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arXiv:1101.2385 · Physical Review A

Atomic matter-wave revivals with definite atom number in an optical lattice

Johannes Schachenmayer, Andrew J. Daley, P. Zoller

Published Apr 18, 2011• 28 citations
arXivHTMLPDF

Abstract

We study the collapse and revival of interference patterns in the momentum distribution of atoms in optical lattices using a projection technique to properly account for the fixed total number of atoms in the system. We consider the common experimental situation in which weakly interacting bosons are loaded into a shallow lattice, which is suddenly made deep. The collapse and revival of peaks in the momentum distribution is then driven by interactions in a lattice with essentially no tunneling. The projection technique allows to us to treat inhomogeneous (trapped) systems exactly in the case that noninteracting bosons are loaded into the system initially, and we use time-dependent density-matrix renormalization group techniques to study the system in the case of finite tunneling in the lattice and finite initial interactions. For systems of more than a few sites and particles, we find good agreement with results calculated via a naive approach, in which the state at each lattice site is described by a coherent state in the particle occupation number. However, for systems on the order of ten lattice sites, we find experimentally measurable discrepancies to the results predicted by this standard approach.

DOI
10.1103/physreva.83.043614

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We study the collapse and revival of interference patterns in the momentum distribution of atoms in optical lattices using a projection technique to properly account for the fixed total number of atoms in the system. We consider the common experimental situation in which weakly interacting bosons are loaded into a shallow lattice, which is suddenly made deep. The collapse and revival of peaks in the momentum distribution is then driven by interactions in a lattice with essentially no tunneling. The projection technique allows to us to treat inhomogeneous (trapped) systems exactly in the case that noninteracting bosons are loaded into the system initially, and we use time-dependent density-matrix renormalization group techniques to study the system in the case of finite tunneling in the lattice and finite initial interactions. For systems of more than a few sites and particles, we find good agreement with results calculated via a naive approach, in which the state at each lattice site is described by a coherent state in the particle occupation number. However, for systems on the order of ten lattice sites, we find experimentally measurable discrepancies to the results predicted by this standard approach.
Quantum-phase collapse and revival are studied in a binary mixture of bosonic atomic species loaded in a one-dimensional optical lattice. Oscillations in the quasimomentum distribution are usually observed after a sudden quench of the lattice parameters from the superfluid regime to the Mott-insulator regime. The influence of one species dynamics on the collapse and revival oscillations in the quasimomentum distribution of the second species is examined. The nontrivial dependencies on the interspecies and intraspecies interactions are illustrated distinguishing between the zero-tunneling and nonzero-tunneling cases.
Motivated by the experimental observation of collapses and revivals of Bose matter wave field, we investigate correlation functions of cold bosons in an optical lattice. Within a simple model we examine two kinds of states: one that employs the commonly used notion of coherent states, and one that obeys the total number of atoms conservation. We identify rare situations at which these states behave differently. Typically, however, their predictions coincide and so: As a function of ``interaction time'' the interference pattern in the density undergoes collapse and revival. Exactly at revival times the system mimics the ideal gas case, in which all correlation functions factorize, while in the collapsed phase of the evolution the system effectively behaves as if initially there was no long-range coherence. Even in the latter case though, an interference pattern should be seen in a single experiment. We stress the role of column averaging, which in fact corresponds to an averaged observation of an ensemble of two-dimensional realizations. We also note that, contrary to the common belief, an interference pattern should also be seen in a single observation of a Mott state.
Using the quantum collapse and revival phenomenon of a Bose--Einstein condensate in three-dimensional optical lattices, the atom number statistics on each lattice site are experimentally investigated. We observe an interaction driven time evolution of on-site number fluctuations in a constant lattice potential with the collapse and revival time ratio as the figure of merit. Through a shortcut loading procedure, we prepare a three-dimensional array of coherent states with Poissonian number fluctuations. The following dynamics clearly show the interaction effect on the evolution of the number fluctuations from Poissonian to sub-Poissonian. Our method can be used to create squeezed states, which are important in precision measurement.
Recent experiments have shown a remarkable number of collapse and revival oscillations of the matter-wave coherence of ultracold atoms in optical lattices [Will et al., Nature (London) 465, 197 (2010)]. Using a mean-field approximation to the Bose-Hubbard model, we show that the visibility of collapse and revival interference patterns reveals number squeezing of the initial superfluid state. To describe the dynamics, we use an effective Hamiltonian that incorporates the intrinsic two-body and induced three-body interactions, and we analyze in detail the resulting complex pattern of collapse and revival frequencies generated by virtual transitions to higher bands, as a function of lattice parameters and mean-atom number. Our work shows that a combined analysis of both the multiband, nonstationary dynamics in the final deep lattice and the number squeezing of the initial superfluid state explains important characteristics of optical lattice collapse and revival physics. Finally, by treating the two- and three-body interaction strengths and the coefficients describing the initial superposition of number states as free parameters in a fit to the experimental data, it should be possible to go beyond some of the limitations of our model and obtain insight into the breakdown of the mean-field theory for the initial state or the role of nonperturbative effects in the final-state dynamics.
We show that collapse and revival experiments with interacting atoms trapped in an optical lattice, based on a suggestion by [1], are a sensitive tool for characterizing its many-body ground state. Collapse and revival experiments involve sudden increases of the lattice depth whereby the system transfers from an initial superfluid ground state to a non-stationary state for a lattice depth where tunneling between sites is negligible. The state then evolves independently in each site for a variable amount of time, after which the momentum distribution is measured. The time evolution of momentum states are then sensitive to the amplitude of the initial atom-number Fock states, and, hence, contain information about the superfluid ground state.
We study the dynamics of spin-1 atoms in a periodic optical-lattice potential and an external magnetic field in a quantum quench scenario where we start from a superfluid ground state in a shallow lattice potential and suddenly raise the lattice depth. The time evolution of the nonequilibrium state shows collective collapse-and-revival oscillations of matter-wave coherence as well as oscillations in the spin populations. We show that the complex pattern of these two types of oscillations reveals details about the superfluid and magnetic properties of the initial many-body ground state. Furthermore, we show that the strengths of the spin-dependent and spin-independent atom-atom interactions can be deduced from the observations. The Hamiltonian that describes the physics of the final deep lattice not only contains two-body interactions but also effective multibody interactions, which arise due to virtual excitations to higher bands. We derive these effective spin-dependent three-body interaction parameters for spin-1 atoms and describe how spin mixing is affected. Spinor atoms are unique in the sense that multibody interactions are directly evident in the in situ number densities in addition to the momentum distributions. We treat both antiferromagnetic (e.g., 23{}^{23}23Na) and ferromagnetic (e.g., 87{}^{87}87Rb and 41{}^{41}41K) condensates.