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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 4 — Feb. 24, 2014
  • pp: 4277–4289
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Phase-controlled localization and directed transport in an optical bipartite lattice

Kuo Hai, Yunrong Luo, Gengbiao Lu, and Wenhua Hai  »View Author Affiliations


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

Quantum control of tunneling processes of particles plays a major role in different areas of physics, optics and chemistry [1

1. H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?” Science 288, 824–828 (2000). [CrossRef] [PubMed]

4

4. J. Thom, G. Wilpers, E. Riis, and A. G. Sinclair, “Accurate and agile digital control of optical phase, amplitude and frequency for coherent atomic manipulation of atomic systems,” Opt. Express 21, 18712–18723 (2013). [CrossRef] [PubMed]

]. 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 [5

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]

], 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

2. M. Grifoni and P. Hänggi, “Driven quantum tunneling,” Phys. Rep. 304, 229–354 (1998). [CrossRef]

, 6

6. F. Grossmann, T. Dittrich, P. Jung, and P. Hänggi, “Coherent destruction of tunneling,” Phys. Rev. Lett. 67, 516–519 (1991). [CrossRef] [PubMed]

, 7

7. F. Grossmann and P. Hänggi, “Localization in a driven two-level dynamics,” Europhys. Lett. 18, 571–576 (1992). [CrossRef]

] at a collapse point of the Floquet quasienergy spectrum [8

8. M. Holthaus, “Collapse of minibands in far-infrared irradiated superlattices,” Phys. Rev. Lett. 69, 351–354 (1992). [CrossRef] [PubMed]

], and has also been observed in different systems [9

9. H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo, “Dynamical control of matter-wave tunneling in periodic potentials,” Phys. Rev. Lett. 99, 220403, 2007). [CrossRef]

12

12. A. Eckardt, M. Holthaus, H. Lignier, A. Zenesini, D. Ciampini, O. Morsch, and E. Arimondo, “Exploring dynamic localization with a Bose-Einstein condensate,” Phys. Rev. A 79, 013611, 2009). [CrossRef]

]. 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

2. M. Grifoni and P. Hänggi, “Driven quantum tunneling,” Phys. Rep. 304, 229–354 (1998). [CrossRef]

, 13

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]

]. Most of the DL and CDT of the electronic systems are generic and also can occur in atomic [8

8. M. Holthaus, “Collapse of minibands in far-infrared irradiated superlattices,” Phys. Rev. Lett. 69, 351–354 (1992). [CrossRef] [PubMed]

, 9

9. H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo, “Dynamical control of matter-wave tunneling in periodic potentials,” Phys. Rev. Lett. 99, 220403, 2007). [CrossRef]

, 14

14. C. Weiss and N. Teichmann, “Differences between mean-field dynamics and N-particle quantum dynamics as a signature of entanglement,” Phys. Rev. Lett. 100, 140408, 2008). [CrossRef] [PubMed]

, 15

15. C. Ding, J. Li, R. Yu, X. Hao, and Y. Wu, “High-precision atom localization via controllable spontaneous emission in a cycle-configuration atomic system,” Opt. Express 20, 7870–7885 (2012). [CrossRef] [PubMed]

] and optical [10

10. G. Della Valle, M. Ornigotti, E. Cianci, V. Foglietti, P. Laporta, and S. Longhi, “Visualization of coherent destruction of tunneling in an optical double well system,” Phys. Rev. Lett. 98, 263601, 2007). [CrossRef] [PubMed]

, 16

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]

20

20. X. B. Luo, Q. T. Xie, and B. Wu, “Nonlinear coherent destruction of tunneling,” Phys. Rev. A 76, 051802(R) (2007). [CrossRef]

] systems.

Recently, different routes to CDT were found by considering, respectively, the priori prescribed number of bosons of a many-boson system [21

21. J. Gong, L. Morales-Molina, and P. Hänggi, “Many-body coherent destruction of tunneling,” Phys. Rev. Lett. 103, 133002, 2009). [CrossRef] [PubMed]

, 22

22. S. Longhi, “Many-body selective destruction of tunneling in a bosonic junction,” Phys. Rev. A 86, 044102, 2012). [CrossRef]

], the distinguishable intersite separations of a bipartite lattice [23

23. C. E. Creffield, “Quantum control and entanglement using periodic driving fields,” Phys. Rev. Lett. 99, 110501, 2007). [CrossRef] [PubMed]

, 24

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]

], the variable driving symmetry of a two-frequency driven particle in a double-well [11

11. E. Kierig, U. Schnorrberger, A. Schietinger, J. Tomkovic, and M. K. Oberthaler, “Single-particle tunneling in strongly driven double-well potentials,” Phys. Rev. Lett. 100, 190405, 2008). [CrossRef] [PubMed]

, 25

25. G. Lu, W. Hai, and H. Zhong, “Quantum control in a double-well with symmetric or asymmetric driving,” Phys. Rev. A 80, 013411, 2009). [CrossRef]

], and the different combined modulations to the different systems [26

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

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]

]. The CDT mechanism has been applied to various physical fields such as the qubit control [28

28. A. Greilich, S. E. Economou, S. Spatzek, D. R. Yakovlev, D. Reuter, A. D. Wieck, T. L. Reinecke, and M. Bayer, “Ultrafast optical rotations of electron spins in quantum dots,” Nat. Phys. 5, 262–266 (2009). [CrossRef]

, 29

29. C. Padurariu and Y. V. Nazarov, “Spin blockade qubit in a superconducting junction,” EPL 100, 57006, 2012). [CrossRef]

], the quantum tunneling switch [30

30. G. Lu and W. Hai, “Quantum tunneling switch in a planar four-well,” Phys. Rev. A 83, 053424, 2011). [CrossRef]

], and the directed transport in a bipartite lattice via the selective CDT to the two different barriers [23

23. C. E. Creffield, “Quantum control and entanglement using periodic driving fields,” Phys. Rev. Lett. 99, 110501, 2007). [CrossRef] [PubMed]

, 24

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]

]. 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 [31

31. K. Xiao, W. Hai, and J. Liu, “Coherent control of quantum tunneling in an open double-well system,” Phys. Rev. A 85, 013410, 2012). [CrossRef]

], optical lattice system [32

32. C. E. Creffield, “Instability and control of a periodically driven Bose-Einstein condensate,” Phys. Rev. A 79, 063612, 2009). [CrossRef]

, 33

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]

] and fiber system [34

34. N. Korneev, V. A. Vysloukh, and E. M. Rodríguez, “Propagation dynamics of weakly localized cnoidal waves in dispersion-managed fiber: from stability to chaos,” Opt. Express 11, 3574–3582 (2003). [CrossRef] [PubMed]

]. 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

32. C. E. Creffield, “Instability and control of a periodically driven Bose-Einstein condensate,” Phys. Rev. A 79, 063612, 2009). [CrossRef]

, 33

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]

]. 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

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

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

]. 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

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

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]

]. Such a system was applied to induce the ratchet-like effect [23

23. C. E. Creffield, “Quantum control and entanglement using periodic driving fields,” Phys. Rev. Lett. 99, 110501, 2007). [CrossRef] [PubMed]

, 24

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]

], to transport quantum information [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]

] and to realize two-qubit quantum gates [38

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]

]. The periodic modulation is usually applied to the potential tilt (bias) between the lattice sites [5

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]

, 39

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

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

] or the tunnel coupling [41

41. F. Massel, M. J. Leskinen, and P. Törmä, “Hopping modulation in a one-dimensional Fermi-Hubbard Hamiltonian,” Phys. Rev. Lett. 103, 066404, 2009). [CrossRef] [PubMed]

44

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]

]. The combined modulations between both have also been adopted to produce the exact solutions for a phase controllable lattice system [26

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

] or for an analytically solvable two-level system [27

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

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]

]. The adjustments of driving parameters can be performed in a nonadiabatic [23

23. C. E. Creffield, “Quantum control and entanglement using periodic driving fields,” Phys. Rev. Lett. 99, 110501, 2007). [CrossRef] [PubMed]

, 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]

] or adiabatic manner [46

46. Y. Qian, M. Gong, and C. Zhang, “Quantum transport of bosonic cold atoms in double-well optical lattices,” Phys. Rev. A 84, 013608, 2011). [CrossRef]

, 47

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

]. On the other hand, the directed transports have been investigated for a single classical particle in a spatially periodic potential [48

48. A. P. Itin and A. I. Neishtadt, “Directed transport in a classical lattice with a high-frequency driving,” Phys. Rev. E 86, 016206, 2012). [CrossRef]

] and for a mean-field treated Bose-Einstein condensate loaded in an optical superlattice [49

49. D. Poletti, T. J. Alexander, E. A. Ostrovskaya, B. Li, and Yuri S. Kivshar, “Dynamics of matter-wave solitons in a ratchet potential,” Phys. Rev. Lett. 101, 150403, 2008). [CrossRef] [PubMed]

, 50

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]

].

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 [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]

]. 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

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

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

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

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]

] on the periodically tilted and shaken optical lattices [26

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

] and could be applied to simulating the different optical systems [16

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

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]

] and solid-state systems [13

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

We consider a driven and tilted bipartite lattice in the form of a train of double wells formed by the tilted laser standing wave
V(x,t)=V1(t)cos(kLxθ)+V2(t)cos(2kLx2θ)+ε(t)x,
which consists of the long lattice of wave-vector kL, the short lattice of wave vector 2kL and the linear potential. Here 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]

], the potential tilt between the lattice sites takes the form [39

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

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

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) = −ε0 cos(ωt) with amplitude ε0 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

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

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]

] Vi(t) = Vi0 + δVi cos(mωtϕ) 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

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

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

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]

], and by moving the position of a retroreflecting mirror which is mounted on a piezoelectric actuator [39

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]

] or by imposing a phase modulation to one of the standing wave component fields [51

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]

]. A single particle of mass 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]

], where we have adopted the spatial coordinate normalized by kL1 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

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

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]

]. Quantum dynamics of such a system is governed by the Hamiltonian [23

23. C. E. Creffield, “Quantum control and entanglement using periodic driving fields,” Phys. Rev. Lett. 99, 110501, 2007). [CrossRef] [PubMed]

, 26

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

, 33

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]

]
H(t)=(i,j)Jij(t)(bibj+H.C.)+ε(t)nxnbnbn.
(1)
Here (i, j) means the nearest-neighbor site pairs. Signs bj and bj are, respectively, the particle creation and annihilation operators in the site j. The spatial location of the nth lattice site reads [5

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]

]
xn=x|w(xxn)|2dx={n(a+b)/2forevenn,(n+1)a/2+(n1)b/2foroddn,
where w(xxi) is the Wannier function. Expressing the lattice depths in terms of the recoil energy Er = (h̄kL)2/(2M), the tunnel coupling is calculated by the formula [5

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]

, 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]

]
Jij(t)=w*(xxi)[d2dx2+V1(t)cos(kLxθ)+V2(t)cos(2kLx2θ)]w(xxj)dx.
Generally, the Jij(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

23. C. E. Creffield, “Quantum control and entanglement using periodic driving fields,” Phys. Rev. Lett. 99, 110501, 2007). [CrossRef] [PubMed]

, 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]

] Jij(t) = J(t) = J0 + δJ cos(mωtϕ), where constant J0 is from the terms of kinetic energy and of Vi0, the shaking intensity δJ is proportional to the driving amplitudes [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]

] δVi. To simplify, we have set = 1 and normalized energy and time by Er and ω01=10(h¯)/Er, 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

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

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]

] λs ∼ 800nm, J0ω0, δJ < J0, ε0λsω ∈ [0, 100](ω0), and a, bλs.

Fig. 1 A single particle is initially placed in the driven bipartite lattice centered at coordinate 0 with two different separations a and b, where the curve denotes the initial potential V(x, t0) = cos(x − 4.6) + 2cos(2x − 9.2). Hereafter all the quantities plotted in the figures are dimensionless.

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 it|ψ(t)=H(t)|ψ(t) we derive the coupled equations of the probability amplitudes [5

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]

, 33

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]

, 47

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

]
ic˙n(t)=J(t)(cn+1+cn1)ε0cos(ωt)xncn,
(2)
where the dot denotes the derivative with respect to time. To solve Eq. (2), we make the function transformation cn(t) = An(t) exp(0ω−1xn sinωt) which leads Eq. (2) to the form
iA˙n(t)=J(t)(An+1eiΔnsinωt+An1eiΔn1sinωt).
(3)
In this equation, we have defined Δn=ε0ω(xn+1xn) such that there are the relations Δn=ε0ωa, Δn1=ε0ωb for even n, and Δn=ε0ωb, Δn1=ε0ωa for odd n.

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 [23

23. C. E. Creffield, “Quantum control and entanglement using periodic driving fields,” Phys. Rev. Lett. 99, 110501, 2007). [CrossRef] [PubMed]

]. 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 coupling
F(t,m,ϕ,Δn)=J(t,m,ϕ)eiΔnsinωt={J0+12δJ[ei(mωtϕ)+ei(mωtϕ)]}l𝒥l(Δn)eilωt
is a rapidly oscillating function which can be replaced by its time-average
F¯(m,ϕ,Δn)=J0𝒥0(Δn)+12δJ[eiϕ+(1)meiϕ]𝒥m(Δn)=J0𝒥0(Δn)+{δJcosϕ𝒥m(Δn)forevenm,iδJsinϕ𝒥m(Δn)foroddm
(4)
with 𝒥mn) = (−1)m𝒥mn) = (−1)m𝒥m(−Δn) being the mth Bessel function of the first kind [33

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]

]. Similarly, the time-average of F(t, ϕ, −Δn−1) = J(t)eiΔn−1sinωt in Eq. (3) reads (m, ϕ, − Δn−1), which is evaluated from Eq. (4) by using −Δn−1 instead of Δn. For an even (odd) n, (m, ϕ, Δn) and (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
iA˙n(t)=F¯(m,ϕ,Δn)An+1+F¯(m,ϕ,Δn1)An1.
(5)
Comparing this equation with Eq. (9) of Ref. [33

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]

], 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

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]

, 33

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]

]
A(k,t)=nAn(t)eink=Ae(k,t)+Ao(k,t)
to transform Eq. (5) into the equations
iA˙e(k,t)=f¯(k)Ao(k,t),iA˙o(k,t)=f¯*(k)Ae(k,t)
with Ae and Ao being the sums of even terms and odd terms respectively in the Fourier series. From the two first order equations of Ae and Ao we derive the second order equation Ä(k, t) = −|(k)|2A(k, t) with the well-known general solution
A(k,t)=α(k)ei|f¯(k)|t+β(k)ei|f¯(k)|t.
Inserting this solution into the inverse Fourier transformation, we immediately obtain the general solution of Eq. (5) as
An(t)=12πππ[α(k)ei|f¯(k)|t+β(k)ei|f¯(k)|t]einkdk.
(6)
Here (k) takes the form
f¯(k)=J+cosk+iJsink,J±=F¯(m,ϕ,Δn)±F¯(m,ϕ,Δn1),
(7)
where |(k)| and *(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

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]

] 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, AnN(0) = 0. Combining the conditions with the discrete Fourier transformation and general solution of A(k, t) produces
A(k,0)=α(k)+β(k)=eiNk,iA˙(k,0)=|f¯(k)|[α(k)β(k)]=inA˙n(0)eink=F¯(m,ϕ,ΔN)ei(N+1)k+F¯(m,ϕ,ΔN1)ei(N1)k.
The final equation is derived from the initial conditions and Eq. (5). Solving the two equations of α(k) and β(k) yields
α(k)=12|f¯(k)|[|f¯(k)|eiNkF¯(m,ϕ,ΔN)ei(N+1)kF¯(m,ϕ,ΔN1)ei(N1)k],β(k)=12|f¯(k)|[|f¯(k)|eiNk+F¯(m,ϕ,ΔN)ei(N+1)k+F¯(m,ϕ,ΔN1)ei(N1)k].
(8)
Given the general solution (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 (m, ϕ, Δn) and (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

23. C. E. Creffield, “Quantum control and entanglement using periodic driving fields,” Phys. Rev. Lett. 99, 110501, 2007). [CrossRef] [PubMed]

, 33

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]

].

Phase-controlled CDT. The CDT conditions mean the zero effective tunneling rates (m, ϕ0, Δn) = (m, ϕ0, −Δn−1) = 0 in Eq. (5), and (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 get
J0𝒥0(Δn)+δJcosϕ0𝒥m(Δn)=J0𝒥0(Δn1)+δJcosϕ0𝒥m(Δn1)=0
for an even m, and
J0𝒥0(Δn)+iδJsinϕ0𝒥m(Δn)=J0𝒥0(Δn1)iδJsinϕ0𝒥m(Δn1)=0
for an odd m. Therefore, we can arrive at or deviate from the CDT conditions by fixing the parameters m, J0, δJ, Δn, Δn−1 and adjusting the phase to arrive at or deviate from the phase ϕ0. In the general case, 𝒥0n) ≠ 𝒥0n−1), for an even m the above CDT conditions imply −δJ cosϕ/J0 = 𝒥0n)/𝒥mn) = 𝒥0n−1)/𝒥mn−1). For example, applying the parameters J0 = 1, δJ = 0.8, m = 2 to the CDT conditions produces the required values Δn ≈ 2.01717, Δn−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 (m, ϕ0, Δn) = (m, ϕ0, −Δn−1) = 0 are established at the phase ϕ0 ≈ 2.4.

Fig. 2 The effective tunneling rates as functions of phase for the parameters J0 = 1, δJ = 0.8, m = 2, Δn = 2.01717, Δn−1 = 5.37977. The solid and dashed curves describe (m, ϕ, Δn) and (m, ϕ, −Δn−1) respectively, which have the same zero point ϕ0 ≈ 2.4.

As a simplest example, we can fix the lattice separations a, b and tune the ratio ε0/ω to obey 𝒥0n) = 𝒥0n−1) = 0 with Δn=ε0ωa2.4048, Δn1=ε0ωb5.5201, then Eq. (4) becomes
F¯(m,ϕ,Δn)={δJcosϕ𝒥m(Δn)forevenm,iδJsinϕ𝒥m(Δn)foroddm,
(9)
Thus we can achieve the CDT by varying value of the phase to ϕ0 = π/2 for an even m or to ϕ0 = 0 for an odd m in a nonadiabatic manner [23

23. C. E. Creffield, “Quantum control and entanglement using periodic driving fields,” Phys. Rev. Lett. 99, 110501, 2007). [CrossRef] [PubMed]

, 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]

].

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

23. C. E. Creffield, “Quantum control and entanglement using periodic driving fields,” Phys. Rev. Lett. 99, 110501, 2007). [CrossRef] [PubMed]

, 24

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]

]. When (m, ϕ, −Δn−1) = 0 and |ψ(0)〉 = |N〉 are set, from Eqs. (7) and (8) we obtain the constant modulus |(k)| = |(m, ϕ, Δn)| and the periodic functions
α(k)=12eiNkF¯(m,ϕ,ΔN)2|F¯(m,ϕ,Δn)|ei(N+1)k,β(k)=12eiNk+F¯(m,ϕ,ΔN)2|F¯(m,ϕ,Δn)|ei(N+1)k
of k. Inserting these into Eq. (6) produces the probability amplitudes
AnN,N+1(t)=0,AN(t)=cos(ω1t),AN+1(t)=iF¯(m,ϕ,ΔN)|F¯(m,ϕ,ΔN)|sin(ω1t).
(10)
They describe Rabi oscillation of the particle between the localized states |N〉 and |N + 1〉 with oscillating frequency ω1 = |(m, ϕ, ΔN)|. Here the selective CDTs between the states |N − 1〉 and |N〉, and between the states |N + 1〉 and |N + 2〉 occur. Similarly, taking (m, ϕ, Δn) = 0 leads to the probability amplitudes
AnN,N1(t)=0,AN(t)=cos(ω2t),AN1(t)=iF¯(m,ϕ,ΔN1)|F¯(m,ϕ,ΔN1)|sin(ω2t),
(11)
which describe Rabi oscillation of the particle between the localized states |N〉 and |N − 1〉 with oscillating frequency ω2 = |(m, ϕ, −ΔN−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
F¯(m,ϕc,ΔN)=F¯(m,ϕc,Δn1)
(12)
for the phase ϕ = ϕc. In fact, when Eq. (12) is satisfied, Eq. (5) can be written as
dAn(τ)/dτ=12[An1(τ)An+1(τ)]
with variable τ = 2iF̄(m, ϕ, ΔN)t being proportional to time, whose general solution is well-known as
An(τ)=Bn𝒥n(τ)+Dn𝒩n(τ),
where 𝒥n(τ) and 𝒩n(τ) are the Bessel and Neuman functions respectively, and Bn, Dn the expansion coefficients determined by means of the initial conditions. For the complex variable τ with nonzero imaginary part, this general solution has the asymptotic property limtAn(τ)~limteτ/2πτ= of the Bessel function, which implies that the initially small deviation δAn(τ) from the given solution An(τ) can grow exponentially fast, meaning the Lyapunov instability of solution An(τ). In fact, by using δAn(τ) + An(τ) instead of An(τ) in the linear Eq. (5), we can find that the deviated solution possesses the same form as that of the given solution,
δAn(τ)=δBn𝒥n(τ)+δDn𝒩n(τ)
with constants δBn and δDn determined by the initial deviation, and has the same asymptotic property, namely limt→∞δAn(τ) tends to infinity exponentially fast. Obviously such an instability can be controlled by tuning the phase to arrive at or deviate from ϕc in the condition (12), as shown in Fig. 3(a).

Fig. 3 The effective tunneling rates versus phase (a) and the time evolutions of the original tunneling rates (b) for the parameters J0 = 1, δJ = 0.8, m = 2, ω = 30, Δn = 2, Δn−1 = 2.2. In (a), the solid curve describes (m, ϕ, Δn) with zero point ϕ2 ≈ 2.49 and the dashed curve labels −(m, ϕ, −Δn−1) with zero point ϕ1 ≈ 1.93. The phase value ϕc ≈ 2.17 corresponds to the cross point of the two curves, where the instability condition (12) holds. In (b), the solid and dashed curves are associated with the original tunneling rates J(t, ϕ1) and J(t, ϕ2), respectively. At the time t = T1 = π/ω1 = 25.2001, the J(t, ϕ1) is nonadiabatically changed to J(t, ϕ2).

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 Hb=12U0nn^(n^1) for n^=bnbn 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

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

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

] r = U0/J0. For U0J0 the ground state of the system describes a superfluid, whereas it has the properties of a Mott insulator for UJ0. 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

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

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

]. Another interesting scheme for controlling the quantum transition has also been suggested in which the periodically modulated interactions were applied [52

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

]. 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, (m, ϕ, Δn) ≠ (m, ϕ, −Δn−1). After transformation into the interaction picture by the unitary operator
U^=exp[iε(t)dtnxnbnbn],
from Eq. (1) and the considered Bose-Hubbard energy we arrive at the transformed interaction Hamiltonian
Hint=U^HIU^=(i,j)F¯(m,ϕ,Δij)(bibj+H.C.)+12U0nn^(n^1)
(15)
in the high-frequency limit. Here we have set
HI=(i,j)J(t,m,ϕ)(bibj+H.C.)+12U0nn^(n1),
and Δij = Δn and −Δn−1 alternately. In such a system, we have two parameter ratios r1 = U0/(m, ϕ, Δn) and r2 = U0/(m, ϕ, −Δn−1), whose values are associated with the following cases:
  • Case 1. The system becomes a Mott insulator with r1rc and r2rc;
  • Case 2. There is a superfluid state with r1rc and r2rc;
  • Case 3. There exist two different states for r1rc, r2 < rc or r1 < rc, r2rc, 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.

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.

Acknowledgments

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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32.

C. E. Creffield, “Instability and control of a periodically driven Bose-Einstein condensate,” Phys. Rev. A 79, 063612, 2009). [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]

34.

N. Korneev, V. A. Vysloukh, and E. M. Rodríguez, “Propagation dynamics of weakly localized cnoidal waves in dispersion-managed fiber: from stability to chaos,” Opt. Express 11, 3574–3582 (2003). [CrossRef] [PubMed]

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]

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]

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]

41.

F. Massel, M. J. Leskinen, and P. Törmä, “Hopping modulation in a one-dimensional Fermi-Hubbard Hamiltonian,” Phys. Rev. Lett. 103, 066404, 2009). [CrossRef] [PubMed]

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]

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]

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]

46.

Y. Qian, M. Gong, and C. Zhang, “Quantum transport of bosonic cold atoms in double-well optical lattices,” Phys. Rev. A 84, 013608, 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]

48.

A. P. Itin and A. I. Neishtadt, “Directed transport in a classical lattice with a high-frequency driving,” Phys. Rev. E 86, 016206, 2012). [CrossRef]

49.

D. Poletti, T. J. Alexander, E. A. Ostrovskaya, B. Li, and Yuri S. Kivshar, “Dynamics of matter-wave solitons in a ratchet potential,” Phys. Rev. Lett. 101, 150403, 2008). [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]

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]

52.

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

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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