Harmonically trapped imbalanced quantum droplets

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Harmonically trapped imbalanced quantum droplets

T. A. Flynn, N. A. Keepfer, N. G. Parker, and T. P. Billam

Phys. Rev. Research 6, 013209 – Published 26 February 2024

Abstract

A two-component quantum droplet is an attractive mixture of ultracold bosons stabilized against collapse by quantum fluctuations. Commonly, two-component quantum droplets are studied within a balanced mixture. However, the mixture can be imbalanced resulting in a lower energy but less stably bound droplet, or even a droplet submerged in a gas. This work focuses on the experimentally relevant question: how are imbalanced droplets modified by harmonic trap potentials? Droplet ground states and breathing modes are analyzed across the two-dimensional parameter space of imbalance and trap strength. The robustness of the droplet imbalance is also studied by releasing the droplet from the trap, demonstrating that this can lead to the creation of free-space, imbalanced droplets.

Received 15 September 2023
Accepted 2 January 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.013209

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)

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

Quantum fluids & solids

Atomic, Molecular & Optical

Authors & Affiliations

T. A. Flynn ^*, N. A. Keepfer , N. G. Parker , and T. P. Billam

Joint Quantum Centre (JQC) Durham-Newcastle, School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom

^*t.flynn@ncl.ac.uk

Article Text

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Vol. 6, Iss. 1 — February - April 2024

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Images

Figure 1
Balanced and imbalanced quantum droplets in isotropic harmonic traps (with, $N_{2} \approx 17027, α \approx 0.00657$ , and $η \approx - 1.11$ ). (a) Droplet ground-state density profiles, of size $\tilde{N} \approx 649$ with balanced components, $δ N_{1} = 0$ , (the light and dark purple dashed lines) and an imbalance of $δ N_{1} \approx 8513$ (majority component, dark orange; minority component, light orange), in a trap of frequency $ω_{r} \approx 0.0442$ . (b) Equivalent balanced and imbalanced droplet to (a), but with a trapping frequency of $ω_{r} \approx 0.353$ . (c) Majority (orange) and minority (purple) chemical potentials across the 2D parameter space of imbalance and trap frequency, i.e., $(δ N_{1}, ω_{r})$ , for the fixed droplet size considered in (a) and (b), at trap frequencies of $ω_{r} \approx {0.00441, 0.0662, 0.128, 0.190}$ , with $0 \leq δ N_{1} ≲ 4257$ . The inset shows the surfaces for the majority (orange) and minority (purple) component chemical potentials across the 2D parameter space for which the curves in (c) are 1D slices of set $ω_{r}$ values. Note that the two surfaces are equal at $δ N_{1} = 0$ imbalance, but diverge for increasing imbalance.
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Figure 2
Breathing modes of trapped imbalanced droplets. (a) Droplet central densities in time for both a self-evaporative, balanced (top) and imbalanced (bottom) droplet (with imbalance $δ N_{1} \approx 4257$ ) in a trap with frequency $ω_{r} \approx 0.00883$ . The droplet is defined with the same parameters as in Fig. 1. Note the recombination events that cause the self-evaporative dynamics to be reinitiated. (b) Droplet central density for the $δ N_{1} \approx 4257$ imbalanced droplet in the bottom panel of (a), in a trap with $ω_{r} \approx 0.00662$ . The trap frequency is sufficiently low such that all three modes, i.e., the intrinsic mode, and the two modes corresponding to the two chemical potentials, can be observed before the first recombination event. (c) A higher trap frequency of $ω_{r} \approx 0.0106$ showing that the shorter period between recombination events no longer allows for the long wavelength oscillation of the majority component. (d) An increased trap frequency of $ω_{r} \approx 0.0309$ . The recombination events occur within such short intervals that the dynamics are dominated by the intrinsic droplet breathing mode as there is not sufficient time for this mode to decay.
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Figure 3
Ground states and breathing modes for high imbalances in a low-frequency trap. (a) Example ground-state density profile in a trap of frequency $ω_{r} \approx 0.00662$ , and an imbalance of $δ N_{1} \approx 16856418$ . (b) Two examples of breathing modes with a smaller imbalance of $δ N_{1} \approx 170267$ in the top panel, and a considerably larger imbalance of $δ N_{1} \approx 16856418$ in the bottom panel. These example imbalances are highlighted in (c) by the vertical dashed lines. (c) Fitted decay rate to the initial droplet breathing mode, with varying size of imbalance.
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Figure 4
Dynamics of imbalanced droplets due to an instantaneous release into free space, with the same parameters as in Figs. 1, 2, 3. (a) Population numbers varying in time following the release into free space at $t = 0$ . The bound, imbalanced droplet has an initial imbalance of $δ N_{1} \approx 596$ . The orange curves correspond to an initial low-frequency trap with $ω_{r} \approx 0.0132$ (dark and light orange corresponding to majority and minority components, respectively), while the purple curves correspond to an initial high-frequency trap, with $ω_{r} \approx 0.261$ (light and dark purple corresponding to majority and minority components, respectively). The insets show the central density differences in time, with the left and right panels corresponding to the low- and high-frequency traps, respectively. These two example simulations are highlighted in the top panel of (b) by the vertical, dashed lines. (b) The equilibrated population numbers and central density differences (inset) for varying initial trap frequencies, in the range $0.00442 ≲ ω_{r} ≲ 0.269$ . The top panel corresponds to a bound, imbalanced droplet ( $δ N_{1} \approx 596$ ) while the bottom panel corresponds to a saturated, imbalanced droplet with an external unbound gas ( $δ N_{1} \approx 4257$ ).
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Figure 5
Dynamics of imbalanced droplets due to a ramped down trap potential, with the same parameters as in Figs. 1, 2, 3, 4. (a) Population numbers varying in time with the orange curves correspond to a relatively slow ramp down of $t_{ramp} \approx 38.3$ (dark and light orange corresponding to majority and minority components, respectively), and purple colors corresponding to a relatively fast ramp down of $t_{ramp} \approx 0.781$ (light and dark purple corresponding to majority and minority components, respectively). The insets show the central density differences in time, with the left and right panels corresponding to the slow and fast ramps, respectively. These two example simulations are highlighted in the top panel of (b), by the vertical, dashed lines. (b) The equilibrated population numbers and central density differences (inset) for varying ramp-down times, $t_{ramp}$ , in the range $0.0 \leq t_{ramp} ≲ 39.1$ . The top panel corresponds to a bound, imbalanced droplet ( $δ N_{1} \approx 596$ ) while the bottom panel corresponds to a saturated, imbalanced droplet with an external unbound gas ( $δ N_{1} \approx 4257$ ).
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