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Scientific Reports

Generation of patterns and higher harmonics in 1D quantum droplets in tilted and driven quasi-periodic confinements

Maitri R. Pathak, Jayanta Bera, Utpal Roy, Ajay Nath

Published Oct 15, 2024• 5 citations
Publisher

Abstract

The generation of patterns by breaking the spatial symmetry in external confinement is a captivating area of physics. The emergence of patterns is a fundamental inquiry spanning various disciplines such as nonlinear optics, condensed matter physics, and fluid dynamics. The article investigates the generation of a variety of patterns in a one-dimensional binary mixture of Bose-Einstein condensate forming quantum droplets. By solving the extended Gross-Pitaevskii equation in the presence of tilted and driven engineered bi-chromatic optical lattices (BOL), the out-of-equilibrium dynamics of droplets under strong dc and ac fields are illustrated. Under the influence of a dc field, a stripe-like pattern emerges in the temporal domain, while the scenario with ac fields demonstrates temporal periodic and bi-periodic oscillations of density waves. The width and period of formed patterns are directly correlated with the strength of ac and dc fields. Moreover, temporal modulation of the BOL potential depth yields various harmonics in the oscillations of the condensate density pattern. Through Fast Fourier Transform (FFT) analysis, it is confirmed that these harmonics encompass multiple and combinational frequencies, suggesting potential applications in generating desired frequency combs within quantum droplets. We have also carried out a thorough numerical stability check of the obtained solutions and found them sufficiently stable.

DOI
10.1038/s41598-024-73319-6

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The generation of patterns by breaking the spatial symmetry in external confinement is a captivating area of physics. The emergence of patterns is a fundamental inquiry spanning various disciplines such as nonlinear optics, condensed matter physics, and fluid dynamics. The article investigates the generation of a variety of patterns in a one-dimensional binary mixture of Bose-Einstein condensate forming quantum droplets. By solving the extended Gross-Pitaevskii equation in the presence of tilted and driven engineered bi-chromatic optical lattices (BOL), the out-of-equilibrium dynamics of droplets under strong dc and ac fields are illustrated. Under the influence of a dc field, a stripe-like pattern emerges in the temporal domain, while the scenario with ac fields demonstrates temporal periodic and bi-periodic oscillations of density waves. The width and period of formed patterns are directly correlated with the strength of ac and dc fields. Moreover, temporal modulation of the BOL potential depth yields various harmonics in the oscillations of the condensate density pattern. Through Fast Fourier Transform (FFT) analysis, it is confirmed that these harmonics encompass multiple and combinational frequencies, suggesting potential applications in generating desired frequency combs within quantum droplets. We have also carried out a thorough numerical stability check of the obtained solutions and found them sufficiently stable.
We consider a quasi-one-dimensional Bose-Einstein condensate with contact and long-range dipolar interactions, under the action of the time-periodic modulation applied to the harmonic-oscillator and optical-lattice trapping potentials. The modulation results in generation of a variety of harmonics in oscillations of the condensate's width and centre-of-mass coordinate. These include multiple and combinational harmonics, represented by sharp peaks in the system's spectra. Approximate analytical results are produced by the variational method, which are verified by systematic simulations of the underlying Gross-Pitaevskii equation. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.
In the paper, we introduce a new model that addresses the generation of quantum droplets (QDs) in the binary Bose–Einstein condensate (BEC) mixture with mutually symmetric spinor components loaded in multi-color optical lattices (MOLs) of commensurate wavelengths and tunable intensities. The considered MOL confinement is the combination of the four-color optical lattice with an exponential periodic trap, which includes the complete set of the Fourier harmonics. Employing the one-dimensional (1D) extended Gross–Pitäevskii equation (eGPE), we calculate the exact analytical form of the wavefunction, MF/BMF nonlinearities, and MOL trap parameters. Utilizing the exact solutions, the formation of supersolid-like spatially periodic matter-wave droplet lattices and superlattices is illustrated under the space-periodic nonlinearity management. The precise positioning of the density maxima/minima of the droplet patterns at the center of the trap and tunable Anderson-like localization are observed by tuning the symmetry and amplitude of the considered MOL trap. The stability of the obtained solution is confirmed using the Vakhitov–Kolokolov (VK) criterion.
Recent theoretical and experimental results show that one-dimensional (1D) weakly interacting atomic Bose-Bose mixtures with repulsive interspecies mean field (MF) interaction are stabilized by attractive quadratic beyond-mean-field (BMF) effects into self-bound quantum droplet (QD) in free space. Here, we construct an exact analytical model to investigate the structure and dynamics of QDs in presence of external harmonic confinement by solving the 1D extended Gross-Pitäevskii equation (eGPE) with temporal variation of MF and BMF interactions. The model provides the analytical form of wavefunction, phase, MF and BMF nonlinearities. The generation of QDs and interesting droplet to soliton transition in presence of regular/expulsive parabolic traps by taking the comparable MF and BMF interactions are illustrated. We derive the phase diagram of the droplet-soliton phase transition between amplitude of MF, BMF interactions and harmonic oscillator frequency. The strength and form of oscillator frequency are identified as key parameter for tuning the compression, fragmentation and transport of droplets. Finally, the stability of the obtained solutions are confirmed from Vakhitov-Kolokolov (VK) criterion and are found stable.
We consider quantum droplets in dipolar Bose-Einstein condensates embedded in optical lattices within the framework of Gross-Pitaevskii equations. In dipolar BECs, the long-range and isotropic dipole-dipole interaction provides an additional mechanism for self-binding. We analyze the linear stability as well dynamics of quantum droplets. We find an effective potential for the width and show that the optimum width for formation of quantum droplet increases as the dipole-dipole interaction increases. We study dynamics of the stable droplets and see that its width oscillates with time, and the amplitude of oscillation increases with the increase of dipole-dipole interaction. In presence of optical lattices, width of a stable droplet changes quasi-periodically while the location of density profile changes periodically in space. The frequency of oscillation depends sensitively on the lattice parameters. We check dynamical stability of a quantum droplet through numerical simulation.
Abstract We consider quantum droplets in dipolar Bose–Einstein condensates embedded in optical lattices (OLs) within the framework of Gross–Pitaevskii equations. In dipolar BECs, the long-range and isotropic dipole–dipole interaction (DDI) provides an additional mechanism for self-binding. We analyze the linear stability as well dynamics of quantum droplets. We find an effective potential for the width and show that the optimum width for formation of quantum droplet increases as the DDI increases. We study dynamics of the stable droplets and see that its width oscillates with time, and the amplitude of oscillation increases with the increase of DDI. In presence of OLs, width of a stable droplet changes quasi-periodically while the location of density profile changes periodically in space. The frequency of oscillation depends sensitively on the lattice parameters. We check dynamical stability of a quantum droplet through numerical simulation.
Abstract We explore the dynamical behavior of one-dimensional quantum droplets (QDs) governed by the extended Gross–Pitaevskii equation, under harmonic confinement supplemented by a static or time-dependent Gaussian spike (Gs) potential. Employing both variational analytical techniques and numerical simulations, we investigate the evolution of the root-mean-square (RMS) size, excitation spectrum, and phase-space dynamics of QDs. Our study reveals that while the harmonic trap sets the primary confinement, the Gs potential enables precise frequency tuning and control over droplet oscillations. A static Gs amplitude modifies the fundamental oscillation frequency depending on its sign, while a time-modulated Gs induces nonlinear dynamics, including higher harmonics and frequency mixing. Our analysis reveals that the resulting frequency spectrum is strongly influenced by inter- and intra-species interactions as well as by the parameters of the external trap. Notably, we establish a relationship between the frequency shift and the amplitude of the Gaussian spike. Wigner phase-space analysis further uncovers coherent rotational behavior, offering insights into hidden phase dynamics not apparent in real-space density profiles. These results highlight the role of time-dependent external potentials in tailoring the excitation landscape of quantum droplets, with implications for quantum simulation, precision control in atomtronics, and probing coherence in low-dimensional quantum fluids.