Quantum Phase Transition From a Superfluid to a Mott Insulator in a Gas of Ultracold Atoms
TL;DR: This work observes a quantum phase transition in a Bose–Einstein condensate with repulsive interactions, held in a three-dimensional optical lattice potential, and can induce reversible changes between the two ground states of the system.
Abstract: For a system at a temperature of absolute zero, all thermal fluctuations are frozen out, while quantum fluctuations prevail. These microscopic quantum fluctuations can induce a macroscopic phase transition in the ground state of a many-body system when the relative strength of two competing energy terms is varied across a critical value. Here we observe such a quantum phase transition in a Bose-Einstein condensate with repulsive interactions, held in a three-dimensional optical lattice potential. As the potential depth of the lattice is increased, a transition is observed from a superfluid to a Mott insulator phase. In the superfluid phase, each atom is spread out over the entire lattice, with long-range phase coherence. But in the insulating phase, exact numbers of atoms are localized at individual lattice sites, with no phase coherence across the lattice; this phase is characterized by a gap in the excitation spectrum. We can induce reversible changes between the two ground states of the system.
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TL;DR: In this article, a review of recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases is presented, focusing on effects beyond standard weakcoupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation.
Abstract: This paper reviews recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BEC crossover.
6,601 citations
Cites background or methods from "Quantum Phase Transition From a Sup..."
...They have found that in the regime beyond a/d & 0.3 where (V0/Er)c ....
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...(11) - and its eventual destruction by strong repulsive interactions at the SF-MI transition was observed experimentally by a time-of-flight method (Greiner et al., 2002a)....
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...(99) leads to critical values (V0/Er)c 1, consistent with the exponential dependence J ∼ Er(V0/Er)3/4 exp−2 √ V0/Er of the hopping matrix element which enters (99) and the truncation of the continuum model to the lowest band....
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...(7) in one dimension reads U1 = √ 2 π ka · ~ω⊥ ( V0 Er )1/4 , (98) the critical value for the ratio V0/Er of the strength of a 1D optical lattice and the recoil energy for the SF-MI transition at unit filling n = n1d = 1 follows from the implicit, transcendental equation (Büchler et al., 2003) (V0/Er)c = 1 4 ln2 [ 4 √ 2πC (V0/Er) 1/2 c /γ ] (99) which replaces Eq....
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...The results obtained by changing both V0/Er and the interaction strength are shown in Fig....
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TL;DR: It is demonstrated that a generic isolated quantum many-body system does relax to a state well described by the standard statistical-mechanical prescription, and it is shown that time evolution itself plays a merely auxiliary role in relaxation, and that thermalization instead happens at the level of individual eigenstates, as first proposed by Deutsch and Srednicki.
Abstract: It is demonstrated that an isolated generic quantum many-body system does relax to a state well described by the standard statistical mechanical prescription The thermalization happens at the level of individual eigenstates, allowing the computation of thermal averages from knowledge of any eigenstate in the microcanonical energy window An understanding of the temporal evolution of isolated many-body quantum systems has long been elusive Recently, meaningful experimental studies1,2 of the problem have become possible, stimulating theoretical interest3,4,5,6,7 In generic isolated systems, non-equilibrium dynamics is expected8,9 to result in thermalization: a relaxation to states in which the values of macroscopic quantities are stationary, universal with respect to widely differing initial conditions, and predictable using statistical mechanics However, it is not obvious what feature of many-body quantum mechanics makes quantum thermalization possible in a sense analogous to that in which dynamical chaos makes classical thermalization possible10 For example, dynamical chaos itself cannot occur in an isolated quantum system, in which the time evolution is linear and the spectrum is discrete11 Some recent studies4,5 even suggest that statistical mechanics may give incorrect predictions for the outcomes of relaxation in such systems Here we demonstrate that a generic isolated quantum many-body system does relax to a state well described by the standard statistical-mechanical prescription Moreover, we show that time evolution itself plays a merely auxiliary role in relaxation, and that thermalization instead happens at the level of individual eigenstates, as first proposed by Deutsch12 and Srednicki13 A striking consequence of this eigenstate-thermalization scenario, confirmed for our system, is that knowledge of a single many-body eigenstate is sufficient to compute thermal averages—any eigenstate in the microcanonical energy window will do, because they all give the same result
2,598 citations
University of Cambridge1, Istituto Italiano di Tecnologia2, Lancaster University3, University of Manchester4, Catalan Institution for Research and Advanced Studies5, Technical University of Denmark6, Nokia7, Queen Mary University of London8, University of Trento9, fondazione bruno kessler10, Technische Universität München11, Polytechnic University of Milan12, Centre national de la recherche scientifique13, University of Trieste14, University of Ioannina15, University of Geneva16, Trinity College, Dublin17, Texas Instruments18, University of Paris19, Spanish National Research Council20, Leiden University21, Delft University of Technology22, University of Patras23, École Normale Supérieure24, Radboud University Nijmegen25, Nest Labs26, Airbus UK27, Seoul National University28, Yonsei University29, University of Oxford30, Chalmers University of Technology31, University of Groningen32, STMicroelectronics33, Chemnitz University of Technology34, Max Planck Society35, Aalto University36
TL;DR: An overview of the key aspects of graphene and related materials, ranging from fundamental research challenges to a variety of applications in a large number of sectors, highlighting the steps necessary to take GRMs from a state of raw potential to a point where they might revolutionize multiple industries are provided.
Abstract: We present the science and technology roadmap for graphene, related two-dimensional crystals, and hybrid systems, targeting an evolution in technology, that might lead to impacts and benefits reaching into most areas of society. This roadmap was developed within the framework of the European Graphene Flagship and outlines the main targets and research areas as best understood at the start of this ambitious project. We provide an overview of the key aspects of graphene and related materials (GRMs), ranging from fundamental research challenges to a variety of applications in a large number of sectors, highlighting the steps necessary to take GRMs from a state of raw potential to a point where they might revolutionize multiple industries. We also define an extensive list of acronyms in an effort to standardize the nomenclature in this emerging field.
2,560 citations
TL;DR: The density-matrix renormalization group (DMRG) as mentioned in this paper is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription.
Abstract: The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG.
2,341 citations
TL;DR: In this paper, the authors give an overview of recent theoretical and experimental progress in the area of nonequilibrium dynamics of isolated quantum systems, particularly focusing on quantum quenches: the temporal evolution following a sudden or slow change of the coupling constants of the system Hamiltonian.
Abstract: This Colloquium gives an overview of recent theoretical and experimental progress in the area of nonequilibrium dynamics of isolated quantum systems There is particularly a focus on quantum quenches: the temporal evolution following a sudden or slow change of the coupling constants of the system Hamiltonian Several aspects of the slow dynamics in driven systems are discussed and the universality of such dynamics in gapless systems with specific focus on dynamics near continuous quantum phase transitions is emphasized Recent progress on understanding thermalization in closed systems through the eigenstate thermalization hypothesis is also reviewed and relaxation in integrable systems is discussed Finally key experiments probing quantum dynamics in cold atom systems are overviewed and put into the context of our current theoretical understanding
2,340 citations
Cites background from "Quantum Phase Transition From a Sup..."
...IV in this Colloquium (Greiner et al., 2002b; Kinoshita et al., 2006)....
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...Seminal work in this direction includes groundbreaking experiments (Greiner et al., 2002a,b) showing both the feasibility of observing a quantum phase transition in cold atoms and the possibility of observing quantum coherent dynamics....
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...In this Colloquium we will discuss both of these questions extensively....
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...In this Colloquium we will concentrate on the simplest paradigm: the study of the nonequilibrium dynamics of closed interacting quantum systems following a change in one of the system parameters (quantum quench)....
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...A particularly dramatic instance of such coherent many-body dynamics was illustrated in the collapse and revival of the matter wave field of a Bose condensate (Greiner et al., 2002b)....
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References
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TL;DR: In this article, the authors reviewed the Bose-Einstein condensation of dilute gases in traps from a theoretical perspective and provided a framework to understand the main features of the condensation and role of interactions between particles.
Abstract: The phenomenon of Bose-Einstein condensation of dilute gases in traps is reviewed from a theoretical perspective. Mean-field theory provides a framework to understand the main features of the condensation and the role of interactions between particles. Various properties of these systems are discussed, including the density profiles and the energy of the ground-state configurations, the collective oscillations and the dynamics of the expansion, the condensate fraction and the thermodynamic functions. The thermodynamic limit exhibits a scaling behavior in the relevant length and energy scales. Despite the dilute nature of the gases, interactions profoundly modify the static as well as the dynamic properties of the system; the predictions of mean-field theory are in excellent agreement with available experimental results. Effects of superfluidity including the existence of quantized vortices and the reduction of the moment of inertia are discussed, as well as the consequences of coherence such as the Josephson effect and interference phenomena. The review also assesses the accuracy and limitations of the mean-field approach.
4,782 citations
"Quantum Phase Transition From a Sup..." refers background in this paper
...This regime is dominated by atom–atom interactions and it is not accessible to theoretical treatments of weakly interacting gases, which have so far proved to be very successful in describing the physics of Bose–Einstein condensate...
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TL;DR: In this paper, the Bose-Hubbard model was used to model the phase transition from the superfluid to the Mott insulator phase induced by varying the depth of the optical potential.
Abstract: The dynamics of an ultracold dilute gas of bosonic atoms in an optical lattice can be described by a Bose-Hubbard model where the system parameters are controlled by laser light We study the continuous (zero temperature) quantum phase transition from the superfluid to the Mott insulator phase induced by varying the depth of the optical potential, where the Mott insulator phase corresponds to a commensurate filling of the lattice (``optical crystal'') Examples for formation of Mott structures in optical lattices with a superimposed harmonic trap and in optical superlattices are presented
2,873 citations
"Quantum Phase Transition From a Sup..." refers background in this paper
...1 atoms, (2) secondorder processes, in which two particle±hole pairs are created simultaneously, with only one in the direction of the applied gradient, and (3) tunnelling processes occurring between lattice sites with n 1 atom next to lattice sites with n 2 atoms....
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TL;DR: It is argued that the superfluid-insulator transition in the presence of disorder may have an upper critical dimension dc which is infinite, but a perturbative renormalization-group calculation wherein the critical exponents have mean-field values for weak disorder above d=4 is also discussed.
Abstract: The phase diagrams and phase transitions of bosons with short-ranged repulsive interactions moving in periodic and/or random external potentials at zero temperature are investigated with emphasis on the superfluid-insulator transition induced by varying a parameter such as the density. Bosons in periodic potentials (e.g., on a lattice) at T=0 exhibit two types of phases: a superfluid phase and Mott insulating phases characterized by integer (or commensurate) boson densities, by the existence of a gap for particle-hole excitations, and by zero compressibility. Generically, the superfluid onset transition in d dimensions from a Mott insulator to superfluidity is ‘‘ideal,’’ or mean field in character, but at special multicritical points with particle-hole symmetry it is in the universality class of the (d+1)-dimensional XY model. In the presence of disorder, a third, ‘‘Bose glass’’ phase exists. This phase is insulating because of the localization effects of the randomness and analogous to the Fermi glass phase of interacting fermions in a strongly disordered potential. The Bose glass phase is characterized by a finite compressibility, no gap, but an infinite superfluid susceptibility. In the presence of disorder the transition to superfluidity is argued to occur only from the Bose glass phase, and never directly from the Mott insulator. This zero-temperature superfluid-insulator transition is studied via generalizations of the Josephson scaling relation for the superfluid density at the ordinary λ transition, highlighting the crucial role of quantum fluctuations. The transition is found to have a dynamic critical exponent z exactly equal to d and correlation length and order-parameter correlation exponents ν and η which satisfy the bounds ν≥2/d and η≤2-d, respectively. It is argued that the superfluid-insulator transition in the presence of disorder may have an upper critical dimension dc which is infinite, but a perturbative renormalization-group calculation wherein the critical exponents have mean-field values for weak disorder above d=4 is also discussed. Many of these conclusions are verified by explicit calculations on a model of one-dimensional bosons in the presence of both random and periodic potentials. The general results are applied to experiments on 4He absorbed in porous media such as Vycor. Some measurable properties of the superfluid onset are predicted exactly [e.g., the exponent x relating the λ transition temperature to the zero-temperature superfluid density is found to be d/2(d-1)], while stringent bounds are placed on others. Analysis of preliminary data is consistent with these predictions.
2,472 citations
"Quantum Phase Transition From a Sup..." refers methods in this paper
...This system was first studied theoretically in the context of superfluid-to-insulator transitions in liquid heliu...
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Book•
01 Apr 2001TL;DR: In this paper, the mapping to classical statistical mechanics: single site models 3. Quantum Ising and Rotor Models: 4. The Ising chain in a transverse field 5. Quantum rotor models: large N limit 6. The d = 1, 0 (N greater than or equal to 3) rotor models 7. Quantum spin chains: bosonization 14. Magnetic ordering transitions of disordered systems 16.
Abstract: Part I. Introduction: 1. Basic concepts 2. The mapping to classical statistical mechanics: single site models 3. Overview Part II. Quantum Ising and Rotor Models: 4. The Ising chain in a transverse field 5. Quantum rotor models: large N limit 6. The d = 1, 0 (N greater than or equal to 3) rotor models 7. The d = 2 (N greater than or equal to 3) rotor models 8. Physics close to and above the upper-critical dimension 9. Transport in d = 2 Part III. Other Models: 10. Boston Hubbard model 11. Dilute Fermi and Bose gases 12. Phase transitions of Fermi liquids 13. Heisenberg spins: ferromagnets and antiferromagnets 14. Spin chains: bosonization 15. Magnetic ordering transitions of disordered systems 16. Quantum spin glasses.
1,870 citations
TL;DR: In this paper, two such resonances have been observed in optically trapped Bose-Einstein condensates of sodium atoms by varying an external magnetic field, which gave rise to enhanced inelastic processes and a dispersive variation of the scattering length by a factor of over ten.
Abstract: It has long been predicted that the scattering of ultracold atoms can be altered significantly through a so-called ‘Feshbach resonance’. Two such resonances have now been observed in optically trapped Bose–Einstein condensates of sodium atoms by varying an external magnetic field. They gave rise to enhanced inelastic processes and a dispersive variation of the scattering length by a factor of over ten. These resonances open new possibilities for the study and manipulation of Bose–Einstein condensates.
1,640 citations
"Quantum Phase Transition From a Sup..." refers methods in this paper
...For example, besides controlling mainly the tunnelling matrix element, as done in this work, it should be possible in future experiments to control the atom–atom interactions via Feshbach resonance...
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