Non-dispersing wave packets in lattice Floquet systems
NNon-dispersing wave packets in lattice Floquet systems
Zhoushen Huang, ∗ Aashish Clerk, † and Ivar Martin ‡ Materials Science Division, Argonne National Laboratory Pritzker School of Molecular Engineering, University of Chicago (Dated: May 20, 2020)We show that in a one-dimensional translationally invariant tight binding chain, non-dispersingwave packets can in general be realized as Floquet eigenstates—or linear combinations thereof—using a spatially inhomogeneous drive, which can be as simple as modulation on a single site. Therecurrence time of these wave packets (their “round trip” time) locks in at rational ratios sT /r of thedriving period T , where s, r are co-prime integers. Wave packets of different s/r can co-exist underthe same drive, yet travel at different speeds. They retain their spatial compactness either infinitely( s/r = 1) or over long time ( s/r (cid:54) = 1). Discrete time translation symmetry is manifestly broken for s (cid:54) = 1, reminiscent of Floquet time crystals. We further demonstrate how to reverse-engineer a driveprotocol to reproduce a target Floquet micromotion, such as the free propagation of a wave packet,as if coming from a strictly linear energy spectrum. The variety of control schemes open up a newavenue for Floquet engineering in quantum information sciences. Introduction—
It is well known that under a time-independent Hamiltonian, quantum wave packets typi-cally spread out due to the presence of dispersion [1].Since the birth of quantum mechanics, the stark contrastbetween the elusiveness of localized quantum entities andthe stability of their classical counterparts has motivatedgenerations of physicists to explore ways to even the dis-parity [2–4]. Schr¨odinger himself had searched for modelswhich can host free traveling wave packets that do notspread, but did not go much beyond harmonic oscillators[5]. Fundamental conceptual interest aside, such dynam-ically stable localized entities, if realizable, could alsohold great technological utility in quantum informationprocessing and computing platforms, since most controltechnologies today are local in nature.Stabilization of non-dispersing wave packets typi-cally requires some form of nonlinearity. For exam-ple, non-linear Sch¨odinger or Gross-Pitaevskii equationsare known to host soliton solutions [6, 7]. An alterna-tive strategy is to invoke Floquet engineering [3, 8–14].This was previously explored in the specific context ofmicrowave-driven Rydberg atoms [3, 10, 15]. There, wavepackets following classical Kepler orbits have been real-ized as Floquet eigenstates. The shape and spread ofthese wave packets is however strongly time-dependent,and the underlying physics can be understood as a stro-boscopic refocusing.In this work, we consider a far more general situa-tion. We explore the creation of non-dispersing, travel-ing wave packets in generic spatially extended systems viaFloquet engineering. Using a homogeneous tight bindingchain as prototype, we discover wave packets that aremanifestly spatially localized Floquet eigenstates (or lin-ear combinations thereof). They maintain their spatialcompactness not only stroboscopically, as in the case ofRydberg atoms, but at all times. We stress that theclass of lattice models we consider are of direct relevanceto several existing quantum information processing plat- forms, e.g. chains of coupled superconducting microwavecavities, or linear arrays of coupled photonic resonators[16–18]. A traveling wave packet on such a device couldconceivably serve as a “bus”, over which quantum infor-mation can be shuttled across the entire chain.A Floquet drive is defined by its period T , and its spa-tial and temporal profiles. We will see that T singles outa series of spectral segments of the undriven system thatare most susceptible to the formation of wave packets, asorganized by their recurrence time (i.e., the time a wavepacket takes to traverse one round trip of the system) T rec = sr T , where s and r are co-prime integers. Thecombination of the drive’s spatial and temporal profilesthen imposes selection rules that determine which wavepackets actualize, as well as their properties such as spa-tial compactness. When s >
1, the Floquet wave packetsmanifestly break the discrete time-translation symmetryof the drive; we will discuss the connection to time crys-tal physics [13, 19–25]. As long as these general rules aresatisfied, the formation of wave packets is robust withrespect to details such as the overall drive strength, theintroduction of spatial or temporal randomness, etc. Thisflexibility also opens up the ability to fine-tune drive pro-tocols for specific applications. As a proof of principle, wewill demonstrate how to design a drive that reproducesa particular target Floquet micromotion.
Floquet wave packets at the primary resonance—
Tobuild intuition, we first discuss the emergence of Floquetwave packets at the primary resonance, that is those witha round trip time equal to the drive period, T rec = T .Consider a time-periodic Hamiltonian ˆ H ( t + T ) = ˆ H ( t ),ˆ H ( t ) = ˆ H + ˆ V ( t ) , ˆ V ( t ) = (cid:88) a g a ˆ V ( a ) e ia Ω t , (1)where ˆ H is the undriven tight-binding Hamiltonian, as-sumed to be spatially homogeneous, ˆ V ( a ) encodes spa-tial dependence of the drive at frequency a Ω, with rela-tive strength g a , and Ω = 2 π/T is the fundamental fre- a r X i v : . [ c ond - m a t . o t h e r] M a y quency. After one drive period, a Floquet eigenstate | ψ (cid:105) returns to itself with an additional phase ( quasienergy ),ˆ U T | ψ (cid:105) = e − iθ | ψ (cid:105) , where ˆ U t = T exp (cid:104) − i (cid:82) t dt (cid:48) ˆ H ( t (cid:48) ) (cid:105) isthe time evolution operator. This state can be lifted to atime-periodic trajectory in Hilbert space, i.e. , a Floquetmicromotion , | ψ ( t ) (cid:105) = | ψ ( t + T ) (cid:105) = e + i θT t ˆ U t | ψ (cid:105) , whichsatisfies the Floquet-Schr¨odinger equation, (cid:104) ˆ H ( t ) − i∂ t (cid:105) | ψ ( t ) (cid:105) = θT | ψ ( t ) (cid:105) . (2)Note that shifting θ → θ + 2 πa ( a ∈ Z ) leads to gaugeequivalent micromotions | ψ ( t ) (cid:105) → | ψ ( t ) (cid:105) e ia Ω t of the same physical time evolution.In the undriven limit, Floquet eigenstates are simplythe energy eigenstates | ε k (cid:105) of ˆ H with integer label k . Atweak drive, thus, most Floquet eigenstates are close to anundriven state and remain spatially extended. However,if the drive frequency Ω = 2 π/T is close to the levelspacing ∆ somewhere in the spectrum of ˆ H , then thedrive can efficiently couple several nearby unperturbedeigenstates. To describe this, one can expand a genericdispersion relation around some k ∗ (not necessarily aninteger), such that ε k = ε ∗ + ( k − k ∗ )Ω + uT ( k − k ∗ ) + · · · , k ∈ Z , (3)and consider a micromotion ansatz | ψ ( t ) (cid:105) = (cid:88) k f k | ε k (cid:105) e − ik Ω t . (4)Note that | ε k (cid:105) e − ik Ω t are Floquet micromotions in theundriven limit, and the gauge ( a = k ) is chosen sothat near resonance, the corresponding quasienergies, θ (0) k = ε k T − πk , are nearly degenerate (in the scale ofΩ T = 2 π ), hence Eq. 4 is akin to degenerate perturbationsolutions. For consistency, the range of the k summationshould be constrained such that { θ (0) k } are roughly withina single Floquet zone.We assume positive u and ∂ k ε k in Eq. 3; the case withone or both of them negative can be similarly handled.Solving Eq. 4 with 2 then leads to an eigenvalue problem (cid:88) k (cid:48) (cid:2) u ( k − k ∗ ) δ k k (cid:48) + g k (cid:48) − k V kk (cid:48) (cid:3) f k (cid:48) = ( θ − θ ∗ ) f k , (5)where V kk (cid:48) = (cid:104) ε k | ˆ V ( k (cid:48) − k ) | ε k (cid:48) (cid:105) T and θ ∗ = ε ∗ T − πk ∗ .Eq. 5 maps our problem to an effective one-dimensional“lattice” with quadratic “on-site potential” u ( k − k ∗ ) and “hopping” g k (cid:48) − k V kk (cid:48) . One thus expects on generalgrounds that its eigenstates will mix different k “sites.”Translating back to the original problem, the Floquetmicromotion | ψ ( t ) (cid:105) is thus a linear superposition of mo-mentum states | ε k (cid:105) with time-independent weights | f k | ,and is therefore a wave packet in coordinate space. 𝑘𝐸 𝑘 Ω Ω Ω ⋮ 𝑘𝐸 𝑘 − 𝑘Ω ⋯⋯ ⋯ ⋯ ⋯ ⋯ ⋯ FIG. 1. Top: schematic of coupled resonators. Temporalmodulation of the first site’s onsite energy induces Floquetwave packets. Such a scenario is modeled by Eq. 6. Middle:At the primary resonance where the drive frequency matchesthe typical level spacing of the undriven problem (left), theeffective model (Eq. 7) is a lattice harmonic oscillator withquadratic on-site energy (right). Bottom: Two (of several)wave packet solutions of Eq. 6 corresponding to the ground(left) and the first excited (right) states of the emergent latticeoscillator. The system size L = 500, drive strength g = 1, anddrive period T = 1005. Note that the wave packets maintaintheir spatial compactness at all time. As a concrete example, we consider an open boundarychain of length L driven on the first site (Fig. 1),ˆ H ( g ) = L − (cid:88) x =1 | x (cid:105)(cid:104) x + 1 | + h.c. + 2 g cos(Ω t ) | (cid:105)(cid:104) | , (6)where the only nonvanishing Fourier components of thedrive are g = g − = g . This limits the effective hoppingin Eq. 5 to nearest neighbor in k , and for simplicity, wewill approximate it as k -independent and evaluate it at k ∗ , writing τ ≡ gV k ∗ ,k ∗ . Eq. 5 then becomes u ( k − k ∗ ) f k + τ ( f k − + f k +1 ) = ( θ − θ ∗ ) f k , (7)and maps to a lattice version of harmonic oscillator, withstiffness u and hopping τ . A similar equation was previ-ously obtained in the context of driven Rydberg atoms[3]. For sufficiently large τ /u , a subset of its eigenstatesare thus Gaussian-like wave packets with 1 , , · · · , D spa-tial peaks, where D counts the number of oscillator-likestates. In Fig. 1, we plot two of the D wave packetsolutions corresponding to the ground and the first ex-cited states of Eq. 7 (and hence with one and two spatialpeaks, respectively). To form a wave packet, the “hop-ping” must be able to efficiently couple several k states,hence D can be estimated as the number of “sites” thatare energetically within one hop’s reach from the poten-tial bottom, uδk ≤ τ ⇒ | δk | ≤ (cid:112) τ /u ⇒ D (cid:39) (cid:112) τ /u ,where δk = k − k ∗ . The crossover drive strength to in-duce any wave packet at all is thus τ c (cid:39) u/
4, althougha substantially stronger drive is needed to produce bet-ter spatial compactness (as D also counts the number ofmomentum constituents in a wave packet). The emer-gent oscillator “frequency,” (cid:36) = 2 √ τ u , is approximatelythe level spacing of the quasienergies { θ } . Physically,thus, if an initial state is a superposition of such wave-packet Floquet eigenstates, it will (approximately) reviveafter 2 π/(cid:36) drive periods. A locality-based measure, suchas the participation ratio (cid:80) x | ψ ( x, t ) | , will then exhibitbeats at frequency ∼ (cid:36) Ω / (2 π ).To evaluate u and τ , we note that the undriven ˆ H (0)has eigenstates (cid:104) x | ε k (cid:105) = (cid:112) / L sin( q k x ) with wave vec-tors q k = πk/ L , and eigenvalues ε k = 2 cos q k . Here L ≡ L +1 and x, k = 1 , , · · · , L . From ∂ k ε k ∗ = Ω and ∂ k ε k ∗ =2 u/T , we get u = π √ T − L / L and τ = 2 g L /T .Parametrizing β = L /T and γ = 1 / (cid:112) − β , one findsthat the emergent “frequency” is (cid:36) = 2 π (cid:112) g/γ L , thenumber of Floquet wave packets is D (cid:39) βπ √ gγ L , andthe crossover drive strength is g c = π / β γ L .It is worth noting that emergence of Floquet wavepackets does not rely on the “on-site potential” beingquadratic. In the SM, we show that they also exist whenthe “on-site potential” becomes cubic, a scenario thatarises when the drive frequency resonates near the inflec-tion point of an undriven spectrum. Floquet wave packets at rational resonances—
Thesame setup can more generally host many series of non-dispersing wave packets that have recurrence times notjust equal to, but rationally commensurate with the driveperiod, T rec = sr T , where s, r are co-prime integers. Thegroup velocity of an ( s, r ) wave packet is v g = 2 L/T rec (2 L being the round trip length), hence it consists mostlyof states from the segment of the undriven energy spec-trum where the typical level spacing is ∆ = rs Ω. We dis-cuss here the more salient features of such wave packetsolutions, and leave mathematical details to the SM.For s > , r = 1, we consider s = 2 as a concrete exam-ple. To leading order, the drive only resonantly coupleswithin even k = 2 κ and odd k = 2 κ + 1, separately. Onecan thus use an ansatz similar to Eq. 4 but restrictedto a given parity, | ψ ( σ ) ( t ) (cid:105) = (cid:80) κ f σκ | ε κ + σ (cid:105) e − iκ Ω t , where σ = 0 , k . Invoking Eq. 2 then leads totwo effective lattice models similar to Eq. 5, one for eachparity, see SM. Similar to the primary resonance case,one then concludes that wave packet solutions generi-cally exist above a crossover drive strength. Crucially,at large L , the two effective chains are essentially iden-tical except for an overall Ω2 shift in the “onsite” energy(Fig. 2 top). This translates to a π gap between theirquasienergy spectra, and is the origin of time-crystallinenature of individual wave packets, as we will see next.In Fig. 2 (center), we plot two ( s, r ) = (2 ,
1) wavepacket solutions resulting from Eq. 6, which correspond 𝑘𝐸 𝑘 Ω /2 Ω Ω /2 ⋮ 𝑘 /2 𝐸 𝑘 − 𝑘 /2 ΩΩ /2 FIG. 2. Top left: when the drive frequency matches twicethe typical level spacing (i.e. an s = 2 , r = 1 resonance),the drive only couples k of the same parity (solid or emptydots). The effective model becomes two independent chainswith an overall Ω / s = 2 , r = 1 wavepacket solutions of Eq. 6 ( L = 500 , g = 1 , T = 225), corre-sponding to the respective ground states of the two effectivechains. Their spatial-temporal patterns are almost indistin-guishable. Their quasienergies are π + δ apart, where numer-ically δ ∼ . × − π . Bottom: dynamical evolution of atime-crystalline wave packet initialized as the sum of the twosolutions, undergoing r = 1 round trip in s = 2 drive periods.After an very long tunneling time of 2 πT /δ ∼ . × T , itwould evolve into the wave packet configuration of the oppo-site linear combination (difference instead of sum). to the “ground state” of the even- and odd-parity effec-tive models, respectively. As shown, both consist of twocounter-propagating wave packets that evolve into eachother after one period. Thus, even though the individualwave packet returns only after 2 T , the Floquet eigenstateremains T -periodic. Individual wave packet can be ob-tained by initializing into the sum (or difference) of thetwo parity ground states. The evolution of one such com-bination is shown in Fig. 2 (bottom). As mentioned be-fore, the quasienergy gap between the two parity-relatedstates is π + δ , where the small deviation δ is due to higherorder effects that mix the two parity sectors. In the limit δ →
0, the individual wave packets are perfectly stable,recurring after 2 T – a manifestation of time-translationsymmetry breaking, analogous to discrete time crystals.A nonzero δ introduces a time scale 2 πT /δ , over whichone time translation symmetry-broken state tunnels intothe other. We find numerically that this tunnelling timecan be indeed very long, reaching thousands of drive peri- FIG. 3. Floquet wave packets at generic ( s, r ) resonances, coexisting under the same drive Eq. 6 with L = 500 , g =1 , T = 1005 (same as Fig. 1). Such Floquet states consist of s traveling wave packets, each traversing r round trips in s drive periods. Left: s = 1 , r = 2. Right: s = 3 , r = 4. ods for reasonable drive strengths and system sizes, and iseasily tunable. For Fig. 2, the tunneling time is ∼ T .The analysis with r > s, r ) Floquet eigenstate consists of s wave packets,each completing a fraction rs of round trip in one driveperiod, see Fig. 3. Like the s = 2 case discussed before, agiven ( s, r ) solution is one of s partners with almost iden-tical spatial-temporal patterns, and their quasienergiesare equally spaced by ∆ θ = 2 π/s to leading order. Theindividual wave packets can be resolved by linear com-binations of the s partners, hence their true recurrencetime is 2 π/ ∆ θ = sT . However, since they completed r round trips in sT , their apparent recurrence time is T rec = sT /r . The rational ratio of T rec /T is suggestive ofa fractional time crystalline order , a notion put forwardvery recently [26, 27]. Floquet drive engineering—
Finally, we discuss how torealize a desired target micromotion through drive engi-neering. Assume the time-dependent Hamiltonian has aform ˆ H ( t ) = (cid:80) n w n Q n ( t ) + h.c. where Q n = ˆ h n e ia n Ω t represent experimentally available Hamiltonian controlsˆ h n at integer harmonics a n , and w n are (generally)complex-valued coefficients. Given a target micromo-tion | ˜ ψ ( t ) (cid:105) and a prescribed set of { Q n } , one can askwhat is the best choice of { w n } to produce a micromotionas close to the target as possible. Writing the Floquet-Schr¨odinger operator as K = ˆ H ( t ) − Q where Q = i∂ t ,the optimal coefficients are those that minimize the vari-ance ∆ = (cid:104) K (cid:105) c , where (cid:104)OO (cid:48) (cid:105) c = (cid:104)OO (cid:48) (cid:105) − (cid:104)O(cid:105)(cid:104)O (cid:48) (cid:105) and (cid:104)O(cid:105) = (cid:82) T dt (cid:104) ˜ ψ |O| ˜ ψ (cid:105) . By construction, ∆ ≥ | ˜ ψ (cid:105) is an exact eigenstate of K [28–30]. De-manding 0 = ∂ ∆ ∂w n = ∂ ∆ ∂w ∗ n then yields the solution ( ww ∗ ) = (cid:0) G FF ∗ G ∗ (cid:1) − (cid:0) JJ ∗ (cid:1) , where G mn = (cid:104) Q m Q n + Q n Q m (cid:105) c , F mn = (cid:104) Q m Q † n + Q † n Q m + h.c. (cid:105) c , J n = (cid:104) Q Q n + Q n Q (cid:105) c ,and ( · ) − is the Moore-Penrose pseudo-inverse. As aproof of principle, we target a non-dispersing Gaus-sian wave packet, | ˜ ψ ( t ) (cid:105) = (cid:80) k ˜ f k | k (cid:105) e − ik Ω t where ˜ f k ∝ k x w (1) / w (2) / 0 0.2 0.4 0.6 0.8 1 t / T FIG. 4. Optimal drive to reproduce a target micromotion | ˜ ψ ( t ) (cid:105) = (cid:80) k e − ( k − k ) / σ − ik Ω t | ε k (cid:105) on a chain of length L =100, with k = 60 and σ = 15. We use static, translation-invariant nearest neighbor hopping, and onsite modulation upto the second harmonic, ˆ H ( t ) = (cid:80) x w (0) | x (cid:105)(cid:104) x +1 | +[ w (1) x e i Ω t + w (2) x e i Ω t ] | x (cid:105)(cid:104) x | + h.c . Left: k -space Gaussian profile of thetarget state. Center: optimal drive strengths; the optimalstatic hopping is w (0) / Ω = 17 .
09. Note that all w numericallyturn out to be real-valued even though they are allowed to becomplex. Right: Fidelity |(cid:104) ˜ ψ | ψ (cid:105)| between reproduced andtarget micromotions over one period. e − ( k − k ) / σ . On a chain of length L = 100, for example,we can realize this wave packet as a Floquet eigenstate(to a high fidelity of > .
99 at all time) using only on-sitedrives and only two frequencies (i.e. the first and secondharmonic); see Fig. 4. In contrast, the static Hamiltoniannecessary to sustain such a dynamically non-dispersingwave packet, (cid:80) k | k (cid:105) k Ω (cid:104) k | , is spatially highly nonlocal.Note that targeting a different micromotion (e.g., onewith different k and σ ) generally results in a differentoptimal drive. Fidelity with the target state can be fur-ther enhanced with more drive terms such as local hops orhigher harmonic modulations. Additional requirementssuch as spatial smoothness of the drive can be imple-mented by including corresponding penalty terms in ∆. Summary and discussion—
We showed that spatiallyinhomogeneous periodic drives applied to a homogeneousquantum system leads to proliferation of stable compactwave packets travelling through the system at a rate com-mensurate with the drive frequency, T rec = rs T . Theemergence of such wave packets can be understood ina reduced variational subspace (i.e., ansatz), which as-cribes to each frequency only one dominant momentumstate, and from which simple effective models like Eq. 5arise. Such effective models allow us to efficiently reasonabout more complicated drives. For example, in Eq. 6,while keeping the temporal profile as cos(Ω t ), one couldreplace the single site modulation with (cid:80) x v ( x ) | x (cid:105)(cid:104) x | ofan arbitrary—potentially fully random—spatial profile v ( x ), yet the product form g k (cid:48) − k V kk (cid:48) in Eq. 5 automat-ically filters out all but one Fourier component in v ( x ),hence its only effect is to renormalize the drive strength.This implies, among other things, that the resulting wavepackets do not percieve any spatial randomness in thedrive. On the other hand, if one fine-tunes v ( x ) suchthat a particular spatial Fourier component vanishes ex-actly, then the corresponding resonance will be fully sup-pressed. We further demonstrated how more refined con-trol over the resulting wave packets, in the form of repro-ducing a target micromotion, can be achieved throughoptimal drive engineering. The flexibility in controllingthese wave packets, via both effective model reasoningand numerical drive engineering, could open up the pos-sibility of using them as encoding bases, with which quan-tum information can be stored and manipulated. Acknowledgment—
We are grateful to Ian Mondragon-Shem, D. Schuster, V. Manucharyan, A. V. Balatsky,A. Saxena, and D. P. Arovas for useful discussions andfeedbacks. This work was supported by Argonne LDRDProject No. 1007112. ∗ [email protected] † [email protected] ‡ [email protected][1] A. Messiah, Quantum Mechanics: Volume I (1964).[2] M. Nauenberg,
Wave packets: Past and present, in ThePhysics and Chemistry of Wave Packets (Wiley NewYork, 2000) pp. 1–30.[3] A. Buchleitner, D. Delande, and J. Zakrzewski, Physicsreports , 409 (2002).[4] M. V. Berry and N. L. Balazs, American Journal ofPhysics , 264 (1979).[5] E. Schr¨odinger, Collected papers on wave mechanics , Vol.302 (American Mathematical Soc., 2003).[6] P. Knight and A. Miller,
Optical solitons: theory andexperiment , Vol. 10 (Cambridge University Press, 1992).[7] Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. ,763 (1989).[8] A. Buchleitner and D. Delande, Physical review letters , 1487 (1995).[9] M. Holthaus, Chaos, Solitons & Fractals , 1143 (1995).[10] H. Maeda and T. F. Gallagher, Physical review letters , 133004 (2004).[11] M. Kalinski, L. Hansen, and D. Farrelly, Physical reviewletters , 103001 (2005).[12] L. V. Vela-Arevalo and R. F. Fox, Physical Review A ,063403 (2005).[13] K. Sacha, Physical Review A , 033617 (2015).[14] A. Goussev, P. Reck, F. Moser, A. Moro, C. Gorini, andK. Richter, Physical Review A , 013620 (2018).[15] F. Dunning, J. Mestayer, C. O. Reinhold, S. Yoshida,and J. Burgd¨orfer, Journal of Physics B: Atomic, Molec-ular and Optical Physics , 022001 (2009).[16] A. Suleymanzade, A. Anferov, M. Stone, R. K. Naik,A. Oriani, J. Simon, and D. Schuster, Applied PhysicsLetters , 104001 (2020).[17] H. Ren, M. H. Matheny, G. S. MacCabe, J. Luo,H. Pfeifer, M. Mirhosseini, and O. Painter, (2019),arXiv:1910.02873 [quant-ph].[18] R. Kuzmin, N. Mehta, N. Grabon, R. Mencia, and V. E.Manucharyan, npj Quantum Information , 1 (2019).[19] F. Wilczek, Phys. Rev. Lett. , 160401 (2012).[20] P. Bruno, Phys. Rev. Lett. , 070402 (2013).[21] H. Watanabe and M. Oshikawa, Phys. Rev. Lett. ,251603 (2015).[22] D. V. Else, B. Bauer, and C. Nayak, Physical ReviewLetters (2016), 10.1103/physrevlett.117.090402.[23] V. Khemani, A. Lazarides, R. Moessner, and S. L.Sondhi, Physical review letters , 250401 (2016).[24] K. Sacha and J. Zakrzewski, Reports on Progress inPhysics , 016401 (2017).[25] Y. Zhang, J. Gosner, S. M. Girvin, J. Ankerhold,and M. I. Dykman, Physical Review A (2017),10.1103/physreva.96.052124.[26] P. Matus and K. Sacha, Physical Review A (2019),10.1103/physreva.99.033626.[27] A. Pizzi, J. Knolle, and A. Nunnenkamp, (2019),arXiv:1910.07539 [cond-mat.other].[28] X.-L. Qi and D. Ranard, Quantum , 159 (2019).[29] E. Chertkov and B. K. Clark, Physical Review X (2018), 10.1103/physrevx.8.031029.[30] M. Greiter, V. Schnells, and R. Thomale, Phys. Rev. B , 081113 (2018). Supplemental Materials
In this note, we provide details on the analysis of r (cid:54) = s (cid:54) = 1 Floquet wave packets. We first discuss analyticallytractable cases where one of r and s is 1. When neither of them is 1, an effective lattice model can still be derived,although it does not yield to analytical solution, and we discuss its qualitative features. We also briefly discuss thespecial case where the Floquet drive resonates with the undriven energy spectrum close to its inflection point, whichleads to an effective lattice model with a cubic “potential”. Effective model for s > , r = 1 In this section, we discuss the effective model for the s > , r = 1 resonance, where the drive frequency matches s times the typical level spacing, Ω (cid:39) s ∆. A generic undriven energy spectrum can be expanded as ( k ∗ not integer ingeneral) ε k = ε k ∗ + ( k − k ∗ ) Ω s + uT ( k − k ∗ ) + · · · . (8)To leading order, the drive only resonantly couples level k to k ± s . This effectively separates the undriven energyeigenstates into s subspaces according to σ = k mod s . For example, when s = 2, σ = 0 , k , and toleading order, the drive does not mix states of different parity. Writing k = κs + σ , (9)then within each subspace σ , the integer κ plays the role of k in primary resonance, hence we can use an ansatz | ψ σ ( t ) (cid:105) = (cid:88) κ f σκ | ε κ,σ (cid:105) e − iκ Ω t . (10)Note that when s = 1, σ can only be 0, and the ansatz above reduces to that of the primary resonance discussed inthe text. Recall that the Floquet-Schrodinger equation is (cid:104) ˆ H ( t ) − i∂ t (cid:105) | ψ ( t ) (cid:105) = θT | ψ ( t ) (cid:105) , (11)where the time-dependent Hamiltonian isˆ H ( t ) = ˆ H + ˆ V ( t ) , ˆ V ( t ) = (cid:88) a g a ˆ V ( a ) e ia Ω t . (12)Solving Eq. 11 with 10 then yields an eigenvalue equation (cid:88) κ (cid:48) (cid:2) s u ( κ − κ σ ∗ ) δ κκ (cid:48) + g κ (cid:48) − κ V σκκ (cid:48) (cid:3) f σκ (cid:48) = ( θ σ − θ σ ∗ ) f σκ , (13)where V σκκ (cid:48) = (cid:104) ε κ,σ | ˆ V ( κ (cid:48) − κ ) | ε κ (cid:48) ,σ (cid:105) T , κ σ ∗ = k ∗ − σs , θ σ ∗ = ε k ∗ T − πκ σ ∗ . (14)The effective model, Eq. 13, thus consists of s independent “chains”, where σ labels the chains, and κ labels “sites”within each chain. The “onsite potential” is quadratic, u ( κ − κ σ ∗ ) , and each chain has its own quasienergy shift (i.e.,a chain-dependent “chemical potential”) θ σ ∗ .In a large system with L physical sites, V σκκ (cid:48) becomes σ -independent (The leading order correction due to finite L is ∼ L − . E.g., in an open boundary chain, it comes from δq k ∂∂q k | ε k (cid:105) where q k = kπL +1 is the wave vector in an openboundary chain, and δq k = q k +1 − q k ∝ L − ). Hence Eq. 13 for different σ have the same set of eigenvalues { θ σ − θ σ ∗ } .The quasienergies { θ σ } from different chains σ are thus “gapped” from each other by θ σ +1 ∗ − θ σ ∗ = πs , but otherwiseidentical. In other words, the i th quasienergy on “chain” σ is θ σi = θ i + πσs , where θ i is the i th quasienergy on “chain” σ = 0. Thus with the same index i , there are s Floquet eigenstates with different σ labels whose quasienergies areequally spaced by ∆ θ = 2 π/s . The time evolution of an arbitrary linear superposition of these Floquet eigenstatesthus have a recurrence time of 2 πT / ∆ θ = sT . Such recombined states manifestly break the time translation symmetryof the driving Hamiltonian, which is periodic in T , and are thus single particle analogues of discrete time crystals.Let us now discuss the spatial feature of these Floquet eigenstates and their time-crystalline linear recombinations,assuming the undriven states are momentum eigenstates of an open boundary chain, (cid:104) x | ε k (cid:105) = (cid:113) L +1 sin( q k x ) wherethe wave vectors are q k = k πL +1 . A Floquet eigenstate | ψ σ (cid:105) of a given σ (Eq. 10) is a linear combination of momentumstates with the same σ , and are therefore invariant under spatial translation by 2 L/s (with phase shift 2 πσ/s ), wherethe system size L is half the round trip length. To conform with this translation symmetry, | ψ σ (cid:105) for any σ mustconsist of s spatial packets equally spaced along the round trip. To resolve these spatial packets, we Fourier transformthe set of {| ψ σ (cid:105)} states at t = 0, | φ λ (cid:105) = s − (cid:88) σ =0 e − i πλσ/s | ψ σ (0) (cid:105) , λ = 0 , , · · · , s − . (15)As discussed before, in the limit where the level spacing of the quasienergies ∆ θ = 2 π/s is exact (i.e., when (1) weignore higher order effect of the drive that mixes different σ sectors, and (2) V σκκ (cid:48) becomes σ -independent at large L ),any linear combination of {| ψ σ (cid:105)} breaks the discrete time translation symmetry of the driving Hamiltonian. For | φ λ (cid:105) ,we have ˆ U T | φ λ (cid:105) = e − iθ | φ λ +1 (cid:105) = ⇒ ˆ U sT | φ λ (cid:105) = e − isθ | φ λ (cid:105) , (16)where ˆ U sT is dynamical time evolution over s drive periods. In other words, the | φ λ (cid:105) states evolve into each otherafter one T , and recur after sT . Physically, each | φ λ (cid:105) corresponds to a single spatial packet that propagates by 2 L/s after T , and completes a round trip of length 2 L after sT .Numerically, the quasienergy spacing among the s partners {| ψ σ (cid:105)} is ∆ θ = 2 π/s + δ , where a small δ originatesfrom higher order effect of the drive that mixes different σ sectors, as well as the σ dependence in V σκκ (cid:48) . Consequently,the | φ λ (cid:105) states will “tunnel” among the s wave packet configurations over a time scale of 2 πT /δ . Numerically, thetunneling time is typically of the order of thousands of drive periods, and may be extended further via parameter finetuning.When ˆ V ( t ) = 2 g cos Ω t | (cid:105)(cid:104) | , i.e., a modulation on the first site at the fundamental frequency, the effective modelEq. 13 of a given σ becomes a lattice version of harmonic oscillator, s u ( κ − κ σ ∗ ) f σκ + τ ( f σκ − + f σκ +1 ) = ( θ σ − θ σ ∗ ) f σκ , (17)where the parameters u and τ can be estimated using ∂ k ε k ∗ = Ω /s and ∂ k ε k ∗ = 2 s u/T , u = π L (cid:112) T − L /s , τ = 2 gs L T . (18)Here L = L + 1. Note that the effective stiffness is now s u . Parametrizing β s = L sT , γ s = 1 (cid:112) − β s , (19)then similar to the primary resonance case, one can estimate the emergent oscillator “frequency” (cid:36) s and the numberof wave packet solutions (per σ ) D s as (cid:36) s = 2 √ s uτ = 2 π (cid:114) gγ s L , D s (cid:39) (cid:114) τs u = 2 β s sπ (cid:112) gγ s L . (20)These reduce to the primary resonance results of the main text when s = 1. The crossover drive strength g ( s ) c toinduce any s > , r = 1 wave packet solution at all is D s ( g ( s ) c ) = 1 = ⇒ g ( s ) c = s π β s γ s L . (21)Thus one generally needs a stronger drive to induce wave packets of larger s . Effective model for r > , s = 1 The situation with r > s = 1, then it takes r drive quanta at frequency Ω toresonantly connect two adjacent energy levels, as they have a spacing ∼ r Ω. As a result, a “degenerate perturbation”ansatz similar to Eq. 4 in the main text would not work: the Floquet-Schr¨odinger operator simply does not havematrix element between | ε k (cid:105) e − irk Ω t and | ε k +1 (cid:105) e − ir ( k +1)Ω t . In principle, one could attempt to derive an effectivecoupling between these levels via an r th order perturbation theory; this is however technically unwieldy.We instead take an alternate route. We are interested in wave packets which traverse r round trips of an L -sitesystem in a single drive period. Heuristically, this can be “unfolded” into one round trip in a system of length rL —much like how the trajectory of a billiard ball bouncing off the pool table can be “unfolded” into a straight line acrossa repetitive tile of tables. This suggests that a proper ansatz should additionally include eigenstates of the unfoldedsystem, truncated to a segment of length L . These correspond to fractional momentum states | k + ρr (cid:105) in the originalsystem ( ρ = 0 , , · · · , r − (cid:104) x | k + ρr (cid:105) ∝ sin( q k + ρr x + ϕ ), where ϕ is a phaseshift depending on how the shorter system is embedded into the longer one. For drives localized on x = 1, as we willshow, ϕ is such that (cid:104) L + 1 | k + ρr (cid:105) = 0.In the remainder of this section, we first justify the use of fractional momentum states from perturbation theory,and then analyze a generalized ansatz that additionally includes these states. The origin of fractional momentum states
We argued that when the drive frequency matches 1 /r of typical level spacing of a tight binding chain of length L , theansatz for Floquet eigenstates should additionally include fractional momentum states, which are energy eigenstates not of a system of length L , but rather of length rL . We now show that such fractional momentum states do emergeas the leading order correction to undriven Floquet eigenstates (the integer momentum states) when the Floquet driveis treated as a perturbation. From the perspective of variational solutions, thus, the purpose of including fractionalmomentum states in the generalized ansatz is so that the variational subspace remains invariant (to leading order)upon the action of the drive.We first note that the Floquet-Schrodinger operator, K = K + ˆ V ( t ) , K = ˆ H − i∂ t , (22)acts on the tensor product space of the physical Hilbert space and the space of periodic functions. The eigenvectorsof K (which are space-time modes) form a complete basis in this space, | k, a (cid:105) ≡ | ε k (cid:105) e − ia Ω t , K | k, a (cid:105) = ε k − a Ω , (23)where | ε k (cid:105) are eigenstates of the undriven Hamiltonian, ˆ H = (cid:80) L − x =1 | x (cid:105)(cid:104) x + 1 | + h.c. ,ˆ H | ε k (cid:105) = ε k | ε k (cid:105) , ε k = 2 cos q k , (24) (cid:104) x | ε k (cid:105) = (cid:114) L sin( q k x ) , q k = k π L , k, x = 1 , , · · · , L , L = L + 1 . (25)Note that the frequency index a in | k, a (cid:105) is independent of the momentum index k . This is unlike the ansatz weused in the main text, which associates to each momentum index a specific frequency index (e.g., a = k for primaryresonance) — that is, the ansatz amounts to a variational solution in a subspace (of the full tensor product space) inwhich frequency and momentum are correlated.Given an operator Q = Q + Q , and unperturbed basis | n (cid:105) with Q | n (cid:105) = λ n | n (cid:105) , the first order correction to theeigenvectors | n (cid:105) are | n (1) (cid:105) = (cid:88) m (cid:54) = n (cid:104) m | Q | n (cid:105) λ n − λ m | m (cid:105) . (26)Now consider a Floquet drive ˆ V ( t ) = 2 g cos(Ω t )ˆ v , (27)Treating ˆ V ( t ) as a perturbation to K , we obtain the first order correction to the undriven modes as | k, a (1) (cid:105) = g (cid:88) ( k (cid:48) ,a (cid:48) ) (cid:54) =( k,a ) (cid:104) ε k (cid:48) | ˆ v | ε k (cid:105)(cid:104) a (cid:48) | t ) | a (cid:105) ε k − ε k (cid:48) + ( a − a (cid:48) )Ω | k (cid:48) , a (cid:48) (cid:105) = ˇ g (cid:88) ρ = ± | χ k,ρ (cid:105) e − i ( a + ρ )Ω t , | χ k,ρ (cid:105) = (cid:88) k (cid:48) (cid:104) ε k (cid:48) | ˆ v | ε k (cid:105) ( ε k − ρ Ω) − ε k (cid:48) | k (cid:48) (cid:105) , (28)where (cid:104) a (cid:48) | f ( t ) | a (cid:105) = (cid:82) T dtT e i ( a (cid:48) − a )Ω t f ( t ). Since we are considering a near resonance where the drive frequency Ωmatches 1 /r of typical level spacing, ε k − ρ Ω is roughly the interpolation of the dispersion relation ε k = 2 cos q k at afractional momentum κ k,ρ , ε k − ρ Ω (cid:39) ε κ k,ρ , κ k,ρ = k + ρr (29)Let us specialize to ˆ v = | L (cid:105)(cid:104) L | , i.e., a drive on the last site on the chain, instead of the first site (as used in the maintext). This choice is for notational convenience only, and we will comment on what changes if the drive is placed onthe first site later. Using (cid:104) ε k (cid:48) | ˆ v | ε k (cid:105) = L ( − k + k (cid:48) sin q k sin q k (cid:48) , we have | χ k,ρ (cid:105) = ( − k +1 sin q k (cid:34) L ( − k (cid:48) (cid:88) k (cid:48) sin q k (cid:48) ε k (cid:48) − ε κ ρ | k (cid:48) (cid:105) (cid:35) . (30)We now show that | χ k,ρ (cid:105) are indeed proportional to fractional momentum states | κ (cid:105) , which are defined as theinterpolation of the integer momentum states (Eq. 25) to non-integer momentum “index” κ , (cid:104) x | κ (cid:105) = (cid:114) L sin( q κ x ) , q κ = κ π L ∀ κ . (31)The overlap of two such states is (cid:104) κ (cid:48) | κ (cid:105) = I ( κ − κ (cid:48) ) − I ( κ + κ (cid:48) ) , (32) I ( η ) ≡ L L (cid:88) x =1 cos ηπx L = 12 L (cid:104) sin( ηπ ) cot ηπ L − cos( ηπ ) − (cid:105) . (33)Setting κ (cid:48) to integer yields the expansion of | κ (cid:105) in the integer momentum basis, (cid:104) ε k | κ (cid:105) = ( − k L sin( κπ ) sin q k cos q k − cos q κ . (34)Comparing with Eq. 30 and noting that cos q κ = ε κ , we find that indeed | χ κ,ρ (cid:105) are fractional momentum states, | χ k,ρ (cid:105) = ( − k +1 sin q k sin( κ k,ρ π ) | κ k,ρ (cid:105) . (35)The effect of the Floquet drive on the undriven modes | k, a (cid:105) = | ε k (cid:105) e − ia Ω t is thus to bring an integer momentum state | k (cid:105) at frequency a to fractional momenta | k ± r (cid:105) at neighboring frequencies a ± x to L +1 − x ,hence the appropriate fractional momentum states | (cid:101) κ (cid:105) are related to | κ (cid:105) (the ones arising from a last site drive) by (cid:104) x | (cid:101) κ (cid:105) = (cid:104) L + 1 − x | κ (cid:105) . This effectively shifts | (cid:101) κ (cid:105) to a different boundary condition, (cid:104) x | (cid:101) κ (cid:105) ∝ sin( q κ x + ϕ ) where ϕ issuch that (cid:104) L + 1 | (cid:101) κ (cid:105) = 0. Generalized ansatz and effective model
We now discuss the effective model for the r > , s = 1 resonance, where the drive frequency matches a fraction ofthe typical level spacing, Ω (cid:39) ∆ r . From a group velocity consideration, in one drive period, a wave packet consistingof states from this part of the undriven spectrum (assuming it can be stabilized) will undergo r round trips (i.e., 2 rL for an open chain of length L ). Earlier in this section, we argued that the r round trips can be “unfolded” into oneround trip in a system of size rL , hence a proper Floquet ansatz should additionally include fractional momentumstates. We also showed that such fractional momentum states naturally emerge as leading order corrections to theinteger momentum states for the r > | ψ ( t ) (cid:105) = (cid:88) k r − (cid:88) ρ =0 f k,ρ | k + ρr (cid:105) e − i ( rk + ρ )Ω t , (36)where | k + ρr (cid:105) are the fractional momentum states Eq. 31. Note that their average energies do not fall on the dispersioncurve of the integer momentum states. Instead, one has ( κ = k + ρr ) (cid:104) κ | ˆ H | κ (cid:105) = 2 L − (cid:88) x =1 (cid:104) κ | x (cid:105)(cid:104) x + 1 | κ (cid:105) = 2 cos q κ (cid:26) L (cid:20) sin(2 q κ − κπ )sin(2 q κ ) − (cid:21)(cid:27) , (37) (cid:104) κ | κ (cid:105) = 1 − sin κπ L sin q κ cos( κπ − q κ ) , (38)(39)hence the energy of | κ (cid:105) is (cid:104) E (cid:105) κ = (cid:104) κ | ˆ H | κ (cid:105)(cid:104) κ | κ (cid:105) = E κ + µ κ , (40) µ κ = 1 L [cos( q κ − κπ ) − cos q κ )] + O ( L − ) , (41)0where E κ = 2 cos q κ is the the dispersion relation of the integer momentum states, and µ κ is the deviation (cid:104) E (cid:105) κ − E κ .Close to resonance, one can expand the (integer- k ) dispersion relation as ε k = ε ∗ + r ( k − k ∗ )Ω + uT ( k − k ∗ ) + · · · . (42)It is useful to simplify µ κ by replacing, in Eq. 41, q κ → q ∗ (where q ∗ = q k ∗ = k ∗ π/ L is the interpolated wave vectorat the resonance center k ∗ ), and 2 κπ → π ρr , yielding µ ρ = 1 L (cid:104) cos( q ∗ − π ρr ) − cos q ∗ (cid:105) , (43)i.e., the deviation µ ρ depends only on the fractional part ρ . Introduce a composite index j = rk + ρ , (44) j labels the integer momentum states in the unfolded system (length rL ). Then invoking Eq. 11 on Eq. 36 leads tothe following eigenvalue problem, (cid:88) j (cid:48) [ ϑ j δ jj (cid:48) + g j − j (cid:48) V jj (cid:48) ] f j (cid:48) = ( θ − θ ∗ ) f j , (45)where ϑ j = µ ρ + u ( jr − k ∗ ) , V jj (cid:48) = (cid:104) k + ρr | ˆ V ( j (cid:48) − j ) | k (cid:48) + ρ (cid:48) r (cid:105) T , θ ∗ = ε ∗ T − πrk ∗ (46)The effective model is thus a 1D “lattice” with “unit cell” label k and “sublattice” label ρ . The “onsite potential” ϑ remains quadratic, but has an additional sublattice-dependent “chemical potential” µ ρ .Before analyzing the effective model, we first discuss why the apparent recurrence time of the wave packet solutionsfor r > T /r . This behavior can be understood from the form of the ansatz. Note that | ψ ( t ) (cid:105) in Eq. 36 canbe separated into “sublattice” contributions, | ψ ( t ) (cid:105) = (cid:80) ρ | ψ ρ ( t ) (cid:105) , where | ψ ρ ( t ) (cid:105) = (cid:80) k f k,ρ | k + ρr (cid:105) e − ij ( k,ρ )Ω t . Sinceby construction, | ψ ρ ( T /r ) (cid:105) = e − i πρ/r | ψ ρ (0) (cid:105) , each “sublattice” recur after a fraction of drive period Tr , but withdifferent phase shift. Thus even though rigorously speaking the full state | ψ ( t ) (cid:105) does not recur after T /r due to thephase shifts (the exact recurrence time is T ), its spatial pattern does approximately return after T /r .We now analyze the effective model assuming the drive has the form ˆ V ( t ) = 2 g cos(Ω t ) | L (cid:105)(cid:104) L | , that is, a modulationon the last site at the fundamental frequency. The reason to modulate the last (instead of the first) site is to simplifythe expression for the fractional momentum states, see discussion at the end of the last section. Then the drive onlycouples j to j ±
1. The effective model becomes (cid:104) µ ρ + ur (cid:105) ( j − j ∗ ) f ( ρ ) j + τ ( f ( ρ +1) j +1 + f ( ρ − j − ) = λf j , (47)where j ∗ = k ∗ /r , λ = ( θ − θ ∗ ), and we have used the approximation that the “hopping” τ is “site”-independent.Note that we have placed a superscript ρ to the coefficients f j , where ρ = j mod r (Eq. 44), and the superscriptsare understood as carrying an implicit mod r (i.e., ρ ± ρ ±
1) mod r , etc.). Let us nowFourier transform the index j into a continuous conjugate variable y , f ( ρ ) j ≡ (cid:90) dy ˇ f ( ρ ) ( y ) e − i ( j − j ∗ ) y . (48)Note that the transformation is performed as if ρ is independent of j . What this means is that if one were given r continuous functions ˇ f ( ρ ) ( y ), ρ = 0 , , · · · , r −
1, then only Fourier components with j ≡ ρ mod k are relevant assolution to Eq. 47. In terms of ˇ f ( ρ ) , Eq. 47 becomes a coupled Mathieu’s equation, (cid:104) ˆ M − ur ∂ y (cid:105) ˇ f = λ ˇ f , (49)where ˆ M = µ τ e − iy τ e iy τ e iy µ τ e − iy τ e iy µ . . .. . . . . . . . .. . . µ r − τ e − iy τ e − iy τ e iy µ r − , ˇ f = ˇ f (0) ˇ f (1) .........ˇ f ( r − (50)1The general strategy is then to solve Eq. 49 in the diagonal basis of the matrix ˆ M .Since a generic ˆ M cannot be diagonalized analytically, we will specialize to r = 2. In this case, one has r = 2 = ⇒ ˆ M = (cid:18) µ τ cos( y )2 τ cos( y ) µ (cid:19) . (51)Denoting the diagonal bases of ˆ M as ˇ f ± ( y ), then Eq. 49 becomes (cid:34) − ur ∂ y ± (cid:114) ∆ µ τ cos y (cid:35) ˇ f ± ( y ) = ( λ ± − ¯ µ ) f ± ( y ) , ∆ µ = µ − µ , ¯ µ = µ + µ . (52)The problem is equivalent to a particle moving in a periodic potential U ± ( y ) = ± (cid:113) ∆ µ + 4 τ cos y . The Floquetwave packets correspond to bound states in one of the two potentials. Near the bottom of either potential, one mayTaylor expand in y and obtain U + ( y ) (cid:39) ∆ µ τ ∆ µ δy + · · · , U − ( y ) (cid:39) − (cid:114) ∆ µ τ + 2 τ (cid:113) ∆ µ + 4 τ δy + · · · . (53)We expect Floquet wave packet solutions to be low-lying states of the effective lattice model Eq. 47 (this is becauseat higher quasienergies, the “hopping” cannot efficiently mix neighboring “sites”, hence the solutions there are closerto single-momentum states, which are spatially extended). This means at a weak drive strength (and hence small τ ),we should choose U − of the two potential branches, as it has a negative overall shift. The effective model is thus a continuum harmonic oscillator of “Hamiltonian” (cid:101) H = − m − ∂ y + qδy − C , (54)where the “mass” m , the “stiffness” q , and the constant shift C are m − = ur , q = 2 τ C , C = (cid:114) ∆ µ τ . (55)The parameters u, τ , and ∆ µ can be estimated as follows. Parametrizing β r = r L T , γ r = 1 (cid:112) − β r , (56)then from ∂ k ε k | k = k ∗ = r Ω and uT = ∂ k ε k | k = k ∗ , we have u = r π β r γ r T . (57)For r = 2, from Eq. 46, we can estimate τ as τ = V j ∗ ,j ∗ +1 = gT (cid:104) k ∗ | L (cid:105)(cid:104) L | k ∗ + 12 (cid:105) = 4 gγ r . (58)Finally, using Eq. 43, we have ∆ µ = µ − µ = 2 T L cos q ∗ = 4 β r γ r . (59)The “frequency” of the emergent harmonic oscillator, Eq. 54, is then (cid:36) = 2 (cid:112) m − q = 8 πgγ r (cid:113) r L (cid:112) g β r . (60)Note that at weak drive, (cid:36) ∝ g . As the drive becomes stronger, (cid:36) ∝ √ g . This is different from the r = 1 cases (witharbitrary s ), where (cid:36) ∝ √ g even at weak drive, see Eq. 20.2 FIG. 5. Non-dispersing wave packets with s = 2 , r = 3 from Hamiltonian ˆ H ( t ) = (cid:80) L − x =1 | x (cid:105)(cid:104) x + 1 | + h.c. + 2 g cos(Ω t ) | (cid:105)(cid:104) | ,with system size L = 500, drive strength g = 1, and drive period T = 1005 (Ω = 2 π/T ). Top: “ground states” of the twoparity effective chains ( σ = 0 , π + δ apart, and numerically δ (cid:39) . × − π . Bottom: Dynamicalevolution of the time-crystalline recombination | φ (cid:105) = | ψ (0) (cid:105) + | ψ (0) (cid:105) . After a tunneling time of 2 πT /δ (cid:39) . × T , itwould evolve into the wave packet configuration of the opposite recombination | ψ (0) (cid:105) − | ψ (1) (cid:105) . General r (cid:54) = s (cid:54) = 1 wave packets We briefly discuss the more general case of r (cid:54) = s (cid:54) = 1. In this case, we can combine the two ansatze above andwrite | ψ σ ( t ) (cid:105) = (cid:88) κ,ρ | k ( κ, ρ, σ ) (cid:105) e − i ( rκ + ρ )Ω t , (61)where k ( κ, ρ, σ ) is a potentially fractional momentum, k ( κ, ρ, σ ) = s ( κ + ρr ) + σ , (62)and κ, ρ, σ are integers, with ρ = 0 , , · · · , r and σ = 0 , , · · · , σ . Thus invoking Eq. 11 on this ansatz will yield aneffective lattice model of s decoupled chains (labeled by σ ), each with r sublattices (labeled by ρ ). Note that eachchain (i.e., a specific σ ) can be analyzed in the same way as the s = 1 , r > j in Eq. 44 is now j = rκ + ρ (i.e., replace k there by κ ). Similar to the r = 1 case, the “onsite” energies of the s chains have an equalspacing of 2 π/s to leading order (with higher order corrections arising from the coupling between different σ sectors),but otherwise essentially identical, hence a given ( s, r ) solution is necessarily one of s partners with almost identicalspatial-temporal patterns, and their quasienergies are equally spaced by ∆ θ = 2 π/s to leading order. Combining theresults of s = 1 and r = 1, one can see that at s (cid:54) = r (cid:54) = 1, an ( s, r ) Floquet eigenstate consists of s wave packets,each completing a fraction rs of round trip in one drive period. The individual wave packets can be resolved by linearrecombinations of the s partners, similar to Eq. 15, hence their true recurrence time is 2 π/ ∆ θ = sT . However, sincethey completed r round trips in sT , their apparent recurrence time is T rec = sT /r . In Fig. 5, we plot the “groundstates” of the two independent effective chains for s = 2 , r = 3, and their time-crystalline recombination. The lattercompletes r = 3 round trips in s = 2 drive periods.3 Emergent lattice model with cubic potential