Long-range doublon transfer in a dimer chain induced by topology and ac fields
LLong-range doublon transfer in a dimer chain induced by topology and ac fields
M. Bello, ∗ C.E. Creffield, and G. Platero Instituto de Ciencias de Materiales, CSIC, Cantoblanco, E-28049, Madrid, Spain Departamento de F´ısica de Materiales, Universidad Complutense de Madrid, E-28040, Madrid, Spain (Dated: November 6, 2018)The controlled transfer of particles from one site of a spatial lattice to another is essential formany tasks in quantum information processing and quantum communication. In this work westudy how to induce long-range transfer between the two ends of a dimer chain, by coupling statesthat are localized just on the chain’s end-points. This has the appealing feature that the transferoccurs only between the end-points – the particle does not pass through the intermediate sites –making the transfer less susceptible to decoherence. We first show how a repulsively bound-pairof fermions, known as a doublon, can be transferred from one end of the chain to the other viatopological edge states. We then show how non-topological surface states of the familiar Shockleyor Tamm type can be used to produce a similar form of transfer under the action of a periodicdriving potential. Finally we show that combining these effects can produce transfer by means ofmore exotic topological effects, in which the driving field can be used to switch the topologicalcharacter of the edge states, as measured by the Zak phase. Our results demonstrate how to inducelong range transfer of strongly correlated particles by tuning both topology and driving.
Recent experimental advances have provided reliableand tunable setups to test and explore the quantum me-chanical world. Paradigmatic examples are ultracoldatomic gases trapped in optical lattices and coherentsemiconductor devices such as quantum dots. Much ofthe interest in the last few years has been focused on thelong-range transfer of particles in these systems, bearingin mind potential applications in the fields of quantuminformation and quantum computing. Several mecha-nisms have been proposed to achieve this aim, includingpropagation along spin chains [1] or a bipartite lattice [2],coherent transport by adiabatic passage (CTAP) [3–6], orthe virtual occupation of intermediate states [7–9]. Har-nessing the effects of topology has also recently becomepossible, in which edge states provide lossless transportthat is protected against disorder. Key to the produc-tion of these topological insulators has been the use oftime-dependent potentials to engineer the tunnelings inthese lattice systems. This has allowed the production ofquantum Hall states [10] [11], and more exotic topologicalsystems such as the Haldane model [12]. It has also beenshown that driving graphene with ac electric fields canbe used to induce a semimetal insulator transition [13].Inspired by these developments, in this work we studyhow the long-range transfer of particles can be achievedby combining these ingredients; topological effects andperiodic driving.Probably the most simple system that can exhibittopological effects is the one-dimensional dimer chain, orone-dimensional Su-Schrieffer-Heeger (SSH) model, orig-inally introduced to describe solitonic effects in polymers[14] [15]. Such a dimer chain supports edge states whenit is in the topologically non-trivial phase. This is de-termined by the ratio between the two hopping rates, J and J (cid:48) , a parameter we will call λ = J (cid:48) /J [14, 16, 17].Recently, many investigations have focused on this model and several results have been confirmed experimentallyusing ultracold atoms trapped in optical lattices [18].Since these edge states form a non-local two-level system,a remarkable dynamics can occur for non-interacting par-ticles moving on such a chain; they can directly passfrom one end to the other without moving through theintermediate sites [19]. This direct transfer of particlesbetween distant sites, which preserves the quantum co-herence of the state, clearly has applications to quantuminformation processing, in which quantum states must becoherently shuttled between quantum gates and registers.In this work we investigate how this long-range trans-fer of particles in a dimer chain can be produced andoptimized in systems of strongly interacting fermions.In general, interactions are known to destroy the topo-logical effects in the non-interacting dimer chain [19].However, by considering the strongly-interacting limit, inwhich fermions form repulsively-bound pairs called “dou-blons” [20–22], we show that the effect can be recoveredby tuning local potentials at the end-points of the lat-tice. We further show that driving the system with ahigh-frequency potential allows the manipulation of thedoublon tunneling rates via the phenomenon known ascoherent destruction of tunneling [23], permitting an al-ternative form of long-range transfer to occur via a non-topological mechanism that we term “Shockley transfer”.Finally we show how combining lattice topology with thedriving potential gives rise to transfer via exotic topolog-ical effects, giving extremely fine control over process oflong-range doublon transfer. a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a r FIG. 1.
Schematic representations a) The full Hamiltonian (1). A chain of M dimers characterized by two hopping rates, J and J (cid:48) , the lattice constant, a , and the intra-dimer distance, b . The other important parameter of the model is the interactionstrength, U , which needs to be large enough with respect to the hoppings for doublons to form. b) Scheme displaying the mainfeatures of the effective model Hamiltonian (2). We consider the doublon as a single quasiparticle which moves through thelattice with hoppings J (cid:48) eff = 2 J (cid:48) /U and J eff = 2 J /U . In a finite system, a chemical potential difference arises between theendpoints and the rest of the lattice sites. c) Scheme showing how to produce topological long-range transfer of doublons. Agate potential at the terminating sites of the chain is needed to restore the lattice periodicity. d) Scheme showing the Shockleylong-range transfer of doublons. The ac-field renormalizes the doublon hoppings to J ( y ) J eff and J ( x ) J (cid:48) eff , where x = Eb ω and y = E ( a − b ) ω , leaving the chemical potential unaffected. RESULTSDoublon dynamics
In the limit of strong interactions, fermions on a lat-tice can pair to form stable bound states known as “dou-blons”, even if the interaction is repulsive. This effect isa consequence of the discretization of space; the kineticenergy of a particle is limited by the width of the energyband, and so if the interaction energy is sufficiently large,the decay of the doublon into free particles is forbiddenon energetic grounds. Doublons have been observed inseveral systems such as ultracold atomic gases [20] andin organic salts [24].The system we have studied can be modelled by a SSH-Hubbard Hamiltonian: H = − J (cid:48) M (cid:88) i =1 ,σ c † i σ c i − σ − J M − (cid:88) i =1 ,σ c † i +1 σ c i σ + H.c. + U M (cid:88) i =1 n i ↑ n i ↓ = H J + H U , (1)where c † i σ ( c i σ ) is the standard creation (annihilation)operator for a fermion of spin σ on site i , and n i σ = c † i σ c i σ is the number operator. The hopping Hamilto- nian H J is parameterized by the two hopping parameters J and J (cid:48) which describe the dimer structure of the lattice(shown schematically in Fig. 1a), while H U accounts forthe interactions between particles by a Hubbard- U term.We study the two fermion case, the smallest number offermions that can form a doublon, and restrict ourselvesto the singlet subspace (one up-spin and one down-spin).In order to obtain an effective Hamiltonian that accu-rately models the dynamics of doublons, we can performa unitary transformation perturbatively in powers of J/U and J (cid:48) /U [19, 25] (see Methods). Assuming we only haveone doublon in the system, we can neglect interactionterms between doublons and the hopping processes ofsingle particles, to arrive at H eff = J (cid:48) eff M (cid:88) i =1 d † i d i − + J eff M − (cid:88) i =1 d † i +1 d i + H.c. + M (cid:88) i =1 µ i n di , (2) µ i = (cid:26) µ bulk = J (cid:48) eff + J eff + U if 1 < i < Mµ edge = J (cid:48) eff + U if i ∈ { , M } (3)Here J eff = 2 J /U and J (cid:48) eff = 2 J (cid:48) /U . d † i = c † i ↑ c † i ↓ , ( d i )is the creation (annihilation) operator for a doublon on E n / J e ff a) b)02468 0 1 2 E n / J e ff λ c) 0 1 2 λ d) FIG. 2.
Energy levels of a 10 dimer chain , b = a / J eff . a) Without a gatepotential, ∆ µ = − J /U , there are no states outside thebulk bands, and therefore no edge states for any value of λ . The interaction destroys the edge states as long as thesystem is in the strongly-interacting regime and no ac fieldis applied. b) Adding a gate potential to compensate for∆ µ , so that ∆ µ + µ g ate = 0. The two states with ener-gies in the gap (red lines) for λ < λ c are the edge statespredicted by the topology of the system. For λ ≥ λ c thegap closes, and the system becomes topologically trivial. c)Driven by an ac-field with intensity and frequency such that J ( Ea /ω ) = 0 .
3, two localized states separate from the bot-tom of the lowest band (red lines). We can see how the acfield renormalizes the hoppings, making the bands narrowerand increasing their separation from the Shockley states. d)Same case with J ( Ea /ω ) = 0 . site i , and n di = d † i d i is the doublon number operator.In the effective model (2), the doublon hopping ratesare smaller than the original ones and positive regard-less of the sign of J or J (cid:48) (see Fig. 1b). Unexpectedly,this transformation also gives rise to a chemical potentialterm, µ i , which depends on the number of neighbors ofsite i . While all sites in the bulk of the chain have twoneighbors, the two end-sites have only one and thus ex-perience a different value of µ i , which breaks the latticeperiodicity. The difference in chemical potential is givenby ∆ µ = µ edge − µ bulk = − J U . In Fig. 2a we showthe energy spectrum for a chain of 10 dimers, where wecan clearly see that even when the system is topologi-cally non-trivial ( λ < λ c , see below), no edge states arevisible. This is a consequence of this finite-size effect;the alteration in chemical potential at the ends of thelattice causes the edge states’ energies to enter the bulkbands. As a consequence the system does not supportedge states for doublons. This corroborates the resultthat interactions destroy topological transfer. Topological transfer
Analogously to the non-interacting case, the topologyin the present case is determined by the ratio betweenthe effective hoppings, given by J (cid:48) eff /J eff = ( J (cid:48) /J ) = λ .For an infinite chain, the system is in the topologicallynon-trivial phase when λ < λ c = 1; for the finite caseof M dimers the critical value of the ratio is given by λ c = (cid:113) − M +1 [16].To obtain topological transfer for doublons, we mustrestore the lattice periodicity by adding a gate voltage, µ gate , to the edge sites to compensate for the differencein chemical potential, such that ∆ µ + µ gate = 0. In thisway we recover edge states for a chain with doublons.We show the result in Fig. 2b, and we can indeed seethat the two edge states lie between the bulk bands for λ < λ c .We show examples of the dynamics in Figs. 3a and 3b;in the topological regime the doublon oscillates betweenthe two edge-sites without passing through intermediatesites, whereas in the trivial regime the doublon simplyspreads over the entire lattice. Interestingly, due to thesublattice symmetry of the system, when the number ofsites is odd, there is one and only one edge state in thechain, localized on one end or the other depending onwhether λ < λ c or λ > λ c [19]. Thus, there is no long-range doublon transfer for systems with an odd numberof sites (half-integer number of dimers). Shockley transfer
The effective Hamiltonian for doublons (2) contains,as we discussed above, a site-dependent chemical poten-tial which breaks translational symmetry. This producesShockley-like surface states [26] if the hopping rates J eff and J (cid:48) eff are smaller than | ∆ µ | . Usually this is not thecase, however there is an efficient way to induce suchstates by driving the system with a high-frequency ac-field. The ac-field renormalizes the hoppings [23] whichbecome smaller than in the undriven case. This cannot beachieved, for example, by simply reducing the hoppings J and J (cid:48) by hand, since this will also affect the effectivechemical potential which still will be of the same orderof J eff and J (cid:48) eff . To model the driven system we add aperiodically oscillating potential that rises linearly alongthe lattice H ( t ) = H J + H U + E cos ωt M (cid:88) i =1 x i ( n i, ↑ + n i, ↓ ) (cid:39) H eff + E cos ωt M (cid:88) i =1 x i ( n i, ↑ + n i, ↓ ) , (4)where E and ω are the amplitude and frequency ofthe driving, and x i is the spatial coordinate along the · T i m e [ J − ] · T i m e [ J − ] · T i m e [ J − ] Lattice site 010 FIG. 3.
Time evolution of the site occupation.
Inall cases U = 16 J and the initial condition consists of twofermions in a singlet state occupying the first site of the chain.The simulations are for a chain containing 5 dimers except for(d). Topological transfer : a) Chain with compensating gatepotentials at the edge-sites and λ = 0 . < λ c (topologicalregime). The doublon oscillates from one edge to the otherwithout occupying intermediate sites, giving an example oflong-range topological transfer in an interacting system. b)As before, but with λ = 1 > λ c (trivial regime). The doublonnow simply spreads over the entire lattice. Shockley trans-fer : c) Chain driven by an ac field with parameters λ = 1, b = a / E/ω = 1 . /a and ω = 2 J . By using the ac field torenormalise the effective hoppings, we can obtain long-rangetransfer without compensating the chemical potentials of theedge points. d) AC driven chain with same parameters, butan odd number of sites. Long-range transfer is mediated bythe Shockley mechanism (no topological transfer would bepossible in this case). AC induced topological transfer : e) ACdriven system with compensating gate potentials, parametersare λ = 1 . b = 0 . a , 2 E/ω = 3 . /a and ω = 2 J (topo-logical regime, red square in Fig. 5b). Long range transferoccurs unlike in the undriven system ( λ = 1 . E/ω = 2 a − (trivial regime, green dot in Fig. 5b).As expected, no long range transfer occurs. chain. Since the Hamiltonian (4) is periodic in time, H ( t ) = H ( t + T ), we can apply Floquet theory andseek solutions of the Schr¨odinger equation of the form | ψ ( t ) (cid:105) = e − i(cid:15) n t | φ n ( t ) (cid:105) , where (cid:15) n are the so called Floquetquasienergies, and | φ n ( t ) (cid:105) are a set of T -periodic func-tions termed Floquet states. Quasienergies play the samerole in the time evolution of the system as conventionalenergies do for a static Hamiltonian. In the strongly in-teracting regime, a perturbative calculation shows thatthe hopping terms are renormalized by the zeroth Besselfunction (see Methods and [19]), J eff → J ( y ) J eff , and J (cid:48) eff → J ( x ) J (cid:48) eff , where y = E ( a − b ) ω and x = Eb ω [17, 21]. We show the effect of this renormalizationin Fig. 2c; as the effective tunneling reduces in magni-tude the bulk bands becomes narrower, and the Shock-ley states are pulled further out of them. The factorof 2 in the argument of J comes from the doublon’stwofold electric charge. The geometry, which so far hasnot played any role, now becomes important in this renor-malization of the hoppings. The simplest case is for b = a /
2, in which both hoppings are renormalized bythe same factor J ( Ea /ω ). An important point is thatthe on-site effective chemical potential, being a local op-erator, commutes with the periodic driving potential, andso is not renormalized. This is the critical reason for usinga periodic driving to modify the tunneling; it renormal-izes the values of J eff while keeping the chemical potentialunchanged.We show in Figs. 2c,2d how varying the hopping rateshas the effect of pulling two energies out of the bulkbands, inducing the presence of localized states at theedges. These edge states occur in pairs and so also forma non-local two level system [26]. Nevertheless they canbe affected by local perturbations and so unlike the pre-vious case, are topologically unprotected [27]. From thestationary eigenstates of Hamiltonian (2) with renormal-ized hoppings, we can define a quantity, G [ | ψ (cid:105) ], that mea-sures the density correlation between the end-sites for agiven eigenstate, | ψ (cid:105) , G [ | ψ (cid:105) ] := |(cid:104) | ψ (cid:105)(cid:104) N | ψ (cid:105)| , G ∈ [0 , / . (5)Here | i (cid:105) = d † i | (cid:105) is the basis of localized doublon states.If the total occupancy at the ends, |(cid:104) | ψ (cid:105)| + |(cid:104) N | ψ (cid:105)| , isa constant then G is maximum when |(cid:104) | ψ (cid:105)| = |(cid:104) N | ψ (cid:105)| .In addition, the energy difference between the two edgestates tells us how fast the doublon transfer time is, T = π/ ∆ (cid:15) . We can see in Fig. 4a that when the valuesof the hoppings are reduced by the ac field, the two lowestenergy eigenstates of Hamiltonian (2) become more local-ized at the edges. Smaller values of λ favor localizationas well. This produces cleaner dynamics with less un-wanted occupancy of the intermediate sites of the chain.On the other hand, we can see in Fig. 4b that the trans-fer time rapidly increases, soon becoming too large toobserve in simulations or in experiment. At larger values | J ( E a / ω ) | λ | J ( E a / ω ) | λ λ λ G [ | ψ (cid:105) ] a)a) > ( T J ) b)b) FIG. 4.
Characterizing Shockley transfer. a) Correla-tion, G , between the edge occupancy of the Shockley-like sur-face states in an ac-driven chain containing 5 dimers. We haveconsidered the case b = a /
2. The long-range transfer oc-curs in the pale region (lower-left) of the parameter space. b)Transfer time, T , computed as π/ ∆ (cid:15) , where ∆ (cid:15) is the energydifference between the two edge states. T tends to infinityas J ( Ea /ω ) or λ go to zero. The black circles correspondto the parameters of the time evolution shown in Fig. 3c; thetransfer time, T ∼ J − , is correctly reproduced. As canbe seen, a slight change in the field parameters can changethe transfer time by several orders of magnitude. of the hoppings the edge states enter the bulk bands, ascan be seen in Fig.2d close to λ = 1, and the long-rangetransfer of doublons is suppressed.We show examples of the dynamics for a periodically-driven system in Figs. 3c, 3d. Since the origin of the edgestates is not topological, long-range transfer can occur viathis mechanism for chains even with an odd number ofsites, as seen in Fig. 3d. AC induced topological transfer
If we combine both methods, adding a gate potentialat the ends and driving the system with an ac-field, itis possible to bring the system into exotic topologicalphases. The effective Hamiltonian is simply given by(11) without the chemical potential term.There is a close connection between the correlation ofthe edge-occupancy, G , for those states which close thegap, and the Zak phase, Z [28]. This is the topologicalinvariant that classifies 1D Hamiltonians with time re-versal, particle-hole and chiral symmetry. The Zak phasehas already been calculated for a driven dimer chain with-out interaction [17], and it is straightforward to extendit to our effective model for doublons Z = π (cid:2) (cid:0) J ( y ) − λ J ( x ) (cid:1)(cid:3) . (6)The argument of the Bessel functions is twice that fora non-interacting system, and the factor λ comes fromthe squared ratio between the effective doublon hoppings.In Figs. 5a, 5b we compare the phase diagram obtained by plotting (6) and the result obtained by computing G [ | ψ (cid:105) ]. We can see that the agreement is excellent, indi-cating that the Zak phase can be directly measured fromthe density correlation function. In Fig. 5c we show thequasienergy spectrum for b = 0 . a , to make a cross-section through the parameter space. It can clearly beseen that when the system is topologically non-trivial,corresponding to G (cid:39) .
5, a pair of edge states emergesfrom the bulk bands and enters the gap. When the sys-tem is topologically trivial they then reenter the bulkagain. The dots in Fig. 5c were obtained from the di-agonalization of the unitary time-evolution operator forone period of the full original Hamiltonian (4) with anadded gate potential, making no approximations. Theagreement between these quasienergies, and those calcu-lated from the effective model (11) is extremely good fordriving parameters 2 Ea /ω ≤
10, indicating that our ap-proximation schemes are valid. For larger values of thedriving parameters our effective model still captures thebehaviour of the quasienergies, but small deviations be-gin to appear as the doublon states begin to couple withother states of the system.In Figs. 3e and 3f we show two examples of the dy-namics corresponding to the two points marked in Fig.5b. When the system is topologically non-trivial (redsquare) the system exhibits long-range doublon transferas expected. In the topologically trivial regime (greendot), however, this does not occur, and the doublon in-stead propagates throughout the whole lattice.
DISCUSSION
We have derived an effective Hamiltonian for two par-ticles in a quantum dimer chain that bind together viaa repulsive interaction. Interestingly this binding pro-duces an effective surface potential, different from thatof the bulk. In general this surface potential preventstopological transfer of particles, but by adding local gatepotentials to compensate for it, topological transfer canbe recovered. We have also shown that by adding a peri-odic driving potential to renormalize the hoppings, whileleaving the surface potential unchanged, we can producelong-range transfer via Shockley states. This transfer, is,however, not topologically protected. Finally, by combin-ing topological transfer with an ac driving field, we canobtain a rich topological phase diagram, in which long-range transfer occurs when the Zak phase is non-zero.It is natural to ask how this long-range transfer phe-nomena depend on the total number of dimers formingthe chain. The transfer time is essentially the inverse ofthe energy difference in the two-level system formed bythe hybridization of the edge states. This energy differ-ence is related to the overlap between the edge states,which are solutions that decay exponentially from thesurface of the lattice. Thus we can conclude that the E a / ω b ( a ) 0 0.5 1 b ( a )0 0.5 1 b ( a )0 0.5 1 b ( a )0 5 10 15 20 (cid:15) n [ J ] Ea /ω a) 00.5 G [ | ψ (cid:105) ] b)b)b) c) FIG. 5.
Exotic topological transfer. a) Plot of Z (Eq. 6)for λ = 1 .
2; green regions are those in which the system is inthe topologically non-trivial phase ( Z = π ). b) Plot of G [ | ψ (cid:105) ]computed for a chain containing 7 dimers with the same λ asin a). The black lines that cross the white regions correspondto the zeros of J (2 Eb /ω ). At those points in parameterspace, the different sites of the chain are uncoupled; the sys-tem is thus degenerate and G is not a well-defined quantity.The yellow line marks the cross-section through parametersspace used in the quasienergy plot. The red square and greendot mark the parameters of the system for the time evolutionsshown in Figs. 3e and f respectively. c) Quasienergies as afunction of the driving amplitude E . The other parametersare set to U = 16 J , b = 0 . a and ω = 2 J . The edge statesappear as predicted by the phase diagram, and correspond tothe highest values of G . For large values of the ac-field inten-sity, the doublon states begin to couple with the other statesof the system and the exact quasienergies ( black dots) divergefrom those predicted by the effective model (blue lines). transfer time increases exponentially when increasing thesize of the chain. Another fact which affects the transfertime is the dependence of J (cid:48) eff and J eff on U . Increasingthe interaction strength has the effect of slowing downthe dynamics.Ultracold atoms confined in optical lattice potentialare extremely clean and only slightly affected by deco-herence. In units of the tunneling time, doublon life timein a three dimensional optical lattice, has been found todepend exponentially on the ratio of the on-site inter-action to the kinetic energy [29]. This is not in generalthe case for electron transfer in semiconductor nanos-tructures where hyperfine or spin-orbit orbit interactionsinduce decoherence, which strongly depends on the mate-rial. However, since we deal with doublons, forming a sin-glet state, spin relaxation and decoherence is suppressedby the energy difference between the intradot singlet and excited triplet states.In summary, we propose three ways for long-rangetransfer of strongly-interacting particles, all mediated byedge states. In the first case, non-trivial topological edgestates are required. In the second, long-range transportis mediated by Shockley states induced by ac driving. Fi-nally, combining both topology and driving allows us totune the range of parameters where long-range transferis achieved. Our proposal could be experimentally con-firmed both in cold atoms and in semiconductor quantumdot arrays. In these last systems either charge detectionby means of a quantum detector, such as a quantum pointcontact or an additional quantum dot, or transport mea-surements are within experimental reach.Our results open new avenues to achieve direct transferof interacting particles between distant sites, an impor-tant issue for quantum information architectures. METHODSEffective Hamiltonian for doublons
The energies of a one-dimensional lattice form a Blochband with a width of 2 J , and thus the maximum kineticenergy carried by two free particles is 4 J . If the particlesare initially prepared in a state with a potential energymuch greater than 4 J , the initial state then cannot de-cay without the mediation of dissipative processes. Weconsider the regime where for doublons to split is ener-getically unfavorable, i.e. U (cid:29) J, J (cid:48) . Following [25] weobtain an effective Hamiltonian just for the doublons in adimer chain by means of a Schrieffer-Wolff (SW) transfor-mation, projecting out the single-occupancy states. Thistransformation is performed perturbatively in powers of
J/U and J (cid:48) /U and up to second order gives rise to theeffective Hamiltonian (2) where the hoppings J and J (cid:48) become renormalized by the interaction. AC driven Hamiltonian: hopping renormalization
If one wants to deal with interactions between particlesas well as interactions with an external driving, and treatboth on an equal footing, a more elaborate procedurethan before is necessary. For a time-periodic Hamiltonian H ( t + T ) = H ( t ) with T = 2 π/ω , the Floquet theoremstates that the time evolution operator U ( t , t ) can bewritten as: U ( t , t ) = e − iK eff ( t ) e − iH eff ( t − t ) e iK eff ( t ) (7)with a time-independent effective Hamiltonian, H eff governing the slow dynamics and a T -periodic op-erator K eff ( t ) that accounts for the fast dynamics;exp( − iK eff ( t )) is also termed the micromotion operator .In the high-frequency limit, by which we mean ω (cid:29) J (cid:48) eff and J eff , these operators can be expanded in powers of1 /ω : H eff = ∞ (cid:88) n =0 H [ n ]eff , H [ n ]eff ∝ (cid:18) ω (cid:19) n (8)For a detailed description of the high-frequency expan-sion (HFE) method see [30–32]. Now we express our pe-riodic Hamiltonian (4) in the rotating frame with respectto both the interaction and the ac field [33]: H int ( t ) = U † ( t ) H ( t ) U ( t ) − i U † ( t ) ∂ t U ( t ) , U ( t ) = exp (cid:0) − iH U t − i (cid:82) H AC ( t ) dt (cid:1) . (9)Here (cid:82) H AC ( t ) dt is just the antiderivative of the operator: H AC ( t ) = E cos ωt M (cid:88) i =1 x i ( n i, ↑ + n i, ↓ ) . (10)To derive the effective Hamiltonian, we perform the HFEof H int ( t ) up to first order in 1 /ω , see [19]. It can be seenthat in the limit U (cid:29) ω the result is the same as theone obtained by first performing the SW transformationand then the hoppings renormalization. Conversely, inthe limit ω (cid:29) U the result is consistent with first doingthe high-frequency hopping renormalization and then theSW transformation. This coincidence can be understoodsince the great difference between U and ω , permits theseparation of the different time scales associated witheach energy in the HFE. The effective Hamiltonians inthe two different regimes are: H U (cid:29) ω eff = J ( x ) J (cid:48) eff M (cid:88) i =1 d † i d i − + J ( y ) J (cid:48) eff M − (cid:88) i =1 d † i +1 d i + H.c. + M (cid:88) i =1 µ i n di .µ i = (cid:26) J (cid:48) eff + J eff + U if 1 < i < MJ (cid:48) eff + U if i ∈ { , M } (11) H ω (cid:29) U eff = J ( x/ J (cid:48) eff M (cid:88) i =1 d † i d i − + J ( y/ J (cid:48) eff M − (cid:88) i =1 d † i +1 d i + H.c. + M (cid:88) i =1 µ i n di .µ i = (cid:26) J ( x/ J (cid:48) eff + J ( y/ J eff + U if 1 < i < M J ( x/ J (cid:48) eff + U if i ∈ { , M } (12)We emphasize that only in the regime U (cid:29) ω the effect ofthe ac field is to renormalize the hopping parameters butnot the effective chemical potential. This is a nontrivialpoint, key to the understanding of the Shockley transferphenomenon. ACKNOWLEDGEMENTS
We acknowledge F. Hofmann for enlightening discus-sions. This work was supported by the Spanish Min-istry through Grants No. MAT2014-58241-P. and No.FIS2013-41716-P. ∗ [email protected][1] Sougato Bose, Quantum communication through an un-modulated spin chain , Phys. Rev. Lett. , 207901(2003).[2] C.E. Creffield, Quantum control and entanglement us-ing periodic driving fields , Phys. Rev. Lett. , 110501(2007).[3] T. Morgan et al. , Coherent transport by adiabatic passageon atom chips , Phys. Rev. A. ,053618 (2013)[4] A. D. Greentree, J. H. Cole, A. R. Hamilton and L. C.Hollenberg, Coherent electronic transfer in quantum dotsystems using adiabatic passage , Phys. Rev. B. ,235317(2004)[5] C. J. Bradly, M. Rab, A. D. Greentree and A. M. Martin, Coherent tunneling via adiabatic passage in a three-wellBose-Hubbard system , Phys. Rev. A. ,053609 (2012)[6] J. Gillet, A. Benseny and T. Busch, Spatial AdiabaticPassage for Interacting Particles , arXiv:1505.03982v1[quant-ph], 15 may 2015.[7] M. Busl et al. , Bipolar spin blockade and coherent spinsuperpositions in a triple quantum dot , Nature Nanotech-nology , 261 (2013).[8] F. R. Braakman, P. Barthelemy, C. Reichl, W. Wegschei-der, and L. M. K. Vandersypen, Long-distance coherentcoupling in a quantum dot array , Nature Nanotechnology , 432 (2013)[9] R. S´anchez et. al. , Long-range spin transfer in triplequantum dots , Phys. Rev. Lett. ,176803 (2014)[10] M. Aidelsburger, M. Atala, M. Lohse, J.T. Barreiro,B. Paredes, and I. Bloch,
Realization of the HofstadterHamiltonian with Ultracold Atoms in Optical Lattices ,Phys. Rev. Lett. , 185301 (2013);[11] H. Miyake, G.A. Siviloglou, C.J. Kennedy, W.C. Burton,and W. Ketterle,
Realizing the Harper Hamiltonian withLaser-Assisted Tunneling in Optical Lattices , Phys. Rev.Lett. , 185302 (2013).[12] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat,T. Uehlinger, D. Greif, and T. Esslinger,
Experimentalrealization of the topological Haldane model with ultra-cold fermions , Nature , 237240 (2014).[13] P. Delplace, A . G´omez-Le´on, and G. Platero,
Mergingof Dirac points and Floquet topological transitions in ac-driven graphene , Phys. Rev. B , 245422 (2013).[14] W. P. Shu, J. R. Schrieffer and A. J. Heeger, Solitons inpolyacetylene , Phys. Rev. Lett. ,1698 (1979)[15] Y. Z. Zhang et al. , Bond order wave and energy gap ina 1D SSH-Hubbard model of conjugated polymers , Synt.Met. 135-136 (2003) 449-450[16] P. Delplace, D. Ullmo and G. Montambaux,
Zak phaseand the existence of edge states in graphene , Phys. Rev.B. ,195452 (2011) [17] A. G´omez-Le´on and G. Platero, Floquet-Bloch theory andtopology in periodically driven lattices , Phys. Rev. Lett. ,200405 (2013).[18] Marcos Atala et al. , Direct measurement of the Zak phasein topological Bloch bands , Nature 10.1038 (2013)[19] See Supplementary Information.[20] K. Winkler et al , Repulsively bound atom pairs in an op-tical lattice , Nature 10.1038 (2006).[21] C. E. Creffield and G. Platero,
Localization of two inter-acting electrons in quantum dot arrays driven by an acfield , Phys. Rev. B. ,165312 (2004)[22] C. E. Creffield and G. Platero, Coherent control of in-teracting particles using dynamical and Aharonov-BohmPhases , Phys. Rev. Lett. ,086804 (2010).[23] F. Grossmann, T. Dittrich, P. Jung, and P. H¨anggi,
Co-herent destruction of tunneling , Phys. Rev. Lett. , 516(1991).[24] S. Wall et al. , Quantum interference between charge ex-citation paths in a solid-state Mott insulator , Nat. Phys. ,114 (2011)[25] Felix Hofmann and Michael Potthoff, Doublon dynam-ics in the extended Fermi-Hubbard model , Phys. Rev. B ,205127 (2012) [26] William Shockley On the surface states associated with aperiodic potential , Phys. Rev. vol. (1939)[27] C.W.J. Beenakker, Search for Majorana Fermions inSuperconductors , Annu. Rev. Con. Mat. Phys. , 113(2013).[28] J. Zak, Berry’s phase for energy bands in solids , Phys.Rev. Lett. ,2747 (1989).[29] Niels Strohmaier et al. , Observation of Elastic DoublonDecay in the Fermi-Hubbard Model , Phys.Rev. Lett. , ,080401 (2010).[30] M. Bukov, L. D’Alessio and A. Plokovnikov,
Universalhigh-frequency behavior of periodically driven systems:from dynamical stabilization to Floquet engineering , Ad-vances in Physics, 2015, Vol. 64, No. 2, 139-226[31] A. Eckardt and E. Anisimovas,
High-frequency approxi-mation for periodically driven quantum systems from aFloquet-space perspective , New J. Phys et al. , Brillouin-Wigner theory forhigh-frequency expansion in periodically driven sys-tems: Application to Floquet topological insulators ,arXiv:1511.00755 [cond-mat.mes-hall][33] M. Bukov, M. Kolodrubetz, A. Polkovnikov,