One-dimensional compression of Bose-Einstein condensates by laser-induced
dipole-dipole interactions
S. Giovanazzi, D. O’Dell, G. Kurizki
Weizmann Institute of Science, 76100 Rehovot, Israel
(February 1, 2008)
We consider a trapped cigar-shaped atomic Bose-Einstein condensate irradiated by a single far-off
resonance laser polarized along the cigar axis. The resulting laser induced dipole-dipole interactions
between the atoms significantly change size of the condensate, and can even cause its self-trapping.
arXiv:cond-mat/0108385v1 23 Aug 2001
PACS: 03.75.Fi, 34.20.Cf, 34.80.Qb, 04.40.-b
The ability to alter the interatomic potential of weakly interacting Bose-Einstein condensates (BECs) by applying
electromagnetic fields affords a high degree of control over the Hamiltonian and thus the possibility to engineer the
macroscopic properties of such many-body systems [1]. The experiment of Inouye et al [2] has demonstrated how the
s-wave scattering length can be changed by magnetic fields via a Feshbach resonance.
We have recently proposed a different avenue: the use of off-resonant lasers to induce long-range dipole-dipole
interactions in atomic BECs [3–6]. These interactions can drastically modify a BEC, causing its self-trapping [3], laser
induced self-”gravity” [4] (for certain laser-beam configurations), “supersolid” structures [5], and peculiar excitation
spectra [6]. At the same time, off-resonant lasers may undergo “superradiant” Rayleigh scattering [7] from the
condensate, concurrently with collective atomic recoil (CARL) [8]. It is important to draw a clear distinction between
such effects and those of laser-induced dipole-dipole forces. The purpose of this paper is to suggest a simple geometry
in which the compression of a trapped condensate and its possible self-trapping are unambiguously caused by laserinduced dipole-dipole interactions, whereas superradiant Rayleigh scattering or CARL are absent.
POLARIZATION
Z
Y
FIG. 1. The laser beam and condensate geometry. By choosing the polarization to be along the long axis of the condensate
superradiant effects are suppressed.
A trapped cigar-shaped BEC that is tightly confined in the radial x-y plane is irradiated by a plane-wave laser at a
wavelength larger than the radial size of the condensate. The laser polarization is chosen to be along the long z-axis
of the condensate in order to suppress “superradiant” Rayleigh scattering [7] or CARL [8] that are forbidden in the
direction of the field polarization. The interatomic dipole-dipole potential induced by far-off resonance electromagnetic
radiation of intensity I, wave-vector q = qŷ (along the y-axis), and polarization ê = ẑ (along the z-axis) is [9]
I
α2 (q)Vzz (q, r) cos(qy).
(1)
U (r) =
4πcε20
Here r is the interatomic axis, α(q) the isotropic, dynamic, polarizability of the atoms at frequency cq, and Vzz is the
appropriate component of the retarded dipole-dipole interaction tensor
Vzz =
i
1h
2
2
2 2
1
−
3
cos
(θ)
cos
qr
+
qr
sin
qr
−
sin
(θ)q
r
cos
qr
r3
(2)
where θ is the angle between the interatomic axis and the z-axis. We note that the far-zone (qr ≫ 1) behavior of the
dipole-dipole potential (1) along the z-axis direction is proportional to − sin(qr)/(qr)2 .
A mean-field description of a BEC with dipole-dipole forces [4,10,11] can be accomplished through the GrossPitaevskii equation [12] for the condensate order parameter
ih̄
δ
∂Ψ
Htot .
=
∂t
δΨ∗
1
(3)
The mean-field energy functional
Htot = Hkin + Hho + Hdd + Hs
contains contributions from the one-body energies and the interatomic potentials: (a) the kinetic energy
Z
Hkin = (h̄2 /2m)|∇Ψ|2 d3 r ;
(4)
(5)
(b) harmonic trapping
Hho =
Z
Vho n(r) d3 r ,
where n(r) is the atomic number density; (c) the dipole-dipole potential
Z
Hdd = (1/2) n(r)n(r′ ) U (r − r′ ) d3 r d3 r′ ;
(6)
(7)
(d) the mean-field energy due to the very short-range (r−6 ) van der Waals interaction, which will be treated, as is
usual, within the delta function pseudo-potential approximation,
Z
Hs = (1/2)(4πah̄2 /m) n(r)d3 r ;
(8)
where a is the s-wave scattering length.
The integrations involved in the energy functionals are simpler in momentum space, for which Hdd = (1/2)(2π)3/2
R
R
e (k) ñ(k) ñ(−k) d3 k. The Fourier transform of the dipole-dipole potential (1), U
e (k) = (2π)−3/2 exp[ik·r] U (r) d3 r,
U
can be shown to be [9,13], for the laser propagation and polarization as in Fig. 1,
kz2 − q 2
kz2 − q 2
2
I α2
e
.
(9)
+ 2
− + 2
U (k) =
3 kx + (ky − q)2 + kz2 − q 2
kx + (ky + q)2 + kz2 − q 2
2(2π)3/2 ǫ20 c
We proceed by adopting a cylindrically symmetric (about the axial ẑ direction) variational ansatz for the density.
For this to hold true requires the tight radial trapping to prevail over the (anisotropic) dipole-dipole forces in the
radial direction. It is then reasonable to approximate the
by a Gaussian profile whose width is the
radial density
variational parameter wr : n(r) ≡ N (πwr2 )−1 nz (z) exp −(x2 + y 2 )/wr2 , where N is the total number of atoms and
nz (z) is the as yet unspecified axial density. Then, in momentum space, the density becomes
N z
w2
f (kz ) exp − r (kx2 + ky2 ) ,
ñ(k) =
n
(10)
(2π)
4
fz (kz ) being the Fourier transform of the axial density nz (z). Assuming tight radial confinement with respect to
n
the laser wavelength, w r ≡ wr /λL ≪ 1, the radial integration in Hdd can be approximately evaluated, so that the
dipole-dipole energy reduces to a one dimensional functional along the axial direction
Z
Z
1
(2π)3/2
fz (k̄z ) dk̄z ,
fz (k̄z )n
fz (−k̄z )U
Hdd =
n
(11)
nz (z)nz (z ′ )U z (z − z ′ ) dz dz ′ =
2
2
where ñ(k) is the Fourier transform of the number density. In this expression the axial momentum has been scaled
by the laser wavenumber k̄z ≡ kz /q, and the one-dimensional (1D) axial potential is given by
with
fz (k̄z ) =
U
I α2 q 3
2
Q(w r , k̄z ) ,
(2π)3/2 8πǫ20 c
2
1
(2πwr )2
2
×
+ 2(k̄z − 1) exp −
Q(w r , k̄z ) ≃ −
3 (2πw r )2
2
2
2
1
1
(k̄z − 1)(2πw r )2
(k̄z − 1)(2πw r )2
2
E1
+
(2πwr ) + exp
4
2
2
2
2
(k̄z2 − 1)(2πwr )2
(k̄z − 1)(2πwr )2
1
2
4
E1
,
(1 − k̄z )(2πw r ) exp
8
2
2
2
(12)
(13)
where E1 is the exponential integral [14].
fz (k̄z ) in Fig. 2. It tends to a positive constant value for large k̄z , as verified by the appropriate limit of
We plot U
(12) and (13):
2
1
1
(2πwr )2
1
I α2 q 3
2
fz (k̄z ) =
−
.
(14)
+
2
exp
−
+
lim U
3 (2πw r )2
2
(2πw r )2
2
(2π)3/2 8πǫ20 c
k̄z →∞
This means that the axial dipole-dipole potential in coordinate space will contain a repulsive delta-function contribution, which can stabilize the trapped condensate in addition to (or instead of) the short-range s-wave scattering.
fz (k̄z ),
Despite this effective short-range repulsion, the overall long-range behavior is dictated by the k̄z → 0 limit of U
which is attractive. When the k̄z = 0 Fourier component of the total 1D-reduced potential (the sum of the s-wave
scattering pseudo-potential and the laser-induced dipole-dipole interaction) becomes negative, the condensate size
decreases drastically until it becomes self-trapped within the laser-wavelength.
0
-20.0
-10.0
0.0
kz
10.0
20.0
fz (k̄z ). The radial width has been set at wr = 1/(4π).
FIG. 2. The k̄z dependence of the one dimensional axial potential U
From Fig. 2, the monotonic form of the axial potential suggests that the simplest ansatz for the axial variational
density is also a Gaussian
1
(2πw z )2 k̄z2
fz (k̄z ) ≡ √ exp −
.
(15)
n
4
2π
Even with this choice, however, the remaining integral in the calculation of Hdd is difficult and will here be evaluated
numerically. In the large N limit, the so-called Thomas-Fermi regime, Hke may be neglected in comparison with the
interaction energies and thus the energy HTF = Hs + Hdd + Hho can be written in the dimensionless form
Z
1
1
m
(2πw r )2
(2πw z )2
2
z
f
HTF = √
,
(16)
+ I n (k̄z )Q(w r , k̄z )dk̄z +
r )4 + 2(ql z )4
2N aq (qlho
q 3 N 2 h̄2 a
2π 2πw z (2πw r )2
ho
p
r
where lho
= h̄/(mωr ) is the harmonic oscillator length due to a radial trap of frequency ωr , and similarly for
z
lho
. Given these trapping frequencies, the real control parameter that allows one to play with the ratio of these
dipole-dipole forces to the s-wave scattering is the dimensionless ‘intensity’
I=
I α2 m
.
8πǫ20 ch̄2 a
(17)
A minimization of HTF brings out the strong dependence of the condensate radius wz and the weaker dependence of
the w r radius upon I as shown in Figures 3 and 4. For I >
∼ 1.1 there is a drastic decrease of w z and self-trapping in
accordance with the discussion following Eq. (14). For I = 3/2 the system collapses and this can be attributed to
the instability caused by the static r−3 part of the dipole-dipole potential (1), as in [10].
3
2.0
wz , wr
1.5
1.0
0.5
0.0
0.0
0.5
1.0
1.5
I
FIG. 3. Condensate size as a function of the scaled laser intensity I. Solid line: wz , dashed line: wr . The trap frequencies
have been chosen so that at I = 0 (i.e. without dipole-dipole forces) the condensate has an aspect ratio of 30:1, with a radial
dimension of wr = 1/(4π).
8.0
7.0
6.0
wz , wr
5.0
4.0
3.0
2.0
1.0
0.0
0.0
0.5
1.0
1.5
I
FIG. 4. Condensate size as a function of the scaled laser intensity I, as in fig. 3. here the trap frequencies are chosen
such that at I = 0 (i.e. without dipole-dipole forces) the condensate has an aspect ratio of 100:1, with a radial dimension of
wr = 1/(4π).
To conclude, we have predicted here a new feature of laser-irradiated cigar-shaped condensates: their strong compression along the soft axial direction. In an experiment this compression would provide a clear signature of laserinduced dipole-dipole forces at work.
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