Quantum Ratchets for Quantum Communication with Optical Superlattices
aa r X i v : . [ qu a n t - ph ] N ov Quantum Ratchets for Quantum Communication with Optical Superlattices
Oriol Romero-Isart
Departament de F´ısica, Grup de F´ısica Te`orica,Universitat Aut`onoma de Barcelona, E-08193 Bellaterra, Spain.
Juan Jos´e Garc´ıa-Ripoll
Facultad de CC. F´ısicas, Universidad Complutense de Madrid,Ciudad Universitaria s/n, Madrid, E-28040, Spain.
We propose to use a quantum ratchet to transport quantum information in a chain of atomstrapped in an optical superlattice. The quantum ratchet is created by a continuous modulation ofthe optical superlattice which is periodic in time and in space. Though there is zero average forceacting on the atoms, we show that indeed the ratchet effect permits atoms on even and odd sitesto move along opposite directions. By loading the optical lattice with two-level bosonic atoms, thisscheme permits to perfectly transport a qubit or entangled state imprinted in one or more atoms toany desired position in the lattice. From the quantum computation point of view, the transport isachieved by a smooth concatenation of perfect swap gates. We analyze setups with noninteractingand interacting particles and in the latter case we use the tools of optimal control to design optimalmodulations. We also discuss the feasibility of this method in current experiments.
I. INTRODUCTION
By a ratchet effect one usually refers to the existenceof directed transport in a system in which there is nonet bias force. Ratchets have been traditionally foundin the study of dissipative systems [1], where externalfluctuations causing Brownian motion cooperate with aperiodic force to bias transport, a mechanism that is inthe basis of some biological motors [2]. Ratchets can alsoappear without dissipation. These Hamiltonian or con-servative ratchets are interesting as they can be extendedto quantum mechanical systems. In this context one findsstudies that relate the existence of transport to classicalproperties of the model, such as some asymmetries of theexternal force [3] or mixing of chaotic and regular phasespace regions [4].Ultracold atoms offer an ideal arena to test this phe-nomenology, as can be seen from the experiments im-plementating both dissipative and conservative ratchets[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. These experimentsrely on the force imparted by near or far from resonancelaser beams which act on the atoms over short periodsof time. These flashing potentials implement variants ofthe δ -kicked rotor model and lead to phenomena such asdynamical localization. On the theoretical side we mustremark the discovery of directed transport on quantummechanical systems whose classical counterpart is com-pletely chaotic [16, 17, 18, 19, 20].The present work aims at exploiting the fact that quan-tum ratchets can be used to transport quantum statesand thus distribute entanglement between distant nodes,a basic task for quantum information and computationpurposes [21]. Within this context, we find two dis-tinct research directions. On the one hand, entanglementswapping and quantum teleportation [22] can be used tobuild efficient quantum repeaters [23, 24] for long dis-tance communication. On the other hand, on a smaller FIG. 1: (Color online)An optical superlattice arises from acombination of two potentials with different periods. By mod-ulating the depths and displacements of these potentials wecan raise and lower the tunneling rates between odd and evenpairs of sites (red arrows). This way, by means of perfectswaps, the state of a particle can be transported along thelattice. scale, a relevant effort has been recently devoted to studyquantum state transfer using static local Hamiltonianswhich act on chains of spins, harmonic oscillators, orbosons in optical lattices [25, 26, 27, 28, 29, 30, 31, 32,33, 34, 35, 36, 37, 38, 39, 40]. Compared to these works,time-dependent (non-adiabatic) models, such as the onepresented here, provide more degrees of freedom to opti-mize the efficiency, speed and robustness of the quantumtransport.Moreover, the scheme we propose here for quantumstate transport can be seen as a quantum ratchet in-duced without breaking translational and time reversalsymmetries in the forces. The transport is just triggeredby the symmetry breaking of the initial state. We illus-trate this with a particular and simple implementation ofquantum state transport based on two-level bosonic ul-tracold atoms in optical superlattices. More specificallythe transport is achieved as follows. All the atoms areinitially prepared in the, say, down state. At a givensite (write port) an arbitrary qubit state is imprintedin the atom. The trapping potential is then modulatedsmoothly and periodically in time and in space. Depend-ing on the initial asymmetric position of the atom, beingin an even or odd lattice site, the qubit state is trans-ported rightwards or leftwards to any desired lattice site(read port) [Fig. 1] without precise individual addressing[53]. We consider both free and interacting particles. Toour knowledge, this is the first time many-body interac-tions are taken into account in Hamiltonian ratchets.From the implementation point of view, this work isinspired by recent advances in the control and manipula-tion of optical superlattices [41, 42, 43, 44, 45]. Presentexperiments can create periodic potentials such as theones depicted in Fig. 1, achieving a great control on thetime-dependence of the potential heights, and even beingable to measure the number of atoms on the even and oddsites [41, 42]. These tools suffice for the protocol devisedin this work.The outline of the paper is as follows. In Sect. II weintroduce our system, made of a chain of atoms in an op-tical superlattice. We state the goal of this work, which isto transfer a quantum state between two arbitrary latticesites using a translationally invariant modulation of thesuperlattice, and we present the Bose-Hubbard Hamilto-nian that models the dynamics of the atoms. In Sect. IIIwe study the case of noninteracting particles and findthat it is possible to induce directed transport with arbi-trary speed. In Sect. IV we discuss a more realistic setupin which atoms interact. We design a ratchet Hamil-tonian with the help of optimal quantum control, andfind that we can still induce perfect quantum transportwith a speed limited by the interaction strength. Thedetails of the optimal quantum control technique are leftto Sect. VI. In Sect. V we study the optical superlatticemodulation in more detail, discussing the experimentalchallenges for implementing these quantum ratchets incurrent experiments. Finally, in Sect. VII we summarizeour results and comment on possible extensions.
II. OPTICAL SUPERLATTICESA. The model
In this work we consider the setting of ultracold neu-tral atoms trapped in an optical superlattice. An opticallattice is a standing wave of coherent off-resonance lightcreated by the interference of two or more laser beams.As explained in Ref. [46], such standing wave behaves as aperiodic potential that confines the atom in the minima or maxima of intensity. The optical superlattice arisesfrom a combination of two potentials with different peri-ods. In the simplest case the superlattice potential canbe written as follows V ( x, y, z, t ) = V x ( t ) cos ( kx ) + V ( t ) cos (2 kx + φ ) ++ V ⊥ [cos ( ky ) + cos ( kz )] . (1)Here V x ( t ) and V ⊥ are the strengths of the lattice alongthe main axis and transversely to it, and k = 2 π/λ is themomentum of the photons. In order to have a configu-ration of decoupled 1D lattices [47], the transverse po-tential V ⊥ has to be much larger than the recoil energy E r = ~ k / m to prevent tunneling between 1D lattices.On top of this potential and along the main axis we findanother lattice of strength V ( t ), with half the period ofthe original one and a possible dephasing φ ∈ Z × π/ H = X σ = ↑ , ↓ L X i =1 (cid:20) − J i ( a † σi a σi +1 + H . c . ) + U a † σi a σi (cid:21) . (2)Here, a σi and a † σi are Fock operators that annihilate andcreate particles on the i -th site of the lattice and in oneof two internal states, σ ∈ {↑ , ↓} . The parameter U is theon-site interaction between atoms, which we assume tobe spin independent, and J i is the tunneling amplitude,which may vary from site to site.In order to describe the dynamics of the atoms, Eq. (2)has to satisfy two conditions. First of all, the tunnel-ing J i between different wells has to be small, so thatthere effectively exists a low energy band formed by thelocalized states on each site. This implies that the su-perlattice cannot drop the barrier between pairs of sitestoo much [Fig. 1], for otherwise we would have to usea multiband model. The second condition is that theon-site interaction U must be small compared to theenergy gap between Bloch bands. Both condition areeasily met in current experiments with optical superlat-tices [41, 43, 44, 45], and have been taken into accountthroughout this work [See Sect. V]. B. The goal: quantum communication
Our objective is as follows. We initially prepare all thetwo-level atoms in the same state, say the down state |↓i . We imprint an arbitrary quantum state on a partic-ular site of the lattice, the write port , in such a way thatthe atom ends up in a superposition of both spin states, α |↑i + β |↓i . The goal is to create, by modulating the en-ergy barriers of the optical superlattice, a ratchet effectthat will transport this state a predetermined distance,to another site, the read port in Fig. 1. The modula-tion of the superlattice will not break the translationalinvariance and it will also be periodic in time (with aperiod equal to 2 T ), alternating the roles of the hoppingbetween even ( J n = J ) and odd ( J n +1 = J ) sites J , ( t + T ) = J , ( t ) . (3)The Hamiltonian in Eq. (2), being translationaly in-variant in time and in space, does not exert any force onthe atoms, as expected from a quantum ratchet. Thiscan be verified by computing the integral of the forcesacting on an atom in a well of the lattice. In particular,the following average is zero Z T dt Z π k dx ddx V ( x, y, z, t ) = 0 . (4)Note however that our superlattice does not have a per-manent asymmetry on the unit cell [0 , π/ k ), unlike otherHamiltonian flashed ratchets with saw-tooth profiles [8].While the average force acting on the atoms is zero,we will show that there is still transport. The reason isthe asymmetry of the initial state | ψ (0) i = ( αa †↑ in + βa †↓ in ) Y j =in a †↓ j | vac i , (5)and the fact that our time-dependent Hamiltonian willperform perfect swaps between neighboring lattice sites.All of this leads, after n swaps of duration T , to a prop-agation of the imprinted qubit rightwards or leftwards | ψ ( nT ) i = ( αa †↑ out + βa †↓ out ) Y j =out a †↓ j | vac i , (6)depending on the starting site. Note also that by design,we will achieve the transport of the quantum state with-out actually moving any atom, thus leaving the numberof atoms per site invariant.Finally, let us mention that this scheme can also beused to distribute entanglement in the same fashion asin [27]. Namely by imprinting a maximally entangledstate in two neighboring lattice sites. Since atoms in oddand even sites move on opposite directions, the atomswill depart from each other and entanglement will getdistributed between arbitrarily distant lattice sites. III. NONINTERACTING CASE
We start with the case of noninteracting particles U = 0. This case is relatively easy to study as all parti-cles can be treated independently and therefore we canfocus on the dynamics of a single particle that starts fromdifferent sites, a problem which is integrable. The basis of states is now denoted by | j i := a † j | vac i , where we nolonger care about the internal state of the particle. Ouronly concern is now to ensure that the state with a par-ticle on the write port | j = in i is mapped after a giventime, t r , to a state with that particle on site | j = out i ,the reading port, in a deterministic fashion. Note alsothat since particles do not interact we do not have toconsider the dynamics of particles on other lattice sites.Nevertheless, as we show below, it is possible to look forsolutions that do not give energy to these other particlesand leave the density profile invariant. If the number ofatoms per lattice site is constant, we can then talk abouttransport of the quantum state and not of particles them-selves [Fig. 1].As described before, we will adopt two restrictions.The first one is that the couplings do not break the trans-lational invariance but are modulated in time, with theroles of J and J being exchanged after a time T , as fromEq. (3). The second one is to assume that the tunnelingcan be brought to zero, so that J ( t ) ≥ , J ( t ) = 0 , t ∈ [0 , T ) ,J , ( t + T ) := J , ( t ) . (7)During the time in which J = 0 we have ( ~ = 1) (cid:18) a n +1 ( t ) a n +2 ( t ) (cid:19) = { cos[ θ ( t )] I + i sin[ θ ( t )] σ x } (cid:18) a n +1 (0) a n +2 (0) (cid:19) , (8)where σ x is a Pauli matrix and θ ( t ) = R t J ( τ ) dτ . Thesimplest solution of this kind is a square signal, J ( t ) = J ,as shown in Fig. 2b. This means that for T = π/ (2 J ) weachieve a perfect swap between even and odd sites upto an irrelevant global phase. In practice the tunnelings J and J will evolve smoothly and require some time toreach a nonzero value. In that case the time for perfectswitch will be given by θ ( T ) = π/ J ∝ /T , so that a fast solution mayrequire a lattice with an unrealistically large hopping. IV. INTERACTING CASE
The case of interacting particles is more realistic butalso more difficult. We can no longer regard the parti-cles as independent and collisions can affect the phase ofthe transported state, as defined by Eq. (5). We willstill look for simple solutions that concatenate a swapdynamics on pairs of lattice sites, with the restrictionsintroduced in Sec. II B.
A. A two-level problem
Let us repeat the calculation of Sect. III but now tak-ing into account the interaction between particles. Since
FIG. 2: (Color online) (a) Model case of two lattice sitesdisconnected from the rest because J = 0. By fixing thehopping between wells to a precise value J for a time T , it ispossible to swap the atoms. Note that the doubly occupiedsites have more energy due to the interaction, U , and a possi-ble inhomogeneity of the lattice potential, ∆. (b) Combiningthe solution for a pair of sites we obtain one possible mod-ulation of the hoppings J (solid) and J (dashed), namely J and J are square waves in antiphase. This produces thequantum transport. (c) Values of J and T for which perfecttransport is achieved. Each circle is a solution; the solid linejoins the solutions with smallest hopping and the dashed lineis the solution for noninteracting particles. J = 0 we focus on the double-well problem with onlytwo distinguishable particles, i.e. in two different inter-nal states. Instead of using second quantization we de-note their state as | i i| j i , where i, j ∈ { , } is the well in which the particle resides. The connections betweenstates are depicted in Fig. 2a. Using Pauli operators theeffective Hamiltonian becomes [54] H = − J ( σ x ⊗ I + I ⊗ σ x ) + U σ z ⊗ σ z + 1) (9)We can further restrict our problem to the only statesthat participate on the dynamics | ψ − i := 1 √ | i − | i ) , (10) | ψ + i := 1 √ | i + | i ) , | φ + i := 1 √ | i + | i ) . Using this basis we obtain the effective Hamiltonian H ( t ) = − J ( t )0 − J ( t ) U , (11)where | ψ − i is shown to be a dark state and the other twoare coupled by a two-level Hamiltonian 2 J ( t ) σ x + U σ z .Our problem is thus to find a hopping J ( t ) such that aftera time T the states above have experienced the followingtransformation | ψ − i → | ψ − i , | ψ + i → −| ψ + i , | φ + i → e iν | φ + i , (12)where the phase ν is unimportant for our purposes. B. The square signal revisited
We will first investigate solutions which are piecewiseconstant, with J ( t ) = J, for t ∈ [0 , T ). For a fixedhopping our Hamiltonian has two nonzero eigenvalues E ± = U ± √ J + U . (13)that contribute to the evolution of the symmetric states.The perfect swap between the atoms takes place at atime T such that the symmetric state | ψ + i changes sign.As | ψ + i can be written as a superposition of the twosymmetric eigenstates with energies E ± , this conditionis satisfied when E ± T = ± (2 n ± + 1) π , where n + and n − are arbitrary integers. Defining x := (2 n + + 1) / (2 n − + 1)we obtain UJ = 2 x − √ x . (14)From this value we can compute the energies E ± andthe time T . As shown in Fig. 2c the minimal value of J is reached for either n + or n − equal to 0 and, unlikein the noninteracting case, there exists a minimal swaptime given by T = 2 π/U . Indeed, using the tools in [48]one can prove that the constant hopping is the fastestsolution and that the gate cannot be performed fasterthan this time. FIG. 3: (Color online)(a) Hopping and (b) fidelity of the gate( F = |h | U ( t ) | i| ) with the perfect transport for half a pe-riod [0 , T ) during which J ( t ) = J ( t ) and J = 0. We findtwo solutions, one for T = 2 π/U (solid) and another one for T = 4 π/U (dashed), with three and two modes, respectively.In (c) we plot the corresponding modulations of the superlat-tice. C. Smooth solutions: optimal quantum control
While optimal, the square signal that we have con-sidered before is probably unrealistic, as in experimentsthe hopping will smoothly increase and decrease as thetunneling barriers are changed. For that reason we haveinvestigated other solutions with continuous derivativesusing the tools of optimal quantum control.The details of the method are left for Sect. VI, but letus sketch the procedure. The first step is to parametrizethe hopping as a linear combination of some functions J ( t ) = X n c n f n ( t ) ≥ f n ( t ) = sin ( πnt/T ) that increase and decrease smoothlyto zero. Since the hopping is positive we have a firstrestriction, c n ≥
0. The second restriction is that thefidelity of the swap procedure has to be one. Both con-straints are imposed to the problem of minimizing the
FIG. 4: (Color online)On the upper figure we plot a solutionof J ( t ) (solid) and J ( t ) (dashed) for a perfect transport ofthe qubit. In the lower figure we plot the average position ofthe qubit on the lattice as transported by these modulations.These plots have been computed using T = 4 π/U . average strength of the hopping, given by E = P n | c n | .This problem is solved numerically with MATLAB’s op-timization toolbox [49] using as aid the derivatives com-puted by means of perturbation theory [Sect. VI].In Fig. 3 we show two instances of the problem, oneoptimized for three modes ( c n = 0 for n >
3) and aduration of T = 2 π/U , the other one for twice the timeand two modes. As it can be appreciated in the picture,we reach perfect fidelity in a rather smooth and robustmanner, so that errors in the timing of the gate will notaffect the process significantly.To further relate these solutions to the notion of aratchet let us look at Fig. 4. There we plot the full timeevolution of the imprinted qubit state for 6 periods of anoptimal modulation. The position may oscillate betweenpairs wells, but there is always a net average transport. D. Dealing with holes
The two solutions derived above, that is the piecewiseconstant and the smooth ones, are designed to inducetransport on a chain of particles. However, in practicesuch a chain will have some endpoints or particles thatstand near an empty site. We can then have three sce-narios: (i) that a particle standing near a hole ends up inthe original site at time t = T ; (ii) that the particle andthe hole are swapped and, more generally, (iii) that theparticle and the hole end up in some coherent superpo-sition of being on each site. Out of these processes, only FIG. 5: (Color online)Parameters of the double well potentialas a function of the superlattice modulation, ∆ V = ( V x + V ) /
2. We plot the effective hopping between wells, J (dash-dot, right axis), the energy gap to higher bands, ∆ E (dashed),and the on-site interaction, u (solid). Everything is expressedin units of the recoil energy, E r . the latter will affect the evolution of the state we want totransport, since there is a small probability that it getsreflected.Holes only have a disturbing effect on the transport if U = 0, since in the noninteracting case surrounding par-ticles are equivalent to holes. Note however that we caneffectively eliminate the scenario (iii) if we impose thatthe hopping on half a period leaves the particle invariant Z T J ( τ ) dτ = 2 π × Z , (16)While this restriction cannot always be achieved for thepiecewise constant profile, it can be easily incorporatedto our optimal control toolbox. V. EXPERIMENTAL IMPLEMENTATION
All the protocols that we have designed can be imple-mented in current experiments with optical superlattices[41, 42, 43, 44, 45]. The implementation should beginby loading the superlattice with approximately one atomper site, all of them in the same internal state. The nextstep is to rotate the state of some atoms using eithermagnetic field gradients, coherent light or a clever com-bination of both [50], in order to imprint the quantumstate we wish to transport. By modulating in time theintensity of the laser beams that participate in the opti-cal lattice one can then achieve a modulation of J ( t ) and J ( t ) that corresponds to the solutions studied above. Af-ter an appropiate time one may retrieve the qubit stateby measuring the lattice sites on which it is expected toarrive.In an experiment one does not directly control the hop-pings J ( t ) and J ( t ), but rather the lattice strengths V x and V [See Eq. (1)]. In order to relate these two quanti-ties we have have performed a band structure calculation.We have focused on the case J ( t ) ≃
0, which correspondsto the first half-period, t ∈ [0 , T ). This hopping can besuppressed by making φ = 0 and V x + V = 70 E r , whichis large enough to effectively suppress hopping every sec-ond site. Given this constraint, any modulation of thelattices depends on a single quantitiy, ∆ V ( t ) ≥
0, suchthat V x = ∆ V ( t ) and V = 70 E r − ∆ V ( t ). It thus remainsto relate ∆ V ( t ) and J ( t ). This is done for each possi-ble value of ∆ V , by computing numerically the groundstate wavefunction of a particle in a double-well, w ( x ),and the first excited state. The energy difference be-tween these states is proportional to the effective tunnel-ing J ( t ), while the on-site interaction energy becomes UE r = a s a u, (17)where a s is the scattering length between atoms, a theperiod of the superlattice and u = R | w ( x ) | .The results are shown in Fig. 5. Since a s /a is verysmall, the on-site interaction energy is smaller than theenergy gap between Bloch bands and the Bose-Hubbardmodel is therefore valid throughout the evolution (2).Furthermore, u does not change very much as a func-tion of the modulation ∆ V , while J decays exponentiallyfast with ∆ V . After an appropiate fit of these quanti-ties, one may convert the time dependence of J ( t ) fromFig. 3a into the associated modulations of the superlat-tice ∆ V ( t ), which are shown in Fig. 3c. VI. COHERENT CONTROL
In this section we present the tools that we have usedto optimize the modulation of the hopping, as they differsignificantly from what is the standard approach in opti-mal quantum control based on a Lagrangian formulation[51].
A. Objective function
Let us formulate our problem: we have a Hamilto-nian, H ( t ; x ) that depends both on time and on someadditional parameters, x . . . x M . Our goal is to find anoptimal set of parameters of the Hamiltonian such thatthe evolution of a number of states is as equivalent aspossible to a transformation given by a specific unitary U g .Stated in a more concrete way, the evolution of an ar-bitrary state | ψ (0) i is given by | ψ ( t ) i = U ( t ; x ) | ψ (0) i ,where the unitary is a solution of the Schr¨odinger equa-tion i ddt U ( t ; x ) = H ( t ; x ) U ( t ; x ) (18)with initial condition U (0; x ) = I [55]. Our goal is tomaximize the fidelity (defined below) of any evolved state U ( t ; x ) | ψ n i with the desired transformed state U g | ψ n i .There are many ways to measure the accuracy of thetransformation. A strict and simple objective function isthe fidelity F = 1 d Re { tr[ U † g U ( T ; x )] } = 1 d Re d X n =1 h ψ n | U † g U ( T ; x ) | ψ n i , (19)where the {| ψ n i} form an orthonormal basis of the d -dimensional Hilbert space H where we want to controlthe evolution [56]. This function is bounded by F ∈ [ − ,
1] and achieves the maximum value for the perfecttransformation, F = 1 ⇔ U ( T ; x ) = U g . (20)The question is thus, how do we maximize F ? A natu-ral way is to compute (when possible) the derivative of F with respect to the parameters x , i.e. ∂F/∂x i , sincethe gradient itself provides a direction along which thefidelity is increased. Indeed, given this derivative thereare multiple optimization algorithms that would allow usto compute the optimal control. B. Formal gradient
In order to obtain the gradient of F with respect tothe parameters x , we straightforwardly obtain from (19) ∂F∂x i = 1 d Re X n (cid:28) ψ n (cid:12)(cid:12)(cid:12)(cid:12) U † g ∂∂x i U ( T ; x ) (cid:12)(cid:12)(cid:12)(cid:12) ψ n (cid:29) , (21)which relates the gradient of F to a derivative of the uni-tary operator. What follows is a simple way to compute ∂U/∂x i which is based on performing a Taylor expansionof the operator U ( t ; x ) with respect to the parameters xU ( t ; x + ǫ ) = U ( t ; x ) + ǫ ∂U∂x ( t ; x ) + O ( ǫ ) . (22)We will obtain this series using time-dependent pertur-bation theory on the Schr¨odinger equation (18), whichwill enable us to identify the derivative of the unitaryoperator.Let us assume that by changing x → x + ǫ our Hamilto-nian decomposes into an unperturbed part, H = H ( t ; x )and a perturbation H ǫH ≡ H ( t ; x + ǫ ) − H = ǫ ∂H∂x ( t ; x ) + O ( ǫ ) . (23)The new unitary operator will satisfy a Schro¨odingerequation with a modified Hamiltonian i ddt U ( t ; x + ǫ ) = ( H + ǫH ) U ( t ; x + ǫ ) , (24) and same initial condition U (0; x + ǫ ) = I . It is now con-venient to move to the interaction picture U ( t ; x + ǫ ) ≡ U ( t ; x ) W ( t ; x ) , (25)which leads to a simpler equation i ddt W ( t ; x ) = ǫU ( t ; x ) † H ( t ; x ) U ( t ; x ) W ( t ; x ) . (26)Integrating formally this differential equation and iterat-ing the resulting formula, one obtains the usual Dysonseries. We only need this series up to first order W ( t ; x ) = I − iǫ Z t dτ U ( τ ; x ) † H ( t ; x ) U ( τ ; x ) + O ( ǫ ) . (27)From (25) combined with (27) we have U ( t ; x + ǫ ) = U ( t ; x ) − (28) − iǫU ( t ) Z t dτ U ( τ ) † H ( τ ) U ( τ ) + O ( ǫ )(In the second term and hereafter we omit the x depen-dence to ease the notation). We can now compare thisexpression with (22) in order to identify the derivative ofthe unitary, that is ∂∂x U ( t ) = − iǫU ( t ) Z t dτ U ( τ ) † ∂H∂x ( τ ) U ( τ ) . (29)Therefore we obtain the formula for the gradient of thefidelity ∂F∂x i = 1 d Re X n Z T dτ (cid:28) U † g U ( T ) U ( τ ) † ∂H∂x i ( τ ) U ( τ ) (cid:29) ψ n . (30) C. Development of the algorithm
Even though we have a closed expression for the deriva-tive of the fidelity, we still need to compute the integralwhich appears in Eq. (30). We have devised a simple,accurate and efficient procedure which is based on solv-ing three sets of ordinary differential equations. The firstone is obtained by transforming the integral in Eq. (30)into d × M ordinary differential equation ddt f n,i ( t ) = 1 d Re (cid:28) ψ n (cid:12)(cid:12)(cid:12)(cid:12) U † g U ( T ) U ( t ) † ∂H∂x i ( t ) U ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ψ n (cid:29) (31)with initial condition f n,i (0) = 0 and final value ∂F∂x i = X n f n,i ( T ) . (32)We now notice that left side of the scalar product inEq. (31) is the state U ( t ) U ( T ) † U g | ψ n i and can be com-puted by solving an ordinary differential equation i ddt | ξ n ( t ) i = H ( t ) | ξ n ( t ) i , (33)with initial condition | ξ n ( T ) i = U g | ψ n i and moving back-wards in time from T to t .We now have all the ingredients to design the protocolthat computes our derivative (30). We summarize it asfollows. First, solve Eq. (33) backwards in time up to t = 0, finding | ξ n (0) i := U ( T ) † U g | ψ n i . (34)We then solve the system of ordinary differential equa-tions i ddt | ψ n ( t ) i := H ( t ) | ψ n ( t ) i (35a) i ddt | ξ n ( t ) i := H ( t ) | ξ n ( t ) i (35b) ddt f n,i := 1 d Im h ξ n ( t ) | ∂H∂x i ( t ) | ψ n ( t ) i . (35c)With the initial conditions | ψ n (0) i being some orthonor-mal basis of our Hilbert space, | ξ n (0) i computed beforeand f n,i (0) = 0. The value of the derivative is then com-puted using Eq. (32).This derivative and the formulas given above can be fedto any optimization package, such as a simple line searchalgorithm or a nonlinear conjugate gradient method. Inparticular we have used Matlab’s nonlinear optimizationtoolbox [49]. VII. CONCLUSIONS
In this work we have proposed to generate quantumtransport using cold atoms in an optical superlattice thatis modulated periodically both in time and in space.Since there is no average force acting on the atoms, we deal with a quantum ratchet effect that makes atoms oneven and odd sites move along opposite directions.We have demonstrated that the ratchet effect can beinduced both for free particles and in the case of nonzeroon-site interactions. The latter represents to our knowl-edge the first time strong many-body interactions havebeen treated exactly in Hamiltonian ratchets [57]This ratchet effect can be used to transport quantuminformation by imprinting a qubit or an entangled stateon one or more atoms of the atomic chain and letting thesystem evolve according to our ratchet potentials. Thedynamics generated with our scheme corresponds to asmooth concatenation of perfect swap gates. Therefore,after a time n × T , we will find that the qubit state hasbeen transported a well determined distance, of order n ,along the lattice.Our ideas could be implemented in current experi-ments with optical superlattices. From the quantum in-formation point of view, compared to other proposalsbased on effective spin interactions between atoms, theratchet effect should be faster and lead to a more flexi-ble dynamics. From the fundamental point of view, webelieve that the modulated superlattices are a rich play-ground in which to study transport phenomena, quantumdiffusion and the influence of noise and of chaos in thetransport of quantum states.We thank J. Calsamiglia for useful discussions. Weare grateful to A. Sanpera, R. Munoz-Tapia, and A. Kayfor a careful reading of the manuscript. O.R.I. acknowl-edges support from the spanish MEC grants AP2005-0595, FIS2005-03169, Consolider-Ingenio2010 CSD2006-00019 QOIT, and Catalan grant SGR-00185. 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