Del Mar Photonics - New Journal of Physics - Axicons in stock

New J. Phys. **8** (2006) 159

doi:10.1088/1367-2630/8/8/159

PII: S1367-2630(06)25646-5

**Long distance transport of ultracold atoms using a 1D optical
lattice**

Stefan Schmid, Gregor Thalhammer, Klaus Winkler, Florian Lang and Johannes Hecker Denschlag

Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria

Email: johannes.denschlag@uibk.ac.at

Received 30 May 2006

Published 30 August 2006

Abstract. We study the horizontal
transport of ultracold atoms over macroscopic distances of up to 20 cm with
a moving 1D optical lattice. By using an optical Bessel beam to form the
optical lattice, we can achieve nearly homogeneous trapping conditions over
the full transport length, which is crucial in order to hold the atoms
against gravity for such a wide range. Fast transport velocities of up to
6 m s^{ - 1} (corresponding to about 1100 photon recoils) and
accelerations of up to 2600 m s^{ - 2} are reached. Even at high
velocities the momentum of the atoms is precisely defined with an
uncertainty of less than one photon recoil. This allows for construction of
an atom catapult with high kinetic energy resolution, which might have
applications in novel collision experiments. |

**Contents**

- 1. Introduction
- 2. Basic principle of transport
- 3. Bessel beams
- 4. Experimental setup
- 5. Transport of ultracold atoms
- 6. Atom catapult
- 7. Conclusion
- Acknowledgments
- Appendix. Transport ramp
- References

**1. Introduction**

Fast, large-distance transport of Bose-Einstein condensates (BEC) from their place of production to other locations is of central interest in the field of ultracold atoms. It allows for exposure of BECs to all different kinds of environments, spawning progress in BEC manipulation and probing.

Transport of cold atoms has already been explored in various approaches using magnetic and optical fields. Magnetic fields have been used to shift atoms, e.g. on atom chips (for a review see [1]) and to move laser-cooled clouds of atoms over macroscopic distances of tens of centimetres, e.g. [2, 3]. By changing the position of an optical dipole trap, a BEC has been transferred over distances of about 40 cm within several seconds [4]. This approach consisted of mechanically relocating the focusing lens of the dipole trap with a large translation stage. A moving optical lattice offers another interesting possibility to transport ultracold atoms. Acceleration of atoms with lattices is intimately connected to the techniques of Raman transitions [5], STIRAP [6, 7] and the phenomenon of Bloch oscillations [8, 9]; (for a recent review on atoms in optical lattices see [10]). Acceleration with optical lattices allows for precise momentum transfer in multiples of two photon recoils to the atoms. Transport of single, laser-cooled atoms in a deep optical lattice over short distances of several mm has been reported in [11]. Coherent transport of atoms over several lattice sites has been described in [12]. Even beyond the field of ultracold atoms, applications of optical lattices for transport are of interest, e.g. to relocate submicron sized polystyrene spheres immersed in heavy water [13].

Here, we experimentally investigate transporting BECs and ultracold thermal
samples with an optical lattice over macroscopic distances of tens of
centimetres. Our method features the combination of the following important
characteristics. The transport of the atoms is in the quantum regime, where all
atoms are in the vibrational ground state of the lattice. With our setup,
mechanical noise is avoided and we achieve precise positioning (on the order of
the imaging resolution of 1 μm). We demonstrate high transport velocities of up
to 6 m s^{ - 1}, which are accurately controlled on the quantum level.
The velocity spread of the atoms is not more than 2 mm s^{ - 1},
corresponding to 1/3 of a photon recoil.

**2. Basic principle of transport**

Horizontal transport of atoms over larger distances holds two challenges: how to move the atoms and how to support them against gravity. Our approach here is to use a special 1D optical lattice trap, which is formed by a Bessel laser beam and a counterpropagating Gaussian beam. The lattice part of the trap moves the atoms axially, whereas the Bessel beam leads to radial confinement holding the atoms against gravity.

In brief, lattice transport works like this. We first load the atoms into a 1D optical lattice, which in general is a standing wave interference pattern of two counterpropagating laser beams far red-detuned from the atomic resonance line (see figure 1). Afterwards the optical lattice is carefully moved, `dragging' along the atoms. Upon arrival at the destination, the atoms are released from the lattice.

Figure 1. Scheme for atom transport. Two
counterpropagating laser beams form a standing wave dipole trap. A BEC is
loaded adiabatically into the vibrational ground state of this 1D optical
lattice. A relative frequency detuning Δ ν between the two laser beams
results in a lattice motion at a velocity v = Δν·λ/2 which drags
along the trapped atoms. We chose the counterpropagating laser beams to
consist of a Gaussian beam with diameter 2w_{0} and a Bessel
beam with a central spot diameter of 2r_{0}. The (in
principle) diffraction-free propagation of the Bessel beam leads to tight
radial confinement of the atoms over long distances, which supports the
atoms against gravity during horizontal transport. |

The lattice motion is induced by dynamically changing the relative frequency detuning Δ ν of the two laser beams, which corresponds to a lattice velocity

where λ is the laser wavelength of the lattice.

In comparison to the classical notion of simply `dragging' along the atoms in
the lattice, atom transport is more subtle on the quantum level. Here, only
momenta in multiples of two photon recoil momenta, 2
*k* = 4 π
/λ
can be transferred to the atoms. This quantized momentum transfer can be
understood in several ways, e.g. based on stimulated Raman transitions or based
on the concept of Bloch-like oscillations in lattice potentials. For a more
thorough discussion in this context, the reader is referred to [14].

In order to prevent the atoms from falling in the gravitational field, the
lattice has to act as an optical dipole trap in the radial direction. It turns
out that for radial trapping, optical lattices formed by Bessel beams have a
clear advantage over Gaussian beam lattices. To make this point clear, we now
show, that a standard optical lattice based on Gaussian beams is not well suited
for long distance transports on the order of 50 cm. During transport, we require
the maximum radial confining force *F*_{max} to be larger than
gravity *mg*, where *m* is the atomic mass and *g*
9.81 m s^{ - 2} is the acceleration due to gravity. For a Gaussian beam
this is

where Γ is the natural linewidth of the relevant atomic transition, Δ the
detuning from this transition, *w*(*z*) the beam radius and *P*_{0}
the total power of the beam. The strong dependence on the beam radius *w*(*z*)
suggests, that
should not vary too much over the transport distance. If we thus require the
Rayleigh range *z*_{R} = π *w*_{0}^{2}/λ to
equal the distance of 25 cm, the waist has to be *w*_{0}
260 μm. For a lattice beam wavelength of e.g. λ = 830 nm, the detuning from
the D-lines of ^{87}Rb is Δ
2π× 130 THz. To hold the atoms against gravity for all *z*, where |*z*| < *z*_{R},
a total laser power of *P*_{0}
10 W is needed, which is difficult to produce. In addition, the spontaneous
photon scattering rate

would reach values on the order of Γ_{scatt} = 2 s^{ - 1}.
For typical transport times of 200 ms, this means substantial heating and atomic
losses.

A better choice for transport are zero order Bessel beams (figure 2).
They exhibit an intensity pattern which consists of an inner intensity spot
surrounded by concentric rings and which does *not* change during
propagation. In our experiments, we have formed a standing light wave by
interfering a Bessel beam with a counterpropagating Gaussian beam, giving rise
to an optical lattice which is radially modulated according to the Bessel beam.^{Note1}
Atoms loaded into the tightly confined inner spot of the Bessel beam can be held
against gravity for moderate light intensities, which minimizes the spontaneous
photon scattering rate. In comparison to the transport with a Gaussian beam, the
scattering rate in a Bessel beam transport can be kept as low as 0.05 s^{ - 1}
by using the beam parameters of our experiment.

Figure 2. Gaussian and Bessel beams. (a)
The radial intensity distribution of a Gaussian beam changes as it
propagates. The smaller the waist w_{0} of the beam, the
higher its divergence (for a given wavelength). (b) Bessel beam: the radial
distribution and in particular the radius of the central spot r_{0}
do not change with z (see equation (8)).
(c) Within a certain axial range z_{max} a Bessel-like beam
can be produced by illuminating an axicon lens with a collimated laser beam. |

**3. Bessel beams**

Bessel beams are a solution of the Helmholtz equation and were first discussed and experimentally investigated about two decades ago [15, 16].

In cylindrical coordinates, the electric field distribution of a Bessel beam
of order *l* is given by

where *J*_{l}(α *r*) is the Bessel function of the
first kind with integer order *l*. The beam is characterized by the
parameters α and β. In the following, we restrict the discussion to order *l*
= 0 which we have used in the experiment. By taking the absolute square of
this expression, one gets the intensity distribution given by

where α determines the radius *r*_{0} of the central spot via
the first zero crossing of *J*_{0}(α *r*)

As pointed out before, *r*_{0} and *I*_{0} do not
change with the axial position *z*. Because of this axial independence, the
Bessel beams are said to be `diffraction-free'.

Bessel-like beams were realized experimentally for the first time by illuminating a circular slit [16]. Since this method is very inefficient, two other ways are common now-a-days. To generate Bessel beams of arbitrary order, holographic elements, such as phase-gratings, are used [17]. In our setup, we use a zero order Bessel beam, which can be produced efficiently by simply illuminating an axicon (conical lens) with a collimated laser beam [18]. How this comes about can be understood by looking at the Fourier transform of the Bessel field

Thus a Bessel beam is a superposition of plane waves with (*k*_{},*k*_{z})
= (α,β). The **k**-vectors of the plane waves all have the same magnitude
and they are forming a cone with radius *k*_{}
and height *k*_{z}. Using an axicon with apex angle δ and
index of refraction *n*, α and β are given by

and

These experimentally produced Bessel beams are not ideal in the sense that
their range *z*_{max} = *kw*_{0}^{in}/α is
limited by the finite size (waist *w*_{0}^{in}) of the beam
impinging on the axicon lens (see figure 2(c)).
Also, the intensity of the Bessel beams might not be independent of the axial
coordinate *z*, as it is also determined by the radial intensity
distribution of the impinging beam (e.g. see figure 4(b)).

**4. Experimental setup**

We work with a ^{87}Rb-BEC in the internal state |*F* = 1, *
m*_{F} = - 1,
initially held in a Ioffe-type magnetic trap with trap frequencies of 2πν_{x,y,z}
= 2π (7, 19 and 20 Hz) [19,
20]. From the magnetic trap, the condensate is
adiabatically loaded in about 100 ms into the inner core of the 1D optical
lattice formed by a Bessel beam of central spot radius *r*_{0} =
36 μm and a counterpropagating Gaussian beam with a waist of *w*_{0}
= 85 μm. About 70 lattice sites are occupied with atoms in the vibrational
ground state. The lattice periodicity is 415 nm, corresponding to the laser
wavelength of 830 nm. For our geometry (see below) the total power needed for
the Bessel beam to support the atoms against gravity is typically 200 mW, since
only a few per cent (
10 mW) of the total power are stored in the central spot. For the Gaussian beam,
a power of roughly 20 mW is chosen, leading to an optical trapping potential at
the centre (*r* = 0) of *U*(*z*) = - *U*_{0} + *U*_{latt}sin^{2}(*kz*),
where the lattice depth (effective axial trap depth) is *U*_{latt}
10*E*_{r} and the total trap depth *U*_{0}
11*E*_{r}. Here, *E*_{r} = (
*k*)^{2}/(2*m*) is the recoil energy.

The corresponding trap frequencies are
in the radial direction and
in the axial direction. In order to better analyse the transport properties, we
mostly perform round trips, where the atoms are first moved to a distance *D*
and then back to their initial spot, which lies in the field of vision of our
CCD camera. Once back, the atoms are adiabatically reloaded into the Ioffe-type
magnetic trap. To obtain the resulting atomic momentum distribution, a standard
absorption imaging picture is taken after sudden release from the magnetic trap
and typically 12 ms of time-of-flight.

The lattice beams for the optical lattice are derived from a Ti:Sapphire-laser
operating at 830 nm. The light is split into two beams, each of which is
controlled in amplitude, phase and frequency with an acousto-optical modulator (AOM).
For both AOMs, the radio-frequency (RF) driver consists of a home-built 300 MHz
programmable frequency generator, which gives us full control over amplitude,
frequency and phase of the radio-wave at any instant of time. The frequency
generator is based on an AD9854 digital synthesizer chip from Analogue Devices
and a 8-bit micro-controller ATmega162 from Atmel, on which the desired
frequency ramps are stored and from which they are sent to the AD9854 upon
request. After passing the AOMs, the two laser beams are mode-cleaned in
single-mode fibres and converted into collimated Gaussian beams. **
One of the Gaussian beams passes the axicon lens (apex
angle = 178°, radius = 25.4 mm, Del Mar Photonics)** with a waist
of *w*_{0}^{in} = 2 mm, producing the Bessel beam. From
there the beam propagates towards the condensate, which - before transport - is
located 5 cm away.

**5. Transport of ultracold atoms**

Figure 3 shows results of a first experiment,
where we have transported atoms over short distances of up to 1 mm (round trip),
so that they never leave the field of view of the camera. The atoms move
perpendicularly to the direction of observation. *In situ* images of the
atomic cloud in the optical lattice are taken at various times during transport
and the centre of mass position of the cloud is determined. As is clear from
figure 3(a), we find very good agreement between the
expected and the measured position of the atoms. In figures 3(b)
and (c), calculations are shown for the corresponding velocity *v*(*t*)
and acceleration *a*(*t*) of the optical lattice, respectively. As
discussed before (see equation (1)), the velocity *
v* of the lattice translates directly into a relative detuning Δ ν of the
laser beam, which we control via the AOMs. In order to suppress unwanted heating
and losses of atoms during transport, we have chosen very smooth frequency ramps
Δν(*t*) such that the acceleration is described by a cubic spline
interpolation curve which is continuously differentiable (details are given in
the appendix). In this way, also the derivative of the acceleration (commonly
called the jerk) is kept small.

Figure 3. (a) Position, (b) velocity and
(c) acceleration of the atomic cloud as a function of time for a typical
transport sequence, here a round-trip over a short distance of 1 mm.
Piecewise defined cubic polynomials are used for the acceleration ramp (see
appendix for an analytical expression). By integrating over time, velocity
and position are obtained. The frequency detuning Δν, which is used to
program the RF synthesizers, corresponds directly to the velocity v
via equation (1). The position ramp is compared
with in situ measurements of the cloud's position (). |

In the next set of experiments, we extended the atomic transport to more
macroscopic distances of up to 20 cm (40 cm round trip), where we moved the
atoms basically from one end of the vacuum chamber to the other and back.
However, the transport distance was always limited by the finite range *z*_{max}
of the Bessel beam (see figures 2(c) and 4).
As shown in figure 4, the total number of atoms
abruptly decreases at the axial position, where the maximum radial force drops
below gravity. It is also clear from the figure how the range of the Bessel beam
is increased by enlarging the waist *w*_{0}^{in} of the
incoming Gaussian beam. Of course, for a given total laser power, the maximum
radial force decreases as the Bessel beam diameter is increased. For the
transport distances of 12 and 20 cm, the total power in the Bessel beam was
approximately 400 mW. For comparison, we have also transported atoms with a
lattice formed by two counterpropagating Gaussian beams (see figure 4(a)).
For this transport, both laser beams have a Rayleigh range of *z*_{R}
2 cm corresponding to a waist of 70 μm. The laser power of the two beams was
130 and
35 mW, respectively. We observe a sudden drop in atom number when the transport
distance exceeds the Rayleigh range. Using the scaling law given in equation (2),
it should be clear that transports of atoms over tens of centimetres with a
Gaussian lattice is hard to achieve.

Figure 4. Long distance transports. (a)
Shown is the number of remaining atoms after a round-trip transport (see
figure 3) over various one-way distances D.
The first two data sets are obtained with two different Bessel beams which
are created by illuminating an axicon with a Gaussian beam with a waist w_{0}^{in}
= 1 and 2 mm, respectively. The transport time T was kept constant
at T = 130 ms and T = 280 ms, respectively. The third data
set (◊) corresponds to a transport in a Gaussian beam lattice (see text).
The calculated maximum radial trapping force of the two Bessel beam lattice
traps is shown in (b) in units of mg, where g
9.81 m s^{ - 2} denotes the gravitational acceleration. The
variation of the trapping force with distance is an imperfection of the
Bessel beam and reflects its creation from a Gaussian beam. When the maximum
radial force drops below 1 g, gravity pulls the atoms out of the
trap, as can be clearly seen in (a). |

Interestingly, the curve corresponding to the Bessel beam with waist *w*_{0}^{in}
= 1 mm in figure 4(a) exhibits a pronounced
minimum in the number of remaining atoms at a distance of about 3 cm. The
position of this minimum coincides with the position, where the lattice depth
has a maximum (see figure 4(b)). This clearly
indicates, that high light intensities adversely affect atom lifetimes in the
lattice. Although we have not studied in detail the origin of the atomic losses
in this work, they should partially originate from spontaneous photon scattering
and three body recombination. In the deep lattice here (60*E*_{r}),
the calculated photon scattering rate is Γ_{scatt} = 0.4 s^{ - 1}.
The tight lattice confinement leads to a high calculated atomic density of *n*_{0}
2 × 10^{14} cm^{ - 3}. Adopting *L* = 5.8× 10^{ - 30} cm^{6} s^{ - 1}
as rate coefficient for the three body recombination [21],
we expect a corresponding loss rate *Ln*_{0}^{2} = 0.3 s^{ - 1}.

In figure 5, we have studied the transport of a
BEC, which is especially sensitive to heating and instabilities. It is important
to determine, whether the atoms are still Bose-condensed after the transport and
what their temperature is afterwards. Figure 5 shows
momentum-distributions for various transport distances *D*, which were
obtained after adiabatically reloading the atoms into the magnetic trap by
ramping down the lattice and subsequent time-of-flight measurements.

Figure 5. Transporting BEC. Shown are the
momentum distributions (thin black lines) of the atoms after a return-trip
transport over various one-way distances D. A bimodal distribution (a
blue parabolic distribution for the condensed fraction and a red Gaussian
distribution for the thermal fraction) is fit to the data. For D
below 10 cm, a significant fraction of the atomic cloud is still condensed.
For D = 18 cm, (
the limit in our experiments) only a thermal cloud remains, however, with a
temperature below the recoil limit (T < 0.2E_{r}/k_{B}
30 nK). |

Before discussing these results, we point out that loading the BEC
adiabatically into the stationary optical lattice is already critical. We
observe a strong dependence of the condensate fraction on the lattice depth. For
too low lattice depths, most atoms fall out of the lattice trap due to the
gravitational field. For too high lattice depths, all atoms are trapped but the
condensate fraction is very small. One explanation for this is that high lattice
depths lead to the regime of 2D pancake shaped condensates where tunnelling
between adjacent lattice sites is suppressed. Relative dephasing of the pancake
shaped condensates will then reduce the condensate fraction after release from
the lattice. We obtain the best loading results for a 11*E*_{r}
deep trap, where we lose about 65% of the atoms, but maximize the condensate
fraction. Because high lattice intensities are detrimental for the BEC, we
readjust the power of the lattice during transport, such that the intensity is
kept constant over the transport range. The adjustments are based on the
calculated axial intensity distribution of the Bessel beam. In this way, we
reach transport distances for BEC of 10 cm. We believe, that more sophisticated
fine tuning of the power adjustments should increase the transport length
considerably. After transport distances of *D* = 18 cm (36 cm round
trip), the atomic cloud is thermal. Its momentum spread, however, is merely 0.3
*k*, which corresponds to a temperature of 30 nK. Additionally, we want to
point out that the loss of atoms due to the transport is negligible ( < 10%)
compared to the loss through loading and simply holding in such a low lattice
potential (
65%).

An outstanding feature of the lattice transport scheme is the precise
positioning of the atomic cloud. Aside from uncontrolled phase shifts due to
residual mechanical noise, such as vibrating optical components, we have perfect
control over the relative phase of the lattice lasers with our RF/AOM setup.
This would in principle result in an arbitrary accuracy in positioning the
optical lattice. We have experimentally investigated the positioning
capabilities in our setup. For this, we measured in many runs the position of
the atomic cloud in the lattice after it had undergone a return trip with a
transport distance of *D* = 10 cm. The position jitter, i.e. the standard
deviation from the mean position, was slightly below 1 μm. For comparison, we
obtain very similar values for the position jitter when investigating BECs in
the lattice before transport. Hence, the position jitter introduced through the
transport scheme is negligible.

Another important property of the lattice transport scheme is its high speed. For example, for a transport over 20 cm (40 cm round trip) with negligible loss, a total transport time of 200 ms turns out to be sufficient. This is more than an order of magnitude faster than in the MIT experiment [4], where an optical tweezer was mechanically relocated. The reason for this speed up as compared to the optical tweezer is mainly the much higher axial trapping frequency of the lattice and the non-mechanical setup.

In order to determine experimentally the lower limit of transportation time,
we have investigated round-trip transports (*D* = 5 mm), where we have
varied the maximum acceleration and the lattice depth (figure 6(a)).
The number of atoms, which still remain in the lattice after transport, is
measured. As soon as the maximum acceleration exceeds a critical value, the
number of atoms starts to drop. For a given lattice depth, we define a critical
acceleration *a*_{crit} as the maximum acceleration of the
particular transport where 50% of the atoms still reach their final destination.
Figure 6(b) shows the critical acceleration *a*_{crit}
as a function of lattice depth. The upper bound on acceleration observed here
can be understood from classical considerations. In our lattice, the maximum
confining force along the axial direction is given by *U*_{latt}*k*,
where *k* is the wave vector of the light field. Thus in order to keep an
atom bound to the lattice, we require the acceleration *a* to be small
enough such that

Our data in figure 6 are in good agreement with
this limit.^{Note2}

Figure 6. Critical acceleration in
lattice. (a) For several round-trip transports with varying maximum
acceleration a and lattice depth (see legend), the number of
remaining atoms after transport is shown. As the maximum acceleration
exceeds a critical value, the number of atoms starts to drop significantly.
We define a critical acceleration as the maximum acceleration for transports
in which 50% of the atoms still reach their final destination. This critical
acceleration is shown as a function of the lattice depth in (b). The
experimentally determined values are compared with the limit expected from
classical considerations: a_{crit} = U_{latt}k/m. |

There is in principle also a lower bound on the acceleration, which is due to
instabilities exhibited by BECs with repulsive interactions loaded into periodic
potentials [23]-[26].
Due to the fact that these instabilities mainly occur at the edge of the
Brillouin zones, the time spent in this critical momentum range should be kept
small. For our lattice parameters, nearly half of the Brillouin zone is an
unstable region, where the lifetime of the BEC is only on the order of 10 ms [25].
Thus we tend to sweep through the Brillouin zone in much less than Δ *t*
= 20 ms, which corresponds to an acceleration of
.
In this way, BECs may be transported without introducing too much heating
through these instabilities.

In contrast to acceleration, the transport velocity in our experiment is only
technically limited due to the finite AOM bandwidth. As discussed before, the
lattice is set in motion by introducing a detuning between the two beams via AOMs
(equation (1)). For detunings exceeding the
bandwidth of the AOM, the diffraction efficiency of the modulator starts to drop
significantly. Consequently the lattice confinement vanishes, and the atoms are
lost. In our setup, we can conveniently reach velocities of up to *v* =
6 m s^{ - 1}
1100*v*_{r}, corresponding to a typical AOM bandwidth of 15 MHz.
This upper bound actually limits the transport time for long distance transports
(*D* > 5 cm).

Finally we have investigated the importance of phase stability of the optical lattice for the transport (see figure 7). For this, we purposely introduced sudden phase jumps during transport to one of the lattice beams. The timescale for the phase jumps, as given by AOM response time of about 100 ns, was much smaller than the inverse trapping frequencies. The phase jumps lead to abrupt displacements of the optical lattice, causing heating and loss of atoms. In figure 7(a), the atomic losses due to a single phase jump during transport are shown. Phase jumps of 60° typically induce a 50% loss of atoms. For continuous phase jitter (see figure 7(b)), the sensitivity is much larger.

Figure 7. Stability requirements for
transport. Sudden phase jumps are introduced in the relative phase of the
two counterpropagating lattice laser beams. The corresponding abrupt
displacements of the optical lattice lead to heating and loss of the atoms.
We measure the number of atoms which remain in the lattice after transport.
(a) Data obtained after a single relative phase jump of variable magnitude.
(b) A phase jitter (200 positive Poissonian-distributed phase jumps with a
variable mean value) is introduced during transport. Mean values on the
order of a few degrees already lead to a severe loss of atoms. |

**6. Atom catapult**

In addition to transport of ultracold atoms, acceleration of atoms to
precisely defined velocities is another interesting application of the moving
optical lattice. For instance, it could be used to study collisions of BECs with
a very high but well-defined relative velocity, similar to the experiments
described in [27, 28].
As already shown above, we have precise control to impart a well-defined number
of up to 1100 photon recoils to the atoms. This corresponds to a large kinetic
energy of *k*_{B}× 200 mK. At the same time, the momentum spread of
the atoms is about 1/3 of a recoil (see figure 5).
To illustrate this, we have performed two sets of experiments, where we
accelerate a cloud of atoms to velocities *v* = 10*v*_{r}
and *v* = 290*v*_{r}
1.6 m s^{ - 1}. After adiabatic release from the lattice, we track their
position in free flight (see figure 8). Initially
the atomic cloud is placed about 8 cm away from the position of the magnetic
trap. It is then accelerated back towards its original location. Before the
atoms pass the camera's field of vision, the lattice beams are turned off within
about 5 ms, to allow a ballistic flight of the cloud. Using absorption imaging,
the position of the atomic cloud as a function of time is determined. The slope
of the straight lines in figure 8(a) corresponds
nicely to the expected velocity. However, due to a time jitter problem,
individual measurements are somewhat less precise than one would expect.^{Note3}
For *v* = 10*v*_{r}, figure 8(b)
shows the trajectory of the ballistic free fall of the atoms in gravity.

Figure 8. Atom catapult. After
acceleration in x-direction and subsequent release from the lattice,
the position of the atomic cloud is tracked as it flies ballistically
through the field of view of the CCD camera. Shown are two data sets where
atoms were accelerated to velocities of either v_{x}
= 10v_{r} or v_{x} = 290v_{r}.
(a) The horizontal position x as a function of time. (b) For the
slower cloud (v_{x} = 10v_{r}) a
parabolic trajectory y = - g/2· (x/v_{x})^{2}
is observed as it falls under the influence of gravity. |

**7. Conclusion**

In conclusion, we have realized a long distance optical transport for ultracold atoms, using a moveable standing wave dipole trap. With the help of a diffraction-free Bessel beam, macroscopic distances are covered for both BEC and ultracold thermal clouds. The lattice transport features a fairly simple setup, as well as a fast transport speed and high positional accuracy. Limitations are mainly technical and leave large room for improvement. In addition to transport, the lattice can also be used as an accelerator to impart a large but well-defined number of photon recoils to the atoms.

**Acknowledgments**

We thank U Schwarz for helpful information on the generation of Bessel beams and for lending us phase-gratings for testing purposes. Furthermore, we thank R Grimm for discussions and support. This work was supported by the Austrian Science fund (FWF) within SFB 15 (project part 17) and the Tiroler Zukunftsstiftung.

**Appendix. Transport ramp**

We give here the analytic expression for the lattice acceleration *a*(*t*)
as a function of time *t* which was implemented in our experiments (see for
example figure 3). *a*(*t*) is a smooth
piecewise defined cubic polynomial,

Here, *D* is the distance over which the lattice is moved and *T*
is the duration of the transport. From *a*(*t*), both the velocity *
v*(*t*) and the location *x*(*t*) may be derived via
integration over time. Our choice for the acceleration *a*(*t*)
features a very smooth transport. The acceleration *a*(*t*) and its
derivative
are zero at the beginning (*t* = 0) and at the end (*t* = *T*)
of the transport. At *t* = *T*/4 and *t* = 3*T*/4, the
absolute value of the acceleration reaches a maximum.

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**Notes**

Note1 In principle, one could also use a pure Bessel lattice (produced by two counterpropagating Bessel beams) for transport. This would improve radial confinement, however, alignment is more involved.

Note2 In the weak lattice
regime (*U*_{latt}
10*E*_{r}) transport losses would be dominated by Landau-Zener
tunnelling, see e.g. [14,
22].

Note3 This is linked to the fact that our clock for the system control is synchronized to the 50 Hz of the power grid. Fluctuations of the line frequency lead to shot to shot variations in the ballistic flight time of the atoms, which translates into an apparent position jitter.

Del Mar Photonics - Axicons in stock

**Axicons in stock**

AX-FS-1-175-2 | Axicon, UV FS, diam. 1", cone angle 175°, BBAR 700-1000 nm | $350.00 | |

AX-FS-1-175-3 | Axicon, UV FS, diam. 1", cone angle 175°, BBAR 800&1064 nm | $350.00 | |

AX-FS-1-175-0 | Axicon, UV FS, diam. 1", cone angle 175°, uncoated | $290.00 | |

AX-FS-1-178-4 | Axicon, UV FS, diam. 1", cone angle 178°, BBAR 1100-1600 nm | $350.00 | |

AX-FS-1-178-1 | Axicon, UV FS, diam. 1", cone angle 178°, BBAR 400-700 nm | $350.00 | |

AX-FS-1-178-2 | Axicon, UV FS, diam. 1", cone angle 178°, BBAR 700-1000 nm | $350.00 | |

AX-FS-1-178-3 | Axicon, UV FS, diam. 1", cone angle 178°, BBAR 800&1064 nm | $350.00 | |

AX-FS-1-179-4 | Axicon, UV FS, diam. 1", cone angle 179°, BBAR 1100-1600 nm | $350.00 | |

AX-FS-1-179-2 | Axicon, UV FS, diam. 1", cone angle 179°, BBAR 700-1000 nm | $350.00 | |

AX-FS-1-179-3 | Axicon, UV FS, diam. 1", cone angle 179°, BBAR 800&1064 nm | $350.00 | |

AX-FS-1-179-0 | Axicon, UV FS, diam. 1", cone angle 179°, uncoated | $290.00 | |

AX-FS-2-140-2 | Axicon, UV FS, diam. 2", cone angle 140°, BBAR 700-1000 nm | $700.00 | |

AX-FS-2-160-C | Axicon, UV FS, diam. 2", cone angle 160°, BBAR 800-1000 nm | $700.00 |