Revisiting a stability problem of two-component quantum droplets

Paweł Zin, Maciej Pylak, and Mariusz Gajda

Phys. Rev. A 103, 013312 – Published 14 January 2021

Abstract

We study the problem of the stability of a two-component droplet. The standard solution [D. S. Petrov, Phys. Rev. Lett. 115, 155302 (2015)] is based on a particular form of the mean field energy functional, in particular on the assumption of vanishing of potentially large hard-mode-source energy contribution. This imposes a constraint on the densities of the two components. The problem is reduced to stability analysis of a one-component system. As opposed to this, we present a two-component approach including possible nonzero hard-mode-source energy. We minimize the energy under conditions corresponding to the experimentally relevant situation where volume is free and atoms can evaporate. For the specific case of a two-component Bose-Bose droplet we find approximate analytic solutions and compare them to the standard result. We show that the densities of a stable droplet are limited to a range depending on interaction strength, in contrast to the original unique solution.

Received 30 September 2020
Revised 30 December 2020
Accepted 5 January 2021

DOI:https://doi.org/10.1103/PhysRevA.103.013312

Physics Subject Headings (PhySH)

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

Atomic, Molecular & Optical

Authors & Affiliations

Paweł Zin ¹, Maciej Pylak ^1,2, and Mariusz Gajda ^2,*

¹National Centre for Nuclear Research, ul. Pasteura 7, PL-02-093 Warsaw, Poland
²Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, PL-02-668 Warsaw, Poland

^*gajda@ifpan.edu.pl

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Vol. 103, Iss. 1 — January 2021

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Images

Figure 1
Solutions of Eq. (14) in the form of contour plots in the $n_{1} − n_{2}$ plane. We show the tip of $p = 0$ isobar where blue color indicates the stable region as given by $μ_{1} < 0$ and $μ_{2} < 0$ constraints, Eqs. (25) and (26). In the inset we show the full zero pressure isobar which has the shape of an elongated loop. The black dot indicates the standard solution to the stability problem according to Eq. (19). We consider two cases: (i) $s = \sqrt{2}$ (top panel). The standard solution is located at the border, but inside the stable region. The solution given by Eq. (32) marked in green is at the center. (ii) $s = 2$ (bottom panel). In this case the standard solution is located outside the stable region. The solution given by Eq. (32) remains well within the limit of stability. Densities are in units of $n_{10} = \frac{25 π}{1024} \frac{δ a^{2}}{a_{11}^{5} s^{2} {(1 + s)}^{5}}$ .
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Figure 2
The total energy as a function of the number of particles in every component for unequal intraspecies interaction, $s = \sqrt{2}$ . Left panel: The colored region corresponds to a composition of the mixture for which the $p = 0$ condition can be met. The isobar $p = 0$ shown in Fig. 1 becomes here the interior of the angular region given by Eq. (16). White lines indicate the edges of the zone of stable droplets where $μ_{1} < 0, μ_{2} < 0$ . The rectangle at the center indicates the region to which we zoom-in in the right panel. Right panel: Zoom of the energy landscape in the $N_{1} − N_{2}$ plane. It illustrates adiabatic evolution of two initial states $(N_{1}^{ini}, N_{2}^{ini})$ marked by black dots. Evolution towards the state of minimal possible energy constrained by initial atom numbers cannot have any positive-valued gradient component of the chemical potential vector $(μ_{1}, μ_{2})$ . The white arrows show trajectories towards the final state $(N_{1}^{fin}, N_{2}^{fin})$ (red dots) of lowest possible energy for the assumed arrangement. Please note that only the edges of the stability region can be reached. Getting into the interior of this region requires increasing the number of atoms of at least one kind. In contrast, all systems having initially a number of particles corresponding to the area between the white lines are stable against small perturbations. The number of atoms is expressed in convenient units of Ref. [1]. Therefore, the “real” number of atoms is $N_{r} = N \times n_{10} {\tilde{r}}^{3} \approx N \times 6300$ , where $\tilde{r} = \sqrt{\frac{3}{2} \frac{s + 1}{4 π | δ a | n_{10}}}$ is the length unit.
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Figure 3
Two-dimensional contour plot of the total energy of the droplet as a function of the number of particles in every component for equal intraspecies interactions, $s = 1$ . Energies were obtained by solving numerically the two-component EGP equations. Black lines indicate previously found analytical boundaries for $μ_{1}, μ_{2} \leq 0$ . Units are the same as in Fig. 2. The black dashed line separates stable droplet solutions from unbound ones. The number of atoms is expressed in convenient units of Ref. [1]. For parameters used in this figure the “real” number of atoms is $N_{r} = N \times 2097$ .
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