Abstract

We explore the use of first and second order same-time atomic spatial correlation functions as a diagnostic for probing the small scale spatial structure of atomic samples trapped in optical lattices. Assuming an ensemble of equivalent atoms, properties of the local wave function at a given lattice site can be measured using same-position first-order correlations. Statistics of atomic distributions over the lattice can be measured via two-point correlations, generally requiring the averaging of multiple realizations of statistically similar but distinct realizations in order to obtain sufficient signal to noise. Whereas two-point first order correlations are fragile due to phase fluctuations from shot-to-shot in the ensemble, second order correlations are robust. We perform numerical simulations to demonstrate these diagnostic tools.

© 1999 Optical Society of America

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1999 (9)

S. K. Dutta, B. K. Teo, and G. Raithel, “Tunneling Dynamics and Guage Potentials in Optical Lattices,” Phys. Rev. Lett. 83, 1093–1936 (1999).
[Crossref]

Gavin K. Brennen, Cartlon M. Caves, Poul S. Jessen, and Ivan H. Deutsch, “Quantum Logic Gates in Optical Lattices,” Phys. Rev. Lett. 82, 1060–1063 (1999).
[Crossref]

D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of Atoms via Cold Controlled Collisions,” Phys. Rev. Lett. 82, 1975–1978 (1999).
[Crossref]

Anders Sorensen and Klaus Molmer, “Spin-Spin Interactions and Spin Squeezing in an Opical Lattice,” Phys. Rev. Lett. 83, 2274–2277 (1999).

S. Lukman Winoto, Marshall T. DePue, Nathan E. Bramall, and David S. Weiss, “Laser cooling at high density in deep far-detuned optical lattices,” Phys. Rev. A 59, R19–R22 (1999).
[Crossref]

Dai-Il Choi and Qian Niu, “Bose-Einstein Condensates in an Optical Lattice,” Phys. Rev. Lett. 82, 2022–2025 (1999).
[Crossref]

C. Orzel, M. Walhout, U. Sterr, P. S. Julienne, and S. L. Rolston, “Spin polarization and quantum-statistical effects in ultracold ionizing collisions,” Phys. Rev. A 59, 1926–1935 (1999).
[Crossref]

M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland, and C. Schönenberger, “The Fermionic Hanbury Brown and Twiss Experiment,” Science 284, 296–298 (1999).
[Crossref] [PubMed]

William D. Oliver, Jungsang Kim, Robert C. Liu, and Yoshihisa Yamamoto, “Hanbury Brown and Twiss-Type Experiment with Electrons,” Science 284, 299–301 (1999).
[Crossref] [PubMed]

1998 (6)

S. E. Hamann, D. L. Haycock, G. Klose, P. H. Pax, I. H. Deutsch, and P. S. Jessen, “Resolved-Sideband Raman Cooling to the Ground State of an Optical Lattice,” Phys. Rev. Lett. 80, 4149–4152 (1998).
[Crossref]

E. V. Goldstein, O. Zobay, and P. Meystre, “Coherence of atom matter-wave fields,” Phys. Rev. A 58, 2373–2384 (1998).
[Crossref]

Kirstine Berg-Sørenson and Klaus Mølmer, “Bose-Einstein condensates in spatially periodic potentials,” Phys. Rev. A 58, 1480–1484 (1998).
[Crossref]

Ivan H. Deutsch and Poul S. Jessen, “Quantum-state control in optical lattices,” Phys. Rev. A 57, 1972–1986 (1998).
[Crossref]

B. P. Anderson and M. A. Kasevich, “Macroscopic Quantum Interference from Atomic Tunnel Arrays,” Science 282, 1686–1689 (1998).
[Crossref] [PubMed]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
[Crossref]

1997 (4)

Klaus Drese and Martin Holthaus, “Exploring a Metal-Insulator Transition with Ultrcold Atoms in Standing Light Waves,” Phys. Rev. Lett. 2932, 2932–2935 (1997).
[Crossref]

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of Interference Between Two Bose Condensates,” Science 275, 637–641 (1997).
[Crossref] [PubMed]

B. Saubaméa, T. W. Hijmans, S. Kulin, E. Rasel, E. Peik, M. Leduc, and C. Cohen-Tannoudji, “Direct Measurement of the Spatial Correlation Function of Ultracold Atoms,” Phys. Rev. Lett. 79, 3146–3149 (1997).
[Crossref]

Hideyuki Kunugita, Tetsuya Ido, and Fujio Shimizu, “Ionizing Collisional Rate of Metastable Rare-Gas Atoms in an Optical Lattice,” Phys. Rev. Lett. 79, 621–624 (1997).
[Crossref]

1996 (5)

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Direct, Nondestructive Observation of a Bose Condensate,” Science 273, 84–87 (1996)
[Crossref] [PubMed]

Masami Yasuda and Fujio Shimizu, “Observation of Two-Atom Correlation of an Ultracold Neon Atomic Beam,” Phys. Rev. Lett. 77, 3090–3093 (1996).
[Crossref] [PubMed]

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

Qian Niu, Xian-Gen Zhao, G. A. Georgakis, and M. G. Raizen, “Atomic Landau-Zener Tunneling and Wannier Stark Ladders in Opical Potenitals,” Phys. Rev. Lett. 76, 4504–4507 (1996).
[Crossref] [PubMed]

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

1995 (3)

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, and W. Ketterle, “Bose-Einstein Condensation in a Gas of Sodium Atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, “Bragg Scattering from Atoms in Optical Lattices,” Phys. Rev. Lett. 75, 2823–2826 (1995).
[Crossref] [PubMed]

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” Science 269, 198–201 (1995).
[Crossref] [PubMed]

1994 (1)

J. E. Thomas and L. J. Wang, “Quantum theory of correlated-atomic-position measurements by resonance imaging,” Phys. Rev. A 49, 558–569 (1994).
[Crossref] [PubMed]

1988 (1)

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[Crossref] [PubMed]

Anderson, B. P.

B. P. Anderson and M. A. Kasevich, “Macroscopic Quantum Interference from Atomic Tunnel Arrays,” Science 282, 1686–1689 (1998).
[Crossref] [PubMed]

Anderson, M. H.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” Science 269, 198–201 (1995).
[Crossref] [PubMed]

Andrews, M. R.

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of Interference Between Two Bose Condensates,” Science 275, 637–641 (1997).
[Crossref] [PubMed]

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Direct, Nondestructive Observation of a Bose Condensate,” Science 273, 84–87 (1996)
[Crossref] [PubMed]

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, and W. Ketterle, “Bose-Einstein Condensation in a Gas of Sodium Atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

Berg-Sørenson, Kirstine

Kirstine Berg-Sørenson and Klaus Mølmer, “Bose-Einstein condensates in spatially periodic potentials,” Phys. Rev. A 58, 1480–1484 (1998).
[Crossref]

Birkl, G.

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, “Bragg Scattering from Atoms in Optical Lattices,” Phys. Rev. Lett. 75, 2823–2826 (1995).
[Crossref] [PubMed]

Bramall, Nathan E.

S. Lukman Winoto, Marshall T. DePue, Nathan E. Bramall, and David S. Weiss, “Laser cooling at high density in deep far-detuned optical lattices,” Phys. Rev. A 59, R19–R22 (1999).
[Crossref]

Brennen, Gavin K.

Gavin K. Brennen, Cartlon M. Caves, Poul S. Jessen, and Ivan H. Deutsch, “Quantum Logic Gates in Optical Lattices,” Phys. Rev. Lett. 82, 1060–1063 (1999).
[Crossref]

Briegel, H.-J.

D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of Atoms via Cold Controlled Collisions,” Phys. Rev. Lett. 82, 1975–1978 (1999).
[Crossref]

Bruder, C.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
[Crossref]

Caves, Cartlon M.

Gavin K. Brennen, Cartlon M. Caves, Poul S. Jessen, and Ivan H. Deutsch, “Quantum Logic Gates in Optical Lattices,” Phys. Rev. Lett. 82, 1060–1063 (1999).
[Crossref]

Choi, Dai-Il

Dai-Il Choi and Qian Niu, “Bose-Einstein Condensates in an Optical Lattice,” Phys. Rev. Lett. 82, 2022–2025 (1999).
[Crossref]

Chu, Benjamin

Benjamin Chu, Laser Light Scattering, Second Edition (Academic Press, San Diego, 1991).

Cirac, J. I.

D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of Atoms via Cold Controlled Collisions,” Phys. Rev. Lett. 82, 1975–1978 (1999).
[Crossref]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
[Crossref]

Cohen-Tannoudji, C.

B. Saubaméa, T. W. Hijmans, S. Kulin, E. Rasel, E. Peik, M. Leduc, and C. Cohen-Tannoudji, “Direct Measurement of the Spatial Correlation Function of Ultracold Atoms,” Phys. Rev. Lett. 79, 3146–3149 (1997).
[Crossref]

Cohen-Tannoudji, Claude

Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë, Quantum MechanicsVol. 1(John Wiley & Sons, New York, 1977).

Cornell, E. A.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” Science 269, 198–201 (1995).
[Crossref] [PubMed]

Davis, K. B.

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, and W. Ketterle, “Bose-Einstein Condensation in a Gas of Sodium Atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

DePue, Marshall T.

S. Lukman Winoto, Marshall T. DePue, Nathan E. Bramall, and David S. Weiss, “Laser cooling at high density in deep far-detuned optical lattices,” Phys. Rev. A 59, R19–R22 (1999).
[Crossref]

Deutsch, I. H.

S. E. Hamann, D. L. Haycock, G. Klose, P. H. Pax, I. H. Deutsch, and P. S. Jessen, “Resolved-Sideband Raman Cooling to the Ground State of an Optical Lattice,” Phys. Rev. Lett. 80, 4149–4152 (1998).
[Crossref]

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

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, “Bragg Scattering from Atoms in Optical Lattices,” Phys. Rev. Lett. 75, 2823–2826 (1995).
[Crossref] [PubMed]

Deutsch, Ivan H.

Gavin K. Brennen, Cartlon M. Caves, Poul S. Jessen, and Ivan H. Deutsch, “Quantum Logic Gates in Optical Lattices,” Phys. Rev. Lett. 82, 1060–1063 (1999).
[Crossref]

Ivan H. Deutsch and Poul S. Jessen, “Quantum-state control in optical lattices,” Phys. Rev. A 57, 1972–1986 (1998).
[Crossref]

Diu, Bernard

Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë, Quantum MechanicsVol. 1(John Wiley & Sons, New York, 1977).

Drese, Klaus

Klaus Drese and Martin Holthaus, “Exploring a Metal-Insulator Transition with Ultrcold Atoms in Standing Light Waves,” Phys. Rev. Lett. 2932, 2932–2935 (1997).
[Crossref]

Durfee, D. S.

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of Interference Between Two Bose Condensates,” Science 275, 637–641 (1997).
[Crossref] [PubMed]

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Direct, Nondestructive Observation of a Bose Condensate,” Science 273, 84–87 (1996)
[Crossref] [PubMed]

Dutta, S. K.

S. K. Dutta, B. K. Teo, and G. Raithel, “Tunneling Dynamics and Guage Potentials in Optical Lattices,” Phys. Rev. Lett. 83, 1093–1936 (1999).
[Crossref]

Ensher, J. R.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” Science 269, 198–201 (1995).
[Crossref] [PubMed]

Ensslin, K.

M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland, and C. Schönenberger, “The Fermionic Hanbury Brown and Twiss Experiment,” Science 284, 296–298 (1999).
[Crossref] [PubMed]

Gardiner, C. W.

D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of Atoms via Cold Controlled Collisions,” Phys. Rev. Lett. 82, 1975–1978 (1999).
[Crossref]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
[Crossref]

Gatzke, M.

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, “Bragg Scattering from Atoms in Optical Lattices,” Phys. Rev. Lett. 75, 2823–2826 (1995).
[Crossref] [PubMed]

Georgakis, G. A.

Qian Niu, Xian-Gen Zhao, G. A. Georgakis, and M. G. Raizen, “Atomic Landau-Zener Tunneling and Wannier Stark Ladders in Opical Potenitals,” Phys. Rev. Lett. 76, 4504–4507 (1996).
[Crossref] [PubMed]

Goldstein, E. V.

E. V. Goldstein, O. Zobay, and P. Meystre, “Coherence of atom matter-wave fields,” Phys. Rev. A 58, 2373–2384 (1998).
[Crossref]

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

Goodman, Joseph W.

Joseph W. Goodman, Statistical Optics (John Wiley & Sons, New York, 1985).

Gould, P. L.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[Crossref] [PubMed]

Hamann, S. E.

S. E. Hamann, D. L. Haycock, G. Klose, P. H. Pax, I. H. Deutsch, and P. S. Jessen, “Resolved-Sideband Raman Cooling to the Ground State of an Optical Lattice,” Phys. Rev. Lett. 80, 4149–4152 (1998).
[Crossref]

Haycock, D. L.

S. E. Hamann, D. L. Haycock, G. Klose, P. H. Pax, I. H. Deutsch, and P. S. Jessen, “Resolved-Sideband Raman Cooling to the Ground State of an Optical Lattice,” Phys. Rev. Lett. 80, 4149–4152 (1998).
[Crossref]

Heinzel, T.

M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland, and C. Schönenberger, “The Fermionic Hanbury Brown and Twiss Experiment,” Science 284, 296–298 (1999).
[Crossref] [PubMed]

Henny, M.

M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland, and C. Schönenberger, “The Fermionic Hanbury Brown and Twiss Experiment,” Science 284, 296–298 (1999).
[Crossref] [PubMed]

Hijmans, T. W.

B. Saubaméa, T. W. Hijmans, S. Kulin, E. Rasel, E. Peik, M. Leduc, and C. Cohen-Tannoudji, “Direct Measurement of the Spatial Correlation Function of Ultracold Atoms,” Phys. Rev. Lett. 79, 3146–3149 (1997).
[Crossref]

Holland, M.

M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland, and C. Schönenberger, “The Fermionic Hanbury Brown and Twiss Experiment,” Science 284, 296–298 (1999).
[Crossref] [PubMed]

Holthaus, Martin

Klaus Drese and Martin Holthaus, “Exploring a Metal-Insulator Transition with Ultrcold Atoms in Standing Light Waves,” Phys. Rev. Lett. 2932, 2932–2935 (1997).
[Crossref]

Ido, Tetsuya

Hideyuki Kunugita, Tetsuya Ido, and Fujio Shimizu, “Ionizing Collisional Rate of Metastable Rare-Gas Atoms in an Optical Lattice,” Phys. Rev. Lett. 79, 621–624 (1997).
[Crossref]

Jaksch, D.

D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of Atoms via Cold Controlled Collisions,” Phys. Rev. Lett. 82, 1975–1978 (1999).
[Crossref]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
[Crossref]

Janicke, U.

Tomographic TOF techniques exist which do not lose this phase information. U. Janicke and M. Wilkens, “Tomography of atom beams,” J. Mod. Opt.42, 2183–2199 (1995). Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature386, 150–153 (1997).
[Crossref]

Jessen, P. S.

S. E. Hamann, D. L. Haycock, G. Klose, P. H. Pax, I. H. Deutsch, and P. S. Jessen, “Resolved-Sideband Raman Cooling to the Ground State of an Optical Lattice,” Phys. Rev. Lett. 80, 4149–4152 (1998).
[Crossref]

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

Jessen, Poul S.

Gavin K. Brennen, Cartlon M. Caves, Poul S. Jessen, and Ivan H. Deutsch, “Quantum Logic Gates in Optical Lattices,” Phys. Rev. Lett. 82, 1060–1063 (1999).
[Crossref]

Ivan H. Deutsch and Poul S. Jessen, “Quantum-state control in optical lattices,” Phys. Rev. A 57, 1972–1986 (1998).
[Crossref]

Julienne, P. S.

C. Orzel, M. Walhout, U. Sterr, P. S. Julienne, and S. L. Rolston, “Spin polarization and quantum-statistical effects in ultracold ionizing collisions,” Phys. Rev. A 59, 1926–1935 (1999).
[Crossref]

Kasevich, M. A.

B. P. Anderson and M. A. Kasevich, “Macroscopic Quantum Interference from Atomic Tunnel Arrays,” Science 282, 1686–1689 (1998).
[Crossref] [PubMed]

Ketterle, W.

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of Interference Between Two Bose Condensates,” Science 275, 637–641 (1997).
[Crossref] [PubMed]

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Direct, Nondestructive Observation of a Bose Condensate,” Science 273, 84–87 (1996)
[Crossref] [PubMed]

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, and W. Ketterle, “Bose-Einstein Condensation in a Gas of Sodium Atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

Kim, Jungsang

William D. Oliver, Jungsang Kim, Robert C. Liu, and Yoshihisa Yamamoto, “Hanbury Brown and Twiss-Type Experiment with Electrons,” Science 284, 299–301 (1999).
[Crossref] [PubMed]

Klose, G.

S. E. Hamann, D. L. Haycock, G. Klose, P. H. Pax, I. H. Deutsch, and P. S. Jessen, “Resolved-Sideband Raman Cooling to the Ground State of an Optical Lattice,” Phys. Rev. Lett. 80, 4149–4152 (1998).
[Crossref]

Kulin, S.

B. Saubaméa, T. W. Hijmans, S. Kulin, E. Rasel, E. Peik, M. Leduc, and C. Cohen-Tannoudji, “Direct Measurement of the Spatial Correlation Function of Ultracold Atoms,” Phys. Rev. Lett. 79, 3146–3149 (1997).
[Crossref]

Kunugita, Hideyuki

Hideyuki Kunugita, Tetsuya Ido, and Fujio Shimizu, “Ionizing Collisional Rate of Metastable Rare-Gas Atoms in an Optical Lattice,” Phys. Rev. Lett. 79, 621–624 (1997).
[Crossref]

Kurn, D. M.

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of Interference Between Two Bose Condensates,” Science 275, 637–641 (1997).
[Crossref] [PubMed]

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Direct, Nondestructive Observation of a Bose Condensate,” Science 273, 84–87 (1996)
[Crossref] [PubMed]

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, and W. Ketterle, “Bose-Einstein Condensation in a Gas of Sodium Atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

Kurtsiefer, Ch.

Tomographic TOF techniques exist which do not lose this phase information. U. Janicke and M. Wilkens, “Tomography of atom beams,” J. Mod. Opt.42, 2183–2199 (1995). Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature386, 150–153 (1997).
[Crossref]

Laloë, Franck

Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë, Quantum MechanicsVol. 1(John Wiley & Sons, New York, 1977).

Leduc, M.

B. Saubaméa, T. W. Hijmans, S. Kulin, E. Rasel, E. Peik, M. Leduc, and C. Cohen-Tannoudji, “Direct Measurement of the Spatial Correlation Function of Ultracold Atoms,” Phys. Rev. Lett. 79, 3146–3149 (1997).
[Crossref]

Lett, P. D.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[Crossref] [PubMed]

Liu, Robert C.

William D. Oliver, Jungsang Kim, Robert C. Liu, and Yoshihisa Yamamoto, “Hanbury Brown and Twiss-Type Experiment with Electrons,” Science 284, 299–301 (1999).
[Crossref] [PubMed]

Loudon, Rodney

Rodney Loudon, The Quantum Theory of Light, Second Edition (Oxford University Press, New York, 1983).

Lukman Winoto, S.

S. Lukman Winoto, Marshall T. DePue, Nathan E. Bramall, and David S. Weiss, “Laser cooling at high density in deep far-detuned optical lattices,” Phys. Rev. A 59, R19–R22 (1999).
[Crossref]

Matthews, M. R.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” Science 269, 198–201 (1995).
[Crossref] [PubMed]

Metcalf, H. J.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[Crossref] [PubMed]

Mewes, M. O.

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, and W. Ketterle, “Bose-Einstein Condensation in a Gas of Sodium Atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

Mewes, M.-O.

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Direct, Nondestructive Observation of a Bose Condensate,” Science 273, 84–87 (1996)
[Crossref] [PubMed]

Meystre, P.

E. V. Goldstein, O. Zobay, and P. Meystre, “Coherence of atom matter-wave fields,” Phys. Rev. A 58, 2373–2384 (1998).
[Crossref]

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

Miesner, H.-J.

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of Interference Between Two Bose Condensates,” Science 275, 637–641 (1997).
[Crossref] [PubMed]

Mlynek, J.

Tomographic TOF techniques exist which do not lose this phase information. U. Janicke and M. Wilkens, “Tomography of atom beams,” J. Mod. Opt.42, 2183–2199 (1995). Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature386, 150–153 (1997).
[Crossref]

Molmer, Klaus

Anders Sorensen and Klaus Molmer, “Spin-Spin Interactions and Spin Squeezing in an Opical Lattice,” Phys. Rev. Lett. 83, 2274–2277 (1999).

Mølmer, Klaus

Kirstine Berg-Sørenson and Klaus Mølmer, “Bose-Einstein condensates in spatially periodic potentials,” Phys. Rev. A 58, 1480–1484 (1998).
[Crossref]

Niu, Qian

Dai-Il Choi and Qian Niu, “Bose-Einstein Condensates in an Optical Lattice,” Phys. Rev. Lett. 82, 2022–2025 (1999).
[Crossref]

Qian Niu, Xian-Gen Zhao, G. A. Georgakis, and M. G. Raizen, “Atomic Landau-Zener Tunneling and Wannier Stark Ladders in Opical Potenitals,” Phys. Rev. Lett. 76, 4504–4507 (1996).
[Crossref] [PubMed]

Oberholzer, S.

M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland, and C. Schönenberger, “The Fermionic Hanbury Brown and Twiss Experiment,” Science 284, 296–298 (1999).
[Crossref] [PubMed]

Oliver, William D.

William D. Oliver, Jungsang Kim, Robert C. Liu, and Yoshihisa Yamamoto, “Hanbury Brown and Twiss-Type Experiment with Electrons,” Science 284, 299–301 (1999).
[Crossref] [PubMed]

Orzel, C.

C. Orzel, M. Walhout, U. Sterr, P. S. Julienne, and S. L. Rolston, “Spin polarization and quantum-statistical effects in ultracold ionizing collisions,” Phys. Rev. A 59, 1926–1935 (1999).
[Crossref]

Pax, P.

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

Pax, P. H.

S. E. Hamann, D. L. Haycock, G. Klose, P. H. Pax, I. H. Deutsch, and P. S. Jessen, “Resolved-Sideband Raman Cooling to the Ground State of an Optical Lattice,” Phys. Rev. Lett. 80, 4149–4152 (1998).
[Crossref]

Peik, E.

B. Saubaméa, T. W. Hijmans, S. Kulin, E. Rasel, E. Peik, M. Leduc, and C. Cohen-Tannoudji, “Direct Measurement of the Spatial Correlation Function of Ultracold Atoms,” Phys. Rev. Lett. 79, 3146–3149 (1997).
[Crossref]

Pfau, T.

Tomographic TOF techniques exist which do not lose this phase information. U. Janicke and M. Wilkens, “Tomography of atom beams,” J. Mod. Opt.42, 2183–2199 (1995). Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature386, 150–153 (1997).
[Crossref]

Phillips, W. D.

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, “Bragg Scattering from Atoms in Optical Lattices,” Phys. Rev. Lett. 75, 2823–2826 (1995).
[Crossref] [PubMed]

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[Crossref] [PubMed]

Raithel, G.

S. K. Dutta, B. K. Teo, and G. Raithel, “Tunneling Dynamics and Guage Potentials in Optical Lattices,” Phys. Rev. Lett. 83, 1093–1936 (1999).
[Crossref]

Raizen, M. G.

Qian Niu, Xian-Gen Zhao, G. A. Georgakis, and M. G. Raizen, “Atomic Landau-Zener Tunneling and Wannier Stark Ladders in Opical Potenitals,” Phys. Rev. Lett. 76, 4504–4507 (1996).
[Crossref] [PubMed]

Rasel, E.

B. Saubaméa, T. W. Hijmans, S. Kulin, E. Rasel, E. Peik, M. Leduc, and C. Cohen-Tannoudji, “Direct Measurement of the Spatial Correlation Function of Ultracold Atoms,” Phys. Rev. Lett. 79, 3146–3149 (1997).
[Crossref]

Rolston, S. L.

C. Orzel, M. Walhout, U. Sterr, P. S. Julienne, and S. L. Rolston, “Spin polarization and quantum-statistical effects in ultracold ionizing collisions,” Phys. Rev. A 59, 1926–1935 (1999).
[Crossref]

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, “Bragg Scattering from Atoms in Optical Lattices,” Phys. Rev. Lett. 75, 2823–2826 (1995).
[Crossref] [PubMed]

Saubaméa, B.

B. Saubaméa, T. W. Hijmans, S. Kulin, E. Rasel, E. Peik, M. Leduc, and C. Cohen-Tannoudji, “Direct Measurement of the Spatial Correlation Function of Ultracold Atoms,” Phys. Rev. Lett. 79, 3146–3149 (1997).
[Crossref]

Schönenberger, C.

M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland, and C. Schönenberger, “The Fermionic Hanbury Brown and Twiss Experiment,” Science 284, 296–298 (1999).
[Crossref] [PubMed]

Shimizu, Fujio

Hideyuki Kunugita, Tetsuya Ido, and Fujio Shimizu, “Ionizing Collisional Rate of Metastable Rare-Gas Atoms in an Optical Lattice,” Phys. Rev. Lett. 79, 621–624 (1997).
[Crossref]

Masami Yasuda and Fujio Shimizu, “Observation of Two-Atom Correlation of an Ultracold Neon Atomic Beam,” Phys. Rev. Lett. 77, 3090–3093 (1996).
[Crossref] [PubMed]

Sorensen, Anders

Anders Sorensen and Klaus Molmer, “Spin-Spin Interactions and Spin Squeezing in an Opical Lattice,” Phys. Rev. Lett. 83, 2274–2277 (1999).

Sterr, U.

C. Orzel, M. Walhout, U. Sterr, P. S. Julienne, and S. L. Rolston, “Spin polarization and quantum-statistical effects in ultracold ionizing collisions,” Phys. Rev. A 59, 1926–1935 (1999).
[Crossref]

Strunk, C.

M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland, and C. Schönenberger, “The Fermionic Hanbury Brown and Twiss Experiment,” Science 284, 296–298 (1999).
[Crossref] [PubMed]

Teo, B. K.

S. K. Dutta, B. K. Teo, and G. Raithel, “Tunneling Dynamics and Guage Potentials in Optical Lattices,” Phys. Rev. Lett. 83, 1093–1936 (1999).
[Crossref]

Thomas, J. E.

J. E. Thomas and L. J. Wang, “Quantum theory of correlated-atomic-position measurements by resonance imaging,” Phys. Rev. A 49, 558–569 (1994).
[Crossref] [PubMed]

Townsend, C. G.

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of Interference Between Two Bose Condensates,” Science 275, 637–641 (1997).
[Crossref] [PubMed]

van Druten, N. J.

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Direct, Nondestructive Observation of a Bose Condensate,” Science 273, 84–87 (1996)
[Crossref] [PubMed]

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, and W. Ketterle, “Bose-Einstein Condensation in a Gas of Sodium Atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

Walhout, M.

C. Orzel, M. Walhout, U. Sterr, P. S. Julienne, and S. L. Rolston, “Spin polarization and quantum-statistical effects in ultracold ionizing collisions,” Phys. Rev. A 59, 1926–1935 (1999).
[Crossref]

Wang, L. J.

J. E. Thomas and L. J. Wang, “Quantum theory of correlated-atomic-position measurements by resonance imaging,” Phys. Rev. A 49, 558–569 (1994).
[Crossref] [PubMed]

Watts, R. N.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[Crossref] [PubMed]

Weiss, David S.

S. Lukman Winoto, Marshall T. DePue, Nathan E. Bramall, and David S. Weiss, “Laser cooling at high density in deep far-detuned optical lattices,” Phys. Rev. A 59, R19–R22 (1999).
[Crossref]

Westbrook, C. I.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[Crossref] [PubMed]

Wieman, C. E.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” Science 269, 198–201 (1995).
[Crossref] [PubMed]

Wilkens, M.

Tomographic TOF techniques exist which do not lose this phase information. U. Janicke and M. Wilkens, “Tomography of atom beams,” J. Mod. Opt.42, 2183–2199 (1995). Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature386, 150–153 (1997).
[Crossref]

Yamamoto, Yoshihisa

William D. Oliver, Jungsang Kim, Robert C. Liu, and Yoshihisa Yamamoto, “Hanbury Brown and Twiss-Type Experiment with Electrons,” Science 284, 299–301 (1999).
[Crossref] [PubMed]

Yasuda, Masami

Masami Yasuda and Fujio Shimizu, “Observation of Two-Atom Correlation of an Ultracold Neon Atomic Beam,” Phys. Rev. Lett. 77, 3090–3093 (1996).
[Crossref] [PubMed]

Zhao, Xian-Gen

Qian Niu, Xian-Gen Zhao, G. A. Georgakis, and M. G. Raizen, “Atomic Landau-Zener Tunneling and Wannier Stark Ladders in Opical Potenitals,” Phys. Rev. Lett. 76, 4504–4507 (1996).
[Crossref] [PubMed]

Zobay, O.

E. V. Goldstein, O. Zobay, and P. Meystre, “Coherence of atom matter-wave fields,” Phys. Rev. A 58, 2373–2384 (1998).
[Crossref]

Zoller, P.

D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of Atoms via Cold Controlled Collisions,” Phys. Rev. Lett. 82, 1975–1978 (1999).
[Crossref]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
[Crossref]

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

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

Phys. Rev. A (7)

Ivan H. Deutsch and Poul S. Jessen, “Quantum-state control in optical lattices,” Phys. Rev. A 57, 1972–1986 (1998).
[Crossref]

Kirstine Berg-Sørenson and Klaus Mølmer, “Bose-Einstein condensates in spatially periodic potentials,” Phys. Rev. A 58, 1480–1484 (1998).
[Crossref]

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

J. E. Thomas and L. J. Wang, “Quantum theory of correlated-atomic-position measurements by resonance imaging,” Phys. Rev. A 49, 558–569 (1994).
[Crossref] [PubMed]

S. Lukman Winoto, Marshall T. DePue, Nathan E. Bramall, and David S. Weiss, “Laser cooling at high density in deep far-detuned optical lattices,” Phys. Rev. A 59, R19–R22 (1999).
[Crossref]

C. Orzel, M. Walhout, U. Sterr, P. S. Julienne, and S. L. Rolston, “Spin polarization and quantum-statistical effects in ultracold ionizing collisions,” Phys. Rev. A 59, 1926–1935 (1999).
[Crossref]

E. V. Goldstein, O. Zobay, and P. Meystre, “Coherence of atom matter-wave fields,” Phys. Rev. A 58, 2373–2384 (1998).
[Crossref]

Phys. Rev. Lett. (15)

Dai-Il Choi and Qian Niu, “Bose-Einstein Condensates in an Optical Lattice,” Phys. Rev. Lett. 82, 2022–2025 (1999).
[Crossref]

B. Saubaméa, T. W. Hijmans, S. Kulin, E. Rasel, E. Peik, M. Leduc, and C. Cohen-Tannoudji, “Direct Measurement of the Spatial Correlation Function of Ultracold Atoms,” Phys. Rev. Lett. 79, 3146–3149 (1997).
[Crossref]

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1988).
[Crossref] [PubMed]

S. E. Hamann, D. L. Haycock, G. Klose, P. H. Pax, I. H. Deutsch, and P. S. Jessen, “Resolved-Sideband Raman Cooling to the Ground State of an Optical Lattice,” Phys. Rev. Lett. 80, 4149–4152 (1998).
[Crossref]

Masami Yasuda and Fujio Shimizu, “Observation of Two-Atom Correlation of an Ultracold Neon Atomic Beam,” Phys. Rev. Lett. 77, 3090–3093 (1996).
[Crossref] [PubMed]

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, “Bragg Scattering from Atoms in Optical Lattices,” Phys. Rev. Lett. 75, 2823–2826 (1995).
[Crossref] [PubMed]

Hideyuki Kunugita, Tetsuya Ido, and Fujio Shimizu, “Ionizing Collisional Rate of Metastable Rare-Gas Atoms in an Optical Lattice,” Phys. Rev. Lett. 79, 621–624 (1997).
[Crossref]

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, and W. Ketterle, “Bose-Einstein Condensation in a Gas of Sodium Atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[Crossref] [PubMed]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
[Crossref]

Klaus Drese and Martin Holthaus, “Exploring a Metal-Insulator Transition with Ultrcold Atoms in Standing Light Waves,” Phys. Rev. Lett. 2932, 2932–2935 (1997).
[Crossref]

Qian Niu, Xian-Gen Zhao, G. A. Georgakis, and M. G. Raizen, “Atomic Landau-Zener Tunneling and Wannier Stark Ladders in Opical Potenitals,” Phys. Rev. Lett. 76, 4504–4507 (1996).
[Crossref] [PubMed]

S. K. Dutta, B. K. Teo, and G. Raithel, “Tunneling Dynamics and Guage Potentials in Optical Lattices,” Phys. Rev. Lett. 83, 1093–1936 (1999).
[Crossref]

Gavin K. Brennen, Cartlon M. Caves, Poul S. Jessen, and Ivan H. Deutsch, “Quantum Logic Gates in Optical Lattices,” Phys. Rev. Lett. 82, 1060–1063 (1999).
[Crossref]

D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of Atoms via Cold Controlled Collisions,” Phys. Rev. Lett. 82, 1975–1978 (1999).
[Crossref]

Anders Sorensen and Klaus Molmer, “Spin-Spin Interactions and Spin Squeezing in an Opical Lattice,” Phys. Rev. Lett. 83, 2274–2277 (1999).

Science (6)

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” Science 269, 198–201 (1995).
[Crossref] [PubMed]

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Direct, Nondestructive Observation of a Bose Condensate,” Science 273, 84–87 (1996)
[Crossref] [PubMed]

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of Interference Between Two Bose Condensates,” Science 275, 637–641 (1997).
[Crossref] [PubMed]

B. P. Anderson and M. A. Kasevich, “Macroscopic Quantum Interference from Atomic Tunnel Arrays,” Science 282, 1686–1689 (1998).
[Crossref] [PubMed]

M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland, and C. Schönenberger, “The Fermionic Hanbury Brown and Twiss Experiment,” Science 284, 296–298 (1999).
[Crossref] [PubMed]

William D. Oliver, Jungsang Kim, Robert C. Liu, and Yoshihisa Yamamoto, “Hanbury Brown and Twiss-Type Experiment with Electrons,” Science 284, 299–301 (1999).
[Crossref] [PubMed]

Other (5)

Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë, Quantum MechanicsVol. 1(John Wiley & Sons, New York, 1977).

Joseph W. Goodman, Statistical Optics (John Wiley & Sons, New York, 1985).

Tomographic TOF techniques exist which do not lose this phase information. U. Janicke and M. Wilkens, “Tomography of atom beams,” J. Mod. Opt.42, 2183–2199 (1995). Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature386, 150–153 (1997).
[Crossref]

Benjamin Chu, Laser Light Scattering, Second Edition (Academic Press, San Diego, 1991).

Rodney Loudon, The Quantum Theory of Light, Second Edition (Oxford University Press, New York, 1983).

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of a TOF experiment. The atomic wave functions initially consists of a superposition of two Gaussians separated by Δξ′ with relative phase ϕ=0 and probability amplitudes c 1=c 2=1/√2. The resulting completely incoherently overlapping fringe pattern, has a fringe spacing of L 2ξ′.

Fig. 2.
Fig. 2.

Schematic of an experiment that measures same-time two-point first order spatial correlations of an atomic field. By measuring the visibility of the fringes as a function of slit spacing one can deduce the atomic distribution.

Fig. 3.
Fig. 3.

Schematic for an experiment that measures same-time second order spatial correlations of an atomic field. The interference arises from the two possible ways that the atoms can be jointly detected, denoted by the solid and dotted paths.

Fig. 4.
Fig. 4.

Atomic distribution bunched around a chosen “seed point”. First column (a)–(c) shows samples with a fixed seed at site N=128. The second column, (d)–(f), shows the atomic distribution for a randomly varying seed point.

Fig. 5.
Fig. 5.

Probability for an atom to be located a site j, Pj(1) , and for two atoms to be separated by j sites, Pj(2) , for a fixed seed point, (a) and (b), and for a randomly varying seed point, (c) and (d).

Fig. 6.
Fig. 6.

Probability for atoms to be separated by j lattice sites, obtained by Fourier transform of simulated coincidence count measurements. Results are shown for atomic distributions which are (a) a random, (b)“bunched”, (c) “anti-bunched”, and (d) macroscopic variation on a “super-lattice”.

Equations (29)

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Ψ ( x , t ) = d x K ( x , t ; x t ) Ψ ( x , t ) .
K ( x , t ; x t ) = i L exp [ i π ( x x L ) 2 ] ,
Ψ ( x , t ) ( 1 2 π σ 2 ) 1 4 e i π ( x x i L ) 2 e ( x x i 2 σ ) 2 .
Ψ ( x , t ) = i L exp [ i 2 π L 2 ( x 2 2 + x x i ) ] F [ Φ ( x ) ] u = x L 2 .
n ( x ) = j = 1 R p j 1 L 2 F [ Φ j ( x ) ] u = x L 2 2 ,
Ψ ( x ) = c 1 Φ 0 ( x Δ ξ 2 ) + e i ϕ c 2 Φ 0 ( x + Δ ξ 2 )
Ψ ( x ) 2 = 1 2 π σ 2 exp ( 2 ( x 2 σ ) 2 ) ( 1 + 2 c 1 c 2 cos ( 2 π x Δ ξ L 2 + ϕ ) ) ,
V ( Δ x ) = g ( 1 ) ( Δ x / 2 , Δ x 2 )
Ψ ( x , t ) e i k j x where k j = j 2 π w L 2 .
b ̂ i = 1 N j a ̂ j e i k j x i .
g ( 1 ) ( x 1 , x 2 ) = G ( 1 ) ( x 1 , x 2 ) G ( 1 ) ( x 1 , x 1 ) G ( 1 ) ( x 2 , x 2 ) ,
G ( 1 ) ( x 1 , x 2 ) = b ̂ 1 b ̂ 2
= 1 N j , e i ( k j x 2 k x 1 ) a ̂ a ̂ j .
g ( 1 ) ( x 1 , x 2 ) = e i k j ( x 2 x 1 ) .
g ( 1 ) = j = 0 N 1 P j ( 1 ) e i ( k j Δ x ) = j = 0 N 1 P j ( 1 ) e i ( j Δ k Δ x ) ,
P j ( 1 ) = 1 2 π N = 0 N 1 g ( 1 ) e i ( j Δ k Δ x ) .
g ( 2 ) ( x 1 , x 2 ) = G ( 2 ) ( x 1 , x 2 ; x 2 , x 1 ) G ( 1 ) ( x 1 , x 1 ) G ( 1 ) ( x 2 , x 2 ) ,
G ( 2 ) ( x 1 , x 2 ; x 2 , x 1 ) = b ̂ 1 b ̂ 2 b ̂ 2 b ̂ 1
= 1 N 2 j , j , , e i ( ( k j k j ) x 1 + ( k k ) x 2 ) a ̂ j a ̂ a ̂ a ̂ j .
R 2 R ( R 1 ) g ( 2 ) 1 = j = 0 N 1 P j ( 2 ) cos ( j Δ k Δ x ) .
g ( 2 ) 1 = j = N N 1 P j ( 2 ) 2 e i j Δ k Δ x
P j ( 2 ) 2 = 1 2 π ( 2 N ) = N N 1 ( g ( 2 ) 1 ) e i j Δ k Δ x .
P ( n ) ( x 1 , x 2 , , x n ) = 1 k 1 , 1 k 2 , , 1 k n ρ ̂ 1 k 1 , 1 k 2 , , 1 k n ,
P j ( 2 ) = = 1 N P ( 2 ) ( x + j w , x ) .
P j ( 2 ) = = 1 N P ( 1 ) P ( x + j w x ) ,
P j ( 2 ) = = 1 N P ( 1 ) P + j ( 1 ) .
P ( x x ) = 1 2 π τ e 2 ( x x 2 τ ) 2 .
P ( x x ) = f ( x x ) .
P j ( 2 ) = P ( 1 ) f ( x + j w x ) = f ( j w ) .

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