Pattern formation in ring condensates subjected to bichromatic driving

TL;DR

This work studies pattern formation in a one-dimensional Bose-Einstein condensate confined to a ring and driven by bichromatic modulation of the interaction strength. The authors derive a generalized Mathieu equation and use Floquet theory to map out instability tongues as functions of the driving amplitude, frequency, and mixing angle, then validate the predictions with time-dependent Gross-Pitaevskii simulations. They show that the mixing angle and frequency ratio can selectively excite higher-order resonances, leading to density patterns at discrete wavenumbers and nonlinear saturation captured by a reduced five-mode model. Together, the analytic, numerical, and reduced-model approaches demonstrate a practical scheme to engineer and control complex nonlinear patterns in ultracold atoms with bichromatic driving.

Abstract

We investigate the dynamical formation of nonlinear patterns in one-dimensional ring condensates under bichromatic periodic modulation of the interaction strength. The stability phase diagram of the condensate's homogeneous density state is analytically derived through a suitable biharmonic variant of the Mathieu equation and computing the associated Floquet spectrum. It reveals the complex interplay between the driving parameters, i.e., amplitude, frequencies, and the so-called frequencies' mixing angle, which dictate the instability onset and the selective enhancement of higher-order resonance tongues, thus offering precise control over the excited modes. These results are in agreement with time-dependent mean-field simulations evidencing the emergence of density wave modulations of specific momenta, while enabling a deeper understanding of the nonlinear stage of the relevant instability. Further insights on the ensuing unstable nonlinear dynamics are provided through a reduced {five-mode} model which captures the instability onset, the oscillatory behavior of the mode populations and the phase-space dynamics, in agreement with the mean-field predictions. Our study highlights the versatility of bichromatic driving to generate and control complex nonlinear patterns that are within reach in present day ultracold atom experiments.

Figures (7)

• Figure 1: Effect of frequency ratio and mixing angle $\chi$ [in Eq. \ref{['eq:interaction_modulation']}] on the stability phase diagram. Floquet resonance tongues (dark shaded regions) across the $\alpha$-$\omega_D$ plane for a 1D BEC under bichromatic interaction driving for unstable wavenumber $k = 20$ and different frequency ratios and mixing angles, $\chi$. Upper (lower) panels refer to frequency ratios $1{:}2$ ($1{:}3$), while each column from left to right corresponds to mixing angle $\chi = 0^\circ, 30^\circ, 45^\circ, 60^\circ, ~\textrm{and}~90^\circ$, respectively. For the $1{:}2$ frequency ratio and $\chi=0^\circ$, the first resonance tongue appears at $\omega_D = 2\Lambda(k)$ (red circles) and the second at $\omega_D = \Lambda(k)$ (cyan plus markers). At $\chi=90^\circ$, the $m=2$ scenario emerges with the dominant resonance now lying at $\omega_D = \Lambda(k)$ (cyan plus markers). For the $1{:}3$ ratios and $\chi=0^\circ$, the first resonance tongue takes place at $\omega_D = 2\Lambda(k)$ (red diamonds) and the third one occurs around $\omega_D = 2 \Lambda(k)/3$ (yellow triangular marker). Again at $\chi=90^\circ$, the $m=3$ setting leads to a dominant peak at $\omega_D = 2 \Lambda(k)/3$ (yellow triangular marker). For intermediate angles, both for $m=2$ and for $m=3$, one sees a combination of the relevant frequencies, progressively transitioning from those of $\chi=0^\circ$ to those of $\chi=90^\circ$.
• Figure 2: Time-evolution of the driven condensate density at $\theta=0$, $|\psi(0,t)|^2$, and its Fourier spectra within different resonance tongues for $k=20$. Panels (a)–(b) show the dynamics for the second resonance tongue ($\omega_D = 2.03$) under a $1{:}2$ double-frequency drive with $\chi = 30^\circ$ and $\chi = 60^\circ$, respectively. The corresponding Fourier spectra are shown in (c)–(d). Dynamics of $|\psi(0,t)|^2$ within the third resonance tongue ($\omega_D = 1.34$) under a $1{:}3$ bichromatic driving for (e) $\chi = 30^\circ$ and (f) $\chi = 60^\circ$. The respective Fourier spectra of panel (e) [(f)] are displayed in (g) [(h)]. Increasing $\chi$ with fixed frequency ratio enhances the instability amplitude, see panels (a) and (b). The temporal period of the patterns is modified by tuning the frequency ratio, e.g. compare panels (c) and (g). Here $\nu/\omega_D$ is the Fourier frequency $\nu$ divided by the external driving frequency $\omega_D$. In all cases, the driving amplitude is fixed at $\alpha = 0.09$.
• Figure 3: (a)-(e) Spatio-temporal density evolution within the GPE following bichromatic driving of the interaction strength [Eq. (\ref{['eq:interaction_modulation']})] with characteristics $\omega_D=2.03$ (second tongue), frequency ratio $1{:}2$, amplitude $\alpha = 0.09$, and wavenumber $k = 20$. Panels correspond (a)–(c) to mixing angles $\chi = 30^\circ$, and (d)–(f) to $\chi = 30^\circ$ (b) [(e)] Density evolution shown in (a) [(d)] but within a specific time-interval visualizing the short time-period resonant patterns. The slow envelope density modulations are evident in panels (a), (d). (c) [(f)] Dynamics of the momentum distribution of the density depicted in (a) [(d)], displaying the temporal evolution of the instability induced excited modes at $p=0$, $p= \pm k$ and $p= \pm 2k$. Here, $\tilde{\psi}(k,t)$ denotes the Fourier transform of $\psi(\theta,t)$
• Figure 4: Density evolution of the ring-shaped driven condensate using the GPE for (a) $\chi = 30^\circ$ and (d) $\chi = 60^\circ$ showing the nucleated nonlinear waves. The bichromatic interaction modulation has frequency $\omega_D = 1.34$, amplitude $\alpha = 0.09$, and wavenumber $k = 20$. (b) [(e)] Spatio-temporal modulation of the density patterns shown in (a) [(d)] within a short time interval depicting the resonance waves. (c) [(f)] Time-evolution of the momentum distribution of the density of panel (a) [(f)] showing the participating excited modes at $p=0$, $p= \pm k$ and $p= \pm 2k$. Here, $\tilde{\psi}(k,t)$ denotes the Fourier transform of $\psi(\theta,t)$
• Figure 5: Comparison between the instability tongues obtained from Floquet theory in the Mathieu equation [Eq. (\ref{['eq:mathieu_equation']})] represented by the shaded light blue areas and the GPE simulations (black dots). We present the second ($\omega_D=2.02 \approx \Lambda(k)$) and third ($\omega_D=1.35 \approx 2\Lambda(k)/3$) tongues shown also in Fig. \ref{['fig:mathieu_double']}. The dynamics is induced by bichromatic interaction driving with a frequency ratio of (a), (b) $1{:}2$ and (c), (d) $1{:}3$. Different mixing angles are considered corresponding to (a), (c) $\chi=30^\circ$ and (b), (d) $\chi=60^\circ$. Excellent agreement between the predictions of the two methods is seen, while relatively small deviations occur for increasing modulation amplitude.