Abstract

We demonstrated trapping single neutral Rb atoms in micro traps of an optical ring lattice formed by superposing the ±l components of the Laguerre-Gaussian mode, and generated by reflecting a single laser beam from a computer controlled spatial light modulator. A single atom in one trap or two atoms with one each in two traps were identified by observing the fluorescence. The trap array loaded with single atoms was rotated by dynamically displaying the hologram animation movie on the modulator. The modulation period in the fluorescence indicates the rotation of one or two single atoms in the lattice.

© 2009 Optical Society of America

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  1. R. Grimm, M. Weidemuller, Y. B. Ovchinnikov, "Optical dipole traps for neutral atoms," Adv. At., Mol., Opt. Phys. 42, 95-170 (2000).
    [CrossRef]
  2. D. Frese, B. Uberholz, S. Kuhr, W. Alt, D. Schrader, V. Gomer, and D. Meschede, "Single atoms in an optical dipole trap: Towards a deterministic source of cold atoms," Phys. Rev. Lett. 85, 3777-3740 (2000).
    [CrossRef] [PubMed]
  3. N. Schlosser, G. Reymond, I. Protsenko, P. Grangier, "Sub-poissonian loading of single atoms in a microscopic dipole trap," Nature (London) 411, 1024-1027 (2001).
    [CrossRef]
  4. S. Bergamini, B. Darquie, M. Jones, L. Jacubowiez, A. Browaeys, 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]
  5. R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, 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. D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, "Neutral Atom Quantum Register," Phys. Rev. Lett. 93, 150501 (2004).
    [CrossRef] [PubMed]
  7. Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, A. Rauschenbeutel, "Quantum engineering: An atom-sorting machine," Nature (London) 442, 151 (2006).
    [CrossRef]
  8. S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, D. Meschede, "Coherence properties and quantum state transportation in an optical conveyor belt," Phys. Rev. Lett. 91, 213002 (2003).
    [CrossRef] [PubMed]
  9. O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hansch, I. Bloch, "Coherent transport of neutral atoms in spin-dependent optical lattice potentials," Phys. Rev. Lett. 91, 010407 (2003).
    [CrossRef] [PubMed]
  10. J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P.A. Jones, G. Messin, A. Browaeys, P. Grangier, "Two-dimensional transport and transfer of a single atomic qubit in optical tweezers," Nat. Phys. 3, 696-699 (2007).
    [CrossRef]
  11. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, A. S. Arnold, "Optical ferris wheel for ultracold atoms," Opt. Express 15, 8619-8625 (2007).
    [CrossRef] [PubMed]
  12. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 - 8189 (1992).
    [CrossRef] [PubMed]
  13. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, "Novel Optical Trap of Atoms with a Doughnut Beam," Phys. Rev. Lett. 78, 4713 - 4716 (1997).
    [CrossRef]
  14. M. Stutz, S. Groblacher, T. Jennewein, A. Zeilinger, "How to create and detect N-dimensional entangled photons with an active phase hologram," Appl. Phys. Lett. 90, 261114 (2007).
    [CrossRef]
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    [CrossRef]
  16. K. Li, L. Deng, E. W. Hagley, M. G. Payne, and M. S. Zhan, "Matter-wave self-imaging by atomic center-of-mass motion induced interference," Phys. Rev. Lett. 101, 250401 (2008).
    [CrossRef] [PubMed]
  17. N. Schlosser, G. Reymond, and P. Grangier, "Collisional Blockade in Microscopic Optical Dipole Traps," Phys. Rev. Lett. 89, 023005 (2002).
    [CrossRef] [PubMed]
  18. H. Hanbury-Brown and R. Q. Twiss, "Correlation between photons in two coherent beams of light," Nature (London) 177, 27-29 (1956).
  19. M. Weber, J. Volz, K. Saucke, C. Kurtsiefer, H. Weinfurter,"Analysis of a single-atom dipole trap," Phys. Rev. A 73, 043406 (2006).
    [CrossRef]
  20. M. P.A. Jones, J. Beugnon, A. Gaetan, J. Zhang, G. Messin, A. Browaeys, and P. Grangier, "Fast quantum state control of a single trapped neutral atom," Phys. Rev. A 75, 040301(R) (2007).
    [CrossRef]
  21. A. Browaeys, H. Haffner, C. McKenzie, S. L. Rolston, K. Helmerson, W. D. Phillips, "Transport of atoms in a quantum conveyor belt," Phys. Rev. A 72, 053605 (2006).
    [CrossRef]

2009 (1)

J. Wang, L. Zhou, R. B. Li, M. Liu, M. S. Zhan, "Cold atom interferometers and their applications in precision measurements," Front. Phys. China 4, 179-189 (2009).
[CrossRef]

2008 (1)

K. Li, L. Deng, E. W. Hagley, M. G. Payne, and M. S. Zhan, "Matter-wave self-imaging by atomic center-of-mass motion induced interference," Phys. Rev. Lett. 101, 250401 (2008).
[CrossRef] [PubMed]

2007 (3)

J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P.A. Jones, G. Messin, A. Browaeys, P. Grangier, "Two-dimensional transport and transfer of a single atomic qubit in optical tweezers," Nat. Phys. 3, 696-699 (2007).
[CrossRef]

M. Stutz, S. Groblacher, T. Jennewein, A. Zeilinger, "How to create and detect N-dimensional entangled photons with an active phase hologram," Appl. Phys. Lett. 90, 261114 (2007).
[CrossRef]

S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, A. S. Arnold, "Optical ferris wheel for ultracold atoms," Opt. Express 15, 8619-8625 (2007).
[CrossRef] [PubMed]

2006 (3)

Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, A. Rauschenbeutel, "Quantum engineering: An atom-sorting machine," Nature (London) 442, 151 (2006).
[CrossRef]

M. Weber, J. Volz, K. Saucke, C. Kurtsiefer, H. Weinfurter,"Analysis of a single-atom dipole trap," Phys. Rev. A 73, 043406 (2006).
[CrossRef]

A. Browaeys, H. Haffner, C. McKenzie, S. L. Rolston, K. Helmerson, W. D. Phillips, "Transport of atoms in a quantum conveyor belt," Phys. Rev. A 72, 053605 (2006).
[CrossRef]

2004 (2)

S. Bergamini, B. Darquie, M. Jones, L. Jacubowiez, A. Browaeys, 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]

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, "Neutral Atom Quantum Register," Phys. Rev. Lett. 93, 150501 (2004).
[CrossRef] [PubMed]

2003 (2)

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, D. Meschede, "Coherence properties and quantum state transportation in an optical conveyor belt," Phys. Rev. Lett. 91, 213002 (2003).
[CrossRef] [PubMed]

O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hansch, I. Bloch, "Coherent transport of neutral atoms in spin-dependent optical lattice potentials," Phys. Rev. Lett. 91, 010407 (2003).
[CrossRef] [PubMed]

2002 (2)

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, 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]

N. Schlosser, G. Reymond, and P. Grangier, "Collisional Blockade in Microscopic Optical Dipole Traps," Phys. Rev. Lett. 89, 023005 (2002).
[CrossRef] [PubMed]

2001 (1)

N. Schlosser, G. Reymond, I. Protsenko, P. Grangier, "Sub-poissonian loading of single atoms in a microscopic dipole trap," Nature (London) 411, 1024-1027 (2001).
[CrossRef]

2000 (2)

R. Grimm, M. Weidemuller, Y. B. Ovchinnikov, "Optical dipole traps for neutral atoms," Adv. At., Mol., Opt. Phys. 42, 95-170 (2000).
[CrossRef]

D. Frese, B. Uberholz, S. Kuhr, W. Alt, D. Schrader, V. Gomer, and D. Meschede, "Single atoms in an optical dipole trap: Towards a deterministic source of cold atoms," Phys. Rev. Lett. 85, 3777-3740 (2000).
[CrossRef] [PubMed]

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, "Novel Optical Trap of Atoms with a Doughnut Beam," Phys. Rev. Lett. 78, 4713 - 4716 (1997).
[CrossRef]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 - 8189 (1992).
[CrossRef] [PubMed]

1956 (1)

H. Hanbury-Brown and R. Q. Twiss, "Correlation between photons in two coherent beams of light," Nature (London) 177, 27-29 (1956).

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 - 8189 (1992).
[CrossRef] [PubMed]

Alt, W.

Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, A. Rauschenbeutel, "Quantum engineering: An atom-sorting machine," Nature (London) 442, 151 (2006).
[CrossRef]

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, D. Meschede, "Coherence properties and quantum state transportation in an optical conveyor belt," Phys. Rev. Lett. 91, 213002 (2003).
[CrossRef] [PubMed]

D. Frese, B. Uberholz, S. Kuhr, W. Alt, D. Schrader, V. Gomer, and D. Meschede, "Single atoms in an optical dipole trap: Towards a deterministic source of cold atoms," Phys. Rev. Lett. 85, 3777-3740 (2000).
[CrossRef] [PubMed]

Arnold, A. S.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 - 8189 (1992).
[CrossRef] [PubMed]

Bergamini, S.

Beugnon, J.

J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P.A. Jones, G. Messin, A. Browaeys, P. Grangier, "Two-dimensional transport and transfer of a single atomic qubit in optical tweezers," Nat. Phys. 3, 696-699 (2007).
[CrossRef]

Birkl, G.

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, 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]

Bloch, I.

O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hansch, I. Bloch, "Coherent transport of neutral atoms in spin-dependent optical lattice potentials," Phys. Rev. Lett. 91, 010407 (2003).
[CrossRef] [PubMed]

Browaeys, A.

J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P.A. Jones, G. Messin, A. Browaeys, P. Grangier, "Two-dimensional transport and transfer of a single atomic qubit in optical tweezers," Nat. Phys. 3, 696-699 (2007).
[CrossRef]

A. Browaeys, H. Haffner, C. McKenzie, S. L. Rolston, K. Helmerson, W. D. Phillips, "Transport of atoms in a quantum conveyor belt," Phys. Rev. A 72, 053605 (2006).
[CrossRef]

S. Bergamini, B. Darquie, M. Jones, L. Jacubowiez, A. Browaeys, 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]

Buchkremer, F. B. J.

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, 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]

Darquie, B.

Deng, L.

K. Li, L. Deng, E. W. Hagley, M. G. Payne, and M. S. Zhan, "Matter-wave self-imaging by atomic center-of-mass motion induced interference," Phys. Rev. Lett. 101, 250401 (2008).
[CrossRef] [PubMed]

Dotsenko, I.

Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, A. Rauschenbeutel, "Quantum engineering: An atom-sorting machine," Nature (London) 442, 151 (2006).
[CrossRef]

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, "Neutral Atom Quantum Register," Phys. Rev. Lett. 93, 150501 (2004).
[CrossRef] [PubMed]

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, D. Meschede, "Coherence properties and quantum state transportation in an optical conveyor belt," Phys. Rev. Lett. 91, 213002 (2003).
[CrossRef] [PubMed]

Dumke, R.

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, 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]

Ellinas, D.

Ertmer, W.

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, 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]

Forster, L.

Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, A. Rauschenbeutel, "Quantum engineering: An atom-sorting machine," Nature (London) 442, 151 (2006).
[CrossRef]

Franke-Arnold, S.

Frese, D.

D. Frese, B. Uberholz, S. Kuhr, W. Alt, D. Schrader, V. Gomer, and D. Meschede, "Single atoms in an optical dipole trap: Towards a deterministic source of cold atoms," Phys. Rev. Lett. 85, 3777-3740 (2000).
[CrossRef] [PubMed]

Gaetan, A.

J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P.A. Jones, G. Messin, A. Browaeys, P. Grangier, "Two-dimensional transport and transfer of a single atomic qubit in optical tweezers," Nat. Phys. 3, 696-699 (2007).
[CrossRef]

Girkin, J. M.

Gomer, V.

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, D. Meschede, "Coherence properties and quantum state transportation in an optical conveyor belt," Phys. Rev. Lett. 91, 213002 (2003).
[CrossRef] [PubMed]

D. Frese, B. Uberholz, S. Kuhr, W. Alt, D. Schrader, V. Gomer, and D. Meschede, "Single atoms in an optical dipole trap: Towards a deterministic source of cold atoms," Phys. Rev. Lett. 85, 3777-3740 (2000).
[CrossRef] [PubMed]

Grangier, P.

J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P.A. Jones, G. Messin, A. Browaeys, P. Grangier, "Two-dimensional transport and transfer of a single atomic qubit in optical tweezers," Nat. Phys. 3, 696-699 (2007).
[CrossRef]

S. Bergamini, B. Darquie, M. Jones, L. Jacubowiez, A. Browaeys, 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]

N. Schlosser, G. Reymond, and P. Grangier, "Collisional Blockade in Microscopic Optical Dipole Traps," Phys. Rev. Lett. 89, 023005 (2002).
[CrossRef] [PubMed]

N. Schlosser, G. Reymond, I. Protsenko, P. Grangier, "Sub-poissonian loading of single atoms in a microscopic dipole trap," Nature (London) 411, 1024-1027 (2001).
[CrossRef]

Greiner, M.

O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hansch, I. Bloch, "Coherent transport of neutral atoms in spin-dependent optical lattice potentials," Phys. Rev. Lett. 91, 010407 (2003).
[CrossRef] [PubMed]

Grimm, R.

R. Grimm, M. Weidemuller, Y. B. Ovchinnikov, "Optical dipole traps for neutral atoms," Adv. At., Mol., Opt. Phys. 42, 95-170 (2000).
[CrossRef]

Groblacher, S.

M. Stutz, S. Groblacher, T. Jennewein, A. Zeilinger, "How to create and detect N-dimensional entangled photons with an active phase hologram," Appl. Phys. Lett. 90, 261114 (2007).
[CrossRef]

Haffner, H.

A. Browaeys, H. Haffner, C. McKenzie, S. L. Rolston, K. Helmerson, W. D. Phillips, "Transport of atoms in a quantum conveyor belt," Phys. Rev. A 72, 053605 (2006).
[CrossRef]

Hagley, E. W.

K. Li, L. Deng, E. W. Hagley, M. G. Payne, and M. S. Zhan, "Matter-wave self-imaging by atomic center-of-mass motion induced interference," Phys. Rev. Lett. 101, 250401 (2008).
[CrossRef] [PubMed]

Hanbury-Brown, H.

H. Hanbury-Brown and R. Q. Twiss, "Correlation between photons in two coherent beams of light," Nature (London) 177, 27-29 (1956).

Hansch, T. W.

O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hansch, I. Bloch, "Coherent transport of neutral atoms in spin-dependent optical lattice potentials," Phys. Rev. Lett. 91, 010407 (2003).
[CrossRef] [PubMed]

Helmerson, K.

A. Browaeys, H. Haffner, C. McKenzie, S. L. Rolston, K. Helmerson, W. D. Phillips, "Transport of atoms in a quantum conveyor belt," Phys. Rev. A 72, 053605 (2006).
[CrossRef]

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, "Novel Optical Trap of Atoms with a Doughnut Beam," Phys. Rev. Lett. 78, 4713 - 4716 (1997).
[CrossRef]

Jacubowiez, L.

Jennewein, T.

M. Stutz, S. Groblacher, T. Jennewein, A. Zeilinger, "How to create and detect N-dimensional entangled photons with an active phase hologram," Appl. Phys. Lett. 90, 261114 (2007).
[CrossRef]

Jones, M.

Jones, M. P .A.

J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P.A. Jones, G. Messin, A. Browaeys, P. Grangier, "Two-dimensional transport and transfer of a single atomic qubit in optical tweezers," Nat. Phys. 3, 696-699 (2007).
[CrossRef]

Khudaverdyan, M.

Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, A. Rauschenbeutel, "Quantum engineering: An atom-sorting machine," Nature (London) 442, 151 (2006).
[CrossRef]

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, "Neutral Atom Quantum Register," Phys. Rev. Lett. 93, 150501 (2004).
[CrossRef] [PubMed]

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, D. Meschede, "Coherence properties and quantum state transportation in an optical conveyor belt," Phys. Rev. Lett. 91, 213002 (2003).
[CrossRef] [PubMed]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, "Novel Optical Trap of Atoms with a Doughnut Beam," Phys. Rev. Lett. 78, 4713 - 4716 (1997).
[CrossRef]

Kuhr, S.

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, D. Meschede, "Coherence properties and quantum state transportation in an optical conveyor belt," Phys. Rev. Lett. 91, 213002 (2003).
[CrossRef] [PubMed]

D. Frese, B. Uberholz, S. Kuhr, W. Alt, D. Schrader, V. Gomer, and D. Meschede, "Single atoms in an optical dipole trap: Towards a deterministic source of cold atoms," Phys. Rev. Lett. 85, 3777-3740 (2000).
[CrossRef] [PubMed]

Kurtsiefer, C.

M. Weber, J. Volz, K. Saucke, C. Kurtsiefer, H. Weinfurter,"Analysis of a single-atom dipole trap," Phys. Rev. A 73, 043406 (2006).
[CrossRef]

Lance, A. M.

J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P.A. Jones, G. Messin, A. Browaeys, P. Grangier, "Two-dimensional transport and transfer of a single atomic qubit in optical tweezers," Nat. Phys. 3, 696-699 (2007).
[CrossRef]

Leach, J.

Lembessis, V. E.

Li, K.

K. Li, L. Deng, E. W. Hagley, M. G. Payne, and M. S. Zhan, "Matter-wave self-imaging by atomic center-of-mass motion induced interference," Phys. Rev. Lett. 101, 250401 (2008).
[CrossRef] [PubMed]

Li, R. B.

J. Wang, L. Zhou, R. B. Li, M. Liu, M. S. Zhan, "Cold atom interferometers and their applications in precision measurements," Front. Phys. China 4, 179-189 (2009).
[CrossRef]

Liu, M.

J. Wang, L. Zhou, R. B. Li, M. Liu, M. S. Zhan, "Cold atom interferometers and their applications in precision measurements," Front. Phys. China 4, 179-189 (2009).
[CrossRef]

Mandel, O.

O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hansch, I. Bloch, "Coherent transport of neutral atoms in spin-dependent optical lattice potentials," Phys. Rev. Lett. 91, 010407 (2003).
[CrossRef] [PubMed]

Marion, H.

J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P.A. Jones, G. Messin, A. Browaeys, P. Grangier, "Two-dimensional transport and transfer of a single atomic qubit in optical tweezers," Nat. Phys. 3, 696-699 (2007).
[CrossRef]

McKenzie, C.

A. Browaeys, H. Haffner, C. McKenzie, S. L. Rolston, K. Helmerson, W. D. Phillips, "Transport of atoms in a quantum conveyor belt," Phys. Rev. A 72, 053605 (2006).
[CrossRef]

Meschede, D.

Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, A. Rauschenbeutel, "Quantum engineering: An atom-sorting machine," Nature (London) 442, 151 (2006).
[CrossRef]

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, "Neutral Atom Quantum Register," Phys. Rev. Lett. 93, 150501 (2004).
[CrossRef] [PubMed]

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, D. Meschede, "Coherence properties and quantum state transportation in an optical conveyor belt," Phys. Rev. Lett. 91, 213002 (2003).
[CrossRef] [PubMed]

D. Frese, B. Uberholz, S. Kuhr, W. Alt, D. Schrader, V. Gomer, and D. Meschede, "Single atoms in an optical dipole trap: Towards a deterministic source of cold atoms," Phys. Rev. Lett. 85, 3777-3740 (2000).
[CrossRef] [PubMed]

Messin, G.

J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P.A. Jones, G. Messin, A. Browaeys, P. Grangier, "Two-dimensional transport and transfer of a single atomic qubit in optical tweezers," Nat. Phys. 3, 696-699 (2007).
[CrossRef]

Miroshnychenko, Y.

J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P.A. Jones, G. Messin, A. Browaeys, P. Grangier, "Two-dimensional transport and transfer of a single atomic qubit in optical tweezers," Nat. Phys. 3, 696-699 (2007).
[CrossRef]

Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, A. Rauschenbeutel, "Quantum engineering: An atom-sorting machine," Nature (London) 442, 151 (2006).
[CrossRef]

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, "Neutral Atom Quantum Register," Phys. Rev. Lett. 93, 150501 (2004).
[CrossRef] [PubMed]

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, D. Meschede, "Coherence properties and quantum state transportation in an optical conveyor belt," Phys. Rev. Lett. 91, 213002 (2003).
[CrossRef] [PubMed]

Muther, T.

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, 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]

Ohberg, P.

Ovchinnikov, Y. B.

R. Grimm, M. Weidemuller, Y. B. Ovchinnikov, "Optical dipole traps for neutral atoms," Adv. At., Mol., Opt. Phys. 42, 95-170 (2000).
[CrossRef]

Padgett, M. J.

Payne, M. G.

K. Li, L. Deng, E. W. Hagley, M. G. Payne, and M. S. Zhan, "Matter-wave self-imaging by atomic center-of-mass motion induced interference," Phys. Rev. Lett. 101, 250401 (2008).
[CrossRef] [PubMed]

Phillips, W. D.

A. Browaeys, H. Haffner, C. McKenzie, S. L. Rolston, K. Helmerson, W. D. Phillips, "Transport of atoms in a quantum conveyor belt," Phys. Rev. A 72, 053605 (2006).
[CrossRef]

Protsenko, I.

N. Schlosser, G. Reymond, I. Protsenko, P. Grangier, "Sub-poissonian loading of single atoms in a microscopic dipole trap," Nature (London) 411, 1024-1027 (2001).
[CrossRef]

Rauschenbeutel, A.

Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, A. Rauschenbeutel, "Quantum engineering: An atom-sorting machine," Nature (London) 442, 151 (2006).
[CrossRef]

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, "Neutral Atom Quantum Register," Phys. Rev. Lett. 93, 150501 (2004).
[CrossRef] [PubMed]

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, D. Meschede, "Coherence properties and quantum state transportation in an optical conveyor belt," Phys. Rev. Lett. 91, 213002 (2003).
[CrossRef] [PubMed]

Reymond, G.

N. Schlosser, G. Reymond, and P. Grangier, "Collisional Blockade in Microscopic Optical Dipole Traps," Phys. Rev. Lett. 89, 023005 (2002).
[CrossRef] [PubMed]

N. Schlosser, G. Reymond, I. Protsenko, P. Grangier, "Sub-poissonian loading of single atoms in a microscopic dipole trap," Nature (London) 411, 1024-1027 (2001).
[CrossRef]

Rolston, S. L.

A. Browaeys, H. Haffner, C. McKenzie, S. L. Rolston, K. Helmerson, W. D. Phillips, "Transport of atoms in a quantum conveyor belt," Phys. Rev. A 72, 053605 (2006).
[CrossRef]

Rom, T.

O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hansch, I. Bloch, "Coherent transport of neutral atoms in spin-dependent optical lattice potentials," Phys. Rev. Lett. 91, 010407 (2003).
[CrossRef] [PubMed]

Rosenfeld, W.

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, D. Meschede, "Coherence properties and quantum state transportation in an optical conveyor belt," Phys. Rev. Lett. 91, 213002 (2003).
[CrossRef] [PubMed]

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, "Novel Optical Trap of Atoms with a Doughnut Beam," Phys. Rev. Lett. 78, 4713 - 4716 (1997).
[CrossRef]

Saucke, K.

M. Weber, J. Volz, K. Saucke, C. Kurtsiefer, H. Weinfurter,"Analysis of a single-atom dipole trap," Phys. Rev. A 73, 043406 (2006).
[CrossRef]

Schlosser, N.

N. Schlosser, G. Reymond, and P. Grangier, "Collisional Blockade in Microscopic Optical Dipole Traps," Phys. Rev. Lett. 89, 023005 (2002).
[CrossRef] [PubMed]

N. Schlosser, G. Reymond, I. Protsenko, P. Grangier, "Sub-poissonian loading of single atoms in a microscopic dipole trap," Nature (London) 411, 1024-1027 (2001).
[CrossRef]

Schrader, D.

Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, A. Rauschenbeutel, "Quantum engineering: An atom-sorting machine," Nature (London) 442, 151 (2006).
[CrossRef]

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, "Neutral Atom Quantum Register," Phys. Rev. Lett. 93, 150501 (2004).
[CrossRef] [PubMed]

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, D. Meschede, "Coherence properties and quantum state transportation in an optical conveyor belt," Phys. Rev. Lett. 91, 213002 (2003).
[CrossRef] [PubMed]

D. Frese, B. Uberholz, S. Kuhr, W. Alt, D. Schrader, V. Gomer, and D. Meschede, "Single atoms in an optical dipole trap: Towards a deterministic source of cold atoms," Phys. Rev. Lett. 85, 3777-3740 (2000).
[CrossRef] [PubMed]

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, "Novel Optical Trap of Atoms with a Doughnut Beam," Phys. Rev. Lett. 78, 4713 - 4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, "Novel Optical Trap of Atoms with a Doughnut Beam," Phys. Rev. Lett. 78, 4713 - 4716 (1997).
[CrossRef]

Sortais, Y. R. P.

J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P.A. Jones, G. Messin, A. Browaeys, P. Grangier, "Two-dimensional transport and transfer of a single atomic qubit in optical tweezers," Nat. Phys. 3, 696-699 (2007).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 - 8189 (1992).
[CrossRef] [PubMed]

Stutz, M.

M. Stutz, S. Groblacher, T. Jennewein, A. Zeilinger, "How to create and detect N-dimensional entangled photons with an active phase hologram," Appl. Phys. Lett. 90, 261114 (2007).
[CrossRef]

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, "Novel Optical Trap of Atoms with a Doughnut Beam," Phys. Rev. Lett. 78, 4713 - 4716 (1997).
[CrossRef]

Tuchendler, C.

J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P.A. Jones, G. Messin, A. Browaeys, P. Grangier, "Two-dimensional transport and transfer of a single atomic qubit in optical tweezers," Nat. Phys. 3, 696-699 (2007).
[CrossRef]

Twiss, R. Q.

H. Hanbury-Brown and R. Q. Twiss, "Correlation between photons in two coherent beams of light," Nature (London) 177, 27-29 (1956).

Uberholz, B.

D. Frese, B. Uberholz, S. Kuhr, W. Alt, D. Schrader, V. Gomer, and D. Meschede, "Single atoms in an optical dipole trap: Towards a deterministic source of cold atoms," Phys. Rev. Lett. 85, 3777-3740 (2000).
[CrossRef] [PubMed]

Volk, M.

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, 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]

Volz, J.

M. Weber, J. Volz, K. Saucke, C. Kurtsiefer, H. Weinfurter,"Analysis of a single-atom dipole trap," Phys. Rev. A 73, 043406 (2006).
[CrossRef]

Wang, J.

J. Wang, L. Zhou, R. B. Li, M. Liu, M. S. Zhan, "Cold atom interferometers and their applications in precision measurements," Front. Phys. China 4, 179-189 (2009).
[CrossRef]

Weber, M.

M. Weber, J. Volz, K. Saucke, C. Kurtsiefer, H. Weinfurter,"Analysis of a single-atom dipole trap," Phys. Rev. A 73, 043406 (2006).
[CrossRef]

Weidemuller, M.

R. Grimm, M. Weidemuller, Y. B. Ovchinnikov, "Optical dipole traps for neutral atoms," Adv. At., Mol., Opt. Phys. 42, 95-170 (2000).
[CrossRef]

Weinfurter, H.

M. Weber, J. Volz, K. Saucke, C. Kurtsiefer, H. Weinfurter,"Analysis of a single-atom dipole trap," Phys. Rev. A 73, 043406 (2006).
[CrossRef]

Widera, A.

O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hansch, I. Bloch, "Coherent transport of neutral atoms in spin-dependent optical lattice potentials," Phys. Rev. Lett. 91, 010407 (2003).
[CrossRef] [PubMed]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 - 8189 (1992).
[CrossRef] [PubMed]

Wright, A. J.

Zeilinger, A.

M. Stutz, S. Groblacher, T. Jennewein, A. Zeilinger, "How to create and detect N-dimensional entangled photons with an active phase hologram," Appl. Phys. Lett. 90, 261114 (2007).
[CrossRef]

Zhan, M. S.

J. Wang, L. Zhou, R. B. Li, M. Liu, M. S. Zhan, "Cold atom interferometers and their applications in precision measurements," Front. Phys. China 4, 179-189 (2009).
[CrossRef]

K. Li, L. Deng, E. W. Hagley, M. G. Payne, and M. S. Zhan, "Matter-wave self-imaging by atomic center-of-mass motion induced interference," Phys. Rev. Lett. 101, 250401 (2008).
[CrossRef] [PubMed]

Zhou, L.

J. Wang, L. Zhou, R. B. Li, M. Liu, M. S. Zhan, "Cold atom interferometers and their applications in precision measurements," Front. Phys. China 4, 179-189 (2009).
[CrossRef]

Adv. At., Mol., Opt. Phys. (1)

R. Grimm, M. Weidemuller, Y. B. Ovchinnikov, "Optical dipole traps for neutral atoms," Adv. At., Mol., Opt. Phys. 42, 95-170 (2000).
[CrossRef]

Appl. Phys. Lett. (1)

M. Stutz, S. Groblacher, T. Jennewein, A. Zeilinger, "How to create and detect N-dimensional entangled photons with an active phase hologram," Appl. Phys. Lett. 90, 261114 (2007).
[CrossRef]

Front. Phys. China (1)

J. Wang, L. Zhou, R. B. Li, M. Liu, M. S. Zhan, "Cold atom interferometers and their applications in precision measurements," Front. Phys. China 4, 179-189 (2009).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nat. Phys. (1)

J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P.A. Jones, G. Messin, A. Browaeys, P. Grangier, "Two-dimensional transport and transfer of a single atomic qubit in optical tweezers," Nat. Phys. 3, 696-699 (2007).
[CrossRef]

Nature (London) (3)

Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, A. Rauschenbeutel, "Quantum engineering: An atom-sorting machine," Nature (London) 442, 151 (2006).
[CrossRef]

N. Schlosser, G. Reymond, I. Protsenko, P. Grangier, "Sub-poissonian loading of single atoms in a microscopic dipole trap," Nature (London) 411, 1024-1027 (2001).
[CrossRef]

H. Hanbury-Brown and R. Q. Twiss, "Correlation between photons in two coherent beams of light," Nature (London) 177, 27-29 (1956).

Opt. Express (1)

Phys. Rev. A (3)

M. Weber, J. Volz, K. Saucke, C. Kurtsiefer, H. Weinfurter,"Analysis of a single-atom dipole trap," Phys. Rev. A 73, 043406 (2006).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 - 8189 (1992).
[CrossRef] [PubMed]

A. Browaeys, H. Haffner, C. McKenzie, S. L. Rolston, K. Helmerson, W. D. Phillips, "Transport of atoms in a quantum conveyor belt," Phys. Rev. A 72, 053605 (2006).
[CrossRef]

Phys. Rev. Lett. (8)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, "Novel Optical Trap of Atoms with a Doughnut Beam," Phys. Rev. Lett. 78, 4713 - 4716 (1997).
[CrossRef]

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, D. Meschede, "Coherence properties and quantum state transportation in an optical conveyor belt," Phys. Rev. Lett. 91, 213002 (2003).
[CrossRef] [PubMed]

O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hansch, I. Bloch, "Coherent transport of neutral atoms in spin-dependent optical lattice potentials," Phys. Rev. Lett. 91, 010407 (2003).
[CrossRef] [PubMed]

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, 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]

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, "Neutral Atom Quantum Register," Phys. Rev. Lett. 93, 150501 (2004).
[CrossRef] [PubMed]

D. Frese, B. Uberholz, S. Kuhr, W. Alt, D. Schrader, V. Gomer, and D. Meschede, "Single atoms in an optical dipole trap: Towards a deterministic source of cold atoms," Phys. Rev. Lett. 85, 3777-3740 (2000).
[CrossRef] [PubMed]

K. Li, L. Deng, E. W. Hagley, M. G. Payne, and M. S. Zhan, "Matter-wave self-imaging by atomic center-of-mass motion induced interference," Phys. Rev. Lett. 101, 250401 (2008).
[CrossRef] [PubMed]

N. Schlosser, G. Reymond, and P. Grangier, "Collisional Blockade in Microscopic Optical Dipole Traps," Phys. Rev. Lett. 89, 023005 (2002).
[CrossRef] [PubMed]

Other (1)

M. P.A. Jones, J. Beugnon, A. Gaetan, J. Zhang, G. Messin, A. Browaeys, and P. Grangier, "Fast quantum state control of a single trapped neutral atom," Phys. Rev. A 75, 040301(R) (2007).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

(color online) The ring lattices of l=[1,2,3] and their holograms. Observed optical intensity distribution of the ring lattices(a) and the corresponding gray-level holograms with 600×600 pixels(b).

Fig. 2.
Fig. 2.

The optical signals of rotating ring lattice (l=±1, two traps) for different rotation rates 60/N Hz with N=10 (a),6(b) and 4(c). The rotation lasts for 1 sec.

Fig. 3.
Fig. 3.

Experimental setup for trapping and rotating single atoms. Both optical ring lattice (solid line) and optical imaging (dotted line) systems are shown (not to scale). The incident linearly polarized light from a single mode laser diode at 830nm is delivered and collimated by a single mode polarization maintaining fiber with an output NA=0.11 and an aspherical lens(AL). A half-wave plate(HWP) rotates the light polarization axis to match the polarization required by the SLM. The laser is then reflected by the SLM and expanded by two doublets and focused onto the MOT region by a triplet. The triplet is also used as an imaging system to collect the fluorescence at 780nm of the atoms. The fluorescence is separated from the dipole laser by a dichroic mirror(DM) and is passed through an interference filter(IF) before entering the spatial filter of a single mode fiber at 1550nm.

Fig. 4.
Fig. 4.

Fluorescence of single atoms measured by SPCM. Each point corresponds to a 20 ms time bin. Shown in (a) is the fluorescence signal from atom in an individual trap. (b) is the measurement of second-order correlation function g (2)(τ) of single atom fluorescence. (c) is the fluorescence signal from the whole two traps in the ring lattice.

Fig. 5.
Fig. 5.

The modulated fluorescence signal of single atoms rotated in the ring lattice. (a) The rotation rate is 15 Hz, viewed to a single atom in a trap; (b) The rotation rate is 6 Hz. In the first second, two atoms are trapped which give fluorescence of 12Hz; in the second second, an atom is lost, thus the remaining atom produces a signal of 6 Hz. (c) Single atom signals recorded at the left trap (solid line) and the upper north-pole (dashed line). The rotation rate of the lattice is 6Hz.

Equations (1)

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

mod [ ( angle [ LG ( x , y , l , z , w 0 , z R ) + LG ( x , y , l , z , w 0 , z R ) ] + x · kx ) , 2 π ] · 256 2 π ,

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