Floquet topological phases in a spin-1/2 double kicked rotor
FFloquet topological phases in a spin- / double kicked rotor Longwen Zhou ∗ and Jiangbin Gong † Department of Physics, National University of Singapore, Singapore 117551, Republic of Singapore (Dated: 2018-03-08)The double kicked rotor model is a physically realizable extension of the paradigmatic kicked rotor modelin the study of quantum chaos. Even before the concept of Floquet topological phases became widely known,the discovery of the Hofstadter butterfly spectrum in the double kicked rotor model [J. Wang and J. Gong,Phys. Rev. A , 031405 (2008)] already suggested the importance of periodic driving to the generation ofunconventional topological matter. In this work, we explore Floquet topological phases of a double kicked rotorwith an extra spin-1 / Rb condensates into a periodically pulsed optical lattice. Under the on-resonancecondition, the spin-1 / π -quasienergy edge states inthe system. Topological phases with arbitrarily large winding numbers can be easily found by tuning the kickingstrength. We discuss an experimental proposal to realize this model in kicked Rb condensates, and suggest todetect its topological invariants by measuring the mean chiral displacement in momentum space.
I. INTRODUCTION
A topological characterization of a quantum chaos modelby Leboeuf et al [1] pioneered the use of periodic drivingfields to create topological phases of matter absent in time-independent systems. The model proposed in Ref. [1] washowever rather abstract because it is quantized on a phasespace torus. By extending the paradigmatic kicked rotormodel in the study of quantum chaos [2–10], Wang andGong proposed a physically realizable double kicked rotormodel [11] and discovered Hofstadter’s butterfly-like Flo-quet spectrum therein [12]. This finding strongly suggestedthat such periodically driven systems are topologically richand should be highly useful as dynamical systems to explorecondensed-matter physics. Indeed, the work by Wang andGong [11] has led to the proposal of a topological Thoulesspump in momentum space [13], the proof of the topologicalequivalence between the double kicked rotor model and thekicked Harper model [14–16], and the identification of manytopological edge states in both of the two models [17].To date, Floquet topological states of matter have been wellrecognized as a promising concept and a fruitful topic. Flo-quet states, being intrinsically out-of-equilibrium, can be en-gineered to carry topological properties that are either anal-ogous to [18–30], or even beyond their static cousins [31–42]. The latter includes, but is not limited to, degenerate π -quasienergy edge states [32–34], counterpropagating [35–37] and anomalous chiral edge states [38, 39] in both insu-lating [40] and superconducting [41] band structures, leadingto new types of topological classification schemes and bulk-boundary relations [42–46]. Accompanying great theoreti-cal e ff orts in exploring these intriguing features [47], Floquettopological states have also been observed in several experi-mental settings, including ultracold atom [48], photonic [49–51], phononic and acoustic systems [52]. ∗ [email protected] † [email protected] Motivated by recent experimental advances, here we con-tinue to explore Floquet topological phases in the context ofdouble kicked rotor model. The Hamiltonian of an earlierquantum double kicked rotor (DKR) [11] model, which wasrealized by cold atoms subjected to pairs of pulses in an opti-cal lattice [53], is given byˆ H = ˆ p + κ cos( ˆ x ) (cid:88) m δ ( t − mT ) + κ cos( ˆ x + β ) (cid:88) m δ ( t − mT − τ ) . (1)The stroboscopic dynamics of the system is governed by itsevolution operator over one δ -kicking period T , i.e. , the Flo-quet operatorˆ F = e − i ( T − τ ) ˆ p (cid:126) e − i κ (cid:126) cos(ˆ x + β ) e − i τ ˆ p (cid:126) e − i κ (cid:126) cos(ˆ x ) . (2)Here all quantities are in dimensionless units. ˆ x and ˆ p areposition and momentum operators for cold atoms. β is thephase shift between two kicking optical lattice potentials ofstrengths κ and κ , separated by a time delay τ ∈ (0 , T ).Due to the spatial periodicity of kicking potentials, the mo-mentum ˆ p take values p = ( n + η ) (cid:126) , where η ∈ (0 ,
1) is theconserved quasimomentum and n ∈ Z . For a Bose-Einsteincondensate (BEC) of large coherence width [8, 54], η can beset to zero, and ˆ p = ˆ n (cid:126) only takes integer multiples of Planckconstant (cid:126) . Then under the on-resonance condition [8, 9, 54] (cid:126) T = π , the quantum DKR has a Hofstadter’s butterfly-like quasienergy spectrum [12], characterized by fruitful topo-logical band / gap structures and consecutive topological phasetransitions versus the change of the system’s e ff ective Planckconstant (cid:126) [11].In this work, we take one step further in the study ofDKR by considering an internal spin-1 / / U = e − i ( T − τ ) ˆ p (cid:126) e − i κ (cid:126) cos(ˆ x + β ) σ y e − i τ ˆ p (cid:126) e − i κ (cid:126) cos(ˆ x ) σ x , (3)where σ x , y , z are Pauli matrices acting on internal spin space a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a r of the rotor. More specifically, in the case of quasimomentum η = τ = T /
2, and under on-resonancecondition [10, 54] (cid:126) τ = π , the Floquet propagator of DKRSreduces to ˆ U = e − iK cos(ˆ x + β ) σ y e − iK cos(ˆ x ) σ x , (4)where K , = κ , / (cid:126) are rescaled kicking strengths. In thefollowing, we will first discuss a possible way of engineer-ing the on-resonance DKRS (ORDKRS) described by Eq. (4)in a periodically pulsed BEC, thanks to a recent experimen-tal realization of quantum walks in momentum space [55, 56].Next, we will explore the rich topological phases of ORD-KRS. Finally, we suggest to probe bulk topological invariantsof ORDKRS by measuring the mean chiral displacement of awave packet over tens of kicks, which is also experimentallyavailable in both photonic [58] and cold atom [59] systems. II. REALIZATION OF THE ORDKRS
The formalism of ORDKRS as described by Eq. (4) is in-spired by a recent experiment, which realizes discrete timequantum walks in momentum space with a BEC of Rb[55, 56]. The experimental platform is sketched in Fig. 1 ofRef. [55]. Each step of the quantum walk is composed of twoconsecutive operations. First, a resonant microwave is appliedto the Rb condensate, which introduces a rotation within thetwo-state space of its ground hyperfine levels 5 S / F = S / F =
2. This realizes a “coin toss” described by [55, 56] M ( α, χ ) = e − i α [sin( χ ) σ x − cos( χ ) σ y ] , (5)where σ x , y , z are Pauli matrices acting on the internal two-statespace, and the rotation angles α, χ are controllable experimen-tally. Next, the BEC is subjected to a short laser pulse, whosefrequency is detuned from the frequency between the twohyperfine levels, realizing the far o ff -resonant condition andproducing periodic potentials. This step employs the atom-optical realization of the quantum kicked rotor (ratchet accel-erator) with a kicking strength k = Ω τ p ∆ , where Ω is the Rabifrequency, τ p is the pulse length, and ∆ is the detuning oflaser light from the atomic transition. Notably, the detuning ∆ is positive for the state 5 S / F = S / F = Rb. Then under the quantum on-resonancecondition [55, 56] (corresponding to the choice (cid:126) τ = π inour model), the second operation in a quantum walk step isdescribed by a propagator [57] T = e − iK cos(ˆ x ) σ z , (6)where K = Ω τ p | ∆ | is the absolute value of kicking strength. Thecoupling between the internal degrees of freedom (hyperfinelevels F = ,
2) and the external motion (hopping in momen-tum space) is realized by the term cos( ˆ x ) σ z .The successful implementations of “coin toss” operation M ( α, χ ) and spin-dependent walk T in kicked BECs set thestarting point for the realization of an ORDKRS as described by Eq. (4). To see this, we rewrite the Floquet operator ofORDKRS as ˆ U = ˆ V ˆ V , (7)where ˆ V = e − iK cos(ˆ x ) σ x and ˆ V = e − iK cos(ˆ x + β ) σ y . Then each ofthese two propagators can be realized by proper combinationsof “coin toss” and spin-dependent walk operations:ˆ V = e − iK cos(ˆ x ) σ x = M (cid:18) − π , (cid:19) T M (cid:18) π , (cid:19) , (8)ˆ V = e − iK cos(ˆ x + β ) σ y = M (cid:18) − π , π (cid:19) T M (cid:18) π , π (cid:19) , (9)where T = e − iK cos(ˆ x ) σ z and T = e − iK cos(ˆ x + β ) σ z are twospin-dependent walks. The di ff erent kicking strengths K , =Ω , τ p / | ∆ , | may be realized by letting the two walks to haveeither a di ff erent Rabi frequency Ω (cid:44) Ω or a di ff erentdetuning | ∆ | (cid:44) | ∆ | . Putting together, the Floquet opera-tor of ORDKRS is realized by a sequence of operations asˆ U = M (cid:16) − π , π (cid:17) T M (cid:16) π , π (cid:17) M (cid:16) − π , (cid:17) T M (cid:16) π , (cid:17) . Sinceeach sub-step in this sequence is already realized in the quan-tum walk experiment of Rb condensates [55, 56], the real-ization of ORDKRS as described by Eq. (4) should alreadybe available in the same experimental setup or other similarplatforms.To further motivate experimental interests, we will analyzethe topological properties of ORDKRS in the following sec-tions. To be more explicit, we choose the phase shift betweenthe two kicks to be β = − π in Eq. (4). This gives us thefollowing Floquet operator of an ORDKRS:ˆ U R = e − iK sin(ˆ x ) σ y e − iK cos(ˆ x ) σ x . (10)As will be shown, this system possesses a fruitful Floquettopological phases, with their topological winding numbersdetectable by measuring momentum distributions of the sys-tem over tens of driving periods.Note in passing that by choosing the initial state to be acoherent superposition of several momentum eigenstates [55,56], the Floquet operator ˆ U R may also be used to engineera split step quantum walk in the momentum space of BECs,whose topological properties have been thoroughly exploredin previous studies [60]. Compared with the split step quan-tum walk, the ORDKRS introduced here admits a richer topo-logical phase diagram, with the possibility to access phaseswith large topological invariants. III. TOPOLOGICAL PHASES OF THE ORDKRS
Similar to their static cousins [61], single-particle Floquettopological phases in one-dimension are all symmetry pro-tected [45]. The Floquet operator ˆ U R , as defined in Eq. (10),possesses a chiral symmetry. Its topological phases are thencharacterized by a pair of integers ( Z × Z ), defined in two com-plementary chiral symmetric time frames [43]. These integerspredict the number of degenerate 0 and π -quasienergy edgestates in the two spectrum gaps of ˆ U R , respectively. Thesewill be demonstrated in the following subsections. A. Chiral symmetric time frame and topological windingnumber
The chiral symmetry of ˆ U R is most clearly seen by trans-forming it into two chiral symmetric time frames [43], inwhich it has the following forms:ˆ U = e − i K cos(ˆ x ) σ x e − iK sin(ˆ x ) σ y e − i K cos(ˆ x ) σ x , (11)ˆ U = e − i K sin(ˆ x ) σ y e − iK cos(ˆ x ) σ x e − i K sin(ˆ x ) σ y . (12)It is seen that both ˆ U and ˆ U are related to ˆ U R by unitarytransformations, meaning that they all share the same Floquetquasienergy spectrum. Furthermore, both ˆ U and ˆ U possessthe chiral symmetry as Γ ˆ U Γ = ˆ U † , Γ ˆ U Γ = ˆ U † , Γ = σ z . (13)Here the chiral symmetry operator Γ is both Hermitian andunitary, i.e. , Γ = Γ † = Γ − . Based on the periodic table of Flo-quet topological states [45], each phase of ˆ U R is then charac-terized by a pair of integer winding numbers ( W , W π ) ∈ Z × Z [43], given by W = W + W , W π = W − W , (14)where W and W are winding numbers of Floquet operatorsˆ U and ˆ U , respectively. The winding numbers ( W , W π ) al-low us to achieve a full classification of the topological phasesof ˆ U R , as will be discussed in Sec. III B.To compute these winding numbers for each Floquet topo-logical phase, we rewrite ˆ U (cid:96) ( (cid:96) = ,
2) by combining its threepieces. In the position representation {| θ (cid:105)| θ ∈ [ − π, π ) } , wethen have ˆ U (cid:96) = (cid:80) θ | θ (cid:105)(cid:104) θ | e − iE ( θ ) n (cid:96) · σ . The dispersion E ( θ ) hasthe form (see Appendix A for more details) E ( θ ) = arccos[cos( K ) cos( K )] , (15)where K ≡ K cos θ and K ≡ K sin θ . The vector ofmatrix σ = ( σ x , σ y ), and the two-component unit vectors n (cid:96) = ( n (cid:96) x , n (cid:96) y ) for (cid:96) = , n x = sin( K ) cos( K ) (cid:113) sin ( K ) cos ( K ) + sin ( K ) , (16) n y = sin( K ) (cid:113) sin ( K ) cos ( K ) + sin ( K ) , (17) and n x = sin( K ) (cid:113) sin ( K ) cos ( K ) + sin ( K ) , (18) n y = sin( K ) cos( K ) (cid:113) sin ( K ) cos ( K ) + sin ( K ) , (19)Using these vectors, the winding number W (cid:96) of Floquet oper-ator ˆ U (cid:96) [58] can be computed as W (cid:96) = ˆ π − π d θ π ( n (cid:96) × ∂ θ n (cid:96) ) z , (cid:96) = , . (20)As evidenced by this expression, the winding number W (cid:96) counts the number of times that the unit vector n (cid:96) rotatesaround the z -axis when θ changes over a period from − π to π . Thanks to the chiral symmetry of ˆ U (cid:96) , the vector n (cid:96) is con-strained to rotate on the x - y plane, ensuring W (cid:96) to be a welldefined integer. Furthermore, the quantization of the windingnumber W (cid:96) is topologically protected, since W (cid:96) cannot changeits value under continuous deformations of the trajectory of n (cid:96) on the x - y plane. The topological property of winding num-bers ( W , W π ) are then carried over from winding numbers W and W through Eq. (14). B. Topological phase diagram
If the trajectory of vector n (cid:96) on the x - y plane happens topass through the origin of z -axis at some critical value θ = θ c ,the dispersion E ( θ ) will become gapless. This situation in-dicates the breakdown of the winding number definition (20)and the existence of a possible topological phase transitionspecified by its corresponding kicking strengths ( K c , K c ).The collection of all these transition points on the plane ofparameter space ( K , K ) forms the boundary between di ff er-ent Floquet topological phases of the ORDKRS.To locate these phase boundaries, we note that being aphase factor defined modulus 2 π , the dispersion E ( θ ) hasin general two gaps at both quasienergies 0 and π , respec-tively. The closure of a spectrum gap in E ( θ ) then corre-sponds to either E ( θ ) = E ( θ ) = π , which means thatcos( K ) cos( K ) = ± K cos θ = µπ and K sin θ = νπ ,where ν, µ are both integers. The combination of these condi-tions yields the following equation for the topological phaseboundaries of ˆ U R : µ K + ν K = π , µ, ν ∈ Z . (21)Following their experimental definitions, we focus on theregime of positive kicking strengths K , K >
0. In thisregime, the phase boundaries can be classified into threegroups based on the value of integers ( µ, ν ).(i) µ =
0: In this case, the phase boundaries K = νπ ( ν = , , , ... ) are straight lines in parallel with the K -axis (1,0)(1,2)(3,2)(3,4)( ,4) (1, -
2) ( , -
2) ( , - ) ( , - )(1,0) (-3,0) (-1,0)(-1,0) (1,0) (1,0) (-1, -4 ) (-1, ) (1, -4 ) (1, ) (-1, - ) (-1, ) ( ,0) (5,0) (- , - ) (- , ) ( -3 , )( -3 , ) ( -3 , )( -3 , ) (1, ) (1, -2 ) (1, -2 )(1, ) ( - , )( - , - ) FIG. 1. (color online) Floquet topological phase diagram of the OR-DKRS ˆ U R versus kicking strengths ( K , K ). Red solid (blue dashed)lines are phase boundaries, where the Floquet spectrum gap close atquasienergy 0 ( π ). Each closed patch corresponds to a unique topo-logical phase, characterized by a pair of winding numbers ( W , W π )deduced from Eq. (14), as denoted in the figure for some representa-tive phases. on the K - K plane. Furthermore, when ν is an odd (even)integer, the Floquet spectrum gap will close at quasienergy π (0). The corresponding topological phase transition is onlyaccompanied by the change of winding number W π ( W ).(ii) ν =
0: In this case, the phase boundaries K = µπ ( µ = , , , ... ) are straight lines in parallel with the K -axison the K - K plane. Furthermore, when µ is an odd (even)integer, the Floquet spectrum gap will close at quasienergy π (0). The corresponding topological phase transition is onlyaccompanied by the change of winding number W π ( W ).(iii) µ, ν (cid:44)
0: In this case, the phase boundary curves are de-scribed by the equation K π = ν (cid:18) − µ K /π (cid:19) − / , with positivesolutions only for K > µπ . Furthermore, when µ, ν have theopposite (same) parities, the Floquet spectrum gap will closeat quasienergy π (0) along the phase boundary curve. The cor-responding topological phase transition is only accompaniedby the change of winding number W π ( W ).Combining points (i-iii) together with winding numberscalculated from Eq. (14), we are able to achieve a full topolog-ical classification of the ORDKRS as described by the Floquetoperator ˆ U R in Eq. (10). A topological phase diagram of thesystem up to K = K = π is shown in Fig. 1. On the phasediagram, each closed patch is characterized by a pair of wind-ing numbers ( W , W π ).In Ref. [17], a phase diagram with similar phase bound-aries is found in a spinless DKR model under a di ff erent on-resonance condition. Notably, the topological phase in eachpatch of that phase diagram is characterized by di ff erent wind-ing numbers from that of the ORDKRS studied here. This dif-ference comes from distinct winding behaviors of the vector n (cid:96) in the two models, even though they share the same Floquet spectrum.Furthermore, in the region ( K , K ) ∈ (0 , π ) × (0 , ∞ )(( K , K ) ∈ (0 , ∞ ) × (0 , π )), the winding numbers W and W π both tend to grow linearly along the direction of K - ( K -)axiswithout bound (see Appendix. B for an illustration). This re-sult mimics the change of quantum Hall resistance (here thewinding number) with the increase of a magnetic field (herethe kicking strength) in quantum Hall e ff ects [5, 62]. A sim-ilar pattern is also observed in the phase diagram of the spin-less DKR studied in Ref. [17]. The possibility of accessingphases with arbitrarily large winding numbers in the ORD-KRS makes it a good candidate to explore Floquet states andphase transitions in the regime of large topological invari-ants, which is usually absent in other experimentally realizedone-dimensional Floquet systems like the split step quantumwalk [60].In the next subsection, we will explore the relation betweenthe winding numbers of ˆ U R and the number of its topologicaledge states in a finite-size momentum space lattice. C. Bulk-boundary correspondence
The Floquet operator ˆ U R = e − iK sin(ˆ x ) σ y e − iK cos(ˆ x ) σ x can bewritten in momentum representation [17] asˆ U R = e − iK (cid:80) n i ( | n (cid:105)(cid:104) n + |− h . c . ) σ y e − iK (cid:80) n ( | n (cid:105)(cid:104) n + | + h . c . ) σ x , (22)where the momentum basis {| n (cid:105)| n ∈ Z } satisfies the eigen-value equation ˆ n | n (cid:105) = n | n (cid:105) , with ˆ n being the dimensionlessmomentum operator as discussed in Sec. I. This result can beobtained, e.g. , by first writing K cos( ˆ x ) σ x in position rep-resentation as K (cid:80) θ e i θ + e − i θ | θ (cid:105)(cid:104) θ | σ x , and then performing aFourier transform from position to momentum representationas | θ (cid:105) = √ N (cid:80) N − n = − N e in θ | n (cid:105) under the periodic boundary con-dition | n (cid:105) = | n + N (cid:105) . Expressed in the form of Eq. (22), ˆ U R admits an interpretation of two consecutive kicks by momen-tum space tight-binding lattices on a spin-1 / K , K ) re-side in a topologically nontrivial patch of the phase diagram.This is guaranteed by the bulk-boundary correspondence ofchiral symmetric Floquet systems [43]. More precisely, theabsolute value of winding number W ( W π ) gives the numberof degenerate edge state pairs at quasienergy 0 ( π ) in the mo-mentum space lattice.An illustration of this bulk-boundary relation is given inFig. 2. The panel (a) of Fig. 2 shows the spectrum of ˆ U R at a fixed value of the first kicking strength K = . π un-der open boundary conditions. With the change of the sec-ond kicking strength K , the system undergoes two topologi-cal phase transitions, with quasienergy gap closing at π (0) for K = π ( K = π ). These transitions separate the system in theconsidered range of parameters into three di ff erent topologicalphases, characterized by winding numbers ( W , W π ) = (1 , ,
2) and (3 ,
2) (See also Fig. 1). These numbers correctly state index -1011 20 40 state index -101 1 20 40 state index -101 (a) (b)(c) ( d ) (1,0) (1,2) ( ,2) FIG. 2. (color online) Bulk-boundary correspondence of the ORD-KRS. Panel (a): Floquet spectrum E of ˆ U R versus K at K = . π ,for a momentum space lattice of N =
20 unit cells under openboundary conditions. Three topological phases with winding num-bers ( W , W π ) = (1 , ,
2) and (3 , K = π, π . Panel (b): Floquet spec-trum E at K = . π , referring to the cut along the red dashed lineat the left end of Panel (a). There is a pair of 0-quasienergy edgestates, corresponding to W =
1. Panel (c): Floquet spectrum E at K = . π , referring to the cut along the black dashed line in the mid-dle of Panel (a). There is a pair of 0- and two pairs of π -quasienergyedge states, corresponding to W = W π =
2. Panel (d): Floquetspectrum E at K = . π , referring to the cut along the blue dashedline at the right end of Panel (a). There are three pairs of 0- andtwo pairs of π -quasienergy edge states, corresponding to W = W π = predict the number of 0- and π -quasienergy edge state pairs inthese three topological phases, as exemplified by panels (b) to(d) of Fig. 2. On the other hand, by counting the number of0 and π edge state pairs in Fig. 2(b-d), we can also obtain thewinding numbers ( W , W π ) for each topological phases. Thisconcludes the verification of bulk boundary correspondencein the chiral symmetric ORDKRS system.As a notable feature of Fig. 2(a), there are regions in whichthe 0 and π quasienergy edge states coexist at the same systemparameters [see Fig. 2(c) or 2(d) as an example]. In a recentstudy [63], it was shown that a superposition of 0 and π edgestates form a new type of symmetry protected discrete timecrystal phase, which is further used to propose a new approachto non-Abelian braiding and topological quantum computingin a superconducting Floquet system. The unbounded growthof winding numbers in the phase diagram Fig. 1 then impliesthe possibility of finding an arbitrarily large number of 0 and π quasienergy edge states at the same parameter of the ORD-KRS, and therefore the potential of engineering many di ff er-ent Floquet time crystal phases [64] in this system by super-posing these edge states.Experimentally, Floquet edge states between systems withdi ff erent bulk topological properties have been observed inphotonic quantum walks [49]. However, for the ORDKRS de- fined in a momentum lattice as Eq. (22), it may not be easy toengineer a boundary between di ff erent momentum space re-gions. In the following section, we discuss an alternative wayof detecting topological winding numbers of the ORDKRS bydirectly imaging the momentum distribution of a wave packet[59], which is available in kicked BEC experimental setups[55]. IV. PROBING BULK TOPOLOGICAL PROPERTIES OFTHE ORDKRS
The topological winding numbers of a one-dimensionalchiral symmetric system can be detected by measuring themean chiral displacement (MCD) of a wave packet [58, 59].Formally, it is the expectation value of chiral displacement op-erator ˆ C ( t ) = ˆ U † ( t )ˆ n Γ ˆ U ( t ) at some time t of the system’s uni-tary evolution ˆ U ( t ). For the ORDKRS, ˆ n and Γ = σ z representthe quantized momentum and chiral symmetry operators, re-spectively. Therefore the MCD of ORDKRS is just a signedmomentum distribution, with the extra sign originating fromthe chiral symmetry. For the system of Rb BECs prepared inthe state | ψ (cid:105) = | n = , S / F = (cid:105) or | n = , S / F = (cid:105) of the n = t =
0, the MCD after t driving periods reads C (cid:96) ( t ) = (cid:104) ψ | ˆ U − t (cid:96) (ˆ n ⊗ σ z ) ˆ U t (cid:96) | ψ (cid:105) , (23)where the Floquet operators ˆ U (cid:96) ( (cid:96) = ,
2) are given byEqs. (11) and (12). Further calculations lead to (see Ap-pendix C for details): C (cid:96) ( t ) = W (cid:96) − ˆ π − π d θ π cos[ E ( θ ) t ]2 ( n (cid:96) × ∂ θ n (cid:96) ) z (cid:96) = , . (24)Here W (cid:96) is the winding number of ˆ U (cid:96) given by Eq. (20). Thedispersion E ( θ ) is given by Eq. (15), and the components ofunit vector n (cid:96) are given by Eqs. (16-19). As can be seen, C (cid:96) ( t )contains a time-independent topological part W (cid:96) and an extratime-dependent oscillating term. For a not-too-flat dispersion E ( θ ), the oscillating term will tend to vanish for large t underthe integral over θ . A bit more rigorously, the C (cid:96) ( t ) averagedover t driving periods, i.e. , C (cid:96) ( t ) ≡ t t (cid:88) t (cid:48) = C (cid:96) ( t (cid:48) ) = W (cid:96) − t t (cid:88) t (cid:48) = ˆ π − π d θ π cos[ E ( θ ) t ]2 ( n (cid:96) × ∂ θ n (cid:96) ) z , (25)will gradually converge to half of the winding number W (cid:96) withthe increase of t . Once W and W are obtained from the timeaveraged MCD, the winding numbers ( W , W π ) characterizingtopological phases of the ORDKRS can be calculated by Eq.(14).In Fig. 3, we present numerical results of C (cid:96) ( t ) along twodi ff erent trajectories in the K - K parameter space, togetherwith theoretical values of half winding numbers W / (a)(b) (1,0) (1,2) (3,2) (1,0) (-3,0) FIG. 3. (color online) Time averaged MCDs. Numerical values of C ( t ) and C ( t ), both averaged over t =
20 driving periods, are shownby blue stars and black circles. Theoretical values of half windingnumbers W and W are shown by red solid and green dashed lines.In Panel (a), the kicking strength K = . π and the three topologi-cal phases, separated by two transitions at K = π, π , have windingnumbers ( W , W π ) = (cid:16) W + W , W − W (cid:17) = (1 , ,
2) and (3 , K = K = √ π have winding numbers( W , W π ) = (cid:16) W + W , W − W (cid:17) = (1 ,
0) and ( − , W /
2. The numerical results at each set of kicking strengths( K , K ) are obtained by directly evolving a wave packet, pre-pared at initial state | n = , F = (cid:105) or | n = , F = (cid:105) ,with propagators ˆ U t and ˆ U t in momentum space to find C ( t )and C ( t ), respectively, and then averaging over the numberof driving periods t . Up to t =
20, we find already verynice convergence of C (cid:96) ( t ) ( (cid:96) = ,
2) to its corresponding halfwinding number W (cid:96) , with the error accounted for by the time-dependent term in Eq. (25). In the setup of Rb BEC, an im-plementation of up to 50 kicks is mentioned to be experimen-tally available [55]. This corresponds to t =
25 driving peri-ods in our double kicked rotor, more then the number neededto see a nice convergence in our numerical simulations.Recently, the measurements of MCD have been achieved inboth photonic [58] and cold atom [59] systems. In Ref. [58],the MCD is extracted from a quantum walk of twisted photonsover 7 steps, and the measured results are robust to dynamicaldisorder. In Ref. [59], Rb condensates are illuminated by apair of o ff -resonant lasers to realize a synthetic lattice in mo-mentum space. The coupling between adjacent momentumsites in this setup is controlled by two-photon Bragg transi-tions, and can be periodically quenched in time. In the high-frequency driving regime, the e ff ective tight-binding Hamilto-nian of the system falls into AIII or BDI topological class [61].Disorder-induced topological phase transitions are then de-tected by meansing the MCD. Based on these facts, we be-lieve that the realization of ORDKRS and measurements ofits topological winding numbers are readily doable under cur- rent experimental conditions. V. CONCLUSIONS
In this work, we proposed a spin-1 / Rbsubjected to pairs of periodic pulses by an optical lattice.The system owns many intriguing Floquet topological phases,each characterized by a pair of winding numbers and pro-tected by the chiral symmetry of the Floquet operator. Us-ing these winding numbers, a full topological phase diagramof the system was established. Under open boundary con-ditions, this pair of winding numbers could also predict thenumber of topologically protected edge state pairs at 0 and π -quasienergies of the Floquet spectrum. Finally, we proposedto detect these topological winding numbers by measuring themean chiral displacement of a wave packet, initially localizedat the center of the momentum space. The numerical valuesof mean chiral displacement, averaged over 20 kicking pe-riods, tend to converge to the theoretical prediction of bulkwinding numbers of the ORDKRS. Recently, the experimen-tal measurements of mean chiral displacements have also beenachieved in other model systems [58, 59].Our choice of the on-resonance condition, i.e. , (cid:126) τ = π with τ = T /
2, makes the free evolution part of the Floquet op-erator to become an identity. Under more general resonanceconditions, the free evolution part can also contribute to thedynamics. The resulting Floquet operators could then possessmore then two Floquet bands and di ff erent types of topolog-ical phases, as already indicated in a previous study of thespinless quantum DKR [17]. Exploring the impact of an extraspin degree of freedom on the topological phases of the DKRunder general resonance conditions is an interesting topic forfuture study.Due to experimental constrains on the detection windowof momentum states, only small to intermediate values ofkicking strength are considered in our numerical simulations.When the kicking strength is large, the dynamics of the spin-1 / W of a peri-odically quenched chiral symmetric Floquet system satisfy aGaussian distribution around W = K = λ K on the phase diagram for any | λ | ∈ (0 , ∞ ). How-ever, for trajectories parallel to K or K axis on the phase dia-gram and constrained within K ∈ (0 , π ) or K ∈ (0 , π ) regions,respectively, the winding numbers W change monotonicallywith the kicking strength and satisfy instead a uniform dis-tribution. The qualitative di ff erence between these two typesof winding number distributions, the transition between them,and its possible connection to the quantum-to-classical transi-tions in ORDKRS also deserve further explorations.Finally, the e ff ect of disorder on Floquet topological phasesis of great theoretical and experimental interests [28, 38, 58,59]. In a chiral symmetric system realized by quantum walkof twisted photons, the Floquet topological phases have beendemonstrated to be robust to weak temporal disorder [58].Furthermore, disorder induced transitions from topologicalAnderson insulator to normal insulator phases, and even thereverse, have also been observed quite recently in the momen-tum space of laser driven ultracold atoms [59]. One limita-tion of the models explored in these experiments is that theirwinding numbers cannot be larger then one. On the contrary,the spin-1 / ACKNOWLEDGEMENT
J.G. is supported by the Singapore NRF grant No. NRF-NRFI2017-04 (WBS No. R-144-000-378-281) and the Singa-pore Ministry of Education Academic Research Fund Tier I(WBS No. R-144-000-353-112).
Appendix A: Expression of ˆ U (cid:96) in position representation In this appendix, we expand a bit more on the derivation ofˆ U (cid:96) ( (cid:96) = ,
2) in the two symmetric time frames used in themain text. In position representation, the Floquet operator ˆ U (cid:96) in symmetric time frame (cid:96) is written as ˆ U (cid:96) = (cid:80) θ | θ (cid:105)(cid:104) θ | U (cid:96) ( θ ),with U ( θ ) = e − i K σ x e − i K σ y e − i K σ x , (A.1) U ( θ ) = e − i K σ y e − i K σ x e − i K σ y , (A.2)where K = K cos θ and K = K sin θ as defined in the maintext. Using the formula e − i γ n · σ = cos( γ ) − i sin( γ ) n · σ , with σ = ( σ x , σ y , σ z ) and n being a unit vector, we can reorganize U ( θ ) and U ( θ ) as U ( θ ) = cos( K ) cos( K ) − i [sin( K ) cos( K ) σ x + sin( K ) σ y ] , (A.3) U ( θ ) = cos( K ) cos( K ) − i [sin( K ) σ x + sin( K ) cos( K ) σ y ] . (A.4)With the identificationscos( E ) = cos( K ) cos( K ) , (A.5)sin( E ) = (cid:113) sin ( K ) cos ( K ) + sin ( K ) , = (cid:113) sin ( K ) + sin ( K ) cos ( K ) , (A.6) W i nd i ng nu m be r s W W
490 500 510490500510490 500 510-510-500-490
FIG. B.1. (color online) The linear growth of winding numbers( W , W π ) versus kicking strength K at a fixed kicking strength K = . π . where E being the dispersion relation, and n x = sin( K ) cos( K )sin( E ) , n y = sin( K )sin( E ) , (A.7) n x = sin( K )sin( E ) , n y = sin( K ) cos( K )sin( E ) , (A.8)we can further express U ( θ ) and U ( θ ) as U (cid:96) ( θ ) = cos( E ) − i sin( E )( n (cid:96) x σ x + n (cid:96) y σ y ) = e − iE ( θ )( n (cid:96) x σ x + n (cid:96) y σ y ) (cid:96) = , n (cid:96) = ( n (cid:96) x , n (cid:96) y ) for (cid:96) = , U (cid:96) = (cid:80) θ | θ (cid:105)(cid:104) θ | e − iE ( θ ) n (cid:96) · σ used in themain text. Appendix B: Linear growth of winding numbers
In this appendix, we give an illustration for the change ofwinding numbers ( W , W π ) along a trajectory in parallel withthe K -axis at a fixed K ∈ (0 , π ) in the phase diagram Fig. 1.From Eq. (20), the winding numbers W , W of Floquet oper-ators ˆ U , ˆ U defined in Eqs. (11,12) are given by W = ˆ π − π d θ π sin( K ) ∂ θ K − sin( K ) cos( K ) cos( K ) ∂ θ K sin ( E ) , (B.1) W = ˆ π − π d θ π sin( K ) cos( K ) cos( K ) ∂ θ K − sin( K ) ∂ θ K sin ( E ) , (B.2)where E = arccos[cos( K ) cos( K )], K = K cos θ and K = K sin θ .In our calculation example, we fix K at 0 . π and scan K from 0 . π to 1000 π . The results of (cid:16) W = W + W , W π = W − W (cid:17) are presented in Fig. B.1. It is clearly seen that both windingnumbers ( W , W π ) grow linearly with the increase of kickingstrength K . Appendix C: Calculation of the mean chiral displacement
In this appendix, we present derivation details of the meanchiral displacement given by Eq. (24) of the main text (seealso Refs. [58, 59]). For the ORDKRS, A time-frame inde-pendent expression of the mean chiral displacement is givenby C ( t ) = (cid:104) | ⊗ (cid:104) F | ˆ U − t (ˆ n ⊗ σ z ) ˆ U t | (cid:105) ⊗ | F (cid:105) , (C.1)where | (cid:105) denotes the 0-momentum eigenvector and | F (cid:105) ( F = ,
2) denotes the eigenvector of hyperfine level F . Note thatfor our choice of initial state, C ( t = = C ( t ) indeedrepresents a displacement over t driving periods. Writing ˆ n inmomentum representation as ˆ n = (cid:80) n n | n (cid:105)(cid:104) n | , we have C ( t ) = (cid:88) n n (cid:104) | ⊗ (cid:104) F | ˆ U − t | n (cid:105)(cid:104) n | ⊗ σ z ˆ U t | (cid:105) ⊗ | F (cid:105) , (C.2)Expanding ˆ U t in position representation as ˆ U t = (cid:80) θ | θ (cid:105)(cid:104) θ | U t ( θ ), we further obtain C ( t ) = (cid:88) θ,θ (cid:48) (cid:88) n n (cid:104) | θ (cid:105)(cid:104) θ (cid:48) | (cid:105)(cid:104) θ | n (cid:105)(cid:104) n | θ (cid:48) (cid:105)×(cid:104) F | U − t ( θ ) σ z U t ( θ (cid:48) ) | F (cid:105) . (C.3)Under periodic boundary conditions, we have the followingFourier transforms between position and momentum basis: | θ (cid:105) = √ N (cid:88) n e i θ n | n (cid:105) , | n (cid:105) = √ N (cid:88) θ e − i θ n | θ (cid:105) , (cid:104) n | θ (cid:105) = √ N e i θ n , (C.4)where n = − N , − N + , ..., N − | n (cid:105) = | n + N (cid:105) and θ = − N π N , − ( N − π N , ..., ( N − π N with | θ (cid:105) = | θ + π (cid:105) . Using theserelation, we can write C ( t ) as C ( t ) = N (cid:88) θ,θ (cid:48) N (cid:88) n ne i ( θ (cid:48) − θ ) n (cid:104) F | U − t ( θ ) σ z U t ( θ (cid:48) ) | F (cid:105) . (C.5)Noting that1 N (cid:88) n ne i ( θ (cid:48) − θ ) n = i ∂ θ N (cid:88) n e i ( θ (cid:48) − θ ) n = i ∂ θ δ θθ (cid:48) , (C.6)the expression of C ( t ) reduces to C ( t ) = N (cid:88) θ,θ (cid:48) i ∂ θ δ θθ (cid:48) (cid:104) F | U − t ( θ ) σ z U t ( θ (cid:48) ) | F (cid:105) . (C.7) To proceed, we need to transform the summation over θ, θ (cid:48) tointegrals by taking the number of unit cells N → ∞ . In thislimit, we have δ θθ (cid:48) → π N δ ( θ − θ (cid:48) ), (cid:80) θ,θ (cid:48) → N ´ π − π d θ π ´ π − π d θ (cid:48) π ,and therefore C ( t ) = ˆ π − π d θ π ˆ π − π (cid:104) F | U − t ( θ ) σ z U t ( θ (cid:48) ) | F (cid:105) (cid:2) i ∂ θ δ ( θ − θ (cid:48) ) (cid:3) d θ (cid:48) . (C.8)Sending i ∂ θ → − i ∂ θ (cid:48) , performing an integration by parts over θ (cid:48) and then integrating out θ (cid:48) , we are left with C ( t ) = ˆ π − π d θ π (cid:104) F | U − t ( θ ) σ z i ∂ θ U t ( θ ) | F (cid:105) . (C.9)According to our discussion in appendix A, U t ( θ ) can be ex-pressed as U t ( θ ) = e − iE ( θ ) t n ( θ ) · σ = cos( Et ) − i sin( Et ) n · σ = [ U − t ( θ )] † , (C.10)where n = ( n x , n y ) represents the unit vector in any chiral sym-metric time frame, and σ = ( σ x , σ y ). Using this expression of U t ( θ ), the operator U − t ( θ ) σ z i ∂ θ U t ( θ ) yields: U − t ( θ ) σ z i ∂ θ U t ( θ ) = it cos(2 Et ) ( ∂ θ E ) ( n x σ y − n y σ x ) + i sin( Et ) cos( Et ) ∂ θ ( n x σ y − n y σ x ) − it (cid:104) ∂ θ sin ( Et ) (cid:105) σ z + sin ( Et )( n x ∂ θ n y − n y ∂ θ n x ) . (C.11)Next, we note that the hyperfine basis | F = , (cid:105) has the fol-lowing vector expressions: | F = (cid:105) = (cid:32) (cid:33) | F = (cid:105) = (cid:32) (cid:33) . (C.12)This means that under the average (cid:104) F | · · · | F (cid:105) , only diagonalelements of the matrix U − t ( θ ) σ z i ∂ θ U t ( θ ) could survive. Fur-thermore, the term ∂ θ sin ( Et ) vanishes after integrating over θ due to the periodicity of E in θ . So we are only left with thelast term of Eq. (C.11) under the θ -integral, i.e. , C ( t ) = ˆ π − π d θ π sin ( Et )( n x ∂ θ n y − n y ∂ θ n x ) . (C.13)Notably, this result is independent of the initial choice of hy-perfine level | F (cid:105) . 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