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arXiv:1109.1012 · Physical Review Letters

Quantum Phase Transition in an Antiferromagnetic Spinor Bose-Einstein Condensate

Eva Bookjans, Anshuman Vinit, Chandra Raman

Published Nov 4, 2011• 129 citations
arXivHTMLPDF

Abstract

We have experimentally observed the dynamics of an antiferromagnetic sodium Bose-Einstein condensate quenched through a quantum phase transition. Using an off-resonant microwave field coupling the F = 1 and F = 2 atomic hyperfine levels, we rapidly switched the quadratic energy shift q from positive to negative values. At q = 0, the system undergoes a transition from a polar to antiferromagnetic phase. We measured the dynamical evolution of the population in the F = 1, mF = 0 state in the vicinity of this transition point and observed a mixed state of all 3 hyperfine components for q < 0. We also observed the coarsening dynamics of the instability for q < 0, as it nucleated small domains that grew to the axial size of the cloud.

DOI
10.1103/physrevlett.107.195306

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We have experimentally observed the dynamics of an antiferromagnetic sodium Bose-Einstein condensate quenched through a quantum phase transition. Using an off-resonant microwave field coupling the F = 1 and F = 2 atomic hyperfine levels, we rapidly switched the quadratic energy shift q from positive to negative values. At q = 0, the system undergoes a transition from a polar to antiferromagnetic phase. We measured the dynamical evolution of the population in the F = 1, mF = 0 state in the vicinity of this transition point and observed a mixed state of all 3 hyperfine components for q < 0. We also observed the coarsening dynamics of the instability for q < 0, as it nucleated small domains that grew to the axial size of the cloud.
This thesis discusses non-equilibrium phenomena in antiferromagnetic spinor quantum gases. We investigate, both experimentally and theoretically, the quench dynamics of antiferromagnetic spinor Bose-Einstein condensates in the vicinity of a zero temperature quantum phase transition. A microwave dressing field was used as a novel tool to access the polar to antiferrmagnetic quantum phase transition at the quadratic Zeeman shift, q=0. By performing instantaneous quenches across the phase transition,a dynamical instability was induced in the atomic cloud. This lead to pair formation and rapid amplification of spin +/-1 atoms through spin-mixing collisions. The spatial ordering kinetics in the vicinity of the phase transition was quantified by performing the local spin measurement. The coarsening dynamics was observed for the instability for q < 0, as it nucleated small domains that grew to the axial size of the cloud. Rich nonequilibrium behavior was observed in the form of growth and decay of both nematic and magnetic spin waves, following the quench. The spatiotemporal evolution was characterized through two particle correlations between atoms in each pair of spin states, which revealed dramatic differences between the dynamics of the spin correlations and those of the spin populations. \nFinally, we demonstrate the remarkable influence of linear Zeeman term on non-equilibrium dynamics of spinor Bose-Einstein condensates. We show that contrary to prior understanding magnetic field gradient actually suppresses the tendency of spin 0 state to separate into spin +/-1 state. This reconciles the Bogoliubov discrepancies and leads to a dramatic sharpening of the transition point, resulting in a resolution at the 1 Hz level. Our results point to the use of dynamics, rather than equilibrium quantities for high precision measurements of phase transitions in quantum gases.
We have experimentally investigated the quench dynamics of antiferromagnetic spinor Bose-Einstein condensates in the vicinity of a zero temperature quantum phase transition at zero quadratic Zeeman shift qqq. The rate of instability shows good agreement with predictions based upon solutions to the Bogoliubov--de Gennes equations. A key feature of this work was removal of magnetic field inhomogeneities, resulting in a steep change in behavior near the transition point. The quadratic Zeeman shift at the transition point was resolved to 250 mHz uncertainty, equivalent to an energy resolution of ${k}_{B}\phantom{\rule{4pt}{0ex}}\ifmmode\times\else\texttimes\fi{}$ (12 pK). A small (2--3 $\ensuremath{\sigma}$) shift of the transition point was observed, from q=0q=0q=0 to q=+650q=+650q=+650 mHz, whose physical mechanism is currently unknown. In this work, we demonstrate a sub-Hz precision measurement of a phase transition in quantum gases. It paves the way toward observing shifts of the transition point due to finite particle number NNN that scale as 1/N1/N1/N, and also to potential Heisenberg limited spectroscopy with antiferromagnetic spinor gases [L.-N. Wu and L. You, Phys. Rev. A 93, 033608 (2016)].
We investigate the phase-transition dynamics of a quasi-two-dimensional antiferromagnetic spin-1 Bose-Einstein condensate from the easy-axis polar phase to the easy-plane polar phase, which is initiated by suddenly changing the sign of the quadratic Zeeman energy qqq. We observe the emergence and decay of spin turbulence and the formation of half-quantum vortices (HQVs) in the quenched condensate. The characteristic time and length scales of the turbulence generation dynamics are proportional to ${|q|}^{\ensuremath{-}1/2}$ as inherited from the dynamic instability of the initial state. In the evolution of the spin turbulence, spin-wave excitations develop from large to small length scales, suggesting a direct energy cascade, and the spin population for the axial polar domains exhibit a nonexponential decay. The final equilibrated condensate contains HQVs, and the number is found to increase and saturate with increasing ∣q∣|q|∣q∣. Our results demonstrate the time-space scaling properties of the phase-transition dynamics near the critical point and the peculiarities of the spin-turbulence state of the antiferromagnetic spinor condensate.
A dynamical quantum phase transition can be characterized by a nonanalytic change in the quench dynamics when a parameter in the governing Hamiltonian is varied. Such a transition typically only shows up in long-time dynamics for extensive systems with short-range couplings. We analyze a model Hamiltonian of spin-1 particles with effectively infinite-range couplings and demonstrate that for this system the nonanalytic transition occurs for local observables in short-time durations even when the system is of a large size. We experimentally realize this model Hamiltonian and observe the dynamical quantum phase transition in an antiferromagnetic spinor Bose-Einstein condensate of around 105{10}^{5}105 sodium atoms. Our observations agree well with the theoretical prediction. We also analyze the scaling exponent near the dynamical phase transition and discuss its relation with the excited-state spectrum of the system.
Spinor Bose-Einstein condensates under external magnetic fields exhibit well-characterized spin domains of its ground state due to spin-dependent interactions. At low temperatures, collision-induced spin-mixing instabilities may promote the condensate to dwell into metastable states occurring near the phase boundaries. In this paper, we study theoretically the metastable spin-phase diagram of a spin-1 antiferromagnetic Bose-Einstein condensate at zero and finite temperatures. The approach makes use of Hartree-Fock theory and exploits the symmetry of the Hamiltonian and of the order parameters yielding a closed system of transcendental equations for the free energy, fully avoiding the use of self-consistency methods. Our results are consistent with recent experiments and allow us to explain qualitatively the different types of observed quench dynamics. In addition, we found that similar phenomena should occur in antiferromagnetic spinor condensates with a sudden change in the temperature. It is shown also that the increase in temperature induces a traceable shift of the ferromagnetic-polar transition boundary, behavior previously not noted by self-consistent mean-field calculations.
The study of quantum degenerate gases has many applications in topics such as condensed matter dynamics, precision measurements and quantum phase transitions. We built an apparatus to create 87Rb Bose-Einstein condensates (BECs) and generated, via optical and magnetic interactions, novel quantum systems in which we studied the contained phase transitions. For our first experiment we quenched multi-spin component BECs from a miscible to dynamically unstable immiscible state. The transition rapidly drives any spin fluctuations with a coherent growth process driving the formation of numerous spin polarized domains. At much longer times these domains coarsen as the system approaches equilibrium. For our second experiment we explored the magnetic phases present in a spin-1 spin-orbit coupled BEC and the contained quantum phase transitions. We observed ferromagnetic and unpolarized phases which are stabilized by the spin-orbit coupling’s explicit locking between spin and motion. These two phases are separated by a critical curve containing both first-order and second-order transitions joined at a critical point. The narrow first-order transition gives rise to long-lived metastable states. For our third experiment we prepared independent BECs in a double-well potential, with an artificial magnetic field between the BECs. We transitioned to a single BEC by lowering the barrier while expanding the region of artificial field to cover the resulting single BEC. We compared the vortex distribution nucleated via conventional dynamics to those produced by our procedure, showing our dynamical process populates vortices much more rapidly and in larger number than conventional nucleation.