## Phase-controlled localization and directed transport in an optical bipartite lattice |

Optics Express, Vol. 22, Issue 4, pp. 4277-4289 (2014)

http://dx.doi.org/10.1364/OE.22.004277

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### Abstract

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

## 1. Introduction

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5. D. H. Dunlap and V. M. Kenkre, “Dynamic localization of a charged particle moving under the influence of an electric field,” Phys. Rev. B **34**, 3625–3633 (1986). [CrossRef]

26. N. Singh, “Phase controllable dynamical localization of a quantum particle in a driven optical lattice,” Phys. Lett. A **376**, 1593–1595 (2012). [CrossRef]

35. J. Sebby-Strabley, B. L. Brown, M. Anderlini, P. J. Lee, W. D. Phillips, J. V. Porto, and P. R. Johnson, “Preparing and probing atomic number states with an atom interferometer,” Phys. Rev. Lett. **98**, 200405, 2007). [CrossRef] [PubMed]

36. S. Trotzky, P. Cheinet, S. Folling, M. Feld, U. Schnorrberger, A. M. Rey, A. Polkovnikov, E. A. Demler, M. D. Lukin, and I. Bloch, “Time-resolved observation and control of superexchange interactions with ultracold atoms in optical lattices,” Science **319**, 295–299 (2008). [CrossRef]

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24. K. Hai, W. Hai, and Q. Chen, “Controlling transport and entanglement of two particles in a bipartite lattice,” Phys. Rev. A **82**, 053412, 2010). [CrossRef]

37. O. Romero-Isart and J. J. García-Ripoll, “Quantum ratchets for quantum communication with optical superlat-tices,” Phys. Rev. A **76**, 052304, 2007). [CrossRef]

38. G. De Chiara, T. Calarco, M. Anderlini, S. Montangero, P. J. Lee, B. L. Brown, W. D. Phillips, and J. V. Porto, “Optimal control of atom transport for quantum gates in optical lattices,” Phys. Rev. A **77**, 052333, 2008). [CrossRef]

5. D. H. Dunlap and V. M. Kenkre, “Dynamic localization of a charged particle moving under the influence of an electric field,” Phys. Rev. B **34**, 3625–3633 (1986). [CrossRef]

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26. N. Singh, “Phase controllable dynamical localization of a quantum particle in a driven optical lattice,” Phys. Lett. A **376**, 1593–1595 (2012). [CrossRef]

27. W. Hai, K. Hai, and Q. Chen, “Transparent control of an exactly solvable two-level system via combined modulations,” Phys. Rev. A **87**, 023403, 2013). [CrossRef]

45. F. T. Hioe and C. E. Carroll, “Two-state problems involving arbitrary amplitude and frequency modulations,” Phys. Rev. A **32**, 1541–1549 (1985). [CrossRef] [PubMed]

**99**, 110501, 2007). [CrossRef] [PubMed]

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*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 [35

35. J. Sebby-Strabley, B. L. Brown, M. Anderlini, P. J. Lee, W. D. Phillips, J. V. Porto, and P. R. Johnson, “Preparing and probing atomic number states with an atom interferometer,” Phys. Rev. Lett. **98**, 200405, 2007). [CrossRef] [PubMed]

39. A. Zenesini, H. Lignier, D. Ciampini, O. Morsch, and E. Arimondo, “Coherent control of sressed matter waves,” Phys. Rev. Lett. **102**, 100403, 2009). [CrossRef]

40. A. Eckardt, C. Weiss, and M. Holthaus, “Superfluid-insulator transition in a periodically driven optical lattice,” Phys. Rev. Lett. **95**, 260404, 2005). [CrossRef]

42. R. Ma, M. E. Tai, P. M. Preiss, W. S. Bakr, J. Simon, and M. Greiner, “Photon-assisted tunneling in a biased strongly correlated Bose gas,” Phys. Rev. Lett. **107**, 095301, 2011). [CrossRef] [PubMed]

44. C. Schori, T. Stöferle, H. Moritz, M. Köhl, and T. Esslinger, “Excitations of a superfluid in a three-dimensional optical lattice,” Phys. Rev. Lett. **93**, 240402, 2004). [CrossRef]

**376**, 1593–1595 (2012). [CrossRef]

16. I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. **518**, 1–79 (2012). [CrossRef]

17. F. Dreisow, Y. V. Kartashov, M. Heinrich, V. A. Vysloukh, A. Tünnermann, S. Nolte, L. Torner, S. Longhi, and A. Szameit, “Spatial light rectification in an optical waveguide lattice,” Europhys. Lett. **101**, 44002, 2013). [CrossRef]

13. J. M. Villas-Boas, S. E. Ulloa, and N. Studart, “Selective coherent destruction of tunneling in a quantum-dot array,” Phys. Rev. B **70**, 041302(R) (2004). [CrossRef]

## 2. General solution in the high-frequency regime

*k*, the short lattice of wave vector 2

_{L}*k*and the linear potential. Here

_{L}*θ*denotes the laser phase [43

43. Y.-A. Chen, S. Nascimbéne, M. Aidelsburger, M. Atala, S. Trotzky, and I. Bloch, “Controlling correlated tunneling and superexchange interactions with ac-driven optical lattices,” Phys. Rev. Lett. **107**, 210405, 2011). [CrossRef] [PubMed]

39. A. Zenesini, H. Lignier, D. Ciampini, O. Morsch, and E. Arimondo, “Coherent control of sressed matter waves,” Phys. Rev. Lett. **102**, 100403, 2009). [CrossRef]

40. A. Eckardt, C. Weiss, and M. Holthaus, “Superfluid-insulator transition in a periodically driven optical lattice,” Phys. Rev. Lett. **95**, 260404, 2005). [CrossRef]

51. K. W. Madison, M. C. Fischer, R. B. Diener, Q. Niu, and M. G. Raizen, “Dynamical Bloch band suppression in an optical lattice,” Phys. Rev. Lett. **81**, 5093–5096 (1998). [CrossRef]

*ε*(

*t*) = −

*ε*

_{0}cos(

*ωt*) with amplitude

*ε*

_{0}and frequency

*ω*, and the time-periodic lattice depths reads [37

37. O. Romero-Isart and J. J. García-Ripoll, “Quantum ratchets for quantum communication with optical superlat-tices,” Phys. Rev. A **76**, 052304, 2007). [CrossRef]

43. Y.-A. Chen, S. Nascimbéne, M. Aidelsburger, M. Atala, S. Trotzky, and I. Bloch, “Controlling correlated tunneling and superexchange interactions with ac-driven optical lattices,” Phys. Rev. Lett. **107**, 210405, 2011). [CrossRef] [PubMed]

44. C. Schori, T. Stöferle, H. Moritz, M. Köhl, and T. Esslinger, “Excitations of a superfluid in a three-dimensional optical lattice,” Phys. Rev. Lett. **93**, 240402, 2004). [CrossRef]

*V*(

_{i}*t*) =

*V*

_{i}_{0}+

*δV*cos(

_{i}*mωt*−

*ϕ*) for

*m*= 0, 1, 2,..., and with constants

*V*

_{i}_{0}and

*δV*. The nonzero

_{i}*m*means the frequency resonance between the modulation components. Such a lattice can be realized experimentally by a periodically shaken optical lattice [37

**76**, 052304, 2007). [CrossRef]

43. Y.-A. Chen, S. Nascimbéne, M. Aidelsburger, M. Atala, S. Trotzky, and I. Bloch, “Controlling correlated tunneling and superexchange interactions with ac-driven optical lattices,” Phys. Rev. Lett. **107**, 210405, 2011). [CrossRef] [PubMed]

**93**, 240402, 2004). [CrossRef]

**102**, 100403, 2009). [CrossRef]

51. K. W. Madison, M. C. Fischer, R. B. Diener, Q. Niu, and M. G. Raizen, “Dynamical Bloch band suppression in an optical lattice,” Phys. Rev. Lett. **81**, 5093–5096 (1998). [CrossRef]

*M*is initially placed near the lattice center, as shown in Fig. 1 [35

35. J. Sebby-Strabley, B. L. Brown, M. Anderlini, P. J. Lee, W. D. Phillips, J. V. Porto, and P. R. Johnson, “Preparing and probing atomic number states with an atom interferometer,” Phys. Rev. Lett. **98**, 200405, 2007). [CrossRef] [PubMed]

*θ*= 4.6, and selected the suitable driving parameters and initial time

*t*

_{0}=

*π*/(2

*ω*) to make

*ε*(

*t*

_{0}) = 0 and

*V*

_{1}(

*t*

_{0}) = 1,

*V*

_{2}(

*t*

_{0}) = 2. The different separations

*a*and

*b*can be adjusted by changing the laser wave vector

*k*and amplitudes

_{L}*V*(

_{i}*t*) [37

**76**, 052304, 2007). [CrossRef]

**107**, 210405, 2011). [CrossRef] [PubMed]

**99**, 110501, 2007). [CrossRef] [PubMed]

**376**, 1593–1595 (2012). [CrossRef]

33. K. Hai, Q. Chen, and W. Hai, “Instability inducing directed tunnelling of a single particle in a bipartite lattice,” J. Phys. B At. Mol. Opt. Phys. **44**, 035507, 2011). [CrossRef]

*i*,

*j*) means the nearest-neighbor site pairs. Signs

*b*are, respectively, the particle creation and annihilation operators in the site

_{j}*j*. The spatial location of the

*n*th lattice site reads [5

**34**, 3625–3633 (1986). [CrossRef]

*w*(

*x*−

*x*) is the Wannier function. Expressing the lattice depths in terms of the recoil energy

_{i}*E*= (

_{r}*h̄k*)

_{L}^{2}/(2

*M*), the tunnel coupling is calculated by the formula [5

**34**, 3625–3633 (1986). [CrossRef]

**107**, 210405, 2011). [CrossRef] [PubMed]

*J*(

_{ij}*t*) for an even

*i*has only a small time-independent difference from that for an odd

*i*and it will henceforth be renormalized, so we can take [23

**99**, 110501, 2007). [CrossRef] [PubMed]

**107**, 210405, 2011). [CrossRef] [PubMed]

*J*(

_{ij}*t*) =

*J*(

*t*) =

*J*

_{0}+

*δJ*cos(

*mωt*−

*ϕ*), where constant

*J*

_{0}is from the terms of kinetic energy and of

*V*

_{i}_{0}, the shaking intensity

*δJ*is proportional to the driving amplitudes [43

**107**, 210405, 2011). [CrossRef] [PubMed]

*δV*. To simplify, we have set

_{i}*h̄*= 1 and normalized energy and time by

*E*and

_{r}*J*

_{0},

*δJ*and (

*ε*

_{0}

*x*) are in units of

_{n}*ω*

_{0}=

*E*/10 with

_{r}*x*being normalized by the wave length

_{n}*λ*=

_{s}*π/k*of the short lattice. Thus all the parameters are dimensionless throughout this paper. The experimentally achievable parameter regions may be selected as [42

_{L}42. R. Ma, M. E. Tai, P. M. Preiss, W. S. Bakr, J. Simon, and M. Greiner, “Photon-assisted tunneling in a biased strongly correlated Bose gas,” Phys. Rev. Lett. **107**, 095301, 2011). [CrossRef] [PubMed]

**93**, 240402, 2004). [CrossRef]

*λ*∼ 800nm,

_{s}*J*

_{0}∼

*ω*

_{0},

*δJ*<

*J*

_{0},

*ε*

_{0}

*λ*∼

_{s}*ω*∈ [0, 100](

*ω*

_{0}), and

*a*,

*b*∼

*λ*.

_{s}*n*〉 be the localized state at the site

*n*, we expand the quantum state |

*ψ*(

*t*)〉 as the linear superposition |

*ψ*(

*t*)〉 = ∑

_{n}*c*(

_{n}*t*)|

*n*〉. Combining this with Eq. (1), from the time-dependent Schrödinger equation

**34**, 3625–3633 (1986). [CrossRef]

**44**, 035507, 2011). [CrossRef]

47. S. Longhi and G. D. Valle, “Quantum transport in bipartite lattices via Landau-Zener tunneling,” Phys. Rev. A **86**, 043633, 2012). [CrossRef]

*c*(

_{n}*t*) =

*A*(

_{n}*t*) exp(

*iε*

_{0}

*ω*

^{−1}

*x*sin

_{n}*ωt*) which leads Eq. (2) to the form In this equation, we have defined

*n*, and

*n*.

*ω*≫ 1. The selective CDT has been illustrated analytically and numerically under this limit for an amplitude modulation [23

**99**, 110501, 2007). [CrossRef] [PubMed]

*A*(

_{n}*t*) may be treated as a set of slowly varying functions of time, and the coupling is a rapidly oscillating function which can be replaced by its time-average

*𝒥*(Δ

_{m}*) = (−1)*

_{n}

^{m}𝒥_{−}

*(Δ*

_{m}*) = (−1)*

_{n}*(−Δ*

^{m}𝒥_{m}*) being the*

_{n}*m*th Bessel function of the first kind [33

**44**, 035507, 2011). [CrossRef]

*F*(

*t*,

*ϕ*, −Δ

_{n}_{−1}) =

*J*(

*t*)

*e*

^{−iΔn−1sinωt}in Eq. (3) reads

*F̄*(

*m*,

*ϕ*, − Δ

_{n}_{−1}), which is evaluated from Eq. (4) by using −Δ

_{n}_{−1}instead of Δ

*. For an even (odd)*

_{n}*n*,

*F̄*(

*m*,

*ϕ*, Δ

*) and*

_{n}*F̄*(

*m*,

*ϕ*, − Δ

_{n}_{−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 Comparing this equation with Eq. (9) of Ref. [33

**44**, 035507, 2011). [CrossRef]

**34**, 3625–3633 (1986). [CrossRef]

**44**, 035507, 2011). [CrossRef]

*A*and

_{e}*A*being the sums of even terms and odd terms respectively in the Fourier series. From the two first order equations of

_{o}*A*and

_{e}*A*we derive the second order equation

_{o}*Ä*(

*k*,

*t*) = −|

*f̄*(

*k*)|

^{2}

*A*(

*k*,

*t*) with the well-known general solution Inserting this solution into the inverse Fourier transformation, we immediately obtain the general solution of Eq. (5) as Here

*f̄*(

*k*) takes the form where |

*f̄*(

*k*)| and

*f̄*

^{*}(

*k*) are the corresponding modulus and complex conjugate,

*α*(

*k*) and

*β*(

*k*) are adjusted by the initial conditions. It should be noticed that the parameters

*J*

_{±}in Eq. (7) may be complex, while the same parameters in Ref. [33

**44**, 035507, 2011). [CrossRef]

*ψ*(

*t*

_{0}= 0)〉 = |

*N*〉 with a fixed integer

*N*, namely the initial conditions read

*A*(0) = 1,

_{N}*A*

_{n}_{≠}

*(0) = 0. Combining the conditions with the discrete Fourier transformation and general solution of*

_{N}*A*(

*k*,

*t*) produces

*α*(

*k*) and

*β*(

*k*) yields

*ψ*(

*t*

_{0})〉 and the different effective tunneling rates

*F̄*(

*m*,

*ϕ*, Δ

*) and*

_{n}*F̄*(

*m*,

*ϕ*, −Δ

_{n}_{−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

*ψ*(

*t*

_{0})〉 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

*J*

_{0}[23

**99**, 110501, 2007). [CrossRef] [PubMed]

**44**, 035507, 2011). [CrossRef]

*Phase-controlled CDT*. The CDT conditions mean the zero effective tunneling rates

*F̄*(

*m*,

*ϕ*

_{0}, Δ

*) =*

_{n}*F̄*(

*m*,

*ϕ*

_{0}, −Δ

_{n}_{−1}) = 0 in Eq. (5), and

*f̄*(

*k*) = 0 in Eq. (7). Substituting the latter into Eq. (6), the probability amplitudes becomes some constants

*A*(

_{n}*t*) =

*A*(

_{n}*t*

_{0}) determined by the initial conditions, which means the occurrence of CDT. Applying the CDT conditions to Eq. (4), we get for an even

*m*, and for an odd

*m*. Therefore, we can arrive at or deviate from the CDT conditions by fixing the parameters

*m*,

*J*

_{0},

*δJ*, Δ

*, Δ*

_{n}

_{n}_{−1}and adjusting the phase to arrive at or deviate from the phase

*ϕ*

_{0}. In the general case,

*𝒥*

_{0}(Δ

*) ≠*

_{n}*𝒥*

_{0}(Δ

_{n}_{−1}), for an even

*m*the above CDT conditions imply −

*δJ*cos

*ϕ/J*

_{0}=

*𝒥*

_{0}(Δ

*)/*

_{n}*𝒥*(Δ

_{m}*) =*

_{n}*𝒥*

_{0}(Δ

_{n}_{−1})/

*𝒥*(Δ

_{m}

_{n}_{−1}). For example, applying the parameters

*J*

_{0}= 1,

*δJ*= 0.8,

*m*= 2 to the CDT conditions produces the required values Δ

*≈ 2.01717, Δ*

_{n}

_{n}_{−1}≈ 5.37977. Adopting these parameters, we plot the effective tunneling rates as functions of phase, as in Fig. 2. It is shown that the effective tunneling rates are tunable by varying the phase, and the CDT conditions

*F̄*(

*m*,

*ϕ*

_{0}, Δ

*) =*

_{n}*F̄*(

*m*,

*ϕ*

_{0}, −Δ

_{n}_{−1}) = 0 are established at the phase

*ϕ*

_{0}≈ 2.4.

*a*,

*b*and tune the ratio

*ε*

_{0}/

*ω*to obey

*𝒥*

_{0}(Δ

*) =*

_{n}*𝒥*

_{0}(Δ

_{n}_{−1}) = 0 with

*ϕ*

_{0}=

*π*/2 for an even

*m*or to

*ϕ*

_{0}= 0 for an odd

*m*in a nonadiabatic manner [23

**99**, 110501, 2007). [CrossRef] [PubMed]

**76**, 052304, 2007). [CrossRef]

*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

**99**, 110501, 2007). [CrossRef] [PubMed]

**82**, 053412, 2010). [CrossRef]

*F̄*(

*m*,

*ϕ*, −Δ

_{n}_{−1}) = 0 and |

*ψ*(0)〉 = |

*N*〉 are set, from Eqs. (7) and (8) we obtain the constant modulus |

*f̄*(

*k*)| = |

*F̄*(

*m*,

*ϕ*, Δ

*)| and the periodic functions of*

_{n}*k*. Inserting these into Eq. (6) produces the probability amplitudes They describe Rabi oscillation of the particle between the localized states |

*N*〉 and |

*N*+ 1〉 with oscillating frequency

*ω*

_{1}= |

*F̄*(

*m*,

*ϕ*, Δ

*)|. Here the selective CDTs between the states |*

_{N}*N*− 1〉 and |

*N*〉, and between the states |

*N*+ 1〉 and |

*N*+ 2〉 occur. Similarly, taking

*F̄*(

*m*,

*ϕ*, Δ

*) = 0 leads to the probability amplitudes which describe Rabi oscillation of the particle between the localized states |*

_{n}*N*〉 and |

*N*− 1〉 with oscillating frequency

*ω*

_{2}= |

*F̄*(

*m*,

*ϕ*, −Δ

_{N}_{−1})

*|*. Here the DL conditions and the different oscillating frequencies are modulated by the phase for a set of other parameters.

*Phase-controlled instability*. Now we prove that the solutions of Eq. (5) are unstable under the condition for the phase

*ϕ*=

*ϕ*. In fact, when Eq. (12) is satisfied, Eq. (5) can be written as with variable

_{c}*τ*= 2

*iF̄*(

*m*,

*ϕ*, Δ

*)*

_{N}*t*being proportional to time, whose general solution is well-known as where

*𝒥*(

_{n}*τ*) and

*𝒩*(

_{n}*τ*) are the Bessel and Neuman functions respectively, and

*B*,

_{n}*D*the expansion coefficients determined by means of the initial conditions. For the complex variable

_{n}*τ*with nonzero imaginary part, this general solution has the asymptotic property

*δA*(

_{n}*τ*) from the given solution

*A*(

_{n}*τ*) can grow exponentially fast, meaning the Lyapunov instability of solution

*A*(

_{n}*τ*). In fact, by using

*δA*(

_{n}*τ*) +

*A*(

_{n}*τ*) instead of

*A*(

_{n}*τ*) in the linear Eq. (5), we can find that the deviated solution possesses the same form as that of the given solution, with constants

*δB*and

_{n}*δD*determined by the initial deviation, and has the same asymptotic property, namely lim

_{n}

_{t}_{→∞}

*δA*(

_{n}*τ*) tends to infinity exponentially fast. Obviously such an instability can be controlled by tuning the phase to arrive at or deviate from

*ϕ*in the condition (12), as shown in Fig. 3(a).

_{c}*Phase-controlled directed transport*. For a set of given parameters

*J*

_{0},

*δJ*, Δ

*and Δ*

_{n}

_{n}_{−1}, we define two different phases

*ϕ*

_{1}and

*ϕ*

_{2}to obey for an even

*m*, which result in the selective CDT between the localized states |

*n*〉 and |

*n*− 1〉 or between the localized states |

*n*〉 and |

*n*+ 1〉, respectively. By nonadiabatically tuning the phase to alternately change between

*ϕ*

_{1}and

*ϕ*

_{2}just after the two different time intervals

*T*

_{1}=

*π/ω*

_{1}and

*T*

_{2}=

*π/ω*

_{2}, the original tunneling rate becomes the continuous and piecewise analytic function with

*T*=

*T*

_{1}+

*T*

_{2}and

*n′*= 0, 1, 2,.... The instability condition (12) will be reached in each process changing phase from

*ϕ*to

_{i}*ϕ*for

_{j}*i*,

*j*= 1, 2. As an example, this is indicated by

*ϕ*in Fig. 3(a) for the parameter set

_{c}*J*

_{0}= 1,

*δJ*= 0.8,

*m*= 2,

*ω*= 30, Δ

*= 2, Δ*

_{n}

_{n}_{−1}= 2.2, where Eqs. (12) and (13) give

*ϕ*≈ 2.17,

_{c}*ϕ*

_{1}≈ 1.93 and

*ϕ*

_{2}≈ 2.49, and the Rabi frequencies and half-periods of Eqs. (10) and (11) read

*ω*

_{1}=

*F̄*(

*m*,

*ϕ*

_{1}, Δ

*) ≈ 0.124666,*

_{n}*ω*

_{2}= |

*F̄*(

*m*,

*ϕ*

_{2}, −Δ

_{n}_{−1})| ≈ 0.140933 and

*T*

_{1}≈ 25.2001,

*T*

_{2}≈ 22.2914, respectively. The corresponding adjustment to the time-dependent tunneling rate

*J*(

*t*,

*ϕ*) = 1 + 0.8cos(60

*t*+

*ϕ*) is exhibited in Fig. 3(b) for the time interval including

*t*=

*T*

_{1}, which only transforms the phase of

*J*(

*t*,

*ϕ*) from

*ϕ*

_{1}to

*ϕ*

_{2}and does not change its magnitude. Clearly, at the time

*t*=

*T*

_{1}+

*T*

_{2}, the tunneling rate will change from

*J*(

*t*,

*ϕ*

_{2}) to

*J*(

*t*,

*ϕ*

_{1}). 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

*ϕ*

_{1}or

*ϕ*

_{2}such that the particle is forced to transit repeatedly between the two stable oscillation states |

*ψ*(

_{n}*t*)〉 =

*A*|

_{n}*n*〉 +

*A*

_{n}_{+1}|

*n*+ 1〉 and |

*ψ*(

_{n′}*t*)〉 =

*A*|

_{n′}*n′*〉 +

*A*

_{n′}_{+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

**99**, 110501, 2007). [CrossRef] [PubMed]

**44**, 035507, 2011). [CrossRef]

*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 [37

**76**, 052304, 2007). [CrossRef]

**99**, 110501, 2007). [CrossRef] [PubMed]

## 4. Extension to a many-particle system

*J*=

*J*

_{0},

*ε*=

*ε*

_{0},

*a*=

*b*and the interaction strength

*U*

_{0}> 0, the characteristic parameter is the ratio [39

**102**, 100403, 2009). [CrossRef]

**95**, 260404, 2005). [CrossRef]

*r*=

*U*

_{0}/

*J*

_{0}. For

*U*

_{0}≪

*J*

_{0}the ground state of the system describes a superfluid, whereas it has the properties of a Mott insulator for

*U*≫

*J*

_{0}. Quantum transition between the superfluid and Mott insulator can occur at the critical value

*r*=

*r*. 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

_{c}*J*

_{0}of the latter being replaced by the effective tunneling rate

*J*=

_{eff}*J*

_{0}

*𝒥*

_{0}(

*ε*

_{0}/

*ω*) for the simple lattice system. Thus one can control the quantum transition and transport by adjusting the driving parameters [39

**102**, 100403, 2009). [CrossRef]

**95**, 260404, 2005). [CrossRef]

52. Á. Rapp, X. Deng, and L. Santos, “Ultracold lattice gases with periodically modulated interactions,” Phys. Rev. Lett. **109**, 203005, 2012). [CrossRef] [PubMed]

*a*and

*b*can be different,

*F̄*(

*m*,

*ϕ*, Δ

*) ≠*

_{n}*F̄*(

*m*,

*ϕ*, −Δ

_{n}_{−1}). After transformation into the interaction picture by the unitary operator from Eq. (1) and the considered Bose-Hubbard energy we arrive at the transformed interaction Hamiltonian in the high-frequency limit. Here we have set and Δ

*= Δ*

_{ij}*and −Δ*

_{n}

_{n}_{−1}alternately. In such a system, we have two parameter ratios

*r*

_{1}=

*U*

_{0}/

*F̄*(

*m*,

*ϕ*, Δ

*) and*

_{n}*r*

_{2}=

*U*

_{0}/

*F̄*(

*m*,

*ϕ*, −Δ

_{n}_{−1}), whose values are associated with the following cases:

*r*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

_{i}*k*, then one can increase the lattice depth to make the atomic sample in the Mott insulating state with a single atom per well [42

_{L}42. R. Ma, M. E. Tai, P. M. Preiss, W. S. Bakr, J. Simon, and M. Greiner, “Photon-assisted tunneling in a biased strongly correlated Bose gas,” Phys. Rev. Lett. **107**, 095301, 2011). [CrossRef] [PubMed]

**107**, 210405, 2011). [CrossRef] [PubMed]

*k*and tilting the double-well train, that achieve the load of single atoms into the “left” sides of tilted double-wells [43

_{L}**107**, 210405, 2011). [CrossRef] [PubMed]

*r*

_{1}<

*r*,

_{c}*r*

_{2}≫

*r*and

_{c}*r*

_{1}≫

*r*,

_{c}*r*

_{2}<

*r*, 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

_{c}**102**, 100403, 2009). [CrossRef]

**95**, 260404, 2005). [CrossRef]

52. Á. Rapp, X. Deng, and L. Santos, “Ultracold lattice gases with periodically modulated interactions,” Phys. Rev. Lett. **109**, 203005, 2012). [CrossRef] [PubMed]

50. D. Poletti, G. Benenti, G. Casati, P. Hänggi, and B. Li, “Steering Bose-Einstein condensates despite time symmetry,” Phys. Rev. Lett. **102**, 130604, 2009). [CrossRef] [PubMed]

50. D. Poletti, G. Benenti, G. Casati, P. Hänggi, and B. Li, “Steering Bose-Einstein condensates despite time symmetry,” Phys. Rev. Lett. **102**, 130604, 2009). [CrossRef] [PubMed]

*s*-wave scattering length to vary the interaction strength.

## 5. Conclusions and discussion

*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.

## Acknowledgments

## References and links

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**OCIS Codes**

(020.0020) Atomic and molecular physics : Atomic and molecular physics

(020.1670) Atomic and molecular physics : Coherent optical effects

(270.0270) Quantum optics : Quantum optics

(020.1475) Atomic and molecular physics : Bose-Einstein condensates

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: November 14, 2013

Revised Manuscript: December 22, 2013

Manuscript Accepted: February 11, 2014

Published: February 18, 2014

**Citation**

Kuo Hai, Yunrong Luo, Gengbiao Lu, and Wenhua Hai, "Phase-controlled localization and directed transport in an optical bipartite lattice," Opt. Express **22**, 4277-4289 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4277

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### References

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