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

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    [CrossRef] [PubMed]
  50. D. Poletti, G. Benenti, G. Casati, P. Hänggi, B. Li, “Steering Bose-Einstein condensates despite time symmetry,” Phys. Rev. Lett. 102, 130604, 2009).
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2013 (3)

J. Thom, G. Wilpers, E. Riis, 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]

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

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

2012 (9)

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

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

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

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

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

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

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

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

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

2011 (6)

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

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

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

K. Hai, Q. Chen, 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]

R.-G. Wan, T.-Y. Zhang, “Two-dimensional sub-half-wavelength atom localization via controlled spontaneous emission,” Opt. Express 19, 25823–25832 (2011).
[CrossRef]

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

2010 (2)

Q. T. Xie, X. B. Luo, B. Wu, “All-optical switch with two periodically modulated nonlinear waveguides,” Opt. Lett. 35, 321–323 (2010).
[CrossRef] [PubMed]

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

2009 (8)

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

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

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

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

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

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

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

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

2008 (5)

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

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

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

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

C. Weiss, 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]

2007 (7)

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

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

P. Král, I. Thanopulos, M. Shapiro, “Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79, 53–77 (2007).
[CrossRef]

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

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

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

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

2005 (1)

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

2004 (2)

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

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

2003 (1)

2000 (1)

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

1998 (2)

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

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

1992 (2)

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

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

1991 (1)

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

1986 (1)

D. H. Dunlap, 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]

1985 (1)

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

Aidelsburger, M.

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

Alexander, T. J.

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

Anderlini, M.

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

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

Arimondo, E.

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

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

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

Atala, M.

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

Bakr, W. S.

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

Bayer, M.

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

Benenti, G.

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

Bloch, I.

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

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

Brown, B. L.

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

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

Calarco, T.

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

Carroll, C. E.

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

Casati, G.

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

Cheinet, P.

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R. Ma, M. E. Tai, P. M. Preiss, W. S. Bakr, J. Simon, M. Greiner, “Photon-assisted tunneling in a biased strongly correlated Bose gas,” Phys. Rev. Lett. 107, 095301, 2011).
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Y. Qian, M. Gong, C. Zhang, “Quantum transport of bosonic cold atoms in double-well optical lattices,” Phys. Rev. A 84, 013608, 2011).
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H. Rabitz, R. de Vivie-Riedle, M. Motzkus, K. Kompa, “Whither the future of controlling quantum phenomena?” Science 288, 824–828 (2000).
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K. W. Madison, M. C. Fischer, R. B. Diener, Q. Niu, M. G. Raizen, “Dynamical Bloch band suppression in an optical lattice,” Phys. Rev. Lett. 81, 5093–5096 (1998).
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Á. Rapp, X. Deng, L. Santos, “Ultracold lattice gases with periodically modulated interactions,” Phys. Rev. Lett. 109, 203005, 2012).
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A. Greilich, S. E. Economou, S. Spatzek, D. R. Yakovlev, D. Reuter, A. D. Wieck, T. L. Reinecke, M. Bayer, “Ultrafast optical rotations of electron spins in quantum dots,” Nat. Phys. 5, 262–266 (2009).
[CrossRef]

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

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S. Trotzky, P. Cheinet, S. Folling, M. Feld, U. Schnorrberger, A. M. Rey, A. Polkovnikov, E. A. Demler, M. D. Lukin, 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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Rodríguez, E. M.

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O. Romero-Isart, J. J. García-Ripoll, “Quantum ratchets for quantum communication with optical superlat-tices,” Phys. Rev. A 76, 052304, 2007).
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Á. Rapp, X. Deng, L. Santos, “Ultracold lattice gases with periodically modulated interactions,” Phys. Rev. Lett. 109, 203005, 2012).
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E. Kierig, U. Schnorrberger, A. Schietinger, J. Tomkovic, M. K. Oberthaler, “Single-particle tunneling in strongly driven double-well potentials,” Phys. Rev. Lett. 100, 190405, 2008).
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S. Trotzky, P. Cheinet, S. Folling, M. Feld, U. Schnorrberger, A. M. Rey, A. Polkovnikov, E. A. Demler, M. D. Lukin, I. Bloch, “Time-resolved observation and control of superexchange interactions with ultracold atoms in optical lattices,” Science 319, 295–299 (2008).
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R. Ma, M. E. Tai, P. M. Preiss, W. S. Bakr, J. Simon, M. Greiner, “Photon-assisted tunneling in a biased strongly correlated Bose gas,” Phys. Rev. Lett. 107, 095301, 2011).
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Phys. Rev. B (2)

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Á. Rapp, X. Deng, L. Santos, “Ultracold lattice gases with periodically modulated interactions,” Phys. Rev. Lett. 109, 203005, 2012).
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A. Zenesini, H. Lignier, D. Ciampini, O. Morsch, E. Arimondo, “Coherent control of sressed matter waves,” Phys. Rev. Lett. 102, 100403, 2009).
[CrossRef]

A. Eckardt, C. Weiss, M. Holthaus, “Superfluid-insulator transition in a periodically driven optical lattice,” Phys. Rev. Lett. 95, 260404, 2005).
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F. Massel, M. J. Leskinen, P. Törmä, “Hopping modulation in a one-dimensional Fermi-Hubbard Hamiltonian,” Phys. Rev. Lett. 103, 066404, 2009).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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Rev. Mod. Phys. (1)

P. Král, I. Thanopulos, M. Shapiro, “Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79, 53–77 (2007).
[CrossRef]

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

S. Trotzky, P. Cheinet, S. Folling, M. Feld, U. Schnorrberger, A. M. Rey, A. Polkovnikov, E. A. Demler, M. D. Lukin, 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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Figures (3)

Fig. 1
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.

Fig. 2
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.

Fig. 3
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).

Equations (31)

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V ( x , t ) = V 1 ( t ) cos ( k L x θ ) + V 2 ( t ) cos ( 2 k L x 2 θ ) + ε ( t ) x ,
H ( t ) = ( i , j ) J i j ( t ) ( b i b j + H . C . ) + ε ( t ) n x n b n b n .
x n = x | w ( x x n ) | 2 d x = { n ( a + b ) / 2 for even n , ( n + 1 ) a / 2 + ( n 1 ) b / 2 for odd n ,
J i j ( t ) = w * ( x x i ) [ d 2 d x 2 + V 1 ( t ) cos ( k L x θ ) + V 2 ( t ) cos ( 2 k L x 2 θ ) ] w ( x x j ) d x .
i c ˙ n ( t ) = J ( t ) ( c n + 1 + c n 1 ) ε 0 cos ( ω t ) x n c n ,
i A ˙ n ( t ) = J ( t ) ( A n + 1 e i Δ n sin ω t + A n 1 e i Δ n 1 sin ω t ) .
F ( t , m , ϕ , Δ n ) = J ( t , m , ϕ ) e i Δ n sin ω t = { J 0 + 1 2 δ J [ e i ( m ω t ϕ ) + e i ( m ω t ϕ ) ] } l 𝒥 l ( Δ n ) e i l ω t
F ¯ ( m , ϕ , Δ n ) = J 0 𝒥 0 ( Δ n ) + 1 2 δ J [ e i ϕ + ( 1 ) m e i ϕ ] 𝒥 m ( Δ n ) = J 0 𝒥 0 ( Δ n ) + { δ J cos ϕ 𝒥 m ( Δ n ) for even m , i δ J sin ϕ 𝒥 m ( Δ n ) for odd m
i A ˙ n ( t ) = F ¯ ( m , ϕ , Δ n ) A n + 1 + F ¯ ( m , ϕ , Δ n 1 ) A n 1 .
A ( k , t ) = n A n ( t ) e i n k = A e ( k , t ) + A o ( k , t )
i A ˙ e ( k , t ) = f ¯ ( k ) A o ( k , t ) , i A ˙ o ( k , t ) = f ¯ * ( k ) A e ( k , t )
A ( k , t ) = α ( k ) e i | f ¯ ( k ) | t + β ( k ) e i | f ¯ ( k ) | t .
A n ( t ) = 1 2 π π π [ α ( k ) e i | f ¯ ( k ) | t + β ( k ) e i | f ¯ ( k ) | t ] e i n k d k .
f ¯ ( k ) = J + cos k + i J sin k , J ± = F ¯ ( m , ϕ , Δ n ) ± F ¯ ( m , ϕ , Δ n 1 ) ,
A ( k , 0 ) = α ( k ) + β ( k ) = e i N k , i A ˙ ( k , 0 ) = | f ¯ ( k ) | [ α ( k ) β ( k ) ] = i n A ˙ n ( 0 ) e i n k = F ¯ ( m , ϕ , Δ N ) e i ( N + 1 ) k + F ¯ ( m , ϕ , Δ N 1 ) e i ( N 1 ) k .
α ( k ) = 1 2 | f ¯ ( k ) | [ | f ¯ ( k ) | e i N k F ¯ ( m , ϕ , Δ N ) e i ( N + 1 ) k F ¯ ( m , ϕ , Δ N 1 ) e i ( N 1 ) k ] , β ( k ) = 1 2 | f ¯ ( k ) | [ | f ¯ ( k ) | e i N k + F ¯ ( m , ϕ , Δ N ) e i ( N + 1 ) k + F ¯ ( m , ϕ , Δ N 1 ) e i ( N 1 ) k ] .
J 0 𝒥 0 ( Δ n ) + δ J cos ϕ 0 𝒥 m ( Δ n ) = J 0 𝒥 0 ( Δ n 1 ) + δ J cos ϕ 0 𝒥 m ( Δ n 1 ) = 0
J 0 𝒥 0 ( Δ n ) + i δ J sin ϕ 0 𝒥 m ( Δ n ) = J 0 𝒥 0 ( Δ n 1 ) i δ J sin ϕ 0 𝒥 m ( Δ n 1 ) = 0
F ¯ ( m , ϕ , Δ n ) = { δ J cos ϕ 𝒥 m ( Δ n ) for even m , i δ J sin ϕ 𝒥 m ( Δ n ) for odd m ,
α ( k ) = 1 2 e i N k F ¯ ( m , ϕ , Δ N ) 2 | F ¯ ( m , ϕ , Δ n ) | e i ( N + 1 ) k , β ( k ) = 1 2 e i N k + F ¯ ( m , ϕ , Δ N ) 2 | F ¯ ( m , ϕ , Δ n ) | e i ( N + 1 ) k
A n N , N + 1 ( t ) = 0 , A N ( t ) = cos ( ω 1 t ) , A N + 1 ( t ) = i F ¯ ( m , ϕ , Δ N ) | F ¯ ( m , ϕ , Δ N ) | sin ( ω 1 t ) .
A n N , N 1 ( t ) = 0 , A N ( t ) = cos ( ω 2 t ) , A N 1 ( t ) = i F ¯ ( m , ϕ , Δ N 1 ) | F ¯ ( m , ϕ , Δ N 1 ) | sin ( ω 2 t ) ,
F ¯ ( m , ϕ c , Δ N ) = F ¯ ( m , ϕ c , Δ n 1 )
d A n ( τ ) / d τ = 1 2 [ A n 1 ( τ ) A n + 1 ( τ ) ]
A n ( τ ) = B n 𝒥 n ( τ ) + D n 𝒩 n ( τ ) ,
δ A n ( τ ) = δ B n 𝒥 n ( τ ) + δ D n 𝒩 n ( τ )
F ¯ ( m , ϕ 1 , Δ n 1 ) = J 0 𝒥 0 ( Δ n 1 ) + δ J cos ϕ 1 𝒥 m ( Δ n 1 ) = 0 , F ¯ ( m , ϕ 2 , Δ n ) = J 0 𝒥 0 ( Δ n ) + δ J cos ϕ 2 𝒥 m ( Δ n ) = 0
J ( t , m , ϕ ) = { J ( t , m , ϕ 1 ) for t [ n T , n T + T 1 ] , J ( t , m , ϕ 2 ) for t [ n T + T 1 , ( n + 1 ) T ]
U ^ = exp [ i ε ( t ) d t n x n b n b n ] ,
H int = U ^ H I U ^ = ( i , j ) F ¯ ( m , ϕ , Δ i j ) ( b i b j + H . C . ) + 1 2 U 0 n n ^ ( n ^ 1 )
H I = ( i , j ) J ( t , m , ϕ ) ( b i b j + H . C . ) + 1 2 U 0 n n ^ ( n 1 ) ,

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