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arXiv:1104.1402 · Physical Review A

Collapse and revival dynamics of number-squeezed superfluids of ultracold atoms in optical lattices

Eite Tiesinga, P. R. Johnson

Published Jun 10, 2011• 31 citations
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Abstract

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.

DOI
10.1103/physreva.83.063609

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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.
The collapses and revivals of a coherent matter wave field of interacting particles can serve as a sensitive interferometric probe of the interactions and the number statistics of the underlying quantum field. Here we show how the ability to observe long time traces of collapse and revival dynamics of a Bose-Einstein condensate loaded into a three-dimensional (3D) optical lattice allowed us to directly reveal the atom number statistics and the presence of effective coherent multi-particle interactions in a lattice. The multi-particle interactions are generated via virtual transitions to higher lattice orbitals and can find use for simulations of effective field theories with ultracold atoms in optical lattices. We measured their absolute strengths up to the case of six-particle interactions and compare our findings with theory.
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.
In this thesis I report on experiments that enter a new regime in the many body physics of ultracold atomic gases. A Bose-Einstein condensate is loaded into a three-dimensional optical lattice potential formed by a standing wave laser light field. In this novel quantum system we have been able to both realize a quantum phase transition from a superfluid to a Mott insulator, and to observe the collapse and revival of a macroscopic matter wave field. Quantum phase transitions are driven by quantum fluctuations and occur, even at zero temperature, as the relative strength of two competing energy terms in the underlying Hamiltonian is varied across a critical value. In the first part of this work I report on the observation of such a quantum phase transition in a Bose-Einstein condensate with repulsive interactions, held in a three-dimensional optical lattice potential. In the superfluid ground state, each atom is spread-out over the entire lattice, whereas in the Mott insulating state, exact numbers of atoms are localized at individual lattice sites. We observed the reversible transition between those states and detected the gap in the excitation spectrum of the Mott insulator. A Bose-Einstein condensate is usually described by a macroscopic matter wave field. However, a quantized field underlies such a "classical" matter wave field of a Bose-Einstein condensate. The striking behavior of ultracold matter due to the field quantization and the nonlinear interactions between the atoms is the focus of the second part of this work. The matter wave field of a Bose-Einstein condensate is observed to undergo a series of collapses and revivals as time evolves. Furthermore, we show that the collisions between individual pairs of atoms lead to a fully coherent collisional phase shift in the corresponding many-particle state, which is a crucial cornerstone of proposed novel quantum computation schemes with neutral atoms. With these experiments we enter a new field of physics with ultracold quantum gases. In this strongly correlated regime, interactions between atoms dominate the behavior of the many-body system such that it can no longer be described by the usual theories for weakly interacting Bose gases. This novel quantum system offers the unique possibility to experimentally address fundamental questions of modern solid state physics, atomic physics, quantum optics, and quantum information.
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.
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.
By loading Bose-Einstein condensates into a three dimensional optical lattice potential we are able to demonstrate several new intriguing regimes in the physics of ultracold atoms. For example by changing the lattice potential depth we have been able to induce a quantum phase transition from a superfluid to a Mott insulating ground state of the system. Furthermore, by rapidly isolating the different potential wells from each other the collapse and revival of the matter wave field of a Bose-Einstein condensate has been observed. 1