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Physical Review A

Signatures of the quantum fluctuations of cold atoms in an optical lattice in the three-body loss rate

Michael W. Jack, Makoto Yamashita

Published Mar 20, 2003• 16 citations
Publisher

Abstract

We consider an optical lattice potential loaded with a Bose-Einstein condensate in the presence of three-body loss. The dynamics of the atoms in the lattice realizes the Bose-Hubbard model of condensed-matter physics, where on-site number fluctuations are reduced by the atomic repulsion energy but increase with increased tunneling between lattice sites. A range of quantum states with different number fluctuations can therefore be created by varying the amplitude of the optical potential, and will be reflected in the values of the on-site correlation functions ${G}_{n}^{i}=〈{b}_{i}^{\ifmmode\dagger\else\textdagger\fi{}n}{b}_{i}^{n}〉,$ where ${b}_{i}$ is the annihilation operator of the $i\mathrm{th}$ lattice site. The three-body loss rate is shown to be proportional to a sum of the on-site three-body correlation functions and therefore acts as a natural probe of the number fluctuations. The three-body correlation function ${G}_{3}^{i}$ is a sensitive measure of the number fluctuations when site occupation is low. In particular, when the average number at each site is less than $3,$ the number fluctuations in the superfluid phase result in ${G}_{3}^{i}\ensuremath{\approx}{n}_{i}^{3}$ and give rise to a finite three-body loss, whereas in the insulator phase, ${G}_{3}^{i}{=n}_{i}{(n}_{i}\ensuremath{-}{1)(n}_{i}\ensuremath{-}2)$ and three-body loss ceases to occur.

DOI
10.1103/physreva.67.033605

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We consider an optical lattice potential loaded with a Bose-Einstein condensate in the presence of three-body loss. The dynamics of the atoms in the lattice realizes the Bose-Hubbard model of condensed-matter physics, where on-site number fluctuations are reduced by the atomic repulsion energy but increase with increased tunneling between lattice sites. A range of quantum states with different number fluctuations can therefore be created by varying the amplitude of the optical potential, and will be reflected in the values of the on-site correlation functions Gni=〈bi\ifmmode†\else\textdagger\finbin〉,{G}_{n}^{i}=〈{b}_{i}^{\ifmmode\dagger\else\textdagger\fi{}n}{b}_{i}^{n}〉,Gni​=〈bi\ifmmode†\else\textdagger\fin​bin​〉, where bi{b}_{i}bi​ is the annihilation operator of the ithi\mathrm{th}ith lattice site. The three-body loss rate is shown to be proportional to a sum of the on-site three-body correlation functions and therefore acts as a natural probe of the number fluctuations. The three-body correlation function G3i{G}_{3}^{i}G3i​ is a sensitive measure of the number fluctuations when site occupation is low. In particular, when the average number at each site is less than 3,3,3, the number fluctuations in the superfluid phase result in G3i\ensuremath≈ni3{G}_{3}^{i}\ensuremath{\approx}{n}_{i}^{3}G3i​\ensuremath≈ni3​ and give rise to a finite three-body loss, whereas in the insulator phase, G3i=ni(ni\ensuremath−1)(ni\ensuremath−2){G}_{3}^{i}{=n}_{i}{(n}_{i}\ensuremath{-}{1)(n}_{i}\ensuremath{-}2)G3i​=ni​(ni​\ensuremath−1)(ni​\ensuremath−2) and three-body loss ceases to occur.
An optical lattice is a periodic potential for atoms, created using a standing wave pattern of light. Due to the interaction between the light and the atoms, the atoms are attracted to either the nodes or the anti-nodes of the standing wave, depending on the exact wave lenght of the light. This means that if such a lattice is loaded with a sufficiently high number of ultracold atoms, a periodic array of atoms is obtained, we an interatomic distance of a few tenths of a micrometer. In order to obtain such a high number of cold atoms, one first has to create a so-called Bose-Einstein condensate. When an optical lattice is loaded from a Bose-Einstein condensate, it is possible to create a system in which every atom is in the lowest band of the lattice and there is on average one atom in each lattice site. Because the lattice potential is created with laser light, the depth of the lattice can easily be tuned by changing the intensity of the laser. When the intensity of the laser light is low, the atoms can tunnel from one site to the next. Due to this tunneling, the gas of atoms in the lattice will remain superfluid. However, if the intensity of the laser light is increased to above a certain critical value, a quantum phase transition occurs to a so-called Mott insulator. In this state, the atoms can no longer tunnel due to the fact that the on-site interaction between atoms becomes more important then the tunneling probability. In this PhD thesis, a description is given of the experimental setup that is being constructed in our group to create these systems in our lab. Also, a theoretical description is given of these systems and several important quantities our derived, such as the gap of the Mott-insulating state. Furthermore, an experiment is proposed that can be used to accurately measure this gap.
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.
We study the one-dimensional Bose-Hubbard model describing the superfluid-Mott insulator quantum phase transition of cold atoms in optical lattices. We show that derivatives of the variance of the on-site atom number occupation, computed with respect to the parameter driving the transition, have extrema that are located off the critical point even in the thermodynamic limit. We discuss whether such extrema provide solid evidence of the quantum Berezinskii-Kosterlitz-Thouless transition taking place in this system. The calculations are done for systems with the mean number of atoms per lattice site equal to either one or two. They also characterize the nearest-neighbor correlation function, which is typically discussed in the context of time-of-flight images of cold atoms.
Quantum gases in optical lattices offer an opportunity to experimentally realize and explore condensed matter models in a clean, tunable system. We used single atom-single lattice site imaging to investigate the Bose-Hubbard model on a microscopic level. Our technique enables space- and time-resolved characterization of the number statistics across the superfluid-Mott insulator quantum phase transition. Site-resolved probing of fluctuations provides us with a sensitive local thermometer, allows us to identify microscopic heterostructures of low-entropy Mott domains, and enables us to measure local quantum dynamics, revealing surprisingly fast transition time scales. Our results may serve as a benchmark for theoretical studies of quantum dynamics, and may guide the engineering of low-entropy phases in a lattice.
We study the one-dimensional Bose-Hubbard model describing the superfluid-Mott insulator quantum phase transition of cold atoms in optical lattices. We show that derivatives of the variance of the on-site atom number occupation, computed with respect to the parameter driving the transition, have extrema that are located off the critical point even in the thermodynamic limit. We discuss whether such extrema provide solid evidence of the quantum Berezinskii-Kosterlitz-Thouless transition taking place in this system. The calculations are done for systems with the mean number of atoms per lattice site equal to either one or two. They also characterize the nearest-neighbor correlation function, which is typically discussed in the context of time-of-flight images of cold atoms.
In a one-dimensional shallow optical lattice, in the presence of both cubic and quintic nonlinearity, a superfluid density wave is identified in a Bose-Einstein condensate. Interestingly, it ceases to exist when only one of these interactions is operative. We predict the loss of superfluidity through a classical dynamical phase transition, where modulational instability leads to the loss of phase coherence. In a certain parameter domain, the competition between lattice potential and the interactions is shown to give rise to a stripe phase, where atoms are confined in finite domains. In a pure two-body case, apart from the known superfluid and insulating phases, a density wave insulating phase is found to exist, possessing two frequency modulations commensurate with the lattice potential.