Abstract

We present a simple method for calculating the binary collision rate between atoms in near-resonant optical lattices. The method is based on Monte Carlo wave-function simulations, and the collision rate is obtained by monitoring the quantum flux beyond the average distance between the atoms. To illustrate the usefulness of the method, we calculate the collision rates for a wide range of occupation densities and various modulation depths of the lattice. The method presented here, combined with the semiclassical calculations accounting for intrawell collisions, can simplify the study of the effects of binary collisions on the dynamics of atomic clouds trapped in near-resonant optical lattices.

© 2003 Optical Society of America

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  1. J. Dalibard and C. Cohen-Tannoudji, “Laser cooling below the Doppler limit by polarization gradients: simple theoretical models,” J. Opt. Soc. Am. B 6, 2023–2045 (1989).
    [CrossRef]
  2. P. J. Ungar, D. S. Weiss, E. Riis, and S. Chu, “Optical molasses and multilevel atoms: theory,” J. Opt. Soc. Am. B 6, 2058–2071 (1989).
    [CrossRef]
  3. P. S. Jessen and I. H. Deutsch, “Optical lattices,” Adv. At., Mol., Opt. Phys. 37, 95–138 (1996).
    [CrossRef]
  4. D. R. Meacher, “Optical lattices–crystalline structures bound by light,” Contemp. Phys. 39, 329–350 (1998).
    [CrossRef]
  5. S. Rolston, “Optical lattices,” Phys. World 11, 27–32 (1998).
  6. L. Guidoni and P. Verkerk, “Optical lattices: cold atoms ordered by light,” J. Opt. B: Quantum Semiclassical Opt. 1, R23–R45 (1999).
    [CrossRef]
  7. H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, Berlin, 1999).
  8. J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys. 71, 1–85 (1999).
    [CrossRef]
  9. K.-A. Suominen, “Theories for cold atomic collisions in light fields,” J. Phys. B 29, 5981–6007 (1996).
    [CrossRef]
  10. J. Lawall, C. Orzel, and S. L. Rolston, “Suppression and enhancement of collisions in optical lattices,” Phys. Rev. Lett. 80, 480–483 (1998).
    [CrossRef]
  11. H. Kunugita, T. Ido, and F. Shimizu, “Ionizing collisional rate of metastable rare-gas atoms in an optical lattice,” Phys. Rev. Lett. 79, 621–624 (1997).
    [CrossRef]
  12. E. V. Goldstein, P. Pax, and P. Meystre, “Dipole–dipole interaction in three-dimensional optical lattices,” Phys. Rev. A 53, 2604–2615 (1996).
    [CrossRef] [PubMed]
  13. C. Boisseau and J. Vigué, “Laser-dressed molecular interactions at long range,” Opt. Commun. 127, 251–256 (1996).
    [CrossRef]
  14. A. M. Guzmán and P. Meystre, “Dynamical effects of the dipole–dipole interaction in three-dimensional optical lattices,” Phys. Rev. A 57, 1139–1148 (1998).
    [CrossRef]
  15. C. Menotti and H. Ritsch, “Mean-field approach to dipole–dipole interaction in an optical lattice,” Phys. Rev. A 60, R2653–R2656 (1999).
    [CrossRef]
  16. C. Menotti and H. Ritsch, “Laser cooling of atoms in optical lattices including quantum statistics and dipole–dipole interactions,” Appl. Phys. B 69, 311–321 (1999).
    [CrossRef]
  17. J. Piilo, K.-A. Suominen, and K. Berg-Sørensen, “Cold collisions between atoms in optical lattices,” J. Phys. B 34, L231–L237 (2001).
    [CrossRef]
  18. J. Piilo, K.-A. Suominen, and K. Berg-Sørensen, “Atomic collision dynamics in optical lattices,” Phys. Rev. A 65, 033411 (2002).
    [CrossRef]
  19. J. Piilo and K.-A. Suominen, “Optical shielding of cold collisions in blue-detuned near-resonant optical lattices,” Phys. Rev. A 66, 013401 (2002).
    [CrossRef]
  20. J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
    [CrossRef] [PubMed]
  21. K. Mølmer, Y. Castin, and J. Dalibard, “Monte Carlo wave-function method in quantum optics,” J. Opt. Soc. Am. B 10, 524–538 (1993).
    [CrossRef]
  22. K. Mølmer and Y. Castin, “Monte Carlo wavefunctions in quantum optics,” Quantum Semiclassic. Opt. 8, 49–72 (1996).
    [CrossRef]
  23. M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
    [CrossRef]
  24. M. J. Holland, K.-A. Suominen, and K. Burnett, “Quantal treatment of cold collisions in a laser field,” Phys. Rev. Lett. 72, 2367–2370 (1994).
    [CrossRef] [PubMed]
  25. M. J. Holland, K.-A. Suominen, and K. Burnett, “Cold collisions in a laser field: quantum Monte Carlo treatment of radiative heating,” Phys. Rev. A 50, 1513–1530 (1994).
    [CrossRef] [PubMed]
  26. This will depend on the particular form of the lattice (see also the discussion on occupation density in Ref. 18).
  27. K.-A. Suominen, Y. B. Band, I. Tuvi, K. Burnett, and P. S. Julienne, “Quantum and semiclasical calculations of cold atom collisions in light fields,” Phys. Rev. A 57, 3724–3738 (1998).
    [CrossRef]
  28. K. I. Petsas, G. Grynberg, and J.-Y. Courtois, “Semiclassical Monte Carlo Approaches for Realistic Atoms in Optical Lattices,” Eur. Phys. J. D 6, 29–47 (1999).
    [CrossRef]
  29. Y. Castin, J. Dalibard, and C. Cohen-Tannoudji, “The limits of Sisyphus cooling,” in Proceedings of Light-Induced Kinetic Effects on Atoms, Ions and Molecules, L. Moi, S. Gozzini, C. Gabbanini, E. Arimondo, and F. Strumia, eds. (ETS Editrice, Pisa, Italy, 1991), pp. 1–24.
  30. Naturally, the methods without the extra potential of Eq. (8) should be used when one wants to calculate the thermodynamical properties of an atomic cloud in the whole lattice. These methods are already well-known; the purpose here is to give a new method for calculation of the collision rate.
  31. See Ref. 18 for the details of the jump operators that are used in our implementation of the method.
  32. It takes a finite length of time before the atoms begin to accumulate in the accumulation region. This is due to the finite distance between the initial lattice site and the accumulation region; see Fig. 1. Thus there is a small time delay in the simulation before the accumulation curve in Fig. 2 begins to increase and achieves nonzero values. We emphasize that it is only the slope β of the accumulation curve that is relevant for the collision-rate calculation here.
  33. This is in agreement with the result given in Ref. 10 where the collisions are also a measure of atomic transport in a lattice.
  34. F. Bardou, J. P. Bouchaud, O. Emile, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling and Lévy flights,” Phys. Rev. Lett. 72, 203–206 (1994).
    [CrossRef] [PubMed]
  35. S. Marksteiner, K. Ellinger, and P. Zoller, “Anomalous diffusion and Lévy walks in optical lattices,” Phys. Rev. A 53, 3409–3430 (1996).
    [CrossRef] [PubMed]
  36. W. Greenwood, P. Pax, and P. Meystre, “Atomic transport on one-dimensional optical lattices,” Phys. Rev. A 56, 2109–2122 (1997).
    [CrossRef]
  37. P. M. Visser and G. Nienhuis, “Quantum transport of atoms in an optical lattice,” Phys. Rev. A 56, 3950–3960 (1997).
    [CrossRef]

2002 (2)

J. Piilo, K.-A. Suominen, and K. Berg-Sørensen, “Atomic collision dynamics in optical lattices,” Phys. Rev. A 65, 033411 (2002).
[CrossRef]

J. Piilo and K.-A. Suominen, “Optical shielding of cold collisions in blue-detuned near-resonant optical lattices,” Phys. Rev. A 66, 013401 (2002).
[CrossRef]

2001 (1)

J. Piilo, K.-A. Suominen, and K. Berg-Sørensen, “Cold collisions between atoms in optical lattices,” J. Phys. B 34, L231–L237 (2001).
[CrossRef]

1999 (5)

C. Menotti and H. Ritsch, “Mean-field approach to dipole–dipole interaction in an optical lattice,” Phys. Rev. A 60, R2653–R2656 (1999).
[CrossRef]

C. Menotti and H. Ritsch, “Laser cooling of atoms in optical lattices including quantum statistics and dipole–dipole interactions,” Appl. Phys. B 69, 311–321 (1999).
[CrossRef]

L. Guidoni and P. Verkerk, “Optical lattices: cold atoms ordered by light,” J. Opt. B: Quantum Semiclassical Opt. 1, R23–R45 (1999).
[CrossRef]

J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys. 71, 1–85 (1999).
[CrossRef]

K. I. Petsas, G. Grynberg, and J.-Y. Courtois, “Semiclassical Monte Carlo Approaches for Realistic Atoms in Optical Lattices,” Eur. Phys. J. D 6, 29–47 (1999).
[CrossRef]

1998 (6)

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
[CrossRef]

K.-A. Suominen, Y. B. Band, I. Tuvi, K. Burnett, and P. S. Julienne, “Quantum and semiclasical calculations of cold atom collisions in light fields,” Phys. Rev. A 57, 3724–3738 (1998).
[CrossRef]

D. R. Meacher, “Optical lattices–crystalline structures bound by light,” Contemp. Phys. 39, 329–350 (1998).
[CrossRef]

S. Rolston, “Optical lattices,” Phys. World 11, 27–32 (1998).

J. Lawall, C. Orzel, and S. L. Rolston, “Suppression and enhancement of collisions in optical lattices,” Phys. Rev. Lett. 80, 480–483 (1998).
[CrossRef]

A. M. Guzmán and P. Meystre, “Dynamical effects of the dipole–dipole interaction in three-dimensional optical lattices,” Phys. Rev. A 57, 1139–1148 (1998).
[CrossRef]

1997 (3)

H. Kunugita, T. Ido, and F. Shimizu, “Ionizing collisional rate of metastable rare-gas atoms in an optical lattice,” Phys. Rev. Lett. 79, 621–624 (1997).
[CrossRef]

W. Greenwood, P. Pax, and P. Meystre, “Atomic transport on one-dimensional optical lattices,” Phys. Rev. A 56, 2109–2122 (1997).
[CrossRef]

P. M. Visser and G. Nienhuis, “Quantum transport of atoms in an optical lattice,” Phys. Rev. A 56, 3950–3960 (1997).
[CrossRef]

1996 (6)

S. Marksteiner, K. Ellinger, and P. Zoller, “Anomalous diffusion and Lévy walks in optical lattices,” Phys. Rev. A 53, 3409–3430 (1996).
[CrossRef] [PubMed]

E. V. Goldstein, P. Pax, and P. Meystre, “Dipole–dipole interaction in three-dimensional optical lattices,” Phys. Rev. A 53, 2604–2615 (1996).
[CrossRef] [PubMed]

C. Boisseau and J. Vigué, “Laser-dressed molecular interactions at long range,” Opt. Commun. 127, 251–256 (1996).
[CrossRef]

K.-A. Suominen, “Theories for cold atomic collisions in light fields,” J. Phys. B 29, 5981–6007 (1996).
[CrossRef]

P. S. Jessen and I. H. Deutsch, “Optical lattices,” Adv. At., Mol., Opt. Phys. 37, 95–138 (1996).
[CrossRef]

K. Mølmer and Y. Castin, “Monte Carlo wavefunctions in quantum optics,” Quantum Semiclassic. Opt. 8, 49–72 (1996).
[CrossRef]

1994 (3)

M. J. Holland, K.-A. Suominen, and K. Burnett, “Quantal treatment of cold collisions in a laser field,” Phys. Rev. Lett. 72, 2367–2370 (1994).
[CrossRef] [PubMed]

M. J. Holland, K.-A. Suominen, and K. Burnett, “Cold collisions in a laser field: quantum Monte Carlo treatment of radiative heating,” Phys. Rev. A 50, 1513–1530 (1994).
[CrossRef] [PubMed]

F. Bardou, J. P. Bouchaud, O. Emile, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling and Lévy flights,” Phys. Rev. Lett. 72, 203–206 (1994).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[CrossRef] [PubMed]

1989 (2)

Aspect, A.

F. Bardou, J. P. Bouchaud, O. Emile, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling and Lévy flights,” Phys. Rev. Lett. 72, 203–206 (1994).
[CrossRef] [PubMed]

Bagnato, V. S.

J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys. 71, 1–85 (1999).
[CrossRef]

Band, Y. B.

K.-A. Suominen, Y. B. Band, I. Tuvi, K. Burnett, and P. S. Julienne, “Quantum and semiclasical calculations of cold atom collisions in light fields,” Phys. Rev. A 57, 3724–3738 (1998).
[CrossRef]

Bardou, F.

F. Bardou, J. P. Bouchaud, O. Emile, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling and Lévy flights,” Phys. Rev. Lett. 72, 203–206 (1994).
[CrossRef] [PubMed]

Berg-Sørensen, K.

J. Piilo, K.-A. Suominen, and K. Berg-Sørensen, “Atomic collision dynamics in optical lattices,” Phys. Rev. A 65, 033411 (2002).
[CrossRef]

J. Piilo, K.-A. Suominen, and K. Berg-Sørensen, “Cold collisions between atoms in optical lattices,” J. Phys. B 34, L231–L237 (2001).
[CrossRef]

Boisseau, C.

C. Boisseau and J. Vigué, “Laser-dressed molecular interactions at long range,” Opt. Commun. 127, 251–256 (1996).
[CrossRef]

Bouchaud, J. P.

F. Bardou, J. P. Bouchaud, O. Emile, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling and Lévy flights,” Phys. Rev. Lett. 72, 203–206 (1994).
[CrossRef] [PubMed]

Burnett, K.

K.-A. Suominen, Y. B. Band, I. Tuvi, K. Burnett, and P. S. Julienne, “Quantum and semiclasical calculations of cold atom collisions in light fields,” Phys. Rev. A 57, 3724–3738 (1998).
[CrossRef]

M. J. Holland, K.-A. Suominen, and K. Burnett, “Cold collisions in a laser field: quantum Monte Carlo treatment of radiative heating,” Phys. Rev. A 50, 1513–1530 (1994).
[CrossRef] [PubMed]

M. J. Holland, K.-A. Suominen, and K. Burnett, “Quantal treatment of cold collisions in a laser field,” Phys. Rev. Lett. 72, 2367–2370 (1994).
[CrossRef] [PubMed]

Castin, Y.

K. Mølmer and Y. Castin, “Monte Carlo wavefunctions in quantum optics,” Quantum Semiclassic. Opt. 8, 49–72 (1996).
[CrossRef]

K. Mølmer, Y. Castin, and J. Dalibard, “Monte Carlo wave-function method in quantum optics,” J. Opt. Soc. Am. B 10, 524–538 (1993).
[CrossRef]

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[CrossRef] [PubMed]

Chu, S.

Cohen-Tannoudji, C.

F. Bardou, J. P. Bouchaud, O. Emile, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling and Lévy flights,” Phys. Rev. Lett. 72, 203–206 (1994).
[CrossRef] [PubMed]

J. Dalibard and C. Cohen-Tannoudji, “Laser cooling below the Doppler limit by polarization gradients: simple theoretical models,” J. Opt. Soc. Am. B 6, 2023–2045 (1989).
[CrossRef]

Courtois, J.-Y.

K. I. Petsas, G. Grynberg, and J.-Y. Courtois, “Semiclassical Monte Carlo Approaches for Realistic Atoms in Optical Lattices,” Eur. Phys. J. D 6, 29–47 (1999).
[CrossRef]

Dalibard, J.

Deutsch, I. H.

P. S. Jessen and I. H. Deutsch, “Optical lattices,” Adv. At., Mol., Opt. Phys. 37, 95–138 (1996).
[CrossRef]

Ellinger, K.

S. Marksteiner, K. Ellinger, and P. Zoller, “Anomalous diffusion and Lévy walks in optical lattices,” Phys. Rev. A 53, 3409–3430 (1996).
[CrossRef] [PubMed]

Emile, O.

F. Bardou, J. P. Bouchaud, O. Emile, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling and Lévy flights,” Phys. Rev. Lett. 72, 203–206 (1994).
[CrossRef] [PubMed]

Goldstein, E. V.

E. V. Goldstein, P. Pax, and P. Meystre, “Dipole–dipole interaction in three-dimensional optical lattices,” Phys. Rev. A 53, 2604–2615 (1996).
[CrossRef] [PubMed]

Greenwood, W.

W. Greenwood, P. Pax, and P. Meystre, “Atomic transport on one-dimensional optical lattices,” Phys. Rev. A 56, 2109–2122 (1997).
[CrossRef]

Grynberg, G.

K. I. Petsas, G. Grynberg, and J.-Y. Courtois, “Semiclassical Monte Carlo Approaches for Realistic Atoms in Optical Lattices,” Eur. Phys. J. D 6, 29–47 (1999).
[CrossRef]

Guidoni, L.

L. Guidoni and P. Verkerk, “Optical lattices: cold atoms ordered by light,” J. Opt. B: Quantum Semiclassical Opt. 1, R23–R45 (1999).
[CrossRef]

Guzmán, A. M.

A. M. Guzmán and P. Meystre, “Dynamical effects of the dipole–dipole interaction in three-dimensional optical lattices,” Phys. Rev. A 57, 1139–1148 (1998).
[CrossRef]

Holland, M. J.

M. J. Holland, K.-A. Suominen, and K. Burnett, “Cold collisions in a laser field: quantum Monte Carlo treatment of radiative heating,” Phys. Rev. A 50, 1513–1530 (1994).
[CrossRef] [PubMed]

M. J. Holland, K.-A. Suominen, and K. Burnett, “Quantal treatment of cold collisions in a laser field,” Phys. Rev. Lett. 72, 2367–2370 (1994).
[CrossRef] [PubMed]

Ido, T.

H. Kunugita, T. Ido, and F. Shimizu, “Ionizing collisional rate of metastable rare-gas atoms in an optical lattice,” Phys. Rev. Lett. 79, 621–624 (1997).
[CrossRef]

Jessen, P. S.

P. S. Jessen and I. H. Deutsch, “Optical lattices,” Adv. At., Mol., Opt. Phys. 37, 95–138 (1996).
[CrossRef]

Julienne, P. S.

J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys. 71, 1–85 (1999).
[CrossRef]

K.-A. Suominen, Y. B. Band, I. Tuvi, K. Burnett, and P. S. Julienne, “Quantum and semiclasical calculations of cold atom collisions in light fields,” Phys. Rev. A 57, 3724–3738 (1998).
[CrossRef]

Knight, P. L.

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
[CrossRef]

Kunugita, H.

H. Kunugita, T. Ido, and F. Shimizu, “Ionizing collisional rate of metastable rare-gas atoms in an optical lattice,” Phys. Rev. Lett. 79, 621–624 (1997).
[CrossRef]

Lawall, J.

J. Lawall, C. Orzel, and S. L. Rolston, “Suppression and enhancement of collisions in optical lattices,” Phys. Rev. Lett. 80, 480–483 (1998).
[CrossRef]

Marksteiner, S.

S. Marksteiner, K. Ellinger, and P. Zoller, “Anomalous diffusion and Lévy walks in optical lattices,” Phys. Rev. A 53, 3409–3430 (1996).
[CrossRef] [PubMed]

Meacher, D. R.

D. R. Meacher, “Optical lattices–crystalline structures bound by light,” Contemp. Phys. 39, 329–350 (1998).
[CrossRef]

Menotti, C.

C. Menotti and H. Ritsch, “Mean-field approach to dipole–dipole interaction in an optical lattice,” Phys. Rev. A 60, R2653–R2656 (1999).
[CrossRef]

C. Menotti and H. Ritsch, “Laser cooling of atoms in optical lattices including quantum statistics and dipole–dipole interactions,” Appl. Phys. B 69, 311–321 (1999).
[CrossRef]

Meystre, P.

A. M. Guzmán and P. Meystre, “Dynamical effects of the dipole–dipole interaction in three-dimensional optical lattices,” Phys. Rev. A 57, 1139–1148 (1998).
[CrossRef]

W. Greenwood, P. Pax, and P. Meystre, “Atomic transport on one-dimensional optical lattices,” Phys. Rev. A 56, 2109–2122 (1997).
[CrossRef]

E. V. Goldstein, P. Pax, and P. Meystre, “Dipole–dipole interaction in three-dimensional optical lattices,” Phys. Rev. A 53, 2604–2615 (1996).
[CrossRef] [PubMed]

Mølmer, K.

K. Mølmer and Y. Castin, “Monte Carlo wavefunctions in quantum optics,” Quantum Semiclassic. Opt. 8, 49–72 (1996).
[CrossRef]

K. Mølmer, Y. Castin, and J. Dalibard, “Monte Carlo wave-function method in quantum optics,” J. Opt. Soc. Am. B 10, 524–538 (1993).
[CrossRef]

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[CrossRef] [PubMed]

Nienhuis, G.

P. M. Visser and G. Nienhuis, “Quantum transport of atoms in an optical lattice,” Phys. Rev. A 56, 3950–3960 (1997).
[CrossRef]

Orzel, C.

J. Lawall, C. Orzel, and S. L. Rolston, “Suppression and enhancement of collisions in optical lattices,” Phys. Rev. Lett. 80, 480–483 (1998).
[CrossRef]

Pax, P.

W. Greenwood, P. Pax, and P. Meystre, “Atomic transport on one-dimensional optical lattices,” Phys. Rev. A 56, 2109–2122 (1997).
[CrossRef]

E. V. Goldstein, P. Pax, and P. Meystre, “Dipole–dipole interaction in three-dimensional optical lattices,” Phys. Rev. A 53, 2604–2615 (1996).
[CrossRef] [PubMed]

Petsas, K. I.

K. I. Petsas, G. Grynberg, and J.-Y. Courtois, “Semiclassical Monte Carlo Approaches for Realistic Atoms in Optical Lattices,” Eur. Phys. J. D 6, 29–47 (1999).
[CrossRef]

Piilo, J.

J. Piilo and K.-A. Suominen, “Optical shielding of cold collisions in blue-detuned near-resonant optical lattices,” Phys. Rev. A 66, 013401 (2002).
[CrossRef]

J. Piilo, K.-A. Suominen, and K. Berg-Sørensen, “Atomic collision dynamics in optical lattices,” Phys. Rev. A 65, 033411 (2002).
[CrossRef]

J. Piilo, K.-A. Suominen, and K. Berg-Sørensen, “Cold collisions between atoms in optical lattices,” J. Phys. B 34, L231–L237 (2001).
[CrossRef]

Plenio, M. B.

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
[CrossRef]

Riis, E.

Ritsch, H.

C. Menotti and H. Ritsch, “Laser cooling of atoms in optical lattices including quantum statistics and dipole–dipole interactions,” Appl. Phys. B 69, 311–321 (1999).
[CrossRef]

C. Menotti and H. Ritsch, “Mean-field approach to dipole–dipole interaction in an optical lattice,” Phys. Rev. A 60, R2653–R2656 (1999).
[CrossRef]

Rolston, S.

S. Rolston, “Optical lattices,” Phys. World 11, 27–32 (1998).

Rolston, S. L.

J. Lawall, C. Orzel, and S. L. Rolston, “Suppression and enhancement of collisions in optical lattices,” Phys. Rev. Lett. 80, 480–483 (1998).
[CrossRef]

Shimizu, F.

H. Kunugita, T. Ido, and F. Shimizu, “Ionizing collisional rate of metastable rare-gas atoms in an optical lattice,” Phys. Rev. Lett. 79, 621–624 (1997).
[CrossRef]

Suominen, K.-A.

J. Piilo and K.-A. Suominen, “Optical shielding of cold collisions in blue-detuned near-resonant optical lattices,” Phys. Rev. A 66, 013401 (2002).
[CrossRef]

J. Piilo, K.-A. Suominen, and K. Berg-Sørensen, “Atomic collision dynamics in optical lattices,” Phys. Rev. A 65, 033411 (2002).
[CrossRef]

J. Piilo, K.-A. Suominen, and K. Berg-Sørensen, “Cold collisions between atoms in optical lattices,” J. Phys. B 34, L231–L237 (2001).
[CrossRef]

K.-A. Suominen, Y. B. Band, I. Tuvi, K. Burnett, and P. S. Julienne, “Quantum and semiclasical calculations of cold atom collisions in light fields,” Phys. Rev. A 57, 3724–3738 (1998).
[CrossRef]

K.-A. Suominen, “Theories for cold atomic collisions in light fields,” J. Phys. B 29, 5981–6007 (1996).
[CrossRef]

M. J. Holland, K.-A. Suominen, and K. Burnett, “Quantal treatment of cold collisions in a laser field,” Phys. Rev. Lett. 72, 2367–2370 (1994).
[CrossRef] [PubMed]

M. J. Holland, K.-A. Suominen, and K. Burnett, “Cold collisions in a laser field: quantum Monte Carlo treatment of radiative heating,” Phys. Rev. A 50, 1513–1530 (1994).
[CrossRef] [PubMed]

Tuvi, I.

K.-A. Suominen, Y. B. Band, I. Tuvi, K. Burnett, and P. S. Julienne, “Quantum and semiclasical calculations of cold atom collisions in light fields,” Phys. Rev. A 57, 3724–3738 (1998).
[CrossRef]

Ungar, P. J.

Verkerk, P.

L. Guidoni and P. Verkerk, “Optical lattices: cold atoms ordered by light,” J. Opt. B: Quantum Semiclassical Opt. 1, R23–R45 (1999).
[CrossRef]

Vigué, J.

C. Boisseau and J. Vigué, “Laser-dressed molecular interactions at long range,” Opt. Commun. 127, 251–256 (1996).
[CrossRef]

Visser, P. M.

P. M. Visser and G. Nienhuis, “Quantum transport of atoms in an optical lattice,” Phys. Rev. A 56, 3950–3960 (1997).
[CrossRef]

Weiner, J.

J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys. 71, 1–85 (1999).
[CrossRef]

Weiss, D. S.

Zilio, S.

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Other (7)

H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, Berlin, 1999).

This will depend on the particular form of the lattice (see also the discussion on occupation density in Ref. 18).

Y. Castin, J. Dalibard, and C. Cohen-Tannoudji, “The limits of Sisyphus cooling,” in Proceedings of Light-Induced Kinetic Effects on Atoms, Ions and Molecules, L. Moi, S. Gozzini, C. Gabbanini, E. Arimondo, and F. Strumia, eds. (ETS Editrice, Pisa, Italy, 1991), pp. 1–24.

Naturally, the methods without the extra potential of Eq. (8) should be used when one wants to calculate the thermodynamical properties of an atomic cloud in the whole lattice. These methods are already well-known; the purpose here is to give a new method for calculation of the collision rate.

See Ref. 18 for the details of the jump operators that are used in our implementation of the method.

It takes a finite length of time before the atoms begin to accumulate in the accumulation region. This is due to the finite distance between the initial lattice site and the accumulation region; see Fig. 1. Thus there is a small time delay in the simulation before the accumulation curve in Fig. 2 begins to increase and achieves nonzero values. We emphasize that it is only the slope β of the accumulation curve that is relevant for the collision-rate calculation here.

This is in agreement with the result given in Ref. 10 where the collisions are also a measure of atomic transport in a lattice.

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Figures (3)

Fig. 1
Fig. 1

Schematic view of the optical potentials for the two ground-state Zeeman sublevels. The initial lattice site of the wave packet is indicated by the arrow, and the modification of the potentials around the points ±za makes the quantum flux unidirectional into the accumulation region (shown as shaded area). The actual length of the accumulation region in the simulations is much longer than shown here.

Fig. 2
Fig. 2

Example (U0=936Er, ρ0=20%) of the accumulation region population |ψa|2 as a function of time. The binary collision rate R is obtained from the slope β of the accumulation curve and the average distance between the atoms za; see text.32

Fig. 3
Fig. 3

Binary collision rate R for three different lattice depths U0 as a function of occupation density ρ0 of the lattice. The points show the simulation results, and the solid curves are the quadratic collision rates averaged from the simulation results for the specific lattice.

Tables (1)

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Table 1 Parameters Used a

Equations (10)

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E(z, t)=E0[exexp(ikr z)-ieyexp(-ikr z)]exp(-iωLt)+c.c.,
U0=-23 δs0,
s0=Ω2/2δ2+Γ2/4.
Hs=p22M-δPe+V.
V=-i Ω2sin(kz)|e3/2g1/2|+13 |e1/2g-1/2|+Ω2cos(kz)|e-3/2g-1/2|+13|e-1/2g-1/2|+H.c.
|ψ(z, t)=j,mψj,m(z, t)|jm.
|ψa|2(t)=|z|>zaψ*(z, t)ψ(z, t)dz.
Hm=0 :|z|<zα-λ/8,-α :|z|>za+λ/8,-α cos22πz+zaλ+1/8 :-za-λ/8z -za+λ/8,-α sin22πz-zaλ+1/8 :za-λ/8z za+λ/8
H=Hs+Hm+Hd,
R=β/za.

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