Extreme (Rogue) Waves: From Theory to Experiments in Ultracold Gases and Beyond

Abstract

In this Chapter, we review key theoretical and experimental advances in the study of extreme nonlinear wave events, called rogue waves (RWs), in both single-component attractively interacting and two-component repulsive mixtures of ultracold quantum gases. Starting from the exact rational solutions of the integrable focusing nonlinear Schroedinger model, the hierarchy of RW solutions is exemplified. These range from the Peregrine soliton (PS) and, related to it, the destabilization into a multi-peak cascade of PSs dubbed "Christmas-tree", to the Akhmediev breather, and Kuznetsov-Ma soliton as well as higher-order RWs. Emphasis is placed on their controllable dynamical emergence and characteristics in non-integrable quantum many-body systems described by Gross-Pitaevskii models and extensions thereof through different protocols such as modulational instability, gradient catastrophe, and dam-break flows. We further discuss how immiscible particle-imbalanced repulsive mixtures can be cast into effective attractive single-component environments capable of hosting RWs. Next, state-of-the-art experimental techniques are summarized within the ultracold realm that can be utilized to realize solitary waves, modulational instability, dispersive shock waves and RWs including the very recent first experimental observation of the PS, enabled through engineered effective focusing interactions and precise dynamical triggering. Observations of these extreme events in water waves, nonlinear optics and beyond are also outlined, highlighting their broader relevance and potential of emergence in disparate physical settings. Our exposition aims at showcasing ultracold atomic gases as versatile platforms for controllably generating and probing extreme nonlinear events, among others, in the quantum realm across integrable and non-integrable settings.

Figures (13)

• Figure 1: Spatiotemporal mean-field density evolution using the Gaussian initial condition of Eq. (\ref{['eq:Gaussian_cond']}) for fixed $\mathcal{A}=1$ and different widths (a) $\sigma=30$, (b) $\sigma=30$, (c) $\sigma=2.5$ and (d) $\sigma=1.3$. Panel (a) is a magnification of (b) depicting the first PS formation before its destabilization due to gradient catastrophe into a Christmas tree configuration. Reducing the width of the Gaussian ansatz yields either (c) a time-periodic KM breather or (d) to a bright soliton. Adapted from charalampidis2016rogue.
• Figure 2: (Upper block of panels) Time-evolution of the mean-field density of a two-component highly imbalanced mixture using an initial Gaussian wave packet having amplitude $\mathcal{A}= 0.2$ and width $\sigma = 50$ presenting the dynamics of the (a) minority and (b) majority species with $\mu = 1$. (c) Evolution of the effective single-component minority species. PS formation occurs at (a) $t_{{\rm two}} = 1846$ and (c) $t_{{\rm single}} = 1808$. (Lower block of panels) Density dynamics of (a), (b) the mixture and (c) its effective reduction for significantly broader Gaussian ansatz with $\mathcal{A}= 0.2$ and width $\sigma = 150$. Here, PS nucleation takes place at $t_{{\rm two}} = 758$ and $t_{{\rm single}} = 716$, respectively before its destabilization into the Christmas-tree configuration. In both cases an overall agreement is evident between the reduced single-component and the actual minority species of the two-component mixture, with the majority exhibiting complementary structures to the minority species. Adapted from Romero_theor.
• Figure 3: Dynamical emergence of the PS within (a)–(h) the experiment and (i)–(p) 3D numerical simulations. (a)–(d) Integrated density profiles of the absorption images shown in (e)–(h) at $t=10, 30, 65, 85$ ms with an additional 9 ms of free expansion. (i)–(l) Integrated density profiles of the 2D density contours depicted in (m)–(p) stemming from the 3D mean-field simulations emulating the experimental process. Adapted from Romero_Ros_2024.
• Figure 4: Nonlinear stage of MI triggered by a repulsive barrier for a particle balanced two-component mixture of $^{87}$Rb atoms in the $|2,0 \rangle$ and $|1,0 \rangle$ hyperfine states. (a) Experimental single-shot absorption image taken after $60$ ms of evolution time, showing the $|2,0 \rangle$ (top) and the $|1,0 \rangle$ (bottom) state. (b) Corresponding integrated cross sections, averaged over 20 independent realizations. (c,d) Time-evolution of the integrated $|2,0 \rangle$ cross sections obtained from the experiment, shown in panel (c), and 3D numerical simulations [Eq. (\ref{['eq:3D_GPEs']})], in panel (d). Dashed lines indicate the experimentally determined speed of sound and solid ones the theoretically predicted MI expansion rate [Eq. (\ref{['eq:MI_Vel']})]. Adapted from Mossman_nonlinearMI.
• Figure 5: (a)-(d) Wave function magnitude, $|\psi_k|$ with $k=1,2,3,4$ of the first four NLS RWs whose solutions are described by Eq. (\ref{['eq:HORW']}). (e), (f) Profiles taken from (b), (d) respectively, depicting HORWs of $2^{{\rm nd}}$ and $4^{{\rm th}}$ order. Dashed lines correspond to 1D NLS simulations and solid lines provide a fit of the relevant analytical solution. Numerical parameters used for panel (e) correspond to trap frequency $\Omega=1.23$, prequench interaction strength $g_i=0.891$ and postquench one $g_f=-0.701$ and chemical potential $\mu=6.19$. Similarly for panel (f), $\Omega=1.93$, $g_i=1.84$, $g_f=-0.483$, $\mu=18.2$. (g) A 2D simulation illustrating the iso-surface of the wave function magnitude of a $3^{{\rm rd}}$ order RW with parameters $\Omega_x = 0.75$, $\Omega_y = 100$, $\mu =71.1$, $g_i = 1$, and $g_f =-0.05$. (h) A 3D simulation depicting the iso-surface of a $2^{{\rm nd}}$ order RW with $\Omega_x = 1$, $\Omega_y = \Omega_z = 100$, $\mu =143$, $g_i = 1$, and $g_f = -0.05$. Adapted from adriazola2024experimentally.