Parametric excitations in a harmonically trapped binary Bose-Einstein condensate

TL;DR

This work investigates parametric excitations and pattern formation in a binary Bose-Einstein condensate near the miscible-immiscible transition using coupled Gross-Pitaevskii equations. By periodically modulating intra-species scattering lengths, the authors induce Faraday patterns in both total-density and spin-density channels, analyzing 1D patterns in elongated traps and 2D, l-fold symmetric patterns in pancake geometries. They identify resonance conditions $\omega_m = 2\omega_d(k)$ for density modes and $\omega_m = 2\omega_s(k)$ for spin modes, estimate key length scales such as the spin healing length $\xi_s$, and map the spin excitation spectrum $\omega(n_r,l)$ via controlled modulations. The study also compares modulation protocols and investigates nonlinear saturation and mode coupling, offering a framework for spectroscopically probing spin dynamics in multi-component BECs with potential experimental relevance.

Abstract

We investigate parametric excitation and pattern formation in a harmonically trapped two-component Bose-Einstein condensate. We assume the condensate to be in the miscible phase, but near the miscible-immiscible phase transition, where total density and spin density excitations are decoupled. By periodically modulating the atomic scattering lengths, Faraday patterns can be generated in both density and spin channels. In an elongated condensate, the pattern in the spin channel corresponds to a one-dimensional standing wave with the two components exhibiting an out-of-phase density oscillation, where the modulation frequency and the oscillation period are related to the velocity of the spin sound. After the spin pattern is fully developed, the system quickly enters a nonlinear destabilization regime. For a pancake-shaped condensate, a two-dimensional Faraday pattern is generated with an interesting l-fold rotational symmetry. The number of nodes along the radial and angular directions increases with larger modulation frequencies. We also compare the growth rates of spin Faraday patterns generated with different modulation protocols, which are accessible to current experiments.

Figures (8)

• Figure 1: Total density Faraday pattern in an elongated two-component BEC generated by an in-phase modulation of the scattering lengths $a_{11}$ and $a_{22}$, starting from the ground state at $t=0$. (a) Total density $n=n_1+n_2$ and spin density $n_s= n_1-n_2$, in the $xy$-plane at $t=495$ ms. (b) Integrated 1D density $\delta n_{1D}(x)=\iint dy dz\ (n-n_{\rm TF})$, after subtracting the Thomas-Fermi equilibrium density. (c) Fourier transform of $\delta n_{1D}(x)$; the side peaks at $k_x\simeq \pm 0.35$$\mu$m$^{-1}$ are associated with the counter-propagating Bogoliubov modes producing the standing density wave. For panels (a-c), the modulation frequency and amplitude are $\omega_{m}/2\pi=384$ Hz and $a_{m}=0.36a$, respectively. (d) Relation between $\omega_m$ and the corresponding side peak wave vector $|k_x|$; the blue circles correspond to the results extracted from the numerical simulations; the solid line is the resonant condition for parametric excitation, $\omega_m=2\omega_d(k_x)$, where $\omega_d(k_x)$ is the dispersion law Eq. (\ref{['eq:Bog']}), which is indistiguishable from the linear dispersion $2c_d k_x$ in this range of $k_x$.
• Figure 2: Spin density Faraday pattern in an elongated two-component BEC generated by an out-of-phase modulation of the scattering lengths $a_{11}$ and $a_{22}$. (a) Total density $n=n_1+n_2$ and spin density $n_s=n_1-n_2$, in the $xy$-plane at $t=219$ ms. (b) Integrated 1D spin density $n_{s,1D}=\iint dy dz\ n_s$. (c) Fourier transform of the $n_{s,1D}$; the side peaks at $k_x\simeq \pm 0.58$$\mu$m$^{-1}$ are associated with the counter-propagating spin waves producing the pattern. For panels (a-c), the modulation frequency and amplitude are $\omega_m/2\pi=195$ Hz and $a_m=0.07a$, respectively. (d) Relation between $\omega_m$ and the corresponding side peak wave vector $|k_x|$; the blue circles correspond to the results extracted from the numerical simulations; the solid line is the resonant condition for parametric excitation, $\omega_m=2\omega_s(k_x)$, where $\omega_s(k_x)$ is the Bogoliubov dispersion Eq. (\ref{['eq:Bog']}) for the spin branch; the dashed line, is the linear dispersion $2c_s k_x$, valid in the long wavelength limit.
• Figure 3: Time evolution of an elongated two-component BEC subject to an out-of-phase modulation of the scattering lengths $a_{11}$ and $a_{22}$, started at $t=0$, with modulation frequency $\omega_{m}=2\pi\times 195$ Hz and amplitude $a_{m}=0.07a$. (a) Integrated 1D spin density $n_{s,1D} (x)$. (b-c) Integrated 1D spin density profiles at $t=170, 219, 270$ ms, and the corresponding Fourier transforms. (d) Time evolution of the integrated power spectrum, $\int_{-\infty }^{+\infty } dk_x |\mathcal{F}(n_{s,1D})|^2$, of the spin density profile.
• Figure 4: Spin Faraday patterns in a two-component pancake-shaped BEC with increasing angular $l$ and radial $n_r$ quantum numbers. In panel (a) we plot the density distributions $n_1$ and $n_2$ for patterns with three radial nodes ($n_r=3$) and increasing values of the angular momentum from $l=0$ to $l=6$, from left to right; these patterns are obtained with modulation frequencies $f\equiv \omega_{m}/2\pi=124, 156, 182, 210, 238, 270, 302$ Hz at $t=320, 956, 1316, 951, 898, 877, 897$ ms. The configurations in panel (b) have $l=3$ and increasing radial quantum number from $n_r=0$ to $n_r=6$, with modulation frequencies $f\equiv\omega_{m}/2\pi=42, 98, 152, 210, 282, 366, 450$ Hz at $t=2895, 1377, 1032, 947, 829, 748, 806$ ms. The modulation amplitude is $a_{m}=0.037a$, except in the first column of the panel (b), where it is $a_{m}=0.07a$.
• Figure 5: Spectrum of spin density excitations, $\omega(n_r,l)$, obtained via the parametric resonance condition $\omega(n_r,l)=\omega_m/2$, where $\omega_m$ is the optimal modulation frequency that produces the corresponding Faraday pattern. Frequency is expressed in units of the radial trapping frequency $\omega_r\equiv \omega_x=\omega_y$. Blue circles are obtained with an out-of-phase modulation (protocol A), while red circles are with an in-phase modulation with initial noise (protocol B). The points along the oblique and vertical dashed lines correspond to the spin density distributions plotted in Fig. \ref{['FIG.4']}(a) and (b), respectively.