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Universality class of a spinor Bose–Einstein condensate far from equilibrium

Nature Physics volume 20, pages 402–408 (2024)Cite this article

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Abstract

Scale invariance and self-similarity in physics provide a unified framework for classifying phases of matter and dynamical properties near equilibrium in both classical and quantum systems. This paradigm has been further extended to isolated many-body quantum systems driven far from equilibrium, for which the physical observables exhibit dynamical scaling with universal scaling exponents. Universal dynamics appear in a wide range of scenarios, including cosmology, quark–gluon matter, ultracold atoms and quantum spin magnets. However, how the universal dynamics depend on the symmetry of the underlying Hamiltonian in non-equilibrium systems remains an outstanding challenge. Here we report on the classification of universal coarsening dynamics in a quenched two-dimensional ferromagnetic spinor Bose gas. We observe spatio-temporal scaling of spin correlation functions with distinguishable scaling exponents that characterize binary and diffusive fluids. The universality class of the coarsening dynamics is determined by the symmetry of the order parameter and the dynamics of the topological defects, such as domain walls and vortices. Our results categorize the universality classes of far-from-equilibrium quantum dynamics based on the symmetry properties of the system.

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Fig. 1: Universal coarsening dynamics and topological defects.
Fig. 2: Dynamic scaling and power-law growth of the domain length.
Fig. 3: Dynamic scaling of the spin structure factor in the easy-axis quench.
Fig. 4: Universal coarsening dynamics in the easy-axis phase.
Fig. 5: Coarsening dynamics in the isotropic ferromagnetic phase.

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Acknowledgements

We acknowledge discussions with I. Bloch, S.-B. Chung, F. Fang, T. Hilker, G. C. Katsimiga, P. G. Kevrekidis, K. Kim, S. K. Kim, S. Majumder, S. M. Reimann and Y. Shin. K.M. acknowledges the PARAM Shakti at the Indian Institute of Technology Kharagpur, a national supercomputing mission, Government of India, for providing computational resources. J.-y.C. is supported by the Samsung Science and Technology Foundation (Grant No. BA1702-06), a National Research Foundation of Korea (NRF) grant (Project Nos. RS-2023-00207974 and 2023M3K5A1094812) and the KAIST UP programme. S.I.M. and H.R.S. acknowledge support from the NSF through a grant to the Institute for Theoretical Atomic Molecular and Optical Physics at Harvard University. K.M. is financially supported by the Knut and Alice Wallenberg Foundation (Grant No. 2018.0217) and the Swedish Research Council and also acknowledges the Ministry of Human Resource Development, Government of India, for a research fellowship at the early stages of this work.

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Authors and Affiliations

  1. Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, South Korea

    SeungJung Huh, Kiryang Kwon, Jihoon Seo, Junhyeok Hur & Jae-yoon Choi

  2. Department of Mathematical Physics and NanoLund, LTH, Lund University, Lund, Sweden

    Koushik Mukherjee

  3. Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur, India

    Koushik Mukherjee

  4. Institute for Theoretical Atomic Molecular and Optical Physics, Center for Astrophysics, Harvard & Smithsonian, Cambridge, MA, USA

    Simeon I. Mistakidis & H. R. Sadeghpour

  5. Department of Physics, Harvard University, Cambridge, MA, USA

    Simeon I. Mistakidis

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Contributions

All authors contributed substantially to the work presented in this manuscript. S.-J.H., K.K., J.S. and J.H. maintained the experimental apparatus and collected the data. S.-J.H., K.K. and J.S. analysed the data. K.M. and S.M. performed the numerical simulations. This work was supervised by S.I.M., H.R.S. and J.-y.C.

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Correspondence to Jae-yoon Choi.

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Extended data

Extended Data Fig. 1 Spin populations after quenching the quadratic Zeeman energy.

a, After quenching to the isotropic ferromagnetic phase (q/h = 0 Hz), the atoms in the spin \(\left\vert 0\right\rangle\) state rapidly decay and create spin \(\left\vert \pm 1\right\rangle\) states. The spin population reach a steady state after 100 ms with equal population (n1, n0, n−1) ≃ (1/3, 1/3, 1/3). During the whole coarsening dynamics, the spin population for all spin states remains constant. In the easy-axis ferromagnetic phase (b,q/h = − 120 Hz and c,q/h = − 200 Hz), the initial \(\left\vert 0\right\rangle\) state rapidly disappear and generate equal population of the spin \(\left\vert \pm 1\right\rangle\) state. The residual spin component in the \(\left\vert 0\right\rangle\) state during the coarsening dynamics is attributed to the spin vector along the horizontal plane at the domain wall (Extended Data Fig. 4). Because of the microwave dressing field, the spin population gradually changes. d, Time evolution of spin population imbalance (\({{{\mathscr{I}}}}={n}_{1}-{n}_{-1}\)) under different quadratic Zeeman energy. The population imbalance is noticeable in the deep easy-axis regime (q/h = − 200 Hz), but its impact on the domain length is not significant as shown in Fig. 2. Each data point is obtained with more than 100 independent experimental runs, and the error bars represent one standard error of the mean.

Source data

Extended Data Fig. 2 Full-time evolution of coarsening dynamics and scaling exponents in the easy-axis ferromagnetic phase.

a, Domain length L(t) in the full time evolution accessible during the experiment. Closed (open) circles represent the domain length after (without) deconvolution. Dashed lines represent the scaling time interval t ∈ [0.2 s, 0.8 s]. The lower bound for the time interval is chosen to ensure that the condensate enters into the coarsening stage after the quench.The upper bound of the time interval is limited by the finite size of the system and lifetime lifetime of the condensate. b, Dependence of the scaling exponents 1/z on the lower bound for the time interval tL. The error bars indicate the 1σ confidence interval of the fit parameters. c, Number of magnetic domains after the quench. The domain number ND is counted by using the Hoshen-Koppelman algorithm (details are available in Supplementary Information). The domain number follows the power law decay (solid line), ND ∼ t−2/z with 1/z = 0.63(4).

Source data

Extended Data Fig. 3 Effect of the microwave dressing on the spinor Bose gas.

a, Long-time evolution of the atom number and b, thermal fraction at q/h = − 120 Hz (square, light blue) and − 200 Hz (circle, dark blue). c, Long-time evolution of the domain length in the easy axis quench q/h = − 200 Hz. After the coarsening terminated (t = 1 s), the domain length is decreased from 61 μm to 57 μm with additional 1 s of hold time. It implies that the domain length during the universal dynamics t ∈ [0.2 s, 0.8 s] could be underestimated by 5%. d,e Absorption images (top) and the magnetization density (bottom) at different hold times (see legends). Domain size is reduced as a result of the atom loss. Each data point is obtained with 40 different experimental realisation, and the error bars denote one standard error of the mean.

Source data

Extended Data Fig. 4 Magnetic domain wall in the easy-axis ferromagnetic phase.

a, Magnetization Fz after 1.5 s of hold time, and b, the cross-section profile across the magnetic domain. The solid line is a fit curve \({F}_{z}(r)={F}_{z0}\tanh (r/{\xi }_{d})\) with ξd = 4.5(2) μm. c,d, Magnetization along the horizontal axis Fx. The spin vectors could be aligned on the same axis (c, Bloch or Neel-type domain wall) or point in opposite directions (d, Bloch line). A long wavelength modulation of the horizontal spin vector could imply the presence of spin wave excitations in the magnetic domain.

Source data

Extended Data Fig. 5 Time evolution of spin domains in the isotropic ferromagnetic phase.

a-e, Longitudinal magnetization Fz (upper) and its two-dimensional spin correlation functions Gz(x, y) (below) at various hold times. f-j, Snapshots of transverse magnetization Fx. and the correlation functions Gx(x, y). In contrast to the easy-axis phase, in the spin isotropic point, coarsening dynamics are observed in both axes, and the domain boundaries are much broader than those of the q < 0 spin domains. The spin correlation functions are averaged over 100 different realizations at a given hold time.

Source data

Extended Data Fig. 6 Numeric simulations of the matter-wave interference for the \({{\mathbb{Z}}}_{2}\) spin vortices.

a, Density profile and b, the argument of the transverse spin vector \(\phi ={\tan }^{-1}({F}_{y}/{F}_{x})\) in the x-y plane after t = 3.5 s of domain coarsening dynamics at the spin isotropic point. The spin vortex can be identified from a phase jump around the vortex core (yellow box). c, Three dimensional spin vectors F = (Fx, Fy, Fz) in the x-y plane near the highlighted region (yellow box). Not only the in-plane spin vector but also the longitudinal spin vector turns around the vortex core, indicating the \({{\mathbb{Z}}}_{2}\) spin vortex. d, Simulated images after the matter-wave interference. The positions of the spin vortices are well-identified from fork-shaped patterns in the spin imbalance image. e–h, Representative experimental images after the matter-wave interference under various hold times (see distinct rows). All images are obtained by independent experimental runs. In the vortex-free region, the magnetization displays a connected stripe pattern, while the spin vortex shows a dislocation of the stripes to form the two-to-one fork-shaped patterns. The vortex positions are highlighted by yellow circles.

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Supplementary information

Supplementary Information

Supplementary discussion with Figs. 1–5.

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Source Data for Fig. 1

Images for longitudinal magnetization and spin correlation function.

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Source Data for Extended Data Fig. 1

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Huh, S., Mukherjee, K., Kwon, K. et al. Universality class of a spinor Bose–Einstein condensate far from equilibrium. Nat. Phys. 20, 402–408 (2024). https://doi.org/10.1038/s41567-023-02339-2

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