SciPost Phys. 15, 095 (2023)

Fermionization of a few-body Bose system immersed
into a Bose-Einstein condensate
Tim Keller⋆ , Thomás Fogarty and Thomas Busch
Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University,
Onna-son, Okinawa 904-0495, Japan⋆ tim.keller@oist.jp

Abstract
We study the recently introduced self-pinning transition [Phys. Rev. Lett. 128, 053401(2022)] in a quasi-one-dimensional two-component quantum gas in the case where thecomponent immersed into the Bose-Einstein condensate has a finite intraspecies interaction strength. As a result of the matter-wave backaction, the fermionization in the limit of
infinite intraspecies repulsion occurs via a first-order phase transition to the self-pinned
state, which is in contrast to the asymptotic behavior in static trapping potentials. The
system also exhibits an additional superfluid state for the immersed component if theinterspecies interaction is able to overcome the intraspecies repulsion. We approximate
the superfluid state in an analytical model and derive an expression for the phase transition line that coincides with well-known phase separation criteria in binary Bose systems.
The full phase diagram of the system is mapped out numerically for the case of two andthree atoms in the immersed component.
Copyright T. Keller et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 06-02-2023
Accepted 20-07-2023
Check forupdatesPublished 14-09-2023
doi:10.21468/SciPostPhys.15.3.095

Contents
1 Introduction

2

2 Model3

3 Fermionization process

4

4 Phase diagram

9

5 Conclusion

11

References

121

SciPost Phys. 15, 095 (2023)

1

Introduction

Quantum phase transitions are one of the most interesting aspects of quantum many-body
physics [1] and they underlie many of the novel states and unique properties that allow neutral atoms in optical lattice potentials to be used as quantum simulators of condensed matter physics [2–5]. A paradigmatic example of this is the superfluid to gapped Mott insulatoror pinned-state transition that happens in a one-dimensional quantum gas that is subject toan external lattice potential of arbitrary small lattice strengths beyond a critical value of theinterparticle repulsion [6, 7]. A mapping between the gas parameters and the Luttinger liquid parameter [8] allows finding this critical value from the renormalization group theory
treatment of the sine-Gordon model [9]. We have recently shown how the same physics canbe observed without the need for a commensurate lattice potential, but by immersing thestrongly-correlated quantum gas into a weakly-correlated background such as a Bose-Einsteincondensate (BEC) instead [10]. The immersed component is able to create its own commensurate matter-wave trapping potential via the backaction with the background gas, leading toan equivalent quantum phase transition.
In both cases the transition is of second order and therefore continuous. First-order ordiscontinuous transitions are a lot less common in cold atomic systems. Such transitions arecharacterized by metastable states in which a system can remain even after crossing the transition [11] and there is a growing interest in the form of proposals for cold atom quantum
simulators of the early universe [12, 13], which might be able to investigate the ‘fate of thefalse vacuum’ [14], i.e. the decay from a metastable state to the true ground state. Metastablestates also give rise to hysteresis and the effects associated with discontinuous phase transitions have already been observed in cold atom experiments, for example for an ultracold
atomic gas in a double-well potential [15], in spinor BECs [16], in a driven one-dimensional(1D) optical lattice [17] and also in imbalanced quasi-1D Bose-Bose mixtures [18].
One-dimensional Bose-Bose mixtures have seen considerable interest, in particular overthe past twenty years [19–21]. If one interprets the two components as representations of thespin-up and spin-down state in a pseudo-spin 1/2 system, it has been shown that they can
support spin-waves [22, 23] and also develop a spontaneous population imbalance, an equivalent of ferromagnetism [24, 25], which is suppressed at finite temperatures [26]. Eisenberg
and Lieb showed that in general the ground state of such interacting bosons with spin is always fully polarized, i.e. ferromagnetic [27]. Furthermore, they can be used to demonstrate
a hallmark effect in 1D electronic systems, which is the separation of single-particle spin andcharge exciations into two distinct collective branches as a result of the dimensionality [28].
A Luttinger liquid approach predicts collapse and pairing phenomena for attractive intercomponent interactions similar to 1D Bose-Fermi mixtures, depending on the density regime
and intracomponent interactions [29, 30], while for repulsive intercomponent interactions a
miscibility criterion identical to the well-known 3D case determines the regime of phase separation [20]. Furthermore, exact solutions have been found for a 1D Bose-Bose mixture on aring with equal densities and identical intraspecies and interspecies interaction strengths [31]and similar to Bose-Fermi mixtures, a supersolid state was also predicted for weak interspecies
but nearly hard-core intraspecies repulsion in a balanced, harmonically trapped Bose mixture
with densities incommensurate with the additional lattice potential [32].
In the few-atom regime, a balanced repulsive binary Bose mixture is characterized bysix different limiting cases of either vanishing (‘BEC limit’) or infinite (‘TG limit’) interactions, quantified by g i j [33]. Along with the phase separated regime there is also ‘composite
fermionization’ [34] (g ii → 0, g i j → ∞) and ‘full fermionization’ (g ii , g i j → ∞), i.e. a singlecomponent Fermi gas equivalent. Finally, more recent developments include the extensionof the famous droplets in 3D Bose-Bose mixtures, stabilized purely by quantum fluctuations,

2

SciPost Phys. 15, 095 (2023)

to the one-dimensional case at zero temperature [35, 36], in the presence of spin-orbit coupling [37], and at finite temperature [38]. There, the self-bound droplets are also predicted
to have a bright solitonic shape, i.e. a density n(x) ∼ cosh−2 (x), similar to the states found in
Ref. [10]. Particularly, for imbalanced mixtures like the one studied in this work, similar structures in the form of composite ‘bright-gray’ solitons [39] or double domain-wall solitons [40]have been predicted.
In this work we show that an imbalanced one-dimensional Bose-Bose mixture consisting ofa few-body bosonic system immersed into a much larger Bose-Einstein condensed backgroundalso gives rise to an apparent first-order phase transition between a coherent superfluid state
and the insulating self-pinned state studied in Ref. [10] as the intraspecies repulsion is increased beyond a critical value going towards the Tonks-Girardeau limit. The paper is organized as follows. In Section 2 we introduce the system and the coupled Schrödinger equations
we use to describe the two components. In Section 3 we numerically study the fermionization process of the immersed component as a function of the intraspecies interaction mainly
in terms of the energy and density of the system. We use the analytical model presented inRef. [10] as benchmark for the limiting cases of vanishing (g → 0) and infinite (g → ∞)intraspecies repulsion and expand the model to the superfluid state which persists for finite
values between 0 < g < ∞. Here we also derive an analytical approximation for the phasetransition line and compare to numerical results. In Section 4 we numerically calculate thephase diagrams in terms of the coherence for two and three atoms in the immersed component
before concluding in Section 5.

2

Model

We consider a strongly imbalanced two-component quantum gas in a quasi-one-dimensionalsetting. The majority component is a Bose-Einstein condensate (BEC) of Nc particles, describedin the mean-field limit by a macroscopic wave function ψ(x). The minority component immersed into the BEC consists of N ≪ Nc particles described by a full many-particle wave function Φ(x = x 1 , x 2 , . . . , x N ). The system is studied at zero temperature and therefore we candescribe all interactions by point-like pseudo-potentials that only depend on the interspecies
and intraspecies scattering lengths. We model the interactions between the two components
by a simple density coupling, which is valid when the interspecies interaction is assumed to
be weak. This leads to the coupled Schrödinger equationsħh2 ∂ 222iħhψ̇(x) = −+ g m |Φ| + g c |ψ| ψ(x) ,(1a)2m ∂ x 2
 N

N
X
X ħ
h2 ∂ 22iħhΦ̇(x) =−+ g m |ψ| + g
δ(|x k − x l |) Φ(x) .
(1b)
2
2m
∂
x
l
l=1
k<l
The intraspecies interaction strengths in the minority and majority component are labeled g
and g c respectively and g m describes the interspecies coupling strength. For simplicity weassume equal masses m for both components and perform numerical simulations by scaling
interaction strengths in units of the BEC interaction strength gc . In all figures presented in thepaper lengths are therefore displayed in units of x 0 = ħ
h2 /mgc , time in units of ω−1 = ħ
h3 /mgc2
22
and energy in units of ħhω = mgc /ħ
h . Choosing for example Rubidium-87 with a mass ofm ≈ 87 mu and a reference value of gc = 7.7 × 10−38 Jm ≈ 2ħhω⊥ as corresponding to theintraspecies interaction of Rubidium-87 in a typical quasi-one-dimensional setup with a radialtrapping frequency of ω⊥ ≈ 2π × 11 kHz and a scattering length of as ≈ 100 aB [41] leads tox 0 ≈ 1 µm. In this work we extend the results of Ref. [10] and go beyond the Tonks-Girardeau
3

SciPost Phys. 15, 095 (2023)(TG) limit of g → ∞ for the minority component by considering finite intraspecies repulsion
0 < g < ∞. Consequently, the single-particle density of the immersed component is now
calculated by tracing out all but one atom from the full many-particle wave functionZρ(x) ≡ |Φ(x)|2 =d x 2 . . . d x N |Φ(x, x 2 , . . . , x N )|2 .

(2)The condensate is assumed to be in free space with an average density nc ≡ Nc /L c = µ0 /g cwith µ0 describing the chemical potential of the condensate in the uncoupled case, whereas
the immersed component is confined to a box potential of width L with V (x) ≡ 0 for |x| ≤ L/2
and V (x) ≡ ∞ otherwise.

3

Fermionization process

While in the TG limit of infinite intraspecies repulsion g → ∞ the immersed component
undergoes a transition to an insulating pinned state, for finite intraspecies interaction strengths
the immersed gas can also exist in a cohesive superfluid state if the interspecies coupling g m
is able to overcome the intraspecies repulsion g. In this section we study the superfluid stateand show how the system fermionizes by transitioning to the pinned state as a function of g.
For finite intraspecies interactions g, numerically solving the coupled equations (1) for morethan a few atoms in the immersed component becomes challenging and we therefore restrict
our consideration to the case of N = 2 and N = 3 immersed particles.
Starting from the limit of vanishing intraspecies interaction g → 0 the immersed species
can be effectively described by N overlapping and identical single-particle wave functions and
we refer to this cohesive state as the superfluid state. We describe this state using the effectivemodel developed in Ref. [10] which is based on the Thomas-Fermi approximation for thecondensate wave function in the regime of heavily imbalanced particle numbers as well as a
weak interspecies interaction g m ≪ µ0 L/N . This leads to a description of the density of theimmersed component in terms of bright solitons according toρsf (x) = N

asf
1
,
2
2 cosh (asf x)(3)

where the peak height asf is given byp
1 + 2N 2 ε − 1
asf (g = 0) = N a0
,
N 2ε

(4)

2
with a0 = mg m/(2g c ħh2 ). The factor ε = 6a02ħ
h2 /(5mµ̃) accounts for the energetic cost of
deforming the BEC background with the modified chemical potential µ̃ = µ0 (1 + g m N /g c Nc ).
It is important to note that the closed expression for asf in Eq. (4) yields the best agreement with
the numerically observed densities in the case of moderate interaction strengths g m , where thewidth of the atomic wave function and the density dip in the condensate are proportional toeach other, whereas for larger values of g m minimizing the energy functional for the total
system energy provides a better match [10].
In order to obtain an approximation to the state of the system for finite g > 0, we use
perturbation theory to calculate the contribution of the intraspecies interaction to the totalenergy. Following Eq. (3) we assume the wave function is a product of single particle solitons
localized at the center of the trap
v
N t
Y
N asf
1
,
(5)
Φ(x) =
2 cosh (asf x n )n=1

4

SciPost Phys. 15, 095 (2023)

and calculate the expectation value of this mean-field wave function according toVintra = g

NX〈Φ(x) |δ(|x − x |)| Φ(x)〉kl〈Φ(x)|Φ(x)〉k<l= N (N − 1) g

2asf
N= (N − 1) g
2
4

Z
dx

1
4

cosh (asf x)

asf
.
6

(6)

p
Using the aforementioned Thomas-Fermi approximation ψ(x, t) = (µ̃ − g m |Φ|2 ) /g c e−i µ̃t/ħh
for the BEC wave function, the total system energy in the superfluid state reads
 2 2

2
2 a2
ħ
h gm
g
µ̃2 L c
a
ħ
h
sf
sf
m
3
Esf = N
N asf+−
N asf + (N − 1) g+,
(7)30mµ̃g c
m 6
6g c
6
2g c
where the bright soliton parameter asf minimizing the energy for finite g > 0 is now modified
to
v



u
gc g
25µ̃2 g c2
5µ̃ g ct 5mµ̃
11−+
asf = −+
1−
4
2
2
3 N gmgm
N
9 N 2 gm
3ħ
h2
r
=N a0



g g
1 + 2N 2 ε 1 − gc2 1 − N1 − 1
m

N 2ε

,

(8)

using the same parameter ε = 6a02ħ
h2 /(5mµ̃) as before. This approximation only gives a valid2
inverse width asf > 0 for g < g ∗ = N g m/[(N − 1)g c ].
Figure 1 shows the energy per immersed particle for increasing intraspecies interaction
g at fixed interspecies coupling (a) g m = 2g c and (b) g m = 3g c . For small values of g, thenumerical values (colored lines) agree very well with the analytical energy for the superfluid
state from Eq. (7) (dot-dashed lines), both for N = 2 and N = 3, before starting to deviate
and very slowly approaching the energy of the pinned state (dashed line) in the large g limit.
The pinned state is a system of individually isolated and localized single particle states which
are separated from each other at particle positions dn and its density is described byρpin (x) =N
apin X2

1

n=1 cosh2

,apin (x − dn )

(9)


p
with the peak height apin = a0 1 + 2ε − 1 /ε. For sufficiently large inter- and intraspecies
interactions the overlap between particles vanishes and they arrange in a periodic structurewith spacing L/N . The loss of coherence and the ordering of the particles has similarities tothe Mott-insulating state in lattice systems [7], albeit in this case triggered by intercomponent
interactions without the need for an external lattice potential. In this self-pinned state thesystem has a total energy ofEpin = N2ħ
h2 g m30mµ̃g c2ħh2 apin
3
−
apin
+

m 6

2
gm

6g c


apin +µ̃2 L c
,
2g c

(10)

and is independent of the interspecies interaction g [10].
The insets of Fig. 1 show a close-up of the point where the numerically obtained energies approach their respective pinned state energies. Interestingly, our results suggest thatthe matter-wave trapping potential created by the background BEC enables a first-order phasetransition in which the immersed component actually reaches the asymptotic state predicted5

(E − EBEC )/N (units of mgc2 /ℏ2 )(E − EBEC )/N (units of mgc2 /ℏ2 )SciPost Phys. 15, 095 (2023)(a) gm = 2gc

0−2−0.662−0.665

−4

−6

N =2
N =3
0

2

−0.668
14
4

15

6
8
g (units of gc )

23
10

27
12

31
14

(b) gm = 3gc

0

−10

−3.32
−3.33
−3.34

−20

−30

N =2
N =3
0

2

4

−3.35
−3.36
6

16

43

16.3

8
10
12
g (units of gc )

14

46
16

49
18

20

Figure 1: Total system energy per immersed particle, adjusted for EBEC = µ̃2 L c /2g c ,as a function of the intraspecies interaction strength g for a system of N = 2 (blueline) and N = 3 (green line) particles at fixed interspecies interaction strengthg m = 2g c (a) and g m = 3g c (b) and density N /L = 1/4 mg c /ħh2 . The dashedline shows the analytical value for the pinned state energy Epin = −0.6656 mg c2 /ħh2(a) and Epin = −3.3482 mg c2 /ħ
h2 (b) for N = 2 from Eq. (10), while the dasheddotted lines show the energy per particle calculated from Eq. (7). The insets show
a zoom of the phase transition point. The colored dotted lines indicate the energy ofthe metastable superfluid state if g is increased beyond the transition point. The
difference in background density of the BEC of µ0 = 103 mg c2 /ħ
h2 (N = 2) and232µ0 = 2/3 × 10 mg c /ħh (N = 3) leads to the slight difference in Epin visible in
the insets.
in the TG limit g → ∞. This is in contrast to the fermionization process in static trapping
potentials, where the system only asymptotically approaches the corresponding energy levelfrom below, but never reaches it, as can be seen for the analytically solvable case of two harmonically trapped atoms [42]. This behavior is also reminiscent of the discontinuous natureof the 1D superfluid-supersolid phase transition in dipolar condensates in the low-density limit
6

SciPost Phys. 15, 095 (2023)

observed in numerical simulations [43] and recently confirmed experimentally [44]. In thethermodynamic limit however, the transition is predicted to be continuous in 1D systems anddiscontinuous in the 2D case [45]. As expected, the intraspecies repulsion in the bigger system
of N = 3 is comparatively larger and leads to a lower value of g at which the system crosses
the transition to the pinned state.
Numerically the data was obtained via a self-consistent imaginary time evolution of thecoupled system of Eqs. (1) using the Fourier split-step method [46]. We calculate the energy ofthe superfluid branch (Esfnum ) by starting from the non-interacting case g = 0 and adiabaticallyincreasing the value of g in small increments while using the previously found ground state
as new initial state for the imaginary time evolution in each interaction step. In the insets ofFig. 1 the colored dotted lines indicate the energy of the then metastable superfluid statenum
if g is increased beyond the transition point. Similarly, for the pinned branch (Epin) westart deep in the strongly interacting regime g ≫ 1 and use the ansatz from Ref. [10] as
the initial state, adiabatically decreasing the value of g in small decrements while again using
the previously found ground state as new initial state for the imaginary time evolution. Finally,
the combined physical branch is obtained from determining the minimum of both branches atnum
each interaction step E(g) = min{Esfnum (g), Epin(g)}.
In panel (a) of Fig. 1 the insets also show that the numerically determined energy of thepinned state is slightly larger than the dashed line obtained from the analytical expression
Eq. (10). This is due to the fact that we are limited to studying a finite system numericallyand that for small values of g m the resulting pinned states are also only weakly localized. The
wave function of the immersed component is influenced by the box potential edges in thatcase, lifting the energy of the state in return. In contrast, for larger values of g m like in panel(b), the immersed particles are localized stronger, therefore not experiencing an influence ofthe system boundaries anymore, and the numerical values lie slightly below the analyticaldashed line as expected from using the closed expression for apin mentioned earlier compared
to minimizing the complete energy functional. In order to minimize these numerical finite
size effects on the results, we determine the phase transition point from the value of g forwhich the numerically obtained energies of the superfluid state intersect with the analyticallydetermined value for the pinned state according to Eq. (10), i.e. Esfnum = Epin , leading to theapparent unphysical discontinuity. As can be seen in the insets of Fig. 1, this has essentially noeffect for larger values of g m while allowing us to obtain a clean and coherent phase transitionline in the region of small g m where these finite size effects play a larger role as shown in thenext section.
In order to obtain an estimate for the phase transition point according to our analyticalmodel, we consider the limit µ̃ → ∞ and ε → 0 in which we can neglect the energetic cost of
deforming the BEC in Eqs. (10) and (7). In that case we haveapin (ε → 0) = a0 ,andasf (ε → 0) = N a0 −

mg
2ħ
h2

(N − 1) ,(11)

and the energy of the pinned state reduces to Epin (ε → 0) = −Nħh2 a02 /6m, while the energy ofthe superfluid state becomes2ħh2 asf





a0
mg
1
Esf (ε → 0) = N−
1− 1−
N 2 asf .
6m
3
N 2a0ħ
h2(12)

It is now easy to check that by choosing
2
gm
ħ
h2
g = 2a0 =
,
m
gc

7

(13)

SciPost Phys. 15, 095 (2023)

0.6

0.8

1

(b) N = 3

4
2

0

0

−2

gm = 2gc
6

gm = 2gc
12

18

24

30

0

5

g (units of gc )
101
100

−2
10

−4

15

g (units of gc )

(c)

101

(d)

100

−1

10−1

10−2

10−2

10

10−3
−4

−2

0

2

−6

4

x (units of ℏ2 /mgc )

−4

−2

0

2

4

6

10−3

x (units of ℏ2 /mgc )x (units of ℏ2 /mgc )0.4(a) N = 220
ρ(x) (units of mgc /ℏ2 )0.2

ρ(x) (units of mgc /ℏ2 )x (units of ℏ2 /mgc )0

Figure 2: Top row: Renormalized minority component density ρ(x) as a functionof the intraspecies interaction strength g for a system of (a) N = 2 and (b) N = 3
particles at fixed interspecies interaction strength g m = 2g c and density N /L = 1/4
mg c /ħ
h2 . Bottom row: The corresponding line densities in the superfluid state atcrit
g = 0 (green line) and right before the transition (red lines) at gnum
≃ 24.9g c and
crit
gnum
≃ 14.2g c are shown to scale in (c) and (d) respectively. The yellow lines show
crit
the density in the pinned state for g > gnum
. The black dash-dotted lines show
the analytical model according to Eq. (3). Other parameters are µ0 = 103 mg c2 /ħh2232(N = 2), µ0 = 2/3 × 10 mg c /ħh (N = 3).

we further have that also asf ε → 0, g = 2ħh2 a0 /m = a0 and that at this point the energies ofthe superfluid and pinned state are identical, Esf = Epin , indicating the point where the phasetransition occurs according to our model. Rewriting the above choice of g asp
crit
= ± gc ggm(14)
shows that the criterion coincides with the miscibility criterion for a two-component BEC [47]and also with a stability criterion for a Bose-Bose mixture derived by Cazalilla and Ho in the
Luttinger liquid framework [29]. Remarkably, Eq. (14) does not depend on the number of particles N and Fig. 1 shows that the intersection point of the analytic curves for Esf and Epin agrees
very well with the predicted values of g crit = 4g c for g m = 2g c [see panel (a)] and g crit = 9g c
for g m = 3g c [see panel (b)]. Even for the largest values of |g m | = 5g c and correspondingly thelargest values of ε considered in this paper, the actual intersection point for the model curves2
differs less than 5% from the analytical criterion, i.e. g crit (|g m | = 5g c ) ≈ 24g c < 25g c = g m/g c .
For the values of g m shown in Fig. 1, there is a large discrepancy between the predicted
critand the numerically observed value of gnum
however. This can be explained by studying thecorresponding minority component densities at fixed interspecies coupling g m = 2g c , which
are plotted in Fig. 2 for both (a) N = 2 and (b) N = 3. The density is renormalized to
its respective maximum at each point for clarity. Starting from line densities at g = 0 thatare well described by Eq. (3), as can be seen in panels (c) and (d), the densities begin to
split into separate branches according to the number of particles and start forming depletions
caused by the intraspecies repulsion, thereby deviating from the ansatz we use to calculate theenergy contribution Vintra . This leads to the observed discrepancy between the analytical and8

SciPost Phys. 15, 095 (2023)

1
gm = 2gc
Coherence

0.8

superfluid
C = 12

0.6
C = 130.4

pinned

0.2
0

N =2
0

5

10

15

N =3

20
25
30
g (units of gc )

35

40

45

50

Figure 3: Coherence C as a function of the intraspecies interaction strength g for asystem of N = 2 (blue line) and N = 3 (green line) particles at fixed interspecies
interaction strength g m = 2g c and density N /L = 1/4 mg c /ħh2 . Other parameters areµ0 = 103 mg c2 /ħ
h2 (N = 2), µ0 = 2/3 × 103 mg c2 /ħ
h2 (N = 3).
numerical values of g crit .
The branching occurs at comparatively lower values of g for N = 3 particles than for
N = 2 particles as one would expect from the larger contribution of the interaction energy
critin the case of more particles. At the numerically obtained critical values of gnum
≃ 24.9g c
crit(N = 2) and gnum ≃ 14.2g c (N = 3) the pinned state becomes energetically favorable in bothcases as can be seen in the insets of Fig. 1, resulting in an abrupt change towards a complete
spatial separation of the atoms in the minority component. Panels (c) and (d) also show thedensities right before and after the transition. The logarithmic scale used in the plots clearly
demonstrates the difference in the central dip between the coherent superfluid state, which isstill largely cohesive and centered around the origin, and the fully separated insulating pinned
state.

4

Phase diagram

In order to distinguish the superfluid from the pinned phase and map out the phase diagramof the system, we calculate the coherence (also known as the condensate fraction) of theimmersed component, C = (maxn λn )/N . It characterizes the off-diagonal long-range order[48–50] and it is defined in terms of the largest eigenvalue λn of the reduced single-particle
density matrix (RSPDM), obtained fromZX
ρ(x, x ′ ) =d x 2 . . . d x N Φ∗ (x, x 2 , . . . , x N )Φ(x ′ , x 2 , . . . , x N ) =λn ϕn∗ (x)ϕn (x ′ ) ,

(15)

n

with the natural orbitals ϕn (x). The eigenvalues λn describe how occupied the natural orbital
states are and therefore are a good indication of the coherence in the system. For instance,
for the superfluid state the lowest orbital is maximally occupied with λ0 ∼ N which entails
a maximal coherence of C → 1. In the pinned state each particle is individually localized
and isolated from one another, the result of which is the minimal coherence of C → 1/N .
The pinned state therefore resembles the reduced state of spinless fermions with λn = 1 for
n = 0, . . . , N − 1 and λn≥N = 0 as the contact interactions between the particles are nullified
by the mean-field potential.

9

0.5
5
sf
2.5

0.6

0.8

0.9

1 0.33

0.5

0.8

0.9

1
5
2.5

(b) N = 3
pin

sf
0

0.7

pin

(a) N = 2

−2.5

0.6

sf

pin

0

−5

0.7

20

40
60
g (units of gc )

80

pin

sf
100 0

0

10

20
30
g (units of gc )gm (units of gc )gm (units of gc )SciPost Phys. 15, 095 (2023)

−2.5
−5
40

Figure 4: Phase diagram of the system for (a) N = 2 and (b) N = 3 as a function of
intraspecies interaction g and interspecies interaction g m for fixed density N /L = 1/4
mg c /ħ
h2 , exhibiting both a superfluid (sf) and pinned (pin) state of the immersed
component. The colormap shows the value of the coherence C while the dashed line2
indicates g m= g c g. The gray shaded areas between |g m | ≲ 1.5g c with C > 1/N
belong to the self-pinned state and are a result of the finite spatial system size, while
the off-white areas in the superfluid phase stem from the finite particle number N (seetext for details). Other parameters are µ0 = 103 mg c2 /ħ
h2 (N = 2), µ0 = 2/3 × 103
mg c2 /ħh2 (N = 3).
In Figure 3 we show the coherence as a function of intraspecies repulsion g for fixed
g m = 2g c and fixed density N /L = 1/4 mg c /ħh2 . For g = 0 the coherence takes its maximum value for both the N = 2 and N = 3 particle systems, however for small finite valuesof g the coherence is smaller than 1. This is a consequence of the RSPDM becoming mixed
when interactions are finite which therefore reduces the coherence of the state and signifies
the presence of quantum correlations
between the particles. In fact, it is worth noting that inp
the Tonks-Girardeau limit λ0 ∼ N as g → ∞ [48, 51] which still characterizes a superfluidlike phase pthat possesses some long range order. The coherence in the TG limit therefore scales
as C → 1/ N , and while this is distinct from a fully incoherent fermionic state with C → 1/N ,this is no longer true in the thermodynamic limit where both values vanish. Care must therefore be taken when considering large systems, however for the finite systems studied here thisis not a concern. Indeed thep coherence we obtain in the superfluid phase for N = 2 and N = 3
is always larger than 1/ N with the transition to the pinned phase signalled by a sudden
decrease in coherence to C = 1/N .
In Figure 4 we show the full phase diagram in terms of the coherence for a system of (a)N = 2 and (b) N = 3 immersed particles as a function of both intraspecies interaction strength
g and interspecies interaction strength g m . In general, for fixed intraspecies interaction g thesystem transitions from the pinned insulating state to the coherent superfluid state if the interspecies interaction strength |g m | is increased until it can counteract the intrinsic repulsion of
the immersed component. Similarly, for fixed interspecies coupling g m the system transitions
from superfluid to pinned if the intraspecies repulsion g is large enough so that the particles
can push themselves apart. This means that for the self-pinning transition the system behavescontrarily to the pinning transition in an external lattice potential, where for finite values of g
the system is pinned for deep lattices but remains superfluid in shallow ones [6, 7].
For small values of |g m | ≲ 1.5g c the phase diagrams both exhibit areas in the pinned region
where C > 1/N . While this suggests the re-emergence of the superfluid phase it is in fact a
result of finite size effects stemming from the particles hitting the edge of the box potential. In
this regime the weak interaction with the BEC is not enough to isolate the particles. Instead,the trap edges keep the particles together, leading to larger values of the coherence. In this10

SciPost Phys. 15, 095 (2023)

region the finite system size also affects the numerically obtained values of the energy in thepinned state as detailed in the previous section and shown in the insets of Fig. 1. Therefore, inorder to obtain a clean and coherent phase transition line in this region, we have determinedthe critical lines in both phase diagrams by checking where the numerically obtained groundstate energy in the superfluid state Esfnum matches the analytical value of Epin according to
Eq. (10) as described earlier.
In general, the phase diagrams are symmetric with respect to the sign of the coupling
parameter g m , as was the case in Ref. [10]. This is a result of the homogeneous BEC density and2is also reflected in the fact that our effective model only depends on g m. For an inhomogeneousbackground gas the phase diagrams are not symmetric in general, as only attractive couplings
g m < 0 might lead to a stable trapped state while repulsive couplings g m > 0 commonly resultin phase separation in that case.
The miscibility criterion derived in Eq. (14) is also shown as a dashed line in both phase
diagrams in Fig. 4. Particularly in the N = 2 case it underestimates the observed numericalcritvalue of gnum
by a large amount due to the branching of the immersed component density
described earlier. For N = 3 the behavior is similar in the region of small g m and small g.
However, for g ≳ 15g c a distinct shoulder appears in the phase transition line, drawing it closer
to the analytical estimate. In this region, the larger interspecies coupling leads to a strong
localization of the immersed component and the formation of the density modulation due tothe intraspecies repulsion, visible in Fig. 2, is suppressed. In other words, the numericallyobtained critical point is close to the prediction Eq. (14) when the line density is still welldescribed by our bright solitonic ansatz.
We expect that in the thermodynamic limit N → ∞ the phase transition line will converge
to our analytical estimate in the region of small g m , leading to a disappearance of the offwhite regions seen in the superfluid phase between the dashed approximation and the actual
transition lines as a result of the finite number N of immersed particles. For larger g m , theeffect of ε > 0, which we have neglected in the derivation, becomes relevant, leading to atransition from superfluid to pinned at weaker intraspecies repulsion than predicted. This isalso in line with and even necessary for our observation in Ref. [10] that in the TG limit g → ∞only the pinned state exists, as the BEC is not able to compress the fermionized bosons for any
value of the interaction strength g m .

5Conclusion

We have studied how a small, initially superfluid, one-dimensional Bose gas immersed intoa Bose-Einstein condensate fermionizes as a function of increasing intraspecies repulsion gand eventually reaches the insulating self-pinned state expected in the Tonks-Girardeau limit
g → ∞. In contrast to static trapping potentials, this asymptotic state is actually crossed
and not just approached from below, implying a first-order phase transition as a result of thematter-wave backaction. We have confirmed this behavior by numerically simulating the system for N = 2 and N = 3 immersed particles and calculating its density, energy and coherence.
We have extended the effective model presented in Ref. [10] to the superfluid state and used it
to derive a phase transition line valid in the mean-field limit N → ∞ and ε → 0, i.e. Nc → ∞.
Finally, we mapped out the phase diagram as a function of interspecies and intraspecies couplings with extensive simulations for the aforementioned N = 2 and N = 3 cases.
Regarding future work, it would be interesting to probe the dynamics of the system byquenching or ramping either the intraspecies or interspecies interaction strengths across thephase transition lines. Furthermore, while we are limited to few-body systems due to numerical restrictions, it would be beneficial to study the system for a larger number of immersed

11

SciPost Phys. 15, 095 (2023)

particles N in order to investigate how the transition from superfluid to pinned scales withsystem size. This will also allow putting the effective model to the test and to extend thephase diagram to finite temperatures by studying the stability of the superfluid phase against
thermal excitations, similar to Ref. [10]. It is also worthwhile to explore the role of finite size
effects on the phase diagram, particularly by considering periodic boundary conditions forboth the immersed gas and the BEC. Already calculating the phase diagram for N = 3 particles with the Fourier split-step method using Ngrid = 512 position grid points at the resolution
in g and g m presented in Fig. 4 (b) required on the order of 106 CPU hours, despite numerous
optimizations. Larger systems therefore need to be studied by alternative techniques such as
world-line [52, 53] and diffusion Monte-Carlo methods [54, 55], multi-configurational timedependent Hartree methods [56], or the density-matrix renormalization group (DMRG) [57]and particularly its continuum extension [58]. All of these techniques are also able to cover
the finite temperature case and specifically the diffusion Monte-Carlo technique has alreadybeen used for studying 1D droplets in binary Bose mixtures [59, 60].

Acknowledgements
The authors would like to thank Mathias Mikkelsen and Tai Tran for fruitful discussions andfor performing supplementary simulations.
Funding information This work has been supported by the Okinawa Institute of Science andTechnology Graduate University and used the computing resources of the Scientific Computing
and Data Analysis section. T. K. acknowledges support from a Research Fellowship for Young
Scientists by the Japan Society for the Promotion of Science under JSPS KAKENHI Grant No.
21J10521. T. F. acknowledges support from JSPS KAKENHI Grant Number JP23K03290. T. F.
and T. B. are also supported by JST Grant Number JPMJPF2221.

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