Approaching the Heisenberg limit in an atom laser
M. Jeppesen,1 J. Dugué,1, 2 G. R. Dennis,1 M. T. Johnsson,1 C. Figl,1 N. P. Robins,1 and J. D. Close1
Australian Research Council Centre Of Excellence for Quantum-Atom Optics,
Department of Physics, The Australian National University, Canberra, ACT 0200, Australia
Laboratoire Kastler-Brossel, 24 rue Lhomond, 75231 Paris Cedex 05, France
We present experimental and theoretical results showing the improved beam quality and reduced
divergence of an atom laser produced by an optical Raman transition, compared to one produced
by an RF transition. We show that Raman outcoupling can eliminate the diverging lens effect that
the condensate has on the outcoupled atoms. This substantially improves the beam quality of the
atom laser, and the improvement may be greater than a factor of ten for experiments with tight
trapping potentials. We show that Raman outcoupling can produce atom lasers whose quality is
only limited by the wavefunction shape of the condensate that produces them, typically a factor of
1.3 above the Heisenberg limit.
PACS numbers: 03.75.Pp,03.75.Mn
Experiments in ultracold dilute atomic gases have had
an enormous impact on physics. The realization of Bose-
Einstein condensates (BECs), degenerate Fermi gases,
BEC-BCS crossover systems, and many others have re-
sulted in many fundamental insights and a wealth of new
results in both experiment and theory. One exciting sys-
tem to emerge from this research is the atom laser, a
highly coherent, directional beam of degenerate atoms,
controllably released from a BEC [1, 2, 3, 4, 5, 6, 7, 8].
The atom lasers demonstrated so far have produced
beams many orders of magnitude brighter than is pos-
sible with thermal atomic beams [9].
Atom laser beams show great promise for studies
of fundamental physics and in high precision measure-
ments [10]. In the future, it will be possible to produce
quadrature squeezing in atoms lasers, to use atom lasers
to produce correlations and entanglement between mas-
sive particles [11], as well as high precision interferome-
ters both on earth and in space [12]. For all these it will
be crucial to develop atom lasers with output modes that
as clean as possible in amplitude and phase, to allow sta-
ble modematching, just as it was crucial for optical lasers.
The beam quality factor M2, introduced for atom lasers
by J.-F. Riou et al. [13, 14], is a measure of how far the
beam deviates from the Heisenberg limit, and is defined
∆x∆px, (1)
where ∆x is the beam width, measured at the waist,
and ∆px is the transverse momentum spread. An ideal
(Gaussian) beam would therefore have M2 = 1 along
both its principal transverse axes. A number of exper-
imental works have shown that the beam quality of an
atom laser is strongly affected by the interaction of the
outcoupled atoms with the BEC from which it is pro-
duced [13, 15, 16, 17, 18]. As the atoms fall through the
condensate, the repulsive interaction acts as a diverging
lens to the outcoupled atoms. This leads to a divergence
in the atom laser beam and (because the BEC is a non-
ideal lens) a poor quality transverse beam profile. Such
behavior may cause problems in mode matching the atom
laser beam to another atom laser, a cavity or to a waveg-
FIG. 1: (color online). Top: Sequence of atom laser beams
showing the improved beam profile of a Raman atom laser.
The atom laser beams were produced using RF (a) and Ra-
man (b and c) transitions. The angle between the Raman
beams (see Fig. 2 (a)) was θ = 30◦ in (b) and θ = 140◦ in
(c), corresponding to a kick of 0.5h̄k (0.3 cm/s) and 1.9h̄k
(1.1 cm/s) respectively. The outcoupling rate differs be-
tween each atom laser. Below: Comparison of experimental
(dashed) and theoretical (solid) beam profiles 500 µm below
the BEC. The height of each theoretical curve has been scaled
to match experimental data.
http://arxiv.org/abs/0704.0291v2
uide. Experiments on atom lasers in waveguides have
produced beams with improved spatial profile [7]. How-
ever, precision measurements with atom interferometry
are likely to require propagation in free space, to avoid
introducing noise from the fluctuations in the waveguide
itself [12].
In a recent Letter [13], it was shown that the quality of
a free space atom laser is improved by outcoupling from
the base of the condensate. Our scheme, however, en-
ables the production of a high quality atom laser while
outcoupling from the center of the condensate. This is
desirable for a number of reasons: First, because the clas-
sical noise level is determined by the outcoupling Rabi
frequency, then outcoupling from the center, where the
density is greatest, gives the highest possible output flux
for a given classical noise level [19]. Second, outcoupling
from the center allows the longest operating time (for a
quasicontinuous atom laser) since the condensate can be
drained completely. Third, outcoupling from the center
minimizes the sensitivity of the output coupling to con-
densate excitations or external fluctuations.
In a recent Letter [9], we have demonstrated a contin-
uously outcoupled atom laser where the output coupler
is a coherent multi-photon (Raman) transition [6, 20]. In
this scheme, the atoms receive a momentum kick from the
absorption and emission of photons. They leave the con-
densate more quickly, so that adverse effects due to the
mean-field repulsion from the condensate are reduced.
In this Letter, we report measurements of a substantial
improvement in the beam quality M2 using this outcou-
pling. In Fig. 1, we show absorption images of atom
laser beams outcoupled from the center of a BEC with
(a) negligible momentum kick, (b) a kick of 0.3 cm/s,
and 1.1 cm/s (c). As the kick increases, the divergence
is reduced and the beam profile improved.
In our experiment, we create 87Rb BECs of
5× 105 atoms in the |F = 1,mF = −1〉 state via stan-
dard runaway evaporation of laser cooled atoms. We
use a highly stable, water cooled QUIC magnetic trap
(axial frequency ωy = 2π × 12 Hz and radial frequency
ωρ = 2π × 128 Hz, with a bias field of B0 = 2 G). We
control drifts in the magnetic bias by using high stability
power supplies and water cooling. This stability allows
us to precisely and repeatably address the condensate.
We produce the atom laser by transferring the atoms to
the untrapped |F = 1,mF = 0〉 state and letting them
fall under gravity. To outcouple atoms with negligible
momentum kick we induce spin flips via an RF field of
a frequency corresponding to the Zeeman shift in the
center of the condensate. Alternatively, we induce the
spin flips via an optical Raman transition. The setup
is shown in Fig. 2 (a). Two optical Raman beams, sep-
arated by an angle θ, propagate in the plane of grav-
ity and the magnetic trap bias field. The momentum
transfer to the atoms through absorption and emission of
the photons is 2h̄k sin(θ/2), with k the wave number of
FIG. 2: (color online) (a) Experimental schematic (not to
scale) showing the BEC, Raman lasers, and trapping coils.
(b) Cross section along the two strong axes of the magnetic
trap, showing the BEC, outcoupling surface, and atom laser
trajectories. Note that the field of view in (b) is rotated 90◦
with respect to (a).
the laser beams. The Raman laser beams are produced
from one 700 mW diode laser. We can turn the laser
power on or off in less than 200 ns using a fast switching
AOM in a double pass configuration. After the switch-
ing AOM, the light is split and sent through two separate
AOMs, again each in a double pass configuration. The
frequency difference between the AOMs corresponds to
the Zeeman plus kinetic energy difference between the
initial and final states of the two-photon Raman transi-
tion. We stabilize the frequency difference by running
the 80 MHz function generators driving the AOMs from
a single oscillator. The beams are then coupled via single
mode, polarization maintaining optical fibers directly to
the BEC through a collimating lens and waveplate, pro-
viding a maximum intensity of 2500 mW/cm2 per beam
at the BEC. The polarization of the beams is optimized
to achieve maximum outcoupling with a downward kick
and corresponds to π polarization for the upper beam
and σ+ for the lower beam.
The outcoupling resonance is set to the center of the
BEC for both RF and Raman outcoupling, as shown
in Fig. 2 (b). This point is found by performing spec-
troscopy on the BEC using 100 ms of weak output cou-
pling at varying RF frequencies, and measuring the num-
ber of atoms remaining in the condensate after the output
coupling time [3]. A typical calibration curve is shown in
Fig. 3 (a), in this case for RF outcoupling. We operate
both RF and Raman output couplers at the point of max-
imum outcoupling rate. We further check this frequency
by ensuring that a continuous beam can still be produced
when the initial condensate is very small, which can only
happen when outcoupling from the center.
We observe the system using standard absorption
imaging along the y (weak trapping) direction, on the
F = 2 → F ′ = 3 transition, with a 200 µs pulse of
repumping light (F = 1 → F ′ = 2) 1 ms prior to imag-
ing. From these images we are able to extract the rms
width of the atom laser as a function of fall distance (see
FIG. 3: (a) Output coupling spectroscopy showing the oper-
ating point at the center of the BEC, solid curve to guide the
eye. (b) The rms beam width for RF and Raman atom lasers.
The dots represent experimental measurements and the solid
curves our theoretical predictions.
Fig. 3 (b)), which we use to calculateM2 (details below).
To model the system, we use a two-step method fol-
lowing [13]. Inside the condensate, we use the WKB
approximation, by integrating the phase along the clas-
sical trajectories of atoms moving in the Thomas-Fermi
potential of the condensate (an inverted paraboloid) [16].
After this, we propagate the atom laser wavefunction us-
ing a Kirchoff-Fresnel diffraction integral over the surface
of the condensate:
ψ(r) =
dS′ · [G∇′ ψ − ψ∇′G], (2)
where G = G(r, r′) is the Green’s function for the Hamil-
tonian in the gravitational potential V (r) = −mgz [21].
Therefore, the model includes only interactions between
condensate atoms and beam atoms; interactions between
atoms within the beam are ignored. The integral in
Eq. (2) is formally a two dimensional surface integral
over the whole condensate. However for simplicity, fol-
lowing [13], we neglect divergence in the weak trapping
axis and only consider cross sections in the plane of the
strong trapping axes, and so the integral becomes one
dimensional. A 3D wavefunction is built up by calculat-
ing the atom laser in a series of planes along the weak
trapping axis.
We ignore the effects of the magnetic field on the atom
laser. The atom laser state |F = 1,mF = 0〉 is unaffected
to first order by the magnetic field, but is weakly anti-
trapped due to the second order Zeeman effect, with an
effective trapping frequency of ω2nd = 2π × 2.6 Hz. The
transverse position of an atom in such a potential is
x(t) = x0 cosh(ω2ndt) ≈ x0(1 + ω
2/2). (3)
For the 1 mm (14 ms) propagation we consider here the
transverse position is affected by less than 3%. We also
ignore the AC Stark effect of the Raman beams on the
atom laser, because the intensity of the beams does not
change significantly over the 1 mm propagation.
We have checked the validity of this model against
a solution of the full 3D Gross-Pitaesvskii (GP) equa-
tion, including beam-beam interactions. To find the
atom laser wavefunction at large distances below the con-
densate (up to 1 mm), we transfer the GP model to a
freely falling frame once the atom laser wavefunction has
reached steady state. The details of the calculation will
be the basis of a future publication. The two models give
good agreement.
Calculating the quality factor M2 of the atom laser
directly from Eq. (1) requires measurement of the beam
width at the waist ∆x0. Because the BEC acts as a
diverging lens on the atom laser, the beam waist is virtual
and located above the BEC, and so it is not possible to
measure the beam quality M2 using Eq. 1 only. For
our simulations, M2 is calculated equivalently from the
wavefunction ψ(x, y, z) at some height z below the BEC
in which the atom laser has reached the paraxial regime:
(M2/2)2 = (∆x(z))2(∆kx(z))
2 − C(z)2, (4)
where ∆x(z) is beam width and C(z) is the curvature-
beam width product [22]:
C(z) =
dx. (5)
In practice it is difficult to measure the wavefunction
phase, and hence C(z). However the beam width, in
the paraxial regime, obeys:
∆(x(t))2 = (∆x0)
2 + (∆vx)
2(t− tw)
2, (6)
where tw is the time when the beam is at its waist, and
∆x0 is the beam waist. In principle M
2 may be deter-
mined simply from measurements of the beam width at
different heights. In our experiment, we can only measure
the beam width in the far field, at distances greater than
300 µm below the condensate (observation at distances
less than 300 µm are prevented by the condensate expan-
sion after trap switchoff.) In the far field the second term
of Eq. 6 dominates, and so only the velocity spread can
be measured. Therefore we calculate ∆x0 and tw from
the model, tw = mC(z)/(h̄∆k
x), with tw negative since
the waist is virtual and located above the BEC. We then
fit to the experimental data to find ∆vx.
In Fig. 4, we present the theoretical and experimental
results. We find that as the kick increases, the beam qual-
ity is improved and the divergence is reduced. For our
parameters, we find that for an RF atom laserM2 = 2.2,
and for a Raman atom laser M2 = 1.4 with the maxi-
mum two photon kick. As the kick increases,M2 contin-
ues to improve, and approaches but does not reach the
Heisenberg limit of one. It asymptotes to a limit slightly
above that, which for our parameters is equal to 1.3. In
this regime of large kick, the interaction of the outcou-
pled atoms with the condensate becomes negligible, and
FIG. 4: (a) Calculated quality factor M2 of an atom laser.
The dots are the experimental measurements, and the solid
line our theoretical predictions. (b) M2 as a function of trap-
ping frequency for an RF atom laser (dashed line), a kick
of 0.5h̄k (0.3 cm/s) (dotted line), and 2h̄k (1.1 cm/s) (solid
line). The condensate number was N = 5 × 105 atoms, and
the aspect ratio ωρ/ωy was 10.
the transverse atom laser wavefunction is approximately
the free space evolution of the condensate wavefunction
(along the outcoupling surface). It is therefore limited
by the non-ideal (non-Gaussian) condensate wavefunc-
tion itself. We calculate the product ∆x∆px for the con-
densate wavefunction (taken through the central horizon-
tal plane of the condensate) to be 1.3. We have therefore
improved the beam quality M2 by 50 percent, down to
a factor of 1.4 above the Heisenberg limit. In addition,
our simulations show that (using the same maximum two
photon kick) it is possible to reach the condensate limit
even for much tighter trapping potentials. In Fig 4 (b),
we show the results of simulations for increasing trap fre-
quencies, up to ω = 2π × 300 Hz. As the trap frequency
increases, theM2 worsens, up toM2 = 14 for RF outcou-
pling from a 2π×300 Hz trap. For the maximum Raman
two photon kick, the increase is only to M2 = 1.7 for
the same 2π × 300 Hz trap. Only for traps of less than
2π×50 Hz is the beam quality of an RF atom laser within
5 percent of that of a Raman atom laser.
With higher order Raman transitions [23], it will be
possible to reach the condensate limit even for exper-
iments with traps of several kilohertz. It will also be
possible to reach the Heisenberg limit by completely re-
moving the atomic interaction, for example by using a
Feschbach resonance. Using Raman lasers phase locked
to the 6.8 GHz hyperfine splitting will prevent populat-
ing the anti-trapped state, and produce a truly two state
atom laser [18, 24]. Such lasers, combined with the high
quality transverse mode of Raman atom lasers, could be
used in a continuous version of the atomic Mach-Zehnder
Bragg interferometer [25], and in the development of
atomic local oscillators.
We thank Ruth Mills for useful discussions. CF ac-
knowledges funding from the Alexander von Humboldt
foundation. This work was financially supported by the
Australian Research Council Centre of Excellence pro-
gram. Numerical simulations were done at the APAC
National Supercomputing Facility.
∗ Electronic address: matthew.jeppesen@anu.edu.au;
URL: http://www.acqao.org
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mailto:matthew.jeppesen@anu.edu.au
http://www.acqao.org

  We present experimental and theoretical results showing the improved beam
quality and reduced divergence of an atom laser produced by an optical Raman
transition, compared to one produced by an RF transition. We show that Raman
outcoupling can eliminate the diverging lens effect that the condensate has on
the outcoupled atoms. This substantially improves the beam quality of the atom
laser, and the improvement may be greater than a factor of ten for experiments
with tight trapping potentials. We show that Raman outcoupling can produce atom
lasers whose quality is only limited by the wavefunction shape of the
condensate that produces them, typically a factor of 1.3 above the Heisenberg
limit.

Approaching the Heisenberg limit in an atom laser
M. Jeppesen,1 J. Dugué,1, 2 G. R. Dennis,1 M. T. Johnsson,1 C. Figl,1 N. P. Robins,1 and J. D. Close1
Australian Research Council Centre Of Excellence for Quantum-Atom Optics,
Department of Physics, The Australian National University, Canberra, ACT 0200, Australia
Laboratoire Kastler-Brossel, 24 rue Lhomond, 75231 Paris Cedex 05, France
We present experimental and theoretical results showing the improved beam quality and reduced
divergence of an atom laser produced by an optical Raman transition, compared to one produced
by an RF transition. We show that Raman outcoupling can eliminate the diverging lens effect that
the condensate has on the outcoupled atoms. This substantially improves the beam quality of the
atom laser, and the improvement may be greater than a factor of ten for experiments with tight
trapping potentials. We show that Raman outcoupling can produce atom lasers whose quality is
only limited by the wavefunction shape of the condensate that produces them, typically a factor of
1.3 above the Heisenberg limit.
PACS numbers: 03.75.Pp,03.75.Mn
Experiments in ultracold dilute atomic gases have had
an enormous impact on physics. The realization of Bose-
Einstein condensates (BECs), degenerate Fermi gases,
BEC-BCS crossover systems, and many others have re-
sulted in many fundamental insights and a wealth of new
results in both experiment and theory. One exciting sys-
tem to emerge from this research is the atom laser, a
highly coherent, directional beam of degenerate atoms,
controllably released from a BEC [1, 2, 3, 4, 5, 6, 7, 8].
The atom lasers demonstrated so far have produced
beams many orders of magnitude brighter than is pos-
sible with thermal atomic beams [9].
Atom laser beams show great promise for studies
of fundamental physics and in high precision measure-
ments [10]. In the future, it will be possible to produce
quadrature squeezing in atoms lasers, to use atom lasers
to produce correlations and entanglement between mas-
sive particles [11], as well as high precision interferome-
ters both on earth and in space [12]. For all these it will
be crucial to develop atom lasers with output modes that
as clean as possible in amplitude and phase, to allow sta-
ble modematching, just as it was crucial for optical lasers.
The beam quality factor M2, introduced for atom lasers
by J.-F. Riou et al. [13, 14], is a measure of how far the
beam deviates from the Heisenberg limit, and is defined
∆x∆px, (1)
where ∆x is the beam width, measured at the waist,
and ∆px is the transverse momentum spread. An ideal
(Gaussian) beam would therefore have M2 = 1 along
both its principal transverse axes. A number of exper-
imental works have shown that the beam quality of an
atom laser is strongly affected by the interaction of the
outcoupled atoms with the BEC from which it is pro-
duced [13, 15, 16, 17, 18]. As the atoms fall through the
condensate, the repulsive interaction acts as a diverging
lens to the outcoupled atoms. This leads to a divergence
in the atom laser beam and (because the BEC is a non-
ideal lens) a poor quality transverse beam profile. Such
behavior may cause problems in mode matching the atom
laser beam to another atom laser, a cavity or to a waveg-
FIG. 1: (color online). Top: Sequence of atom laser beams
showing the improved beam profile of a Raman atom laser.
The atom laser beams were produced using RF (a) and Ra-
man (b and c) transitions. The angle between the Raman
beams (see Fig. 2 (a)) was θ = 30◦ in (b) and θ = 140◦ in
(c), corresponding to a kick of 0.5h̄k (0.3 cm/s) and 1.9h̄k
(1.1 cm/s) respectively. The outcoupling rate differs be-
tween each atom laser. Below: Comparison of experimental
(dashed) and theoretical (solid) beam profiles 500 µm below
the BEC. The height of each theoretical curve has been scaled
to match experimental data.
http://arxiv.org/abs/0704.0291v2
uide. Experiments on atom lasers in waveguides have
produced beams with improved spatial profile [7]. How-
ever, precision measurements with atom interferometry
are likely to require propagation in free space, to avoid
introducing noise from the fluctuations in the waveguide
itself [12].
In a recent Letter [13], it was shown that the quality of
a free space atom laser is improved by outcoupling from
the base of the condensate. Our scheme, however, en-
ables the production of a high quality atom laser while
outcoupling from the center of the condensate. This is
desirable for a number of reasons: First, because the clas-
sical noise level is determined by the outcoupling Rabi
frequency, then outcoupling from the center, where the
density is greatest, gives the highest possible output flux
for a given classical noise level [19]. Second, outcoupling
from the center allows the longest operating time (for a
quasicontinuous atom laser) since the condensate can be
drained completely. Third, outcoupling from the center
minimizes the sensitivity of the output coupling to con-
densate excitations or external fluctuations.
In a recent Letter [9], we have demonstrated a contin-
uously outcoupled atom laser where the output coupler
is a coherent multi-photon (Raman) transition [6, 20]. In
this scheme, the atoms receive a momentum kick from the
absorption and emission of photons. They leave the con-
densate more quickly, so that adverse effects due to the
mean-field repulsion from the condensate are reduced.
In this Letter, we report measurements of a substantial
improvement in the beam quality M2 using this outcou-
pling. In Fig. 1, we show absorption images of atom
laser beams outcoupled from the center of a BEC with
(a) negligible momentum kick, (b) a kick of 0.3 cm/s,
and 1.1 cm/s (c). As the kick increases, the divergence
is reduced and the beam profile improved.
In our experiment, we create 87Rb BECs of
5× 105 atoms in the |F = 1,mF = −1〉 state via stan-
dard runaway evaporation of laser cooled atoms. We
use a highly stable, water cooled QUIC magnetic trap
(axial frequency ωy = 2π × 12 Hz and radial frequency
ωρ = 2π × 128 Hz, with a bias field of B0 = 2 G). We
control drifts in the magnetic bias by using high stability
power supplies and water cooling. This stability allows
us to precisely and repeatably address the condensate.
We produce the atom laser by transferring the atoms to
the untrapped |F = 1,mF = 0〉 state and letting them
fall under gravity. To outcouple atoms with negligible
momentum kick we induce spin flips via an RF field of
a frequency corresponding to the Zeeman shift in the
center of the condensate. Alternatively, we induce the
spin flips via an optical Raman transition. The setup
is shown in Fig. 2 (a). Two optical Raman beams, sep-
arated by an angle θ, propagate in the plane of grav-
ity and the magnetic trap bias field. The momentum
transfer to the atoms through absorption and emission of
the photons is 2h̄k sin(θ/2), with k the wave number of
FIG. 2: (color online) (a) Experimental schematic (not to
scale) showing the BEC, Raman lasers, and trapping coils.
(b) Cross section along the two strong axes of the magnetic
trap, showing the BEC, outcoupling surface, and atom laser
trajectories. Note that the field of view in (b) is rotated 90◦
with respect to (a).
the laser beams. The Raman laser beams are produced
from one 700 mW diode laser. We can turn the laser
power on or off in less than 200 ns using a fast switching
AOM in a double pass configuration. After the switch-
ing AOM, the light is split and sent through two separate
AOMs, again each in a double pass configuration. The
frequency difference between the AOMs corresponds to
the Zeeman plus kinetic energy difference between the
initial and final states of the two-photon Raman transi-
tion. We stabilize the frequency difference by running
the 80 MHz function generators driving the AOMs from
a single oscillator. The beams are then coupled via single
mode, polarization maintaining optical fibers directly to
the BEC through a collimating lens and waveplate, pro-
viding a maximum intensity of 2500 mW/cm2 per beam
at the BEC. The polarization of the beams is optimized
to achieve maximum outcoupling with a downward kick
and corresponds to π polarization for the upper beam
and σ+ for the lower beam.
The outcoupling resonance is set to the center of the
BEC for both RF and Raman outcoupling, as shown
in Fig. 2 (b). This point is found by performing spec-
troscopy on the BEC using 100 ms of weak output cou-
pling at varying RF frequencies, and measuring the num-
ber of atoms remaining in the condensate after the output
coupling time [3]. A typical calibration curve is shown in
Fig. 3 (a), in this case for RF outcoupling. We operate
both RF and Raman output couplers at the point of max-
imum outcoupling rate. We further check this frequency
by ensuring that a continuous beam can still be produced
when the initial condensate is very small, which can only
happen when outcoupling from the center.
We observe the system using standard absorption
imaging along the y (weak trapping) direction, on the
F = 2 → F ′ = 3 transition, with a 200 µs pulse of
repumping light (F = 1 → F ′ = 2) 1 ms prior to imag-
ing. From these images we are able to extract the rms
width of the atom laser as a function of fall distance (see
FIG. 3: (a) Output coupling spectroscopy showing the oper-
ating point at the center of the BEC, solid curve to guide the
eye. (b) The rms beam width for RF and Raman atom lasers.
The dots represent experimental measurements and the solid
curves our theoretical predictions.
Fig. 3 (b)), which we use to calculateM2 (details below).
To model the system, we use a two-step method fol-
lowing [13]. Inside the condensate, we use the WKB
approximation, by integrating the phase along the clas-
sical trajectories of atoms moving in the Thomas-Fermi
potential of the condensate (an inverted paraboloid) [16].
After this, we propagate the atom laser wavefunction us-
ing a Kirchoff-Fresnel diffraction integral over the surface
of the condensate:
ψ(r) =
dS′ · [G∇′ ψ − ψ∇′G], (2)
where G = G(r, r′) is the Green’s function for the Hamil-
tonian in the gravitational potential V (r) = −mgz [21].
Therefore, the model includes only interactions between
condensate atoms and beam atoms; interactions between
atoms within the beam are ignored. The integral in
Eq. (2) is formally a two dimensional surface integral
over the whole condensate. However for simplicity, fol-
lowing [13], we neglect divergence in the weak trapping
axis and only consider cross sections in the plane of the
strong trapping axes, and so the integral becomes one
dimensional. A 3D wavefunction is built up by calculat-
ing the atom laser in a series of planes along the weak
trapping axis.
We ignore the effects of the magnetic field on the atom
laser. The atom laser state |F = 1,mF = 0〉 is unaffected
to first order by the magnetic field, but is weakly anti-
trapped due to the second order Zeeman effect, with an
effective trapping frequency of ω2nd = 2π × 2.6 Hz. The
transverse position of an atom in such a potential is
x(t) = x0 cosh(ω2ndt) ≈ x0(1 + ω
2/2). (3)
For the 1 mm (14 ms) propagation we consider here the
transverse position is affected by less than 3%. We also
ignore the AC Stark effect of the Raman beams on the
atom laser, because the intensity of the beams does not
change significantly over the 1 mm propagation.
We have checked the validity of this model against
a solution of the full 3D Gross-Pitaesvskii (GP) equa-
tion, including beam-beam interactions. To find the
atom laser wavefunction at large distances below the con-
densate (up to 1 mm), we transfer the GP model to a
freely falling frame once the atom laser wavefunction has
reached steady state. The details of the calculation will
be the basis of a future publication. The two models give
good agreement.
Calculating the quality factor M2 of the atom laser
directly from Eq. (1) requires measurement of the beam
width at the waist ∆x0. Because the BEC acts as a
diverging lens on the atom laser, the beam waist is virtual
and located above the BEC, and so it is not possible to
measure the beam quality M2 using Eq. 1 only. For
our simulations, M2 is calculated equivalently from the
wavefunction ψ(x, y, z) at some height z below the BEC
in which the atom laser has reached the paraxial regime:
(M2/2)2 = (∆x(z))2(∆kx(z))
2 − C(z)2, (4)
where ∆x(z) is beam width and C(z) is the curvature-
beam width product [22]:
C(z) =
dx. (5)
In practice it is difficult to measure the wavefunction
phase, and hence C(z). However the beam width, in
the paraxial regime, obeys:
∆(x(t))2 = (∆x0)
2 + (∆vx)
2(t− tw)
2, (6)
where tw is the time when the beam is at its waist, and
∆x0 is the beam waist. In principle M
2 may be deter-
mined simply from measurements of the beam width at
different heights. In our experiment, we can only measure
the beam width in the far field, at distances greater than
300 µm below the condensate (observation at distances
less than 300 µm are prevented by the condensate expan-
sion after trap switchoff.) In the far field the second term
of Eq. 6 dominates, and so only the velocity spread can
be measured. Therefore we calculate ∆x0 and tw from
the model, tw = mC(z)/(h̄∆k
x), with tw negative since
the waist is virtual and located above the BEC. We then
fit to the experimental data to find ∆vx.
In Fig. 4, we present the theoretical and experimental
results. We find that as the kick increases, the beam qual-
ity is improved and the divergence is reduced. For our
parameters, we find that for an RF atom laserM2 = 2.2,
and for a Raman atom laser M2 = 1.4 with the maxi-
mum two photon kick. As the kick increases,M2 contin-
ues to improve, and approaches but does not reach the
Heisenberg limit of one. It asymptotes to a limit slightly
above that, which for our parameters is equal to 1.3. In
this regime of large kick, the interaction of the outcou-
pled atoms with the condensate becomes negligible, and
FIG. 4: (a) Calculated quality factor M2 of an atom laser.
The dots are the experimental measurements, and the solid
line our theoretical predictions. (b) M2 as a function of trap-
ping frequency for an RF atom laser (dashed line), a kick
of 0.5h̄k (0.3 cm/s) (dotted line), and 2h̄k (1.1 cm/s) (solid
line). The condensate number was N = 5 × 105 atoms, and
the aspect ratio ωρ/ωy was 10.
the transverse atom laser wavefunction is approximately
the free space evolution of the condensate wavefunction
(along the outcoupling surface). It is therefore limited
by the non-ideal (non-Gaussian) condensate wavefunc-
tion itself. We calculate the product ∆x∆px for the con-
densate wavefunction (taken through the central horizon-
tal plane of the condensate) to be 1.3. We have therefore
improved the beam quality M2 by 50 percent, down to
a factor of 1.4 above the Heisenberg limit. In addition,
our simulations show that (using the same maximum two
photon kick) it is possible to reach the condensate limit
even for much tighter trapping potentials. In Fig 4 (b),
we show the results of simulations for increasing trap fre-
quencies, up to ω = 2π × 300 Hz. As the trap frequency
increases, theM2 worsens, up toM2 = 14 for RF outcou-
pling from a 2π×300 Hz trap. For the maximum Raman
two photon kick, the increase is only to M2 = 1.7 for
the same 2π × 300 Hz trap. Only for traps of less than
2π×50 Hz is the beam quality of an RF atom laser within
5 percent of that of a Raman atom laser.
With higher order Raman transitions [23], it will be
possible to reach the condensate limit even for exper-
iments with traps of several kilohertz. It will also be
possible to reach the Heisenberg limit by completely re-
moving the atomic interaction, for example by using a
Feschbach resonance. Using Raman lasers phase locked
to the 6.8 GHz hyperfine splitting will prevent populat-
ing the anti-trapped state, and produce a truly two state
atom laser [18, 24]. Such lasers, combined with the high
quality transverse mode of Raman atom lasers, could be
used in a continuous version of the atomic Mach-Zehnder
Bragg interferometer [25], and in the development of
atomic local oscillators.
We thank Ruth Mills for useful discussions. CF ac-
knowledges funding from the Alexander von Humboldt
foundation. This work was financially supported by the
Australian Research Council Centre of Excellence pro-
gram. Numerical simulations were done at the APAC
National Supercomputing Facility.
∗ Electronic address: matthew.jeppesen@anu.edu.au;
URL: http://www.acqao.org
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mailto:matthew.jeppesen@anu.edu.au
http://www.acqao.org
