Quantum droplets of dipolar mixtures
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Quantum droplets of dipolar mixtures
R. N. Bisset,1, 2 L. A. Peña Ardila,1 and L. Santos1
1
Institut für Theoretische Physik, Leibniz Universität Hannover, Germany
2
Institut für Experimentalphysik, Universität Innsbruck, Innsbruck, Austria
Recently achieved two-component dipolar Bose-Einstein condensates open exciting possibilities for the study
of mixtures of ultradilute quantum liquids. While nondipolar self-bound (without external confinement) mix-
tures are necessarily miscible with an approximately fixed ratio between the two densities, the density ratio
for the dipolar case is free. Therefore, self-bound dipolar mixtures present qualitatively novel and much richer
arXiv:2007.00404v2 [cond-mat.quant-gas] 12 Mar 2021
physics, characterized by three possible ground-state phases: miscible, symmetric immiscible and asymmetric
immiscible, which may in principle occur at any population imbalance. Self-bound immiscible droplets are
possible due to mutual nonlocal intercomponent attraction, which results in the formation of a droplet molecule.
Moreover, our analysis of the impurity regime, shows that quantum fluctuations in the majority component cru-
cially modify the miscibility of impurities. Our work opens intriguing perspectives for the exploration of spinor
physics in ultradilute liquids, which should resemble to some extent that of 4 He-3 He droplets and impurity-
doped helium droplets.
Introduction.– Helium droplets have been a major focus for
many years [1–4]. They remain liquid at low pressures, even
at zero temperature, constituting an extraordinary scenario for
the study of superfluidity down to nanoscopic scales [5]. In-
terestingly, helium has two stable isotopes, bosonic 4 He and
fermionic 3 He, allowing for self-bound droplet mixtures. Un-
der a typical experimentally achievable temperature of 0.15K,
4
He is a superfluid, whereas 3 He remains a normal fluid [6].
Moreover, due to its smaller mass and limited solubility in
4
He, 3 He resides at the droplet surface surrounding the 4 He
component [3]. Droplets of helium mixtures are hence charac-
teristically phase separated in a core-shell structure, although
droplets under rotation may display more intricate distribu-
tions [7]. Helium droplets can also be doped with other ele-
ments or molecules, which may remain at the surface or sink
to the core. These crucial properties have been extensively ex-
plored, both in what concerns the use of embedded dopants to
prove superfluidity [5], and helium-nanodroplet spectroscopy, FIG. 1. Ground-state phase diagram for Dy-Dy mixtures – in the
absence of external confinement – with aaa = abb = 70a0 and
i.e. the use of the pristine low-temperature environment pro-
Na = Nb = N/2 as a function of total particle number N and aab .
vided by the helium droplet for spectroscopic studies of im- The shaded regions indicate self-bound droplet solutions, whereas
purities [8–11]. below these the solutions are unbound. The dashed curve indicates
Helium droplets constituted up until very recently the only the prediction obtained using the Gaussian Ansatz (6). The insets
show isodensity surface examples for (a) a miscible and (b) an asym-
example of a self-bound quantum liquid, confined in the
metric immiscible self-bound droplet.
absence of external trapping. New developments in the
field of ultracold atoms have, however, changed this picture.
Quantum droplets have been observed both in dipolar Bose-
Einstein condensates (BECs) made of highly magnetic lan-
thanide atoms [12–14], and in binary (nondipolar) homonu- The recently observed, ultradilute self-bound mixtures dif-
clear [15, 16] and heteronuclear [17] Bose mixtures. Strik- fer in a crucial way to helium droplet mixtures: they must
ingly, these droplets are orders of magnitude more dilute than remain miscible. Moreover, to a good approximation such ul-
helium droplets. They are kept self-bound by a mechanism tradilute droplets must keep a fixed ratio between the particle
known as quantum stabilization [18]: an almost complete can- number in each component, and deviations from this ratio are
cellation of the various mean-field forces results in a small evaporated before the droplet sets in. As a result, the spin
residual attraction which is compensated by the repulsive Lee- degree of freedom (i.e. the population difference) remains to
Huang-Yang (LHY) energy induced by quantum fluctuations. a large extent frozen, and the mixture behaves as a single-
In a dipolar BEC, the mean-field forces are given by the dipo- component BEC [18]. Bose-Fermi mixtures must remain mis-
lar and contact interactions [19], whereas in nondipolar binary cible as well [20].
mixtures a similar role is played by inter- and intracomponent In this Letter, we show that recently realized mixtures of
interactions [18]. two dipolar species [21, 22] open new perspectives for the2
study of self-bound mixtures in which the spin degree of free- which converges for na = 0 or nb = 0 to the expression for a
dom is genuinely free. Self-bound dipolar mixtures may be single-component dipolar BEC [33], and for µa,b = 0 to that
miscible but, crucially, also immiscible (Fig. 1). In the latter for a nondipolar mixture [18] (see [26]).
scenario, which to the best of our knowledge is unique to dipo- From the form of V± (θk ) it is easy to see that LHY =
lar mixtures, the two components phase separate while still n5/2 F (P ), where n = na + nb and F is a function of the
being self-bound due to the interplay between quantum sta- polarization P = nb /na . A similar form occurs as well in
bilization and intercomponent dipole-dipole attraction. More- nondipolar binary mixtures. However, for the latter, P is ho-
over, in contrast to experimentally achieved 3 He-4 He droplets, mogeneously fixed at approximately (gaa /gbb )1/2 in the self-
both components should remain superfluid in Bose droplet bound regime [18]. nondipolar self-bound mixtures are hence
mixtures under typical experimental conditions. We identify necessarily miscible, the LHY energy just depends on the total
three different ground-state phases for self-bound dipolar mix- density, and the system is well approximated by an effective
tures: miscible, symmetric immiscible, and asymmetric im- single-component model [18]. In contrast, as discussed be-
miscible. In contrast to nondipolar mixtures, droplets with any low, in a dipolar mixture the polarization is neither fixed nor
population imbalance (polarization) are possible, all the way homogeneous, resulting in rich spinor physics, including the
from the fully balanced case to the impurity limit [23]. We possibility of immiscible droplets. The problem is thus inher-
show that impurity solubility in a dipolar droplet is crucially ently a two-component one. In particular, the LHY energy is
affected by quantum fluctuations in the majority component. a function of the local densities of both components, and not
Although we illustrate the possible physics for the case of Dy- only of the total density.
Dy mixtures [24], the qualitative features are generally valid Formalism.– We are interested in the ground state of self-
for other dipolar mixtures (in particular Er-Dy [21, 22]), open- bound dipolar mixtures. From Eq. (4), we evaluate the LHY
(σ)
ing intriguing perspectives for the study of spinor physics and contribution to the chemical potentials, µLHY ({na,b }) =
impurities in ultradilute dipolar liquids. ∂LHY /∂nσ . As with single-component dipolar BECs [19]
LHY energy.– We first consider a homogeneous binary con- and nondipolar mixtures [18], we study spatially inhomoge-
densate of components σ = a, b, with densities nσ , charac- neous dipolar mixtures by applying a local-density approxi-
terized by the intracomponent scattering lengths aσσ , the in- (σ)
mation (LDA) [34] to the LHY term, µLHY [{na,b (~r)}], ob-
tercomponent scattering length aab , and the magnetic dipole taining two coupled Gross-Pitaevskii (GP) equations which
moments µσ (our theory is equally valid for electric dipoles). incorporate the effect of quantum fluctuations:
All dipole moments are oriented by an external field along the
same direction, z. For simplicity we consider equal masses ∂ h −~2 ∇2 X Z
i~ ψσ (~r) = + d3 r0 Vσσ0 (~r − ~r0 )nσ0 (~r0 )
ma,b = m, although the formalism can be easily extended to ∂t 2m
σ0
unequal masses (for the experimentally relevant Er-Dy mix- i
(σ)
X
tures [21], the masses are approximately equal). + gσσ0 nσ0 (~r) + µLHY [{na,b (~r)}] ψσ (~r), (5)
Using Hugenholz-Pines formalism [25, 26], we obtain the σ0
equation for the LHY energy density correction, LHY [27]:
where nσ (~r) ≡ |ψσ (~r)|2 and Vσσ0 (~r) = µ04πr µσ µσ0
3 (1 −
1X ∂ 2
3 cos θ), with θ the angle between ~r and the dipole moments.
LHY (na , nb )− nσ LHY (na , nb ) = χ(na , nb ), (1)
2 σ ∂nσ In addition to numerically intensive 3D simulations of
Eqs. (5), we employ a simple variational approximation in the
with miscible regime using a Gaussian Anstatz:
d3 k X [ξλ (~k) − E(k)]3
Z
1 !1/2
χ(na , nb ) = − , (2)
ρ2 2
(2π)3 Nσ − 21 + zl2
2
λ=± 4ξλ (~k)E(k) ψσ (~r; lρ , lz ) = e l2
ρ z , (6)
π lρ2 lz
3/2
where ξ± (~k) = [E(k)(E(k) + V± (θk ))]1/2 are the Bogoli-
ubov modes of the mixture, E(k) = ~2 k 2 /2m, and where lρ.z are determined from energy minimization [26].
X q Ansatz (6) is, however, inappropriate for immiscible
V± (θk )= 2 n n . (3)
ησσ nσ ± (ηaa na − ηbb nb )2 + 4ηab a b droplets (see [26] for an alternative ansatz in that regime).
σ Impurity limit.– The limit Nb Na transparently illus-
trates the possible ground states of a dipolar mixture. The
Above, θk is the angle between ~k and the dipole moments, majority component is to a first approximation a single-
d 2
ησσ0 (cos θk ) = gσσ0 + gσσ 0 (3 cos θk − 1), with gσσ 0 =
component dipolar BEC, which remains self-bound for suf-
4π~ aσσ0 /m, gσσ0 = µ0 µσ µσ0 /3 = 4π~2 adσσ0 /m, and µ0
2 d
ficiently large Na and low aaa /adaa [14, 35, 36]. Within the
is the vacuum permeability. The solution of Eq. (1) is given self-bound regime, the minority component experiences an ef-
by [29]: fective potential induced by the majority component:
8 m 32Z X 5 Z
LHY (na , nb ) = √ dθk sin θk Vλ (θk )2 , 3
15 2π 4π~ 2 µab (~r) ' gab na (~r) + d3 r0 Vab (~r −~r0 )na (~r0 ) + γab na (~r) 2 ,
λ=±
(4) (7)3
32 R 1 1
where γab = 332 √ m
π 4π~2 0
du ηaa (u) 2 ηab (u)2 . The last 4
term in Eq. (7) is the zero-momentum beyond-mean-field cor-
rection of the polaron energy resulting from the interaction of 3
the impurity with the elementary excitations of the majority
component. This repulsive term is crucial for the miscibility 2
of the mixture. It favors immiscibility, reducing the critical
1
aab by tens of a0 . Take the example of Na = 1270, Nb → 0,
and aaa = 70a0 . When γab is properly included we find that 0
immiscibility occurs at aab ' 75a0 , whereas excluding γab
pushes the immiscibility threshold up to aab ' 115a0 . -1
Dipolar attraction dominates for small-enough gab >
-2
0, resulting in a minimum of µab (~r) at the droplet cen-
ter, see Fig. 2(a). Component b then remains within the -3
droplet and the mixture is miscible. In contrast, for large-
-100 -50 0
enough gab , µab (~r) develops a maximum at the droplet cen- -4
ter (Figs. 2(b,c)). In the absence of dipolar interactions the
-1 0 1 -1 0 1 -1 0 1
minority component would be ejected. However, crucially,
the partially attractive and long-range character of the dipolar
interaction results in two potential minima, along the dipole FIG. 2. Effective potential µab (x, y = 0, z) [arb. unit] experienced
direction, z, which extend outside the a droplet (Figs. 2(b,c)). in the impurity limit by the minority component in (a) miscible, (b)
With increasing gab , component b is pushed away from the asymmetric immiscible, and (c) symmetric immiscible regimes. The
droplet center, first developing two µab (~r) minima while still majority component (Na = 2000) is represented by a black density
miscible, and is eventually positioned outside component a in contour, while the impurity component (Nb = 20) contour is white-
complete immiscibility. A sufficiently large gbb > 0 favors black dotted – both are drawn at 10% of the respective peak densities.
an equal occupation of both minima (Fig. 2(c)), whereas for
smaller gbb the b component will be biased towards one of
the minima, spontaneously breaking the discrete Z2 symme-
try (Fig. 2(b)). As shown below, although the energy scales
interplay differently for more balanced populations, the same
three self-bound ground states still occur: miscible, symmet-
ric immiscible and asymmetric immiscible.
Self-bound miscible and immiscible droplets.– Figures 1
and 3 summarize our GP results of the ground-state physics
for a Dy-Dy mixture (adaa,bb = 129.2a0 , with a0 the Bohr ra-
dius), for equal intracomponent interactions, aaa,bb = 70a0 .
Figure 1 shows the phase diagram for a fully balanced mix-
ture (Na,b = N/2), as a function of the total particle number
N and aab . The self-bound–unbound transition is marked by a
solid curve. Within the self-bound regime, the system experi-
ences an abrupt phase transition (dotted line) from a miscible
regime at low aab [see Fig. 1(a)] to an asymmetric immisci- FIG. 3. Instability threshold as a function of particle number in each
ble regime for large aab [Fig. 1(b)]. For the particular case component for a Dy-Dy mixture with aaa = abb = 70a0 , and
of Figs. 1 and 3, where the intracomponent interactions and aab = 50a0 , 70a0 and 90a0 . The mixture remains self-bound for
the dipole moments are equal, the miscible-immiscible thresh- the parameter region above the curves. The inset shows the results
obtained using the variational ansatz (6) for aab = 50a0 and 70a0 .
old is approximately independent of the number of atoms. In The subplots show the 3D probability contour for the a (red) and
more general cases, as illustrated below, the transition may be b (blue) component, drawn at 10% of the respective peak densities.
driven by changing the particle number.
While in the impurity limit the droplet stability only de-
pends on the intracomponent interactions of the majority com-
ponent, independently of the miscibility or immiscibility of line in the figure shows the instability boundary predicted by
the mixture, in the balanced case there is a marked interplay the Gaussian ansatz (6), which reproduces well the GP results
between droplet stability and miscibility. When decreasing within the miscible regime.
aab into the miscible regime, the droplet becomes significantly The instability threshold presents a marked dependence on
more stable. In particular, the critical total number of parti- the polarization Na /Nb of the mixture. In Fig. 3, we depict the
cles for self-binding falls considerably, see Fig. 1. The dashed stability threshold as a function of Na and Nb , for aab = 50a0 ,4
FIG. 5. Symmetric immiscible-to-miscible crossover. Contrast
∆ (see text) for different (Na , Nb ) going from (2000, 0) to
(2000, 2000), and then from (2000, 2000) to (0, 2000), for im-
balanced interactions (aaa , aab , abb )/a0 = (65, 70, 75). Subplots
show the 3D density contours for a (red) and b (blue) components for
FIG. 4. Asymmetric immiscible-to-symmetric immiscible transition. the impurity limits, (1000, 2000), (2000, 2000), and (2000, 1000).
Energy of the symmetric (dashed) and asymmetric (solid) immis- All contours are drawn at 5% of the respective peak densities.
cible phase as a function of (abb − aaa ) for aab = 85a0 , and
Na,b = 2000. The subplots show 3D contours for the a (red) and
b (blue) components, drawn at 5% of the respective peak densities.
two droplets that would be individually unstable. The insta-
bility threshold flattens within the immiscible regime (Fig. 1),
70a0 and 90a0 for the same case as Fig. 1. In the impurity due to the drastic reduction of the intercomponent overlap-
limit, as mentioned above, the stability does not depend on ping, but the non-negligible dependence on aab shows that the
aab and all curves converge to the critical number for a single width of the domain wall remains finite.
component. Deep within the miscible regime (aab = 50a0 ), While the cases considered above display a miscible-to-
balanced droplets have a much lower critical total number, asymmetric immiscible transition, an immiscible-immiscible
Ncr , for stability compared to the single-component case. For transition may also occur between a symmetric and asymmet-
aab = 50a0 , Ncr ' 700 for Na = Nb , i.e. just 350 parti- ric configuration, as illustrated in Fig. 4, where we consider
cles in each component, whereas Ncr ' 1270 for Na = 0 or Na = Nb = 2000, aab = 85a0 , and (aaa + abb )/2 = 70a0 .
Nb = 0, showing that the mutual confinement strongly rein- This figure shows that the population distribution may be
forces self-binding. changed not only by modifying aab but also by changing the
In the immiscible regime, a droplet molecule forms, i.e. a ratio aaa /abb . While for aaa = abb the asymmetric configu-
self-bound solution of two attached droplets. The repulsion ration has a lower energy compared to the symmetric one, the
resulting from the intercomponent mean-field contact term symmetric configuration becomes the ground-state at a criti-
and the LHY energy [37] results in phase separation. For cal abb − aaa , marking the onset of a first order phase tran-
the particular cases in Figs. 1 and 3, this separation is al- sition. The symmetric immiscible solution can be the ground
ways asymmetric, see Fig. 1(b) and Fig. 3(a) (the latter should state – overcoming the cost of two domain walls – because the
be compared to Fig. 2(b) in the impurity limit). In more component with the smaller intraspecies contact interactions
general scenarios, as illustrated below, the interplay between forms a narrower droplet (see Fig. 4 insets). Not only does this
intra- and intercomponent interactions may favor a symmet- reduce its internal dipolar energy, it also creates deeper attrac-
ric configuration with two domain walls (as in Fig. 2(c) in tive potential pockets at both ends, within which the second
the impurity limit). In any case, as in the impurity limit, the component can equally divide itself to minimize energy.
droplets remain attached despite their phase separation due to Finally, the symmetric immiscible configuration may
the intercomponent dipole-dipole interactions. Each compo- crossover to a miscible phase, as illustrated in Fig. 5, where
nent creates at its borders (and beyond) an attractive poten- we consider (aaa , aab , abb )/a0 = (65, 70, 75). We monitor
tial pocket in which the other component is trapped, leading the crossover by considering the contrast, ∆ ≡ |na0 /nam −
to mutual attachment. The attractive interactions exerted by nb0 /nbm |, where nσm is the maximal density of component
the other component lead not only to attachment, but also to σ, and nσ0 is its density at the droplet center [38]. The sys-
reinforced stability. As shown in Fig. 3, for the immiscible tem undergoes a symmetric immiscible-to-miscible crossover
regime (aab = 90a0 ), in contrast to the miscible case, Ncr when Nb /Na grows. This is because in the impurity limit,
grows when the mixture is more balanced (Ncr ' 1500 for Nb → 0, (aaa , aab )/a0 = (65, 70) leads to an immiscible
Na = Nb ). Even so, only Na,b = 750 particles in each com- mixture [ca. Fig. 2(c)], whereas for Na → 0, (aab , abb )/a0 =
ponent are necessary for self-binding – compared to ' 1270 in (70, 75) results in miscibility. Note that component a always
the single-component case – showing that despite phase sep- remains at the center since aaa is the lowest. Furthermore,
aration, the mutual attachment allows for the stabilization of we should point out that more generally all possible transi-5
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sions. We acknowledge support of the Deutsche Forschungs- [24] We take the atomic mass for both components to be 161.9u and
the dipole moments as 9.93µB .
gemeinschaft (DFG, German Research Foundation) under
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Germany’s Excellence Strategy – EXC-2123 QuantumFron- [26] See the Supplementary Material for more details concerning the
tiers – 390837967, and FOR 2247. RNB acknowledges the derivation of the LHY correction and the variational calcula-
European Union’s Horizon 2020 research and innovation pro- tions.
gramme under the Marie Skłodowska-Curie grant agreement [27] See [28] for a discussion of other effects of quantum fluctua-
No. 793504 (DDQF). tions in dipolar Bose mixtures.
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Note added: After the completion of this work we became
[29] See Ref. [30] for an alternative derivation, which results in an
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complementary to ours. 3D dipolar Bose mixture. The formalism we employ provides
the explicit expression (4) and does not require us to cure di-
vergences. The latter makes our formalism better suited to treat
lower- and cross-dimensional problems [31, 32].
[30] A. Boudjemaa, Phys. Rev. A 98, 033612 (2018).
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Supplementary Material
DERIVATION OF THE LHY CORRECTION
We briefly discuss further details on the derivation of the LHY correction of Eq. (4) of the main text. The Hugenholz-
Pines (HP) formalism may be easily extended to two-component condensates. As discussed in the main text, the LHY energy
density, LHY obeys the differential equation:
1X ∂
LHY (na , nb )− nσ LHY (na , nb ) = χ(na , nb ). (S1)
2 σ ∂nσ
2
χ(na , nb ), which is given by Eq. (2) of the main text, can be re-written in the form: χ(na , nb ) = ~m (na aaa )5/2 G(P ), where
2
G(P ) is a function of the polarization P = nb /na . We employ then the ansatz LHY = ~m (na aaa )5/2 G̃(P ). Note that
∂ ∂ 5
P P
σ nσ ∂nσ P = 0, and hence σ nσ ∂nσ LHY = 2 LHY . As a result, the HP equation is greatly simplified: LHY (na , nb ) =
−4χ(na , nb ), and hence
d3 k X (ξν (k, θk ) − E(k))3
Z
LHY (na , nb ) = 2
(2π)3 ν 4ξν (k, θk )E(k)
p 3
2m
3/2
1
Z π X Z ∞ q2 + 1 − q
= dθk sin θk Vλ (θk )5/2 dqq 2
~2 8π 2 0
p
λ=± 0 q2 + 1
8 m 32 Z X 5
= √ 2
dθk sin θk Vλ (θk ) 2 , (S2)
15 2π 4π~ λ=±
as in Eq. (4) of the main text. For a single-component dipolar BEC (nb = 0, aaa = a, adaa /a = dd ), Eq. (S2) becomes of the
form:
1/2 Z 1
na3
ELHY 64 2
= gn du(1 + dd (3u2 − 1))5/2 , (S3)
V 15 π 0
recovering the result of Ref. [33]. For nondipolar binary mixtures (adaa = adbb = 0), Eq. (S2) becomes
2
ELHY 8 m 3/2 5/2 aab abb nb
= (g n
aa a ) f , , (S4)
V 15π 2 ~2 aaa abb aaa na
p 5/2
1
P
with f (x, y) = √
4 2 σ=± 1+y± (1 − y)2 + 4xy , recovering the result of Ref. [18].
VARIATIONAL CALCULATIONS
Gaussian ansatz
2
Assuming miscibility of the mixture, we may consider a Gaussian ansatz, nσ (~r; lρ , lz ) = |ψ(~r, lρ , lz )| (see Eq. (6) of the
main text),
ρ2 2
Nσ − l2
+ zl2
nσ (~r; lρ , lz ) = 3/2 2 e ρ z . (S5)
π lρ lz7
Using this ansatz we may evaluate the total energy as a function of the variational widths lρ,z :
~2 2
1 X
E[lρ , lz ] = + 2 Nσ
4m lρ2 lz σ
1 X d
+ 3/2 2
Nσ Nσ0 gσσ0 + gσσ 0 f (κ)
2(2π) lρ lz σ,σ0
!32
XZ 1
32 m 5
+ √ Qλ (u)2 du, (S6)
75 5π 4π ~2 lρ2 lz
5/2
λ=± 0
2
with gσσ0 and gσσd
0 the contact and dipolar coupling strengths defined in the main text. In addition, f (κ) = 2κ +1
κ2 −1 −
2
√
3κ arctan κ2 −1 l P p
(κ2 −1)3/2
with κ = lρz the aspect ratio, and Q± (u) = σ Nσ ησσ (u) ± (Na ηaa (u) − Nb ηbb (u))2 + 4ηab (u)2 Na Nb ,
and the functions ησσ0 are defined in the main text.
Flat-top ansatz
The Gaussian ansatz discussed previously is not suitable for treating immiscible mixtures. For this purpose we employ an
alternative ansatz, where we assume that the density profile of the droplet is Gaussian radially and flat-top axially:
2
Nσ − z + zσ
ρ
nσ (~r; Lρ , Lσ ) = e Lρ
Π , (S7)
πL2ρ Lσ Lσ
where Π(x) = 1 if |x| < 1/2 and zero otherwise. Note that in this ansatz, we allow for different axial lengths Lσ – where
σ = {a, b} – and center-of-mass (COM) positions of the components, zσ . Miscibility with an axial flat-top density profile is
captured by this ansatz when za,b = 0. Energy is minimized with respect to four variational parameters: Lρ , La , Lb , and the
displacement ∆zσ,σ0 = |zσ − zσ0 |. The energy as a function of the variational parameters is of the form:
~2 X
E[Lρ , Lσ ] = Nσ
2mL2ρ σ
1 X Nσ Nσ 0
gσσ0 Λcσσ0 + gσσ
d d
+ 2
√ 0 Λσσ 0
4πLρ Lσ Lσ0
σ,σ 0
!3/2 Z Z
4 m X 5/2
+ √ 2 2
√ dz dθ k sin θ k [Sλ (cos θk )] . (S8)
2
75π 2 π~ Lρ La Lb λ=±
Here we employ the auxiliary functions
r r !
c 1 Lσ Lσ0 ∆zσ,σ0
Λσσ0 = + −√ (S9)
2 Lσ0 Lσ Lσ Lσ0
Z
1 kz Lσ kz Lσ0 ∆zσ,σ0
Λdσσ0 = dkz hσσ0 (kz )sinc sinc exp −ikz √ , (S10)
2π 2 2 Lσ Lσ0
√
2
3k2
h i
where hσσ0 (kz ) = dkρ kρ (kρ /κ 0z)2 +k2 − 1 exp − 21 kρ2 = −1 − 3ueu Ei − kz√ κσσ0
R
2
, with κσσ0 = Lρ / Lσ Lσ0 , and
σσ z
q q q
z 2 2 nz nz , with nz = Lb z+za z La z+zb
S± (cos θk ) = ησσ nσ ± (ηaa na − ηbb nzb ) + 4ηab
P z
a b a La Π La and n b = Lb Π Lb .
Variational results
In Fig. S1, we plot the radial and axial density profiles for Na,b = 2000 and aab = 64.5a0 (miscible regime) and aab =
85a0 (strongly-immiscible regime). In the former case, both the fully-Gaussian ansatz and the flat-top ansatz are compared8
10 21 z = -2.22 m 10 21 = 0.037 m 10 21 z = -2.22 m 10 21 = 0.037 m
4
4 5
FT FT FT FT
2.5 Gaussian Gaussian GP GP
GP 3 GP 4
2 3
density( )
density( )
density(z)
density(z)
3
1.5 2 2
1 2
1 1
0.5 1
0 0
0 0.1 0.2 0.3 0.4 0.5 -10 -5 0 5 10 0 0.1 0.2 0.3 0.4 0.5 -10 -5 0 5 10
( m) z ( m) ( m) z ( m)
10 21 z = 2.22 m 10 21 = 0.037 m 10 21 z = 2.22 m 10 21 = 0.037 m
4
4 5
FT FT GFT FT
2 Gaussian Gaussian GP GP
GP 3 GP 4
3
density( )
density( )
density(z)
density(z)
1.5
3
2 2
1
2
1 1
0.5 1
0 0
0 0.1 0.2 0.3 0.4 0.5 -10 -5 0 5 10 0 0.1 0.2 0.3 0.4 0.5 -10 -5 0 5 10
( m) z ( m) ( m) z ( m)
FIG. S1. Radial and axial density profiles for the case, (Na , Nb ) = (2000, 2000) and aaa = abb = 70a0 . The (left) miscible regime
aab = 64.5a0 and (right) immiscible regime aab = 85a0 . We compare the results of the fully-Gaussian ansatz, the flat-top (FT) ansatz, and
the GP calculations.
3000
Gaussian
2500 GP
FT
total atom number (N)
2000
1500
1000
500
30 40 50 60 70 80 90 100
aab/a 0
FIG. S2. Critical number of particles for stability of the fully balanced (Na = Nb ) self-bound solution for aaa = abb = 70a0 . We compare
the results of the fully-Gaussian ansatz, the flat-top (FT) ansatz, and the GP calculations.
against the GP prediction. Qualitatively, both ansatzes give a good description of the radial density profile, but the fully-Gaussian
density profile gives a better quantitative agreement, especially at the self-bound/unbound transition (not explicitly shown, but
see the good agreement with the full GP solutions for the stability boundary in Fig. S2).
For the strongly immiscible case, the fully Gaussian ansatz is no longer adequate, since the axially displaced Gaussians do not
provide a good description of the the domain-wall region, which is typically characterized by a sharply-changing density (see
the GP results in Fig. S1, right). In contrast, the flat-top ansatz captures well the qualitative features of the domain-wall region.
Note, however, that the flat-top ansatz is only suitable for the asymmetric immiscible case.
As shown in Fig. S2, the flat-top ansatz qualitatively reproduces the miscible/immiscible transition (see the kink at acr ab '
80a0 ), which is moderately shifted compared to the GP result (acr ab ' 70a0 ). For the flat-top ansatz, the immiscible solution
is always fully immiscible, and the critical number of particles remains constant for aab > acr ab . However, the flat-top ansatz
significantly overestimates the critical number of particles in the immiscible regime, by close to a factor of 2.You can also read
