Abstract

We study, analytically and numerically, the dynamics of interband transitions in two-dimensional hexagonal periodic photonic lattices. We develop an analytical approach employing the Bragg resonances of different types and derive the effective multi-level models of the Landau-Zener-Majorana type. For two-dimensional periodic potentials without a tilt, we demonstrate the possibility of the Rabi oscillations between the resonant Fourier amplitudes. In a biased lattice, i.e., for a two-dimensional periodic potential with an additional linear tilt, we identify three basic types of the interband transitions or Zener tunnelling. First, this is a quasi-one-dimensional tunnelling that involves only two Bloch bands and occurs when the Bloch index crosses the Bragg planes away from one of the high-symmetry points. In contrast, at the high-symmetry points (i.e., at the M and Γ points), the Zener tunnelling is essentially two-dimensional, and it involves either three or six Bloch bands being described by the corresponding multi-level Landau-Zener-Majorana systems. We verify our analytical results by numerical simulations and observe an excellent agreement. Finally, we show that phase dislocations, or optical vortices, can tunnel between the spectral bands preserving their topological charge. Our theory describes the propagation of light beams in fabricated or optically-induced two-dimensional photonic lattices, but it can also be applied to the physics of cold atoms and Bose-Einstein condensates tunnelling in tilted two-dimensional optical potentials and other types of resonant wave propagation in periodic media.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Bloch, "Uber die Quantenmechanik der Elektronen in Kristallgittern," Z. Phys. 52, 555-600 (1928).
  2. C. Zener, "Non-Adiabatic Crossing of Energy Levels," Proc. R. Soc. London A 137, 696-702 (1932).
  3. G. H. Wannier, "Wave Functions and Effective Hamiltonian for Bloch Electrons in an Electric Field," Phys. Rev. 117, 432-439 (1960).
    [CrossRef]
  4. L. D. Landau, "On the theory of transfer of energy at collisions II," Phys. Z. Sowjetunion 2, 46 (1932).
  5. E. Majorana, "Orientated atoms in a variable magnetic field," Nuovo Cimento 9, 43-50 (1932).
    [CrossRef]
  6. L. Esaki, "Long journey into tunneling," Rev. Mod. Phys. 46, 237-244 (1974).
    [CrossRef]
  7. B. Bourlon, D. C. Glattli, B. Placais, J. M. Berroir, C. Miko, L. Forro, and A. Bachtold, "Geometrical Dependence of High-Bias Current in Multiwalled Carbon Nanotubes," Phys. Rev. Lett. 92, 026804 (2004).
    [CrossRef] [PubMed]
  8. A. Sibille, J. F. Palmier, and F. Laruelle, "Zener Interminiband Resonant Breakdown in Superlattices," Phys. Rev. Lett. 80, 4506-4509 (1998).
    [CrossRef]
  9. M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J. H. M¨uller, E. Courtade, M. Anderlini, and E. Arimondo, "Asymmetric Landau-Zener Tunneling in a Periodic Potential," Phys. Rev. Lett. 91, 230406 (2003).
    [CrossRef] [PubMed]
  10. M. Jona-Lasinio, O. Morsch, M. Cristiani, E. Arimondo, and C. Menotti, "Nonlinear Effects for BoseEinstein Condensates in Optical Lattices," Laser Phys. 15, 1180-1188 (2005).
  11. G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo, "Theory of photon Bloch oscillations in photonic crystals," Phys. Rev. B 63, 035108 (2001).
  12. V. Agarwal, J. A. del Rıo, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
    [CrossRef] [PubMed]
  13. H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual Observation of Zener Tunneling," Phys. Rev. Lett. 96, 023901 (2006).
    [CrossRef] [PubMed]
  14. H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices," Phys. Rev. Lett. 96, 053903 (2006).
    [CrossRef] [PubMed]
  15. M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch Oscillations of Atoms in an Optical Potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
    [CrossRef] [PubMed]
  16. E. Peik, M. B. Dahan, I. Bouchoule, Y. Castin, and C. Salomon, "Bloch oscillations of atoms, adiabatic rapid passage, and monokinetic atomic beams," Phys. Rev. A 55, 2989-3001 (1997).
  17. B. P. Anderson and M. A. Kasevich, "Macroscopic Quantum Interference from Atomic Tunnel Arrays," Science 282, 1686 (1998).
    [CrossRef] [PubMed]
  18. F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, "Josephson Junction Arrays with Bose-Einstein Condensates," Science 293, 843 (2001).
    [CrossRef] [PubMed]
  19. L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford: Clarendon Press, 2003).
  20. L. Santos, M. A. Baranov, J. I. Cirac, H.-U. Everts, H. Fehrmann, and M. Lewenstein, "Atomic Quantum Gases in Kagome Lattices," Phys. Rev. Lett. 93, 030601 (2004).
    [CrossRef] [PubMed]
  21. A. A. Burkov and E. Demler, "Vortex-Peierls States in Optical Lattices," Phys. Rev. Lett. 96, 180406 (2006).
    [CrossRef] [PubMed]
  22. L. Sanchez-Palencia and L. Santos, "Bose-Einstein condensates in optical quasicrystal lattices," Phys. Rev. A 72, 053607 (2005).
  23. V. S. Shchesnovich, S. B. Cavalcanti, J. M. Hickmann, and Yu. S. Kivshar, "Zener tunneling in two-dimensional photonic lattices," Phys. Rev. E. 74, 056602 (2006).
  24. A. S. Desyatnikov, Yu. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, "Resonant Zener tunneling in two-dimensional periodic photonic lattices," Opt. Lett. 32, 325-327 (2007).
    [CrossRef] [PubMed]
  25. N. W. Ashcroft and N. D. Mermin, Solid state physics (NewYork: Holt, Rinehart and Winston, 1976).
  26. A. R. Kolovsky and H. J. Korsch, "Bloch oscillations of cold atoms in two-dimensional optical lattices," Phys. Rev. A 67, 063601 (2003).
  27. D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, "Bloch oscillations in two-dimensional lattices," New J. Phys. 6, 41 (2004).
    [CrossRef]
  28. B. Wu and Q. Niu, "Nonlinear Landau-Zener tunneling," Phys. Rev. A 61, 023402 (2000).
  29. O. Zobay and B. M. Garraway, "Time-dependent tunneling of Bose-Einstein condensates," Phys. Rev. A 61, 033603 (2000).
  30. V. A. Brazhnyi, V. V. Konotop, and V. Kuzmiak, "Nature of the Intrinsic Relation between Bloch-Band Tunneling and Modulational Instability," Phys. Rev. Lett. 96, 150402 (2006).
    [CrossRef] [PubMed]
  31. V. V. Konotop, P. G. Kevrekidis, and M. Salerno, "Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice," Phys. Rev. A 72, 023611 (2005).
  32. V. A. Brazhnyi, V. V. Konotop, V. Kuzmiak, and V. S. Shchesnovich, "Nonlinear tunneling in two-dimensional lattices," Phys. Rev. A 76, 023608 (2007).
  33. V. S. Shchesnovich and V. V. Konotop, "Nonlinear tunneling of Bose-Einstein condensates in an optical lattice: Signatures of quantum collapse and revival," Phys. Rev. A 75, 063628 (2007).
  34. V. S. Shchesnovich and V. V. Konotop, "Nonlinear intraband tunneling of a Bose-Einstein condensate in a cubic three-dimensional lattice," Phys. Rev. A 77, 013614 (2008).
  35. S. Brundobler and V. Elser, "S-matrix for generalized Landau-Zener problem," J. Phys. A: Math. Gen. 26, 1211- 1227 (1993).
    [CrossRef]
  36. Yu. N. Demkov and V. N. Ostrovsky, "Multipath interference in a multistate Landau-Zener-type model," Phys. Rev. A. 61, 032705 (2000).
    [CrossRef]
  37. A. V. Shytov, "Landau-Zener transitions in a multilevel system: An exact result," Phys. Rev. A 70, 052708 (2004).
  38. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (San Diego: Academic, 2003).
  39. V. V. Konotop and M. Salerno, "Modulational instability in Bose-Einstein condensates in optical lattices," Phys. Rev. A 65, 021602 (2002).
  40. F. Keck and H. J. Korsch, "Infinite-variable Bessel functions in two-dimensional Wannier-Stark systems," J. Phys. A: Math. Gen. 35, L105-L116 (2002).
    [CrossRef]
  41. G. A. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, "Wannier functions analysis of the nonlinear Schrdinger equation with a periodic potential," Phys. Rev. E 66, 046608 (2002).
  42. V. S. Shchesnovich and S. Chavez-Cerda, "Bragg-resonance-induced Rabi oscillations in photonic lattices," Opt. Lett. 32, 1920-1922 (2007).
    [CrossRef] [PubMed]
  43. W. V. Houston, "Acceleration of Electrons in a Crystal Lattice," Phys. Rev. 57, 184-186 (1940).
    [CrossRef]
  44. V. S. Shchesnovich and S. B. Cavalcanti, "Finite-dimensional model for the condensate tunnelling in an accelerating optical lattice," J. Phys. B: At. Mol. Opt. Phys. 39, 1997-2011 (1997).
    [CrossRef]
  45. A. S. Desyatnikov, N. Sagemerten, R. Fischer, B. Terhalle, D. Tr¨ager, D. N. Neshev, A. Dreischuh, C. Denz, W. Krolikowski, and Yu. S. Kivshar, "Two-dimensional self-trapped nonlinear photonic lattices," Opt. Express 14, 2851-2863 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-7-2851.
    [CrossRef] [PubMed]
  46. M. S. Soskin and M. V. Vasnetsov, "Singular Optics," Progress in Optics 42, 219 (2001).
    [CrossRef]
  47. T. J. Alexander, A. S. Desyatnikov, and Yu. S. Kivshar, "Multivortex solitons in triangular photonic lattices," Opt. Lett. 32, 1293-1295 (2007).
    [CrossRef] [PubMed]
  48. A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical Vortices and Vortex Solitons," Progress in Optics 47, 291-391 (2005).
    [CrossRef]

2008 (1)

V. S. Shchesnovich and V. V. Konotop, "Nonlinear intraband tunneling of a Bose-Einstein condensate in a cubic three-dimensional lattice," Phys. Rev. A 77, 013614 (2008).

2007 (5)

2006 (6)

A. S. Desyatnikov, N. Sagemerten, R. Fischer, B. Terhalle, D. Tr¨ager, D. N. Neshev, A. Dreischuh, C. Denz, W. Krolikowski, and Yu. S. Kivshar, "Two-dimensional self-trapped nonlinear photonic lattices," Opt. Express 14, 2851-2863 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-7-2851.
[CrossRef] [PubMed]

A. A. Burkov and E. Demler, "Vortex-Peierls States in Optical Lattices," Phys. Rev. Lett. 96, 180406 (2006).
[CrossRef] [PubMed]

V. S. Shchesnovich, S. B. Cavalcanti, J. M. Hickmann, and Yu. S. Kivshar, "Zener tunneling in two-dimensional photonic lattices," Phys. Rev. E. 74, 056602 (2006).

V. A. Brazhnyi, V. V. Konotop, and V. Kuzmiak, "Nature of the Intrinsic Relation between Bloch-Band Tunneling and Modulational Instability," Phys. Rev. Lett. 96, 150402 (2006).
[CrossRef] [PubMed]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual Observation of Zener Tunneling," Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

2005 (4)

M. Jona-Lasinio, O. Morsch, M. Cristiani, E. Arimondo, and C. Menotti, "Nonlinear Effects for BoseEinstein Condensates in Optical Lattices," Laser Phys. 15, 1180-1188 (2005).

V. V. Konotop, P. G. Kevrekidis, and M. Salerno, "Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice," Phys. Rev. A 72, 023611 (2005).

L. Sanchez-Palencia and L. Santos, "Bose-Einstein condensates in optical quasicrystal lattices," Phys. Rev. A 72, 053607 (2005).

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical Vortices and Vortex Solitons," Progress in Optics 47, 291-391 (2005).
[CrossRef]

2004 (5)

L. Santos, M. A. Baranov, J. I. Cirac, H.-U. Everts, H. Fehrmann, and M. Lewenstein, "Atomic Quantum Gases in Kagome Lattices," Phys. Rev. Lett. 93, 030601 (2004).
[CrossRef] [PubMed]

D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, "Bloch oscillations in two-dimensional lattices," New J. Phys. 6, 41 (2004).
[CrossRef]

A. V. Shytov, "Landau-Zener transitions in a multilevel system: An exact result," Phys. Rev. A 70, 052708 (2004).

V. Agarwal, J. A. del Rıo, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

B. Bourlon, D. C. Glattli, B. Placais, J. M. Berroir, C. Miko, L. Forro, and A. Bachtold, "Geometrical Dependence of High-Bias Current in Multiwalled Carbon Nanotubes," Phys. Rev. Lett. 92, 026804 (2004).
[CrossRef] [PubMed]

2003 (2)

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J. H. M¨uller, E. Courtade, M. Anderlini, and E. Arimondo, "Asymmetric Landau-Zener Tunneling in a Periodic Potential," Phys. Rev. Lett. 91, 230406 (2003).
[CrossRef] [PubMed]

A. R. Kolovsky and H. J. Korsch, "Bloch oscillations of cold atoms in two-dimensional optical lattices," Phys. Rev. A 67, 063601 (2003).

2002 (3)

V. V. Konotop and M. Salerno, "Modulational instability in Bose-Einstein condensates in optical lattices," Phys. Rev. A 65, 021602 (2002).

F. Keck and H. J. Korsch, "Infinite-variable Bessel functions in two-dimensional Wannier-Stark systems," J. Phys. A: Math. Gen. 35, L105-L116 (2002).
[CrossRef]

G. A. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, "Wannier functions analysis of the nonlinear Schrdinger equation with a periodic potential," Phys. Rev. E 66, 046608 (2002).

2001 (3)

F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, "Josephson Junction Arrays with Bose-Einstein Condensates," Science 293, 843 (2001).
[CrossRef] [PubMed]

G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo, "Theory of photon Bloch oscillations in photonic crystals," Phys. Rev. B 63, 035108 (2001).

M. S. Soskin and M. V. Vasnetsov, "Singular Optics," Progress in Optics 42, 219 (2001).
[CrossRef]

2000 (3)

Yu. N. Demkov and V. N. Ostrovsky, "Multipath interference in a multistate Landau-Zener-type model," Phys. Rev. A. 61, 032705 (2000).
[CrossRef]

B. Wu and Q. Niu, "Nonlinear Landau-Zener tunneling," Phys. Rev. A 61, 023402 (2000).

O. Zobay and B. M. Garraway, "Time-dependent tunneling of Bose-Einstein condensates," Phys. Rev. A 61, 033603 (2000).

1998 (2)

B. P. Anderson and M. A. Kasevich, "Macroscopic Quantum Interference from Atomic Tunnel Arrays," Science 282, 1686 (1998).
[CrossRef] [PubMed]

A. Sibille, J. F. Palmier, and F. Laruelle, "Zener Interminiband Resonant Breakdown in Superlattices," Phys. Rev. Lett. 80, 4506-4509 (1998).
[CrossRef]

1997 (2)

E. Peik, M. B. Dahan, I. Bouchoule, Y. Castin, and C. Salomon, "Bloch oscillations of atoms, adiabatic rapid passage, and monokinetic atomic beams," Phys. Rev. A 55, 2989-3001 (1997).

V. S. Shchesnovich and S. B. Cavalcanti, "Finite-dimensional model for the condensate tunnelling in an accelerating optical lattice," J. Phys. B: At. Mol. Opt. Phys. 39, 1997-2011 (1997).
[CrossRef]

1996 (1)

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch Oscillations of Atoms in an Optical Potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

1993 (1)

S. Brundobler and V. Elser, "S-matrix for generalized Landau-Zener problem," J. Phys. A: Math. Gen. 26, 1211- 1227 (1993).
[CrossRef]

1974 (1)

L. Esaki, "Long journey into tunneling," Rev. Mod. Phys. 46, 237-244 (1974).
[CrossRef]

1960 (1)

G. H. Wannier, "Wave Functions and Effective Hamiltonian for Bloch Electrons in an Electric Field," Phys. Rev. 117, 432-439 (1960).
[CrossRef]

1940 (1)

W. V. Houston, "Acceleration of Electrons in a Crystal Lattice," Phys. Rev. 57, 184-186 (1940).
[CrossRef]

1932 (3)

L. D. Landau, "On the theory of transfer of energy at collisions II," Phys. Z. Sowjetunion 2, 46 (1932).

E. Majorana, "Orientated atoms in a variable magnetic field," Nuovo Cimento 9, 43-50 (1932).
[CrossRef]

C. Zener, "Non-Adiabatic Crossing of Energy Levels," Proc. R. Soc. London A 137, 696-702 (1932).

1928 (1)

F. Bloch, "Uber die Quantenmechanik der Elektronen in Kristallgittern," Z. Phys. 52, 555-600 (1928).

Agarwal, V.

V. Agarwal, J. A. del Rıo, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Alexander, T. J.

Alfimov, G. A.

G. A. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, "Wannier functions analysis of the nonlinear Schrdinger equation with a periodic potential," Phys. Rev. E 66, 046608 (2002).

Anderlini, M.

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J. H. M¨uller, E. Courtade, M. Anderlini, and E. Arimondo, "Asymmetric Landau-Zener Tunneling in a Periodic Potential," Phys. Rev. Lett. 91, 230406 (2003).
[CrossRef] [PubMed]

Anderson, B. P.

B. P. Anderson and M. A. Kasevich, "Macroscopic Quantum Interference from Atomic Tunnel Arrays," Science 282, 1686 (1998).
[CrossRef] [PubMed]

Arimondo, E.

M. Jona-Lasinio, O. Morsch, M. Cristiani, E. Arimondo, and C. Menotti, "Nonlinear Effects for BoseEinstein Condensates in Optical Lattices," Laser Phys. 15, 1180-1188 (2005).

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J. H. M¨uller, E. Courtade, M. Anderlini, and E. Arimondo, "Asymmetric Landau-Zener Tunneling in a Periodic Potential," Phys. Rev. Lett. 91, 230406 (2003).
[CrossRef] [PubMed]

Bachtold, A.

B. Bourlon, D. C. Glattli, B. Placais, J. M. Berroir, C. Miko, L. Forro, and A. Bachtold, "Geometrical Dependence of High-Bias Current in Multiwalled Carbon Nanotubes," Phys. Rev. Lett. 92, 026804 (2004).
[CrossRef] [PubMed]

Baranov, M. A.

L. Santos, M. A. Baranov, J. I. Cirac, H.-U. Everts, H. Fehrmann, and M. Lewenstein, "Atomic Quantum Gases in Kagome Lattices," Phys. Rev. Lett. 93, 030601 (2004).
[CrossRef] [PubMed]

Berroir, J. M.

B. Bourlon, D. C. Glattli, B. Placais, J. M. Berroir, C. Miko, L. Forro, and A. Bachtold, "Geometrical Dependence of High-Bias Current in Multiwalled Carbon Nanotubes," Phys. Rev. Lett. 92, 026804 (2004).
[CrossRef] [PubMed]

Bloch, F.

F. Bloch, "Uber die Quantenmechanik der Elektronen in Kristallgittern," Z. Phys. 52, 555-600 (1928).

Bouchoule, I.

E. Peik, M. B. Dahan, I. Bouchoule, Y. Castin, and C. Salomon, "Bloch oscillations of atoms, adiabatic rapid passage, and monokinetic atomic beams," Phys. Rev. A 55, 2989-3001 (1997).

Bourlon, B.

B. Bourlon, D. C. Glattli, B. Placais, J. M. Berroir, C. Miko, L. Forro, and A. Bachtold, "Geometrical Dependence of High-Bias Current in Multiwalled Carbon Nanotubes," Phys. Rev. Lett. 92, 026804 (2004).
[CrossRef] [PubMed]

Brauer, A.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual Observation of Zener Tunneling," Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

Brazhnyi, V. A.

V. A. Brazhnyi, V. V. Konotop, V. Kuzmiak, and V. S. Shchesnovich, "Nonlinear tunneling in two-dimensional lattices," Phys. Rev. A 76, 023608 (2007).

V. A. Brazhnyi, V. V. Konotop, and V. Kuzmiak, "Nature of the Intrinsic Relation between Bloch-Band Tunneling and Modulational Instability," Phys. Rev. Lett. 96, 150402 (2006).
[CrossRef] [PubMed]

Brundobler, S.

S. Brundobler and V. Elser, "S-matrix for generalized Landau-Zener problem," J. Phys. A: Math. Gen. 26, 1211- 1227 (1993).
[CrossRef]

Burger, S.

F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, "Josephson Junction Arrays with Bose-Einstein Condensates," Science 293, 843 (2001).
[CrossRef] [PubMed]

Burkov, A. A.

A. A. Burkov and E. Demler, "Vortex-Peierls States in Optical Lattices," Phys. Rev. Lett. 96, 180406 (2006).
[CrossRef] [PubMed]

Castin, Y.

E. Peik, M. B. Dahan, I. Bouchoule, Y. Castin, and C. Salomon, "Bloch oscillations of atoms, adiabatic rapid passage, and monokinetic atomic beams," Phys. Rev. A 55, 2989-3001 (1997).

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch Oscillations of Atoms in an Optical Potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

Cataliotti, F. S.

F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, "Josephson Junction Arrays with Bose-Einstein Condensates," Science 293, 843 (2001).
[CrossRef] [PubMed]

Cavalcanti, S. B.

A. S. Desyatnikov, Yu. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, "Resonant Zener tunneling in two-dimensional periodic photonic lattices," Opt. Lett. 32, 325-327 (2007).
[CrossRef] [PubMed]

V. S. Shchesnovich, S. B. Cavalcanti, J. M. Hickmann, and Yu. S. Kivshar, "Zener tunneling in two-dimensional photonic lattices," Phys. Rev. E. 74, 056602 (2006).

V. S. Shchesnovich and S. B. Cavalcanti, "Finite-dimensional model for the condensate tunnelling in an accelerating optical lattice," J. Phys. B: At. Mol. Opt. Phys. 39, 1997-2011 (1997).
[CrossRef]

Chavez-Cerda, S.

Cirac, J. I.

L. Santos, M. A. Baranov, J. I. Cirac, H.-U. Everts, H. Fehrmann, and M. Lewenstein, "Atomic Quantum Gases in Kagome Lattices," Phys. Rev. Lett. 93, 030601 (2004).
[CrossRef] [PubMed]

Courtade, E.

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J. H. M¨uller, E. Courtade, M. Anderlini, and E. Arimondo, "Asymmetric Landau-Zener Tunneling in a Periodic Potential," Phys. Rev. Lett. 91, 230406 (2003).
[CrossRef] [PubMed]

Cristiani, M.

M. Jona-Lasinio, O. Morsch, M. Cristiani, E. Arimondo, and C. Menotti, "Nonlinear Effects for BoseEinstein Condensates in Optical Lattices," Laser Phys. 15, 1180-1188 (2005).

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J. H. M¨uller, E. Courtade, M. Anderlini, and E. Arimondo, "Asymmetric Landau-Zener Tunneling in a Periodic Potential," Phys. Rev. Lett. 91, 230406 (2003).
[CrossRef] [PubMed]

Dahan, M. B.

E. Peik, M. B. Dahan, I. Bouchoule, Y. Castin, and C. Salomon, "Bloch oscillations of atoms, adiabatic rapid passage, and monokinetic atomic beams," Phys. Rev. A 55, 2989-3001 (1997).

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch Oscillations of Atoms in an Optical Potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

Demkov, Yu. N.

Yu. N. Demkov and V. N. Ostrovsky, "Multipath interference in a multistate Landau-Zener-type model," Phys. Rev. A. 61, 032705 (2000).
[CrossRef]

Demler, E.

A. A. Burkov and E. Demler, "Vortex-Peierls States in Optical Lattices," Phys. Rev. Lett. 96, 180406 (2006).
[CrossRef] [PubMed]

Denz, C.

Desyatnikov, A. S.

Di Carlo, A.

G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo, "Theory of photon Bloch oscillations in photonic crystals," Phys. Rev. B 63, 035108 (2001).

Dreischuh, A.

Elser, V.

S. Brundobler and V. Elser, "S-matrix for generalized Landau-Zener problem," J. Phys. A: Math. Gen. 26, 1211- 1227 (1993).
[CrossRef]

Esaki, L.

L. Esaki, "Long journey into tunneling," Rev. Mod. Phys. 46, 237-244 (1974).
[CrossRef]

Everts, H.-U.

L. Santos, M. A. Baranov, J. I. Cirac, H.-U. Everts, H. Fehrmann, and M. Lewenstein, "Atomic Quantum Gases in Kagome Lattices," Phys. Rev. Lett. 93, 030601 (2004).
[CrossRef] [PubMed]

Fehrmann, H.

L. Santos, M. A. Baranov, J. I. Cirac, H.-U. Everts, H. Fehrmann, and M. Lewenstein, "Atomic Quantum Gases in Kagome Lattices," Phys. Rev. Lett. 93, 030601 (2004).
[CrossRef] [PubMed]

Fischer, R.

Forro, L.

B. Bourlon, D. C. Glattli, B. Placais, J. M. Berroir, C. Miko, L. Forro, and A. Bachtold, "Geometrical Dependence of High-Bias Current in Multiwalled Carbon Nanotubes," Phys. Rev. Lett. 92, 026804 (2004).
[CrossRef] [PubMed]

Fort, C.

F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, "Josephson Junction Arrays with Bose-Einstein Condensates," Science 293, 843 (2001).
[CrossRef] [PubMed]

Garraway, B. M.

O. Zobay and B. M. Garraway, "Time-dependent tunneling of Bose-Einstein condensates," Phys. Rev. A 61, 033603 (2000).

Glattli, D. C.

B. Bourlon, D. C. Glattli, B. Placais, J. M. Berroir, C. Miko, L. Forro, and A. Bachtold, "Geometrical Dependence of High-Bias Current in Multiwalled Carbon Nanotubes," Phys. Rev. Lett. 92, 026804 (2004).
[CrossRef] [PubMed]

Hickmann, J. M.

A. S. Desyatnikov, Yu. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, "Resonant Zener tunneling in two-dimensional periodic photonic lattices," Opt. Lett. 32, 325-327 (2007).
[CrossRef] [PubMed]

V. S. Shchesnovich, S. B. Cavalcanti, J. M. Hickmann, and Yu. S. Kivshar, "Zener tunneling in two-dimensional photonic lattices," Phys. Rev. E. 74, 056602 (2006).

Houston, W. V.

W. V. Houston, "Acceleration of Electrons in a Crystal Lattice," Phys. Rev. 57, 184-186 (1940).
[CrossRef]

Inguscio, M.

F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, "Josephson Junction Arrays with Bose-Einstein Condensates," Science 293, 843 (2001).
[CrossRef] [PubMed]

Jona-Lasinio, M.

M. Jona-Lasinio, O. Morsch, M. Cristiani, E. Arimondo, and C. Menotti, "Nonlinear Effects for BoseEinstein Condensates in Optical Lattices," Laser Phys. 15, 1180-1188 (2005).

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J. H. M¨uller, E. Courtade, M. Anderlini, and E. Arimondo, "Asymmetric Landau-Zener Tunneling in a Periodic Potential," Phys. Rev. Lett. 91, 230406 (2003).
[CrossRef] [PubMed]

Kasevich, M. A.

B. P. Anderson and M. A. Kasevich, "Macroscopic Quantum Interference from Atomic Tunnel Arrays," Science 282, 1686 (1998).
[CrossRef] [PubMed]

Kavokin, A.

G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo, "Theory of photon Bloch oscillations in photonic crystals," Phys. Rev. B 63, 035108 (2001).

Keck, F.

D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, "Bloch oscillations in two-dimensional lattices," New J. Phys. 6, 41 (2004).
[CrossRef]

F. Keck and H. J. Korsch, "Infinite-variable Bessel functions in two-dimensional Wannier-Stark systems," J. Phys. A: Math. Gen. 35, L105-L116 (2002).
[CrossRef]

Kevrekidis, P. G.

V. V. Konotop, P. G. Kevrekidis, and M. Salerno, "Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice," Phys. Rev. A 72, 023611 (2005).

G. A. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, "Wannier functions analysis of the nonlinear Schrdinger equation with a periodic potential," Phys. Rev. E 66, 046608 (2002).

Kivshar, Yu. S.

T. J. Alexander, A. S. Desyatnikov, and Yu. S. Kivshar, "Multivortex solitons in triangular photonic lattices," Opt. Lett. 32, 1293-1295 (2007).
[CrossRef] [PubMed]

A. S. Desyatnikov, Yu. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, "Resonant Zener tunneling in two-dimensional periodic photonic lattices," Opt. Lett. 32, 325-327 (2007).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

V. S. Shchesnovich, S. B. Cavalcanti, J. M. Hickmann, and Yu. S. Kivshar, "Zener tunneling in two-dimensional photonic lattices," Phys. Rev. E. 74, 056602 (2006).

A. S. Desyatnikov, N. Sagemerten, R. Fischer, B. Terhalle, D. Tr¨ager, D. N. Neshev, A. Dreischuh, C. Denz, W. Krolikowski, and Yu. S. Kivshar, "Two-dimensional self-trapped nonlinear photonic lattices," Opt. Express 14, 2851-2863 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-7-2851.
[CrossRef] [PubMed]

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical Vortices and Vortex Solitons," Progress in Optics 47, 291-391 (2005).
[CrossRef]

Kolovsky, A. R.

A. R. Kolovsky and H. J. Korsch, "Bloch oscillations of cold atoms in two-dimensional optical lattices," Phys. Rev. A 67, 063601 (2003).

Konotop, V. V.

V. S. Shchesnovich and V. V. Konotop, "Nonlinear intraband tunneling of a Bose-Einstein condensate in a cubic three-dimensional lattice," Phys. Rev. A 77, 013614 (2008).

V. A. Brazhnyi, V. V. Konotop, V. Kuzmiak, and V. S. Shchesnovich, "Nonlinear tunneling in two-dimensional lattices," Phys. Rev. A 76, 023608 (2007).

V. S. Shchesnovich and V. V. Konotop, "Nonlinear tunneling of Bose-Einstein condensates in an optical lattice: Signatures of quantum collapse and revival," Phys. Rev. A 75, 063628 (2007).

V. A. Brazhnyi, V. V. Konotop, and V. Kuzmiak, "Nature of the Intrinsic Relation between Bloch-Band Tunneling and Modulational Instability," Phys. Rev. Lett. 96, 150402 (2006).
[CrossRef] [PubMed]

V. V. Konotop, P. G. Kevrekidis, and M. Salerno, "Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice," Phys. Rev. A 72, 023611 (2005).

V. V. Konotop and M. Salerno, "Modulational instability in Bose-Einstein condensates in optical lattices," Phys. Rev. A 65, 021602 (2002).

G. A. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, "Wannier functions analysis of the nonlinear Schrdinger equation with a periodic potential," Phys. Rev. E 66, 046608 (2002).

Korsch, H. J.

D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, "Bloch oscillations in two-dimensional lattices," New J. Phys. 6, 41 (2004).
[CrossRef]

A. R. Kolovsky and H. J. Korsch, "Bloch oscillations of cold atoms in two-dimensional optical lattices," Phys. Rev. A 67, 063601 (2003).

F. Keck and H. J. Korsch, "Infinite-variable Bessel functions in two-dimensional Wannier-Stark systems," J. Phys. A: Math. Gen. 35, L105-L116 (2002).
[CrossRef]

Krolikowski, W.

A. S. Desyatnikov, N. Sagemerten, R. Fischer, B. Terhalle, D. Tr¨ager, D. N. Neshev, A. Dreischuh, C. Denz, W. Krolikowski, and Yu. S. Kivshar, "Two-dimensional self-trapped nonlinear photonic lattices," Opt. Express 14, 2851-2863 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-7-2851.
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Kuzmiak, V.

V. A. Brazhnyi, V. V. Konotop, V. Kuzmiak, and V. S. Shchesnovich, "Nonlinear tunneling in two-dimensional lattices," Phys. Rev. A 76, 023608 (2007).

V. A. Brazhnyi, V. V. Konotop, and V. Kuzmiak, "Nature of the Intrinsic Relation between Bloch-Band Tunneling and Modulational Instability," Phys. Rev. Lett. 96, 150402 (2006).
[CrossRef] [PubMed]

Landau, L. D.

L. D. Landau, "On the theory of transfer of energy at collisions II," Phys. Z. Sowjetunion 2, 46 (1932).

Laruelle, F.

A. Sibille, J. F. Palmier, and F. Laruelle, "Zener Interminiband Resonant Breakdown in Superlattices," Phys. Rev. Lett. 80, 4506-4509 (1998).
[CrossRef]

Lederer, F.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual Observation of Zener Tunneling," Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Lewenstein, M.

L. Santos, M. A. Baranov, J. I. Cirac, H.-U. Everts, H. Fehrmann, and M. Lewenstein, "Atomic Quantum Gases in Kagome Lattices," Phys. Rev. Lett. 93, 030601 (2004).
[CrossRef] [PubMed]

M¨uller, J. H.

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J. H. M¨uller, E. Courtade, M. Anderlini, and E. Arimondo, "Asymmetric Landau-Zener Tunneling in a Periodic Potential," Phys. Rev. Lett. 91, 230406 (2003).
[CrossRef] [PubMed]

Maddaloni, P.

F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, "Josephson Junction Arrays with Bose-Einstein Condensates," Science 293, 843 (2001).
[CrossRef] [PubMed]

Majorana, E.

E. Majorana, "Orientated atoms in a variable magnetic field," Nuovo Cimento 9, 43-50 (1932).
[CrossRef]

Malossi, N.

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J. H. M¨uller, E. Courtade, M. Anderlini, and E. Arimondo, "Asymmetric Landau-Zener Tunneling in a Periodic Potential," Phys. Rev. Lett. 91, 230406 (2003).
[CrossRef] [PubMed]

Malpuech, G.

G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo, "Theory of photon Bloch oscillations in photonic crystals," Phys. Rev. B 63, 035108 (2001).

Menotti, C.

M. Jona-Lasinio, O. Morsch, M. Cristiani, E. Arimondo, and C. Menotti, "Nonlinear Effects for BoseEinstein Condensates in Optical Lattices," Laser Phys. 15, 1180-1188 (2005).

Michaelis, D.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual Observation of Zener Tunneling," Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

Miko, C.

B. Bourlon, D. C. Glattli, B. Placais, J. M. Berroir, C. Miko, L. Forro, and A. Bachtold, "Geometrical Dependence of High-Bias Current in Multiwalled Carbon Nanotubes," Phys. Rev. Lett. 92, 026804 (2004).
[CrossRef] [PubMed]

Minardi, F.

F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, "Josephson Junction Arrays with Bose-Einstein Condensates," Science 293, 843 (2001).
[CrossRef] [PubMed]

Morsch, O.

M. Jona-Lasinio, O. Morsch, M. Cristiani, E. Arimondo, and C. Menotti, "Nonlinear Effects for BoseEinstein Condensates in Optical Lattices," Laser Phys. 15, 1180-1188 (2005).

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J. H. M¨uller, E. Courtade, M. Anderlini, and E. Arimondo, "Asymmetric Landau-Zener Tunneling in a Periodic Potential," Phys. Rev. Lett. 91, 230406 (2003).
[CrossRef] [PubMed]

Mossmann, S.

D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, "Bloch oscillations in two-dimensional lattices," New J. Phys. 6, 41 (2004).
[CrossRef]

Neshev, D. N.

A. S. Desyatnikov, N. Sagemerten, R. Fischer, B. Terhalle, D. Tr¨ager, D. N. Neshev, A. Dreischuh, C. Denz, W. Krolikowski, and Yu. S. Kivshar, "Two-dimensional self-trapped nonlinear photonic lattices," Opt. Express 14, 2851-2863 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-7-2851.
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Niu, Q.

B. Wu and Q. Niu, "Nonlinear Landau-Zener tunneling," Phys. Rev. A 61, 023402 (2000).

Ostrovsky, V. N.

Yu. N. Demkov and V. N. Ostrovsky, "Multipath interference in a multistate Landau-Zener-type model," Phys. Rev. A. 61, 032705 (2000).
[CrossRef]

Palmier, J. F.

A. Sibille, J. F. Palmier, and F. Laruelle, "Zener Interminiband Resonant Breakdown in Superlattices," Phys. Rev. Lett. 80, 4506-4509 (1998).
[CrossRef]

Panzarini, G.

G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo, "Theory of photon Bloch oscillations in photonic crystals," Phys. Rev. B 63, 035108 (2001).

Peik, E.

E. Peik, M. B. Dahan, I. Bouchoule, Y. Castin, and C. Salomon, "Bloch oscillations of atoms, adiabatic rapid passage, and monokinetic atomic beams," Phys. Rev. A 55, 2989-3001 (1997).

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch Oscillations of Atoms in an Optical Potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

Pertsch, T.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual Observation of Zener Tunneling," Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Peschel, U.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual Observation of Zener Tunneling," Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

Placais, B.

B. Bourlon, D. C. Glattli, B. Placais, J. M. Berroir, C. Miko, L. Forro, and A. Bachtold, "Geometrical Dependence of High-Bias Current in Multiwalled Carbon Nanotubes," Phys. Rev. Lett. 92, 026804 (2004).
[CrossRef] [PubMed]

Reichel, J.

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch Oscillations of Atoms in an Optical Potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

Sagemerten, N.

Salerno, M.

V. V. Konotop, P. G. Kevrekidis, and M. Salerno, "Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice," Phys. Rev. A 72, 023611 (2005).

V. V. Konotop and M. Salerno, "Modulational instability in Bose-Einstein condensates in optical lattices," Phys. Rev. A 65, 021602 (2002).

G. A. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, "Wannier functions analysis of the nonlinear Schrdinger equation with a periodic potential," Phys. Rev. E 66, 046608 (2002).

Salomon, C.

E. Peik, M. B. Dahan, I. Bouchoule, Y. Castin, and C. Salomon, "Bloch oscillations of atoms, adiabatic rapid passage, and monokinetic atomic beams," Phys. Rev. A 55, 2989-3001 (1997).

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch Oscillations of Atoms in an Optical Potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

Sanchez-Palencia, L.

L. Sanchez-Palencia and L. Santos, "Bose-Einstein condensates in optical quasicrystal lattices," Phys. Rev. A 72, 053607 (2005).

Santos, L.

L. Sanchez-Palencia and L. Santos, "Bose-Einstein condensates in optical quasicrystal lattices," Phys. Rev. A 72, 053607 (2005).

L. Santos, M. A. Baranov, J. I. Cirac, H.-U. Everts, H. Fehrmann, and M. Lewenstein, "Atomic Quantum Gases in Kagome Lattices," Phys. Rev. Lett. 93, 030601 (2004).
[CrossRef] [PubMed]

Shchesnovich, V. S.

V. S. Shchesnovich and V. V. Konotop, "Nonlinear intraband tunneling of a Bose-Einstein condensate in a cubic three-dimensional lattice," Phys. Rev. A 77, 013614 (2008).

V. S. Shchesnovich and V. V. Konotop, "Nonlinear tunneling of Bose-Einstein condensates in an optical lattice: Signatures of quantum collapse and revival," Phys. Rev. A 75, 063628 (2007).

A. S. Desyatnikov, Yu. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, "Resonant Zener tunneling in two-dimensional periodic photonic lattices," Opt. Lett. 32, 325-327 (2007).
[CrossRef] [PubMed]

V. A. Brazhnyi, V. V. Konotop, V. Kuzmiak, and V. S. Shchesnovich, "Nonlinear tunneling in two-dimensional lattices," Phys. Rev. A 76, 023608 (2007).

V. S. Shchesnovich and S. Chavez-Cerda, "Bragg-resonance-induced Rabi oscillations in photonic lattices," Opt. Lett. 32, 1920-1922 (2007).
[CrossRef] [PubMed]

V. S. Shchesnovich, S. B. Cavalcanti, J. M. Hickmann, and Yu. S. Kivshar, "Zener tunneling in two-dimensional photonic lattices," Phys. Rev. E. 74, 056602 (2006).

V. S. Shchesnovich and S. B. Cavalcanti, "Finite-dimensional model for the condensate tunnelling in an accelerating optical lattice," J. Phys. B: At. Mol. Opt. Phys. 39, 1997-2011 (1997).
[CrossRef]

Shytov, A. V.

A. V. Shytov, "Landau-Zener transitions in a multilevel system: An exact result," Phys. Rev. A 70, 052708 (2004).

Sibille, A.

A. Sibille, J. F. Palmier, and F. Laruelle, "Zener Interminiband Resonant Breakdown in Superlattices," Phys. Rev. Lett. 80, 4506-4509 (1998).
[CrossRef]

Smerzi, A.

F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, "Josephson Junction Arrays with Bose-Einstein Condensates," Science 293, 843 (2001).
[CrossRef] [PubMed]

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, "Singular Optics," Progress in Optics 42, 219 (2001).
[CrossRef]

Streppel, U.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual Observation of Zener Tunneling," Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

Sukhorukov, A. A.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Terhalle, B.

Torner, L.

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical Vortices and Vortex Solitons," Progress in Optics 47, 291-391 (2005).
[CrossRef]

Tr¨ager, D.

Trombettoni, A.

F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, "Josephson Junction Arrays with Bose-Einstein Condensates," Science 293, 843 (2001).
[CrossRef] [PubMed]

Trompeter, H.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual Observation of Zener Tunneling," Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, "Singular Optics," Progress in Optics 42, 219 (2001).
[CrossRef]

Wannier, G. H.

G. H. Wannier, "Wave Functions and Effective Hamiltonian for Bloch Electrons in an Electric Field," Phys. Rev. 117, 432-439 (1960).
[CrossRef]

Witthaut, D.

D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, "Bloch oscillations in two-dimensional lattices," New J. Phys. 6, 41 (2004).
[CrossRef]

Wu, B.

B. Wu and Q. Niu, "Nonlinear Landau-Zener tunneling," Phys. Rev. A 61, 023402 (2000).

Zener, C.

C. Zener, "Non-Adiabatic Crossing of Energy Levels," Proc. R. Soc. London A 137, 696-702 (1932).

Zobay, O.

O. Zobay and B. M. Garraway, "Time-dependent tunneling of Bose-Einstein condensates," Phys. Rev. A 61, 033603 (2000).

J. Phys. A: Math. Gen. (2)

S. Brundobler and V. Elser, "S-matrix for generalized Landau-Zener problem," J. Phys. A: Math. Gen. 26, 1211- 1227 (1993).
[CrossRef]

F. Keck and H. J. Korsch, "Infinite-variable Bessel functions in two-dimensional Wannier-Stark systems," J. Phys. A: Math. Gen. 35, L105-L116 (2002).
[CrossRef]

J. Phys. B: At. Mol. Opt. Phys. (1)

V. S. Shchesnovich and S. B. Cavalcanti, "Finite-dimensional model for the condensate tunnelling in an accelerating optical lattice," J. Phys. B: At. Mol. Opt. Phys. 39, 1997-2011 (1997).
[CrossRef]

Laser Phys. (1)

M. Jona-Lasinio, O. Morsch, M. Cristiani, E. Arimondo, and C. Menotti, "Nonlinear Effects for BoseEinstein Condensates in Optical Lattices," Laser Phys. 15, 1180-1188 (2005).

New J. Phys. (1)

D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, "Bloch oscillations in two-dimensional lattices," New J. Phys. 6, 41 (2004).
[CrossRef]

Nuovo Cimento (1)

E. Majorana, "Orientated atoms in a variable magnetic field," Nuovo Cimento 9, 43-50 (1932).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. (2)

W. V. Houston, "Acceleration of Electrons in a Crystal Lattice," Phys. Rev. 57, 184-186 (1940).
[CrossRef]

G. H. Wannier, "Wave Functions and Effective Hamiltonian for Bloch Electrons in an Electric Field," Phys. Rev. 117, 432-439 (1960).
[CrossRef]

Phys. Rev. A (11)

E. Peik, M. B. Dahan, I. Bouchoule, Y. Castin, and C. Salomon, "Bloch oscillations of atoms, adiabatic rapid passage, and monokinetic atomic beams," Phys. Rev. A 55, 2989-3001 (1997).

B. Wu and Q. Niu, "Nonlinear Landau-Zener tunneling," Phys. Rev. A 61, 023402 (2000).

O. Zobay and B. M. Garraway, "Time-dependent tunneling of Bose-Einstein condensates," Phys. Rev. A 61, 033603 (2000).

L. Sanchez-Palencia and L. Santos, "Bose-Einstein condensates in optical quasicrystal lattices," Phys. Rev. A 72, 053607 (2005).

A. V. Shytov, "Landau-Zener transitions in a multilevel system: An exact result," Phys. Rev. A 70, 052708 (2004).

V. V. Konotop, P. G. Kevrekidis, and M. Salerno, "Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice," Phys. Rev. A 72, 023611 (2005).

V. A. Brazhnyi, V. V. Konotop, V. Kuzmiak, and V. S. Shchesnovich, "Nonlinear tunneling in two-dimensional lattices," Phys. Rev. A 76, 023608 (2007).

V. S. Shchesnovich and V. V. Konotop, "Nonlinear tunneling of Bose-Einstein condensates in an optical lattice: Signatures of quantum collapse and revival," Phys. Rev. A 75, 063628 (2007).

V. S. Shchesnovich and V. V. Konotop, "Nonlinear intraband tunneling of a Bose-Einstein condensate in a cubic three-dimensional lattice," Phys. Rev. A 77, 013614 (2008).

A. R. Kolovsky and H. J. Korsch, "Bloch oscillations of cold atoms in two-dimensional optical lattices," Phys. Rev. A 67, 063601 (2003).

V. V. Konotop and M. Salerno, "Modulational instability in Bose-Einstein condensates in optical lattices," Phys. Rev. A 65, 021602 (2002).

Phys. Rev. A. (1)

Yu. N. Demkov and V. N. Ostrovsky, "Multipath interference in a multistate Landau-Zener-type model," Phys. Rev. A. 61, 032705 (2000).
[CrossRef]

Phys. Rev. B (1)

G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo, "Theory of photon Bloch oscillations in photonic crystals," Phys. Rev. B 63, 035108 (2001).

Phys. Rev. E (1)

G. A. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, "Wannier functions analysis of the nonlinear Schrdinger equation with a periodic potential," Phys. Rev. E 66, 046608 (2002).

Phys. Rev. E. (1)

V. S. Shchesnovich, S. B. Cavalcanti, J. M. Hickmann, and Yu. S. Kivshar, "Zener tunneling in two-dimensional photonic lattices," Phys. Rev. E. 74, 056602 (2006).

Phys. Rev. Lett. (10)

L. Santos, M. A. Baranov, J. I. Cirac, H.-U. Everts, H. Fehrmann, and M. Lewenstein, "Atomic Quantum Gases in Kagome Lattices," Phys. Rev. Lett. 93, 030601 (2004).
[CrossRef] [PubMed]

A. A. Burkov and E. Demler, "Vortex-Peierls States in Optical Lattices," Phys. Rev. Lett. 96, 180406 (2006).
[CrossRef] [PubMed]

V. A. Brazhnyi, V. V. Konotop, and V. Kuzmiak, "Nature of the Intrinsic Relation between Bloch-Band Tunneling and Modulational Instability," Phys. Rev. Lett. 96, 150402 (2006).
[CrossRef] [PubMed]

V. Agarwal, J. A. del Rıo, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual Observation of Zener Tunneling," Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch Oscillations of Atoms in an Optical Potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

B. Bourlon, D. C. Glattli, B. Placais, J. M. Berroir, C. Miko, L. Forro, and A. Bachtold, "Geometrical Dependence of High-Bias Current in Multiwalled Carbon Nanotubes," Phys. Rev. Lett. 92, 026804 (2004).
[CrossRef] [PubMed]

A. Sibille, J. F. Palmier, and F. Laruelle, "Zener Interminiband Resonant Breakdown in Superlattices," Phys. Rev. Lett. 80, 4506-4509 (1998).
[CrossRef]

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J. H. M¨uller, E. Courtade, M. Anderlini, and E. Arimondo, "Asymmetric Landau-Zener Tunneling in a Periodic Potential," Phys. Rev. Lett. 91, 230406 (2003).
[CrossRef] [PubMed]

Phys. Z. Sowjetunion (1)

L. D. Landau, "On the theory of transfer of energy at collisions II," Phys. Z. Sowjetunion 2, 46 (1932).

Proc. R. Soc. London A (1)

C. Zener, "Non-Adiabatic Crossing of Energy Levels," Proc. R. Soc. London A 137, 696-702 (1932).

Progress in Optics (2)

M. S. Soskin and M. V. Vasnetsov, "Singular Optics," Progress in Optics 42, 219 (2001).
[CrossRef]

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical Vortices and Vortex Solitons," Progress in Optics 47, 291-391 (2005).
[CrossRef]

Rev. Mod. Phys. (1)

L. Esaki, "Long journey into tunneling," Rev. Mod. Phys. 46, 237-244 (1974).
[CrossRef]

Science (2)

B. P. Anderson and M. A. Kasevich, "Macroscopic Quantum Interference from Atomic Tunnel Arrays," Science 282, 1686 (1998).
[CrossRef] [PubMed]

F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, "Josephson Junction Arrays with Bose-Einstein Condensates," Science 293, 843 (2001).
[CrossRef] [PubMed]

Z. Phys. (1)

F. Bloch, "Uber die Quantenmechanik der Elektronen in Kristallgittern," Z. Phys. 52, 555-600 (1928).

Other (3)

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford: Clarendon Press, 2003).

N. W. Ashcroft and N. D. Mermin, Solid state physics (NewYork: Holt, Rinehart and Winston, 1976).

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (San Diego: Academic, 2003).

Supplementary Material (6)

» Media 1: AVI (2621 KB)     
» Media 2: AVI (841 KB)     
» Media 3: AVI (1716 KB)     
» Media 4: AVI (1722 KB)     
» Media 5: AVI (2124 KB)     
» Media 6: AVI (3015 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1.
Fig. 1.

(a) The first Brillouin zone of the hexagonal lattice with b=|b l |=2, where the vectors b l correspond to translations between the points M, M′ and M″. (b) The hexagonal cos-lattice Eq. (5) and (c) corresponding “triangular” sin-lattice. The vectors d 1 and d 2 give two fundamental periods of the lattice. (d) The Bloch band structure (the first 9 bands) of the hexagonal cos-lattice in (b) with I 0=0.1.

Fig. 2.
Fig. 2.

Rabi oscillations between two X-points, see also movie Rabi.avi. Intensity of the initially Gaussian beam after propagation of t=6 in real (a) and Fourier (b) spaces. The dynamics of the powers P 1,2 of two oscillating beams is shown in (c), see text for the details. [Media 1]

Fig. 3.
Fig. 3.

Quasi-one-dimensional tunnelling through the X-point with the tilt directed along b 1, see also movie 1D.avi. (a,b) Intensity in the real (top) and Fourier (bottom) spaces are shown for t=0 in (a) and t=10 in (b), corresponding dynamics of beam powers is shown in (d, solid lines). Solutions to the LZM system Eq. (13) are shown in (c) and (d, dashed lines). See text for the details and parameter values. [Media 2]

Fig. 4.
Fig. 4.

(a) The structure of the Bloch bands along the ΓM-direction in a neighborhood of the M-point. The distance between the quasi-degenerate Bloch bands (E 2-E 1 for -t≫1 and E 3-E 1 for t≫1) is equal to 3ε 1 I 0/4. (b–d) Symmetric tunnelling through the M-point [24] with the tilt directed along b 1+b 2, see text and the movie . [Media 3]

Fig. 5.
Fig. 5.

Tunnelling of the Rabi oscillations. Solutions to the LZM system in (a) and (d) are compared to numerical simulations in (b), (c), and (e); see also movie 3D3.avi. [Media 4]

Fig. 6.
Fig. 6.

(a) The six resonant Γ-points (indicated by Γ j with j =1, …, 6) forming an extended Brillouin zone (solid hexagon). The reciprocal lattice vectors relating the Γ1-point to the other resonant Γ-points are indicated. (b–d) solutions to the LZM systems: (b) corresponds to Eq. (32), (c) to system 1 of (35) and (d) to system 2 of (35). The dashed lines show the analytical results; c 1(0)=1 in all cases.

Fig. 7.
Fig. 7.

Symmetric tunnelling through the Γ-point with the tilt directed along b 1. As before and in Fig. 6 we use ε 1=3/2, ε 2=1/4, ε 3=1/2, I 0=0.1, and | α |=0.05. [Media 5]

Fig. 8.
Fig. 8.

Three-fold symmetric Zener tunnelling of a vortex beam, see the movie V-1.avi. (a) Intensity and (b) phase of the initial beam with the topological charge m=-1 in the vicinity of the M-point, see (c). Intensities in the Fourier domain (c,d) are identical for both, vortex, m=+1, and antivortex, m=-1, while they differ significantly in the real space, compare (e,g) and corresponding phases (f,h). [Media 6]

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

i t Ψ + 1 2 2 Ψ + ϰ Δ n Ψ = 0 ,
i t Ψ = 1 2 2 Ψ + [ V ( x ) + α x ] Ψ ,
Ψ ( x , 0 ) = D d k C ( k ) e i kx ,
V = I 0 { ε 1 Σ l = 1 3 cos ( b l x ) + ε 2 Σ l = 1 3 cos ( 2 b l x ) +
ε 3 [ cos ( [ b 1 + b 2 ] x ) + cos ( [ b 2 + b 3 ] x ) + cos ( [ b 3 b 1 ] x ) ] + } ,
V = 8 I 0 cos 2 ( b 1 x 2 ) cos 2 ( b 2 x 2 ) cos 2 ( b 3 x 2 )
V ( x ) = Q V ̂ Q e i Q x ,
i t C ( q Q ) = E 0 ( q Q ) C ( q Q ) + Q V ̂ Q Q C ( q Q ) ,
i t C ( δ q + Q 2 ) = E 0 ( δ q + Q 2 ) C ( δ q + Q 2 ) + V ̂ Q C ( δ q Q 2 ) ,
i t C ( δ q Q 2 ) = E 0 ( δ q Q 2 ) C ( δ q Q 2 ) + V ̂ Q * C ( δ q + Q 2 ) ,
Ψ ( x , t ) = D d k C ( k , t ) e i ( k α t ) x .
i t C ( q Q ) = E 0 ( q Q α t ) C ( q Q ) + Q V ̂ Q Q C ( q Q ) .
V j = ε j I 0 2 ( e i Q j x + e i Q j x ) , Ψ j = C 1 ( t ) e i q ( t ) x + C 2 ( t ) e i [ q ( t ) Q j ] x .
i c ˙ 1 = ( Q j α ) t 2 c 1 + ε j I 0 2 c 2 , i c ˙ 2 = ( Q j α ) t 2 c 2 + ε j I 0 2 c 1 .
P = exp { 2 π V ̂ Q 2 Q α } ,
V res = ε 1 I 0 2 [ e i ( q M q M ) x + e i ( q M q M ) x + e i ( q M q M ) x + c . c . ] ,
Ψ = C 1 ( t ) e i q ( t ) x + C 2 ( t ) e i [ q ( t ) + q M q M ] x + C 3 ( t ) e i [ q ( t ) + q M q M ] x ,
i c ˙ 1 = ( v 1 t ) c 1 + ε 1 I 0 2 ( c 2 + c 3 ) ,
i c ˙ 2 = ( v 2 t ) c 2 + ε 1 I 0 2 ( c 1 + c 3 ) ,
i c ˙ 3 = ( v 3 t ) c 3 + ε 1 I 0 2 ( c 1 + c 2 ) ,
E res ( M ) = b 2 6 + [ ε 1 I 0 2 ; ε 1 I 0 2 ; ε 1 I 0 ] ,
P m m = e x p { 2 π j m Δ m j 2 v m v j } ,
P 1 1 = e x p { π ε 1 2 I 0 2 2 [ 1 v 1 v 2 + 1 v 1 v 3 ] } ,
P 3 3 = e x p { π ε 1 2 I 0 2 2 [ 1 v 1 v 3 + 1 v 2 v 3 ] } .
i b ˙ 1 = ( v t ε 1 I 0 2 ) b 1 ,
i b ˙ 2 = ( v t + ε 1 I 0 2 ) b 2 + ε 1 I 0 2 b 3 ,
i b ˙ 3 = 2 v t b 3 + ε 1 I 0 2 b 2 .
Ψ = C 1 ( t ) e i q ( t ) x + l = 2 6 C l ( t ) e i [ q ( t ) + q Γ l q Γ 1 ] x ,
i C ˙ = ( Λ t + I 0 2 H ) C , H = ( A B B A ) ,
E res ( Γ ) = b 2 2 + I 0 2 [ Δ 1 ; Δ 1 ; Δ 2 ; Δ 2 ; Δ 3 ; Δ 4 ] ,
P 1 1 = P 4 4 = e x p { π I 0 2 2 [ ε 1 2 λ 1 λ 2 + ε 1 2 λ 1 λ 3 + ε 3 2 λ 1 λ 3 + ε 3 2 λ 1 λ 2 + ε 2 2 2 λ 1 ] } ,
P 2 2 = P 5 5 = e x p { π I 0 2 2 [ ε 1 2 λ 2 λ 1 + ε 1 2 λ 2 λ 3 + ε 3 2 λ 1 λ 2 + ε 3 2 λ 2 λ 3 + ε 2 2 2 λ 2 ] } .
i B 0 ˙ = ( Λ 0 t + I 0 H 0 ) B 0 , H 0 ( 0 ε 1 2 ε 3 2 ε 2 2 ε 1 2 ε 3 2 ε 1 + ε 2 2 ε 3 2 ε 3 2 ε 1 + ε 2 2 ε 3 2 ε 1 2 ε 2 2 ε 3 2 ε 1 2 0 )
i b 5 ˙ = ( λ 1 t 2 ε 3 I 0 2 ) b 5 + I 0 2 ( ε 1 ε 2 ) b 6 ,
i b 6 ˙ = ( λ 1 t 2 ε 3 I 0 2 ) b 6 + I 0 2 ( ε 1 ε 2 ) b 5 .
i B j ˙ = ( Λ j z + I 0 2 H j ) B j , j = 1 , 2 ,
P ( j ) = exp { π I 0 2 2 λ 1 [ ( ε 1 ± ε 3 ) 2 + 1 2 ( ε 2 ± ε 3 ) 2 ] } ,
Ψ n = [ ( x x 0 ) + i ( y y 0 ) ] n 2 π σ exp { i k 0 ( x x 0 ) ( x x 0 ) 2 2 σ 2 } .
F { Ψ n } ( k ) = [ i k λ ( x n + iy n ) ] n F { Ψ 0 } ( k ) ,
F { Ψ n } = 2 π σ ( i σ 2 ) n [ ( k k 0 ) + i ( λ λ 0 ) ] n exp { i x 0 ( k k 0 ) σ 2 2 ( k k 0 ) 2 } ,

Metrics