Exactly-solvable system of one-dimensional trapped bosons with short and long-range interactions

M. Beau,1, 2 S. M. Pittman,3 G. E. Astrakharchik,4 and A. del Campo5, 6, 7, 8

1
Department of Physics, University of Massachusetts, Boston, Massachusetts 02125, USA
2
Dublin Institute for Advanced Studies, School of Theoretical Physics, 10 Burlington Road, Dublin 4, Ireland
3
Department of Physics, Harvard University, Cambridge, MA 02138, USA
4
Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4-B5, E-08034, Barcelona, Spain
5
Donostia International Physics Center, E-20018 San Sebastián, Spain
6
IKERBASQUE, Basque Foundation for Science, E-48013 Bilbao, Spain
7
Department of Physics, University of Massachusetts, Boston, MA 02125, USA
8
Theory Division, Los Alamos National Laboratory, MS-B213, Los Alamos, NM 87545, USA
arXiv:2009.03319v1 [cond-mat.quant-gas] 7 Sep 2020

We consider trapped bosons with contact interactions as well as Coulomb repulsion or gravitational attraction
in one spatial dimension. The exact ground state energy and wave function are identified in closed form to-
gether with a rich phase diagram, unveiled by Monte Carlo methods, with crossovers between different regimes.
A trapped McGuire quantum soliton describes the attractive case. Weak repulsion results in an incompressible
Laughlin-like fluid with flat density, well reproduced by a Gross-Pitaevskii equation with long-range interac-
tions. Higher repulsion induces Friedel oscillation and the eventual formation of a Wigner crystal.

Low-dimensional quantum gases can be engineered with The LL model is known to admit an exact treatment by
ultracold atomic vapors by freezing the dynamics along a Bethe ansatz in homogeneous external potentials such as a
given axis under tight confinement. In one spatial dimension, ring [3], a box [33–35], or the continuum [36]. While the
quantum fluctuations are enhanced, and yet, the stability of Bethe ansatz method has proved to be very powerful and use-
the system extends to strongly-interacting regimes, in contrast ful [18, 19], it cannot account exactly for the presence of a
with the three-dimensional case. Effectively one-dimensional non-uniform confinement that is ubiquitous in experimental
quantum gases offer a test-bed for integrability and provide a settings with trapped ultracold gases. In the presence of a har-
faithful implementation of a range of exactly solvable models monic trap, away from the Tonks-Girardeau limit [37], the un-
[1, 2]. Among them, the homogeneous gas with contact in- derstanding of the LL gas relies on a combination of approxi-
teractions described by a δ-function pseudopotential, exactly mate, effective theories and numerical methods [1, 38, 39]. In
solved in 1963 by Lieb and Liniger (LL model) [3, 4], is a this context, it is thus of great importance to find other exactly-
paradigmatic reference model. Olshanii showed that ultracold solvable many-body models in a trapped geometry.
atomic gases confined in tight waveguides experience reso- In this work we introduce a novel exact solution for a one-
nance in the effective one-dimensional interactions and can be dimensional Bose gas in the presence of a harmonic trap and
used to realize the LL model [5, 6]. Following this observa- with both short-range contact and long-range linear pairwise
tion, the LL model has been routinely realized in the labora- inter-particle interactions. The long-range contribution corre-
tory [7–17]. sponds physically to one-dimensional Coulomb repulsion or
gravitational attraction. The exact ground state is found in
From the theoretical point of view, a rich variety of pow-
closed form and is shown to reduce to the McGuire soliton
erful mathematical methods such as the Bethe ansatz is avail-
solution for attractive interactions when the frequency of the
able to analyze LL model, making it a favorite test-bed for
harmonic confinement vanishes. We characterize the ground-
these techniques [18, 19]. The limit of strong interactions,
state properties and demonstrate that a wide variety of regimes
known as the Tonks-Girardeau gas, corresponds to hard-core
can be accessed in this system, including soliton-like, ideal
bosons [20] and has helped to elucidate the fact that quantum
Bose gas, incompressible fluid and Wigner-crystal regimes.
exchange statistics is ill-defined as an independent concept in
We start by considering a very general system consisting of
one dimension, being inextricably woven to inter-particle in-
N atoms of mass m confined in a harmonic trap of frequency
teractions. Its study revealed the existence of Bose-Fermi du-
ω, interacting via both the contact interaction with coupling
ality and its generalizations [21, 22]. More recently, it has
strength g and a long-range potential of strength σ. The sys-
been pointed out that the LL gas constitutes the universal non-
tem Hamiltonian is
relativistic limit of a variety of integrable quantum field the-
N 
 ~2 ∂2  N h
 X
ories [23]. In nonlinear physics, the quantum version of both X 1 i
bright and dark solitons has been found in the LL model. In H= − + mω i+
2 2
x gδ(xi j ) + σ|xi j | , (1)
2m ∂xi
2

i=1
2 i< j
particular, the cluster solution of the attractive LL gas was ex-
plicitly found by McGuire via Bethe ansatz [24]. For repulsive where xi are particle coordinates and xi j = xi − x j the rela-
interactions, many-body states describing gray solitons, char- tive distances. The long-range interaction potential in Eq. (1)
acterized by a dip in the density profile of finite depth, have corresponds to the solution of the Poisson equation for a point
been discussed in [25–30]. Dark solitons with vanishing den- source distribution of mass or charge ∆V(r) = κδ(r). For any
sity at the dip have been also found in the Tonks-Girardeau dimension d ≥ 1 the Fourier transform of the Laplacian equals
regime that describes hard-core bosons [29, 31, 32]. to −k2 , where k ∈ Rd is the wave vector. For a point source
 2

distribution of mass or charge, the Fourier transform of the state that describes a bright quantum soliton, and thus, the
static potential V is given by ±Q/k2 , where Q is the electric thermodynamic limit does not exist [24].
charge or mass of the particles. Taking the inverse Fourier To include a harmonic trap, it is convenient to introduce
transform yields V = ∓Q|x| in one-dimension (the familiar analogues of the creation and annihilation operators
case V = ±Q/|r| is found in d = 3).
πi − imωxi † πi + imωxi
The exchange operator formalism introduced by Poly- ai = √ , ai = √ , (7)
chronakos [40] has brought great insight into one-dimensional 2mω~ 2mω~
quantum integrable models and we use it to analyze Hamilto- satisfying
nian (1). We show that the latter admits a decoupled form, in mg X
terms of phase-space variables. To do this, we consider the [ai , a†i ] = 1 + |xi j |Mi j . (8)
~2 j,i
Hermitian generalized momenta of N particles

i {ai , ai } = HLL + i 2 mω xi and the identity

†
X P 1 2 2
πi = pi + i Vi j Mi j ,
P
(2) The relation ~ω 2
j,i ai ai = 2 {ai , ai } − 2 [ai , ai ] allows us to derive the Hamiltonian
† 1 † 1 †

of the system embedded in a trap,

defined in terms of the canonical momenta p j = −i~ ∂x∂ j for
particles j = 1, · · · , N. The particle permutation operator Mi j
X
H = ~ω a†i ai + E0 (9)
is idempotent Mi2j = 1, symmetric Mi j = M ji and acts on an i
arbitrary operator A j as Mi j A j = Ai Mi j , Mi j Ak = Ak Mi j . We XN 
 ~2 ∂2 mω2 xi2  X
 "
mωg|xi j |
#
choose the so-called prepotential function Vi j = V(xi j ) to be = − + + gδ(xi j )− .
2m ∂xi2
 
2 ~
proportional to the sign function i=1 i< j

~ This is an instance of the Hamiltonian class (1) in which the

Vi j = − sgn (xi j ), (3) trap frequency ω, the coupling constant g and the strength of
as
the long-range interaction σ satisfy the following relation
with a s being a real constant (negative or positive) with units
of length. Later, we shall see that a s is physically equivalent σ = −mωg/~. (10)
to a s-wave scattering length.
First we consider a homogeneous case and construct a Thus, for repulsive contact interactions (g > 0), the linear
purely kinetic Hamiltonian of the form, long-range term corresponds to Coulomb repulsion between
equal charges in one dimension. Similarly, for attractive con-
1 X 2 tact interactions (g < 0) the linear long-range term describes
H0 = π (4)
2m i i gravitational attraction between equal masses in d = 1. Thus,
X p2
  short and long-range interactions are either both attractive or
1 X X 
= i
+  (~Vi j Mi j + Vi j ) − 2
0 2
Vi jk Mi jk  , both repulsive.
2m 2m i< j
i i< j<k The value of the coupling α = ωg/~ depends on the fre-
where Vi jk = Vi j V jk + V jk Vkl + Vkl Vi j , Mi jk = Mi j M jk , and quency of the trap, the mass of the particle and the coupling
the prime denotes the spatial derivative. Explicit computation constant g. Using ultracold gases as a platform, one could
shows that Vi2j = ~2 /a2s and Vi0j = −(2~/a s )δ(xi j ) for the two- adjust the value of g independently of the mass of the parti-
body term, while the three-body term reduces to a constant cles using a Feshbach resonance. This is tantamount to tuning
Vi jk = −~2 /a2s . This yields the many-body Hamiltonian of a the effective gravity/anti-gravity acceleration α or electrostatic
one-dimensional gas with pairwise contact interactions charge ~ωc/(4π0 r⊥2 ).
The exact expression for the ground-state energy of Hamil-
N
~2 X ∂2 X tonian (9) can be written explicitly as
HLL = H0 + E0 = − + g δ(xi j )Mi j , (5)
2m j=1 ∂x j
2
i< j N~ω mg2 N(N 2 − 1)
E0 = − 2 , (11)
where the coupling constant is related to the one-dimensional 2 ~ 24
s-wave scattering length as g = −2~2 /(ma s ) and which is independent of the sign of the coupling constant g,
mg N(N − 1) 2 2 and scales as ∝ N 3 for large atom number as will be com-
E0 = − . (6) mented in more details later. For bosons (Mi j = 1), the
~2 24
ground-state wave function satisfies ai Ψ0 (x) = 0, which can
For Bose statistics, Mi j reduces to the identity and Hamil- be written in terms of the logarithmic derivative,
tonian (5) describes N one-dimensional bosons subject to
s-wave pairwise interactions, i.e., Lieb-Liniger [3, 4] and ∂ xi Ψ0 xi X sgn(xi j )
=− 2 − ,
McGuire [24] systems. It is known that for repulsive inter- Ψ0 aho j,i
as
actions, g > 0, only scattering states are possible and the ho- √
mogeneous gas is stable. On the contrary, for attractive inter- where aho = ~/(mω) is the harmonic oscillator length.
actions, g < 0, the system collapses into a many-body bound Hence, it follows that the exact ground-state wave function
 3

of Hamiltonian (1) is of the free energy functional obtained for√this field leads to a
non-linear Schrödinger equation for Φ = Nφ(x) [43]
 1 x2 
" #  
Y |xi j | Y
Ψ0 (x) = N −1 exp − exp − 2i  , (12) ~2 ∂2 mω2 2
!
i< j
as i
2 aho − + x + g|Φ(x)| Φ(x)
2
2m ∂x2 2
Z
where N −1 denotes the normalization factor. As expected − mα dx0 |x − x0 ||Φ(x0 )|2 φ(x) = µΦ(x) , (17)
from Kohn’s theorem, it is possible to factorize the center of
mass R = N1 i xi [41],
P
where µ is the chemical potential, α = ωg/~ is an ef-
Rfective coupling constant, and the normalization condition
 (|xi j | + Na2ho /a s )2 
 
NR2
!Y
0 −1
Ψ0 (x) = N exp − exp −  , dx n(x) = N is given in terms of the local density of par-
2aho2 i< j 2Na2ho ticle is n(x) = |Φ(x)|2 . This equation can be recognized as the
(13) Gross-Pitaevskii equation modified with an additional non-
where we used the identity, iN xi2 = NR2 + N1 i< j (xi j )2 , and linear long-range potential
P P
N 0 = N exp{4a2s /[a2ho N 2 (N − 1)]}. Z
For g > 0 (i.e. a s < 0) the ground state thus describes a V(x) = −mα dx0 |x − x0 ||Φ(x0 )|2 ,
crystal-like order in the sense that the wave function is maxi-
mal for |xi j | = Na2ho /a s . which is a solution of the one-dimensional Poisson equation
In order to analyze different physical regimes, it is conve- ∆V(x) = −2αρ(x) ,
nient to recast the Hamiltonian in dimensionless form (9)
where the local mass distribution
N 
 1 ∂2 x̃i2 
X  Xh i
H̃ = − + + c δ( x̃i j ) − | x̃i j | , (14) ρ(x) = mn(x) = m|Φ(x)|2 ,
2 ∂ x̃i2
 
2

i=1 i< j
∆ = d2 /dx2 denotes the Laplacian in d = 1, and α is the cou-
where tilde symbols denote that ~ω is used as a unit of energy pling strength with dimensional units of acceleration. This
and harmonic oscillator length aho as a unit of distance. Sys- long-range potential can be interpreted as a gravity and anti-
tem properties are governed by two dimensionless parameters gravity interaction between particles depending on whether
which are number of particles N and dimensionless interac- the sign of α is negative or positive, respectively. For α > 0
tion strength c defined as the potential can also describe electrostatic interaction, after
p rewriting the coupling strength |α| = ~ωc, which is indepen-
gm1/2 σ |gσ| 2aho dent of the mass. In this case, by tuning the inverse length
c = 3/2 1/2 = − 1/2 1/2 3/2 = sgn(g) = − .
~ ω ~ m ω ~ω as constant c, we can vary the value of the effective charge q of
(15) the particle ~ωc = 4π0 r⊥2 |q|, where 0 is the vacuum permi-
Its value quantifies the relative strength of the interaction po- tivity and r⊥ is a dimension reduction characteristic length.
tential (contact and gravitational/Coulomb) with respect to It follows that Eq. (17) can be interpreted as a mean-field
the trapping potential: c > 0 refers to repulsive contact and equation for a one-dimensional system of N particles inter-
Coulomb potentials, while c < 0 corresponds to attractive acting with a short-range δ- function pseudopotential and a
short- and long-range interactions. For strong short-range at- long-range gravity (α < 0) or Coulomb/anti-gravity (α > 0)
traction, c → −∞, the solution for the Hamiltonian (5) re- potential. In the Thomas-Fermi regime, where the kinetic en-
duces to McGuire bound-state solution ergy is negligible compared to the energy of the short-range
! interaction between particles, the coupling strength is greater
Y |xi j |
Ψ0 (x) = exp − , (16) than a critical value g > gc ≡ ~ωa0 /(2N), assuming that
i< j
as Ekin ∼ ~2 /(2ma20 ) and Eshort ∼ gN/a0 . In this critical regime,
the energy of the long-range interaction is of the same order as
describing a quantum bright soliton. Indeed, the system that of the short-range potential, Elong ∼ mωgNa0 /~ = gN/a0 ,
energy
√ (11) is similar to that of a bright soliton for c  hence both short and long-range contributions are significant.
− 24/N. In this regime, Eq. (14) can be identified as By neglecting the kinetic term in Eq. (17) we obtain the inte-
the (parent) Hamiltonian with a trapped McGuire soliton as gral equation
ground state [42].
mω2 2
! Z
Mean-field theory and the Gross-Pitaevskii equation.— In 1 mω
nMTF (x) = µ− x + dx0 |x − x0 |nMTF (x0 )
one spatial dimension, the mean-field limit is reached as the g 2 ~
density is increased, when the distance between particles is = nTF (x) + δn(x) , (18)
small with respect to the scattering length. The ground state
of a large number of particles is then well described by where nTF (x) is the standard Thomas-Fermi density function
the Hartree-Fock approximation, Ψ(x1 , · · · , xN ) = i=1 φ(xi ) obtained by taking α = 0 in Eq. (17), and δn(x) is the ad-
QN
where φ(xi ) is a single-particle wave function. The variation ditional long-local term. Fortunately, Eq. (18) can be solved
 4

Knowledge of the exact wave function and its simple

pair-product form facilitates the analysis of the ground
state correlations. The local density profile n(x) =
dx2 . . . dxN |Ψ0 (x1 = x, x2 , . . . xN )|2 Rand density-density cor-
R

relation function g2 (x) = N(N − 1) dx1 . . . dxN δ(x − |x1 −

x2 |)|Ψ0 (x1 , . . . xN )|2 can be numerically obtained by Monte
Carlo integration of the square of the exact ground-state wave
function. The obtained results are shown in Fig. 1 for different
characteristic values of the interaction strength c and number
of particles N.
In the non-interacting case, c = 0 shown with black lines
in in Fig. 1, the density profile has a Gaussian shape typical
for a harmonic oscillator while the pair-correlation function
is continuous at the contact position, x = 0. In the repul-
sive case the probability of finding two particles at the same
position is reduced and a kink is formed at x = 0 with its am-
plitude increasing with c (compare c = 0.1 for N = 100 and
FIG. 1. Density profile n(x) (upper row), pair distribution g2 (x) (bot- c = 0.5; 0.8 for N = 5 in Fig. 1). There are two types of corre-
tom row) for N = 5 (left column) and N = 100 (right column) par- lations contributing to the shape of g2 (x): the large x envelope
ticles and for different values of interaction strength. Solid lines are
results of Monte Carlo sampling, while dashed lines show the mean-
is dominated by the one-body density profile, while the two-
field profile, Eq. (19). Harmonic oscillator units are used. body correlations provide an important contribution to the the
behavior in the vicinity of x = 0. For weak repulsion g2 (r) is
dominated by the one-body rather than two-body correlations
and the mean-field approach is applicable. The density pro-
using standard techniques, as shown in the Supplemental Ma- file decreases monotonously from the center to the edges and
terial [41]. This yields the local density profile for large number of particles is well predicted by the mean-
  √  field prediction of Eq. (19), shown in dashed lines. A distin-
1
1 − cosh( 2x/aho )
, |x| ≤ L guishing feature is the formation of a flat-topped mesa profile,

 √
n(x)aho =  ,

 2c cosh( 2L/aho ) (19)

0,
 |x| > L typical of incompressible liquids where the addition of more
particles does not change the density of a droplet but rather
where L is the size of the density profile determined by the increases its size. According to Eq. (19) the density of the
equation n(x) = 0 together with the normalization condition plateau, n0 aho = 1/(2c), is fixed by the interaction strength
RL
dx nMTF (x) = N. After numerical computation we find c and the system size L is directly proportional to N for the
−L large number of particles. Yet, the system differs from the
these values and show the density profile in Fig. 1. We also
find the value of the chemical potential using Eq. (18). The usual liquids as the external potential is present. The origin
normalization condition imposes that for g  gc ≡ ~ω/N of its incompressibility resembles that of Laughlin states[44]
as the exact solution (12) is analogous to the Laughlin’s wave
the density function is homogeneous √ nMTF (x) ≈ ~ω/(2g) for function and it allows Laughlin’s plasma analogy [45].
|x| ≤ L with L ≈ gN/(~ω) + a0 / 2. In the opposite case
1/3
For strong repulsion, c ≈ 1, two-body correlations become

with g  gc , the cloud radius equals L ≈ 3gN/2mω2 ,
which corresponds to the standard Thomas-Fermi spread in important in g2 (x) and result in Friedel oscillations observed
one dimension. in the density profile (see c = 0.5, 0.8 and N = 5). The mean-
Discussion.— A remarkable feature of the linear long-range field approach is then no longer applicable. Finally, for even
potential is that a negative coefficient in front of absolute stronger repulsion, c  1, there is vanishing probability of
value of separation |xi j | in Hamiltonians (9,14) actually re- finding two particles in the same position, g(0) → 0. The
sults in repulsion rather than attraction. Indeed, upon identify- strong Coulomb interaction leads to formation of a Wigner
ing infinitely-separated particles as non-interacting, the term crystal which is seen as a periodic modulation of the total den-
−|xi j | results in a higher potential energy at small separations. sity (see c = 3 and N = 5). Different regimes are reached via
Further, the value of the ground-state energy (11) is in- continuous crossovers in accordance to what is expected in
dependent of the sign of c. This property stems from one-dimensional and finite-size system.
condition (10) which allow us to make the system solv- In conclusion, we have introduced a model of a trapped
able. Nonetheless, opposite signs describe drastically differ- one-dimensional Bose or Fermi atoms interacting via both
ent regimes. For example, large value values of |c| corre- contact and long-range interactions. The latter can account
spond to a bright soliton in the attractive (gravitational) case for either a Coulomb potential or a gravitational interaction,
while they correspond to a Wigner crystal in the repulsive in the repulsive and attractive case, respectively. Our main re-
(Coulomb) case. sults are the Hamiltonian (9), its ground state energy (11) and
 5

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from the John Templeton Foundation and UMass Boston to the strongly interacting regime,” Physical Review Letters 106
(project P20150000029279) and the Spanish Ministerio de (2011), 10.1103/physrevlett.106.230405.
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 6

dimensional bose system with the repulsive delta-function inter- GROUND-STATE WAVEFUNCTION
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Let us consider the definition of the collective coordinate
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1 X
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250602 (2019). 1 X 1 X 2
[33] M. Gaudin, “Boundary energy of a bose gas in one dimension,” (xi j )2 = (x + x2j − 2xi x j ) (24)
Phys. Rev. A 4, 386–394 (1971). N i< j N i< j i
[34] M. T. Batchelor, X. W. Guan, N. Oelkers, and C. Lee, “The 1d N−1X 2 2 X
interacting bose gas in a hard wall box,” Journal of Physics A: = xi + xi x j . (25)
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N i< j
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Petković, and Zoran Ristivojevic, “Exact results for the bound- In short,
ary energy of one-dimensional bosons,” Phys. Rev. Lett. 123, 1
!
1 X 2 1 X
250602 (2019). R2 = 1 − + x − (xi j )2 (26)
[36] Michel Gaudin, The Bethe Wavefunction, edited by Jean- N N i i N i< j
Sébastien Caux (Cambridge University Press, 2014).
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N i< j
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As a result, one can rewrite the ground-state wavefunction as
[39] A. Iu. Gudyma, G. E. Astrakharchik, and Mikhail B. Zvonarev, follows
“Reentrant behavior of the breathing-mode-oscillation fre- Y " #
|xi j | Y  1 x2 
 
quency in a one-dimensional bose gas,” Phys. Rev. A 92, Ψ0 (x) = N −1 exp − exp − 2i  (28)
021601 (2015). i< j
as i
2 aho
[40] Alexios P. Polychronakos, “Exchange operator formalism for  
 1 R2  Y

 |xi j |

1
= N −1 exp − 2  2
.

integrable systems of particles,” Phys. Rev. Lett. 69, 703–705 exp − − |x |
ij 
2 aho i< j as 2
(1992). 2Naho
[41] See the Supplemental Material.
(29)
[42] Adolfo del Campo, “Exact ground states of quantum
many-body systems under confinement,” arXiv e-prints , Completing the square one finds
arXiv:2005.03904 (2020), arXiv:2005.03904 [quant-ph].  2 
[43] Lev Pitaevskii and Sandro Stringari, Bose-Einstein Condensa- Na2ho  
  
 1 R2  Y  1
Ψ0 (x) = N exp − 2 
−1 |xi j | +

tion and Superfluidity (Oxford University Press, Oxford, 2016) exp −  
p. 576. 2 aho i< j 2Na2ho as
[44] Elliott H. Lieb, Nicolas Rougerie, and Jakob Yngvason,  2 
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“Rigidity of the laughlin liquid,” Journal of Statistical Physics × exp  ho  . (30)
172, 544–554 (2018). i< j
2a2s
[45] R. B. Laughlin, “Anomalous quantum hall effect: An incom-
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Phys. Rev. Lett. 50, 1395–1398 (1983). ground-state wavefunction
 2 
Na2ho  
  
0 −1
 1 R2  Y  1 
Ψ0 (x) = N exp − 2  exp − |xi j | +   ,
2 aho i< j 2Na2ho as
(31)
 7

with where A, B, C are three real constants. The constant C rep-

 2 resents the particular solution of Eq. (33), which is found to
 N (N − 1)a2ho 

N 0 −1 = N −1 exp   . (32) be C = ~ω/(2g). The constant B is obviously zero as the den-
4a2s sity profile is symmetric with respect to the origin x = 0. To
find the constant A, we have to use the boundary condition
determined by the position x = L (or x = −L, equivalently) at
EXACT SOLUTION OF MEAN-FIELD THEORY
which the density profile vanishes. After solving the equation
n(L) = 0 for A, we find
In the main body of the paper, we show that the density
profile satisfies the following integral equation

mω2 2
! ~ω/(2g)
A=− .
Z
1 mω
nMTF (x) = µ− x + dx0 |x − x0 |nMTF (x0 ) q
2mω
g 2 ~ cosh ~ L
where nMTF (x) is the modified Thomas-Fermi density func-
tion. In order to solve this integral equation, it suffices to take
the second-order derivative on both sides of Eq. (33) with re- The density function now reads
spect to the variable x, which yields

d2 mω2 mω 2
Z
0 d
  q 
nMTF (x) = − + dx |x − x0 |nMTF (x0 ) 
  Cosh 2mω
x 
dx2 dx2

g ~ω  ~
  , |x| ≤ L(33a)
~ 
  
 1 −
nMTF (x) = 

 q
mω2 2mω
Z 2g  2mω
Cosh ~ L

=− + dx0 δ(x − x0 )|nMTF (x0 )




g ~ 

, |x| > L(33b)

0
mω2 2mω The position L, which determine half of the width of the den-
=− + nMTF (x) ,
g ~ sity profile, can be found using the normalization condition
RL
dx nMTF (x) = N, where N is the number of particles. Per-
where the commutation of the integral and the derivative is −L
d2 forming the integral, we find that L satisfies following equa-
2 |x − x | =
0
justified by the Lesbegue theorem and we used dx
0 tion
2δ(x − x ). Therefore, we obtain a second order ordinary dif-
ferential equation
 
d2 mω2 2mω  r 
2
nMTF (x) = − + nMTF (x) . N=
~ωL  
1 − q
1  2mω 

tanh  L ,

dx g ~ g  ~ 
2mω
~ L
Using standard techniques, we find that the general solution
of Eq. (33) is
r 
 2mω 
r 
 2mω  which is a transcendental equation for L that can be solved,
nMTF (x) = A cosh  x + B sinh  x + C , e.g., numerically.
~  ~ 