Theoretical and numerical evidence for the potential realization of the Peregrine soliton in repulsive two-component Bose-Einstein condensates

The present work is motivated by the recent experimental realization of the Townes soliton in an effective two-component Bose-Einstein condensate by B. Bakkali-Hassan et al. [Phys. Rev. Lett. 127, 023603 (2021)]. Here, we use a similar multicomponent platform to exemplify theoretically and numerically, within the mean-field Gross-Pitaevskii framework, the potential toward the experimental realization of a different fundamental wave structure, namely the Peregrine soliton. Leveraging the effective attractive interaction produced within the mixture's minority species in the immiscible regime, we illustrate how initialization of the condensate with a suitable power-law decaying spatial density pattern yields the robust emergence of the Peregrine wave in the absence and in the presence of a parabolic trap. We then showcase the spontaneous emergence of the Peregrine soliton via a suitably crafted wide Gaussian initialization, again both in the homogeneous case and in the trap scenario. It is also found that narrower wave packets may result in periodic revivals of the Peregrine soliton, while broader ones give rise to a cascade of Peregrine solitons arranged in a so-called Christmas-tree structure. Strikingly, the persistence of these rogue-wave structures is demonstrated in certain temperature regimes as well as in the presence of transversal excitations through three-dimensional computations in a quasi-one-dimensional regime. This proof-of-principle illustration is expected to represent a practically feasible way to generate and observe this rogue wave in realistic current ultracold atom experimental settings.

Figure 1

Density evolution of the Peregrine initial condition [Eq. (4)] in the absence of a trap for (a) the minority and (b) the majority species of the two-component setting with ${\mu }_u=1$. Equation (4) is initialized with $P_o=0.01$ and $t_o=1000$. (c) Dynamics of the density within the effective single-component model. (d) Density difference between the (a) two- and the (c) single-component dynamics (the color bar is rescaled by a factor of ${10}^3$). (e) Density profile of the $v$ component for the two- and single-component setups at the time instant of Peregrine formation, namely $t_{\text{two}}^{\prime }=1019$ and $t_{\text{single}}^{\prime }=1000$. (f) Temporal evolution of the Peregrine wave emerging in the $v$ component of both the two- and single-component setups at $x=0$ (see legend). In both panels (e) and (f), the corresponding analytical Peregrine solution of Eq. (4) is provided. Note that length and time are measured in units of $a_{\perp }$ and ${\omega }_{\perp }^{-1}$, respectively.

Figure 2

Spatiotemporal density evolution of a Peregrine initial condition in the presence of a trap with $\mathrm{\Omega }=0.002$ for (a) the minority and (b) the majority species of the two-component case with ${\mu }_u=1$. Equation (4) is initialized with $P_o=0.01$ and $t_o=1000$ on top of the ground state obtained for the super-Gaussian potential of Eq. (5) using ${\mu }_v=P_o$ (see text). (c) Density snapshot of the $v$ component capturing the instantaneous formation, $t_{\text{two}}^{\prime }=1053$, of the Peregrine soliton. (d) The temporal density profile of the $v$ component at the fixed spatial location of $x=0$. In both panels (c) and (d), the relevant theoretical prediction of Eq. (4) is provided for a direct comparison (see legend). Length (time) is given in terms of $a_{\perp }$ (${\omega }_{\perp }^{-1}$).

Figure 3

Time evolution of the density of a Gaussian wave packet with $A=0.2$ and $w=50$ depicting the dynamics of the (a) minority and (b) majority species of the two-component system with ${\mu }_u=1$. (c) Evolution of the minority $v$ component within the effective single-component description. (d) Density difference between the (a) two- and the (c) single-component dynamics. (e) Density profile of the $v$ component for the two- and single-component setups at the times of their largest amplitudes, namely at $t_{\text{two}}^{\prime }=1846$ and $t_{\text{single}}^{\prime }=1808$, respectively. (f) Evolution of the center position $x=0$ of the density of the $v$ component for the two- and the single-component setups. In both panels (e) and (f), the corresponding analytical solution of Eq. (4) is given, with its peak fitted to the single-component case (see text). Length and time are expressed in units of $a_{\perp }$ and ${\omega }_{\perp }^{-1}$, respectively.

Figure 4

Same as Fig. 3 but considering a significantly broader Gaussian initial condition with $w=150$. In this case, the Peregine soliton formation within the two models occurs at $t_{\text{two}}^{\prime }=758$ and $t_{\text{single}}^{\prime }=716$, respectively. Notice also the subsequent emergence of the Christmas-tree structure. Recall that length and time are given in units of $a_{\perp }$ and ${\omega }_{\perp }^{-1}$, respectively.

Figure 5

Evolution of the density of a Gaussian wave packet with $A=0.2$ and $w=150$ in the presence of a harmonic trap with $\mathrm{\Omega }=0.002$. (a) Temporal evolution of the minority $v$ component. (b) A magnification of (a) is provided. (c) Effective single-component case. (d) Same as (b) but upon considering a mass-imbalanced BEC mixture (see text). (e) Density profiles of the $v$ component for the single-component and mass-balanced and mass-imbalanced two-component setups at the instant of formation of the Peregrine soliton $t_{\text{single}}^{\prime }=516$, $t_{\text{two}}^{\prime }=800$, and $t_{\text{two}}^{\prime (1:3)}=795$, respectively. All Peregrine amplitudes are normalized to unity for direct comparison. (f) Density profile at $t_{\text{two}}^{\prime (1:3)}=1851$, depicting the pairwise DB soliton nucleation that follows the decay of the Christmas-tree pattern, for the mass-imbalanced mixture. In all cases, the majority $u$ component complements its relevant minority one, and thus it is omitted. The length and time units shown are in terms of $a_{\perp }$ and ${\omega }_{\perp }^{-1}$, respectively.

Figure 6

Dynamical evolution of the density of the minority $v$ component for $A=0.2$ and [(a), (b)] $w=50$ and [(c), (d)] $w=150$. The dissipation strength is [(a), (c)] $\gamma ={10}^{-4}$ and [(b), (d)] $\gamma =5\times {10}^{-4}$. Other parameters used are ${\mu }_u=1$, ${\mu }_v=0.01$, $m_u=m_v$, and $\mathrm{\Omega }=0.002$. Note that length (time) is in units of $a_{\perp }$ (${\omega }_{\perp }^{-1}$).

Figure 7

(a) [(b)] Spatiotemporal evolution of the integrated along the $yz$ direction density, of the minority $v$ component for $A=0.5\mu {\mathrm{m}}^{-1/2}$ and $w=0.7\mu \mathrm{m}$ $[w=3.5\mu \mathrm{m}]$. Insets in panel (a) depict density snapshots of the (top) $v$ and (bottom) $u$ components, rescaled by a factor of $1/3$ and $1/7$, respectively, in the $x\text{−}y$ plane at $t_0\approx 13.5\mathrm{ms}$ when the Peregrine initially forms. Panels (c)–(e) [(f)–(h)]: Integrated density profiles at selective time instants during evolution of the Peregrine and its revivals [of the Christmas tree]. A comparison with the relevant analytical prediction of Eq. (4) is also provided (see legends). For this quasi-1D evolution, the corresponding trapping frequencies are $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (3.06,400,400)\mathrm{Hz}$ and $N=7\times {10}^5$.

Figure 8

(a) Density evolution of the Peregrine initial condition of Eq. (4), for $P_o=0.01$ and $t_o=1000$, in the homogeneous two-component setup. The majority $u$ component is complementary to the minority $v$ component and thus omitted. White lines, corresponding to Eq. (A1), designate the borders of the wedge-like pattern formed. (b) Enlargement of panel (a). (c) Density profile of the minority $v$ component at the time instant of Peregrine soliton formation, namely at $t_0^{\prime }=1023$, $t_2^{\prime }=5148$, and $t_4^{\prime }=8052$. (d) Temporal evolution of the modulation wave at $x=0$. In both panels (c) and (d), the corresponding analytical solutions of Eq. (4) are provided for the chosen Peregrine solitons. Length and time units are expressed in terms of $a_{\perp }$ and ${\omega }_{\perp }^{-1}$, respectively.