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Physical Review A

Patterns and excitations in antiferromagnetic spinor condensates

Michał Matuszewski, Kai Bongs, Jochen Kronjäger

Published Feb 27, 2012• 1 citation
Publisher

Abstract

We consider the phenomenon of spin-pattern formation reported in recent experiments in spin-2 rubidium condensate [J. Kronj\"ager, C. Becker, P. Soltan-Panahi, K. Bongs, and K. Sengstock, Phys. Rev. Lett. 105, 090402 (2010)]. To understand the mechanism underlying the process and explain the parameters such as pattern period and growth rate, we employ the Bogoliubov--de Gennes equations and an alternative method based on energy conservation and the uncertainty principle. While the two methods agree in the spin-1 case, only the second method can provide analytical results in the spin-2 case. Using these results, we explain the occurrence of two regimes, the spin-dominated and the Zeeman-dominated, with the prevalence of spin-wave modes and quadrupole modes, respectively. We obtain very good agreement between theory and experiment for the pattern period and growth rate over several orders of magnitude of the magnetic field strength.

DOI
10.1103/physreva.85.023635

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We consider the phenomenon of spin-pattern formation reported in recent experiments in spin-2 rubidium condensate [J. Kronj"ager, C. Becker, P. Soltan-Panahi, K. Bongs, and K. Sengstock, Phys. Rev. Lett. 105, 090402 (2010)]. To understand the mechanism underlying the process and explain the parameters such as pattern period and growth rate, we employ the Bogoliubov--de Gennes equations and an alternative method based on energy conservation and the uncertainty principle. While the two methods agree in the spin-1 case, only the second method can provide analytical results in the spin-2 case. Using these results, we explain the occurrence of two regimes, the spin-dominated and the Zeeman-dominated, with the prevalence of spin-wave modes and quadrupole modes, respectively. We obtain very good agreement between theory and experiment for the pattern period and growth rate over several orders of magnitude of the magnetic field strength.
We show that metastable phases of an antiferromagnetic spin-1 condensate in a simple model with pure contact interactions can exhibit a rotonlike minimum in the excitation spectrum. The introduction of a magnetic field gives rise to the instability of roton modes, which can lead to spontaneous emergence of regular periodic, polygonal, polyhedral, or crystalline patterns, as shown in numerical simulations within the truncated Wigner approximation. An explanation of the occurrence of rotonlike instability is given based on the energy and spin conservation laws.
We present an analytical model for the theoretical analysis of spin dynamics and spontaneous symmetry breaking in a spinor Bose-Einstein condensate (BEC). This allows for an excellent intuitive understanding of the processes and provides good quantitative agreement with the experimental results of Scherer et al. [Phys. Rev. Lett. 105, 135302 (2010)]. It is shown that the dynamics of a spinor BEC initially prepared in an unstable Zeeman state mF=0{m}_{F}=0mF​=0 ($\left|0\right\ensuremath{\rangle}$) can be understood by approximating the effective trapping potential for the state $\left|\ifmmode\pm\else\textpm\fi{}1\right\ensuremath{\rangle}$ with a cylindrical box potential. The resonances in the creation efficiency of these atom pairs can be traced back to excitation modes of this confinement. The understanding of these excitation modes allows for a detailed characterization of the symmetry-breaking mechanism, showing how a twofold spontaneous breaking of spatial and spin symmetry can occur. In addition, a detailed account of the experimental methods for the preparation and analysis of spinor quantum gases is given.
We establish a theoretical correspondence between spin-one antiferromagnetic spinor condensates in an external magnetic field and quantum rotor models in an external potential. We show that the rotor model provides a conceptually clear picture of the possible phases and dynamical regimes of the antiferromagnetic condensate. We also show that this mapping simplifies calculations of the condensate's spectrum and wave functions. We use the rotor mapping to describe the different dynamical regimes recently observed in 23Na^{23}\mathrm{Na}23Na condensates [Phys. Rev. Lett. 102, 125301 (2009); Phys. Rev. Lett. 102, 225301 (2009)]. We also suggest a way to experimentally observe quantum-mechanical effects (collapse and revival) in spinor condensates.
We study the formation of transient Faraday patterns and spin textures in driven quasi-one-dimensional and quasi-two-dimensional spin-1 Bose-Einstein condensates under the periodic modulation of sss-wave scattering lengths a0a_0a0​ and a2a_2a2​, starting from the anti-ferromagnetic phase. This phase is characterized by a Bogoliubov spectrum consisting of three modes: one mode is gapped, while the other two are gapless. When a0a_0a0​ is modulated and half of the modulation frequency lies below the gapped mode, density and spin Faraday patterns emerge. In that case, in quasi-one-dimension, the spin texture is characterized by periodic domains of opposite zzz-polarizations. When driven above the gap, the spin texture is characterized by random orientations of spin vectors along the condensate axis. Qualitatively new features appear in the driven quasi-two-dimensional condensate. For instance, when driven above the gap, the spin textures are characterized by anomalous vortices and antivortices that do not exhibit phase winding in individual magnetic components. Below the gap, the spin texture exhibits irregular ferromagnetic patches with opposite polarizations. The spatial spin-spin correlations in quasi-one-dimension exhibit a Gaussian envelope, whereas they possess a Bessel function dependence in quasi-two-dimension. Under the a2a_2a2​-modulation, the density patterns dominate irrespective of the driving frequency, unless the spin-dependent interaction strength is sufficiently smaller than that of the spin-independent interaction. The intriguing scenario of competing instability can emerge when both scattering lengths are simultaneously modulated. Finally, we show that the competing instabilities result in a complex relationship between the population transfer and the strength of the quadratic Zeeman field, while keeping all other parameters constant.
We study the spin dynamics of a spin-1 ferromagnetic Bose-Einstein condensate with magnetic dipole-dipole interaction (MDDI) based on the Gross-Pitaevskii and Bogoliubov theories. We find that various magnetic structures such as checkerboards and stripes emerge in the course of the dynamics due to the combined effects of spin-exchange interaction, MDDI, quadratic Zeeman and finite-size effects, and nonstationary initial conditions. However, the short-range magnetic order observed by the Berkeley group [Phys. Rev. Lett. 100, 170403 (2008)] is not fully reproduced in our calculations; the periodicity of the order differs by a factor of 3 and the checkerboard pattern eventually dissolves in the course of time. Possible reasons for the discrepancy are discussed.
We study experimentally the equilibrium phase diagram of a spin-1 Bose-Einstein condensate with antiferromagnetic interactions, in a regime where spin and spatial degrees of freedom are decoupled. For a given total magnetization mz{m}_{z}mz​, we observe for low magnetic fields an ``antiferromagnetic'' phase where atoms condense in the $m=\ifmmode\pm\else\textpm\fi{}1$ Zeeman states, and occupation of the m=0m=0m=0 state is suppressed. Conversely, for large enough magnetic fields, a phase transition to a ``broken-axisymmetry'' phase takes place: The m=0m=0m=0 component becomes populated and rises sharply above a critical field Bc(mz){B}_{c}({m}_{z})Bc​(mz​). This behavior results from the competition between antiferromagnetic spin-dependent interactions (dominant at low fields) and the quadratic Zeeman energy (dominant at large fields). We compare the measured Bc{B}_{c}Bc​ as well as the global shape of the phase diagram with mean-field theory, and find good quantitative agreement.