arXiv:0803.2922v2 [cond-mat.other] 12 Jul 2008Theory ofcorrelations between ultra-cold bosons released from an optical latticeE. Toth1, A. M. Rey2and P. B. Blakie11Jack Dodd Centre for Quantum Technology, Department ofPhysics, University ofOtago, Dunedin, New Zealand and2Institute for Theoretical Atomic, Molecular and Optical Physics, Cambridge, MA, 02138.(Dated: July 12, 2008)In this paper we develop a theoretical description of the correlations between ultra-cold bosons after freeexpansion from confinement in an optical lattice. We consider the system evolution during expansion and givecriteria for a far field regime. We develop expressions for first and second order two-point correlations basedon a variety of commonly used approximations to the many-body state of the system including Bogoliubov,meanfield decoupling, and particle-hole perturbative solution about the perfect Mott-insulator state. Using theseapproaches we examine the effects of quantum depletion and pairing on the system correlations. Comparisonwith the directly calculated correlation functions is used to justify a Gaussian form of our theory from whichwe develop a general three-dimensional formalism for inhomogeneous lattice systems suitable for numericalcalculations of realistic experimental regimes.PACS numbers: 03.75.-b, 03.75.HhI.INTRODUCTIONIn the past decade noise correlation analysis, analogous to the pho-ton correlations observed in the landmark experiments of Hanbury-Brown and Twiss [1], has been applied to ultra-cold atom experi-ments. In atomic systems such measurements can be used to revealinformation about interaction-induced (i.e. many-body) correlationsbetween the atoms. In addition, the dramatic differences betweenBose and Fermi statistics have been clearly demonstrated with neu-tral atoms (e.g. see [2]).A wide range of atomic physics experiments examining variousaspects of correlations have been conducted. The first experimentsby Yasuda et al. [3] observed atom bunching using a neutral beamof (bosonic) neon atoms. In quantum degenerate Bose gases localthird-order correlations have been inferred by measuring three-bodydecay rates [4, 5], and first order coherence has been studied usingmatter wave interference [6, 7] and Bragg spectroscopy [8, 9]. Ofmost interest for the investigation in this paper has been the recentexperimental progress in the spatially resolved measurement of sec-ond order correlations in both bosonic and fermionic ultra-cold gases[2, 10, 11, 12, 13, 14]. Two general approaches are used to makethese measurements. One approach involves directly counting atoms[2, 10, 11], while the other uses absorption imaging to measure thedensity [12, 13, 14, 15, 16]. The applications of these measurementshave included Bose and Fermi gases inharmonic traps [2, 10, 11, 13],and in optical lattices [12, 14, 15, 16].In this work we are concerned with the theoretical formalism forthe spatial noise correlations of an ultra-cold Bose gas after expan-sion from an optical lattice, relevant to the experiments reported inRefs. [12, 15, 16]. Initial theoretical work on this subject was pro-vided by Altman et al. [17], who predicted the noise correlationsusing a perfect Mott-insulator approximation (i.e. neglecting tunnel-ing) and assuming a simplified form for the single particle expan-sion. Subsequent experiments in the Mott-insulator regime [12, 15]verified those predictions, in particular the periodic bunching peaksin the noise correlations. Several recent theoretical proposals havebuilt on that framework and investigated the use of noise correla-tions in characterizing many-body states produced in optical lattices[18, 19, 20, 21, 22, 23]. This line of research provides an interestingnew avenue for investigating the effects of interactions which com-pliments the other techniques available such as direct density imag-ing [15, 24, 25], Bragg [26, 27, 28, 29, 30, 31, 32] and Raman spec-troscopy [33, 34, 35].The basic organisation ofthe paper is as follows. In the remainderof this section we introduce the system of interest and give an intro-duction to how the far-field correlations are determined. In Sec. II wederive the properties ofsingle particle expansion from the lattice anduse this to derive a simplified far field form and its validity condi-tions. We discuss the correlation function formalism and its relation-ship to the Bose-Hubbard Hamiltonian in Sec. III. The main resultsfor expanded correlations functions of a 1D lattice system are devel-oped in Sec. IV using a variety of theoretical approaches. In Sec. Vwe introduce a Gaussian approach which we justify by comparisonto the earlier results. In Sec. VI we extend this Gaussian approach toa general 3D theory for the inhomogeneous lattice system, and thenconclude.A.Optical latticeConsider a system ofbosonic atoms in an optical lattice, describedby the HamiltonianH =Zd3xˆψ†(x) p22m+«3Xj=1V0sin2(kxj) + Vext(x)+U02ˆψ†(x)ˆψ(x)ˆψ(x),(1)whereˆψ(x) is a bosonic field operator, Vext describes any externalpotential (typically harmonic) and U0 = 4πas2/m characterisesthe binary interactions between the particles, with asthe s-wave scat-tering length. The lattice is taken to be simple cubic, with a = π/kand b = 2k the lengths of the direct and reciprocal lattice vectorsalong each direction, where k is the wavelength of light used toproduce the lattice. This lattice is hence of separable form, of thetype produced in experiments with three sets of orthogonal counter-propagating light beams, and the depth along each direction (i.e. V0)is assumed to be the same. The theory we present here can be easilyextended to more general lattice configurations, but we restrict ourattention to this case for notational simplicity. We define the quan-tities ωR = k2/2m and ER = ωR as the recoil frequency andenergy respectively.
