Edge Localized Schrödinger Cat States in Finite Lattices via Periodic Driving
EEdge Localized Schr¨odinger Cat States in Finite Lattices via Periodic Driving
Asadullah Bhuiyan and Frank Marsiglio
Department of Physics, University of Alberta, Edmonton, Alberta, Canada, T6G 2E1 (Dated: October 2, 2020)Floquet states have been used to describe the impact of periodic driving on lattice systems,either using a tight-binding model, or by using a continuum model where a Kronig-Penney-likedescription has been used to model spatially periodic systems in one dimension. A number ofthese studies have focused on finite systems, and results from these studies are distinct from thoseof infinite lattice systems as a consequence of boundary effects. In the case of a finite system,there remains a discrepancy in the results between tight-binding descriptions and continuous latticemodels. Periodic driving by a time-dependent field in tight-binding models results in a collapseof all quasienergies within a band at special driving amplitudes. In the continuum model, on theother hand, a pair of nearly-degenerate edge bands emerge and remain gapped from the bulk bandsas the field amplitude increases. We resolve these discrepancies and explain how these edge bandsrepresent Schr¨odinger cat-like states with effective tunneling across the entire lattice. Moreover,we show that these extended cat-like states become perfectly localized at the edge sites when theexternal driving amplitude induces a collapse of the bulk bands.
I. INTRODUCTION
Periodically driven solid-state systems have been asubject of intense study over the past three decades. Theinteraction of lattice potentials and time-dependent peri-odic driving has led to a number of theoretical proposalsfor coherent control of quantum dynamics. In particu-lar, non-local engineering of quantum tunneling in drivenquantum well systems can be realized by tuning externaldriving parameters. Two seminal results that exemplifythis statement are dynamic localization of (DL) and co-herent destruction of tunneling (CDT), both of whichcan be realized through the application of an AC fieldto a multi-stable quantum system. DL refers to the to-tal suppression of wave packet diffusion in a periodicallydriven infinite tight-binding lattice, while CDT refers tothe complete suppression of tunneling across a single bar-rier in a double-well potential. These phenomena pre-serve quantum coherence and localization through non-local controls, potentially having applications in quan-tum information processing.
As such, CDT and DLhave experienced numerous theoretical extensions and experimental investigations with Bose-Einsteincondensates in periodically driven optical lattices.Both CDT and DL rely on the crossings of quasiener-gies, which are the pertinent eigenvalues in the Floquetformalism. Additionally, CDT requires that the Flo-quet eigenstates are well localized throughout a singleperiod of its oscillation cycle. For sinusoidal driving F ( t ) = F sin( ωt ), the condition for DL in an infiniteone-dimensional tight-binding lattice is met when thedriving parameters satisfy J ( aF /ω ) = 0 ⇒ aF /ω = β ,n , (1)where J ( x ) is the zeroth order Bessel function, F isthe driving amplitude, ω is the driving frequency, a isthe nearest neighbour (n.n.) distance, and β ,n is the n th root of J ( x ). Here and henceforth (cid:126) = 1. Sub-sequent studies on driven two-site tight-binding modelshave shown that Eq. (1) is precisely the condition that must be satisfied for CDT to occur. The conditionEq. (1) works well in the limit that ω is much greaterthan the width of the lowest tunnel split energy spec-trum, which is the frequency regime considered in thiswork. These similarities point to an intimate connectionbetween DL and CDT; DL can be interpreted as a gen-eralized CDT for an infinite chain of quantum wells. Accordingly, CDT occurs when the quasienergies of thetwo-state system form an exact crossing, while DL iscaused by the exact crossing (collapse) of all quasiener-gies within a tight-binding band.In the realm between driven two-site models anddriven infinite tight-binding chains are driven finite lat-tice models, whose finite size profoundly affects thestructure of quasienergy crossings. In the n.n. tight-binding limit, periodically driving a finite lattice resultsin quasienergy bands that pseudo -collapse; rather thanshowing an exact intersection of all quasienergies in theband, the quasienergy band displays an intricate set ofquasienergy crossings and anti-crossings within a minutebut finite width near collapse points. Quasienergies be-longing to differing dynamical symmetries are allowed tocross and each pair of exact crossings results in CDTbetween a specific pair of n.n. quantum wells. These re-sults were observed for four and six site systems and thephenomenon was dubbed selective CDT by Villa-Bˆoaset al. and stimulated a number of theoretical studieson driven finite lattice systems. Well before these finite size effective model studies,numerical calculations of quasienergy bands from finitelattice systems were performed by Holthaus and cowork-ers.
Holthaus had aimed to mimic the behaviourof quasienergy bands of infinite lattices by periodicallydriving a finite sized system of many quantum wellsmodeled by a Kronig-Penney-like potential. While theconcept of band collapse was established, two nearly-degenerate bands deviate from the collapse for varyingdriving amplitude. These anomalous edge bands are aconsequence of the boundary effects of the finite lattice,but are unseen in the finite size effective model studies a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p previously mentioned.Both DL and CDT are sensitive to tiny perturbationsthat may break symmetries of the system. Perturba-tions that hardly effect properties of a time-independentsystem can significantly alter their driven counterparts,such as the appearance of anomalous edge bands seenin the work of Holthaus. This amplified edge behaviourrepresents the interaction of the periodic driving withan effective work function, represented by the particularboundary condition at the edge of the sample. On theother hand, these amplified edge effects are unseen in thetight-binding model calculations.In this work, we investigate the appearance of theseedge bands and reconcile results for complete Hilbertspace and effective (tight-binding) model calculations.To achieve this, a perturbative modification to the stan-dard tight-binding model is proposed. We find that sucha perturbation has little effect on the non-driven system,but becomes amplified for non-zero driving amplitude.Moreover, the eigenstates of these edge bands are stud-ied and it is found that perfect edge localization unseenin the non-driven system can be generated. This impliesthat the driving perturbation can be utilized as a switchto induce complete edge localization. The eigenstatesthemselves form Schr¨odinger cat-like states at the quan-tum wells nearest to the boundaries, resulting in effectivetunneling across the entire lattice. The amplified edgebehaviour examined here may be experimentally realizedin driven solid-state systems with a high work function,or through modulating the boundaries of a driven opticallattice. II. FLOQUET FORMALISM
An understanding of the Floquet formalism is essentialfor the study of periodically driven quantum systems. Inan effort to keep this work self-contained, we begin withan overview of key results and terminology of the Floquetformalism.Consider a particle of charge q and mass m present ina one dimensional (1D) finite lattice potential consistingof N quantum wells described by the time-independentHamiltonian H and immersed in monochromatic radia-tion. Within the dipole approximation, the single parti-cle Schr¨odinger equation for such a system is given by i∂ t | Ψ( t ) (cid:105) = [ H − xq E sin( ωt )] | Ψ( t ) (cid:105) = H ( t ) | Ψ( t ) (cid:105) , (2)where E is the electric field strength. The position-coupled driving term is a gauge choice for this systemand is commonly referred to as the length gauge . Sincethe Hamiltonian above is periodic with time over T ≡ (2 π ) /ω , we can utilize Floquet’s theorem to decomposethe Schr¨odinger solution into a discrete set of states givenby | Ψ α ( t ) (cid:105) = e − iε α t/ (cid:126) | Φ α ( t ) (cid:105) (3) where | Φ α ( t + T ) (cid:105) = | Φ α ( t ) (cid:105) is the time-periodic Floquetstate and ε α is the corresponding quasienergy, both ofwhich are indexed by the Floquet state quantum number α = 1 , , , ... . Substituting Eq. (3) into Eq. (2) leads tothe following eigenvalue problem for the quasienergies H F | Φ α ( t ) (cid:105) = ε α | Φ α ( t ) (cid:105) (4)where H F ( t ) ≡ H ( t ) − i∂ t is the Floquet Hamiltonian.This eigenvalue problem serves as an auxiliary equationto obtain the physical Schr¨odinger solutions defined byEq. (3). Due to the periodicity of a Floquet state, wecan see that it is not uniquely defined, since | Φ α (cid:48) ( t ) (cid:105) = | Φ α ( t ) (cid:105) e inωt , n ∈ Z (5)yields an identical Schr¨odinger solution, but with itsquasienergy shifted by an integer multiple of the pho-ton energy ω ε α (cid:48) = ε α + nω. (6)Just as the quasimomentum of Bloch’s theorem isuniquely defined up to integer multiples of its reciprocallattice vector, ε α is uniquely defined up to integer multi-ples of ω ( ω ≡ π/T ). Thus, the α th Floquet state actu-ally represents an entire class of infinitely many Floquetstate solutions that all correspond to a single identicalSchr¨odinger solution. Accordingly, α should be replacedby a double index α → ( α, n ), | Φ α,n ( t ) (cid:105) = | Φ α, ( t ) (cid:105) e inωt , ε α,n = ε α, + nω (7)where n can be referred to as the transition index ,which adjusts the Floquet-Brillouin zone in which thequasienergy resides. Because of this index, the orderingof quasienergies can become ill-defined compared to theunperturbed eigenenergies.Floquet-Brillouin zones are defined relative to a firstBrillouin zone. For the α th quasienergy, there existssome transition index n Fα that maps ε α, onto a firstBrillouin zone of width ω , in which the following holds: ε α,n Fα ∈ [ − ω/ , ω/ . (8)Note the dependence of this transition index on α , sincequasienergy eigenvalues can be separated by gaps largerthan ω .When dealing with quasienergies however, the princi-ple Brillouin zone tends to be more intuitive. We definethe α th quasienergy of transition index n = 0 to belongto its characteristic principle Brillouin zone. In this zone,the α th quasienergy approaches the α th eigenenergy ofthe unperturbed system in the limit that the driving am-plitude E goes to zero ε α, −→ E → E α . (9)Working with quasienergies in their principle zones al-lows us to quasi-order them relative to the unperturbedenergies from which they emerge as we increase the driv-ing amplitude from zero. In general, the exact limits ofthe principle Brillouin zone for a quasienergy ε α dependson α . For this work, we consider a frequency regimewhere intra-band transitions are not allowed.Due to the periodicity of the Floquet states, it is usefulto introduce a composite Hilbert space S = R⊗T , where R is the space of square-integrable functions, and T isthe space of T -periodic functions. T is spanned by theorthogonal Fourier basis (cid:104) t | m (cid:105) ≡ e imωt . It follows thatFloquet states in the space S obey an extended innerproduct (cid:104)(cid:104) Φ α (cid:48) ,n (cid:48) | Φ α,n (cid:105)(cid:105) ≡ T (cid:90) T d t (cid:104) Φ α (cid:48) ,n (cid:48) ( t ) | Φ α,n ( t ) (cid:105) = δ α (cid:48) α δ n (cid:48) n , (10)where the double braket notation (cid:104)(cid:104) . . . | . . . (cid:105)(cid:105) is used toemphasize that this inner product is distinct from the in-ner product for states in R . All non-redundant phys-ical Schr¨odinger solutions can be obtained from Floquetstates corresponding to quasienergies of the same n andit is this these states that form a complete set in R . Utilizing the Floquet formalism, we calculate theeigenvalues (quasienergies) of H F via basis expansion inthe composite Hilbert space S . Defining the Floqueteigenstate (cid:104) t | Φ α (cid:105)(cid:105) ≡ | Φ α ( t ) (cid:105) , we expand it as | Φ α (cid:105)(cid:105) = (cid:88) m,n C ( α ) m,n | m, n (cid:105)(cid:105) , (11)where (cid:104) t | m, n (cid:105)(cid:105) ≡ e imωt | n (cid:105) and | n (cid:105) ∈ R is some spatialbasis vector indexed by a quantum number n . Actingon this expansion with Eq. (2) and taking the extendedinner product of both sides with some arbitrary bra state (cid:104)(cid:104) m (cid:48) , n (cid:48) | we get the following eigenvalue problem (cid:88) m,n (cid:104)(cid:104) m (cid:48) , n (cid:48) |H F | m, n (cid:105)(cid:105) C ( α ) m,n = ε α C ( α ) m (cid:48) ,n (cid:48) . (12)Evaluation of the matrix elements leads to the following (cid:104)(cid:104) m (cid:48) , n (cid:48) |H F | m, n (cid:105)(cid:105) = δ m (cid:48) m ( H ) n (cid:48) n + δ m (cid:48) m δ n (cid:48) n ωm + ix n (cid:48) n q E δ m (cid:48) ,m +1 − δ m (cid:48) ,m − ] . (13)It is clear that the Floquet Hamiltonian possesses a blocktridiagonal structure in the composite space time basis | m, n (cid:105)(cid:105)H F = . . . . . .. . . H − ω i q E x − i q E x H i q E x − i q E x H + ω . . .. . . . . . . (14)As such, we can implement a matrix continued fractionmethod to efficiently determine the α th quasienergy inits principle Brillouin zone (see Appendix A). III. DRIVEN TIGHT-BINDING MODEL
We first consider the effect of periodic driving on a fi-nite chain of N quantum wells within the standard n.n.tight-binding approximation. In this work, we will fo-cus on the case of N = 8 quantum wells. The time-independent component of the Hamiltonian is H = U N (cid:88) j =1 n j − t N − (cid:88) j =1 (cid:104) c † j +1 c j + h.c. (cid:105) (15)where U is the on-site interaction energy, t is the n.n.hopping amplitude, c † j ( c j ) is the creation (annihilation)operator for site j , and n j = c † j c j is the standard numberoperator. As can be inferred from the form of Eq. (15),open boundary condition are enforced at the systemedges. For Eq. (15) an analytic solution can be obtainedfor the energy spectrum E (anal.) n = U − t cos( ka ) ,ka = πnN + 1 , n = 1 , , ..., N (16)where a is the n.n. distance. Thus, for a monochromati-cally driven finite chain of quantum wells, we can expressthe Floquet Hamiltonian in a basis of localized states as H F = H − i∂ t − xq E sin( ωt ) ,x = a N (cid:88) j =1 n j . (17)Expansion of the Floquet eigenstate in a compos-ite space-time basis allows us to carry out a matrix-continued-fraction method to diagonalize H F and ac-quire its eigenvalues (quasienergies).In Fig. 1 we plot the quasienergies as a function ofthe dimensionless driving amplitude, ¯ E ≡ q E a/t , for ω/t = 50. Since our choice of driving frequency clearlyplaces our results in the high-frequency regime ω >> t ,we expect the quasienergies to approximately obey ε (anal.) n ≈ U − J ( q E a/ω ) t cos( ka ) ,ka = πnN + 1 , n = 1 , , ..., N (18)to first order in inverse frequency. Such an approx-imation correctly predicts a Bessel function envelopebut does not capture the fine set of crossings and anti-crossing or the finite width of the band pseudo-collapse. a) b) FIG. 1. a) Quasienergies of Eq. (17), describing a tight-binding model, vs. dimensionless driving strength ¯ E for an N = 8 site system with open boundary conditions. Verticaldashed lines mark collapse points as predicted by Eq. (1).We have set the interaction energy U = 0 for convenience. b) Expanded view of a) at the first collapse point. Thequasienergy band forms a specific pattern of crossings andanti-crossings near the first collapse point as dictated by theirsymmetry classes. These results were obtained in the highfrequency regime with ω/t = 50. Observing Fig. 1 a), we see that quasienergies varywith ¯ E as expected with a clearly depicted Bessel func-tion envelope as predicted by Eq. (18). In Fig. 1 b) how-ever, we see that near collapse points, the quasienergyband pinches into a set of fine crossing and anti-crossings.This is not predicted within a first order correction;higher order analysis is necessary. Fig. 1 serves as the reference point for the resultsdiscussed in this work. In particular, we will show howthese same results can be obtained from a continuousfinite tight-binding potential.
IV. CONTINUOUS FINITE LATTICE
In this section, we consider the effect of periodic driv-ing on a continuous finite lattice system and compareresults to our corresponding discrete tight-binding re-sults. We begin by briefly studying the system withoutperiodic driving to determine the potential parametersnecessary to reproduce tight-binding-like behaviour.
A. Time-Independent System
For a continuous finite lattice, the time-independentHamiltonian of interest is given by H = p m + V ( x ) . (19)For our finite lattice, we choose a sequence of N potentialbarriers of identical height V , with a distance a betweenn.n. cells. As such, the total system size is exactly N a .At the edges of our lattice, at x = 0 and at x = N a , weplace infinitely high potential walls. These infinitely highwalls serve as a crude model of a work function for finitesized systems. Naturally, this implies that we enforce“open” boundary condition at the edges for our wavefunction, i.e. the wave function must go to zero at theboundaries. A potential V ( x ) = V P ( x ) that describessuch a finite lattice is given by the analytical expression V P ( x ) = (cid:40) V L ( x ) + V Bulk ( x ) + V R ( x ) , x ∈ [0 , N a + 2 δb ] ∞ , otherwise , (20)where the set of bulk barrier potentials are described by ( θ [ x ] is the Heaviside step function) V Bulk ( x ) = V N − (cid:88) j =1 θ [ x − ( δb + ja − b/ θ [( δb + ja + b/ − x )] (21)and the left “plateau” (given with δb = 0 in Fig. 2) is described by V L ( x ) = V θ [ x ] θ [ δb + b/ − x ] (22)and the right “plateau” (again given with δb = 0 in Fig. 2) is described by V R ( x ) = V θ [ x − ( N a + δb − b/ θ [ N a + 2 δb − x ] . (23)The parameters b , a = w + b , and w ≡ a − b are the barrier width, the unit cell length, and the well width,respectively. This finite lattice has N potential minimaand is illustrated in Fig. 2 a) with δb = 0; extensionswith δb (cid:54) = 0 will be discussed below. The dynamics of aparticle in such a potential was studied by Holthaus; this model resulted in edge bands when a driving per-turbation is introduced.Using a basis expansion method, we calculate theeigenenergies for the unperturbed lattice. A convenientbasis set that satisfies the boundary conditions of thispotential (with δb = 0) is given by (cid:104) x | n (cid:105) = (cid:114) N a sin (cid:16) nπxN a (cid:17) ( δb = 0) . (24)Such a calculation method has been used before to calcu-late the eigenenergies of a finite lattice. Before turningon a driving perturbation, we must establish that thispotential demonstrates tight-binding behaviour. To dothis, we set our lattice parameters such that the resultingenergy spectra properly mimic the tight-binding energyband. Numerical results for eigenenergies are shownin Fig. 2 b). We use dimensionless units of energy byadopting the unit of energy as E = π / (2 ma ). Nat-urally, this leads to length units of the n.n. distance a . Hereafter, a quantity Q will have its dimensionlessanalogue denoted by ˜ Q .In Fig. 2 b), we see that our lowest band of eigenen-ergies approximately obey E n = U − t cos( ka ) ,ka = πnN , n = 0 , , ..., N − . (25)While the behaviour is cosine-like, there is a discrep-ancy between the effective wave vectors of the analyticaltight-binding result ka = πn/ ( N + 1) and our numeri-cal results ka = πn/N . This discrepancy in wave vectorsuggests that the boundary conditions adopted for thesquare well system do not quite match those implied intight-binding.As drawn, and typically used, the boundary condi-tions represented by Fig. 2 and given by Eqs. (20 - 23)with δb = 0 imply a certain amount of quantum pressurearising from the effective work function. For this reasonwe generalized our lattice potential to allow an arbitrarydistance between the potential well nearest the surfaceand the surface itself, represented by a non-zero valuefor δb . For example, increasing this distance reduces thequantum pressure due to the boundaries, which mini-mizes the impact of the effective work function. Ourmodified finite lattice potential is shown in Fig. 3 a) andis achieved by taking δb to be positive (in Fig. 3 a), δb = 5 a ).In fact the parameter δb can range from − b/ N a → N a + 2 δb , and so, for purposes of calculation, ourspatial basis function | n (cid:105) defined in Eq. (24) for δb = 0is extended accordingly: (cid:104) x | n (cid:105) = (cid:114) N a + 2 δb sin (cid:18) nπxN a + 2 δb (cid:19) . (26) With this adjustment to our continuous potential ( δb =5 a instead of zero), the results for the lowest band areshown in Fig. 3 for the particular value of δb = 5 a . a)b) FIG. 2. a) Schematic of finite lattice potential normalizedto barrier height V for N = 8 quantum wells, with unitcell length a . The total system size is Na . We can tunethe width of each quantum well by adjusting the parameter w . Note that at the edges, there are potential plateaus withwidth b/ a − w ) / N cells. Open boundary conditions are implementedat the edges of the system, i.e. the wave function must go tozero at x = 0 and at x = Na . b) The lowest 8 (dimensionless) numerically exact eigenener-gies (shown with blue circles) plotted vs. eigenvalue number n . The system parameters are ˜ V = 15, ˜ w = 0 . N = 8.Also shown is a pink curve fitted to a n.n. tight-bindingcosine function. We fit the 3-parameter tight-binding band U − t cos( k n ) to ensure that our results are within thetight-binding regime. a)b) FIG. 3. a) Schematic of extended finite lattice potential,Eq. (20) with δb = 5 a , normalized to barrier height V for N = 8 quantum wells, each with width w separated by barri-ers of width b . The unit cell distance between n.n. potentialwell centres is a . The total system size is Na + 2 δb . Openboundary conditions are implemented at the edges of the sys-tem for purposes of calculation. b) Lowest 8 (dimensionless) numerically exact eigenenergies(blue circles) plotted against eigenvalue number n . The sys-tem parameters are identical to those of Fig. 2, except now δb = 5 a . Also shown (pink curve) is the result of a n.n.tight-binding cosine fit. The 3-parameter tight-binding band U − t cos( k n ) fits the numerical results very accurately,clearly illustrating that we are in the tight-binding regime. As opposed to what we have seen in Fig. 2, the fit wavevector parameter k is now in agreement with the ana-lytic tight-binding solution given by Eq. (16). Clearly, this indicates that the standard finite chain tight-bindingapproximation models an “isolated” set of quantum wellsin free space rather than a “confined” set of quantumwells. While this change does lead to slightly differingresults in the energy spectrum, this difference is quitesubtle. As seen in Figs. 2 and 3, the only fitting pa-rameter that differs significantly in these two figures isthe (dimensionless) wave vector k . But even this differ-ence is relatively small, and would be even smaller in thelimit of large N , where k would be practically identicalin both potentials. Though this difference is slight inthe time-independent case, we will see in the next sec-tion that this slight difference in eigenvalue behaviourcaused by tuning the degree of quantum pressure at theboundaries is amplified by periodic driving. This leadsto radically different quasienergy spectra in the presenceof a driving field. B. Periodically Driven System
We now turn on a periodic driving perturbation andcalculate the quasienergy spectra of the finite lattice with N wells. The Floquet Hamiltonian of interest is givenby H F = H − i∂ t − ( x − x ) q E sin( ωt ) , (27)where H ≡ p / (2 m ) + V P ( x ) and x ≡ ( N a + 2 δb ) / δb = 0, Eq. (27) represents a periodically driven latticedescribed by the “natural” choice of boundary condi-tions with plateaus at either end of width b/
2, so thatthe total sample length is precisely
N a . Expanding theFloquet eigenstate of Eq. (27) in a composite space-timebasis with spatial basis states given by Eq. (26), we use amatrix-continued-fraction method to calculate the low-est eight quasienergies for varying ˜ E ≡ q E a/E in an N = 8 quantum well system. This is done for both δb = 0 (Fig. 2 a)) and δb = 5 a (Fig. 3 a)). We chooseour driving frequency to be much greater than the widthof the lowest unperturbed energy band. Given the nu-merical results in Figs. 2 b) and 3 b), a dimensionlessdriving frequency ˜ ω = 5 is clearly sufficient. Numer-ical results for quasienergies vs. dimensionless drivingstrength ˜ E for both δb = 0 and δb = 5 a are shown inFig. 4. a.1) a.2) b.1) b.2) FIG. 4. a.1)
Lowest band of quasienergies vs. dimensionless driving amplitude ˜ E for δb = 0 and ˜ ω = 5. All other systemparameters are identical to the ones used for Fig. 2. The quasienergies are plotted in their principle Brillouin zone and thus areordered by their values at ˜ E = 0. Vertical dashed lines mark values of ˜ E for which ˜ E / ˜ ω = β ,n . These values are preciselywhere the lowest 6 quasienergy quasienergies pseudo-collapse, in accordance with Eq. (1). The highest two quasienergiesconverge into each other and become nearly-degenerate as E is increased. These edge bands do not participate in the pseudo-collapses. a.2) Expanded view of a.1) near the first collapse point, as indicated by the arrow. The lowest 6 quasienergies pseudo-collapseand form an intricate set of crossings and anti-crossings near ˜ E / ˜ ω = β ,n . However, as is evident from the figure, this conditiondoes not predict the exact location of the set of crossings near the pseudo-collapse. b.1) Lowest band of quasienergies vs. dimensionless driving amplitude ˜ E for δb/a = 5 and ˜ ω = 5. All other system parametersare identical to the ones used for Fig. 2. The quasienergies are plotted in their principle Brillouin zone and thus are orderedby their values at ˜ E = 0. Vertical dashed lines mark values of ˜ E for which ˜ E / ˜ ω = β ,n . b.2) Expanded view of b.1) near the first collapse point, as indicated by the arrow. All quasienergies participate in thepseudo-collapse and form an intricate set of crossings and anti-crossings near ˜ E / ˜ ω = β ,n . We see in Fig. 4 a.1) that only the lowest 6 quasiener-gies participate in pseudo-collapses for δb = 0. Thehighest two quasienergies are gapped for low drivingstrength, and in addition become nearly-degenerate for˜ E >
2. The emergence of this pair of edge bands persistsfor systems with N ≥ E , changing values from ∼ − to ∼ − .These abrupt decreases in difference may be indicativeof exact crossings occurring between the the edge bands.In Fig. 4 a.2), we see that the band collapse is not exact. Instead, the quasienergies of differing states formdistinct crossings and anti-crossings near ˜ E / ˜ ω = β , .While this is expected for finite-sized systems, the loca-tions of the crossings do not obey any symmetries withrespect to the center of the quasienergy spectrum (hori-zontal mirror symmetry), in contrast to the system stud-ied by Villa-Bˆoas. In Fig. 4 b.1), where δb = 5 a , we see that the be-haviour of the quasienergy band resembles the behaviourseen in the tight-binding regime much more than Fig. 4a.1). The emergence of nearly-degenerate edge bandsno longer occurs. Instead, all eight quasienergy bandsnow participate in a pseudo-collapse at collapse points,in accordance with the tight-binding result Fig. 1 a). Itshould be noted that the driving frequency used for Fig. 1a) is ω/t = 50. For our continuous lattice, we foundthat ˜ t ∼ . t ≡ t / ( π / (2 ma )). Thus, ourcontinuous lattice has an equivalent driving frequency of ω/t = 5 / . ∼ ω/t = 50 is already in the high frequencyregime, and we already establish qualitative agreementbetween Fig. 4 b.1) and Fig. 1 a), so our specific choiceof frequency does not appear to matter.In Fig. 4 b.2), the pattern of crossings and anti-crossings are clearly more orderly than the equivalentplot in Fig. 4 a.2), displaying a beautiful symmetry aboutthe collapse point. That said, the actual patterns ofcrossings and anti-crossings are quite different from whatis seen in Fig. 1 b). Since these patterns emerge oversuch a tiny resolution, minute differences in system con-figuration affects their behaviour greatly. Clearly, thequasienergies are incredibly sensitive quantities. Thefinite width and depth of the quantum wells in V P ( x )clearly lead to differences compared to the results of thetight-binding approximation at this resolution. Regard-less, qualitative agreement between Fig. 1 a) and Fig. 4b.2) clearly establishes that our extended finite latticepotential V P ( x ) with δb = 5 a (or higher) leads to re-sults much closer in agreement with the results of thetight-binding model used in Eq. (17). C. Perfect Edge Localization
While our original finite lattice system may not re-produce the standard tight-binding behaviour, it is stillrepresentative of an (idealized) physical system. As men-tioned before, the potential described by Eq. (20) with δb = 0 models a chain of N atoms in the presence ofan effective work function at the edges. Thus, our un-usual tight-binding fit in Fig. 2 b) is not an artifact of our theoretical setup. Rather, it is an accurate descrip-tion for a finite chain of atoms that faithfully considersedge effects. Though this difference is quite subtle inthe time-independent system, the failure of the standardfinite chain tight-binding approximation to address edgeeffects becomes truly apparent when we look at the pe-riodically driven system. Periodic driving amplifies edgeeffects, leading to the emergence of nearly-degenerateedge bands in Fig. 2 a.1).The Floquet edge bands deviate greatly from the be-haviour of the bulk bands with increasing ˜ E . It is thusof interest to examine how the edge band Floquet proba-bility densities behave in comparison to probability den-sities of the bulk band. In Fig. 5, we plot the low-est 8 time-averaged Floquet state probability densities T (cid:82) T | Φ α ( x, t ) | dt of this system for ˜ E = 12 (near thefirst collapse point). Since we are working in a high-frequency regime, the inherent oscillation amplitude of aFloquet state Φ α ( x, t ) will be much less than the quan-tum well width ˜ w . As such, we are not losing essentialinformation about the Floquet probability density afterperforming a time-average. We also plot the non-drivenprobability density | Ψ α ( x ) | ( ˜ E = 0) to provide a com-parison with the corresponding driven Floquet state. Incontrast to the lower quasienergy Floquet states, the Flo-quet probabilities of the edge bands appear to be com-pletely localized at the quantum wells of the edges. Thisis unseen in the corresponding non-driven probabilitydensities. Moreover, the edge band Floquet probabil-ity densities ( α = 7 ,
8) are essentially identical to oneanother. This is again reminiscent of the split groundstate doublet seen in the time-independent solution fora double well potential. It appears that the AC fieldcreates a double well-like potential with the quantumwells nearest to the system boundaries when the drivingstrength is near collapse points, causing complete edgelocalization for the highest doublet of Floquet states.
FIG. 5. Lowest eight time-averaged Floquet probability densities T (cid:82) T | Φ α ( x, t ) | dt (pink curve with circular points) vs lowesteight non-driven probability densities | Ψ α ( x, t ) | (blue curves with diamond points) for driving strength near a collapse point˜ E = 12 (˜ ω = 5 here). Complete edge localization is observed for Floquet states α = 7 ,
8, which correspond to quasienergiesof the edge band. Also plotted is the finite lattice potential V P ( x ), with δb = 0, normalized to barrier height V . All othersystem parameters are identical to those used in Fig. 2. For reference, the same plots are given in Appendix B for δb = 5 a . We can further quantify the degree to which edge lo-calization occurs by performing a spatial integration ofthe time-averaged Floquet states over the unit cells ofthe edges. Utilizing the spatial symmetry of our sys-tem about x = N a/ δb = 0), we thus define thequantity ρ edgeα = 2 (cid:90) a T (cid:90) T | Φ α ( x, t ) | d t d x (28)as a measure of edge population. In Fig. 6, we plotEq. (28) for the lowest eight Floquet states vs. dimen-sionless driving amplitude ˜ E . Consistent with the result of Fig. 5, we observe thatthe highest Floquet doublet (states α = 7 ,
8) completelypopulate the quantum wells of the edges precisely atthe collapse points. In contrast, the lower states showzero edge population at the collapse points. It is atthese collapse points at which the gap between theedge bands and the bulk bands reaches a maximum.Consequently, the collapse points form a set of “resonantdriving amplitudes” at which perfect edge localizationcan occur. This behaviour is unseen in the non-drivensystem ( ˜ E = 0) and so can be switched on and offthrough the action of non-local periodic driving. It it0possible that this phenomenon may be utilized to realizerobust, chiral edge states in semi-finite 2D lattices. FIG. 6. Edge population ρ edgeα vs ˜ E for the lowest eightFloquet states of our system. Dashed vertical lines markvalues of ˜ E at which collapse points occur in the Floquetband spectra. One should note that perfect edge localization is sen-sitive to the δb parameter that extends the length of theedge barriers. For δb < ∼ w , where w is the width of aquantum well, we have observed that perfect edge local-ization can be maintained. Beyond this regime however,the standard behaviour of effective tight-binding modelsis recovered.As shown throughout this section, the quantum pres-sure of the boundaries dramatically affects the Floquetbands for high driving amplitude. An effective doublewell potential is created between the quantum wells nearthe boundaries, leading to a pair of split Schr¨odinger cat-like edge bands that deviate from the standard collapsebehaviour seen by the lower bands. When driving pa-rameters are set to induce band collapse, the edge bandsbecome maximally gapped from the lower bands thatparticipate in the pseudo-collapse, resulting in perfectedge localization. We have also shown that the standardtight-binding approximation cannot reproduce these re-sults. So the question remains: can we modify the tight-binding approximation such that we may observe theemergence of gapped edge bands in the quasienergy spec-trum? This question is addressed in the forthcomingsection. V. MODIFIED FINITE CHAINTIGHT-BINDING MODEL
We have now established what sort of “ab-initio” po-tential a standard driven tight-binding model representsin the previous section. However, there is no reason whyour original finite lattice potential, Eq. (20) with δb = 0,cannot be described by a modified effective tight-binding model. As such, we modify the standard finite tight-binding model (Eq. (15) with U = 0) by adding a per-turbative term U to the on-site energy of the edge sites H mod ( U ) = U [ n + n N ] − t N − (cid:88) j =1 (cid:104) c † j +1 c j + h.c. (cid:105) . (29)It is clear that this Hamiltonian reduces to Eq. (15) for U = 0. Our hope is that the addition of U will leadto results that mimic the behaviour seen from calcula-tions with our original finite lattice in Fig. 2. The ideais that the presence of a (positive) U will model theadditional quantum pressure that the nearby walls exerton a particle to vacate the boundary well sites. To in-vestigate what value of U will achieve this, we calculatethe eigenenergy spectrum ( E = 0) for various values of U in units of the hopping amplitude t . We also plotEq. (16) and Eq. (25) for reference. Numerical resultsare shown in Fig. 7. FIG. 7. Eigenenergies of H mod ( U ) (Eq. 29) for varying U ∈ [0 , t ] in a N = 8 site system. The analytic finite chain tight-binding result Eq. (16) (blue, lower curve) and the numericalfit Eq. (25) found for the eigenenergies of our original finitelattice potential V P ( x ; δb = 0) (pink, upper curve) are plottedas references. The on-site energy U is set to zero here for bothequations. This figure shows that by setting U = t , the eigenen-ergies of H mod and the cosine behaviour seen in Fig. 2are in clear agreement. In fact this result is exact, as firstshown by Goodwin in 1939. Goodwin had investigatedthe tight-binding approximation for finite atomic chainsbut avoided the assumption that the on-site energy atthe edge sites and the bulk sites were identical. He wenton to analytically determine a set of equations that givethe exact eigenenergies of the system for general U , butonly two values of U lead to wave vectors that are exactratio multiples of π . For U = 0, one obtains the wavevector ka = πn/ ( N + 1) and Eq. (16) is obtained. Onthe other hand, if the edge site energy is greater thanthe bulk sites by precisely a n.n. hopping amplitude1 U = t , the wave vector is given by ka = πn/N , andthus Eq. (25) is obtained. But this is precisely the be-haviour followed by the eigenenergies in Fig. 2 b)! Thus,we proceed with U = t to address the edge effects ofour original lattice potential V P ( x ; δb = 0). a)b)c) FIG. 8. a) Quasienergies obtained from Eq. (30), a modifiedtight-binding model, vs. dimensionless driving amplitude ¯ E for ω/t = 50. Vertical dashed lines mark values of ¯ E atwhich collapse points occur. b) Expanded view of a) at the first collapse point, as indi-cated by the arrow. c) Expanded view of the lower four quasienergy bands seenin b), as indicated by the arrow.
The pertinent Floquet Hamiltonian is given by H F = H mod ( U = t ) − i∂ t − q E a sin( ωt ) N (cid:88) j =1 n j . (30)We numerically obtain the quasienergies for this systemvia construction and diagonalization of the propagatorover one period U (0 , T ). Results are shown in Fig. 8,where the quasienergy spectra now resemble the spectraseen in Fig. 2.We see the emergence of edge bands that branch fromthe highest pair of unperturbed eigenenergies, while thelower six participate in a pseudo-collapse at collapsepoints predicted by Eq. (1). Fig. 8 b) reveals the cross-ing pattern at the first collapse point, which is distinctfrom the crossing patterns seen in earlier quasienergyspectra figures. While the lowest four quasienergies con-verge further at the collapse point, the next highest twodeviate from the lower levels, instead becoming a nearly-degenerate pair. Of these four lowest quasienergies thatappear to converge, we see in Fig. 8 c) that they actuallysplit into a lower pair and higher pair of bands. Eachpair exhibit an exact crossing at the collapse point, butdisplay an anti-crossing relative to each other.Since we are able to observe the emergence of edgebands in our modified effective model, we expect thatperfect edge localization should occur at the collapsepoints. To measure the degree of edge localization inour discrete tight-binding potential, we redefine ρ edgeα asfollows ρ edgeα = | ˜ C | + | ˜ C N | , (31)where | ˜ C n | = 1 T (cid:90) T | (cid:104) n | Φ α ( t ) (cid:105) | dt, (32)and | n (cid:105) is a localized state and | Φ α ( t ) (cid:105) is the α th Floquetstate. We plot ρ edgeα against varying driving amplitudein Fig. 9. FIG. 9. Edge population for all eight states vs. driving ampli-tude ¯ E for the system described by Eq. (30). Vertical dashedlines mark collapse points. The edge population measureused here is defined by Eq. (31). We have used ω/t = 50. α = 7 , ρ edgeα with vary-ing ¯ E resembles its continuous analogue in Fig. 6. Whileedge localization appears to be nearly perfect here ( ρ edgeα peaks at ∼ .
996 at collapse points), this can be affectedby the driving frequency. In Appendix C, we reproduceFigs. 1, 8, and 9 for ω/t = 5. It turns out that raising ω decreases the width of a pseudo-collapse point withoutactually changing the fine pattern of crossings, therebydecreasing the gap between the edge states and the sixthquasienergy at a collapse point. This decrease in gap sizecauses greater deviation in ρ edgeα from unity at a collapsepoint.For example, we see in Appendix C (Fig. 13) that ρ edgeα peaks at ∼ .
97 at a collapse point. “High” drivingfrequency means essentially that the driving frequencyexceeds the unperturbed energy bandwidth. In this casethe latter is 4 t , so ω = 5 t is near the lower limit. Theresults in Appendix D along with the insensitivity of ourresults at even higher frequency than ω = 50 t bothillustrate that the magnitude of frequency is immaterial,provided it is in the high frequency regime. VI. CONCLUSION
In summary, we have studied the impact of time-periodic fields on finite lattices, both in the tight-bindingframework, and with Kronig-Penney-like arrays of wells,described by a continuum model. The implicit assump-tion of “open” boundary conditions for the tight-bindingmodels is that there is no hopping from the edge sitesinto the vacuum. On the other hand, continuum modelsgenerally consist of an assembly of N unit cells consist-ing of wells separated by barriers. For a system length of N a , where a is the unit cell length, this naturally leadsto edges with a barrier of width equal to half the regu-lar barrier width followed by the infinite barrier to thevacuum (see Fig. 2 a)). These two configurations leadto slightly different electronic properties, barely notice-able in the time-independent problem. In particular, thecontinuum model singles out the edge wells as “different”than the bulk wells, since the nearby wall (half a barrierwidth away) exerts a “quantum pressure” on a particleresiding in that well (and slightly in the neighbouringwell also), while no such “quantum pressure” is includedin the standard tight-binding description.The difference, between these two configurations, how-ever, is amplified when one increases the amplitude ofthe time-periodic applied field. At certain special fieldamplitudes the quasienergy bands all collapse, almost toa point, with the exception, in the continuum model, oftwo nearly-degenerate edge bands that remain gappedfrom the lower bands. In contrast, in the tight-bindingmodel there are no edge bands, and all the bands collapseat these special field amplitudes. We have shown how analteration of the boundaries, represented by an infinite potential at either edge, brings the continuum model re-sults into agreement with those from tight-binding. Inparticular, as we move the boundaries further away fromthe last well, the two results come into qualitative agree-ment with one another, and the edge quasienergy bands,previously seen as distinct in the continuum model re-sults, behave in a manner like the rest of the bands whenthis boundary alteration is applied.Conversely, the tight-binding results can be made tomimic those of the continuum model by increasing thebase level of the two end sites by an amount preciselyequivalent to the n.n. hopping amplitude. Our inter-pretation is that this increased base level at these twosites represent the increased quantum pressure in thecontinuum model exerted by the nearby boundaries. In-stead, implementation of the increased base level at theboundary sites means it simply becomes less favourableto occupy these two sites. In the presence of the time-periodic field, however, states develop that have exclu-sive occupation in these two sites (see Fig. 5), and theseform Schr¨odinger cat-like edge states across the entirearray of sites. We also defined and studied a quan-tity ρ edgeα which illustrates the localization of these edgestates as the field amplitude is varied. We anticipatethat these edge states can be exploited for applicationsusing time-periodic driving fields. In particular, we ex-pect that these gapped, edge localized states will lead torobust, chiral edge currents in an analogously driven 2Dsemi-finite lattice system that can be ‘switched’ on byinitializing the non-local drive. ACKNOWLEDGMENTS
This work was supported in part by the NaturalSciences and Engineering Research Council of Canada(NSERC) and by a MIF from the Province of Alberta.We also appreciate support from the Department ofPhysics at the University of Alberta.3
Appendix A: Matrix-Continued-Fraction Method
In this Appendix we describe in some detail the contin-ued fraction method used to solve Eq. (12). This followsthe description given in Refs. [17 and 27].We begin by substituting Eq. (13) into Eq. (12). Defin-ing i q E x n (cid:48) n ≡ D n (cid:48) n , it follows (cid:88) n (cid:104) H n (cid:48) n C ( α ) m (cid:48) ,n + D n (cid:48) n C ( α ) m (cid:48) − ,n + D ∗ n (cid:48) n C ( α ) m (cid:48) +1 ,n (cid:105) = ( ε α − (cid:126) ωm (cid:48) ) C ( α ) m (cid:48) ,n (A1)Following this last expression, we will drop the ‘prime’in front of the dummy index m for simplicity. It isnow useful to switch to the matrix representation inthe spatial basis | n (cid:105) , thus, H n (cid:48) n → ˆH , D n (cid:48) n → ˆD , and C ( α ) m,n → C ( α ) m . Defining ˆG m ( ε α ) ≡ ˆH − ( ε α − (cid:126) ωm ) ˆ1 , weobtain the following tri-diagonal recursive relation ˆG m ( ε α ) C ( α ) m + ˆDC ( α ) m − + ˆD † C ( α ) m +1 = 0 . (A2)Such an equation allows us to make use of the matrix-continued-fraction method to efficiently solve for thequasienergy eigenvalues ε α . To begin, we assume thatthere exists invertible raising and lower operators ˆS m , ˆT m such that ˆS m C m = C m +1 (A3) ˆT m C m = C m − . (A4)The Floquet state quantum number α has been droppedfor simplicity. We can express both these operators ascontinued fractions if we adopt the fraction notation ˆQ − ≡ ˆQ for inversion. For m = 0, one finds that theraising and lowering operators have the following form ˆS = − ˆG ( ε ) + ˆD † ˆS ˆD = − ˆG ( ε ) + ˆD † − ˆG ( ε )+ ˆD † − ˆG ε )+ ˆD † ... ˆD ˆD , (A5) ˆT = − ˆG − ( ε ) + ˆD ˆT − ˆD † = − ˆG − ( ε ) + ˆD − ˆG − ( ε )+ ˆD − ˆG − ε )+ ˆD ... ˆD † ˆD † . (A6)In practice, we must truncate these continued fractionsat some finite m = M , such that ˆS M = ˆT − M = 0.Substituting the above two operators into Eq. (A2) for m = 0, and reminding ourselves that ˆG m ( ε ) ≡ ˆH − ( ε − m (cid:126) ω ) ˆ1 , we acquire the follow matrix equation (cid:104) ˆH − ε ˆ1 + ˆD ˆT + ˆD † ˆS (cid:105) C = 0 . (A7) The raising and lower operators dependence on ε is im-plicit through its dependence on the operator ˆG m ( ε ).We make this dependence explicit by defining the oper-ator ˆQ ( ε ) ≡ ˆD ˆT + ˆD † ˆS . This leads to (cid:104) ˆH − ε ˆ1 + ˆQ ( ε ) (cid:105) C = 0 . (A8)With the above, we see that the numerical determinationof the quasienergies is reduced to calculating the rootsof the matrix determinantdet (cid:104) ˆH − ε ˆ1 + ˆQ ( ε ) (cid:105) = 0 . (A9)4 Appendix B: Floquet States vs Non-Driven States of Extended Continuous Lattice
We show here the figure corresponding to Fig. 5, but with δb = 5 a . Note that no edge states arise in this case. FIG. 10. Lowest eight time-averaged Floquet probability densities T (cid:82) T | Φ α ( x, t ) | dt (pink curves with circles) vs lowest eightnon-driven probability densities | Ψ α ( x, t ) | (blue curves with diamonds) for driving strength near a collapse point ˜ E = 12(˜ ω = 5 here). Also plotted is our extended lattice potential V P ( x ) for δb/a = 5, normalized to barrier height V . All othersystem parameters are identical to those used in Fig. 2. Appendix C: Driven Tight-Binding Results: ω/t = 5 In this Appendix we provide results for the “high”frequency ω/t = 5, which is essentially at the lowerlimit for what is deemed high. Results in the text for ω = 50 t are representative of results for ω > ∼ t andfor much higher frequency.Note that the widths of the pseudo-collapse points inFig. 11 b) and Fig. 12 b), c) are much greater than whatis seen in the analogous figures in the text, where ω/t =50 is used (Fig. 1 b) and Fig. 8 b), c)). a)b) FIG. 11. a) Quasienergies of Eq. (17), describing a tight-binding model, vs. dimensionless driving strength ¯ E for an N = 8 site system with open boundary conditions. Verticaldashed lines mark collapse points as predicted by Eq. (1).We have set the interaction energy U = 0 for convenience. b) Expanded view of a) at the first collapse point. Thequasienergy band forms a specific pattern of crossings andanti-crossings near the first collapse point as dictated by theirsymmetry classes. These results were obtained in the highfrequency regime with ω/t = 5. a)b)c) FIG. 12. a) Quasienergies obtained from Eq. (30), a modifiedtight-binding model, vs. dimensionless driving amplitude ¯ E for ω/t = 5. Vertical dashed lines mark values of ¯ E at whichcollapse points occur. b) Expanded view of a) at the first collapse point, as indi-cated by the arrow. c) Expanded view of the lower four quasienergy bands seenin b), as indicated by the arrow. FIG. 13. Edge population for all eight states vs. drivingamplitude ¯ E for the system described by Eq. (30). Verticaldashed lines mark collapse points. The edge population mea-sure used here is defined by Eq. (31). We have used ω/t = 5.Note that this localization measure peaks at ∼ .
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