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Physics Today

A Critical Point

Nikolay V. Prokof'ev, Boris V. Svistunov

Published Jul 1, 2002• 1 citation
Publisher

Abstract

We read with great interest the illuminating review article by Barbara Goss Levi (Physics Today, Physics Today 0031-9228 55 3 2002 18 https://doi.org/10.1063/1.1472381. March 2002, page 18 ), in which she discusses the recent pioneering experiment on superfluid-Mott insulator transition in a system of ultracold atoms in an optical lattice. However, we disagree with the description of the data shown in figure 1, in particular that “the phase transition occurs somewhere between (f) and (g).” The Garching–Munich group, which performed the experiment Levi describes, associates the quantum critical point with the case (e). We also disagree with the interpretation given by the Garching–Munich group of the interference peaks seen in their experiment. They associate the disappearance of Bragg reflection peaks with the phase transition of the Bose-Einstein condensate from a coherent, superfluid phase to a Mott insulating phase. We contend, however, that one cannot interpret the fading of Bragg peaks as a signature of a phase transition. The appearance of a Mott phase will be seen only in the fine structure of peaks.The interference pattern of Bragg peaks results from the periodic lattice structure of the system and the phase coherence between lattice sites. As long as phase coherence exists on length scales of several lattice sites, one should see a clear picture of narrow interference peaks, with peak positions being in one-to-one correspondence with the reciprocal lattice space—that is, the Bragg peaks simply reflect the underlying periodic lattice. In a Mott phase, such a coherence is still present near the critical point (when insulating gaps are small) due to quantum hopping of loosely localized atoms to the neighboring sites; the coherence extends over a large correlation length. In this case, the interference signals only the essential quantum nature of the ground state. The coherence disappears gradually as the tunneling strength grows weaker, as seen between (f) and (g), but this fading of Bragg peaks occurs beyond the critical point. For quantitative details and an extended discussion, see our precise numeric simulation of the experimental situation at http://arXiv.org/abs/cond-mat/0202510.© 2002 American Institute of Physics.

DOI
10.1063/1.1506772

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We read with great interest the illuminating review article by Barbara Goss Levi (Physics Today, Physics Today 0031-9228 55 3 2002 18 https://doi.org/10.1063/1.1472381. March 2002, page 18 ), in which she discusses the recent pioneering experiment on superfluid-Mott insulator transition in a system of ultracold atoms in an optical lattice. However, we disagree with the description of the data shown in figure 1, in particular that “the phase transition occurs somewhere between (f) and (g).” The Garching–Munich group, which performed the experiment Levi describes, associates the quantum critical point with the case (e). We also disagree with the interpretation given by the Garching–Munich group of the interference peaks seen in their experiment. They associate the disappearance of Bragg reflection peaks with the phase transition of the Bose-Einstein condensate from a coherent, superfluid phase to a Mott insulating phase. We contend, however, that one cannot interpret the fading of Bragg peaks as a signature of a phase transition. The appearance of a Mott phase will be seen only in the fine structure of peaks.The interference pattern of Bragg peaks results from the periodic lattice structure of the system and the phase coherence between lattice sites. As long as phase coherence exists on length scales of several lattice sites, one should see a clear picture of narrow interference peaks, with peak positions being in one-to-one correspondence with the reciprocal lattice space—that is, the Bragg peaks simply reflect the underlying periodic lattice. In a Mott phase, such a coherence is still present near the critical point (when insulating gaps are small) due to quantum hopping of loosely localized atoms to the neighboring sites; the coherence extends over a large correlation length. In this case, the interference signals only the essential quantum nature of the ground state. The coherence disappears gradually as the tunneling strength grows weaker, as seen between (f) and (g), but this fading of Bragg peaks occurs beyond the critical point. For quantitative details and an extended discussion, see our precise numeric simulation of the experimental situation at http://arXiv.org/abs/cond-mat/0202510.© 2002 American Institute of Physics.
Quantum phases of matter are routinely identified by coherence features, with interference patterns being one of the most directly observable quantities. In lattices, the superfluid-to-Mott-insulator (SF-MI) transition is commonly viewed as a change from wave-like coherence to particle-like localization: interference peaks are taken as a hallmark of superfluidity, whereas their disappearance is used to diagnose insulating behavior. Here, we challenge this picture for one-dimensional (1D) strongly interacting gases subject to a lattice potential. We realize a gapped Mott insulator through pinning in a shallow lattice and find that pronounced interference peaks persist deep in the insulating regime. Strikingly, the interference becomes stronger as the Mott fraction increases, demonstrating that a certain degree of coherence still exists in the insulator state. Measurements of the one-body correlation function reveal an oscillatory, exponentially decaying coherence pattern across several lattice sites, in quantitative agreement with quantum Monte Carlo (QMC) simulations. Our work shows that interference does not uniquely diagnose superfluidity and it exposes the unexpected wave nature of a 1D Mott insulator.
We review the nature of superfluid ground states and the universality of their properties with emphasis to Bose Einstein Condensate systems in atomic physics. We then study the superfluid Mott transition in such systems. We find that there could be two types of Mott transitions and phases. One of them was described long ago and corresponds to suppression of Josephson tunneling within superfluids sitting at each well. On the other hand, the conditions of optical lattice BEC experiments are such that either the coherence length is longer than the interwell separation, or there is too small a number of bosons per well. This vitiates the existence of a superfluid order parameter within a well, and therefore of Josephson tunneling between wells. Under such conditions, there is a transition to a Mott phase which corresponds to suppression of individual boson tunneling among wells. This last transition is in general discontinuous and can happen for incommensurate values of bosons per site. If the coherence length is small enough and the number of bosons per site large enough, the transition studied in the earlier work will happen.
We review the nature of superfluid ground states and the universality of their properties with emphasis to the Bose Einstein condensate (BEC) systems in atomic physics. We then study the superfluid Mott transition in such systems. We find that there could be two types of Mott transitions and phases. One of them was described long ago and corresponds to the suppression of Josephson tunneling within superfluids sitting at each well. On the other hand, the conditions of optical lattice BEC experiments are such that either the coherence length is longer than the interwell separation, or there is too small a number of bosons per well. This vitiates the existence of a superfluid order parameter within a well, and therefore of Josephson tunneling between wells. Under such conditions, there is a transition to a Mott phase which corresponds to the suppression of individual boson tunneling among wells. This last transition is in general discontinuous and can happen for incommensurate values of bosons per site. If the coherence length is small enough and the number of bosons per site is large enough, the transition studied in the earlier work will happen.
Confinement of atoms by laser technology has opened up a new field of physics. Following the realization of Bose– Einstein condensation (BEC) of bosonic atoms, fermionic atom systems and even boson–fermion mixture systems were successfully trapped and cooled. Application of a periodic potential, so called optical lattice, could also unveil interesting nature of the atoms. One of the interesting phenomena of the atoms on optical lattice is Mott transition. Although there is a long history of the study on bosons or fermions on lattice, coexistence of bosons and fermions on lattice is a rather new topic to study. In particular, we have little knowledge about strongly-correlated bosons and fermions on lattice. In our previous works we performed numerical simulations of Bose–Fermi mixtures to clarify the phase diagram and the effect of the confinement potential. When the total number of bosonic and fermionic atoms per site takes an integer value and interatomic interactions are sufficiently strong, the system would undergo Mott transition where the local atom density is fixed to the integer value. A characteristic of this Mott insulating phase is that a boson and fermion located next to each other can exchange their positions by going through a virtual doubly-occupied state with the atom exchange energy t ’ tbtf=Ubf where tb and tf are the hopping energy of the bosons and fermions respectively and Ubf denotes the on-site interaction between the boson and the fermion. Although there still remain a number of open questions on the Mott state of Bose–Fermi mixtures, the property of superfluid state of the strongly-correlated bosons and fermions has yet to be revealed. In this short note we would like to focus on strongly-correlated Bose–Fermi mixtures at commensurate filling in a three-dimensional periodical potential and see characteristic behaviors, if any, of the superfluid phase close to the phase boundary to the Mott insulating phase. In the following analysis we assume uniformity of the system and thus neglect the possibility of the emergence of so-called supersolid where superfluidity and structural long-range order coexist. For simplicity we ignore all the internal degree of freedom of the atoms and the influence of the trap potential which in fact creates a nonuniform density profile of the bosons and fermions. For the system at commensurate filling on optical lattice, we assume that the interatomic interactions are weak enough to avoid formation of Mott state but strong enough to allow us to expect that there are only few empty or doublyoccupied sites and most sites are singly occupied. Namely we analyze the system that stays in a superfluid phase but lies close to the boundary to the Mott phase. Since a boson and fermion located next to each other can exchange their positions as stated above, the energy of the superfluid state of the mixture system is given mostly by the kinetic energy of the bosons and fermions exchanging their positions. We therefore express the Hamiltonian in the following form, ignoring the kinetic energy of the atoms at the empty sites and the interaction energy at the doubly-occupied sites:
The first successful experimental realization of a Bose-Einstein condensate (BEC) in 1995 sparked a still growing research activity that in the meantime has led to beautiful and surprising results, such as the interference of matter waves [4], the Mott-superfluid phase transition [49] or the direct observation of Anderson localization of matter waves [10]. The strong interest in BECs arises, among others, from the possibility to investigate known but hard to study quantum systems on a rather macroscopic scale: for example one can load ultra-cold bosons into an optical lattice created by a standing light wave, and thereby simulate a solid state device. When it comes to a measurement on the BEC, however, essentially all measurement techniques result in the destruction of the condensate: for example in the so called “time of flight method” the optical lattice is switched off to observe the expansion of the trapped atoms and thereby directly visualize their momentum distribution, which highlights the condensed component [11]. In this thesis we elaborate on a means to probe a Bose-Einstein condensate from outside, with a scattering particle. The interaction between the probe particle and the bosons allows energy exchange between them, i.e., this setup includes inelastic scattering. After the scattering process, the condensate is still confined in the optical lattice and can therefore be used for further investigations. Of course, this method of probing a BEC is only then a serious alternative if the properties of the BEC can be inferred from the scattering signal. Besides outlining the scattering formalism, the central aim of this work is to show how the properties of the BEC can be deduced from those quantities accessible in a scattering experiment, like for example the transmission or the inelastic cross section. The other, equally strong motivation of this work concerns the signature of the interaction between the bosons in the scattering quantities. The dynamics of an analog classical system depends strongly on the interaction strength, which can induce a complex structure of the underlying phase space like, e.g., the appearance of a separatrix or even the emergence of chaotic dynamics in a globally mixed regular-chaotic phase space. We are interested to which extent such phase-space metamorphosis is reflected in the properties of the scattering particle. Currently light scattering (Bragg spectroscopy) on an optical lattice [78, 77] or rather on an optical lattice embedded into a cavity [62,68,19] is under intense investigation. These efforts predominantly aim at a characterization of the Mott-superfluid transition by specific scattering properties. In the present thesis we focus on condensates loaded in two and three lattice sites, with a large number of bosons. These systems exhibit a diversity of important physical effects (as we will see below) and thus are of paradigmatic relevance for a detailed understanding of multiparticle quantum dynamics in periodic potentials.
We present a study of the superfluid properties of atomic Bose gases in optical lattice potentials using the Bose-Hubbard model. To do this, we use a microscopic definition of the superfluid fraction based on the response of the system to a phase variation imposed by means of twisted boundary conditions. We compare the superfluid fraction to other physical quantities, i.e., the interference pattern after ballistic expansion, the quasimomentum distribution, and number fluctuations. We have performed exact numerical calculations of all these quantities for small one-dimensional systems. We show that the superfluid fraction alone exhibits a clear signature of the Mott-insulator transition. Observables like the fringe visibility, which probe only ground-state properties, do not provide direct information on superfluidity and the Mott-insulator transition.