Spinor Condensates and Light Scattering from Bose-Einstein Condensates

Stamper-Kurn, Dan; Ketterle, Wolfgang

doi:10.1007/3-540-45338-5_2

Cited by 51 publications

(90 citation statements)

References 109 publications

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“…One may also regard this instability as the phase separation of a two-component condensate [17,18], recalling that the |m z = 0 state represents an equal superposition of the |m φ = ±1 eigenstates of any transverse spin operator,F φ =F x cos φ +F y sin φ. Additionally, since c 2 < 0, the ±1 eigenstates of any spin component are immiscible [19]. Ferromagnetism thus arises by the spinodal decomposition of a binary gaseous mixture into neighboring regions of oppositely oriented transverse magnetization.…”

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confidence: 99%

Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose–Einstein condensate

Sadler

Higbie²,

Leslie³

et al. 2006

Nature

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941

1,084

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A central goal in condensed matter and modern atomic physics is the exploration of manybody quantum phases and the universal characteristics of quantum phase transitions in so far as they differ from those established for thermal phase transitions. Compared with condensedmatter systems, atomic gases are more precisely constructed and also provide the unique opportunity to explore quantum dynamics far from equilibrium. Here we identify a second-order quantum phase transition in a gaseous spinor BoseEinstein condensate, a quantum fluid in which superfluidity and magnetism, both associated with symmetry breaking, are simultaneously realized. 87 Rb spinor condensates were rapidly quenched across this transition to a ferromagnetic state and probed using in-situ magnetization imaging to observe spontaneous symmetry breaking through the formation of spin textures, ferromagnetic domains and domain walls. The observation of topological defects produced by this symmetry breaking, identified as polar-core spin-vortices containing non-zero spin current but no net mass current, represents the first phase-sensitive in-situ detection of vortices in a gaseous superfluid.Most ultracold atomic gases consist of atoms with nonzero total angular momentum denoted by the quantum number F , which is the sum of the total electronic angular momentum and nuclear spin. In spinor atomic gases, such as F = 1 and F = 2 gases of 23 Na [1, 2] and 87 Rb [3,4], all magnetic sublevels representing all orientations of the atomic spin may be realized [5]. The phase coherent portion of a Bose-Einsein condensed spinor gas is described by a vector order parameter and therefore exhibits spontaneous magnetic ordering. Nevertheless, considerable freedom remains for the type of ordering that can occur. For 87 Rb F = 1 spinor gases, the spindependent energy per particle in the condensate is the sum of two terms, c 2 n F 2 + q F 2 z , where F denotes the dimensionless spin vector operator. The first term describes spin-dependent interatomic interactions, with n being the number density and c 2 = (4π 2 /3m)(a 2 − a 0 ) depending on the atomic mass m and the s-wave scattering lengths a f for collisions between pairs of particles with total spin f [6,7]. Given c 2 < 0 for our system [3,4,8,9], the interaction term alone favors a ferromagnetic phase with broken rotational symmetry. The second term describes a quadratic Zeeman shift in our experiment, with q = (h × 70 Hz/G 2 )B 2 at a magnetic field of magnitude B [10]. This term favors instead a scalar phase with no net magnetization, i.e. a condensate in the |m z = 0 magnetic sublevel. These phases are divided by a second-order quantum phase transition at q = 2|c 2 |n.This article describes our observation of spontaneous symmetry breaking in a 87 Rb spinor BEC that is rapidly quenched across this quantum phase transition. Nearlypure spinor Bose-Einstein condensates were prepared in the scalar |m z = 0 phase at a high quadratic Zeeman shift (q ≫ 2|c 2 |n). By rapidly reducing the magnitude of the applied magneti...

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Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose–Einstein condensate

Sadler

Higbie²,

Leslie³

et al. 2006

Nature

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941

1,084

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“…The hyperfine degree of freedom in these systems allows complex ordered states, distinct from those occurring in more conventional systems of spinless bosons, and combining superfluidity with different types of magnetic behavior. The dynamics and topological defects of such states can be observed either destructively or in situ [3] [4]. In particular, the experimental study of two-dimensional condensates has been of special interest recently for at least three reasons: the superfluid transition in two dimensions (2D) is of unconventional Kosterlitz-Thouless (KT) type [5] (i.e., driven by unbinding of vortex defects); twodimensional superfluids have power-law correlations of the quantum phase, rather than true long-range order; and two-dimensional models are appropriate for some current experiments [4] [6].…”

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Topological Defects and the Superfluid Transition of thes=1Spinor Condensate in Two Dimensions

2006

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The s = 1 spinor Bose condensate at zero temperature supports ferromagnetic and polar phases that combine magnetic and superfluid ordering. We analyze the topological defects of the polar condensate, correcting previous studies, and show that the polar condensate in two dimensions is unstable at any finite temperature; instead there is a nematic or paired superfluid phase with algebraic order in exp(2iθ), where θ is the superfluid phase, and no magnetic order. The KosterlitzThouless transition out of this phase is driven by unbinding of half-vortices (the spin-disordered version of the combined spin and phase defects found by Zhou), and the anomalous universal 8Tc/π stiffness jump at the transition is confirmed in numerical simulations. The anomalous stiffness jump is a clear experimental signature of this phase and the corresponding phase transition. . In particular, the experimental study of two-dimensional condensates has been of special interest recently for at least three reasons: the superfluid transition in two dimensions (2D) is of unconventional Kosterlitz-Thouless (KT) type [5] (i.e., driven by unbinding of vortex defects); twodimensional superfluids have power-law correlations of the quantum phase, rather than true long-range order; and two-dimensional models are appropriate for some current experiments [4][6].Topological defects and ordered phases in the s = 1 (i.e., total spin F = 1) spinor condensate have been discussed theoretically in many papers since the work of Ho [7] and Ohmi and Machida [8]. Experimentally, s = 1 systems are realized using atoms of 23 Na, 39 K and 87 Rb with nuclear spin I = 3/2. In this letter, we resolve the nature of the topological defects in the polar or antiferromagnetic phase by explicitly obtaining the order parameter manifold and its first homotopy group, then show that a new phase results in any polar s = 1 condensate at finite temperature in 2D. For the polar phase, Zhou [9] found the order parameter manifold (U (1)×S 2 )/Z 2 by noticing an Z 2 symmetry omitted in earlier work of Ho [7], but obtained an incorrect first homotopy group [10]; that this homotopy group was incorrect was shown by Makëlä et al. [11], but these authors claimed that Zhou's order parameter manifold was also wrong. We show that Zhou's order parameter manifold is correct but has the homotopy groups obtained indirectly in Ref. 11. These topological defects are crucial because they create a new phase in two dimensions: we find that the 2D polar condensate is unstable to a paired or nematic phase at any finite temperature. This phase has algebraic order in 2θ where θ is the superfluid phase, no spin order, and a Kosterlitz-Thouless superfluid transition driven by unconventional topological defects (half-vortices). It has a clear experimental signature: an anomalous superfluid stiffness jump, which can be observed in an optically trapped condensate by the approach used by Hadzibabic et. al [12] to measure the conventional stiffness jump.The Hamiltonian for the s = 1 spinor condensate is [7] H = d...

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“…Most studies were thus focused on scalar models, i.e., single-component quantum degenerate gases [12]. One of the most important recent developments in BECs was the study of spin-1 condensates (of atoms with hyperfine quantum number F = 1) [17,27,34,10,31], and they were realized in experiments recently using both 23 Na and 87 Rb [24,35]. In fact, the emergence of the spin-1 BEC [19,20,24] has created opportunities for understanding degenerate gases with internal degrees of freedom [21,22,17,18,14,25,26,32,37].…”

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A Mass and Magnetization Conservative and Energy-Diminishing Numerical Method for Computing Ground State of Spin-1 Bose–Einstein Condensates

Bao¹,

Wang²

2007

SIAM J. Numer. Anal.

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Abstract. In this paper, a mass (or normalization) and magnetization conservative and energydiminishing numerical method is presented for computing the ground state of spin-1 (or F = 1 spinor) Bose-Einstein condensates (BECs). We begin with the coupled Gross-Pitaevskii equations, and the ground state is defined as the minimizer of the energy functional under two constraints on the mass and magnetization. By constructing a continuous normalized gradient flow (CNGF) which is mass and magnetization conservative and energy-diminishing, the ground state can be computed as the steady state solution of the CNGF. The CNGF is then discretized by the Crank-Nicolson finite difference method with a proper way to deal with the nonlinear terms, and we prove that the discretization is mass and magnetization conservative and energy-diminishing in the discretized level. 1. Introduction. Since its realization in dilute bosonic atomic gases [2, 13, 9], the atomic Bose-Einstein condensate (BEC) has been produced and studied extensively in the laboratory [28,29,16] and has provided a successful testing ground of theoretical studies of quantum many-body systems [28,29]. In earlier BEC experiments, atoms were spatially confined with magnetic traps, which essentially freeze the atomic internal degrees of freedom [2,13,9]. Most studies were thus focused on scalar models, i.e., single-component quantum degenerate gases [12]. One of the most important recent developments in BECs was the study of spin-1 condensates (of atoms with hyperfine quantum number F = 1) [17,27,34,10,31], and they were realized in experiments recently using both 23 Na and 87 Rb [24,35]. In fact, the emergence of the spin-1 BEC [19,20,24] has created opportunities for understanding degenerate gases with internal degrees of freedom [21,22,17,18,14,25,26,32,37].At temperature T much smaller than the critical condensate temperature T c [23], a spin-1 BEC is well described by the three-component wave function Ψ = (ψ 1 (x, t), ψ 0 (x, t), ψ −1 (x, t)) T whose evolution is governed by the coupled Gross-Pitaevskii equations (GPEs) [23,17,18,38,36]:

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Spinor Condensates and Light Scattering from Bose-Einstein Condensates

Cited by 51 publications

References 109 publications

Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose–Einstein condensate

Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose–Einstein condensate

Topological Defects and the Superfluid Transition of thes=1Spinor Condensate in Two Dimensions

A Mass and Magnetization Conservative and Energy-Diminishing Numerical Method for Computing Ground State of Spin-1 Bose–Einstein Condensates

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