## Spatial correlation diagnostics for atoms in optical lattices

Optics Express, Vol. 5, Issue 11, pp. 249-261 (1999)

http://dx.doi.org/10.1364/OE.5.000249

Acrobat PDF (463 KB)

### 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.

© Optical Society of America

## 1 Introduction

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]

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]

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, and W. Ketterle, “Direct, Nondestructive Observation of a Bose Condensate,” Science **273**, 84–87 (1996) [CrossRef] [PubMed]

*ħk*)

^{2}/(2

*M*) (

*ħk*is the photon momentum and

*M*is the mass of the atom). For cold atomic samples, these “recoil kicks” generally heat the sample very quickly, although in the case of off-resonance spectroscopy this heating has been suppressed by a factor of order 100, allowing for multiple images before the sample is destroyed [18

18. 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]

19. 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]

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]

*t*, after which they are counted with point-like detectors, such as a microchannel plate array (MCP) in the case of meta-stable noble gas atoms [21

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]

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, and C. Schönenberger, “The Fermionic Hanbury Brown and Twiss Experiment,” Science **284**, 296–298 (1999). [CrossRef] [PubMed]

27. 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]

## 2 Spatial Correlation Functions

### 2.1 Local Wave Function via Atomic Density Measurements

*g*

^{(1)}(

*x*,

*x*), is a measure of the atomic density

*n*(

*x*). Such correlations can be measured using the well known time of flight (TOF) technique in which a trapped atomic sample is “suddenly” released from an optical lattice in a time short compared to the characteristic period of oscillation of a trapped atom. The atoms then expand ballistically in free space until they reach the detection plane. The arrival times of the atoms are measured and the initial momentum distribution of the atoms is inferred [23

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]

*x*,

*t*), at the detection plane from the initial wave function, Ψ(

*x′*,

*t′*), with unprimed variables (

*x*,

*t*) denoting space-time coordinates of the detection plane and the primed variables (

*x′*,

*t′*) denoting the optical lattice plane,

*K*(

*x*,

*t*;

*x′*,

*t′*) is given by

*σ*=

*L*

^{2}/(4

*πσ′*) is the width of the Gaussian in the detection plane.

*i*in the state, Φ(

*x′*-

*x′*

_{i},

*t′*=0). The wave function in the far field is,

*u*is the reciprocal coordinate. The mean location of the initial wave packet,

*x′*

_{i}, is mapped onto the phase of the final wave function as a consequence of the well known shift theorem from Fourier analysis. Thus, in the far field the detected signal is proportional to the absolute value squared of the momentum space wave function and is insensitive to the initial position of the atomic wave packet. For a large collection of incoherent atoms, each in an arbitrary mixed state, the measured TOF signal is given by a statistical average,

*p*

_{j}is the classical probability for the

*j*

^{th}local wave function to occur. If the atomic sample consists of

*R*localized atoms in quantum mechanical pure states which differ only by translation to a given lattice site, as recently demonstrated by Hamman

*et al*. [29

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]

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,” Nature386, 150–153 (1997). [CrossRef]

*N*double wells using a configuration of counterpropagating lasers with wavelength λ whose linear polarization have a relative angle

*θ*[2

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

*i*

^{th}double well is located at

*x′*

_{i}and the well separation is set by Δ

*ξ′*=(λ/(2π)) tan

^{-1}(tan(

*θ*)/2). With the appropriate cooling and preparation of the initial state, the wave packet dynamics is essentially restricted to a two dimensional Hilbert space spanned by two macroscopically separated Gaussians Φ

_{0}of width,

*σ′*, centered about, ±Δ

*ξ′*/2. A general state is given by the wave packet,

*c*

_{1}and

*c*

_{2}are the (real) probability amplitudes and

*ϕ*is the relative phase between the Gaussians. If this wave packet is allowed to freely expand, the atomic density at the detector plane will have a Gaussian envelope with fringes whose spacing is given by,

*d*

_{f}=

*L*

^{2}/Δ

*ξ′*, (See Fig. 1),

*σ*is defined above. Observation of the fringes requires that the size of the sample,

*S*, be much smaller than the fringe separation,

*S*≪

*d*

_{f}(i. e. we are sufficiently in the far field). In the absence of decoherence the complete initial wave function can be inferred. The relative phase can be deduced from the shift of the center of the diffraction pattern with respect to the fringe envelope and the probability amplitudes can be obtained from the visibility of the fringes as a function of time. In the presence of decoherence the visibility of these fringes will decay with time and will not exhibit recurrences characteristic of the macroscopic superposition state. This could be a useful diagnostic to measure the decohering effects of the lattice environment [2

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

32. 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]

### 2.2 Spatial Distribution via First Order Visibility Measurements

*g*

^{(1)}(

*x*

_{1},

*x*

_{2}), defined as the

*complex degree of coherence*[30]. The modulo of this function,

*x*=

*x*

_{2}-

*x*

_{1}. In Michelson stellar interferometry one measures the visibility of the interference pattern as a function of Δ

*x*to deduce the spatial intensity distribution of the source [30]. We consider the atom-optic version here as a diagnostic of the distribution of atoms throughout the lattice (i.e. the small scale structure).

*N*sites with lattice constant

*w′*. For simplicity we take the wells to be harmonic at each site and the atomic state to be thermal so that the local wave function is Gaussian. An atom in the vibrational ground state, initially located at

*x′*

_{j}=

*jw′*(

*j*=0, …,

*N*-1), will expand into an approximate plane wave in the far field (See Fig. 2),

*N*discrete plane waves with a mode spacing of Δ

*k*=2

*πw′*/

*L*

^{2}. We define creation and annihilation operators,

*â*

_{j}, for the

*j*

^{th}mode. The position space annihilation operator

*b̂*

_{i}at detector position

*x*

_{i}is given by,

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

*z*-direction divided by the group velocity. Furthermore, if we assume that an equal flux impinges on each slit, then the complex degree of coherence is [34],

*x′*

_{j}. The state of this system is described by the one-atom Fock state,

*exactly*reproduced for each run of the experiment. In general, however, the atomic distribution will vary from shot-to-shot, causing uncontrollable phase shifts in the fringe pattern which will wash out the spatial information, even for atomic samples that are statistically similar. By contrast, the second order correlation function is more robust because it is insensitive to this phase. An analogous situation exists in optics. The Michelson stellar interferometer which measures first-order field correlations is very sensitive to atmospheric fluctuations whereas the Hanbury-Brown Twiss interferometer which measures intensity correlations is more stable [30]. For this reason we consider higher order spatial correlations.

### 2.3 Spatial Distribution via Coincidence Count Measurements

*x*to deduce the spatial intensity distribution of the source [30]. The normalized same-time second order spatial correlation function at different positions,

*g*

^{(2)}(

*x*

_{1},

*x*

_{2}), corresponds to atom coincidence counts between two point-like detectors located at

*x*

_{1}and

*x*

_{2}in the detection plane [34] (See Fig. 3),

*x′*

_{j}and

*x′*

_{j}+

*w′*, with state vector,

*k*=2

*πw′*/

*L*

^{2}. Inserting this in Eq. (15) gives, 2

*g*

^{(2)}(

*x*

_{1},

*x*

_{2})-1=cos(Δ

*k*(

*x*

_{2}-

*x*

_{1})). The difference in position between the two atoms maps onto the spatial period of the cosinusoidially varying coincidence counts. For Fermions, the fringes receive a

*π*phase shift due to their anticommutation relations. The interference exists even with

*no quantum entanglement*between the two atoms. The interference term arises from the two possible indistinguishable paths that lead to joint detection (See Fig. 3). This relation can be generalized for field states of

*R*atoms distributed throughout

*N*lattice sites. We define

*ℓ*Δ

*x*(

*ℓ*=0,…,

*N*-1), given the state

*j*lattice constants

*jw′*(

*j*=0,…,

*N*-1), or equivalently, for two atomic plane waves to impinge the detector plane (2) with a mode spacing of

*j*Δ

*k*. We see that

*R*=2, reflecting the perfect second-order coherence for the state of exactly two atoms and

*R*→∞, which is the usual bunching factor associated with a highly chaotic macroscopic distribution [34]. Note,

*R*and analytically extending the sum to negative values of

*j*(

*j*=-

*N*,…,

*N*-1), since

*g*

^{(2)}is symmetric under reflections about zero, we obtain the following discrete Fourier relation,

*w′*in the optical lattice plane will result in a coincidence count period of, Λ=

*L*

^{2}/

*w′*. The smallest atomic separation in the lattice plane that can be resolved is determined by the width of the Gaussian envelope in the detection plane which modulates the interference fringes. For the situation of cesium atomic wave packets freely expanding for ~1 sec,

*w′*⋍0.5

*µm*, and

*σ′*⋍30 nm, then Λ⋍

*σ*⋍1 cm (where

*σ′*and

*σ*were defined in Sec. 2.1). Also, for any case of practical significance, the small scale information will be contained in the “wings” of the Gaussian in the detection plane where the phase varies quadratically. The above Fourier relations will still be valid, however, if the detectors are located on the wave front of constant phase, since the Van-Cittert Zernike theorem is valid under a Fresnel approximation [30]. This can be accomplished with a curved paraboloid detection surface, symmetric coincidence counts, or a flat detection surface with appropriate electronic time delays introduced. Finally these relations are completely separable and can be extended to three dimensions. For the

*z*-direction (the direction that the atoms are falling) same-time spatial coincidences are replaced by same-position, temporal coincidences, with the transformation given by the appropriate dynamical equations. If the atoms are falling under the influence of gravity, then

*x*=

*gt*

^{2}/2, and a detector spacing of Δ

*x*corresponds to a detection delay time of

*gt*Δ

*t*.

## 3 Results

*ρ̂*, offers a complete description of the atomic distribution. For our assumptions about the nature of the sample with no coherences between different lattice site, we need only consider diagonal matrix elements,

*g*

^{(1)}(

*x*

_{1},

*x*

_{2}) allows for a direct measure of

*x′*

_{j}, and

*g*

^{(2)}(

*x*

_{1},

*x*

_{2}) provides for a direct measure of the

*j*lattice spacings,

*w′*, independent of the absolute location of the pair,

*P*(

*x′*

_{ℓ}+

*jw′*|

*x′ℓ*) is the conditional probability for an atom to be located at

*x′*

_{ℓ}+

*jw′*given that an atom is at

*x′*

_{ℓ}. Only in the special case that the lattice sites are statistically independent so that

*P*(

*x′*

_{ℓ}+

*jw′*|

*x′*

_{ℓ})=

*x′*

_{ℓ}was picked and then the atomic distribution was conditioned on it in such a way as to cluster around it,

*N*=128 for each run of the gedenken experiment and in the second we let the initial seed point vary randomly (See Fig. 4).

*g*

^{(1)}(

*x*

_{1},

*x*

_{2}) depend on the absolute locations of atoms and on average it sees a randomly filled lattice. In contrast,

*g*

^{(2)}(

*x*

_{1},

*x*

_{2}) measures only the relative locations of atoms independent of the absolute location of the cluster.

*g*

^{(2)}(

*x*

_{1},

*x*

_{2}), is a direct measure of the relative conditional probabilities of the lattice, independent of global properties of the lattice such as intensity or magnetic field inhomogeneities.

## 4 Summary

*g*

^{(1)}(

*x*

_{1},

*x*

_{2}). In contrast,

*g*

^{(2)}(

*x*

_{1},

*x*

_{2}) is more robust and can be a more useful diagnostic. We have performed computer simulations of these detection schemes which illustrate the salient features.

## Acknowledgements

## References and links

1. | P. S. Jessen and I. H. Deutsch, “Optical Lattices,” Adv. At. Mol. Opt. Phys. |

2. | Ivan H. Deutsch and Poul S. Jessen, “Quantum-state control in optical lattices,” Phys. Rev. A |

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. |

4. | S. K. Dutta, B. K. Teo, and G. Raithel, “Tunneling Dynamics and Guage Potentials in Optical Lattices,” Phys. Rev. Lett. |

5. | Gavin K. Brennen, Cartlon M. Caves, Poul S. Jessen, and Ivan H. Deutsch, “Quantum Logic Gates in Optical Lattices,” Phys. Rev. Lett. |

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. |

7. | Anders Sorensen and Klaus Molmer, “Spin-Spin Interactions and Spin Squeezing in an Opical Lattice,” Phys. Rev. Lett. |

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 |

9. | B. P. Anderson and M. A. Kasevich, “Macroscopic Quantum Interference from Atomic Tunnel Arrays,” Science |

10. | Dai-Il Choi and Qian Niu, “Bose-Einstein Condensates in an Optical Lattice,” Phys. Rev. Lett. |

11. | Kirstine Berg-Sørenson and Klaus Mølmer, “Bose-Einstein condensates in spatially periodic potentials,” Phys. Rev. A |

12. | E. V. Goldstein, P. Pax, and P. Meystre, “Dipole-dipole in three-dimensional optical lattices,” Phys. Rev. A |

13. | D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. |

14. | Klaus Drese and Martin Holthaus, “Exploring a Metal-Insulator Transition with Ultrcold Atoms in Standing Light Waves,” Phys. Rev. Lett. |

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 |

16. | J. E. Thomas and L. J. Wang, “Quantum theory of correlated-atomic-position measurements by resonance imaging,” Phys. Rev. A |

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. |

18. | 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 |

19. | 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 |

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. |

21. | Hideyuki Kunugita, Tetsuya Ido, and Fujio Shimizu, “Ionizing Collisional Rate of Metastable Rare-Gas Atoms in an Optical Lattice,” Phys. Rev. Lett. |

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 |

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. |

24. | Benjamin Chu, |

25. | Masami Yasuda and Fujio Shimizu, “Observation of Two-Atom Correlation of an Ultracold Neon Atomic Beam,” Phys. Rev. Lett. |

26. | M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland, and C. Schönenberger, “The Fermionic Hanbury Brown and Twiss Experiment,” Science |

27. | William D. Oliver, Jungsang Kim, Robert C. Liu, and Yoshihisa Yamamoto, “Hanbury Brown and Twiss-Type Experiment with Electrons,” Science |

28. | Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë, |

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. |

30. | Joseph W. Goodman, |

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,” Nature386, 150–153 (1997). [CrossRef] |

32. | 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. |

33. | E. V. Goldstein, O. Zobay, and P. Meystre, “Coherence of atom matter-wave fields,” Phys. Rev. A |

34. | Rodney Loudon, |

**OCIS Codes**

(020.0020) Atomic and molecular physics : Atomic and molecular physics

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(110.0110) Imaging systems : Imaging systems

(140.3320) Lasers and laser optics : Laser cooling

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 8, 1999

Published: November 22, 1999

**Citation**

John Grondalski, Paul Alsing, and Ivan Deutsch, "Spatial correlation diagnostics for atoms in optical lattices," Opt. Express **5**, 249-261 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-5-11-249

Sort: Journal | Reset

### References

- 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).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.