We investigate coherent control of a single atom interacting with an optical bipartite lattice via a combined high-frequency modulation. Our analytical results show that the quantum tunneling and dynamical localization can depend on phase difference between the modulation components, which leads to a different route for the coherent destruction of tunneling and a convenient phase-control method for stabilizing the system to implement the directed transport of atom. The similar directed transport and the phase-controlled quantum transition are revealed for the corresponding many-particle system. The results can be referable for experimentally manipulating quantum transport and transition of cold atoms in the tilted and shaken optical bipartite lattice or of analogical optical two-mode quantum beam splitter, and also can be extended to other optical and solid-state systems.
© 2014 Optical Society of America
Quantum control of tunneling processes of particles plays a major role in different areas of physics, optics and chemistry [1–4]. As early as 1986, Dunlap and Kenkre studied theoretically the quantum motion of a charged particle on a discrete lattice driven by an ac field , and found the surprising result that particle transport can be completely suppressed when ratio of the strength and the frequency of the ac field takes some special values. This effect of extreme localization was later found to be associated with the coherent destruction of tunneling (CDT) [2, 6, 7] at a collapse point of the Floquet quasienergy spectrum , and has also been observed in different systems [9–12]. Generally, the dynamical localization (DL) refers to the phenomenon wherein a particle initially localized in a lattice can transport within a fi-nite distance and periodically return to its original state. There has been growing interest in the quantum control of electrons in semiconductor superlattices or arrays of coupled quantum dots from both theoretical and experimental sides [2, 13]. Most of the DL and CDT of the electronic systems are generic and also can occur in atomic [8, 9, 14, 15] and optical [10, 16–20] systems.
Recently, different routes to CDT were found by considering, respectively, the priori prescribed number of bosons of a many-boson system [21, 22], the distinguishable intersite separations of a bipartite lattice [23, 24], the variable driving symmetry of a two-frequency driven particle in a double-well [11, 25], and the different combined modulations to the different systems [26, 27]. The CDT mechanism has been applied to various physical fields such as the qubit control [28, 29], the quantum tunneling switch , and the directed transport in a bipartite lattice via the selective CDT to the two different barriers [23, 24]. It is worth noting that the CDT and DL mechanisms can also be applied to coherently control instability of the periodically driven double-well system , optical lattice system [32, 33] and fiber system . In the sense of Lyapunov, by the instability we mean that the small initial deviation from a given solution grows without upper limit, which could lead to destruction of the solution behavior. It was found that instability of the bipartite lattice systems depends on different signs of the effective tunneling rates of two nearest-neighbor barriers [32, 33]. Therefore, one can stabilize the systems by tuning the effective tunneling rates. For a single particle held in a simple lattice with a single intersite separation, the nearest-neighbor barriers are the same such that the instability cannot be shown. For a bipartite lattice system, such an instability may be induced and suppressed alternately by adjusting the driving parameters, resulting in the directed transport of particles. Here our aim is finding a new route of CDT and supplying a simple stabilization method to realize the directed transport and quantum transition of cold atoms in a driven bipartite lattice.
The coherent control of an ac driven particle in a single-band lattice with a single intersite separation has been investigated widely in the nearest-neighbor tight binding (NNTB) approximation [5, 26]. More recently, a bipartite optical lattice or double-well train with two different intersite separations has been realized experimentally by superimposing two laser beams with two different wavelengths [35, 36]. Such a system was applied to induce the ratchet-like effect [23, 24], to transport quantum information  and to realize two-qubit quantum gates . The periodic modulation is usually applied to the potential tilt (bias) between the lattice sites [5, 39, 40] or the tunnel coupling [41–44]. The combined modulations between both have also been adopted to produce the exact solutions for a phase controllable lattice system  or for an analytically solvable two-level system [27, 45]. The adjustments of driving parameters can be performed in a nonadiabatic [23, 37] or adiabatic manner [46, 47]. On the other hand, the directed transports have been investigated for a single classical particle in a spatially periodic potential  and for a mean-field treated Bose-Einstein condensate loaded in an optical superlattice [49, 50].
In this work, we firstly consider a single atom held in an optical bipartite lattice with two different separations a and b and driven by a combined modulation of two resonant external fields with a phase difference between the bias and coupling. Such a system can also be regarded as an atomic analog of the optical two-mode quantum beam splitter . In the high-frequency regime and NNTB approximation, we derive an analytical general solution for the probability amplitude of the particle in any localized state in which the quantum tunneling and stability can depend on the phase difference between the two modulation components. A new route of CDT and a simple method for stabilizing the system to perform the directed transport are found by adjusting the phase difference nonadiabatically. Such a phase-adjustment may be more convenient in experiments compared to the usual amplitude- and interaction-modulations. Finally, we suggest a scheme for extending the results to the phase-controlled directed transport and quantum transition between the superfluid and Mott insulator for the corresponding many-particle system. The results can be readily amenable with existing experimental setups [39, 40, 42–44] on the periodically tilted and shaken optical lattices  and could be applied to simulating the different optical systems [16, 17] and solid-state systems .
2. General solution in the high-frequency regime
We consider a driven and tilted bipartite lattice in the form of a train of double wells formed by the tilted laser standing wave43], the potential tilt between the lattice sites takes the form [39, 40, 51] ε(t) = −ε0 cos(ωt) with amplitude ε0 and frequency ω, and the time-periodic lattice depths reads [37, 43, 44] Vi(t) = Vi0 + δVi cos(mωt − ϕ) for m = 0, 1, 2,..., and with constants Vi0 and δVi. The nonzero m means the frequency resonance between the modulation components. Such a lattice can be realized experimentally by a periodically shaken optical lattice [37, 43, 44], and by moving the position of a retroreflecting mirror which is mounted on a piezoelectric actuator  or by imposing a phase modulation to one of the standing wave component fields . A single particle of mass M is initially placed near the lattice center, as shown in Fig. 1 , where we have adopted the spatial coordinate normalized by and the phase θ = 4.6, and selected the suitable driving parameters and initial time t0 = π/(2ω) to make ε(t0) = 0 and V1(t0) = 1, V2(t0) = 2. The different separations a and b can be adjusted by changing the laser wave vector kL and amplitudes Vi(t) [37, 43]. Quantum dynamics of such a system is governed by the Hamiltonian [23, 26, 33] 5] 5, 43] 23, 43] Jij(t) = J(t) = J0 + δJ cos(mωt − ϕ), where constant J0 is from the terms of kinetic energy and of Vi0, the shaking intensity δJ is proportional to the driving amplitudes  δVi. To simplify, we have set h̄ = 1 and normalized energy and time by Er and , which are determined by the laser wave vector and atomic mass. The parameters J0, δJ and (ε0xn) are in units of ω0 = Er/10 with xn being normalized by the wave length λs = π/kL of the short lattice. Thus all the parameters are dimensionless throughout this paper. The experimentally achievable parameter regions may be selected as [42–44] λs ∼ 800nm, J0 ∼ ω0, δJ < J0, ε0λs ∼ ω ∈ [0, 100](ω0), and a, b ∼ λs.
Letting |n〉 be the localized state at the site n, we expand the quantum state |ψ(t)〉 as the linear superposition |ψ(t)〉 = ∑n cn(t)|n〉. Combining this with Eq. (1), from the time-dependent Schrödinger equation we derive the coupled equations of the probability amplitudes [5, 33, 47]Eq. (2), we make the function transformation cn(t) = An(t) exp(iε0ω−1xn sinωt) which leads Eq. (2) to the form
We focus our attention on the situation of high-frequency regime with ω ≫ 1. The selective CDT has been illustrated analytically and numerically under this limit for an amplitude modulation . Here we shall give a general analytical solution of the system, which reveals the phase-controlled CDT and directed transport for the considered combined modulation, then extend the results to a many-particle system. Note that in Eq. (3), An(t) may be treated as a set of slowly varying functions of time, and the coupling33]. Similarly, the time-average of F(t, ϕ, −Δn−1) = J(t)e−iΔn−1sinωt in Eq. (3) reads F̄(m, ϕ, − Δn−1), which is evaluated from Eq. (4) by using −Δn−1 instead of Δn. For an even (odd) n, F̄(m, ϕ, Δn) and F̄(m, ϕ, − Δn−1) are associated with the effective tunneling rates of the lattice separations a (b) and b (a), respectively. Clearly, they may be real or complex, corresponding to the even or odd m. Given Eq. (4), Eq. (3) is transformed to Eq. (9) of Ref. , we find that the former can become the latter by using the new effective tunneling rates instead of the old. Thus we can construct the exact general solution of Eq. (5) by applying the same discrete Fourier transformation [5, 33] Eq. (5) into the equations Eq. (5) as Eq. (7) may be complex, while the same parameters in Ref.  are real. Such a difference may result in some different properties of the solutions. Without loss of generality, let the initially occupied state be |ψ(t0 = 0)〉 = |N〉 with a fixed integer N, namely the initial conditions read AN(0) = 1, An≠N(0) = 0. Combining the conditions with the discrete Fourier transformation and general solution of A(k, t) produces Eq. (5). Solving the two equations of α(k) and β(k) yields (6) with Eqs. (7) and (8), we can investigate the general transport characterization for the different initial conditions |ψ(t0)〉 and the different effective tunneling rates F̄(m, ϕ, Δn) and F̄(m, ϕ, −Δn−1). In the general cases, the particle may be in the expanded states or localized states, depending on the system parameters.
3. Unusual transport phenomena
We are interested in the unusual transport phenomena such as the DL, CDT, instability and directed transport. It will be found that such unusual phenomena can be controlled under the initial condition |ψ(t0)〉 and for some special parameter sets with different phases. The routes for implementing the phase-controlled transport are very different from that of the previously considered case δJ = 0 with a constant tunneling rate J0 [23, 33].
Phase-controlled CDT. The CDT conditions mean the zero effective tunneling rates F̄(m, ϕ0, Δn) = F̄(m, ϕ0, −Δn−1) = 0 in Eq. (5), and f̄(k) = 0 in Eq. (7). Substituting the latter into Eq. (6), the probability amplitudes becomes some constants An(t) = An(t0) determined by the initial conditions, which means the occurrence of CDT. Applying the CDT conditions to Eq. (4), we getFig. 2. It is shown that the effective tunneling rates are tunable by varying the phase, and the CDT conditions F̄(m, ϕ0, Δn) = F̄(m, ϕ0, −Δn−1) = 0 are established at the phase ϕ0 ≈ 2.4.
As a simplest example, we can fix the lattice separations a, b and tune the ratio ε0/ω to obey 𝒥0(Δn) = 𝒥0(Δn−1) = 0 with , , then Eq. (4) becomes23, 37].
Phase-controlled DL and selective CDT. The DL conditions mean one of the two effective tunneling rates vanishing, which leads to selective CDT to the two different barriers [23, 24]. When F̄(m, ϕ, −Δn−1) = 0 and |ψ(0)〉 = |N〉 are set, from Eqs. (7) and (8) we obtain the constant modulus |f̄(k)| = |F̄(m, ϕ, Δn)| and the periodic functionsEq. (6) produces the probability amplitudes
Phase-controlled instability. Now we prove that the solutions of Eq. (5) are unstable under the conditionEq. (12) is satisfied, Eq. (5) can be written as Eq. (5), we can find that the deviated solution possesses the same form as that of the given solution, (12), as shown in Fig. 3(a).
Phase-controlled directed transport. For a set of given parameters J0, δJ, Δn and Δn−1, we define two different phases ϕ1 and ϕ2 to obey(12) will be reached in each process changing phase from ϕi to ϕj for i, j = 1, 2. As an example, this is indicated by ϕc in Fig. 3(a) for the parameter set J0 = 1, δJ = 0.8, m = 2, ω = 30, Δn = 2, Δn−1 = 2.2, where Eqs. (12) and (13) give ϕc ≈ 2.17, ϕ1 ≈ 1.93 and ϕ2 ≈ 2.49, and the Rabi frequencies and half-periods of Eqs. (10) and (11) read ω1 = F̄(m, ϕ1, Δn) ≈ 0.124666, ω2 = |F̄(m, ϕ2, −Δn−1)| ≈ 0.140933 and T1 ≈ 25.2001, T2 ≈ 22.2914, respectively. The corresponding adjustment to the time-dependent tunneling rate J(t, ϕ) = 1 + 0.8cos(60t + ϕ) is exhibited in Fig. 3(b) for the time interval including t = T1, which only transforms the phase of J(t, ϕ) from ϕ1 to ϕ2 and does not change its magnitude. Clearly, at the time t = T1 + T2, the tunneling rate will change from J(t, ϕ2) to J(t, ϕ1). By repeatedly using such operations, the initially stable Rabi oscillation is broken under the conditions (12), then the instability is suppressed by the conditions (13) with phase ϕ1 or ϕ2 such that the particle is forced to transit repeatedly between the two stable oscillation states |ψn(t)〉 = An|n〉 + An+1|n + 1〉 and |ψn′ (t)〉 = An′ |n′〉 + An′+1|n′ + 1〉 with the amplitudes and frequencies of Eqs. (10) and (11) for (n, n′) = (N, N +1);(N +1, N +2);... that lead to the directed motion toward the right [23,33]. Because the phase-transformation does not change the magnitude of J(t, ϕ) at the operation moment, it supplies a more convenient method to manipulate the directed transport, compared to the previous amplitude-modulation schemes to the coupling  or to the tilt .
4. Extension to a many-particle system
It is worth noting that the method realizing the directed transport can be extended to controlling the transport of a many-particle system in an optical bipartite lattice, where a Bose-Hubbard interaction energy for should be added into Eq. (1). It is well known that for an undriven simple lattice system with J = J0, ε = ε0, a = b and the interaction strength U0 > 0, the characteristic parameter is the ratio [39, 40] r = U0/J0. For U0 ≪ J0 the ground state of the system describes a superfluid, whereas it has the properties of a Mott insulator for U ≫ J0. Quantum transition between the superfluid and Mott insulator can occur at the critical value r = rc. It has been argued that in the presence of a high-frequency driving the system behaves similar to the undriven system, but with the tunneling rate J0 of the latter being replaced by the effective tunneling rate Jeff = J0𝒥0(ε0/ω) for the simple lattice system. Thus one can control the quantum transition and transport by adjusting the driving parameters [39, 40]. Another interesting scheme for controlling the quantum transition has also been suggested in which the periodically modulated interactions were applied . In the Mott insulating state, average atomic current along any direction approximately vanishes.
However, in the case of many particles held in the considered optical bipartite lattice, the effective tunneling rates of the lattice separations a and b can be different, F̄(m, ϕ, Δn) ≠ F̄(m, ϕ, −Δn−1). After transformation into the interaction picture by the unitary operatorEq. (1) and the considered Bose-Hubbard energy we arrive at the transformed interaction Hamiltonian
- Case 1. The system becomes a Mott insulator with r1 ≫ rc and r2 ≫ rc;
- Case 2. There is a superfluid state with r1 ≪ rc and r2 ≪ rc;
- Case 3. There exist two different states for r1 ≫ rc, r2 < rc or r1 < rc, r2 ≫ rc, respectively.
We call the two states the local superfluid states, which may be gone through in the process of transformations between case 1 and case 2.
When the ratios ri are changed between the case 1 and case 2, the system undergoes a quantum transition between the superfluid and Mott insulator. Because the ratios depends on the effective tunneling rates and the latter can be tuned by varying the driving phase, we can control the quantum transition by the phase-modulation. On the other hand, to realize the directed transport of the many-particle system, the possible experiment can begin by loading a Bose-Einstein condensate into the long lattice of wave-vector kL, then one can increase the lattice depth to make the atomic sample in the Mott insulating state with a single atom per well [42,43]. Further one can divide every well into a double-well by ramping up the short lattice of wave-vector 2kL and tilting the double-well train, that achieve the load of single atoms into the “left” sides of tilted double-wells . In such a state with a single atom per double-well such that the interatomic interaction could be negligible and the phase-modulation method for the single atom system becomes valid for the many-body system. Therefore, we can tune the driving phase according to Eqs. (13) and (14) such that the parameter ratios change alternately between r1 < rc, r2 ≫ rc and r1 ≫ rc, r2 < rc, leading to the nonzero average atomic current along a sin-gle direction. Such a phase-controlled directed transport of many particles will form a stronger particle current compared to the single particle case. Particularly, the dynamics of the super-fluid and Mott insulator is very different from that of the single-atom case. The scheme of the phase-controlled quantum transition and directed transport through the local superfluid states also differs from the previous amplitude-modulation [39, 40] and interaction-modulation  proposals on controlling quantum transitions and transports in simple optical lattices. In the latter case, the local superfluid states and the directed motion cannot exist for a symmetric driving. It is interesting for us to compare our results with those of a repulsive Bose-Einstein condensate loaded in the spatially asymmetric optical superlattice without tilt potential and treated in mean-field approximation . Differing from the quantum states governed by the spatially discrete and linear Eq. (2), the governing Schrödinger equation of the latter system are spatially continuous and nonlinear. Therefore, quantum transition and directed transport of the former system can be described by the ratios of the interaction strength and the effective tunneling rates associated with the time-periodic lattice tilt, while the directed atomic current of the latter system is expressed by the time-averaged momentum expectation and this transport phenomenon can occur only if the interaction strength exceeds a critical value for a symmetric driving . Thus one can steer the directed current by adjusting the s-wave scattering length to vary the interaction strength.
5. Conclusions and discussion
We have investigated the coherent control of a single atom held in the optical bipartite lattice with two different separations a and b and driven by a combined modulation of two resonant external fields with a phase difference between the bias and coupling. In the high-frequency regime and NNTB approximation, we derive an analytical general solution of the time-dependent Schrödinger equation, which quantitatively describes the dependence of the tunneling dynamics on the phase difference between the modulation components. It is demonstrated that a new route of CDT (or DL) can be formed by tuning the phase to make two (or one) of the effective tunneling rates of the lattice separations a and b vanishing. When the two effective tunneling rates are adjusted to go through the values of the same magnitude and opposite signs, the system loses its stability. The phase-controlled selective CDT enables the system to be stabilized and the directed tunneling of the particle to be coherently manipulated. In the process of control, the appropriate operation times are fixed by the two tunneling half-periods. The theoretical results have also been extended to the phase-controlled directed transport and quantum transition between the superfluid and Mott insulator for the corresponding many-particle system. The analytic results based on the high-frequency approximation should have an advantage over a direct numerical integration of the Schröedinger equation for transparently predicting or explaining the experimental results.
The obtained results can be tested by employing the current accessible experimental setups [39, 40, 42, 43], and may be applicable for investigating atomic transport, quantum transition and quantum information processing [37, 38]. In experiments, the periodic modulations of the tunneling rate [37, 43, 44] and the potential tilt [12, 39, 51] have been realized through different methods. In order to experimentally implement the directed transport of the bipartite lattice systems, the effective tunneling rate must be changed alternately between two different values. The previous schemes [23,37] suggested that one can achieve the modulation by rapidly changing the amplitudes of driving fields. In such a process, the deviation from the probability amplitude of the initial Rabi-oscillation state crossing the separation a linearly grows in time, then the system is stabilized to the final oscillation state crossing the separation b. This mechanism is similar to the resonance transition in standard quantum mechanics . In contrast, when the phase-modulation is adopted, the deviation from the the initial state can grow exponentially fast, as indicated below Eq. (12), that may result in faster transition to the final state. The difference between both implies that the phase-modulation offers new dynamics of tunneling. In addition, in Fig. 3 we show that the phase-transformation does not change the magnitude of the original tunneling rate at the operation moment. Therefore, the phase-modulation scheme of the periodic shaking may be more convenient and experimentally feasible compared to the previous amplitude modulations. The directed tunneling is related to the ratchetlike effect of quantum particles in the periodically tilted and shaken optical bipartite lattices and could be well suited to simulating the different optical systems [16,17] and solid-state systems . Particularly, the optical analog of such a system may be realized by an optical two-mode quantum beam splitter .
This work was supported by the NNSF of China under Grant Nos. 11204027, 11175064 and 11205021, the Construct Program of the National Key Discipline of China, the Hunan Provincial NSF ( 11JJ7001) and the Scientific Research Fund of Hunan Provincial Education Department ( 12B082).
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