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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Hohenberg, P. C. & Halperin, B. I. Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977).
Polkovnikov, A., Sengupta, K., Silva, A. & Vengalattore, M. Colloquium: nonequilibrium dynamics of closed interacting quantum systems. Rev. Mod. Phys. 83, 863–883 (2011).
Eisert, J., Friesdorf, M. & Gogolin, C. Quantum many-body systems out of equilibrium. Nat. Phys. 11, 123–130 (2015).
Ueda, M. Quantum equilibration, thermalization and prethermalization in ultracold atoms. Nat. Rev. Phys. 2, 669–681 (2020).
Makotyn, P., Klauss, C. E., Goldberger, D. L., Cornell, E. A. & Jin, D. S. Universal dynamics of a degenerate unitary Bose gas. Nat. Phys. 10, 116–119 (2014).
Eigen, C. et al. Universal prethermal dynamics of Bose gases quenched to unitarity. Nature 563, 221–224 (2018).
Gałka, M. et al. Emergence of isotropy and dynamic scaling in 2D wave turbulence in a homogeneous Bose gas. Phys. Rev. Lett. 129, 190402 (2022).
Prüfer, M. et al. Observation of universal dynamics in a spinor Bose gas far from equilibrium. Nature 563, 217–220 (2018).
Erne, S., Bücker, R., Gasenzer, T., Berges, J. & Schmiedmayer, J. Universal dynamics in an isolated one-dimensional Bose gas far from equilibrium. Nature 563, 225–229 (2018).
Glidden, J. A. P. et al. Bidirectional dynamic scaling in an isolated Bose gas far from equilibrium. Nat. Phys. 17, 457–461 (2021).
Fontaine, Q. et al. Kardar–Parisi–Zhang universality in a one-dimensional polariton condensate. Nature 608, 687–691 (2022).
Zu, C. et al. Emergent hydrodynamics in a strongly interacting dipolar spin ensemble. Nature 597, 45–50 (2021).
Wei, D. et al. Quantum gas microscopy of Kardar–Parisi–Zhang superdiffusion. Science 376, 716–720 (2022).
Joshi, M. K. et al. Observing emergent hydrodynamics in a long-range quantum magnet. Science 376, 720–724 (2022).
Baier, R., Mueller, A. H., Schiff, D. & Son, D. T. ‘Bottom-up’ thermalization in heavy ion collisions. Phys. Lett. B 502, 51–58 (2001).
Berges, J., Rothkopf, A. & Schmidt, J. Nonthermal fixed points: effective weak coupling for strongly correlated systems far from equilibrium. Phys. Rev. Lett. 101, 041603 (2008).
Schole, J., Nowak, B. & Gasenzer, T. Critical dynamics of a two-dimensional superfluid near a nonthermal fixed point. Phys. Rev. A 86, 013624 (2012).
Schmidt, M., Erne, S., Nowak, B., Sexty, D. & Gasenzer, T. Non-thermal fixed points and solitons in a one-dimensional Bose gas. New J. Phys. 14, 075005 (2012).
Berges, J., Boguslavski, K., Schlichting, S. & Venugopalan, R. Turbulent thermalization process in heavy-ion collisions at ultrarelativistic energies. Phys. Rev. D 89, 074011 (2014).
Berges, J., Boguslavski, K., Schlichting, S. & Venugopalan, R. Universality far from equilibrium: from superfluid Bose gases to heavy-ion collisions. Phys. Rev. Lett. 114, 061601 (2015).
Piñeiro Orioli, A., Boguslavski, K. & Berges, J. Universal self-similar dynamics of relativistic and nonrelativistic field theories near nonthermal fixed points. Phys. Rev. D 92, 025041 (2015).
Kudo, K. & Kawaguchi, Y. Magnetic domain growth in a ferromagnetic Bose–Einstein condensate: effects of current. Phys. Rev. A 88, 013630 (2013).
Hofmann, J., Natu, S. S. & Das Sarma, S. Coarsening dynamics of binary Bose condensates. Phys. Rev. Lett. 113, 095702 (2014).
Williamson, L. A. & Blakie, P. B. Universal coarsening dynamics of a quenched ferromagnetic spin-1 condensate. Phys. Rev. Lett. 116, 025301 (2016).
Williamson, L. A. & Blakie, P. B. Coarsening dynamics of an isotropic ferromagnetic superfluid. Phys. Rev. Lett. 119, 255301 (2017).
Bray, A. J. Theory of phase-ordering kinetics. Adv. Phys. 43, 357–459 (1994).
Bertini, B., Collura, M., De Nardis, J. & Fagotti, M. Transport in out-of-equilibrium XXZ chains: exact profiles of charges and currents. Phys. Rev. Lett. 117, 207201 (2016).
Castro-Alvaredo, O. A., Doyon, B. & Yoshimura, T. Emergent hydrodynamics in integrable quantum systems out of equilibrium. Phys. Rev. X 6, 041065 (2016).
Ljubotina, M., Žnidarič, M. & Prosen, T. Spin diffusion from an inhomogeneous quench in an integrable system. Nat. Commun. 8, 16117 (2017).
Gopalakrishnan, S. & Vasseur, R. Kinetic theory of spin diffusion and superdiffusion in XXZ spin chains. Phys. Rev. Lett. 122, 127202 (2019).
Ljubotina, M., Žnidarič, M. & Prosen, T. Kardar–Parisi–Zhang physics in the quantum Heisenberg magnet. Phys. Rev. Lett. 122, 210602 (2019).
Kardar, M., Parisi, G. & Zhang, Y.-C. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986).
Mikheev, A. N., Schmied, C.-M. & Gasenzer, T. Low-energy effective theory of nonthermal fixed points in a multicomponent Bose gas. Phys. Rev. A 99, 063622 (2019).
Huh, S., Kim, K., Kwon, K. & Choi, J.-y. Observation of a strongly ferromagnetic spinor Bose–Einstein condensate. Phys. Rev. Res. 2, 033471 (2020).
Ho, T.-L. Spinor Bose condensates in optical traps. Phys. Rev. Lett. 81, 742–745 (1998).
Ohmi, T. & Machida, K. Bose–Einstein condensation with internal degrees of freedom in alkali atom gases. J. Phys. Soc. Jpn 67, 1822–1825 (1998).
Kawaguchi, Y. & Ueda, M. Spinor Bose–Einstein condensates. Phys. Rep. 520, 253–381 (2012).
Sadler, L. E., Higbie, J. M., Leslie, S. R., Vengalattore, M. & Stamper-Kurn, D. M. Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose–Einstein condensate. Nature 443, 312–315 (2006).
Blakie, P. B., Bradley, A. S., Davis, M. J., Ballagh, R. J. & Gardiner, C. W. Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques. Adv. Phys. 57, 363–455 (2008).
Kim, K., Hur, J., Huh, S., Choi, S. & Choi, J.-y. Emission of spin-correlated matter-wave jets from spinor Bose–Einstein condensates. Phys. Rev. Lett. 127, 043401 (2021).
Huse, D. A. Corrections to late-stage behavior in spinodal decomposition: Lifshitz–Slyozov scaling and Monte Carlo simulations. Phys. Rev. B 34, 7845–7850 (1986).
Bourges, A. & Blakie, P. B. Different growth rates for spin and superfluid order in a quenched spinor condensate. Phys. Rev. A 95, 023616 (2017).
De, S. et al. Quenched binary Bose–Einstein condensates: spin-domain formation and coarsening. Phys. Rev. A 89, 033631 (2014).
Gauthier, G. et al. Giant vortex clusters in a two-dimensional quantum fluid. Science 364, 1264–1267 (2019).
Johnstone, S. P. et al. Evolution of large-scale flow from turbulence in a two-dimensional superfluid. Science 364, 1267–1271 (2019).
Karl, M. & Gasenzer, T. Strongly anomalous non-thermal fixed point in a quenched two-dimensional Bose gas. New J. Phys. 19, 093014 (2017).
Schmied, C.-M., Prüfer, M., Oberthaler, M. K. & Gasenzer, T. Bidirectional universal dynamics in a spinor Bose gas close to a nonthermal fixed point. Phys. Rev. A 99, 033611 (2019).
Fujimoto, K., Hamazaki, R. & Ueda, M. Flemish strings of magnetic solitons and a nonthermal fixed point in a one-dimensional antiferromagnetic spin-1 Bose gas. Phys. Rev. Lett. 122, 173001 (2019).
Bhattacharyya, S., Rodriguez-Nieva, J. F. & Demler, E. Universal prethermal dynamics in Heisenberg ferromagnets. Phys. Rev. Lett. 125, 230601 (2020).
Rodriguez-Nieva, J. F., Orioli, A. P. & Marino, J. Far-from-equilibrium universality in the two-dimensional Heisenberg model. Proc. Natl Acad. Sci. USA 119, 2122599119 (2022).
Dzyaloshinsky, I. A thermodynamic theory of ‘weak’ ferromagnetism of antiferromagnetics. J. Phys. Chem. Solids 4, 241–255 (1958).
Moriya, T. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120, 91–98 (1960).
Guzman, J. et al. Long-time-scale dynamics of spin textures in a degenerate F = 1 87Rb spinor Bose gas. Phys. Rev. A 84, 063625 (2011).
Hild, S. et al. Far-from-equilibrium spin transport in Heisenberg quantum magnets. Phys. Rev. Lett. 113, 147205 (2014).
Neely, T. W., Samson, E. C., Bradley, A. S., Davis, M. J. & Anderson, B. P. Observation of vortex dipoles in an oblate Bose–Einstein condensate. Phys. Rev. Lett. 104, 160401 (2010).
Kwon, W. J., Moon, G., Seo, S. W. & Shin, Y. Critical velocity for vortex shedding in a Bose–Einstein condensate. Phys. Rev. A 91, 053615 (2015).
Choi, J.-y., Kwon, W. J. & Shin, Y.-i. Observation of topologically stable 2D skyrmions in an antiferromagnetic spinor Bose–Einstein condensate. Phys. Rev. Lett. 108, 035301 (2012).
Inouye, S. et al. Observation of vortex phase singularities in Bose–Einstein condensates. Phys. Rev. Lett. 87, 080402 (2001).
Mukherjee, K., Mistakidis, S. I., Kevrekidis, P. G. & Schmelcher, P. Quench induced vortex-bright-soliton formation in binary Bose–Einstein condensates. J. Phys. B: At. Mol. Opt. Phys. 53, 055302 (2020).
Kwon, K. et al. Spontaneous formation of star-shaped surface patterns in a driven Bose–Einstein condensate. Phys. Rev. Lett. 127, 113001 (2021).
Crank, J. & Nicolson, P. A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Math. Proc. Camb. Philos. Soc. 43, 50–67 (1947).
Antoine, X., Bao, W. & Besse, C. Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations. Comput. Phys. Commun. 184, 2621–2633 (2013).
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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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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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.
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).
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.
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.
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.
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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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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DOI: https://doi.org/10.1038/s41567-023-02339-2
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