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Birkhäuser-Verlag eBooks

Bose-Einstein Quantum Phase Transition in an Optical Lattice Model

Published Dec 22, 2005• 0 citations
Publisher

Abstract

Bose-Einstein condensation (BEC) in cold gases can be turned on and off by an external potential, such as that presented by an optical lattice.We present a model of this phenomenon which we are able to analyze rigorously.The system is a hard core lattice gas at half-filling and the optical lattice is modeled by a periodic potential of strength λ.For small λ and temperature, BEC is proved to occur, while at large λ or temperature there is no BEC.At large λ the low-temperature states are in a Mott insulator phase with a characteristic gap that is absent in the BEC phase.The interparticle interaction is essential for this transition, which occurs even in the ground state.Surprisingly, the condensation is always into the p = 0 mode in this model, although the density itself has the periodicity of the imposed potential.

DOI
10.1007/3-7643-7337-7_11

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Bose-Einstein condensation (BEC) in cold gases can be turned on and off by an external potential, such as that presented by an optical lattice.We present a model of this phenomenon which we are able to analyze rigorously.The system is a hard core lattice gas at half-filling and the optical lattice is modeled by a periodic potential of strength λ.For small λ and temperature, BEC is proved to occur, while at large λ or temperature there is no BEC.At large λ the low-temperature states are in a Mott insulator phase with a characteristic gap that is absent in the BEC phase.The interparticle interaction is essential for this transition, which occurs even in the ground state.Surprisingly, the condensation is always into the p = 0 mode in this model, although the density itself has the periodicity of the imposed potential.
Bose-Einstein condensation (BEC) in cold gases can be turned on and off by an external potential, such as that presented by an optical lattice. We present a model of this phenomenon which we are able to analyze rigorously. The system is a hard core lattice gas at half of the maximum density and the optical lattice is modeled by a periodic potential of strength $\ensuremath{\lambda}$. For small $\ensuremath{\lambda}$ and temperature, BEC is proved to occur, while at large $\ensuremath{\lambda}$ or temperature there is no BEC. At large $\ensuremath{\lambda}$ the low-temperature states are in a Mott insulator phase with a characteristic gap that is absent in the BEC phase. The interparticle interaction is essential for this transition, which occurs even in the ground state. Surprisingly, the condensation is always into the p=0\mathbf{p}=\mathbf{0}p=0 mode in this model, although the density itself has the periodicity of the imposed potential.
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.
An important new development in the field of ultracold atomic gases is the study of the properties of these gases in a so-called optical lattice. An optical lattice is a periodic trapping potential for the atoms that is formed by the interference pattern of a few laser beams. A reason for the interest in these systems is that the effects of the interatomic interactions can be strongly enhanced. More specifically, it has been shown in a beautiful experiment by Greiner et al. in 2002 that by loading a Bose-Einstein condensate into an optical lattice it is possible for the system to undergo a quantum phase transition to a new quantum phase of matter, the so-called Mott insulator phase. Within this Mott insulator phase each lattice site is occupied by exactly one atom. This makes the Mott insulator phase especially well suited for applications in the field of quantum computation and quantum information processing. We have theoretically investigated the above mentioned quantum phase transition and our formalism allows for a description of the Mott insulator phase at nonzero temperatures. Another important experimental development in the field of ultracold atomic gases is the use of Feshbach resonances to control the interatomic interactions. Such a resonance occurs whenever two colliding atoms form a long-lived molecule for some time. The crucial point of a Feshbach resonance is that the above mentioned molecule has a magnetic moment that is not equal to twice the magnetic moment of the atom. As a consequence the energy difference between the two atoms and the molecule and hence the interactions between the atoms can be controlled by using an external magnetic field. By combining these two techniques, i.e, by trapping ultracold atomic gases in an optical lattice and by tuning a magnetic field near a Feshbach resonance there can be a new quantum phase transition between two superfluid phases. We gave derived the the theory for the description of these Feshbach resonances in optical lattices and applied it to various systems. To be a bit more precise, if we tune the external magnetic field such that the energy difference between a molecule and two atoms is sufficiently negative, then the gas consists of a Bose-Einstein condensate of molecules. In contrast, if the energy difference is large enough and positive we have a gas that consists primarily of a Bose-Einstein condensate of atoms. It turns out that these two limits are separated by an Ising-like quantum phase transition. By using atomic Bose gases near a Feshbach resonance detailed experimental studies which test the theoretical predictions of the statistical and dynamical properties of these quantum phase transitions can be made.
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
We present a general lattice model for a multicomponent atomic Bose-Einstein system in an optical lattice. Using the model, we analytically study the quantum phase transition between the Mott insulator and a superfluid. A mean-field theory is developed from the Mott-insulator ground state. When the interspecies interactions are strong enough, the Mott insulator demonstrates the phase separation behavior. For weak interspecies interactions, the multispecies system is miscible. Finally, the phase diagram is discussed with emphasis on the role of interspecies interactions. The tip and the shape of the Mott insulator lobes do not depend on the interspecies interactions, but the latter indeed modify the position of the phase boundaries.