Universal Dynamics of Rogue Waves in a Quenched Spinor Bose Condensate

Ido Siovitz, Stefan Lannig, Yannick Deller, Helmut Strobel, Markus K. Oberthaler, and Thomas Gasenzer

Phys. Rev. Lett. 131, 183402 – Published 2 November 2023

Abstract

Isolated many-body systems far from equilibrium may exhibit scaling dynamics with universal exponents indicating the proximity of the time evolution to a nonthermal fixed point. We find universal dynamics connected with the occurrence of extreme wave excitations in the mutually coupled magnetic components of a spinor gas which propagate in an effectively random potential. The frequency of these rogue waves is affected by the time-varying spatial correlation length of the potential, giving rise to an additional exponent $δ_{c} ≃ 1 / 3$ for temporal scaling, which is different from the exponent $β_{V} ≃ 1 / 4$ characterizing the scaling of the correlation length $ℓ_{V} \sim t^{β_{V}}$ in time. As a result of the caustics, i.e., focusing events, real-time instanton defects appear in the Larmor phase of the spin-1 system as vortices in space and time. The temporal correlations governing the instanton occurrence frequency scale as $t^{δ_{I}}$ . This suggests that the universality class of a nonthermal fixed point could be characterized by different, mutually related exponents defining the evolution in time and space, respectively. Our results have a strong relevance for understanding pattern coarsening from first principles and potential implications for dynamics ranging from the early Universe to geophysical dynamics and microphysics.

Received 2 May 2023
Accepted 10 October 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.183402

Physics Subject Headings (PhySH)

Bose gases Bose-Einstein condensates Cold atoms & matter waves Dynamic critical phenomena Nonequilibrium statistical mechanics Nonlinear dynamics in fluids Pattern formation Quantum field theory Quantum quench Spinor Bose-Einstein condensates Topological defects

1-dimensional systems Nonequilibrium systems Random & disordered media Ultracold gases

Critical exponents Semiclassical methods

Atomic, Molecular & OpticalStatistical Physics & ThermodynamicsFluid DynamicsNonlinear Dynamics

Authors & Affiliations

Ido Siovitz , Stefan Lannig , Yannick Deller, Helmut Strobel, Markus K. Oberthaler , and Thomas Gasenzer ^*

Kirchhoff-Institut für Physik, Ruprecht-Karls Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany

^*t.gasenzer@uni-heidelberg.de

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Vol. 131, Iss. 18 — 3 November 2023

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Images

Figure 1
Characteristics of caustics in the system after a quench. Units are set by the spin healing length $ξ_{s} = (2 M n | c_{1} |)^{- 1 / 2}$ and its corresponding spin-changing collision time $t_{s} = 2 π / (n | c_{1} |)$ . (a) Excerpt of the space-time evolution of phase defects in the system. The phase gradients $v_{1} = \partial_{x} φ_{1}$ (purple to green) and $v_{- 1}$ (red to blue) show the formation of rogue-wave-like excitations in the condensate which focus on a singular point marked by the cross. (b) The scintillation index $S_{v} (t)$ , Eq. (2), around a rogue wave at $t = 0$ . The blue solid line shows the scintillation profile averaged over $\sim 10^{3}$ (not normalized) rogue waves. The dashed line depicts $S_{v} (t)$ for the single truncated Wigner run in (a). (c) PDF of the local Larmor velocity $v_{L} = \partial_{x} φ_{L} = \partial_{x} (φ_{1} - φ_{- 1})$ for different times. The PDF takes the form of a Rayleigh exponential distribution (gray dashed line fit) with a heavy tail. The extreme events are characterized as those with an amplitude larger than $2 v_{c}$ , where $v_{c}$ is the scale velocity, representing the mean of the upper tertile of events. The inset shows that $v_{L} > 2 v_{c}$ at the focusing time $t = 0$ . (d) The probability of finding an extreme event as a function of time. A power law decay $t^{- δ_{c}}$ , with $δ_{c} = 0.332 (3)$ is found. The inset shows the deviation of the fit from the data divided by the data point error.
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Figure 2
Structures and defects in the time evolution of the Larmor phase after a quench (units chosen as in Fig. 1). (a) Time evolution of the winding number $Q_{w}$ for the run shown in panel (b). (b) Space-time evolution of the Larmor phase of the transversal spin $F_{⊥} = | F_{⊥} | \exp [i φ_{L}]$ across the entire system in a single truncated-Wigner (TW) run, with the spin speed of sound $c_{s} = \sqrt{n | c_{1} | / 2 M}$ (dashed line). In the strongly fluctuating system, vortex structures in space and time are observed, as the phase wraps around one point [cf. zoom in panel (c)]. Instantons (orange) and anti-instantons (black), each cause an integer jump in the winding number $Q_{w} (t)$ . (c) Structure of the real-time instanton. In the upper panel, the averaged $| F_{⊥} |$ profile of a defect located at $x_{0}$ at time $t_{0}$ is depicted. The lower panel shows the vortexlike nature of the defect in more detail, around which the Larmor phase winds by $2 π$ . (d) The lower panel shows the corresponding intersection of two rogue waves in $v_{\pm 1}$ at the position of the instanton, recall Fig. 1. The upper panel exhibits the temporal evolution (bright to dark pink) of the $F_{⊥}$ field configuration in spin space, within the window shown in the lower panels. The outer circle represents a histogram (black to bright red color code) of spin orientations in the $F_{x} − F_{y}$ plane averaged over 100 TW runs.
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Figure 3
Statistics of the instantons after a quench. (a) Short-time Fourier transform (STFT) of the winding number $Q_{w} (t)$ (main panel, color scale), exhibiting a Gaussian fall-off for small frequencies (up to the gray dashed line in the right inset) $STFT [Q_{w}] (t) \sim \exp {- ω^{2} / [2 Γ^{2} (t)]}$ (right inset), with width decreasing as $Γ (t) \sim t^{- δ_{I}}$ , $δ_{I} = 0.34 (1)$ (left inset and red line in main panel), confirming power-law coarsening dynamics of the governing timescale of $Q_{w}$ . The lower panel shows the time evolution of the winding number $Q_{w}$ for a single realization. (b) The PDF of spatial defect separation is found to fall off exponentially, $P (r, t) \sim A (t) \exp [- r / ζ (t)]$ , with mean distance $⟨ r ⟩ (t) \sim t^{β_{I}}$ , increasing from early times (yellow) to later times (black), exhibiting power-law coarsening with exponent $β_{I} = 0.26 (1)$ (lower inset). The upper inset shows the chosen threshold of the current $J (x, t) = | \partial_{x} φ_{L} | \cdot (⟨ | F_{⊥} | ⟩_{x} - | F_{⊥} |)$ for defect detection which corresponds to the top decile of amplitude. The lower panel shows the difference of the data to the fit function divided by the standard deviation of each data point.
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