# Generation of optical potentials for ultracold atoms using a
superluminescent diode

Aaron Smith  School of Physics and Astronomy, University of Birmingham,
Edgbaston, Birmingham, B15 2TT, UK  Thomas Easton  School of Physics and
Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK  Vera
Guarrera  School of Physics and Astronomy, University of Birmingham,
Edgbaston, Birmingham, B15 2TT, UK  Giovanni Barontini<EMAIL_ADDRESS>School of Physics and Astronomy, University of Birmingham, Edgbaston,
Birmingham, B15 2TT, UK

###### Abstract

We report on the realization and characterisation of optical potentials for
ultracold atoms using a superluminescent diode. The light emitted by this
class of diodes is characterised by high spatial coherence but low temporal
coherence. On the one hand, this implies that it follows Gaussian propagation
similar to lasers, allowing for high intensities and well-collimated beams. On
the other, it significantly reduces those interference effects that lead to
severe distortions in imaging. By using a high-resolution optical setup, we
produce patterned optical potentials with a digital micromirror device and
demonstrate that the quality of the patterns produced by our superluminescent
diode is consistently and substantially higher than those produced by our
laser. We show that the resulting optical potentials can be used to arrange
the atoms in arbitrary structures and manipulate them dynamically. Our results
can open new opportunities in the fields of quantum simulations and
atomtronics.

##  I Introduction

Quantum simulations with cold atoms is one of the most productive areas in
quantum technology, and has provided us with important insight in condensed
matter physics [1, 2], the physics of disordered systems including the
observation of Anderson localisation [3, 4], low dimensional systems [5], the
use of synthetic fields [6] and dimensions [7], high energy physics including
Higgs modes [8] and black holes [9], topological effects [10] and many more.
The main feature of quantum simulators based on cold atoms is the exquisite
control that is nowadays possible to achieve on these systems, either by
engineering the atomic internal states or by producing potentials to alter the
shape, position, temperature and momentum of the samples [11]. This has been
largely enabled by the rapid development of laser technology, that provides
light with the ideal combination of high spatial and temporal coherence, high
intensity and narrow line width. Such a combination allows one to precisely
address atomic transitions with one or more photons, implement atoms traps
with a large variety of shapes, produce optical lattices and dynamical
potentials, and even address and manipulate single atoms [12, 13].

Lately, laser light combined with spatial light modulators is also emerging as
a new avenue for the production of arbitrary and even time varying potentials
[14, 15, 16, 17], including non-trivial phase engineering of Bose-Einstein
condensates [18] and applications in atomtronics [19, 20]. While the coherence
of laser light is a key feature for the vast majority of the applications
discussed, it could represent a hindrance in the generation of complex
patterns using digital micromirror devices (DMDs). In these kinds of systems,
the pattern produced on the DMD is imaged on the atoms and interference
effects caused by the large temporal coherence can cause aberrations in the
image. To mitigate this, one could use very complex and expensive optical
systems or implement a feedback loop [21], modifying the pattern in real time
on the DMD so as to minimise the difference between the projected image and
the desired one.

The methods described above attempt to mitigate the detrimental effects of
temporal coherence. An alternative approach is to eradicate the problem by
using light sources with reduced coherence properties. Since high spectral
specifications are not required to produce optical potentials, as long as the
whole spectrum of the source is far detuned, one could choose to use
incoherent light sources such as LEDs to obtain pristine patterns. The main
issue in this case is that the absence of spatial coherence makes it very
difficult to reach intensities that allow one to produce sufficiently high
optical potentials to manipulate the atomic samples. In this work, we solve
this problem and produce optical potentials with significantly reduced
aberrations by using super luminescent diodes (SLDs) in combination with a
DMD. A SLD is similar to a standard laser diode, except that it strongly
suppresses feedback through a combination of a tilted waveguide and anti-
reflection coatings applied to the facets. This lack of feedback means that
lasing is not achieved and instead the diode performs amplified spontaneous
emission, with a spectrum that is not narrowed by the effects of a cavity.
SLDs are therefore temporally incoherent[22, 23] but feature high spatial
coherence [24]. We show that the low temporal coherence of SLDs allows us to
produce images with significantly reduced interference effects compared to a
laser. At the same time, due to its high spatial coherence, the device is
capable of producing sufficiently high intensities in a well behaved Gaussian
beam shape, allowing us to manipulate ultracold atomic clouds with arbitrary
patterns.

Figure 1: a) Schematic representation of our optical system. The SLD light is
first amplified by a TA and then injected into an optical fibre to clean the
mode. An AOM is used to control the power injected in the fibre. The output of
the fibre is reflected by a DMD that imprints arbitrary patterns on the
intensity profile. The patterned intensity profile is then imaged on the atoms
using a 100$\times$ demagnification system that includes a 50$\times$
objective. The plane where the image is produced, that corresponds to the
plane where the atoms are trapped, is then imaged on a CCD camera using a
20$\times$ magnification system. We can use the same path for the laser source
by activating a flip mirror. b) Emission spectrum of the SLD (blue) and
spectrum after the SLD emission has been amplified by a TA (black). The
spectra are normalised to have the same area. c) Image produced by the laser
at the position of the atoms with the DMD fully _on_. d) Same as c) but with a
2$\times$ zoom on a pattern containing an irregular mesh, a star and an arrow.
Both the bars in c) and d) correspond to 50 $\mu$m. e) and f), same as c) and
d) but using the SLD as light source.

The paper is organised as follows: in Sec. II we describe our experimental
system and discuss the characteristics of our SLD; in Sec. III we show that
the SLD allows us to produce images with higher quality than with a laser, in
Sec. IV we demonstrate the use of the potential created by the DMD and SLD to
arrange ultracold atoms in arbitrary 1d potentials, that we also modify
dynamically. In Sec. V we report our conclusions.

##  II The system

Details about our experimental sequence and methods can be found in [18, 25].
In brief, we load 87Rb atoms from a 3d MOT into a crossed dipole trap and then
perform evaporative cooling for 6 seconds to produce a pure Bose-Einstein
condensate (BEC). For this work the final trapping frequencies are $f_{r}=150$
Hz and $f_{x}=10$ Hz, resulting in an elongated BEC with $N=2\times 10^{4}$
atoms at $\simeq$20 nK.

Figure 2: a) Image produced on the plane of the atoms using the laser source
and a square-wave pattern on the DMD with period 1512 $\mu$m, corresponding to
a period of 15.12 $\mu$m in the imaging plane. b) The same as a) but using the
amplified SLD as the source. c) And d) are the residuals obtained by fitting
with a square-wave grating the regions within the red rectangles in a) and b)
respectively, which correspond to the beams peak intensity regions. The colour
bar indicates the values of the residuals. The green curves in e) and f)
correspond to 1d cuts along the green lines in a) and b) respectively. The
magenta curves are the fits to the data using eq. (1).

To manipulate the atomic sample, we use the optical setup of Fig. 1a. Our SLD
light source is a Thorlabs SLD830s-A20, emitting 23 mW over a spectrum with
full-width at half maximum of $\Delta\lambda=$20.3 nm, centred at
$\lambda_{c}=$836 nm, as shown in Fig. 1b. From the spectrum we estimate the
coherence length of the SLD to be $l=2\ln
2\lambda_{c}^{2}/n\pi\Delta\lambda\simeq 15$ $\mu$m. The light produced by the
SLD is then amplified to 300 mW using a tapered amplifier (TA) with gain
centered at 840 nm (Eagleyard EYP-TPA-0830). The spectrum of the amplified
light is a convolution between the gain curve of the TA and the SLD spectrum.
This is shown in Fig. 1b, where it is possible to appreciate that after the
amplification the spectrum is centered at $\lambda_{c}^{\prime}=$840 nm and
its full-width at half maximum is $\Delta\lambda^{\prime}=$9.4 nm, therefore
smaller than that of the bare SLD. This results in a doubly longer coherence
length of $l^{\prime}\simeq$ 33 $\mu$m. Care must be taken to provide 30 dB of
isolation between the TA and the SLD to prevent damage to the latter with the
TA back emission. The amplified light passes through an acousto-optic
modulator that allows us to control its power and then is injected into a
single mode optical fibre. A flip mirror allows us to use the same acousto-
optic modulator and fibre with a laser tuned at 800 nm, enabling a direct
comparison between the two light sources 111We do not observe any difference
in the acousto-optic modulator efficiency between the two light sources.. The
light coming out from the optical fibre is reflected by a DMD and then sent
onto the atoms along the vertical direction, demagnified by a factor 100 using
an optical setup that includes a 50$\times$ high-resolution optical microscope
objective. The DMD has $1920\times 1080$ micromirrors of size 10.8 $\mu$m.
Each micromirror can be individually set _on_ or _off_ every 100 $\mu$s. After
reflection by the DMD and transmission through the optics, we typically obtain
1 mW of power at the position of the atoms 222We note that the reflection from
our DMD is only $\simeq$10%, although we expected $\simeq$50 % reflectivity
according to the specifications of the device..

Figure 3: a) Fidelity (circles) and contrast (squares) of the SLD images (open
symbols) and laser images (filled symbols) as a function of the semi-
periodicity of the square wave $L$. b) Quality of the image as a function of
$L$ for the SLD images (open symbols) and the laser images (filled symbols).

The images produced by our system at the position of the atoms are imaged on a
CCD mounted in the vertical direction, using a 20$\times$ magnifying system,
resulting in a resolution of $\simeq 2$ $\mu$m. The same imaging system is
used to perform high-resolution absorption imaging of the atoms. In Fig 1c-e
we show two examples of the images obtained using each of the two light
sources. The SLD-generated images presents sensibly reduced interference
fringes, are overall more uniform and feature finer details.

##  III Image characterisation

In this section we quantitatively compare the images produced using the SLD
and the laser. For simplicity and clarity, we constrain our analysis to one-
dimensional structures, as our findings can be easily extended to two
dimensions. In particular, we investigate the ability of the two sources to
produce square-wave gratings with our optical system, as in the examples in
Fig. 2a and b. In order to quantitatively compare the images produced by the
two sources, we use two parameters: the _fidelity_ and the _contrast_. We have
chosen to separately evaluate these two parameters to highlight the superior
ability of SLDs to produce square-wave patterns, regardless of their amplitude
or residual background. To compute the fidelity, we perform two-dimensional
fits of the images produced on the CCD $I$ with the “target image”
$I_{0}(i,j)=A+B\vartheta[\cos(\pi id/L+\phi)]$, i.e., a perfect square-wave
grating, with $\\{i,j\\}$ the pixel coordinates in the two directions, $d$ the
pixel size and $L$ the semi-period of the square-wave. $A,B$ and $\phi$ are
free fitting parameters. The fidelity is then defined as:

$F=1-\frac{1}{BN}\sqrt{\sum_{i,j}(I(i,j)-I_{0}(i,j))^{2}}$ (1)

where $N$ is the number of CCD pixels in the region of interest. As an
example, in Fig. 2c and d we report the residuals $I(i,j)-I_{0}(i,j)$ obtained
by fitting the outlined regions in Fig. 2a and b with the same square-wave
function. We consistently observe that for SLD-generated images the values of
the residuals are lower all across the images, and also that the images have
fewer aberrations. This is also evident in Fig. 2e and f, where we show the 1d
cuts across the green lines in Fig. 2a and b together with the fits done using
eq. (1). To evaluate the contrast, we integrate the images along the $y$
direction obtaining one-dimensional arrays $\tilde{I}$ and then compute

$C=\frac{\text{max}(\tilde{I})-\text{min}(\tilde{I})}{\text{max}(\tilde{I})}.$
(2)

The fidelity quantitatively estimates how close the image produced is to the
target image, while the contrast accounts for the ability of a certain source
to effectively produce dark regions, indeed it varies linearly with
$\text{min}(\tilde{I})$. For the implementation of accurate optical potentials
for cold atoms we require both these quantities to be as close to 1 as
possible. Indeed, low values of $F$ indicate aberrations in the optical
potential and therefore a limited capacity to accurately manipulate the atoms.
Low values of $C$ imply instead that higher intensities need to be used to
reach the desired potential depth, causing an increase in the scattering rate
that makes the potential increasingly less conservative. To account for both
these effects we use the _quality of the image_ $Q=F\times C$ as our figure of
merit. An image with $Q=1$ would perfectly produce the desired optical
potential.

In Fig. 3 we report the behaviour of $F$, $C$ and $Q$ as a function of $L$.
With $L$ we indicate the semi-periodicity in the imaging plane, therefore $L$
is a factor 100 smaller than the semi-periodicity of the pattern on the DMD.
The SLD images are characterised by a slightly lower contrast than the laser
images, especially for low values of $L$. Our measurements therefore confirm
that temporal coherence plays a minor role in the contrast of an image [28],
but also suggest that as the size of the features is reduced, such role
becomes increasingly important. Notably, SLD images consistently feature a
substantially higher fidelity, as the reduced temporal coherence of the SLD
almost completely eliminates interference effects. The main source of loss of
fidelity for the SLD images is the underlying Gaussian distribution of the
intensity, as can be seen for example in Fig. 1e and 2b,d. The overall effect
is that the SLD generates images with values of $Q$ higher by at least 0.1
than those generated by the laser over the whole range explored. Our imaging
system does not allow us to perform quantitative measurements for $L<5$
$\mu$m. To provide a qualitative estimate below the limits of our imaging
system, we fit $C$ and $F$ with second order polynomial functions. By doing
this we infer that the SLD images should feature a higher $Q$ down to
sub-$\mu$m structures.

Figure 4: Quality of the image generated by the SLD (open symbols) and the
laser (filled symbols) for $L$=7.56 $\mu$m a function of the distance from the
focal plane $D$. For SLD-generated images $Q$ is a smooth function of $D$,
while for laser-generated images $Q$ is modulated by the Talbot effect, that
every $\simeq$80$\mu$m generates images with semi-periodicity $L/2$, as shown
in the inset.

Another important feature stemming from the partial incoherence of SLD-
generated light is that it is free from the Talbot effect, that alters the
periodicity of regular structures as the beam propagates. This can be observed
in Fig. 4, where we report the quality of the image $Q$ as a function of the
distance from the focal plane $D$ for $L=7.56$ $\mu$m. We observe that for
SLD-generated images $Q$ is a smooth function of $D$, while for laser-
generated images $Q$ presents some local maxima and minima. The minima are due
to the Talbot effect, that at integer fractions of the Talbot length
$\Lambda=2(2L)^{2}/\lambda$ causes the appearance of images with $L/2$ semi-
periodicity. This in turn causes a degradation both in $F$ and $C$. SLD-
generated images feature also higher values of $Q$ over a broad range of $D$.
The fact that the images produced by the SLD are free from the Talbot effect
can be relevant for the implementation of high-quality 3d-patterns, lattices
and holograms using two perpendicular beams that carry periodic or near-
periodic structures.

##  IV SLD-generated optical potentials for cold atoms

Besides the low temporal coherence that enables the generation of optical
potentials with higher quality than lasers, SLDs feature high spatial
coherence, that in turn implies Gaussian propagation. On the one hand this
represents a problem in reaching very high values of $Q$, as demonstrated in
the previous section. On the other, this is a crucial feature to reach
sufficiently high intensities and generate optical potentials that can be used
to manipulate ultracold atoms.

Figure 5: In-situ absorption imaging of our ultracold atoms under the action
of two distinct ‘disordered’ potentials. The temperature of the atoms is 20
nK. The patterns imprinted on the DMD are reported in the back panels, and we
indicate with black (white) the regions of the DMD where the micromirrors are
set to off (on). Regions where the micromirrors are on correspond to regions
with full intensity on the atoms, and therefore to attractive potentials. The
depth of the potential is 90 nK. The atoms re-arrange in such ‘disordered’
potentials, nicely retracing the patterns set on the DMD.

Using the SLD, with our setup we can achieve 1 mW of power over an area of
207$\times$117 $\mu$m at the position where the atoms are trapped. To evaluate
the corresponding optical potential, we fit the amplified spectrum of Fig. 1b
with a double-Gaussian obtaining the power spectrum $G(\lambda)$, with
$\lambda$ the wavelength, and then we calculate

$U(x,y)=-\frac{\pi
c^{2}}{2}\Gamma\int\left(\frac{2}{\Delta_{2}\omega_{2}^{3}}+\frac{1}{\Delta_{1}\omega_{1}^{3}}\right)J(x,y,\lambda)d\lambda$
(3)

with $\Gamma$ the linewidth and $\omega_{1,2}$ the frequencies of the D1 and
D2 lines of 87Rb, $\Delta_{1,2}=\omega_{1,2}-2\pi c/\lambda$ the corresponding
detunings, and

$J(x,y,\lambda)=\frac{2P}{\pi w^{2}}\frac{G(\lambda)}{\int
G(\lambda)d\lambda}e^{-2\frac{x^{2}+y^{2}}{w^{2}}},$ (4)

where $P$ is the total power at the position of the atoms and $w$ is the waist
of the beam obtained with a Gaussian fit of the intensity profile of Fig. 1e.
Similarly, we calculate the heating rate as

$\Gamma_{h}(x,y)=\frac{\pi
c^{2}\Gamma^{2}}{2\hbar}\int\frac{E_{r}}{k_{B}}\left(\frac{2}{\Delta_{2}^{2}\omega_{2}^{3}}+\frac{1}{\Delta_{1}^{2}\omega_{1}^{3}}\right)J(x,y,\lambda)d\lambda,$
(5)

with $E_{r}=2\pi^{2}\hbar^{2}/m\lambda^{2}$ the recoil energy, where $m$ is
the mass of the atoms, and $k_{B}$ the Boltzmann constant. With our setup we
can obtain a maximum intensity of $2P/(\pi w^{2})=$14.3 W$/$cm2, corresponding
to a maximum potential depth $U/k_{B}\simeq 90$ nK. This red-detuned potential
is therefore sufficiently strong to efficiently manipulate our ultracold
sample. The associated maximum heating rate is instead $\simeq$1 nK/s 333For
comparison, the heating rate for a laser at 840 nm with the same intensity is
$\simeq 0.5$ nK/s.. Indeed, we could not observe any measurable heating in the
typical time scale of our experiments (hundreds of ms) even when illuminating
the atomic cloud with the DMD fully on.

Figure 6: a) Evolution of the atomic column density, integrated along the $y$
direction, as a function of time under the action of the dynamical potential
(also integrated along the $y$ direction) displayed in panel b). The time
frames are taken every 8 ms. The black (white) regions in b) correspond to
micromirrors set off (on) on the DMD.

To give an example, in Fig. 5 we demonstrate that the combination of arbitrary
patterns produced with a DMD and the good quality of the images produced by
the SLD can be used to arrange ultracold atoms in non-periodic structures. We
have implemented two arbitrary ‘disordered’ 1d potentials by setting unequally
spaced gratings on the DMD, as displayed in the back panels. The white
vertical lines, corresponding to regions of high intensity, cross with the
dipole trap potential creating a series of neatly-separated, sharp squared
dimples of different sizes and with different spacing. The dimples are
sufficiently deep to trap the atoms, indeed their density profile in
absorption nicely retraces the patterns imprinted on the DMD, including the
sharp edges. Evaluating the fidelity and contrast directly on the the atomic
sample can therefore also be used to characterise the potential.

We finally demonstrate the implementation of dynamical potentials with SLDs.
The possibility to rapidly reconfigure the pattern on the DMD is particularly
appealing as it allows us to dynamically and arbitrarily modify the potential
on the atoms a fast as one frame every 100 $\mu$s. For dynamical potentials,
good image quality is especially desirable as it ensures a good uniformity of
the potential across the whole illuminated area, potentially reducing losses
of atoms, spurious excitations and heating. In Fig. 5a we report the evolution
of the density profile of our ultracold atomic sample under the action of the
dynamically varying ‘disordered’ potential of Fig. 5b. For every frame we have
integrated the density profiles along the $y$ direction. In this case, the
pattern on the DMD was modified every 500 $\mu$s. We observe that the atomic
density is dynamically modified following the trajectory of the optical
potential. After 200 ms of evolution, we retain $\simeq$75% of the atoms.

##  V Conclusion

In conclusion, we have investigated the use of a SLD in combination with a DMD
to produce optical potentials for ultracold atoms. We have compared the
ability of a SLD and a laser to produce regular square-wave gratings with
different periodicity. We have shown that by using a SLD we consistently
obtain images with higher fidelity than and comparable contrast to a laser,
resulting in an overall superior quality over the whole range of parameters
explored. We have used the patterns generated with the SLD to produce
arbitrary 1d disordered potential for our ultracold atoms. This has resulted
in patterned atomic densities that nicely retrace the optical potential. We
have further demonstrated the implementation of dynamical 1d potentials with
our system. Notably, the setup studied here is simple, robust, cost-effective,
and, as such, can easily find application in many ultracold atoms experiments
worldwide. Additionally, SLDs are available in the near infrared and across
most of the visible spectrum, making them suitable for a wide range of
possible applications in quantum technology. Our results can open new avenues
in the study of 1d disordered systems and transport phenomena in 1d. The
extension to two-dimensional patterns in combination with a flat trap could be
relevant for applications in atomtronics, phase imprinting and quantum
simulations experiments. The further extension to three-dimensional patterns
using two or more DMDs has the potential of enabling more pristine control in
quantum simulators.

Acknowledgments. We acknowledge financial support by the EPSRC (Grant No.
EP/R021236/1) and fruitful discussions with the members of the Cold Atoms
Group at the University of Birmingham.

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