Revisiting a stability problem of two-component quantum droplets

In the general approach sketched above a kinetic energy was included. This way we accounted for surface tension providing a necessary pressure to stabilize the system. Unfortunately including kinetic energy leads to differential equations which cannot be treated analytically in more detail in the general case.

To get some better insight into the problem of stability of a droplet we simplify our analysis and assume that the system is large and the surface energy is much smaller than the interaction energy so that it can be neglected. This approximation is known as the Thomas-Fermi approximation. It amounts to assuming that ${\cal E}=\varepsilon(n_{1},n_{2})$. Such a system is uniform, has well defined volume $V$ and constant densities $n_{i}=N_{i}/V$. The energy density of a mixture of two quantum-degenerate Bose gases is of the form:

where $a_{ij}>0$ are scattering lengths. The first term describes repulsive interspecies interactions while the second one corresponds to interspecies attraction. The last term is the LHY energy contribution, $c=64/(15\sqrt{\pi})$ By introducing the parameter $\delta a=-(a_{12}+\sqrt{a_{11}a_{22}})>0$ the energy density can be brought to the form

where ”hard mode” and ”soft mode” are clearly visible. We assume that $\delta a\ll a_{11},a_{22}$, i.e. that the collapse instability is weak and a small LHY term can balance it. This assumption ensures that the system is weakly interacting.

The total energy of the uniform system is $E_{u}(N_{1},N_{2},V)=V\cdot\varepsilon(N_{1}/V,N_{2}/V)$. It depends not only on the number of particles $N_{i}$ but also on the volume $V$. Differential change of energy due to infinitesimal change of volume and particle number is:

where

is a pressure, while

are chemical potentials of the species. Note that we used different symbols from those in Eq.(5).

For a uniform free system, as opposed to a system with a surface, we get an additional constraint: a droplet will stabilize its volume if internal pressure vanishes:

Equation (14) allows to find the volume of a droplet as a function of particle number $V=V_{u}(N_{1},N_{2})$.

Physical solutions (i.e. solutions with real and positive $V_{u}(N_{1},N_{2})$) of Eq. (14) exist if:

The first important observation is that the right hand of inequality Eq.(16) significantly reduces the possible variation of the ratio $N_{1}/N_{2}$, because $\delta a\ll a_{11},a_{22}$. Thus a term $\sqrt{a_{11}N_{1}}-\sqrt{a_{22}N_{2}}$ must be very small. To quantify this difference we introduce a small parameter $\delta b=\frac{\delta a}{\sqrt{a_{11}a_{22}}}\ll 1$, and a variable $\xi$ being a scaled ratio of atom numbers (or atomic densities), $\xi=\frac{n_{2}\sqrt{a_{22}}}{n_{1}\sqrt{a_{11}}}$. After neglecting corrections of higher order in $\delta b$, Eq.(16), can be brought to the form:

where $\delta\xi=\xi-1$. Obviously $\delta\xi$ is the second small parameter of the theory.

In view of Eq.(17) it is reasonable to assume that at equilibrium $\delta\xi\simeq 0$ so the ratio of atom numbers (and the ratio of equilibrium densities) is approximately equal to:

This is a basic assumption of the analysis in Petrov (2015). Note that condition Eq.(18) eliminates the hard mode contribution to the energy density Eq.(10). Only soft mode and LHY energies remain. Using Eqs.(18) and (15) the equilibrium densities $n^{0}_{i}$ of a droplet can be well approximated by:

If $\delta\xi=0$ then the ratio of densities of the components is equal to the ’magic‘ value $s$ at which the hard mode contribution to the mean field energy vanishes. Therefore $\delta\xi$ measures a deviation of a droplet‘s density from this ratio. On the other hand it is easy to check that this parameter equals to fluctuations of density of the hard mode. If we define relative fluctuations of the densities $\delta_{i}$ in each mode via the relation: $n_{i}=n^{0}_{i}(1+\delta_{i})$, then $\delta\xi$ measures the difference between these fluctuations:

So far we have only made use of the condition that a stable droplet has a vanishing pressure $p=0$ which stabilizes its volume $V=V_{u}(N_{1},N_{2})$. The total energy of droplets, $E(N_{1},N_{2})=E_{u}(N_{1},N_{2},V_{u}(N_{1},N_{2}))$, becomes a function of number of atoms only. We are thus on the same footing as in the situation of a droplet having a nonuniform density profile where there is no need to introduce a volume as a free parameter.

Let us observe that the two functions $E(N_{1},N_{2})$ and $E_{u}(N_{1},N_{2},V)$ are different because in the latter case $V$ is an independent variable as opposed to the previous case where volume is a well-defined function of $N_{1},N_{2}$, Eq.(15). This leads to two different definitions of chemical potentials. One is the $\mu_{i}$ given by Eq. (5), $\mu_{i}=\partial E/\partial N_{i}$, and the second one is given by Eq.(13), $\mu_{i,u}=\partial E_{u}/\partial N_{i}$. The relation between these two is given by

where $p_{u}=\frac{\partial E_{u}(N_{1},N_{2},V)}{\partial V}|_{V=V_{u}}$ is pressure of a droplet with volume $V_{u}(N_{1},N_{2})$. For a stable droplet this pressure vanishes $p_{u}=0$, therefore there is no additional energy cost related to change of volume. Both definitions of chemical potential are equivalent, $\mu_{i}=\mu_{i,u}$. The stability conditions Eqs.(7), (8) are valid also in the case when droplet densities are approximated by constant functions and volume is introduced as an additional free parameter.

Utilizing the explicit form of the energy density functional Eq.(10), stability conditions of a quantum Bose-Bose droplet can be written as follows:
(i) the pressure vanishes , $p(n_{1},n_{2})=0$, Eq.(14):

(ii) both chemical potentials are negative:

where we introduced: $e_{0}=\frac{4\pi\hbar^{2}}{m}\sqrt{a_{11}a_{22}}n_{1}n_{2}$, and $\eta=\frac{3c}{2}\left(n_{1}a^{3}_{11}n_{2}a^{3}_{22}\right)^{1/4}$. By expanding the above equations to leading order in the small parameter $\delta\xi=\xi-1$ we get the equation corresponding to the $p=0$ isobar in the $n_{1}-n_{2}$ plane:

Similar expansion allows for approximate but analytic determination of conditions limiting the region of stability of a quantum droplet with respect to evaporation, Eqs. (24,25). The region of corresponding parameters forms a segment of $p=0$ isobar where the ratio of densities are limited as follows:

Equations (26),(27) summarize the main results of our paper. They specify the region in the density-density plane where stable droplets exist. These results should be compared to other approximate expressions existing in the literature.

In this subsection we compare our results with previous formulae existing in the literature. Historically the first expression for droplets density was presented in Petrov (2015). Based on: (i) assumption that hard mode contribution to the system energy vanishes at equilibrium, and (ii) condition of vanishing pressure, the equilibrium densities are found - see Egs.(19),(20). The solution gives $n_{1}=n^{0}_{1}$, $n_{2}=n^{0}_{2}$. Because all approaches discussed below use condition of vanishing pressure therefore a value of the parameter $\delta\xi$ is a perfect measure of differences between various results. Approximations of Petrov (2015) give:

This is the simplest but very good estimation of ratio of densities of large droplets confirmed in experiments Cheiney et al. (2018); Cabrera et al. (2018); Semeghini et al. (2018).

Another solution is given in Rakshit et al. (2019a, b). Although results obtained there are dedicated to a Bose-Fermi mixture, but by following the main lines of the approach one can easily adapt the solutions of Rakshit et al. (2019a, b) to a Bose-Bose system. The energy functional of a Bose-Fermi mixture is not a quadratic form of densities and splitting of energy into hard and soft modes is not possible. Thus the approach of Ref.Petrov (2015) cannot be used there. Instead, the results are obtained in the limit of infinite uniform system. The issue of a finite volume does not have to be addressed at all then. In such a situation only intensive quantities make sense. These are the energy density and pressure.

The stability problem is defined as a problem of finding a constrained minimum of the energy density under condition of vanishing pressure. This condition ensures that there are no net internal forces acting on a fictitious surface inside the bulk of a droplet. This is the same condition which fixes the volume of a finite homogeneous droplet, Eq.(14). Additionally, at a minimum of energy density $\varepsilon(n_{1},n_{2})$ any infinitesimally small variation of atomic densities cannot change the energy:

To stay on the $p(n_{1},n_{2})=0$ isobar the variations of both densities $dn_{1}$ and $dn_{2}$ must be related:

Combining condition Eq.(30) with Eq.(29) the following equation for the constrained minimum of energy density, $\epsilon(n_{1},n_{2})$, can be found:

When accompanied by $p(n_{1},n_{2})=0$ equation the densities of both species can be found.

Eq.(31) accompanied by $p(n_{1},n_{2})=0$ condition can be applied to the Bose-Bose mixtures. Again, expressing derivatives of pressure contributing to the above equation in terms of $\xi$ and expanding Eq.(31) in the small parameter $\delta\xi$ the approximate solution for densities of a stationary droplet is:

This solution meets the stability criteria defined here, Eq.(27). Independently of the value of the parameter $s$, Eq.(32) predicts droplet densities very close to the center of the stability region. This solution is marked by a green dot in Fig. (1).

Evidently both formulae Eq. (28) and Eq.(32) are equivalent if intraspecies interactions are equal, i.e. if $s=1$. Note that Eq.(32) confirms small contribution of hard mode excitations to the densities of a stable droplet.

The situation is different for $s>3/2$. Then the standard result $\delta\xi=0$ given by Eq. (28) is outside the stability region given by Eq. (27). Thus, for sufficiently strong asymmetric intraspecies interaction the standard solution does not support a stable droplet.

The results are illustrated in Fig.(1) where we show the stability diagram in a plane of atomic densities, $n_{1}$ and $n_{2}$. For comparison we present the two cases: $s=\sqrt{2}$ and $s=2$. The $p=0$ isobar has the form of a closed loop originating at the center of the coordinate system – see inset in Fig. (1). The region which is stable with respect to atom losses, ($\mu_{1},\mu_{2}\leq 0$), Eqs. (24,25), is located close to the tip of the loop which we zoom-in in the main frame. This is the part of the isobar marked in blue. By green dot we mark the solution corresponding to the global minimum of an infinite system as suggested in Rakshit et al. (2019a, b) and given by Eq. (32). This result is well in the stable part of the diagram regardless the interactions. The standard solution of Petrov (2015), Eq.(18), is indicated by a black dot. We stress that when the disproportion of intraspecies interactions is too large ($s=2$) the standard solution of Ref.Petrov (2015) is out of the stability region.

In the last part of the paper we go back to the problem which was the inspiration for our study. We address the question asked at the beginning of this work, i.e. we are going to show what is a minimal energy state which can be reached having at disposal $N_{1}^{ini}$ atoms of the first kind and $N_{2}^{ini}$ atoms of the second kind allowing for throwing away some of them.

The solution to this problem is illustrated in Fig. (2) which shows the total energy of the system $E(N_{1},N_{2},V(N_{1},N_{2}))=V(N_{1},N_{2})\varepsilon(N_{1}/V,N_{2}/V)$ in the plane of extensive quantities $N_{1},N_{2}$. If one has initially a two component mixture with $(N_{1}^{ini},N_{2}^{ini})$ atoms then the droplet formed would be in general a mixture of $(N_{1}^{fin},N_{2}^{fin})$ atoms of both kinds. To find the droplet of the lowest energy among all possible final states of droplets composed with number of atoms limited by the initial values, $N_{1}^{fin}\leq N_{1}^{ini}$ and $N_{2}^{fin}\leq N_{2}^{ini}$ we directly examine the region of energies in the relevant rectangular domain in $N_{1}-N_{2}$ plane:

In Fig. (2) the initial composition of droplet is marked by a black dot. White vertical and horizontal arrows point to the final states $(N_{1}^{fin},N_{2}^{fin})$ which minimize the energy constrained according to the previous discussion. We consider two situations. The first one is that $N_{2}^{ini}/N_{1}^{ini}$ is so large that $-\delta\xi/\delta b>1-5/3(1/(1+s))$, i.e. the second component of the mixture is a strongly excessive one. In such a case the excessive atoms simply evaporate until the system reaches the boundary of the stable region, vertical arrow in the figure. It is worth mentioning that the number of minority atoms is conserved. Further evaporation stops when the border of the stability sector is reached. This is because equilibrium results from a competition of the two tendencies: the system tends to decrease the chemical potential (the energy per particle) as much as possible and simultaneously to keep as many atoms as possible, since total energy is extensive.

The second case shown in Fig. (2) relates to the situation where the first component dominates, i.e. if $-\delta\xi/\delta b<-1+5/3(s/(1+s))$. The scenario described above repeats. Excessive atoms of the first component evaporate, while the number of atoms in the second component remains constant (horizontal white arrow in figure). This process stops while reaching the border of the stable sector.

If initially the system is prepared in the stable zone, i.e. in the area limited by the two white lines in Fig.(2) it will not evaporate atoms at all. We have to add that the present discussion is based on stationary stability analysis and no time dynamics was considered at all. Therefore all our conclusions, in particular these invoking dynamic processes such as evaporation, implicitly assume that the system remains at equilibrium and adiabatically follows the state determined by external parameters and temporal number of atoms. For the same reason we are not able to discuss a situation when the initial number of atoms is outside of the coloured angular sector in Fig.(2) indicating $p=0$ zone. The outer part is a totally unstable sector and releasing atoms from the trap while parameters are in this range will trigger a violent dynamics. Only dynamical studies of the process of droplet formation might give the state of an eventually formed droplet.