Revisiting a stability problem of two-component droplets
RRevisiting a stability problem of two-component quantum droplets
Paweł Zin, Maciej Pylak,
1, 2 and Mariusz Gajda National Centre for Nuclear Research, ul. Pasteura 7, PL-02-093 Warsaw, Poland Institute of Physics, Polish Academy of Sciences,Aleja Lotników 32/46, PL-02-668 Warsaw, Poland
We study the problem of the stability of a two-component droplet. The standard solution Ref. [1]is based on a particular form of the mean field energy functional, in particular on the assumptionof vanishing large energy hard mode contribution. This imposes a constraint on the densities ofthe two components. The problem is reduced to stability analysis of a one component system. Asopposed to this, we present a two component approach including possible hard mode excitations.We minimize the energy under conditions corresponding to the experimentally relevant situationwhere volume is free and atoms can evaporate. For the specific case of a two component Bose-Bosedroplet we find approximate analytic solutions and compare them to the standard result. We showthat the densities of a stable droplet are limited to a range depending on interaction strength, incontrast to the original unique solution.
I. INTRODUCTION
Quantum droplets, as predicted in the seminal paperby D. Petrov [1], are self bound systems of a mixtureof two Bose-Einstein condensates under such conditionsthat interspecies attraction drives them towards collapse.The stabilizing agent is the Lee-Huang-Young (LHY) en-ergy [2] originating from quantum fluctuations of the Bo-goliubov vacuum. The stability analysis presented in [1]is based on the observation that a stable droplet can beformed if interactions are chosen in such a way that themean field energy almost vanishes. In fact the systemshould be effectively very weakly attractive, so that theinstability is suppressed by a small contribution of quan-tum fluctuations – the LHY term.Soon experiments came. The first experiments withmixtures of Potassium atoms in two different internalstates [3–5] as well as the recent achievement of quan-tum droplets in heteronuclear bosonic mixtures [6] wellagree with the theoretical predictions. Moreover, a sim-ilar stabilization mechanism, originating from quantumfluctuations [7], occurred to be responsible for stabiliza-tion of elongated dipolar condensates of Dysprosium [8–11] and Erbium [12] atoms. For a review on the presentstate of quantum droplet physics see [13].Standard stability analysis of a two component Bose-Bose mixture presented in Ref. [1] is based on the ob-servation that the mean field energy density ε ( n , n ) isa quadratic form of densities of both components n , n ,and can be brought to the diagonal form ε ( n , n ) = λ + n + λ − n − . Explicit form of n ± and λ ± is not neededhere, it can be found in [1]. n + and n − are densitiesof hard and soft modes respectively, both being linearcombinations of n and n . The dominant contributionto the energy density comes from the hard mode, whichowes its name to the hard mode interaction strength λ + which significantly dominates the soft mode strength, λ − ,i.e. λ + (cid:29) | λ − | . The soft mode is unstable λ − < butthe instability is weak and can be tamed by the smallLHY energy. This energy is negligible in typical experi- mental arrangements but becomes crucial when the meanfield energy nearly vanishes. This can happen if the hardmode contribution is very small. In [1] the approxima-tion n + ( n , n ) = 0 is used. The densities of both speciesbecome dependent then.This way the problem of a mixture can be reduced toan effectively single component case – the energy densitybecomes a functional of a single function only. To get theequilibrium densities of a self bound droplet it is enoughto notice that in a free uniform system the pressure van-ishes. This condition determines unambiguously the den-sity of the droplet. Although dilute ( ∼ − cm − ),quantum droplets behave like liquids. Their densities arefixed by interaction and in the limit of an infinite systemthey do not depend on atom number.In this paper we want to address the issue of stabilityof a two component droplet going beyond approxima-tions which rely on the distinction between hard and softmodes and neglect the former one. The two componentapproach shows the standard result from a broader per-spective. Moreover, it allows to find a contribution ofhard mode excitations to the ground state energy of thestable droplet. We want to add that the hard mode con-tribution to the energy may be important not only in theground state. It reveals itself especially in collisions ofdroplets [14].We address the issue of the stability of the mixture ina quite general setting, however, we have in mind a twocomponent system of ultracold bosonic atoms. The sta-bility conditions are formulated in a form not assumingany particular energy density functional. These condi-tions are determined by constraints imposed by the typ-ical physical situation. The question of a global uncon-strained minimum has a simple but trivial answer if theanalysis is limited to the mean field approach – (i) in aneffectively attractive case the system collapses and bothdensities become infinite, (ii) or on the other hand if thesystem is a repulsive one, the atoms expand to infinityand their densities vanish. The collapse predicted on themean field level in fact signifies that the description useddoes not account for physical processes in this situation. a r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p Formation of bound molecules, larger complexes or solid-ification is expected then.We study a typical experimental situation where ini-tially N atoms of the first component are mixed with N atoms of the second component in an external trap.After tuning the interactions to the region in which a sta-ble droplet is expected, the external potential is removed.Eventually a droplet is formed. This is a scenario whichdefines plausible physical constraints. Our goal is to findthe densities of a stable system formed this way and/orthe final number of atoms in each component. Note thatthe final number of atoms need not be necessarily thesame as the number of atoms mixed initially, as somemay evaporate. II. CONSTRAINTS
In this section we formulate constraints defining a sta-ble self bound system taking into account the typicalphysical situation. Although we explicitly refer to a Bose-Bose mixture we want to stress that the approach pre-sented here can be, after minor adjustments, applied toother mixtures provided that their potential energy de-pends on densities of both species only. In particular wehave in mind Bose-Fermi mixtures as studied in [15, 16].The interaction energy density of a mixture of Bose-Einstein condensates, for fixed interaction strength, isa function of densities of the two components n , n , ε = ε ( n , n ) . The densities are related to correspond-ing wavefunctions n i = | ψ i | which are normalized to thenumber of particles (cid:82) d r | ψ i ( r ) | = N i . The total energydensity with kinetic energy included is: E = − (cid:126) m (cid:88) i =1 ∇ ψ ∗ i ∇ ψ i + ε ( n , n ) , (1)where we assumed equal masses m of the two kind ofatoms involved. The choice of equal masses has one seri-ous advantage – it allows for (to a large extent) analyticaltreatment. The entire procedure is also valid for differ-ent masses of both species, however, numerics is neededthen.For a droplet to be formed the energy functional ε should contain a positive contribution from repulsive in-traspecies interactions as well as a negative contributionfrom attractive interaspecies interactions.A corresponding time-dependent set of two coupledGross-Pitaevskii (GP) equations describing the dynamicsof both components can be easily obtained by minimizingthe action S = (cid:82) d r (cid:82) dt L , where the Lagrangian densityis L = (cid:126) R e ( i (cid:80) j ψ ∗ j ∂ t ψ j ) − E : i (cid:126) ∂∂t ψ i ( r ) = (cid:20) − (cid:126) m ∆ + δε ( n , n ) δn i (cid:21) ψ i ( r ) (2)By the standard substitution ψ i ( r , t ) = e − iµ i t/ (cid:126) ψ i ( r ) thetime dependent equations lead to a set of two coupled stationary GP equations: (cid:20) − (cid:126) m ∆ + δε ( n , n ) δn i (cid:21) ψ i ( r ) = µ i ψ i ( r ) (3)where the eigenvalues µ i are chemical potentials.We assume that interactions are tuned in such a waythat the system is effectively very weakly attractive andis on the collapse side of the stability diagram [17, 18]. Asshown by D. Petrov [1], if the energy of quantum fluctua-tions is included in ε ( n , n ) in addition to the aforemen-tioned mean-field interaction energy, the collapse may beavoided and a liquid droplet of volume V can be formed.This however can only happen if the interactions are ap-propriately tuned. Moreover, the numbers of availableatoms in every component must be in a right proportion.In general the number of particles forming a droplet isdifferent from the number of atoms N ini , N ini preparedinitially in the trap and used in the formation process.After the trapping potential is removed the system is free.There are no external mechanisms of controlling the vol-ume or number of atoms in the system. The excessiveparticles will be ejected and will not contribute to thetotal energy. We do not assume interaction of the sys-tem with any external reservoir of particles, the numberof particles forming a droplet may not grow larger thanthe initial N ini and N ini . These are the only physicalconstraints we impose on the system.The question we want to answer here is: which stablesystem (for fixed interactions) can be formed having atdisposal N ini atoms of the first kind and N ini atoms ofthe second kind? Stable solutions of GP equations Eq.(2) should corre-spond to a minimum of energy. If it is a global minimumthe system is absolutely stable. Metastable states corre-spond to a local minimum of energy. A potential barrierseparates the system from the global minimum. The to-tal energy of the system is: E ( N , N ) = (cid:90) d r E ( r ) . (4)Chemical potentials µ i appear in Eqs.(2, 3) as eigenen-ergies of stationary solutions of the GP equations. It is asimple exercise to verify that these eigenenergies µ i , as itshould be in the case of true chemical potentials, describea response of the total energy of the system to a changeof particle number: ∂E∂N i = µ i (5)Because the energy density functional, Eq.(1), accountsfor the kinetic energy, the density of a droplet decays ex-ponentially at its surface. Density profiles follow directlyfrom solutions of the stationary GP equations Eq.(3) andvolume of the droplet is not an additional parameter.If the system is stable, i.e. if its energy E ( N , N ) corresponds to some minimum, then infinitesimally smallchange of atom number in any component must increaseits energy: dE = ∂E∂N dN + ∂E∂N dN = µ dN + µ dN > (6)In a typical experimental situation the number of atomsmay only decrease, i.e. dN i < . If any of the chemicalpotentials were positive the system would decrease its en-ergy by evaporating some particles of the correspondingkind. Therefore the constraints we impose on a stabledroplet are: µ < , (7) µ < . (8)If in a given state both chemical potentials are negativethen there is no state of lower energy in its close neigh-bourhood. We are going to exploit these conditions inthe following.Note, however, that in the above we analyzed only sta-tionary states i.e. states being the solution of stationaryGP equation (3). We did not consider dynamical stabil-ity of the solution against some small perturbations. It isknown that there exists stationary localized droplet solu-tions which are however dynamically unstable - a smallinitial perturbation grows exponentially in time [19]. Inwhat follows we do not consider the issue of dynamicalstability. III. BOSE-BOSE DROPLETSA. Region of stability
In the general approach sketched above a kinetic en-ergy was included. This way we accounted for surfacetension providing a necessary pressure to stabilize thesystem. Unfortunately including kinetic energy leads todifferential equations which cannot be treated analyti-cally in more detail in the general case.To get some better insight into the problem of stabil-ity of a droplet we simplify our analysis and assume thatthe system is large and the surface energy is much smallerthan the interaction energy so that it can be neglected.This approximation is known as the Thomas-Fermi ap-proximation. It amounts to assuming that E = ε ( n , n ) .Such a system is uniform, has well defined volume V andconstant densities n i = N i /V . The energy density of amixture of two quantum-degenerate Bose gases is of theform: ε ( n , n ) / (cid:18) π (cid:126) m (cid:19) = 12 (cid:88) i =1 a ii n i − a n n ++ c ( a n + a n ) / (9)where a ij > are scattering lengths. The first term de-scribes repulsive interspecies interactions while the sec-ond one corresponds to interspecies attraction. The last term is the LHY energy contribution, c = 64 / (15 √ π ) Byintroducing the parameter δa = − ( a + √ a a ) > the energy density can be brought to the form ε ( n , n ) / (cid:18) π (cid:126) m (cid:19) = 12 ( √ a n − √ a n ) − δan n + c ( a n + a n ) / (10)where ”hard mode” and ”soft mode” are clearly visible.We assume that δa (cid:28) a , a , i.e. that the collapseinstability is weak and a small LHY term can balanceit. This assumption ensures that the system is weaklyinteracting.The total energy of the uniform system is E u ( N , N , V ) = V · ε ( N /V, N /V ) . It dependsnot only on the number of particles N i but also onthe volume V . Differential change of energy due toinfinitesimal change of volume and particle number is: dE u ( N , N , V ) = − pdV + µ ,u dN + µ ,u dN (11)where p = − ∂E u ∂V , (12)is a pressure, while µ i,u = ∂E u ∂N i = ∂ε∂n i , (13)are chemical potentials of the species. Note that we useddifferent symbols from those in Eq.(5).For a uniform free system, as opposed to a system witha surface, we get an additional constraint: a droplet willstabilize its volume if internal pressure vanishes: p = − ∂E u ∂V = µ ,u n + µ ,u n − ε ( n , n ) = 0 (14)Equation (14) allows to find the volume of a droplet as afunction of particle number V = V u ( N , N ) . V u ( N , N ) / = 3 c ( a N + a N ) / δaN N − ( √ a N − √ a N ) . (15)Physical solutions (i.e. solutions with real and positive V u ( N , N ) ) of Eq. (14) exist if: |√ a N − √ a N | < (cid:112) δaN N (16)The first important observation is that the right handof inequality Eq.(16) significantly reduces the possiblevariation of the ratio N /N , because δa (cid:28) a , a .Thus a term √ a N − √ a N must be very small. Toquantify this difference we introduce a small parameter δb = δa √ a a (cid:28) , and a variable ξ being a scaled ratio ofatom numbers (or atomic densities), ξ = n √ a n √ a . Afterneglecting corrections of higher order in δb , Eq.(16), canbe brought to the form: δξ < δb (17)where δξ = ξ − . Obviously δξ is the second smallparameter of the theory.In view of Eq.(17) it is reasonable to assume that atequilibrium δξ (cid:39) so the ratio of atom numbers (andthe ratio of equilibrium densities) is approximately equalto: N N = n n = (cid:114) a a = s (18)This is a basic assumption of the analysis in [1]. Note thatcondition Eq.(18) eliminates the hard mode contributionto the energy density Eq.(10). Only soft mode and LHYenergies remain. Using Eqs.(18) and (15) the equilibriumdensities n i of a droplet can be well approximated by: n a = (cid:18) c (cid:19) δb (1 + s ) (19) n a = (cid:18) c (cid:19) δb (1 + s ) (20)If δξ = 0 then the ratio of densities of the componentsis equal to the ‘magic’ value s at which the hard modecontribution to the mean field energy vanishes. Therefore δξ measures a deviation of a droplet’s density from thisratio. On the other hand it is easy to check that thisparameter equals to fluctuations of density of the hardmode. If we define relative fluctuations of the densities δ i in each mode via the relation: n i = n i (1 + δ i ) , then δξ measures the difference between these fluctuations: δξ = δ − δ (21)So far we have only made use of the condition thata stable droplet has a vanishing pressure p = 0 whichstabilizes its volume V = V u ( N , N ) . The total energy ofdroplets, E ( N , N ) = E u ( N , N , V u ( N , N )) , becomesa function of number of atoms only. We are thus onthe same footing as in the situation of a droplet havinga nonuniform density profile where there is no need tointroduce a volume as a free parameter.Let us observe that the two functions E ( N , N ) and E u ( N , N , V ) are different because in the latter case V is an independent variable as opposed to the pre-vious case where volume is a well-defined function of N , N , Eq.(15). This leads to two different definitionsof chemical potentials. One is the µ i given by Eq. (5), µ i = ∂E/∂N i , and the second one is given by Eq.(13), µ i,u = ∂E u /∂N i . The relation between these two is givenby µ i ≡ ∂E ( N , N ) ∂N i = ∂E u ( N , N , V u ) ∂N i − p u ∂V u ∂N i = ∂E u ( N , N , V u ) ∂N i ≡ µ i,u , (22)where p u = ∂E u ( N ,N ,V ) ∂V | V = V u is pressure of a dropletwith volume V u ( N , N ) . For a stable droplet this pres-sure vanishes p u = 0 , therefore there is no additional en-ergy cost related to change of volume. Both definitions of chemical potential are equivalent, µ i = µ i,u . The stabil-ity conditions Eqs.(7), (8) are valid also in the case whendroplet densities are approximated by constant functionsand volume is introduced as an additional free parameter. FIG. 1. Solutions of Eq. (14) in the form of contour plots inthe n − n plane. we show the tip of p = 0 isobar where byblue color we indicate the stable region as given by µ < and µ < constraints, Eqs.(24,25). In the inset we show thefull zero pressure isobar which has a shape of the elongatedloop. By the black dot we indicate the standard solutionto the stability problem according to Eq.(18). We considertwo cases: (i) s = √ (left panel). The standard solutionis located at the border, but inside the stable region. Thesolution given by Eq.(31) marked in green is at the centre;(ii) s = 2 (right panel). In this case the standard solutionis located outside the stable region. The solution given byEq.(31) remains well within the limit of stability. Densitiesare in units of n = π δa a s (1+ s ) Utilizing the explicit form of the energy density func-tional Eq.(10), stability conditions of a quantum Bose-Bose droplet can be written as follows:(i) the pressure vanishes , p ( n , n ) = 0 , Eq.(14): pe = − δb + 12 δξ ξ + η (cid:18) √ ξs + (cid:112) ξs (cid:19) / = 0 (23)(ii) both chemical potentials are negative: µ n e = − δb − δξξ + 53 η (cid:18)(cid:112) ξs + 1 √ ξs (cid:19) / √ ξs < (24) µ n e = − δb + δξ + 53 η (cid:18)(cid:112) ξs + 1 √ ξs (cid:19) / (cid:112) ξs < (25) where we introduced: e = π (cid:126) m √ a a n n , and η = c (cid:0) n a n a (cid:1) / . By expanding the above equationsto leading order in the small parameter δξ = ξ − weget the equation corresponding to the p = 0 isobar in the n − n plane: η = 3 c (cid:0) n a n a (cid:1) / = δb − δξ (cid:16) √ s + √ s (cid:17) / (26)Similar expansion allows for approximate but analytic de-termination of conditions limiting the region of stabilityof a quantum droplet with respect to evaporation, Eqs.(24,25). The region of corresponding parameters forms asegment of p = 0 isobar where the ratio of densities arelimited as follows: − s s < − δξδb < −
53 11 + s (27)Equations (26),(27) summarize the main results of ourpaper. They specify the region in the density-densityplane where stable droplets exist. These results shouldbe compared to other approximate expressions existingin the literature. B. Comparison with previous results
In this subsection we compare our results with previ-ous formulae existing in the literature. Historically thefirst expression for droplets density was presented in [1].Based on: (i) assumption that hard mode contribution tothe system energy vanishes at equilibrium, and (ii) condi-tion of vanishing pressure, the equilibrium densities arefound - see Egs.(19),(20). The solution gives n = n , n = n . Because all approaches discussed below usecondition of vanishing pressure therefore a value of theparameter δξ is a perfect measure of differences betweenvarious results. Approximations of [1] give: δξ = (cid:18) n n − n n (cid:19) = 0 (28)This is the simplest but very good estimation of ratio ofdensities of large droplets confirmed in experiments [3–5].Another solution is given in [15, 16]. Although resultsobtained there are dedicated to a Bose-Fermi mixture, but by following the main lines of the approach one caneasily adapt the solutions of [15, 16] to a Bose-Bose sys-tem. The energy functional of a Bose-Fermi mixture isnot a quadratic form of densities and splitting of energyinto hard and soft modes is not possible. Thus the ap-proach of Ref.[1] cannot be used there. Instead, the re-sults are obtained in the limit of infinite uniform system.The issue of a finite volume does not have to be addressedat all then. In such a situation only intensive quantitiesmake sense. These are the energy density and pressure.The stability problem is defined as a problem of find-ing a constrained minimum of the energy density undercondition of vanishing pressure. This condition ensuresthat there are no net internal forces acting on a ficti-tious surface inside the bulk of a droplet. This is thesame condition which fixes the volume of a finite homo-geneous droplet, Eq.(14). Additionally, at a minimum ofenergy density ε ( n , n ) any infinitesimally small varia-tion of atomic densities cannot change the energy: dε = µ dn + µ dn = 0 (29)To stay on the p ( n , n ) = 0 isobar the variations of bothdensities dn and dn must be related: dp = ∂p∂n dn + ∂p∂n dn = 0 . (30)Combining condition Eq.(30) with Eq.(29) the followingequation for the constrained minimum of energy density, (cid:15) ( n , n ) , can be found: µ ∂p∂n − µ ∂p∂n = 0 . (31)When accompanied by p ( n , n ) = 0 equation the densi-ties of both species can be found.Eq.(31) accompanied by p ( n , n ) = 0 condition canbe applied to the Bose-Bose mixtures. Again, expressingderivatives of pressure contributing to the above equa-tion in terms of ξ and expanding Eq.(31) in the smallparameter δξ the approximate solution for densities of astationary droplet is: δξ = (cid:18) n n − n n (cid:19) = δb (cid:18) − s s (cid:19) . (32)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 centerof the stability region. This solution is marked by a greendot in Fig. (1).Evidently both formulae Eq. (28) and Eq.(32) areequivalent if intraspecies interactions are equal, i.e. if s = 1 . Note that Eq.(32) confirms small contribution ofhard mode excitations to the densities of a stable droplet.The situation is different for s > / . Then the stan-dard result δξ = 0 given by Eq. (28) is outside the sta-bility region given by Eq. (27). Thus, for sufficientlystrong asymmetric intraspecies interaction the standardsolution does not support a stable droplet. FIG. 2. The total energy as a function of the number of particles in every component for unequal intraspecies interaction, s = √ . Left panel: Coloured region corresponds to such a composition of the mixture for which 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 theedges of the zone of stable droplets where µ < , µ < . The rectangle at the center indicates the region which we zoom-in inthe right panel. Right panel: Zoom of the energy landscape in N − N plane. It illustrates adiabatic evolution of two initialstates ( N ini , N ini ) marked by black dots. Evolution towards the state of minimal possible energy constrained by initial atomnumbers cannot have any positive-valued gradient component of the chemical potential vector ( µ , µ ) . The white arrows showtrajectories towards the final state ( N fin , N fin ) (red dots) of lowest possible energy for the assumed arrangement. Please notethat only the edges of the stability region can be reached. Getting into the interior of this region requires increasing the numberof atoms of at least one kind. On the contrary, all systems having initially a number of particles corresponding to the areabetween 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 equal to N r = N · n ˜ r ≈ N · , where ˜ r = (cid:113) s +14 π | δa | n is the length unit The results are illustrated in Fig.(1) where we show thestability diagram in a plane of atomic densities, n and n . For comparison we present the two cases: s = √ and s = 2 . The p = 0 isobar has the form of a closedloop originating at the center of the coordinate system– see inset in Fig. (1). The region which is stable withrespect to atom losses, ( µ , µ ≤ ), Eqs. (24,25), is lo-cated close to the tip of the loop which we zoom-in inthe main frame. This is the part of the isobar marked inblue. By green dot we mark the solution correspondingto the global minimum of an infinite system as suggestedin [15, 16] and given by Eq. (32). This result is well inthe stable part of the diagram regardless the interactions.The standard solution of [1], Eq.(18), is indicated by ablack dot. We stress that when the disproportion of in-traspecies interactions is too large ( s = 2 ) the standardsolution of Ref.[1] is out of the stability region. C. Stable equilibrium for given initial particlenumber
In the last part of the paper we go back to the problemwhich was the inspiration for our study. We address thequestion asked at the beginning of this work, i.e. we are going to show what is a minimal energy state which canbe reached having at disposal N ini atoms of the first kindand N ini atoms of the second kind allowing for throwingaway some of them.The solution to this problem is illustrated inFig. (2) which shows the total energy of the system E ( N , N , V ( N , N )) = V ( N , N ) ε ( N /V, N /V ) inthe plane of extensive quantities N , N . If one has ini-tially a two component mixture with ( N ini , N ini ) atomsthen the droplet formed would be in general a mixtureof ( N fin , N fin ) atoms of both kinds. To find the dropletof the lowest energy among all possible final states ofdroplets composed with number of atoms limited by theinitial values, N fin ≤ N ini and N fin ≤ N ini we directlyexamine the region of energies in the relevant rectangulardomain in N − N plane: ≤ N ≤ N ini , (33) ≤ N ≤ N ini . (34)In Fig. (2) the initial composition of droplet is markedby a black dot. White vertical and horizontal arrowspoint to the final states ( N fin , N fin ) which minimizethe energy constrained according to the previous discus-sion. We consider two situations. The first one is that N ini /N ini is so large that − δξ/δb > − / / (1 + s )) ,i.e. the second component of the mixture is a stronglyexcessive one. In such a case the excessive atoms simplyevaporate until the system reaches the boundary of thestable region, vertical arrow in the figure. It is worthmentioning that the number of minority atoms is con-served. Further evaporation stops when the border ofthe stability sector is reached. This is because equilib-rium results from a competition of the two tendencies:the system tends to decrease the chemical potential (theenergy per particle) as much as possible and simulta-neously to keep as many atoms as possible, since totalenergy is extensive.The second case shown in Fig. (2) relates to the sit-uation where the first component dominates, i.e. if − δξ/δb < − / s/ (1 + s )) . The scenario describedabove repeats. Excessive atoms of the first componentevaporate, while the number of atoms in the second com-ponent remains constant (horizontal white arrow in fig-ure). This process stops while reaching the border of thestable 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) itwill not evaporate atoms at all. We have to add that thepresent discussion is based on stationary stability analy-sis and no time dynamics was considered at all. Thereforeall our conclusions, in particular these invoking dynamicprocesses such as evaporation, implicitly assume that thesystem remains at equilibrium and adiabatically followsthe state determined by external parameters and tempo-ral number of atoms. For the same reason we are not ableto discuss a situation when the initial number of atoms isoutside of the coloured angular sector in Fig.(2) indicat-ing p = 0 zone. The outer part is a totally unstable sectorand releasing atoms from the trap while parameters are in this range will trigger a violent dynamics. Only dy-namical studies of the process of droplet formation mightgive the state of an eventually formed droplet. IV. CONCLUSIONS
In this paper we specified stability conditions for aself-bound two-component droplet. The conditions arenot related to any particular form of the energy den-sity functional, and can be applied to other systems likefor instance Bose-Fermi mixtures. The case of Bose-Bosedroplets was studied in detail. In contrast to the standardsolution of Ref.[1] we show that stable droplets’ densities,for fixed values of interactions strength, can take valuesfrom some finite range of parameters, thus there is nounique droplet solution. This regime of allowed densitiesis however rather small and deviations from the standardsolution are limited particularly for similar strengths ofintraspecies interactions. In the limit of large droplets,when kinetic energy can be neglected, we found a veryuseful analytic expression for the boundaries of the stabil-ity zone. We have shown that if intraspecies interactionsare very different from each other then the prediction ofRef.[1] is outside the stability sector.
ACKNOWLEDGMENTS
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