Tailored particle current in an optical lattice by a weak time-symmetric harmonic potential
Julio Santos, Rafael A. Molina, Juan Ortigoso, Mirta Rodríguez
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J un Tailored particle current in an optical lattice by a weak time-symmetric harmonicpotential
Julio Santos, Rafael A. Molina, Juan Ortigoso and Mirta Rodr´ıguez
Instituto de Estructura de la Materia, CSIC, Serrano 121-123, 28006 Madrid, Spain. (Dated: June 8, 2011)Quantum ratchets exhibit asymptotic currents when driven by a time-periodic potential of zeromean if the proper spatio-temporal symmetries are broken. There has been recent debate on whetherdirected currents may arise for potentials which do not break these symmetries. We show here that,in the presence of degeneracies in the quasienergy spectrum, long-lasting directed currents can beinduced, even if the time reversal symmetry is not broken. Our model can be realized with ultracoldatoms in optical lattices in the tight-binding regime, and we show that the time scale of the averagecurrent can be controlled by extremely weak fields.
PACS numbers: 05.60.Gg, 03.75.Kk, 37.10.Jk, 67.85.Hj
Brownian motors or ratchets are spatially periodic sys-tems with noise and/or dissipation in which a directedcurrent of particles can emerge from an unbiased zero-mean external force [1, 2]. Models for biological enginesthat transform chemical energy into unidirectional me-chanical motion behave as Brownian motors [3]. Ex-tensive studies of the ratchet effect in classical systems[4] stated the relation between symmetry breaking po-tentials and the existence of the asymptotic current [5].For a system driven by a flashing potential of the form V ( x, t ) = V x ( x ) V t ( t ) with V t ( t ) = V t ( t + T ) of zero meanand V x ( x ) = V x ( x + L x ) there are up to four differentsymmetries in the classical system that must be brokenin order to generate an asymptotic current [6]. A ratchetcurrent arises if one breaks the relevant spatio-temporalsymmetries, here denoted by S x , and the time-reversalsymmetry S t : ( x, p, t ) → ( x, − p, − t + 2 t s ). Lately therehas been an increasing interest in the coherent ratcheteffect in Hamiltonian quantum systems [7]. It has beenshown that the same symmetry requirements apply tothem [6], i.e. if the Hamiltonian preserves any of thesymmetries, no asymptotic current is possible.Experimentally, directed current generation was firststudied in solid state devices, quantum dots and Joseph-son junctions [8]. More recently, the precise controlachievable in cold atom experiments opened up the pos-sibility of realizing directed atomic currents for Hamil-tonian systems with controllable or no dissipation in thetime scale of the measurements [9–12]. Recently, a veryclean realization of a coherent quantum ratchet was ex-perimentally demonstrated in a Bose-Einstein conden-sate exposed to a sawtooth potential realized with anoptical lattice which was periodically modulated in time[13]. Directed transport of atoms was observed whenthe driving lattice potential broke the spatio-temporalsymmetries. The current oscillations and the dependenceof the current on the initial time and the resonant fre-quencies [14] were measured, demonstrating the quantumcharacter of the ratchet.Although the generation of an asymptotic directed cur- rent needs the breaking of the symmetries S x and S t si-multaneously for unbiased potentials, there has been a re-cent discussion on the possibility of obtaining long-lastingdirected currents without it [15–19]. Many-body effects[15, 19] with the proper choice of the initial state [15]or an accidental degeneracy in the quasienergy spectrum[18] may result in a directed current without breaking thetime-reversal symmetry. In contrast to previous workswe show here that one can exploit a quasi-degeneracy,present for a wide range of parameters, in the quasienergyspectrum in order to generate a long-lasting directed av-erage current in a weakly driven system where we canachieve full control over its magnitude and time scale.Previous work on quantum accelerators [12] has shownthat in the presence of quantum resonances one can ob-tain large currents without breaking the time-reversalsymmetry using a delta-kicked potential in time. Essen-tially, for particular values of the Hamiltonian parame-ters the spectrum of the Floquet operator becomes con-tinuous due to quantum resonances. Under such circum-stances one can obtain a linear increase in momentumwith time which has been claimed as a true ratchet effectdriven by resonances instead of noise [20]. However, thereis no formal proof that the dynamics show unbounded ac-celeration for times longer that those that are computed[12]. Nonetheless, a significant difference between quan-tum accelerator ratchets and our system is the existenceof a constant component in the delta potential.One useful way of treating time-periodic quantumHamiltonians, H ( t ) = H ( t + T ), is the Floquet formal-ism [22]. The cyclic states | φ j ( t + T ) i = e − iε j T | φ j ( t ) i are the eigenstates of the evolution operator for oneperiod while the quasienergies ε j are the eigenvalues.The solution to the time-dependent Schr¨odinger equa-tion H ( t ) | ψ ( t ) i = i ~ ∂ | ψ ( t ) i /∂t can be spanned in thecyclic eigenbasis ( ~ = 1) | ψ ( t ) i = X j e − iε j t c j | φ j ( t ) i (1)where c j = h φ j (0) | ψ (0) i . The average current generatedduring n cycles is given by I ( t = nT ) = n P nm =1 I m with I m = 1 T Z mT ( m − T h ψ ( t ) | p | ψ ( t ) i dt, (2)where p is the momentum operator. Note that due tothe periodicity of the cyclic states the average currentduring n cycles can be simplified in terms of integrals ofthe cyclic states during the first period I ( nT ) = 1 n X j,j ′ c j c ∗ j ′ h p i jj ′ − e − inT ( ε j − ε j ′ ) − e iT ( ε j − ε j ′ ) , (3)which is valid for a discrete Floquet spectrum andwhere h p i jj ′ := T R T h φ j ( t ) | p | φ j ′ ( t ) i e − it ( ε j ′ − ε j ) dt .Our model, that represents well optical lattices, has byconstruction a pure point spectrum. In the limit of aninfinite number of states the single-band tight-bindingmodel remains integrable [23]. However, if more bandsare added the spectrum could become absolutely contin-uous (in resonance) or singular continuous. Our guess,based on our previous experience in related problems [24],is that except at very long times even if the spectrum issingular continuous the time evolution will be similar tothat given by a pure point spectrum. In general, a sumof oscillatory off-diagonal terms with arbitrary exponentsdecays rapidly [25] and for long times only the diagonalterms in Eq. (3) remain. If both S x and S t are broken, thecyclic eigenstates desymmetrise and carry net momen-tum, i.e. h p i jj = 0 [6]. In such case, the asymptotic av-erage current at n → ∞ is nonzero I ( ∞ ) = P j | c j | h p i jj .Correspondingly, if either of the relevant symmetries isnot broken h p i jj = 0 and thus I ( ∞ ) = 0. Note, how-ever, that the off-diagonal terms in Eq. (3) become rele-vant if the initial state projects mainly into degenerate orquasidegenerate cyclic states with h p i jj ′ = 0. If one in-duces a resonance between the proper quasienergy statesat low driving, the average current contains only a smallnumber of terms in the sum and the exponents can bevery small, leading to very slow oscillations whose pe-riod can be fitted by tunning the driving. In order tomaximise the average current one should then optimiseboth the projection into the initial states c j c ∗ j ′ and the h p i jj ′ . We illustrate this here and show that it is possibleto populate a high average momentum superposition ofcyclic eigenstates for times which can be tuned up to thelifetime of the experiment.We consider a driven system of non-interacting bosonswith H ( t ) = H + V ( t ) in a lattice of L sites with periodicboundary conditions [26] and H = − J L X l =1 | l i h l + 1 | + | l + 1 i h l | , (4) V ( l, t ) = V sin( ωt ) (cid:20) sin( M πlL ) + α sin( M πlL + φ ) (cid:21) , (5) FIG. 1: (color online) a) Thick solid line shows the aver-age current, I ( t ) in Eq. (3), in recoil units ( k recoil = 2 π/L )for an initial zero momentum state. Thin solid line is thecurrent average per cycle I m in Eq. (2) obtained with a nu-merical integration of the Schr¨odinger equation for M = 5, L = 41, J/ω = 1 . V /ω = 0 . φ/π = 0 . α = 1 . ω is tuned to the resonance conditionin Eq.(6). Dashed line corresponds to the current of the ef-fective 3-level system in Eq. (7) and Eq. (8). The averagecurrent attains a maximum M . b) Numerical calculation of I ( t ) for same parameters except for solid line V /ω = 0 . φ/π = 0 . V /ω = 10 − √ φ/π = − . V /ω = 10 − , φ/π = 0 . V − ω . with M integer, where J is the tunneling probability and | l i represents the state of a boson located on site l . Theeigenenergies of H are E k = − J cos(2 πk/L ) with inte-ger k = [ − k max , k max ] where k max = ( L − / L oddand the corresponding momentum eigenvectors h l | k i = e i k πlL / √ L are degenerate for ± k . For convenience onecan introduce the basis h l | s k i = p /L cos( k πl/L ) and h l | a k i = p /L sin( k πl/L ) which are symmetric or an-tisymmetric under the inversion of k . We add a timeand spatially modulated periodic function V ( l, t ) withfrequency ω = 2 π/T tuned to the M -dependent resonantcondition [16, 27]2 ω = E M − E = 2 J (1 − cos(4 πM/L )) (6)and consider the zero momentum | i as initial state.Note that in contrast to [16, 27] we add the parameters M and φ to the driving potential in Eq.(5). Parameter M ≤ k max / | M i and | M i , and theparameter φ , key to our model, allows for the couplingbetween the | s k i and | a k i basis states.Our choice of a two-harmonic spatial potential timesa monochromatic time-dependent potential implies thatsymmetry S t (labelled S in [6]) is not broken for t s = π/ (2 ω ). Therefore h p i jj = 0 and no asymptotic currentis possible for our system. The average current generated FIG. 2: (color online) Sketch of the second order processin Eq. (7) that takes place for weak coupling
V /J < δ indicates the quasienergy shifts induced by the poten-tial in Eq. (5). Arrows represent the couplings Ω between theFloquet basis states induced by the driving. The optimal cur-rent shown in Fig. 1 is obtained when the mixing between thesymmetric and antisymmetric states is maximized and bothcouplings in this diagram have the same weight. after n cycles arises only from crossed terms between thecyclic states. Our aim is to maximise the average cur-rent during any experimental time t e . The key ingredi-ents are to keep few terms in the sum in Eq.(3), withsmall exponents and relevant prefactors. The two firstare achieved by tuning the resonance in Eq. (6) withweak driving V /J <
1. We show that the prefactorscan be successfully optimized if the quasiresonant cyclicstates that have non-zero projection into the initial zeromomentum state mix the symmetric and antisymmetricmomentum states, which is obtained for φ = l π for l integer. For φ = lπ , accidental degeneracies could inprinciple allow to obtain a small non asymptotic currentfor some specific parameters and a particular value ofthe coupling [18]. In contrast, we show that for φ = lπ/ M , φ and α for any L and J set by H , to obtain an aver-age current that can be tuned up to near optimal value I ( t e ) ≃ M ≃ k max /
4, for a time interval [0 , t e ] where t e can be independently tuned by adjusting the drivingstrength V /J .We show in fig 1a) the average current I and the os-cillating average current per cycle I m as a function ofthe number of cycles. We note that the average currentachieves a maximum M in recoil units and, as expected,vanishes for long times. We observe in fig 1b) that thecurrent changes direction with a sign change in φ and t e scales with V − ω . For the weak driving strength usedhere the current is nearly zero ( I ≤ − ) for φ = 0.The previous results are obtained from a full numer-ical calculation. In order to further understand the ef- fect we follow the methodology of [27] and develop anapproximate perturbative model. We find that for ourpotential this simplified model also explains the mainfeatures and gives very accurate results for low driving.Close to the resonance Eq. (6) and for weak driving V /J < {| s M , i , | , i , | a M , i} , where | j i = | k, n i with h t | n i = e − inωt . We apply time-independent perturbation theory in Floquet space, usingthe T -matrix approach T ( ǫ ) = V + V G ( ǫ ) T [28], where G ( ǫ ) = P j | j ih j | ǫ − ε j and ε j ≡ E k − nω . Around the groundstate quasienergy ǫ = ε , the first non-zero term connect-ing the three states is given by the second order in theexpansion T ( ε ) ≃ V G ( ε ) V which reduces to T ≃ V δ s ( α, φ ) Ω M Ω M ( α, φ )Ω M δ ( α ) 0Ω ∗ M ( α, φ ) 0 δ a ( α, φ ) ! . (7)A sketch of the relevant processes is depicted in Fig.2and the exact values of the matrix elements, inverse ofquasienergy differences which depend on M/L , can befound in the complementary material. The energy shiftsare δ and the couplings Ω that correspond to each partof the potential in Eq. (5) are indicated by the subindex M or 2 M . Remarkably, Ω M ∝ α sin(2 φ ) and thus thecoupling between the symmetric and antisymmetric basisstates requires φ = 0. The other effect of φ is to bringthose states closer in energy δ s − δ a ∝ α cos(2 φ ). Theoptimal current is obtained when the cyclic states (re-lated to eigenvectors of the above matrix) mix the threebasis states on an equal foot, corresponding to the twosecond order processes sketched in Fig. 2 being of thesame order. Due to the structure of the spectrum, thisoptimal mixing can be reached for M ≤ k max /
4. Then allthe matrix elements in Eq. (7) are of order
O ≃ /ω andthe quasienergies ε j ≃ V /ω , leading to the time scale ofthe dynamics shown in Fig. 1b).For an initial state | ψ (0) i = | i , the average currentat cycle m after evolution with the effective HamiltonianEq. (7) reduces to a sum of 3 oscillatory terms with dif-ferent frequencies and the same weight I m = C ( α, φ ) X j 4, the current is linearwith M . Thus, we can set M opt as the closest integerto k max / I ≃ M opt . As shown in figure 1(dashed line), the three-mode approximation Eq. (8) fitsperfectly the exact numerical results.We show the current amplitude C in the left panel offigure 3 for different parameters α and φ . As explained α φ / π φ / π FIG. 3: (color online) Left panel: Amplitude C / M of thecurrent in Eq.(8) for different values of the potential parame-ters φ and α for an initial zero momentum state and L = 41, M = 5, J/ω = 1 . T -matrix in Eq.(7) in units of V / (2 ω ) as a functionof φ for α = 1 . 2. For fixed values φ and α the average currentper cycle in Eq. (8) is a sum of sinusoidals with an ampli-tude C shown in the left panel and frequencies given by thequasienergy differences. above, it attains its maximum at α opt ≃ . φ with π periodicity and has vanishing valuesfor φ = lπ/ l integer. For φ opt ≃ ± π/ ε f = 0 . V /ω in the sum in Eq. (8) while theother sinusoidals oscillate with half this frequency. Onecan then average I m over different periods to obtain I which achieves its maximum I ( t e = n e T ) ≃ M after n e = 1 . V /ω ) − cycles as shown in fig. 1 b).In the context of cold atoms it may be of interest notonly the generation of a current from an initial zero mo-mentum state, but also the control of the quantum stateof the system. We plot in fig. 4 the particle state inthe momentum basis and the average current per cycleand the average kinetic energy R T dt h H i /T . We observethat the zero momentum state can be indeed convertedinto an almost pure momentum state | ± M i . One couldthen switch off the driving, thus breaking the time re-versal symmetry S t , and use this scheme to generate anasymptotic current. This is an example of the high con-trollability of our system.Finally, let us analyze the feasibility of the model.We can summarise our findings in a simple recipe. H sets the energy scale J and the length L of the sys-tem. We can obtain a final average current I of nearly( L − / , t e ]if one tunes a driving potential in Eq. (5) with param-eters ω from Eq. (6), M opt , α opt , φ opt defined aboveand V = p . J (1 − cos(4 πM opt /L )) /t e with the con-straint that V /J ≤ 1. We show in fig 3 that small changesin α and φ around optimal values only slightly affect thecurrent. Smaller M would reduce the maximum averagecurrent attained and require adjustment of the resonancecondition in Eq.(6). Thus the only actual requirementsof our model are that the system is tuned to resonance FIG. 4: Upper panel: Real (thick bar) and imaginary part(thin bar) of the {h− M | , h | , h M |} | ψ ( t ) i at t = 0 and atthe times showed by vertical lines in the lower panel. Lowerpanel: Average current per cycle I m /M in recoil units andaverage tunneling energy per cycle (dashed line) in units of1 /ω as a function of time. Same parameters as in fig. 1 a). 20 40 60 80 100 120−4−20 t/T c u rr en t FIG. 5: Average current in recoil units in Eq. (3) as a functionof time for an initial zero momentum state. Parameters M =5 , L = 41, J = 1 . φ/π = 0 . α = 1 . 2. Dashedline corresponds to V = 0 . ω = 1 .