Dynamical formation of quantum droplets in a $^{39}\mathrm{K}$ mixture

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Dynamical formation of quantum droplets in a ${}^{39}K$ mixture

G. Ferioli, G. Semeghini, S. Terradas-Briansó, L. Masi, M. Fattori, and M. Modugno

Phys. Rev. Research 2, 013269 – Published 12 March 2020

Abstract

We report on the dynamical formation of self-bound quantum droplets in attractive mixtures of ${}^{39}K$ atoms. Considering the experimental observations of Semeghini et al. [Phys. Rev. Lett. 120, 235301 (2018)], we perform numerical simulations to understand the relevant processes involved in the formation of a metastable droplet from an out-of-equilibrium mixture. We first analyze the so-called self-evaporation mechanism, where the droplet dissipates energy by releasing atoms, and then we consider the effects of losses due to three-body recombinations and to the balancing of populations in the mixture. We discuss the importance of these three mechanisms in the observed droplet dynamics and their implications for future experiments.

Received 19 December 2019
Accepted 14 February 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.013269

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Atomic & molecular clusters Bose-Einstein condensates Cold atoms & matter waves Mixtures of atomic and/or molecular quantum gases

Quantum fluids & solids

Atomic, Molecular & Optical

Authors & Affiliations

G. Ferioli^1,2, G. Semeghini ^3,*, S. Terradas-Briansó ⁴, L. Masi^1,2, M. Fattori^1,2, and M. Modugno ^5,6

¹LENS and Dipartimento di Fisica e Astronomia, Universitá di Firenze, 50019 Sesto Fiorentino, Italy
²CNR Istituto Nazionale Ottica, 50019 Sesto Fiorentino, Italy
³Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
⁴Instituto de Ciencia de Materiales de Aragón and Departamento de Física de la Materia Condensada, CSIC–Universidad de Zaragoza, 50009 Zaragoza, Spain
⁵Departamento de Física Teórica e Historia de la Ciencia, Universidad del Pais Vasco UPV/EHU, 48080 Bilbao, Spain
⁶IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain

^*Corresponding author: semeghini@fas.harvard.edu

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Vol. 2, Iss. 1 — March - May 2020

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Figure 1
Evolution of the droplet size $σ (τ)$ , plotted in terms of its relative variation with respect to the equilibrium value $σ_{0}$ , for small initial excitations, for ${(\tilde{N} - \tilde{N_{c}})}^{1 / 4} ≃ 3.5$ (a) and ${(\tilde{N} - \tilde{N_{c}})}^{1 / 4} ≃ 7$ (b) [1]. Purple dots correspond to the values extracted from the numerical simulation, while the green solid lines are the result of damped sinusoidal fits on separate time intervals (a) and of a pure sinusoidal fit (b).
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Figure 2
(a) Plot of the monopole frequency $ω_{0}$ (obtained from the numerical simulation) as a function of ${(\tilde{N} - {\tilde{N}}_{c})}^{1 / 4}$ [1]. In the self-evaporation region (shaded area) the range of variation of $ω_{0}$ is shown by (purple) arrows (see Fig. 1). Out of the self-evaporation regime, where the droplet size performs a sinusoidal oscillation as in Fig. 1, the value of $ω_{0}$ is represented by solid circles. The lines correspond to the theoretical predictions of Ref. [1] for $- \tilde{μ} (\tilde{N})$ (green dashed line) and ${\tilde{ω}}_{0} (\tilde{N})$ (red dotted line). (b) Values of $ω_{0}$ (purple circles) and $1 / \tilde{τ}$ (green squares) fitted on consecutive time intervals of length $Δ τ = 25$ (indicated as error bars), extracted from the damped oscillation of Fig. 1. The black dashed line corresponds to $- \tilde{μ}$ . (c),(d) Variation of the parameter $\tilde{N}$ and of the droplet energy $E$ in each time interval (see text), respectively.
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Figure 3
Evolution of the droplet width $σ (t)$ (a), its energy $E_{d} (t)$ (b), the ratio $N_{1}^{d} (t) / N_{2}^{d} (t)$ (c), and the running value of ${\tilde{N}}_{R} (t)$ [(d), see text], for $ω / 2 π = 200$ Hz (purple) and $ω / 2 π = 600$ Hz (green). The insets in the top row show the total density of the binary mixture, $n (r, t) = \sum_{i = 1}^{2} N_{i} {| ψ_{i} (r, t) |}^{2}$ , at different evolution times, corresponding to the red circles in (a). As a reference, the profiles of a droplet with $\tilde{N} = 200$ (dot-dashed line, $t = 0, 4$ ms), $\tilde{N} = 193$ and $\tilde{N} = 159$ (dotted and dashed line respectively, for $t = 12$ ms), are also shown in the insets. In (a), (b) the dashed (dotted) horizontal lines represent the equilibrium values of the rms size [in (a)] and the energy for a droplet with $\tilde{N} = 193$ (159) [in (b)]. The horizontal line in (c) represent the equilibrium value $N_{1} / N_{2} = \sqrt{a_{22} / a_{11}} ≃ 0.698$ , and the dashed area the corresponding tolerance (see text).
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Figure 4
Numerical results for the dynamical formation of the droplet in presence of strong 3BL. (a) Evolution of the droplet size $σ (t)$ , (b) of the number of atoms in the droplet (solid) and in the unbound cloud (dashed), (c) of the number of atoms lost due to 3BL, (d) of the population ratio in the droplet $N_{1} / N_{2}$ , and (e) of the droplet energy. In (b),(c) we plot the atom numbers separately for the two atomic species $| 1 〉$ (black) and $| 2 〉$ (brown). In (d) we compare the population ratio with the nominal range where the interactions energies in the droplet are properly compensated, $N_{1} / N_{2} ≃ \sqrt{a_{22} / a_{11}}$ (shaded area) [1]. In (e) the droplet energy is shown along with the ground state energy for the corresponding atom number $E_{e q} (N (t))$ (black dashed line).
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Figure 5
Plot of the total density $n (r, t) = \sum_{i = 1}^{2} N_{i} {| ψ_{i} (r, t) |}^{2}$ (a),(c) and of the quantity $\bar{n} (r, t) \equiv r^{2} n (r, t)$ , at different evolution times ( $t = 4, 12$ ms; here for the case $ω / 2 π = 200$ Hz of the simulations discussed in Sec. 3b). The abscissa of the red dot (see text) represents the value of the droplet radius $R_{d} (t)$ .
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Physical Review Research

Dynamical formation of quantum droplets in a K39 mixture

G. Ferioli, G. Semeghini, S. Terradas-Briansó, L. Masi, M. Fattori, and M. Modugno

Phys. Rev. Research 2, 013269 – Published 12 March 2020

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Dynamical formation of quantum droplets in a ${}^{39}K$ mixture