Abstract

We demonstrate an experimental technique for high-resolution, high-speed spatial manipulation of atom clouds. By combining holographically engineered laser beams from a spatial light modulator with off-axis shear mode acousto-optic deflectors, we manipulate 1 × 3 arrays of cold atoms with individual site addressability. Additionally, we demonstrate smooth 2-dimensional motion of atomic ensembles, and the ability to guide multiple atomic ensembles independently.

© 2007 Optical Society of America

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  1. J. M. McGuirk, M. J. Snadden, and M. A. Kasevich, "Large area light-pulse Atom Interferometry," Phys. Rev. Lett. 85, 4498-4501 (2000).
    [CrossRef] [PubMed]
  2. N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, "Guiding neutral atoms on a chip," Phys. Rev. Lett. 84, 1124-1127 (2000).
    [CrossRef] [PubMed]
  3. Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, "Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate," Phys. Rev. Lett. 94, 090405 (2005).
    [CrossRef] [PubMed]
  4. Y. Shin, C. Sanner, G.-B. Jo, T. A. Pasquini, M. Saba, W. Ketterle, and D. E. Pritchard, "Interference of Bose-Einstein condensates split with an atom chip," Phys. Rev. A 72, 021604(R) (2005).
  5. R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer, "Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits," Phys. Rev. Lett. 89, 097903 (2002).
    [CrossRef] [PubMed]
  6. R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, "Interferometer-type structures for guided atoms," Phys. Rev. Lett. 89, 220402 (2002).
    [CrossRef] [PubMed]
  7. A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, I. Bloch, "Coherent collisional spin dynamics in optical lattices," Phys. Rev. Lett. 95, 190405 (2005).
    [CrossRef] [PubMed]
  8. T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, "Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps," Phys. Rev. A 61, 022304 (2000).
    [CrossRef]
  9. D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, "Applications of spatial light modulators in atom optics," Opt. Express 11, 158-166 (2003).
    [CrossRef]
  10. F. K. Fatemi and M. Bashkansky, "Cold atom guidance using a binary spatial light modulator," Opt. Express 14,1368-1375 (2006).
    [CrossRef] [PubMed]
  11. N. Chattrapiban, E. A. Rogers, I. V. Arakelyan, R. Roy, and W. T. Hill, "Laser beams with embedded vortices: tools for atom optics," J. Opt. Soc. Am. B 23, 94-103 (2006).
    [CrossRef]
  12. D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Optics 53, 547-556 (2006).
    [CrossRef]
  13. S. Bergamini, B. Darquie, M. Jones, L. Jacubowiez, A. Browaeys, and P. Grangier, "Holographic generation of microtrap arrays for single atoms by use of a programmable phase modulator," J. Opt. Soc. Am. B 21, 1889-1894 (2004).
    [CrossRef]
  14. F. K. Fatemi and M. Bashkansky, "Generation of hollow beams by using a binary spatial light modulator," Opt. Lett. 31, 864-866 (2006).
    [CrossRef] [PubMed]
  15. V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, "Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator," Phys. Rev. A 73, 031402(R) (2006).
    [CrossRef]
  16. W. J. Hossack, E. Theofanidou, J. Crain, K. Heggarty, and M. Birch, "High-speed holographic optical tweezers using a ferroelectric liquid crystal microdisplay," Opt. Express 11, 2053-2059 (2003).
    [CrossRef] [PubMed]
  17. V. Boyer, C. M. Chandrashekar, C. J. Foot, and Z. J. Laczik, "Dynamic optical trap generation using FLC SLMs for the manipulation of cold atoms," J. Mod. Opt. 51, 2235 (2004).
    [CrossRef]
  18. Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, "Atom Interferometry with Bose-Einstein condensates in a double-well potential," Phys. Rev. Lett. 92, 050405 (2002).
    [CrossRef]
  19. C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, "Direct observation of sub-Poissonian number statistics in a degenerate bose gas," Phys. Rev. Lett. 95, 260403 (2005).
    [CrossRef]
  20. T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, "Bose-Einstein condensate in a box," Phys. Rev. A 71, 041604R (2005).
    [CrossRef]
  21. D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, "Fast ground state manipulation of neutral atoms in microscopic optical traps," Phys. Rev. Lett. 96, 063001 (2006).
    [CrossRef] [PubMed]
  22. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).
  23. R. Ozeri, L. Khaykovich, and N. Davidson, "Long spin relaxation times in a single-beam blue-detuned optical trap," Phys. Rev. A 59, R1750-R1753 (1999).
    [CrossRef]
  24. T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).

2006 (6)

F. K. Fatemi and M. Bashkansky, "Cold atom guidance using a binary spatial light modulator," Opt. Express 14,1368-1375 (2006).
[CrossRef] [PubMed]

N. Chattrapiban, E. A. Rogers, I. V. Arakelyan, R. Roy, and W. T. Hill, "Laser beams with embedded vortices: tools for atom optics," J. Opt. Soc. Am. B 23, 94-103 (2006).
[CrossRef]

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Optics 53, 547-556 (2006).
[CrossRef]

F. K. Fatemi and M. Bashkansky, "Generation of hollow beams by using a binary spatial light modulator," Opt. Lett. 31, 864-866 (2006).
[CrossRef] [PubMed]

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, "Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator," Phys. Rev. A 73, 031402(R) (2006).
[CrossRef]

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, "Fast ground state manipulation of neutral atoms in microscopic optical traps," Phys. Rev. Lett. 96, 063001 (2006).
[CrossRef] [PubMed]

2005 (4)

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, I. Bloch, "Coherent collisional spin dynamics in optical lattices," Phys. Rev. Lett. 95, 190405 (2005).
[CrossRef] [PubMed]

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, "Direct observation of sub-Poissonian number statistics in a degenerate bose gas," Phys. Rev. Lett. 95, 260403 (2005).
[CrossRef]

T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, "Bose-Einstein condensate in a box," Phys. Rev. A 71, 041604R (2005).
[CrossRef]

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, "Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate," Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef] [PubMed]

2004 (2)

S. Bergamini, B. Darquie, M. Jones, L. Jacubowiez, A. Browaeys, and P. Grangier, "Holographic generation of microtrap arrays for single atoms by use of a programmable phase modulator," J. Opt. Soc. Am. B 21, 1889-1894 (2004).
[CrossRef]

V. Boyer, C. M. Chandrashekar, C. J. Foot, and Z. J. Laczik, "Dynamic optical trap generation using FLC SLMs for the manipulation of cold atoms," J. Mod. Opt. 51, 2235 (2004).
[CrossRef]

2003 (2)

2002 (3)

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, "Atom Interferometry with Bose-Einstein condensates in a double-well potential," Phys. Rev. Lett. 92, 050405 (2002).
[CrossRef]

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer, "Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits," Phys. Rev. Lett. 89, 097903 (2002).
[CrossRef] [PubMed]

R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, "Interferometer-type structures for guided atoms," Phys. Rev. Lett. 89, 220402 (2002).
[CrossRef] [PubMed]

2000 (3)

J. M. McGuirk, M. J. Snadden, and M. A. Kasevich, "Large area light-pulse Atom Interferometry," Phys. Rev. Lett. 85, 4498-4501 (2000).
[CrossRef] [PubMed]

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, "Guiding neutral atoms on a chip," Phys. Rev. Lett. 84, 1124-1127 (2000).
[CrossRef] [PubMed]

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, "Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps," Phys. Rev. A 61, 022304 (2000).
[CrossRef]

1999 (1)

R. Ozeri, L. Khaykovich, and N. Davidson, "Long spin relaxation times in a single-beam blue-detuned optical trap," Phys. Rev. A 59, R1750-R1753 (1999).
[CrossRef]

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).

1972 (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Anderson, D. Z.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, "Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate," Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef] [PubMed]

Arakelyan, I. V.

Bashkansky, M.

Bergamini, S.

Birch, M.

Birkl, G.

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer, "Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits," Phys. Rev. Lett. 89, 097903 (2002).
[CrossRef] [PubMed]

R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, "Interferometer-type structures for guided atoms," Phys. Rev. Lett. 89, 220402 (2002).
[CrossRef] [PubMed]

Bloch, I.

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, I. Bloch, "Coherent collisional spin dynamics in optical lattices," Phys. Rev. Lett. 95, 190405 (2005).
[CrossRef] [PubMed]

Boyer, V.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, "Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator," Phys. Rev. A 73, 031402(R) (2006).
[CrossRef]

V. Boyer, C. M. Chandrashekar, C. J. Foot, and Z. J. Laczik, "Dynamic optical trap generation using FLC SLMs for the manipulation of cold atoms," J. Mod. Opt. 51, 2235 (2004).
[CrossRef]

Bright, V. M.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, "Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate," Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef] [PubMed]

Browaeys, A.

Buchkremer, F. B. J.

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer, "Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits," Phys. Rev. Lett. 89, 097903 (2002).
[CrossRef] [PubMed]

Calarco, T.

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, "Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps," Phys. Rev. A 61, 022304 (2000).
[CrossRef]

Cassettari, D.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, "Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator," Phys. Rev. A 73, 031402(R) (2006).
[CrossRef]

Chandrashekar, C. M.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, "Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator," Phys. Rev. A 73, 031402(R) (2006).
[CrossRef]

V. Boyer, C. M. Chandrashekar, C. J. Foot, and Z. J. Laczik, "Dynamic optical trap generation using FLC SLMs for the manipulation of cold atoms," J. Mod. Opt. 51, 2235 (2004).
[CrossRef]

Chattrapiban, N.

Chuu, C.-S.

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, "Direct observation of sub-Poissonian number statistics in a degenerate bose gas," Phys. Rev. Lett. 95, 260403 (2005).
[CrossRef]

T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, "Bose-Einstein condensate in a box," Phys. Rev. A 71, 041604R (2005).
[CrossRef]

Cirac, J. I.

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, "Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps," Phys. Rev. A 61, 022304 (2000).
[CrossRef]

Cornell, E. A.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, "Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate," Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef] [PubMed]

Crain, J.

Darquie, B.

Davidson, N.

R. Ozeri, L. Khaykovich, and N. Davidson, "Long spin relaxation times in a single-beam blue-detuned optical trap," Phys. Rev. A 59, R1750-R1753 (1999).
[CrossRef]

Deb, A. B.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, "Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator," Phys. Rev. A 73, 031402(R) (2006).
[CrossRef]

Dekker, N. H.

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, "Guiding neutral atoms on a chip," Phys. Rev. Lett. 84, 1124-1127 (2000).
[CrossRef] [PubMed]

Dholakia, K.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Optics 53, 547-556 (2006).
[CrossRef]

D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, "Applications of spatial light modulators in atom optics," Opt. Express 11, 158-166 (2003).
[CrossRef]

Diot, Q.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, "Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate," Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef] [PubMed]

Drndic, M.

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, "Guiding neutral atoms on a chip," Phys. Rev. Lett. 84, 1124-1127 (2000).
[CrossRef] [PubMed]

Dumke, R.

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer, "Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits," Phys. Rev. Lett. 89, 097903 (2002).
[CrossRef] [PubMed]

R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, "Interferometer-type structures for guided atoms," Phys. Rev. Lett. 89, 220402 (2002).
[CrossRef] [PubMed]

Ertmer, W.

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer, "Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits," Phys. Rev. Lett. 89, 097903 (2002).
[CrossRef] [PubMed]

R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, "Interferometer-type structures for guided atoms," Phys. Rev. Lett. 89, 220402 (2002).
[CrossRef] [PubMed]

Fatemi, F. K.

Folling, S.

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, I. Bloch, "Coherent collisional spin dynamics in optical lattices," Phys. Rev. Lett. 95, 190405 (2005).
[CrossRef] [PubMed]

Foot, C. J.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, "Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator," Phys. Rev. A 73, 031402(R) (2006).
[CrossRef]

V. Boyer, C. M. Chandrashekar, C. J. Foot, and Z. J. Laczik, "Dynamic optical trap generation using FLC SLMs for the manipulation of cold atoms," J. Mod. Opt. 51, 2235 (2004).
[CrossRef]

Freegarde, T.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Optics 53, 547-556 (2006).
[CrossRef]

Gerbier, F.

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, I. Bloch, "Coherent collisional spin dynamics in optical lattices," Phys. Rev. Lett. 95, 190405 (2005).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Gericke, T.

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, I. Bloch, "Coherent collisional spin dynamics in optical lattices," Phys. Rev. Lett. 95, 190405 (2005).
[CrossRef] [PubMed]

Gherardi, D. M.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Optics 53, 547-556 (2006).
[CrossRef]

Godun, R. M.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, "Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator," Phys. Rev. A 73, 031402(R) (2006).
[CrossRef]

Grangier, P.

Hanssen, J. L.

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, "Direct observation of sub-Poissonian number statistics in a degenerate bose gas," Phys. Rev. Lett. 95, 260403 (2005).
[CrossRef]

T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, "Bose-Einstein condensate in a box," Phys. Rev. A 71, 041604R (2005).
[CrossRef]

Heggarty, K.

Henage, T.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, "Fast ground state manipulation of neutral atoms in microscopic optical traps," Phys. Rev. Lett. 96, 063001 (2006).
[CrossRef] [PubMed]

Hill, W. T.

Hinds, E. A.

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, "Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps," Phys. Rev. A 61, 022304 (2000).
[CrossRef]

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).

Hossack, W. J.

Jacubowiez, L.

Jaksch, D.

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, "Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps," Phys. Rev. A 61, 022304 (2000).
[CrossRef]

Johnson, T. A.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, "Fast ground state manipulation of neutral atoms in microscopic optical traps," Phys. Rev. Lett. 96, 063001 (2006).
[CrossRef] [PubMed]

Jones, M.

Kasevich, M. A.

J. M. McGuirk, M. J. Snadden, and M. A. Kasevich, "Large area light-pulse Atom Interferometry," Phys. Rev. Lett. 85, 4498-4501 (2000).
[CrossRef] [PubMed]

Ketterle, W.

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, "Atom Interferometry with Bose-Einstein condensates in a double-well potential," Phys. Rev. Lett. 92, 050405 (2002).
[CrossRef]

Khaykovich, L.

R. Ozeri, L. Khaykovich, and N. Davidson, "Long spin relaxation times in a single-beam blue-detuned optical trap," Phys. Rev. A 59, R1750-R1753 (1999).
[CrossRef]

Kishimoto, T.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, "Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate," Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef] [PubMed]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).

Kulatunga, P. B.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, "Fast ground state manipulation of neutral atoms in microscopic optical traps," Phys. Rev. Lett. 96, 063001 (2006).
[CrossRef] [PubMed]

Laczik, Z. J.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, "Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator," Phys. Rev. A 73, 031402(R) (2006).
[CrossRef]

V. Boyer, C. M. Chandrashekar, C. J. Foot, and Z. J. Laczik, "Dynamic optical trap generation using FLC SLMs for the manipulation of cold atoms," J. Mod. Opt. 51, 2235 (2004).
[CrossRef]

Leanhardt, A. E.

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, "Atom Interferometry with Bose-Einstein condensates in a double-well potential," Phys. Rev. Lett. 92, 050405 (2002).
[CrossRef]

Lee, C. S.

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, "Guiding neutral atoms on a chip," Phys. Rev. Lett. 84, 1124-1127 (2000).
[CrossRef] [PubMed]

Livesey, J.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Optics 53, 547-556 (2006).
[CrossRef]

Lorent, V.

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, "Guiding neutral atoms on a chip," Phys. Rev. Lett. 84, 1124-1127 (2000).
[CrossRef] [PubMed]

Mandel, O.

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, I. Bloch, "Coherent collisional spin dynamics in optical lattices," Phys. Rev. Lett. 95, 190405 (2005).
[CrossRef] [PubMed]

McGloin, D.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Optics 53, 547-556 (2006).
[CrossRef]

D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, "Applications of spatial light modulators in atom optics," Opt. Express 11, 158-166 (2003).
[CrossRef]

McGuirk, J. M.

J. M. McGuirk, M. J. Snadden, and M. A. Kasevich, "Large area light-pulse Atom Interferometry," Phys. Rev. Lett. 85, 4498-4501 (2000).
[CrossRef] [PubMed]

Melville, H.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Optics 53, 547-556 (2006).
[CrossRef]

D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, "Applications of spatial light modulators in atom optics," Opt. Express 11, 158-166 (2003).
[CrossRef]

Meyrath, T. P.

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, "Direct observation of sub-Poissonian number statistics in a degenerate bose gas," Phys. Rev. Lett. 95, 260403 (2005).
[CrossRef]

T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, "Bose-Einstein condensate in a box," Phys. Rev. A 71, 041604R (2005).
[CrossRef]

Muther, T.

R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, "Interferometer-type structures for guided atoms," Phys. Rev. Lett. 89, 220402 (2002).
[CrossRef] [PubMed]

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer, "Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits," Phys. Rev. Lett. 89, 097903 (2002).
[CrossRef] [PubMed]

Ozeri, R.

R. Ozeri, L. Khaykovich, and N. Davidson, "Long spin relaxation times in a single-beam blue-detuned optical trap," Phys. Rev. A 59, R1750-R1753 (1999).
[CrossRef]

Pasquini, T. A.

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, "Atom Interferometry with Bose-Einstein condensates in a double-well potential," Phys. Rev. Lett. 92, 050405 (2002).
[CrossRef]

Prentiss, M.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, "Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate," Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef] [PubMed]

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, "Guiding neutral atoms on a chip," Phys. Rev. Lett. 84, 1124-1127 (2000).
[CrossRef] [PubMed]

Price, G. N.

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, "Direct observation of sub-Poissonian number statistics in a degenerate bose gas," Phys. Rev. Lett. 95, 260403 (2005).
[CrossRef]

Pritchard, D. E.

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, "Atom Interferometry with Bose-Einstein condensates in a double-well potential," Phys. Rev. Lett. 92, 050405 (2002).
[CrossRef]

Proite, N.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, "Fast ground state manipulation of neutral atoms in microscopic optical traps," Phys. Rev. Lett. 96, 063001 (2006).
[CrossRef] [PubMed]

Raizen, M. G.

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, "Direct observation of sub-Poissonian number statistics in a degenerate bose gas," Phys. Rev. Lett. 95, 260403 (2005).
[CrossRef]

T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, "Bose-Einstein condensate in a box," Phys. Rev. A 71, 041604R (2005).
[CrossRef]

Rhodes, D. P.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Optics 53, 547-556 (2006).
[CrossRef]

Rogers, E. A.

Roy, R.

Saba, M.

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, "Atom Interferometry with Bose-Einstein condensates in a double-well potential," Phys. Rev. Lett. 92, 050405 (2002).
[CrossRef]

Saffman, M.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, "Fast ground state manipulation of neutral atoms in microscopic optical traps," Phys. Rev. Lett. 96, 063001 (2006).
[CrossRef] [PubMed]

Saravanan, R. A.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, "Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate," Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef] [PubMed]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Schmiedmayer, J.

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, "Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps," Phys. Rev. A 61, 022304 (2000).
[CrossRef]

Schreck, F.

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, "Direct observation of sub-Poissonian number statistics in a degenerate bose gas," Phys. Rev. Lett. 95, 260403 (2005).
[CrossRef]

T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, "Bose-Einstein condensate in a box," Phys. Rev. A 71, 041604R (2005).
[CrossRef]

Segal, S. R.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, "Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate," Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef] [PubMed]

Shin, Y.

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, "Atom Interferometry with Bose-Einstein condensates in a double-well potential," Phys. Rev. Lett. 92, 050405 (2002).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).

Sibbett, W.

Smirne, G.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, "Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator," Phys. Rev. A 73, 031402(R) (2006).
[CrossRef]

Smith, S. P.

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, "Guiding neutral atoms on a chip," Phys. Rev. Lett. 84, 1124-1127 (2000).
[CrossRef] [PubMed]

Snadden, M. J.

J. M. McGuirk, M. J. Snadden, and M. A. Kasevich, "Large area light-pulse Atom Interferometry," Phys. Rev. Lett. 85, 4498-4501 (2000).
[CrossRef] [PubMed]

Spalding, G.

Theofanidou, E.

Thywissen, J. H.

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, "Guiding neutral atoms on a chip," Phys. Rev. Lett. 84, 1124-1127 (2000).
[CrossRef] [PubMed]

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).

Urban, E.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, "Fast ground state manipulation of neutral atoms in microscopic optical traps," Phys. Rev. Lett. 96, 063001 (2006).
[CrossRef] [PubMed]

Volk, M.

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer, "Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits," Phys. Rev. Lett. 89, 097903 (2002).
[CrossRef] [PubMed]

R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, "Interferometer-type structures for guided atoms," Phys. Rev. Lett. 89, 220402 (2002).
[CrossRef] [PubMed]

Walker, T. G.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, "Fast ground state manipulation of neutral atoms in microscopic optical traps," Phys. Rev. Lett. 96, 063001 (2006).
[CrossRef] [PubMed]

Wang, Y.-J.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, "Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate," Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef] [PubMed]

Westervelt, R. M.

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, "Guiding neutral atoms on a chip," Phys. Rev. Lett. 84, 1124-1127 (2000).
[CrossRef] [PubMed]

Widera, A.

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, I. Bloch, "Coherent collisional spin dynamics in optical lattices," Phys. Rev. Lett. 95, 190405 (2005).
[CrossRef] [PubMed]

Wu, S.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, "Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate," Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef] [PubMed]

Yavuz, D. D.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, "Fast ground state manipulation of neutral atoms in microscopic optical traps," Phys. Rev. Lett. 96, 063001 (2006).
[CrossRef] [PubMed]

Zoller, P.

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, "Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps," Phys. Rev. A 61, 022304 (2000).
[CrossRef]

J. Mod. Opt. (1)

V. Boyer, C. M. Chandrashekar, C. J. Foot, and Z. J. Laczik, "Dynamic optical trap generation using FLC SLMs for the manipulation of cold atoms," J. Mod. Opt. 51, 2235 (2004).
[CrossRef]

J. Mod. Optics (1)

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, "Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation," J. Mod. Optics 53, 547-556 (2006).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Express (3)

Opt. Lett. (1)

Optik (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Phys. Rev. A (4)

R. Ozeri, L. Khaykovich, and N. Davidson, "Long spin relaxation times in a single-beam blue-detuned optical trap," Phys. Rev. A 59, R1750-R1753 (1999).
[CrossRef]

T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, "Bose-Einstein condensate in a box," Phys. Rev. A 71, 041604R (2005).
[CrossRef]

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, "Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator," Phys. Rev. A 73, 031402(R) (2006).
[CrossRef]

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, "Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps," Phys. Rev. A 61, 022304 (2000).
[CrossRef]

Phys. Rev. Lett. (10)

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, "Atom Interferometry with Bose-Einstein condensates in a double-well potential," Phys. Rev. Lett. 92, 050405 (2002).
[CrossRef]

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, "Direct observation of sub-Poissonian number statistics in a degenerate bose gas," Phys. Rev. Lett. 95, 260403 (2005).
[CrossRef]

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer, "Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits," Phys. Rev. Lett. 89, 097903 (2002).
[CrossRef] [PubMed]

R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, "Interferometer-type structures for guided atoms," Phys. Rev. Lett. 89, 220402 (2002).
[CrossRef] [PubMed]

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, I. Bloch, "Coherent collisional spin dynamics in optical lattices," Phys. Rev. Lett. 95, 190405 (2005).
[CrossRef] [PubMed]

J. M. McGuirk, M. J. Snadden, and M. A. Kasevich, "Large area light-pulse Atom Interferometry," Phys. Rev. Lett. 85, 4498-4501 (2000).
[CrossRef] [PubMed]

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, "Guiding neutral atoms on a chip," Phys. Rev. Lett. 84, 1124-1127 (2000).
[CrossRef] [PubMed]

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, "Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate," Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef] [PubMed]

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, "Fast ground state manipulation of neutral atoms in microscopic optical traps," Phys. Rev. Lett. 96, 063001 (2006).
[CrossRef] [PubMed]

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).

Other (1)

Y. Shin, C. Sanner, G.-B. Jo, T. A. Pasquini, M. Saba, W. Ketterle, and D. E. Pritchard, "Interference of Bose-Einstein condensates split with an atom chip," Phys. Rev. A 72, 021604(R) (2005).

Supplementary Material (5)

» Media 1: AVI (48 KB)     
» Media 2: AVI (297 KB)     
» Media 3: AVI (335 KB)     
» Media 4: AVI (109 KB)     
» Media 5: AVI (85 KB)     

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Figures (7)

Fig. 1.
Fig. 1.

(a). Hollow beams formed by the SLM. (b). 1 × 4 arrays formed by driving the AOD with four frequencies simultaneously. c) 1 × 4 array created solely by the SLM.

Fig. 2.
Fig. 2.

1 × 4 array of n=1 hollow laser beams. The two central beams are modulated by a triangle wave with periods of 20 and 40 ms as the 1 × 4 array is scanned from left to right across the frame. Time between frames is ∼3 msec. (Movie: 0.28 Mb) [Media 1]

Fig. 3.
Fig. 3.

Optical layout for the crossed hollow beams. (a). Setup for single beam deflection (Movie: 0.24 Mb) [Media 2] (b). Multiple drive frequencies (Movie: 0.38 Mb) [Media 3]

Fig. 4.
Fig. 4.

(a). Image of n=4 hollow beam at the MOT location and cross sectional profile. (b). Fluorescence images of cold 85Rb atoms confined in an n=4 crossed hollow beam that is moving sinusoidally with period 50 ms. Time between frames is 2.5 ms, and the deflection amplitude is 0.9 mm.

Fig. 5.
Fig. 5.

Atoms remaining in the crossed blue detuned trap as a function of time. Fits use data after 10 msec to eliminate atoms starting outside the trap. For clarity, error bars are not displayed, but are approximately 10% of the measured values, arising from variations in initial MOT number.

Fig. 6.
Fig. 6.

2D motion of atoms. Circle diameter is ∼500 microns. Time between frames is 10 msec. (Movie: 0.6 MB) [Media 4]

Fig. 7.
Fig. 7.

Control of multiple atoms clouds with the AOD. Maximum distance between top and bottom cloud is 2 mm. Time between frames is ~3 msec. (Movie 0.3 MB) [Media 5]

Equations (3)

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

U ( r ) = ħ Δ 2 ln [ 1 + I ( r ) / I o 1 + 4 ( Δ / Γ ) 2 ]
U ( r ) = ħ Γ 8 [ ( Γ Δ ) I ( r ) I o ]
Γ sp ( r ) = Γ 8 [ ( Γ Δ ) 2 I ( r ) I o ]

Metrics