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arXiv:cond-mat/0305696 · Modern Physics Letters B

Quantum Phase Transition and Engineering in Two-Component BEC in Optical Lattices

Yong-Shi Wu

Published Jul 10, 2003• 1 citation
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Abstract

In this paper we review recent progress in studying quantum phase transitions in one- and two-component Bose–Einstein condensates (BEC) in optical lattices. These phase transitions involve the emergence and disappearance of quantum coherence over the whole optical lattice and of linear superposition of macroscopic quantum states. The latter may provide new means to engineer and manipulate novel macroscopic quantum states and novel coherent atomic beams for quantum information processing, quantum computing and other purposes.

DOI
10.1142/s0217984903005846

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arXiv:1109.6045•Physical Review A•Oct 17, 2016•3 citations
In this paper we review recent progress in studying quantum phase transitions in one- and two-component Bose–Einstein condensates (BEC) in optical lattices. These phase transitions involve the emergence and disappearance of quantum coherence over the whole optical lattice and of linear superposition of macroscopic quantum states. The latter may provide new means to engineer and manipulate novel macroscopic quantum states and novel coherent atomic beams for quantum information processing, quantum computing and other purposes.
Recently, experiments on Bose\|Einstein condensates (BECs) confined in an optical lattice have made breakthroughs. A research group in Munich has demonstrated for the first time the quantum phase transition of a superfluid state to the Mott insulator state of a BEC confined in a 3D optical lattice. Such a quantum phase transition is not only fundamentally profound, but also provides new access to applications of BECs. This paper reports our recent theoretical work on the spin dynamics and magnetism of BECs confined in an optical lattice, and explores their possible applications in atomic magnetic sensors and quantum computation.
Two component Bose-Einstein condensates (BECs) have been recently shown to be viable systems for storing and manipulating quantum information. Unlike standard single-particle qubits, the quantum information is duplicated in a large number of identical bosonic particles, thus can be considered to be a macroscopic qubit. The duplication of the quantum information makes them potentially more robust than conventional qubits, where all the quantum information is lost with a single error. It has been shown theoretically and experimentally that such ensembles can be used in many ways the same way as a standard qubit: they can be visualized on the Bloch sphere, and can be manipulated analogously to standard qubits. On the other hand, the BEC qubits do not have genuine interaction between each other and one of the main difficulties with such a system is how to effectively interact them together in order to transfer quantum information and create entanglement. Furthermore, the larger Hilbert space of the macroscopic bosonic system does not allow for unique mapping of standard quantum algorithms. However, in a few past years the main building blocks of quantum information processing and several quantum algorithms were translated to the BEC qubits. In this paper we give a brief colloquium of the current achievements and outline new perspectives of the use of BEC qubits and spin-coherent ensembles for quantum technologies.
Bose–Einstein condensates (BEC) of sodium atoms are transferred into one-dimensional (1D) optical lattice potentials, formed by two laser beams with a wavelength of 1064 nm, in a shallow optical trap. The phase coherence of the condensate in the lattice potential is studied by changing the lattice depth. A qualitative change in behavior of the BEC is observed at a lattice depth of ∼ 13.7 E r, where the quantum gas undergoes a transition from a superfluid state to a state that lacks well-to-well phase coherence.
Phase coherence and localization of Bose-Einstein Condensates (BEC) in optical lattices is a notable property that is changing under the superfluid-Mott insulator phase transition. For a single-component BEC, long-range phase coherence across lattice wells is lost as the lattice height is sufficiently increased. Phase coherences of two-component overlapping mixture of BECs, however, are nontrivial because of the interspecies interaction between components, affecting localization of condensates. For such cases, the BECs form two interdependent mean-field lattices made of each component in addition to optical lattices. We prepare initial BEC clouds as a collection of small Bogoliubov quasiparticles added to classical mean-fields, and obtain the dynamics under phase space representations, where the density operators are projected onto phase space. Among various phase space representations, we focus on a Truncated Wigner Approximation (TWA) and a positive P representation, and we construct a hybrid model to combine them. Since a high volume of numerical integration is required for sufficient convergence from ensembles, we construct efficient numerical methods to reduce the sample variance. We demonstrate time evolution of phase coherence of one-component and two-component BECs in state-dependent optical lattices. For one-component BECs, we show that phase coherence loss depends on lattice heights. In the cycling process of lattice height changes, we investigate nonadiabatic effects depending on initial ramp-up speeds. The hybrid model compared with the TWA exhibits temporal fluctuations in coherence and sudden phase diffusion at the end of ramp-up of a lattice. For two-component BECs, we consider that the two wavefunctions are localized at periodic potential wells whose positions differ by a half-period between the two. Dependence on population imbalance between two components becomes another factor on coherence loss. We show that under gradual loading of an optical lattice effective only to first component, loss of phase coherence of first atoms is enhanced and that of second atoms is weakened as the fraction of second atoms is increased in the mixture. We also demonstrate effects of localizing mean-field lattices on coherence loss under various lattice heights.
Bose-Einstein condensate (BEC) is a new state of matter, experimentally realized in a dilute alkali gas of weakly interacting bosons. The gas is cooled to very low temperatures, of the order of nano-Kelvin, at which point a large fraction of the atoms occupy the lowest quantum state. Under such conditions, the quantum effects become apparent on a macroscopic scale. Many fundamental ideas of quantum mechanics have been realized and proven to be true in experiments, by studying the Bose-Einstein condensates. Amongst them, the observation of macroscopic matter wave interference, the creation of vortices, dark and bright solitons etc, find special mention. The dynamics of this system in presence of an optical lattice reveals a superfluid to Mott insulator quantum phase transition. This has opened a new avenue for research on BEC and its applications. From a theoretical standpoint, the dynamics of BEC can be described by means of an effective mean-field theory. This formalism is much simpler than the many-body Schr¨odinger equation and based on a classical nonlinear evolution equation, known as Gross-Pitaevskii equation. The nonlinear nature of the GPE leads to several interesting phenomena like soliton propagation, vortices, four wave mixing etc., which are closely related to nonlinear fiber optics. The present thesis deals with nonlinear excitations and different phases of BEC and is organized as follows: after a brief introduction, in chapter two, we start with BEC in a one-dimensional shallow optical lattice, experiencing both cubic and quintic nonlinearity. A superfluid density wave, of twice the periodicity of lattice potential, is identified. This indicates matter redistribution, where atoms from neighbouring sites have been depleted leading to a density wave behaviour. 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 presence of pure two-body interaction, the superfluid density shows a periodicity commensurate with the lattice potential. Apart from the superfluid state, a density wave insulating phase is also found to exist, possessing two frequency modulations commensurate with the lattice potential. A dynamical phase transition connects the superfluid phase with an insulating phase, which in turn, is connected with the density wave insulating phase. The solutions are found to be marginally stable, as per the Vakhitov-Kolokolov (VK) criterion. Chapter three describes the dynamics of the atomic-molecular Bose-Einstein condensate (AMBEC) by mean-field Gross-Pitaevskii equation (GPE), taking into account the atom-molecular interaction, as well as atom-molecule inter-conversion. The two-body atom-atom, atom-molecule and molecule-molecule interactions are analogous to the cubic nonlinearity of the standard GPE, whereas, the quadratic nonlinear term arising from the inter-conversion between atoms and molecules is identical to the scenario in strongly coupled BEC in one dimension. The presence of both weak and strong interactions adds further richness to this system. We find the molecular BEC possesses a complex dark (grey) soliton, while the atomic condensate is a bright one. The energy and momentum of this system is computed in order to see the effect of the both weak and strong interactions on Lieb mode profile and compare it with the pure atomic condensate profile. Another class of solutions exist in this system, where both atomic and molecular condensates possess bright solitons. In chapter four, we investigate the effect of a harmonic trap on the dynamics of grey solitons in Bose-Einstein condensates. It is observed that, these nonlinear excitations of the cigar-shaped Bose-Einstein condensate have strong coupling with the trap at low momenta and hence can be effectively isolated from the Bogoliubov sound modes, which respond weakly to harmonic confinement. This strong coupling with the trap also makes the grey soliton amenable for control and manipulation through trap modulation and temporal variation of the two-body interaction. We have shown that in presence of an expulsive oscillator trap, the grey soliton gets amplified, while having a compression in its width. Exact solutions are also obtained in case of the two component Bose-Einstein condensates (TBECs), where one component is a grey soliton and the other a bright one. The presence of the trap and inter-species interaction significantly affects the grey soliton dynamics. Similar to the single component, in the present case, we have shown that both grey and bright soliton can be compressed simultaneously. A detailed study of the energy and momentum of the soliton is reported, in order to find the effect of the trap on the Lieb mode. In chapter five, we study a BEC-defect complex in a quasi-one dimensional scenario, where the defect atom moves in the condensate with constant velocity. Explicit solutions of the coupled mean-field equations, describing defect-grey soliton dynamics are obtained, demonstrating the coexistence of grey soliton and a localized defect. Unlike the case of dark soliton, where the defect trapping center has vanishing superfluid density, the moving grey soliton necessarily possesses a finite superfluid component at the defect location. The presence of the defect can lower the grey soliton’s maximum velocity, as compared to the defect free case, where the maximum velocity is the sound velocity. We have also carried out the stability analysis of the obtained solutions using renormalized momentum formalism and Vakhitov-Kolokolov criteria and find that the obtained solutions are stable through both the methods. The grey soliton’s energy gets substantially modified through its interaction with the defect, opening up the possibility of its control through defect dynamics. Chapter six describes a parametrically forced Bose-Einstein condensate in the presence of a general moving optical lattice. The interaction between the atoms in the condensate and the time dependent lattice potential leads to a novel propagating superfluid matter wave, which can be controlled through chirp management. This system, when placed in a trap, accelerates and undergoes rapid nonlinear compression, controlled by the chirp. The density achieves its maximum, when the matter wave changes direction. It is also shown that this system exhibits dynamical superfluid insulator transition, where superfluidity breaks down and the condensate transits to an insulating phase. The stability has been investigated using Vakhitov-Kolokolov criterion and found that it depends on the chirp, as well as on the nature of the atom-atom interaction. The exact expression for energy is obtained and analyzed in detail to gain physical understanding of the chirp management of the sinusoidal excitations and also the dynamical phase transition.
A binary mixture of interacting Bose-Einstein condensates (BEC) in the presence of fragmentation-driving external lattice potentials forms two interdependent mean-field lattices made of each component. These effective mean-field lattices, like ordinary optical lattices, can induce additional fragmentation and phase coherence loss of BECs between lattice sites. In this study, we consider the nonequilibrium dynamics of two hyperfine states of one-dimensional Bose-Einstein condensates, subjected to state-dependent optical lattices. Our numerical calculations using the truncated Wigner approximation (TWA) show that phase coherence in a mixture of two-component BECs can be lost not just by optical lattices, but by mean-field lattices gradually formed by other components, and we reveal that such an effect of internal mean-field lattices, however, is limited, contrary to external optical lattices, in regard to phase decoherence.