Townes soliton and beyond Non-miscible Bose mixtures in 2D Brice Bakkali-Hassani1and Jean Dalibard2

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Townes soliton and beyond:

Non-miscible Bose mixtures in 2D

Brice Bakkali-Hassani1and Jean Dalibard2

1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

2Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Uni-

versité, 11 Place Marcelin Berthelot, 75005 Paris, France

Summary. — In these lecture notes, we discuss the physics of a two-dimensional

binary mixture of Bose gases at zero temperature, close to the point where the two

ﬂuids tend to demix. We are interested in the case where one of the two ﬂuids (the

bath) ﬁlls the whole space, while the other one (the minority component) contains a

ﬁnite number of atoms. We discuss under which condition the minority component

can form a stable, localized wave packet, which we relate to the celebrated "Townes

soliton". We discuss the formation of this soliton and the transition towards a

droplet regime that occurs when the number of atoms in the minority component

is increased. Our investigation is based on a macroscopic approach based on cou-

pled Gross-Pitaevskii equations, and it is complemented by a microscopic analysis

in terms of bath-mediated interactions between the particles of the minority com-

ponent.

Binary mixtures of low-temperature Bose gases can lead to a large variety of phenom-

ena, depending on the strength and nature – repulsive or attractive – of intraspecies and

interspecies interactions [1, 2]. Even when each component is individually stable at the

mean-ﬁeld level, i.e., it has a positive scattering length, a demixing instability, resp. a

collapse, may occur if the interspecies interaction is suﬃciently large and repulsive, resp.

©Società Italiana di Fisica 1

arXiv:2210.14045v1 [cond-mat.quant-gas] 25 Oct 2022

2Brice Bakkali-Hassani1and Jean Dalibard2

g12

−√g11g22 0+√g11g22

Quantum droplets

N1∼N2

This lecture

Inﬁnite bath

of component 1

mixed regimecollapse demixtion

Figure 1. – Generic phase diagram for a binary mixture as function of the intercomponent

interaction strength g12 , assuming repulsive intraspecies (gii >0). This lecture is devoted

to the 2D case. The study of [3] regarding the generation of quantum droplets stabilized by

beyond-mean-ﬁeld forces addresses the 3D case.

suﬃciently large and attractive (see ﬁgure 1).

The vicinity of these two singular points is particularly interesting. For example,

the regime of quantum droplets in three dimensions predicted in [3] occurs close to the

collapse threshold. It takes advantage of the smallness of the mean-ﬁeld interaction

energy at this point: Beyond-mean-ﬁeld corrections can play a signiﬁcant role and they

give access to a liquid state of matter, although the density remains several orders of

magnitude lower than in usual liquids.

In this lecture, we focus on the vicinity of the demixing point. We consider the case

where one component forms an inﬁnite bath ﬁlling the entire space, while the second

component contains only a ﬁnite number of particles N2. By contrast to [3], all the

eﬀects considered in these notes originates from mean-ﬁeld interactions. In addition, we

suppose that the system is at zero temperature. The state of the binary mixture is thus

well described by two coupled Gross-Pitaevskii equations.

We wish to address the general question of the equilibrium shape of the minority

component in these conditions: does it ﬁll the whole space as the majority component,

or does it form a stable, localized wave packet immersed in the bath? In the latter case,

is it possible to describe this localized state using a (modiﬁed) Gross-Pitaevskii equation

involving only the minority component?

More speciﬁcally, we are interested here in the two-dimensional (2D) case. For given

interaction strengths, we show that there exists a threshold value NTfor the number of

particles in the minority component, below which this component tends to ﬁll the whole

space. For any N2> NT, the minority component can form a stable wave packet with

a deﬁnite size. When N2decreases to N+

T, the minority component converges towards a

stationary wave packet that constitutes a realization of the celebrated "Townes soliton"

[4]. This solitary wave is a remarkable example of a scale-invariant object, which can

Townes soliton and beyond: Non-miscible Bose mixtures in 2D 3

exist in principle with an arbitrary size. On the contrary, as N2increases towards values

much larger than NT, the wave packet gradually evolves towards a droplet-like object

with a density imposed by the bath.

The outline of this lecture is the following. In § 1, we brieﬂy review how solitons can

be formed in a ﬂuid described by the Gross-Pitaevski energy functional. We explain the

special feature of the 2D case in relation with scale invariance. Then, we brieﬂy describe

two recent experiments in which Townes solitons were observed with a 2D cold atom

setup [5, 6]. In § 2, we present a theoretical modeling of the binary mixture starting

with the coupled Gross-Pitaevskii equations associated to each component. We discuss

various situations in which the degrees of freedom of the bath can be eliminated to the

beneﬁt of a single equation for the minority component. We explain how the system

evolves when the number of particles is increased above the Townes threshold NTand

we describe the transition towards the droplet regime, where the minority component

sits in a localized region of space where the bath is fully depleted. In § 3, we turn to

another point of view on this system and study the interactions between the particles

of the minority component that are mediated by the bath. We ﬁrst address the case

where the mass of a particle of the minority component m2is much larger than the

mass of a bath particle m1. We show that in this case, the mediated interactions are

well described by a Yukawa potential, with a range related to the bath healing length.

Then, for momenta kof the minority component much smaller than 1/ξ, we use Born

approximation to simplify the description of these mediated interactions, and show that

the result can be extended to the case m1=m2. This allows us to recover the results

obtained in § 2by the macroscopic approach. Finally, we draw in § 4some conclusions

and perspectives regarding this two-dimensional binary mixture of Bose gases.

1. – Solitons in two dimensions

1.1. The Gross-Pitaevskii energy functional . – A soliton is an emblematic object of

non-linear wave physics [7]. It is deﬁned as a wave packet that maintains its shape over

time, as a result of the competition between non-linear and dispersive eﬀects. In this

lecture, we consider a wave packet ψ(r, t)whose energy at time tis described by the

Gross-Pitaevskii functional in Dspatial dimensions:

(1) E[ψ] = 1

2Z|∇ψ(r, t)|2+g|ψ(r, t)|4dDr.

This energy functional is relevant in optics (for the propagation of laser beams in a

non-linear medium), in atomic physics (for the classical-ﬁeld description of a weakly-

interacting Bose gas) and in condensed matter (e.g., as an order parameter for superﬂuid

liquid helium). Note that we have set here ~= 1 and m= 1 for the mass of the

particles, in the case where ψdescribes a matter-wave ﬁeld. The ﬁrst contribution

to (1) corresponds to diﬀraction or kinetic energy, and the second one to the (cubic)

non-linearity of the medium or to the interactions between particles. The parameter g

characterizes the strength of the non-linear coupling.

4Brice Bakkali-Hassani1and Jean Dalibard2

The dynamics associated with the energy functional (1) is described by the Lagrangian

(2) L[ψ]=iZψ∗(r, t)∂tψ(r, t) dDr−E[ψ].

The Euler-Lagrange equations then lead to the time-dependent Gross-Pitaevskii equa-

tion:

(3) i∂tψ(r, t) = −1

2∇2ψ(r, t) + g|ψ(r, t)|2ψ(r, t).

1.2. Dimensional analysis for the soliton size. – Here we consider essentially the

physics of an atomic Bose gas with the number of particles Ndeﬁned as

(4) Z|ψ(r, t)|2dDr=N.

The non-linear coeﬃcient gin (1) describes the strength of the interactions between the

particles at the mean-ﬁeld level. Here, we suppose that g < 0, corresponding to an

attractive interaction.

In order to obtain some intuition on the existence and stability of a soliton, we consider

a stationary wave packet of size `, with a central density N/`D, and we perform a

dimensional analysis of the energy per particle deduced from (1):

(5) E(`)

N∼1

`2−N|g|

`D,

up to a numerical factor of order unity in front of each contribution.

The variation of E(`)deduced from (5) is shown in Fig. 2 for dimensions D= 1 and

D= 3. The solution in the one-dimensional case is well known: the attractive interaction

term dominates at large `and the kinetic energy dominates at small `. Therefore, there

exists a size `∗∼1/N|g|which corresponds to a stable equilibrium. In the context of

cold atomic gases, such solitons were ﬁrst observed in [8, 9].

In 3D, the extremum of the energy functional, occurring for a length `∗∼N|g|is

unstable. The expansion term due to the kinetic energy dominates for ` > `∗, and the

contraction term due to the interaction energy leads to a collapse for `<`∗. For small

atom numbers, the ﬂuid can be stabilized using an additional harmonic conﬁnement,

i.e. by adding a component ∝`2to (5) [10]. For large atom numbers, it leads to the

celebrated phenomenon of "Bose Nova" [11].

By contrast to 1D and 3D, the two-dimensional case is critical, in the sense that no

length scale emerges when we try to minimize (5). This is a consequence of the fact that

the parameter gis dimensionless in 2D, while it has the dimension of a length (resp.

the inverse of a length) in 3D (resp. 1D). This absence of length scale associated with

interactions in 2D is an illustration of the scale invariance of the action (2) in this case.

Townes soliton and beyond: Non-miscible Bose mixtures in 2D 5

ℓ∗

ℓ

E(ℓ)

ℓ∗

ℓ

Figure 2. – Sketch of the energy per particle (5) for a wave packet of size `in 1D (left) and 3D

(right).

1.3. The 2D case: Townes proﬁle. – For a more quantitative analysis of the extrema

of the energy functional (1) in two dimensions, we turn to the time-independent Gross-

Pitaevskii equation deduced from the minimization of (1):

(6) −1

2∇2φ(r) + g|φ(r)|2φ(r) = µ φ(r)Z|φ(r)|2d2r=N.

Here the chemical potential µis the Lagrange parameter introduced to take into account

the constraint on the normalization of φ. Once φ(r)is known, the function ψ(r, t) =

φ(r) e−iµt is a solution of the time-dependent Gross Pitaevskii equation (3).

It can been shown that (6) has physically acceptable solutions only for well-deﬁned

values of the product Ng. More precisely, these solutions have an isotropic density

distribution and a phase dependence of the form exp(isθ),θbeing the polar angle and

s∈Zthe embedded vorticity [12]. For s= 0, the node-less solution of (6) exists only for

Ng =−5.85... [4, 13], whereas solutions with one and two nodes require Ng =−38.6...

and Ng =−97.9... respectively [14, 15]. The solution with no node (except in r= 0)

and with embedded vorticity s= 1 (resp. s= 2) requires Ng =−24.1... (resp. N g =

−44.86...) [16].

The s= 0 node-less solution corresponds to the celebrated Townes soliton. Its radial

proﬁle is given (after an arbitrary rescaling) by the positive solution R(r)of the radial

equation

(7) R00 +R0

r+R3=RZ+∞

R2(r) 2πr dr= 2 ×5.85... ,

and it is represented in ﬁgure 3. Here, we have set R(r) = p2|g|φ(r)in (6) and we have

chosen the value µ=−1/2for the chemical potential (as we show below, this value can

be chosen arbitrarily).

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摘要：

Townessolitonandbeyond:Non-miscibleBosemixturesin2DBriceBakkali-Hassani1andJeanDalibard21DepartmentofPhysics,HarvardUniversity,Cambridge,Massachusetts02138,USA2LaboratoireKastlerBrossel,CollègedeFrance,CNRS,ENS-PSLUniversity,SorbonneUni-versité,11PlaceMarcelinBerthelot,75005Paris,FranceSummary.Inth...

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Townes soliton and beyond Non-miscible Bose mixtures in 2D Brice Bakkali-Hassani1and Jean Dalibard2

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