Spontaneous Formation of Star-Shaped Surface Patterns in a Driven Bose-Einstein Condensate

K. Kwon, K. Mukherjee, S. J. Huh, K. Kim, S. I. Mistakidis, D. K. Maity, P. G. Kevrekidis, S. Majumder, P. Schmelcher, and J.-y. Choi

Phys. Rev. Lett. 127, 113001 – Published 10 September 2021

Abstract

We observe experimentally the spontaneous formation of star-shaped surface patterns in driven Bose-Einstein condensates. Two-dimensional star-shaped patterns with $l$ -fold symmetry, ranging from quadrupole ( $l = 2$ ) to heptagon modes ( $l = 7$ ), are parametrically excited by modulating the scattering length near the Feshbach resonance. An effective Mathieu equation and Floquet analysis are utilized, relating the instability conditions to the dispersion of the surface modes in a trapped superfluid. Identifying the resonant frequencies of the patterns, we precisely measure the dispersion relation of the collective excitations. The oscillation amplitude of the surface excitations increases exponentially during the modulation. We find that only the $l = 6$ mode is unstable due to its emergent coupling with the dipole motion of the cloud. Our experimental results are in excellent agreement with the mean-field framework. Our work opens a new pathway for generating higher-lying collective excitations with applications, such as the probing of exotic properties of quantum fluids and providing a generation mechanism of quantum turbulence.

Received 20 May 2021
Accepted 3 August 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.113001

Physics Subject Headings (PhySH)

Bose-Einstein condensates Cold atoms & matter waves Pattern formation

Floquet systems

Atomic, Molecular & Optical

Authors & Affiliations

K. Kwon ¹, K. Mukherjee ², S. J. Huh¹, K. Kim ¹, S. I. Mistakidis ³, D. K. Maity ², P. G. Kevrekidis ⁴, S. Majumder ², P. Schmelcher ^3,5, and J.-y. Choi ^1,*

¹Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea
²Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India
³Center for Optical Quantum Technologies, Department of Physics,University of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
⁴Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
⁵The Hamburg Centre for Ultrafast Imaging, University of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany

^*jae-yoon.choi@kaist.ac.kr

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Vol. 127, Iss. 11 — 10 September 2021

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Images

Figure 1
Experimental observation of star-shaped condensates. (a) A BEC of ${}^{7}{Li}$ atoms is prepared in a pancake-shaped trap consisting of a red-detuned optical trap for the axial confinement and a magnetic trap, induced by Feshbach magnetic field curvature, for the radial confinement. The scattering length is modulated by oscillating the magnetic field near the Feshbach resonance. The mean modulation amplitude ${\bar{a}}_{m}$ is defined as the average of upper ( $a_{m}^{+}$ ) and lower ( $a_{m}^{-}$ ) modulation peaks, ${\bar{a}}_{m} = (a_{m}^{+} + a_{m}^{-}) / 2$ . (b)–(g) Representative in trap images (single shots) of the ensuing regular polygons with $D_{l}$ symmetry triggered by the periodic modulation. The modulation frequencies are 84 Hz ( $D_{2}$ ), 104 Hz ( $D_{3}$ ), 119 Hz ( $D_{4}$ ), 132 Hz ( $D_{5}$ ), 147 Hz ( $D_{6}$ ), and 161 Hz ( $D_{7}$ ), respectively. The mean modulation amplitude ${\bar{a}}_{m} = 19 a_{B}$ ( $a_{m}^{+} = 21 a_{B}$ and $a_{m}^{-} = 17 a_{B}$ ). The scale bar in the $l = 2$ mode represents $100 μ m$ . (h) The orientation angle $ϕ^{*}$ for pentagon-shaped BECs, defined in (e), is measured with 400 consecutive experimental runs. The histogram displays the corresponding occurrence of the angle with bin size of 3°. (i) Hexagon and (j) heptagon-shaped patterns obtained by solving the 3D GPE (see main text).
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Figure 2
Hydrodynamic instability of the surface excitations. (a) Spectral peak of various $l$ -fold star patterns, created after 1 s of modulation, as a function of the modulation frequency. The spectral peak for the hexagon ( $l = 6$ mode) is taken after $t = 0.3 s$ due to the involved dipole instability (see main text). The filled circles designate the experimental results and solid lines refer to the GPE predictions; notice the very good agreement between the two. Each data point is averaged over 5–10 independent experimental realizations, and the error bars denote the standard deviation of the mean. (b) Floquet stability tongues for different modulation strengths ${\bar{a}}_{m} / a_{b g}$ and frequencies $ω_{m} / (2 π)$ characterizing distinct $l$ -fold patterns.
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Figure 3
Dispersion relation of the surface modes for superfluids in a harmonic trap. The spectral peak value for each $l$ mode is obtained by fitting a Gaussian function to each resonance spectrum with small driving amplitude. Solid line represents a guide to the eye for the dispersion law of the surface mode in trapped superfluids. The inset shows the ratio of the measured resonance frequency and the hydrodynamic predictions. The error bars indicate a 95% confidence interval of the fit. Uncertainty of the radial trap frequency is marked by the shaded region.
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Figure 4
Time evolution of the triangular surface mode. (a) Dynamics of the growth rate of the triangular mode under a 104 Hz modulation. The fluctuations of the spectral peak ( $l = 3$ ) increase exponentially in the course of time. Each data point is a single experimental realization. Inset: the averaged spectral peak $⟨ F_{3} ⟩$ over a single oscillation time interval, $[t, t + 9.5] ms$ . Dashed line designates an exponential fit to the data (dark blue circle), and the solid line is obtained from the GPE simulation. (b) Zooming in at 1 s reveals the oscillation of the surface mode with 104 Hz driving frequency. Each data point corresponds to the mean over four independent experimental runs, and the error bars indicate the standard deviation of the mean. Inset: absorption images during the first oscillation period. The oscillation frequency is subharmonic, a result that is confirmed by the GPE calculations, see also Ref. [29].
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