without decoherence are presentedin Fig. 3(b), where the spatial probability distribution isshown as a function of position x and number of steps n , P ( x ; n ) = (cid:80) s ∈{↑ , ↓} |(cid:104) x , s | ψ n (cid:105)| . Because the initial statehas a large overlap with the TP edge state ( (cid:39) . forthe example shown in Fig. 3), the walker is trapped atthe boundary with a high probability, yielding a peakedposition distribution around the origin even in the longtime limit.In the 2D walk protocol, the boundary between twodistinct topological domains describes a 1D contour.Along this boundary, which can have in general anyshape, TP edge states are expected to exist [61]. How-ever, unlike in the 1D split-step walk, the wavefunction p = 0.05 S p = 0.2 S p = 0.5 S p = 0.05 P p = 0.2 P p = 0.5 P spin decoherence ba spatial decoherence c n p r o b a b i l i t y P ( x ; n ) p r o b . P ( ; n ) step n i n c r e a s i n g d e c o h e r e n c e r a t e c o i n a n g l e θ -2 24- 4 θ = – = const.π2position x position x ( , ) ( , ) (ν , ν π ) = (0,0) (ν , ν π ) = (1,0) Figure 3. ( a ) Position dependency of the coin angle in the 1D split-step walk given by Eq. (9) realizing two spatiallyadjacent, distinct topological domains with invariants ( ν , ν π ) = (0 , for x (cid:28) and (1 , for x (cid:29) . We use a smoothcrossover transition corresponding to the diffraction-limited optical resolution of our imaging system (see Sec. IV for details).( b ) Decoherence-free evolution of the spatial density distribution P ( x ; n ) as a function of the number of steps n for a walkerinitially prepared in the single site state | , ↓(cid:105) . The narrow peak located at the boundary near x = 0 indicates the component ofthe walker populating the TP edge state. ( c ) The same walk is subject to pure spin decoherence and pure spatial decoherencewith increasing decoherence probabilities p S , p P . Insets: time dependence of the walker’s probability P ( x = 0; n ) to be at theorigin x = 0 in logarithmic scale. It exhibits an exponential decay for small amounts of decoherence, while stays constant forthe decoherence-free evolution. The time evolution is calculated for a large number of lattice sites (201) to prevent the walkerfrom reaching the boundaries in the given maximum number of steps. q u a s i e n e r g y ε quasimomentum k x xy π0-π gapFloquet gapFloquet gap -π/a π/a0 +1+1–1 Figure 4. Quasienergy spectrum of an inhomogeneous 2DDTQW with a horizontal strip geometry. The horizontalstrip, 40 sites wide along the y -direction, is associated withRudner invariant − , whereas the rest of the bulk with +1 (refer to Fig. 2(b) for the phase diagram). Unidirectionaledge modes are visible in the gaps (thick lines), with the blueand red color denoting each edge of the strip. For any givenquasienergy (cid:15) in the gaps, two TP edge modes exist per edge,as expected from the bulk-boundary correspondence princi-ple. The spectrum is computed numerically using 100 sites inthe y -direction. of the TP edge states is delocalized in space, extend-ing along the whole length of the boundary. As a resultof that, a walker in a superposition of TP edge statesis no longer confined in the vicinity of the initial site,but can propagate along the whole boundary. We gatherfurther insight into the transport dynamics along edgesby studying the propagation of a wave packet along astraight boundary, which we assume oriented along, say,the x -direction. The flatness of the boundary ensuresthat the quasimomentum in the boundary’s direction, k x , is preserved, so that it can be used to derive the en-ergy dispersion relation of the edge modes. Fig. 4 showsthe quasienergies as a function of the quasimomentum k x computed from the effective Hamiltonian for the case ofhorizontal boundaries between topological domains. Thequasienergy spectrum shows edge modes present in thegaps of the bulk phases. Recalling the expression of thegroup velocity, v g ( k ) = ∂(cid:15) ( k x ) /∂k x , characterizing themotion of a wave packet, we realize from the the slope ofthe dispersion relations that the TP edge modes trans-port currents in a unidirectional manner. Moreover, forthe specific situation of a straight horizontal boundaryas considered in Fig. 4, it appears that the group veloc-ity does not depend on k (i.e., dispersionless transport),being equal to ± site per step. We remark that disper-sionless transport is not a topological feature, but rathera quantum transport property of the specific DTQW pro-tocol defined in Eq. (6).To give evidence of the robustness of TP edge modesagainst deformations of the boundary’s shape, we havechosen the boundary to form a closed topological island with a droplet shape, with the coin angles being definedas ( θ , θ ) = (cid:40) ( π/ , π/
5) ( x, y ) ∈ inside, (4 π/ , π/
5) ( x, y ) ∈ outside. (10)With reference to the phase diagram in Fig. 2(b), thischoice of angles is associated with Rudner invariants − inside and +1 outside. We have chosen to add a sharpcorner on top of the topological island to test the robust-ness of the TP edge modes against irregularities of theboundary. As in the 1D case, we again consider a contin-uous variation of the coin angle at the boundary. Anglesat the crossover between the inside and outside regionsare varied along the line marked in the phase diagramin Fig. 2(b). Fig. 5(a) again shows the spatial proba-bility density distribution P ( x ; n ) as a function of posi-tion x and number of steps n . We initialize the walkerin a single site near the boundary, so that its state hasa significant overlap with the TP edge states, leadingto a unidirectional propagation around the island. Inthe absence of decoherence effects, we observe that theedge current persists even after many revolutions aroundthe island, indicating the presence of metallic edge statesdelocalized along the whole contour of the island. How-ever, unlike for the straight boundary discussed in Fig. 4,which exhibits dispersionless transport, we observe forthe droplet-shaped island that the wave packet’s proba-bility distribution spreads along the entire border afterseveral revolutions. We attribute the observed disper-sion to the short radius of curvature associated with theborder. III. DECOHERENCE EFFECTS ONTOPOLOGICALLY PROTECTED EDGE STATESA. Stroboscopic decoherence model
Quantum superposition states are fragile against deco-herence that is, disturbances caused by the surround-ing environment onto the quantum system. The effectof decoherence on the quantum evolution can be effec-tively described as the projection of quantum states ontoa particular basis of so-called pointer states [63], whichare robust against decoherence. In quantum-walk experi-ments with neutral atoms, the pointer states are the spin | s (cid:105) , s ∈ {↑ , ↓} , and the position states | x (cid:105) , x ∈ Z N [43].Assuming a small amount of decoherence per step, wecan approximate the continuous-time decoherence pro-cess through a series of discrete measurement operations,which are applied stroboscopically after each unitary stepof the walk. We assume that each measurement only re-solves the walker’s state with a certain decoherence prob-ability ≤ p ≤ . The walk’s evolution is coherent for p = 0 , while it describes a classical random walk for p = 1 . Our model relies on the assumption of small de-coherence to be accurate, p (cid:28) . Henceforth, we denote ab n =10 p o s i t i o n y position x probabillity p o s i t i o n y n =30 n =60 n =80 n =500 Figure 5. ( a ) Color-coded spatial probability distribution P ( x ; n ) of a decoherence-free two-dimensional DTQW. The coinangles depend on the position as specified by Eq. (10), creating a droplet-shaped topological island with Rudner invariants − inside and +1 outside of the island. The width of the transition is limited by the optical resolution of our experimental setup withAbbe radius R A (cid:39) . a (see Sec. IV for details). The walker is initially prepared in the single site state | ( x = − , y = 0) , ↓(cid:105) near the phase boundary and shows a unidirectional moving population of edge states around the boundary as time evolves. In( b ) the same walk is subject to spin decoherence under realistic experimental conditions ( p S = 0 . ), exhibiting a slow decayof the edge current over time. An animation showing the evolution over steps is provided online [62]. by p = p S and p = p P the decoherence probability relatedto the spin and position states, respectively.We follow Ref. 43 to describe the non-unitary timeevolution of the walker by means of the reduced densitymatrix formalism. As the walker is initially preparedin a pure state | ψ (cid:105) , the initial density matrix is ρ = | ψ (cid:105)(cid:104) ψ | . The density matrix ρ n +1 describing the walkerat time t = ( n + 1) T depends only on the state of thewalker at time t = n T (Markovian assumption). Hence, ρ n +1 is obtained through the repetitive application of thelinear superoperator E , which accounts for the effect ofenvironment-induced decoherence at each step [64]: ρ n +1 = E n +1 ( ρ ) = E ( ρ n ) == (1 − p ) W ρ n W † + p (cid:88) i P i ( W ρ n W † ) P † i , (11)where i ∈ {↑ , ↓} for pure spin and i ∈ { x } for pure posi-tion decoherence. The projectors P i are defined as P x = (cid:88) s | x , s (cid:105)(cid:104) x , s | , P s = (cid:88) x | x , s (cid:105)(cid:104) x , s | . (12)We found in a previous study that this simple modelreproduces in a satisfactory manner the effects of deco-herence occurring in our experiments with neutral atoms[43]. In particular, our previous analysis revealed thatspin decoherence is the main mechanism responsible forthe loss of coherence in the current 1D quantum-walksetup. We therefore focus in this work primarily on de-coherence by spin dephasing. In addition, our numeric analyses assume a conservative decoherence probabilityof p S (cid:39) . per step, which is based on previous exper-imental results [43]. However, the construction of a newquantum-walk setup for 2D DTQWs is underway thatpromises decoherence probabilities as low as p S < . owing to a number of technical improvements, including,among others, shielding of stray magnetic fields and sup-pression of polarization distortions of the optical latticelaser beams. B. Decoherence effects on TP edge states in 1D
We illustrate the effect of decoherence by analyzing thewalk evolution of a 1D DTQW with two adjacent bulkswith coin angles defined by Eq. (9). We again initializethe walker in a single site state | , ↓(cid:105) near the boundary,so that the walker is able to populate the TP edge state.In Fig. 3(c) we show the spatial probability distribution P ( x ; n ) = (cid:80) s ∈{↑ , ↓} (cid:104) x , s | ρ n | x , s (cid:105) obtained numericallyusing Eq. (11). The resulting distribution of the walk re-flects two phenomena. First, the walker occupies the TPedge state, resulting in a narrow probability peak locatedaround the crossover point at x = 0 . Second, this peakstays nearly constant in position and shape, but decaysover time with a rate increasing with the decoherencestrength, p . On the other hand, the component of thewalker’s wave function that has no overlap with the TPedge state expands in the bulk. For small decoherence,the expansion preserves a ballistic-like behavior for manysteps, resulting in the characteristic distribution with off-center peaks. The number of peaks and the direction ofpropagation depends on the initial state of the walker.For stronger decoherence, this expansion exhibits a dif-fusive behavior [43], with a distribution centered aroundthe starting point, thus overlapping with the TP edgestate. From our simulations, it results that experimentsmust be conducted under small decoherence conditions, p < . , in order to be able to detect the persistence ofa sharply peaked distribution at the boundary a signa-ture of the TP edge state. It should be noted that thedecoherence rate determines the point in time where theexpansion changes from a ballistic spreading on a shorttime scale to a diffusive behavior for longer times [43].The probability for the walker to remain in the origin, P ( x = 0; n ) , is an indicator for the robustness of the TPedge state, see the insets in Fig. 3. It shows an oscilla-tory evolution for a short transient due to the dynamics ofthe walker’s component overlapping with the bulk states,which is free to expand into the bulk. For longer times,the probability stays constant for the decoherence-freeevolution, while decays nearly exponentially for small de-coherence rates. In case of strong decoherence, the pop-ulation of the TP edge state deviates from a simple ex-ponential decay. In this regime, however, the assumptionunderlying our stroboscopic decoherence model, p (cid:28) ,does not hold anymore, see Sec. III A. A more detaileddiscussion based on an analytic model is presented inSec. III D. C. Decoherence effects on TP edge states in 2D
The evolution of the 2D walk revolving around thedroplet-shaped topological island in the presence of weakspin decoherence is presented in Fig. 5(b). The prob-ability current along the boundary shows a slow decayover time. As an indicator for the population of the TPedge modes, we study the probability P ( x ∈ F; n ) forthe walker to be situated in a small band, F , aroundthe edge, as shown in Fig. 6(a). For an initial tran-sient period of (cid:39) steps, the edge probability showsa decrease which is nearly independent of the decoher-ence probability, and is attributed to the non-vanishingprojection of the initial single-site state onto the bulkstates. For the decoherence-free evolution, the probabil-ity tends, in the long time limit, to a constant value, P ( x ∈ F; n (cid:29)
1) = 0 . . It is worth emphasizing thatsuch a high probability is favorable to future experiments,which aim to detect matter waves trapped at the bound-ary. In the presence of decoherence, instead, we ob-serve an approximately exponential decay in qualitativeagreement with the results obtained in the 1D walk (seeSec. III B).While decoherence reduces the probability current, ithas no discernible effect on the propagation velocity ofa wave packet along the boundary. The comparison be-tween Fig. 5(a) and 5(b) shows, in fact, that the front a L -1 +1-1 +1 p r o b . P ( x ∈ F ; n ) P ( x ∈ L ; n ) / P ( x ∈ F ; n ) p s = 0.00 p s = 0.05 step n
200 400 1000600 8005000 100000.20.70.5 b F Figure 6. ( a ) Probability P ( x ∈ F; n ) for the walker to beinside the grayed region F as a function of the number of steps, n , in logarithmic scale. ( b ) Probability P ( x ∈ L; n ) for thewalker to be inside the grayed region L near the lower half ofthe phase boundary, normalized to the population probability P ( x ∈ F; n ) . The probabilities are shown for the unitary walkevolution (dashed curves) and for a decoherence rate p S =0 . (solid curves). Inset: close-up view in the long time limitfor the evolution without decoherence. of the wave packet moves, in both cases, with a speedof approximately one lattice site per step, irregardlessof whether the walker is subject to decoherence. Thisvelocity is also in good agreement with that computedin Sec. II C from the energy dispersion relation of aflat boundary. Interestingly, the propagation along theboundary attains the highest velocity, one site per step,allowed by the 2D quantum walk protocol defined in Eq. 6(i.e., attains the effective speed of light for the DTQWprotocol).To gain further insight into the dynamics of the walkerrevolving around the island, we display in Fig. 6(b) theprobability P ( x ∈ L; n ) for the walker to be in the lowerhalf, L, of the boundary. This probability exhibits peri-odic oscillations in time with a period that is indepen-dent of the decoherence rate, and approximately equal,in units of steps, to the length of the contour of the topo-logical island. The period, in particular, corroborates ourprevious observation that the wave packet moves unidi-rectionally along the boundary with a velocity of nearlyone site per step. We also observe that the oscillation am-plitude is damped after several revolutions. We explainthis damping as the result of the group velocity disper-sion of the TP edge states, which make the wave packetspread along the entire boundary. In the presence of de-coherence, the damping occurs on a much shorter timescale, presumably due to the walker’s component that isdiffused into the bulk, but yet located inside the band L . For the unitary evolution, however, oscillations per-sist with the same periodicity for long times, as shown inthe inset of Fig. 6(b). The modulation of the oscillation0amplitude over long time scales is attributed to partialcollapses and revivals, since the time evolution is unitaryand the edge of the topological island constitutes a finiteHilbert space with a discrete spectrum [65]. A detailedstudy of the residual oscillations would require furtherinvestigation. D. Analytical model of the decay of TP edge states
We consider the 1D split-step walk protocol to derivea simple analytical model predicting the decay rate ofTP edge states in the presence of decoherence. Assum-ing that the walker is initially in a TP edge state | E (cid:105) , wecompute the probability Π( n ) that it remains in the samestate after n steps. Due to decoherence, the walker’s wavefunction acquires a non-vanishing overlap with the con-tinuum of the bulk states. In order to carry out the com-putation analytically, we assume that the walker’s com-ponent coupled to the bulk rapidly leaves the boundarybecause of the nearly ballistic expansion, without everrepopulating the TP edge state. Under this assumption,which is well justified in the regime of weak decoherence p (cid:28) , we obtain in Appendix A that the probability ofoccupying the edge state is Π( n ) = tr ( | E (cid:105)(cid:104) E | ρ n ) (cid:39) (1 − γ ) n , (13)where the decay rate γ depends on | E (cid:105) and is linear in p . For pure spin decoherence, the decay rate is given by γ S = p S (cid:20) − (cid:88) s (cid:16)(cid:88) x |(cid:104) x , s | E (cid:105)| (cid:17) (cid:21) . (14)A similar expression for the decay rate γ P for pure posi-tion decoherence is provided in Appendix A. Moreover,the expression in Eq. (14) can be written in a more com-pact form as γ S = p S (1 − (cid:80) s |(cid:104) s | s E (cid:105)| ) by exploiting thefactorization of 1D TP edge states into a position andspin component, | E (cid:105) = | χ (cid:105) ⊗ | s E (cid:105) , as ensured by chiralsymmetry (see Sec. II B).This simple model predicts an exponential decay of theedge state population, which agrees well with the numer-ical simulations for short times and small decoherence,as shown in Fig. 7. In addition, we attribute deviationsfrom the exponential decay model, observed for longertimes, to a non-negligible probability that decoherencetransfers the walker from the bulk states back to the TPedge state. E. Limits of the stroboscopic decoherence model
In Sec. III A, we have modeled the effect of decoherencethrough a single measurement operation, of either thespin or the position of the particle, applied after eachcoherent step of the walk W . This constitutes, in general,a good approximation of the actual dynamics, provided n T P e d g e s t a t e p r o b a b i l i t y Π ( n ) p S =0.64 Figure 7. Probability of populating a TP edge state as afunction of the number of steps n for different amounts of spindecoherence p S (semi-logarithmic scale). The data points arecalculated numerically for the 1D split-step walk with the coinangles as defined in Eq. (9), and with the initial state beingthe TP edge state with quasienergy (cid:15) = 0 . The solid linesrepresent an exponential decay as predicted by the analyticalmodel in Eq. (13). that the amount of decoherence is small ( p (cid:28) ), as isthe case of ultracold atom experiments (see Sec. IV).However, situations exist where the stroboscopic ap-plication of decoherence can completely fail to describethe decay of a TP edge state. We would like to cautionthe reader about that by providing an explicit example,which is constructed ad hoc to prove the existence of aTP edge state that is robust against any amount of stro-boscopic spin decoherence. Such a situation can occurwhen the quantum walk possesses a special symmetry(for example, chiral symmetry) that forces the spin com-ponent of the TP edge state to be oriented along a givendirection, for example, along the z -direction. It is evi-dent in this case that spin measurements in the z -basisleave the TP edge state unperturbed. This is confirmedby Eq. (14), predicting in this case a decay rate γ S = 0 for any p S .This can be realized by considering a unitary trans-formation of the walk operator in Eq. (7), ˜ W = C ( π/ W (cid:48) C ( − π/ . This transformation is equivalentto a cyclic permutation, and it does not change the walkevolution in the bulk as well as the corresponding topo-logical invariants. The chiral symmetry operator of thetransformed walk is σ z since σ z ˜ W σ z = ˜ W † . Sincethe TP edge states are eigenstates of the symmetry opera-tor (see Sec. II B), their spin must be either |↑(cid:105) or |↓(cid:105) , anda projective measurements of the spin in the z -basis leavethe TP edge state unaffected. We note that an analogoussituation can be reproduced in the Su-Schrieffer-Heegertopological model, where it is known that the sublatticesymmetry (tantamount to chiral symmetry) forces theTP edge state to lie on either one of the two sublattices[9]. Hence, a quantum non-demolition measurement ofthe sublattice would leave, in like manner, the TP edgestate unaffected.1A remedy to avoid such seemingly paradoxical sit-uations, where TP edge states are left unmodified byenvironment-induced decoherence, consists in modifyingEq. (11) to allow the decoherence Kraus operators to actafter each discrete operation of the single step. Further-more, identifying the exact operator-sum representationin terms of Kraus operators of the decohered coin opera-tion would ultimately provide the most accurate model-ing of decoherence effects [66]. IV. EXPERIMENTAL PROPOSAL WITHNEUTRAL ATOMS IN OPTICAL LATTICESA. Optical lattice experimental setup
We have shown in previous experiments [36] that anatomic quantum walk can be realized employing a sin-gle neutral cesium atom in an optical lattice at a spe-cific wavelength λ L = 866 nm. The outermost hyper-fine ground states, |↑(cid:105) = | F = 4 , m F = 4 (cid:105) and |↓(cid:105) = | F = 3 , m F = 3 (cid:105) , define the pseudo spin-1/2 states of thequantum walker. Due to their different ac-polarizability,each of these states experiences, to a large extent, onlythe trapping potential of either one of two distinct σ + -and σ − -circularly polarized optical lattices. The setupfor spin-dependent shift operations in one dimension isdepicted in Fig. 8(a), where two counter-propagatinglaser beams of linear polarization form a 1D optical lat-tice along the direction of the quantization axis. Spin-dependent shift operations are then realized by control-ling the polarization and phase of just one of the two op-tical lattice beams (beam 1 in the figure). A rotation ofits linear polarization, which is achieved through a shiftof the relative phase between circular polarization com-ponents, displaces into opposite directions the two circu-larly polarized optical lattices and, thereby, atoms in dif-ferent internal state. Previous implementations [67, 68]of this concept based on an electro-optic device sufferfrom the shortcoming that shift operations are limitedto a maximum distance of about one lattice site at atime and, most importantly, to only relative displace-ments between |↑(cid:105) and |↓(cid:105) spin components. Sole rela-tive displacements are not sufficient to realize the S ↓ x and S ↑ x operations, which are required by the split-step walkprotocol in Eq. (5). However, we recently demonstrateda different technique for precision polarization synthesis,which overlaps two fully independent laser beams withopposite polarizations to form a beam of arbitrary po-larization and phase [69]. The new implementation ofspin-dependent transport allows us to independently shifteach individual spin component by an arbitrary distance,ultimately limited by the Rayleigh length.We propose to extend the concept of spin-dependenttransport, which has hitherto been demonstrated only inone dimension, to a square lattice in two dimensions. Weemploy three interfering laser beams with linear polar-ization, as illustrated in Fig. 8(b). With reference to the
00 1/21/2 1/4 1/4 c ϕ /π ϕ /π b ϕ ϕ BB a beam 3 π/4-π/4 ϕ ϕ beam 1 beam 2beam 1 beam 2 Figure 8. ( a ) One-dimensional lattice potentials created bytwo linearly polarized beams. A polarization rotation by φ leads to a relative displacement of the two optical potentials(orange and blue curves), which spin-dependently trap atomsin either the |↑(cid:105) or |↓(cid:105) internal state. The vector B repre-sents the direction of the external magnetic field, which fixesthe quantization axis. ( b ) Two-dimensional lattice potentialscreated by three interfering laser beams for spin-dependenttransport on a square lattice. The polarization of beam 3points out of the plane, whereas the polarization of beam1 and 2 can rotate, producing spin-dependent displacementsalong two diagonal directions at ± ◦ relative to the quan-tization axis. Two counter-propagating beams (not shown)orthogonal to the plane provide the confinement in the thirddirection. ( c ) Potential depth of the two spin-dependent opti-cal lattices (orange and blue) for different polarization angles, φ and φ . figure, the polarization of beam 1 and 2 can be rotatedin time by an angle φ and φ , respectively, employingour recently developed polarization-synthesis setup foreach of the two beams. The polarization of beam 3 isinstead fixed and orthogonal to the quantization axis,which is chosen along the direction of beam 1 and 2. Inessence, a rotation of the two polarization angles resultsin a spin-dependent shift operation along one of the two2 SLMRamanlaser telescope lens objective lens opticallattice f f + f f Figure 9. The intensity of Raman lasers, utilized to imple-ment the coin operation, is modulated in space to give riseto sharp topological phase boundaries. A spatial light mod-ulator (SLM) creates a structured intensity pattern, which isimaged onto the optical lattice by a high-numerical-aperture(NA=0.92) objective lens mounted in a 4 f optical system. diagonal directions, as shown in Fig. 8(c). This novel ex-perimental scheme allows the precise control of discrete-time spin-dependent shift operations along the two maindirections of a square lattice. We note that our schemediffers substantially from other experimental schemes forcontinuous-time spin-orbit coupling, which are based oneither a dynamical rotation of the magnetic field (i.e., ofthe quantization axis) [70] or a dynamical modulation ofa magnetic field gradient [71, 72].The geometric arrangement of laser beams in Fig. 8(b)increases the spacing between adjacent lattice sites by afactor √ (thus, a = √ λ L / ) compared to the 1D lat-tice presented in Fig. 8(a), constituting an advantage tooptically address each lattice site individually. In addi-tion, the concurrent interference of the all three beamsyields a trap depth that is / times as deep as thatobtained by a 1D lattice for the same optical power.The construction of the experimental apparatus is cur-rently underway. An objective lens with large numericalaperture (NA), which is placed at µm in front of the2D lattice, allows us to detect the location of atoms withsingle site resolution by fluorescence imaging on the D2line at λ f = 852 nm [73], as well as to project a structuredintensity pattern for local, optical control of the coin op-eration. The coin operation can be implemented eitherthrough microwave radiation resonant with the hyperfinesplitting at . GHz, or through a pair of Raman laserbeams with wavelength λ C = 894 nm slightly detunedfrom the D1 line. Microwave pulses are most suited fordriving coin operations with position-independent coinangles, while Raman laser pulses allow spatial variationsof the coin angles by modulating their intensity. For thelocal control of the Raman laser intensity with single siteresolution, we propose the 4f optical system illustratedin Fig. 9. The coin rotation angle at a certain latticesite depends linearly on the intensity of Raman lasersilluminating that given site. B. Realization of topological phase boundaries
In the experiments, sharp crossovers between topolog-ical phases are preferable because their TP edge statesare strongly localized in the proximity of the boundary,thereby avoiding slowly decaying tails in the direction ofthe bulk. This ensures a relatively high probability thatan atom originally prepared in a single lattice site nextto the boundary populates the edge state. Additionally,sharp boundaries make it less demanding for experimentsto realize coherence lengths [74] longer than the size ofTP edge states.However, there is a limit on how sharp crossoversbetween different topological domains can be, which isdetermined by diffraction in the optical system. Fordiffraction-limited optical systems, the sharpness of thephase crossover depends on the numerical aperture NA of the objective lens, the lattice constant a , and thewavelength λ C of the Raman lasers. Mathematically,the intensity profile experienced by atoms results fromthe convolution of the profile generated by the spatiallight modulator (see Fig. 9) with the point spread func-tion (PSF) of the imaging system [73]. In the numeri-cal simulations presented in this work, we approximatedthe experimentally measured Airy-disc-like PSF with aGaussian function with standard deviation ( √ /π ) R A , R M S s i z e e d g e s t a t e p r o b . P i n i t a /R A NA=0.22 NA=0.90 ab position x c o i n a n g l e θ p r o b a b i l i t y p r o b a b i l i t y c o i n a n g l e θ position x Figure 10. Analysis of a TP edge state | E (cid:105) in the 1D split-step DTQW with coin angles given by Eq. (9) for differentslopes of the phase crossover, as determined by the diffractionparameter a/R A . ( a ) RMS size of the TP edge state (black,dashed) and overlap probability of the initial state | x = 0 , s E (cid:105) with the TP edge state | E (cid:105) = | χ (cid:105) ⊗ | s E (cid:105) (red, solid). Thetwo vertical arrows indicate the values corresponding to the1D and 2D quantum-walk setups. ( b ) Coin angles θ (blackcircles) and position distribution (cid:80) s | (cid:104) E | x, s (cid:105) | of the TPedge state (red lines) computed for the current 1D (left) andthe new 2D experimental setups (right). R A = λ C / (2 NA) is the Abbe radius. Hence, theunit step profile with coin angles θ L for x ≤ and θ R for x > , which we considered for the 1D simulations,results after the convolution in θ ( x ) = θ L + θ R − θ L (cid:20) (cid:18) a π R A x (cid:19)(cid:21) , (15)where erf is the Gaussian error function. The present 1Dquantum-walk setup with NA = 0 . [73] and a = λ L / allows only moderately sharp boundaries, R A (cid:39) . a .The new 2D quantum-walk setup, instead, features anobjective lens with a higher numerical aperture, NA =0 . , and a longer lattice constant, a = √ λ L / , re-sulting in R A (cid:39) . a . This permits nearly abrupt phaseboundaries, where the coin angle is varied across just (cid:39) lattice site.In order to obtain a quantitative relation between theoptical resolution of the optical system and the shapeof TP edge states, we numerically studied the phasecrossover in the 1D protocol as a function of the ratio a/R A . As shown in Fig. 10, the size of the TP edge statedecreases monotonically with the optical resolution untilit attains a constant value around one lattice site. Thefigure also displays the probability P init = |(cid:104) E | x , s (cid:105)| to populate the TP edge state | E (cid:105) from the initial state | x , s (cid:105) . In the experiments, it is important to maximizethis probability by choosing a sharp boundary and theinitial spin, | s (cid:105) , such that it coincides with the spin ofthe edge state at position x . The initial spin can beeasily prepared by applying a suitable microwave pulse. V. OUTLOOK AND DISCUSSION
In this paper, we have studied the robustness of TPagainst environment-induced decoherence, which causesdephasing of the quantum-walk states. We have analyzedthe effect of decoherence on the existence and form of TPedge states. We have found that decoherence of spin andposition states leads, in both cases, to an approximatelyexponential decay of the TP edge state into the bulkstates. A study of phase coherence properties of matterwaves propagating along a quantum circuit of TP edgestates will be the subject of future work, similar to thatpursued by Ref. 75 with IQHE solid-state devices [13]. The novel scheme for 2D spin-dependent transportcombined with Raman laser pulses to drive the coin op-eration will allow us to realize arbitrary topological do-mains in 1D and 2D quantum walks under realistic deco-herence conditions. Owing to a high numerical aperture,the diffraction-limited optical system utilized to projectthe Raman pulses reduces the size of the TP edge statesto a minimum, yielding a high probability to populatethem from a single site.Exploring the limits of the stroboscopic decoherencemodel revealed that specific TP edge states can be un-affected by decoherence. In the future, we plan to buildupon this result to construct Kraus operators that canpump the walker into a TP edge state when applied pe-riodically in time. This would allow us to engineer dis-sipation to protect TP edge states not only from staticdisorder, but also from a weak amount of environmentaldecoherence [76].As yet, only little is known about the role of inter-actions in topological insulators [77, 78]. While topo-logical phases of non-interacting systems are relativelywell understood, the classification of interacting topo-logical phases is in its infancy. The most promising di-rection of future quantum-walk experiments with neutralatoms consists in exploiting the strong, controllable inter-actions between atoms in order to understand topologicalphases with interacting particles. Atoms have in fact thepotential to shed new light on topological phases withstrongly correlated particles, which go beyond a purelywave-mechanical picture as that of non-interacting topo-logical phases [31, 34, 35].
ACKNOWLEDGMENTS
We thank Carsten Robens, Geol Moon, and MichaelFleischhauer for insightful discussions. We acknowledgefinancial support from the Deutsche Forschungsgemein-schaft SFB project Oscar, the ERC grant DQSIM, andthe EU project SIQS. We acknowledge support by theHungarian Scientific Research Fund (OTKA) under Con-tract No. NN109651, the Deutscher Akademischer Aus-tauschdienst (TempusDAAD Project No. 65049). T.G.was supported by the Studienstiftung des deutschenVolkes. J.K.A. was supported by the Janos Bolyai Schol-arship of the Hungarian Academy of Sciences. [1] M. Z. Hasan and C. L. Kane, “Colloquium : Topologicalinsulators,” Rev. Mod. Phys. , 3045 (2010).[2] B. A. Bernevig and T. L. Hughes, Topological insula-tors and topological superconductors (Princeton Univer-sity Press, 2013).[3] R. J. Haug, “Edge-state transport and its experimen-tal consequences in high magnetic fields,” Semicond. Sci.Technol. , 131 (1993). 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Fleischhauer, “Topo-logical Edge States in the One-Dimensional Superlat-tice Bose-Hubbard Model,” Phys. Rev. Lett. , 260405(2013). Appendix A: Analytical decay model of TP edgestate under decoherence, Eqs. (13) and (14)
We derive an analytical model describing the decayof the TP edge state under pure spin decoherence. Amodel describing the decay under decoherence affectingthe position states only, can be derived analogously.Let | E (cid:105) be a TP eigenstate of the walk operator W with quasienergy (cid:15) . The corresponding density matrix ρ = | E (cid:105)(cid:104) E | is then invariant under application of thewalk operator W : W | E (cid:105)(cid:104) E | W † = e − i (cid:15) | E (cid:105)(cid:104) E | e i (cid:15) = ρ . (A1)We consider the 1D walk evolution of this state underspin decoherence as defined by Eq. (11). After one step,the walker’s state is described by ρ = (1 − p S ) ρ + p S (cid:88) s ∈{↑ , ↓} P s ρ P † s , (A2)where P s is the projector onto the spin state s , as definedin Eq. (12). The probability Π(1) to find the walker inthe same state | E (cid:105) is given by Π(1) = tr ( | E (cid:105)(cid:104) E | ρ )= (1 − p S ) tr (cid:0) ρ (cid:1) + p S (cid:88) s tr (cid:0) ρ P s ρ P † s (cid:1) = (1 − p S ) + p S (cid:88) s (cid:88) x , x (cid:48) (cid:104) x (cid:48) , s | ρ | x , s (cid:105)(cid:104) x , s | ρ | x (cid:48) , s (cid:105) = (1 − p S ) + p S (cid:88) s (cid:88) x , x (cid:48) |(cid:104) x , s | ρ | x (cid:48) , s (cid:105)| = (1 − p S ) + p S (cid:88) s (cid:16)(cid:88) x |(cid:104) x , s | E (cid:105)| (cid:17) , (A3)where we used the orthogonality of the basis states | x , s (cid:105) as well as the purity of the initial state, tr( ρ ) = 1 .Hence, we obtain ρ = Π(1) ρ + (1 − Π(1)) ˜ ρ , (A4)where ˜ ρ describes a statistical mixture with no overlapwith the initial state, tr ( | E (cid:105)(cid:104) E | ˜ ρ ) = 0 . Assuming that | E (cid:105) will never be populated by the time evolution of ˜ ρ , tr ( | E (cid:105)(cid:104) E | E n (˜ ρ )) = 0 ∀ n > , (A5)the probability Π( n ) to find the walker at time t = n T in the initial state is given by Π( n ) = tr ( | E (cid:105)(cid:104) E | ρ n ) = tr (cid:0) ρ E n − ( ρ ) (cid:1) = Π(1) tr (cid:0) ρ E n − ( ρ ) (cid:1) + (1 − Π(1)) tr (cid:0) ρ E n − (˜ ρ ) (cid:1) = Π(1) n tr (cid:0) ρ (cid:1) = (1 − γ S ) n , (A6)where the decay rate γ S is defined as γ S = 1 − Π(1) = p S (cid:34) − (cid:88) s (cid:16)(cid:88) x |(cid:104) x , s | E (cid:105)| (cid:17) (cid:35) . (A7)For pure position decoherence, one analogously obtains Π( n ) = (1 − γ P ) n , (A8)where γ P = p P (cid:34) − (cid:88) x (cid:16)(cid:88) s |(cid:104) x , s | E (cid:105)| (cid:17) (cid:35) ..