Semidiscrete Quantum Droplets and Vortices

Xiliang Zhang, Xiaoxi Xu, Yiyin Zheng, Zhaopin Chen, Bin Liu, Chunqing Huang, Boris A. Malomed, and Yongyao Li

Phys. Rev. Lett. 123, 133901 – Published 27 September 2019

Abstract

We consider a binary bosonic condensate with weak mean-field (MF) residual repulsion, loaded in an array of nearly one-dimensional traps coupled by transverse hopping. With the MF force balanced by the effectively one-dimensional attraction, induced in each trap by the Lee-Hung-Yang correction (produced by quantum fluctuations around the MF state), stable on-site- and intersite-centered semidiscrete quantum droplets (QDs) emerge in the array, as fundamental ones and self-trapped vortices, with winding numbers, at least, up to five, in both tightly bound and quasicontinuum forms. The application of a relatively strong trapping potential leads to squeezing transitions, which increase the number of sites in fundamental QDs and eventually replace vortex modes by fundamental or dipole ones. The results provide the first realization of stable semidiscrete vortex QDs, including ones with multiple vorticity.

Received 23 June 2019
Revised 1 August 2019

DOI:https://doi.org/10.1103/PhysRevLett.123.133901

Physics Subject Headings (PhySH)

Solitons

Nonlinear Dynamics

Authors & Affiliations

Xiliang Zhang^1,*, Xiaoxi Xu^1,*, Yiyin Zheng¹, Zhaopin Chen ², Bin Liu¹, Chunqing Huang¹, Boris A. Malomed^2,1, and Yongyao Li ^1,2,†

¹School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
²Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and the Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978, Israel

^*These two authors contributed equally to this Letter.
^†yongyaoli@gmail.com

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Vol. 123, Iss. 13 — 27 September 2019

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Images

Figure 1
Density profiles of coexisting stable OC and IC semidiscrete QDs with zero vorticity and the number of sites from 1 to 4 in (a1)–(d1), respectively. Parameters are $ω = 0$ and $(g, C, N) = (1, 0.01, 100)$ . (a2)–(d2) QDs produced from those displayed in (a1)–(d1) by the squeezing transition imposed by the trap with $ω = 0.004$ , 0.011, 0.015, and 0.021, in Eq. (1) respectively. The corresponding transition points are $ω_{c} = 0.003$ , 0.010, 0.014, and 0.020, and the ratio of the QD’s longitudinal size to the trapping length $L_{ω} = ω_{c}^{- 1 / 2}$ is $L_{z} / L_{ω} = 10.20$ , 8.11, 6.74, and 5.93.
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Figure 2
(a),(b) Dependences $μ (C)$ and $\sqrt{ϵ (C)}$ for the OC and IC (red solid and black dotted lines, respectively) QDs with zero vorticity, built of $2 J - 1$ or $2 J$ sites, vs $J$ , at $g = 1$ and $ω = 0$ . The dashed line in (a) is $μ_{0} (g = 1) \approx - 0.222$ . These branches exist up to terminal values of $J$ , viz., $J_{T}^{(on, in)} = 5$ and 4, respectively. The subplot in (a) shows critical values $C_{J}$ , at which the respective branches terminate, vs $J$ . The subplot in (b) displays the switch of the single eventually surviving on-site-centered mode from $J = 5$ to 6 with the subsequent increase of $C$ . The green point $C_{B}$ , at which $ϵ = 1$ , is the boundary between the QDs with the TB and QC structure for $g = 1$ . (c) The TB-QC boundary, $C_{B} (g)$ .
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Figure 3
(a) The stability area (purple) for IC semidiscrete QDs with zero vorticity, in the $(C, g)$ parameter plane. Unstable modes, such as the one corresponding to point $A$ , with $(g, C) = (0.9, 0.15)$ , transform into robust breathers, as shown in Fig. 4. (b) Stability areas for OC and IC vortex QDs with $S = 1$ (yellow and red regions, and the yellow-only one, respectively). (a),(b) The fixed normalization is $N = 100$ . (c) Stability areas for OC vortices with $S = 2$ (all colored regions), 3 ( $orange + brown + green$ ), 4 ( $brown + green + blue$ ), and 5 ( $green + blue + dark gray$ ). For the convenience of plotting, the normalization for $S = 2 - 5$ is fixed as $N = 400$ , 900, 2500, and 4500, respectively. (d1),(d2) Examples of stable on-site- and intersite-centered vortices, for $(g, C) = (0.48, 0.1)$ and (0.77,0.15), which correspond, respectively, to points $B$ and $C$ in (c). The insets display the respective phase profiles. (d3),(d4) Fundamental and dipole-mode QDs, into which the squeezing transforms the on-site- and intersite-centered vortices at $ω = 0.024$ and 0.006. The corresponding critical values are $ω_{c}^{'} = 0.023$ and 0.005, respectively.
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Figure 4
Examples of the evolution of unstable semidiscrete QDs. (a) The OC QD with $S = 0$ , denoted by point $A$ in Fig. 3, transforms into a breather that performs shuttle motion. (b)–(d) Vortex QDs with $(g, C) = (0.55, 0.3)$ , (0.65,0.6), and (0.75,0.18), which are denoted by points $D - F$ in Fig. 3.
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