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.

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  1. P. S. Jessen and I. H. Deutsch, "Optical Lattices," Adv. At. Mol. Opt. Phys. 37, 95-138 (1996).
    [CrossRef]
  2. Ivan H. Deutsch and Poul S. Jessen, "Quantum-state control in optical lattices," Phys. Rev. A 57, 1972-1986 (1998).
    [CrossRef]
  3. 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]
  4. 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]
  5. 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]
  6. 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]
  7. Anders Sorensen and Klaus Molmer, "Spin-Spin Interactions and Spin Squeezing in an Opical Lattice," Phys. Rev. Lett. 83, 2274-2277 (1999).
  8. 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]
  9. B. P. Anderson and M. A. Kasevich, "Macroscopic Quantum Interference from Atomic Tunnel Arrays," Science 282, 1686-1689 (1998).
    [CrossRef] [PubMed]
  10. Dai-Il Choi and Qian Niu, "Bose-Einstein Condensates in an Optical Lattice," Phys. Rev. Lett. 82, 2022-2025 (1999).
    [CrossRef]
  11. Kirstine Berg-Sorenson and Klaus Molmer, "Bose-Einstein condensates in spatially periodic potentials," Phys. Rev. A 58, 1480-1484 (1998).
    [CrossRef]
  12. E. V. Goldstein, P. Pax, and P. Meystre, "Dipole-dipole in three-dimensional optical lattices," Phys. Rev. A 53, 2604-2615 (1996).
    [CrossRef] [PubMed]
  13. 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]
  14. 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]
  15. 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]
  16. 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]
  17. 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]
  18. M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, "Direct, Nondestructive Observation of a Bose Condensate," Science 273, 84-87 (1996).
    [CrossRef] [PubMed]
  19. M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, W. Ketterle, "Observation of Interference Between Two Bose Condensates," Science 275, 637-641 (1997).
    [CrossRef] [PubMed]
  20. 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]
  21. 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]
  22. 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]
  23. 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]
  24. Benjamin Chu, Laser Light Scattering, Second Edition (Academic Press, San Diego, 1991).
  25. 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]
  26. M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland, C. Schonenberger, "The Fermionic Hanbury Brown and Twiss Experiment," Science 284, 296-298 (1999).
    [CrossRef] [PubMed]
  27. William D. Oliver, Jungsang Kim, Robert C. Liu, Yoshihisa Yamamoto, "Hanbury Brown and Twiss-Type Experiment with Electrons," Science 284, 299-301 (1999).
    [CrossRef] [PubMed]
  28. Claude Cohen-Tannoudji, Bernard Diu, Franck Lalo�e, Quantum Mechanics Vol. 1 (John Wiley & Sons, New York, 1977).
  29. 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]
  30. Joseph W. Goodman, Statistical Optics (John Wiley & Sons, New York, 1985).
  31. 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," Nature 386, 150-153 (1997).
    [CrossRef]
  32. B. Saubamea, 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]
  33. E. V. Goldstein, O. Zobay, and P. Meystre, "Coherence of atom matter-wave fields," Phys. Rev. A 58, 2373-2384 (1998).
    [CrossRef]
  34. Rodney Loudon, The Quantum Theory of Light, Second Edition (Oxford University Press, New York, 1983).

Other

P. S. Jessen and I. H. Deutsch, "Optical Lattices," Adv. At. Mol. Opt. Phys. 37, 95-138 (1996).
[CrossRef]

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

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]

B. P. Anderson and M. A. Kasevich, "Macroscopic Quantum Interference from Atomic Tunnel Arrays," Science 282, 1686-1689 (1998).
[CrossRef] [PubMed]

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

Kirstine Berg-Sorenson and Klaus Molmer, "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]

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]

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]

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]

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]

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, 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, W. Ketterle, "Observation of Interference Between Two Bose Condensates," Science 275, 637-641 (1997).
[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]

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]

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]

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

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]

M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland, C. Schonenberger, "The Fermionic Hanbury Brown and Twiss Experiment," Science 284, 296-298 (1999).
[CrossRef] [PubMed]

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

Claude Cohen-Tannoudji, Bernard Diu, Franck Lalo�e, Quantum Mechanics Vol. 1 (John Wiley & Sons, New York, 1977).

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]

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," Nature 386, 150-153 (1997).
[CrossRef]

B. Saubamea, 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]

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

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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