This document discusses using a Bose-Einstein condensate for quantum simulation by applying laser fields with arbitrary amplitude and phase control. It describes producing a BEC using evaporative cooling in a magneto-optical trap. Computer-generated holograms can create arbitrary laser fields by imprinting a phase pattern on a spatial light modulator. The document focuses on using a conjugate gradient minimization algorithm to optimize the phase pattern to produce a desired laser intensity profile. Debugging the existing code improved amplitude and phase control by applying Fourier shifts and normalizing terms. Various error metrics were used to quantify the quality of generated patterns compared to targets.

1. 1
Toward Quantum Simulation in Ultracold Gases:
Conjugate-Gradient Algorithm for Arbitrary Laser Fields
Caroline de Groot
Rector’s Scholar 2016
Supervisors: Dr Donatella Cassetari, Dr Graham Bruce and David Bowman

2. 2
Contents
Introduction................................................................................................................................ 3
Quantum simulation................................................................................................................... 4
Producing and Using a Bose-Einstein Condensate ......................................................................4
Magneto Optical Trap ............................................................................................................... 4
Evaporative cooling .................................................................................................................. 5
The need for arbitrary laser fields............................................................................................... 5
Creating computer-generated holograms..................................................................................... 6
Conjugate-Gradient Minimisation.............................................................................................. 7
Method ....................................................................................................................................7
Amplitude and phase control .....................................................................................................8
Cost functions......................................................................................................................... 10
Limitations ............................................................................................................................. 11
Error Metrics............................................................................................................................ 9
Patterns of interest.................................................................................................................. 12
Conclusion................................................................................................................................ 15
Bibliography............................................................................................................................. 16

3. 3
Introduction
Quantum simulations through an experimental analogue are a tenable method for grasping
hard-to-solve problems, especially in astrophysics or condensed matter physics. One such
analogue is the Bose-Einstein Condensate (BEC), which could act as an artificial gauge field
[1] or experimentally simulate gravitational waves [2]. Once the technical process of creating
a BEC has been accomplished, quantum simulations can be done by applying a particular laser
field. It is therefore highly useful to have arbitrary control of the laser field’s properties:
amplitude and phase.
A particular 2D intensity pattern creates the desired optical trap, and phase is equivalent to
angular momentum of the photons, hence causing motion within the BEC when the laser and
the atoms interact. To create an arbitrary hologram, a Gaussian laser beam can be sent through
a Spatial Light Modulator (SLM). An algorithm determines the required phase distribution on
the SLM to produce the desired pattern. Possibilities for the algorithm include Mixed Region
Amplitude Freedom (MRAF) [3] and the powerful Conjugate-Gradient (CG) minimisation
approach [4], the latter of which was used in this project. The work done this summer involved
optimising the existing codes through explorations of the cost function, addition of smoothing
terms and using regional freedom.
Figure 1: A Bose-Einstein Condensate [10]

4. 4
Quantum simulation
Computer simulation allows one to visualize abstract concepts in physics, but quantum physics
can only be simulated by a quantum computer, Feynman once wrote [5]. It follows that
quantum simulation is therefore a useful technique to explore complex physical systems that
can otherwise be very difficult, if not impossible, to
investigate directly. Quantum systems could be
modelled by a corresponding quantum analogue1, such
as superconducting circuits used by IBM [6], quantum
dots or ultracold atoms [7]. Through table-top
experiments, elusive phenomena such as black-holes
[8], superconductivity and superfluidity [2], and many
others can be studied with relative ease.
This summer project focussed on using ultracold atoms
as a resource, and in particular a BEC, which the Cold
Atoms Group in St Andrews is currently working on
producing.
Producing and Using a Bose-Einstein Condensate
Atoms can be cooled and trapped via a choice of laser cooling methods, such as Sisyphus
cooling and Doppler cooling [9]. However, every experimental group has a different set-up,
and a couple of the most important stages in St Andrews are the evaporative cooling and the
magneto-optical trap (MOT) [9]. The resource being used is Rubidium, chosen partly because
it has a transition close to the laser frequency in the red-detuned part of the spectrum2, and
because it is, of course, bosonic.
Magneto Optical Trap
The MOT is a vacuum chamber with a spatially-varying magnetic field within. Due to spin-
orbit coupling, atomic energy levels are effected by the presence of electric and magnetic fields,
the latter of which is called the Zeeman Effect [9]. A quadrupole magnetic field is generated
by an anti-Helmholtz configuration of coils, shown in Fig 3a. In the centre of the trap the
energy levels are not shifted, and are maximally shifted on the edges, as demonstrated in Fig
3c. Since the system uses red-detuned light, the closer an atom is to the edge of the trap, the
closer its resonant frequency will be to that of the laser (see Fig 3b); in other words, the
detuning is less, and there is a much larger probability that an atom will absorb a photon. If an
atom absorbs a photon there is a momentum transfer to it, and the atom will be pushed towards
the centre of the trap. This happens over as many as 105 atoms, so on average in this process,
atoms will be shifted towards the centre of the trap, decreasing in velocity as they do so.
Velocity is related to temperature via the Equipartition Theorem, so this results in cooling [12].
1
i.e. with the same Hamiltonian
2
Red-detuned meaning that thelaser has a lower frequency than the resonant frequency of theatom.
Figure 2: Optical lattice of cold atoms [11]

5. 5
However, like any cooling process, there is a minimum temperature associated, which is in the
mK regime. This is due to spontaneous emission of photons in random directions, which
contributes to a broadening in momentum space [13], the opposite desired effect. Cooling
results in a high density in momentum space.
Evaporative cooling
Evaporative cooling is the following phase in the process of creating a BEC. This method
allows atoms with the largest velocities (and therefore highest temperatures) to escape, leaving
behind slower (cooler) atoms. By adjusting the magnetic field of the MOT so that the potential
well is lowered as shown in Figure 4, the escape velocity of the trap is effectively lowered at
the edges. This ensures that cold atoms already in the centre of the trap remain, while hot atoms
can escape [14]. The Boltzmann distribution of the atoms in the trap demonstrates that as the
trap depth is decreased, and the hotter atoms leave, the distribution of velocities gets narrower.
A BEC is successfully produced when all the atoms reach the same velocity; they all fall into
the lowest energy state3 and form a quantum macrostate, or a “super atom” [15]. It is described
by a single wave function, and has superfluid and superconducting properties.
The need for arbitrary laser fields
With appropriate adjustment to the system, a BEC can have the same Hamiltonian as another
system that is interesting but harder to study, creating a quantum simulation or an artificial
gauge field [1]. Light-matter interaction applied to ultracold atoms by means of a laser can
create the desired effects, but this requires control of phase and amplitude.
For example, say one wanted to study the effect of the Lorentz force of a charged particle in a
magnetic field. However, neutral atoms don’t experience a Lorentz force, so to study them,
3
Fermions don’t display this behaviour since they obey Fermi-Dirac statistics, which don’t allow sharing of energy states.
The most densely packed fermions fill up all states up to the Fermi Energy and form a degenerate gas [15].
Figure 3 a. The magnetic field is generated by a pair of coils with current in opposing directions. 3b. In the frame of an atom
moving left, it is closer to resonance with the photon from the left. 3c. Atomic energy level splitting as a function of z-axis in
the trap. [13]

6. 6
artificial magnetic fields can be created by engineering laser fields with a varying phase [1]. A
phase gradient in light, on interaction with matter, causes motion. Considering a classical
electromagnetic wave, phase difference between neighbouring wave fronts means that one
wave may peak before the other, at neighbouring sites. Due to the quantum nature of light, a
photon has momentum, and so each site of light-matter interaction experiences a time-changing
impulse. This causes an unbalanced force which causes the atom to move. Phase control is
clearly crucial to creating an artificial gauge field, as well as amplitude.
Creating computer-generated holograms
Computer-generated holograms (CGH) are promising in this area because the optical traps they
form allow microscopic manipulation and trapping of atoms in arbitrary geometries, as well as
real-time updating [16]. However, research into algorithms so far has only succeeded in
modulating the amplitude of the wave fronts keeping phase arbitrary, while recent research
considers modulating both the amplitude and phase which would provide additional degrees of
freedom to the trap, as done through MRAF [3] and possibly CG minimisation, which is the
focus of this project.
A CGH can best be created4 by a laser sent through a spatial light modulator (SLM) with a
particular phase pattern imprinted on it that is set by the result of the algorithm, be it CG or
something else [17]. The SLM used by the St Andrews group is a nematic liquid crystal device,
which has 256 x 256 pixels, each a square with side lengths 6.14mm. The SLM uses the
principle of birefringence, so that
under influence of an electric field, the
liquid crystal particles reorient
themselves to align with the field.
Each pixel can therefore produce a
phase retardation between 0 and 2π, or
one full rotation of the long axis of the
liquid crystals.
4 Can also use a diffraction grating but this is difficult and expensive to manufacture [16]
Figure 4 Lowering the trap depth keeps the coldest atoms stable in the centre, but gradually releases the hot atoms [15].
Figure 5 Fourier Optics [16].

7. 7
After imparting a phase on the incident laser, usually Gaussian and with constant phase, the
next step is to consider the far-field diffraction limit. This can be forced by sending the plane
wave Ein through a lens, and which results in a laser beam described by a field Eout which is the
Fourier transform of Ein. This final field Eout holds the phase and amplitude properties of the
laser pattern desired for use. Therefore the key to a successful CGH is producing a very good
SLM phase plating, by optimising the algorithm which selects it.
Conjugate-Gradient Minimisation
This summer project aimed to improve the existing hologram calculation algorithms of the St
Andrews group, with phase modulation included, and work towards experimental
implementation of holographic optical traps using CG minimisation.
Method
To understand how the CG method works, it is simpler to conceptualise the Steepest Descent
(SD) method on a contour plot. The first step is choosing a starting point x(0) at which the
largest gradient is calculated. The second step is to move on the line until reaching the first
minimum. The third step is to move orthogonal to the previous direction until another minimum
is reached. The previous steps are repeated until the function finds a local minimum. This is an
inefficient procedure, as moving orthogonal and parallel to the original steepest gradient will
not arrive at the minimum very quickly. A zig-zag-like pattern is often characteristic of the SD
method’s path, as shown in Fig 5a. In higher dimensional problems, the zig-zagging slows
down the algorithm so much that it is really not feasible. Visualising the minimum for the SLM
phase output is more complex than a simple 2D contour, as it is a hyperspace of 256 x 256
dimensions with phase and amplitude5. The minimum must be found for each pixel collectively
[18].
5
It’s not possible to separateinto two functions for amplitude and phasewhich can be minimised separately, so this adds
another 256 x 256 dimensions to the problem.
Figure 6a. Steepest Descent method [18] 6b. Conjugate Gradient descent method [18]

8. 8
CG minimisation is much more powerful and applicable, though less easy to visualise. It
minimises a quadratic function6 𝑓(𝑥) =
1
2
𝑥 𝑇
𝐴𝑥 + 𝑥 𝑇
𝑏 + 𝑐, where 𝑥 is an N-dimensional
vector, 𝑏 and 𝑐 are constants, and 𝐴 is a matrix whose significance will become apparent. 𝐴 is
defined to be positive-definite such that 𝑥 𝑇
𝐴𝑥 > 0, which means that the contour has a
minimum and not a maximum [18]. Rather than searching in orthogonal directions as in the
SD method, the search directions are A-orthogonal, which stretches “normal” orthogonality,
so that 𝑓(𝑥) is an ellipsoid.
By construction the CG method looks for a minimum of the cost function, which quantifies the
error between the intensity distribution of the calculated output, and that of the target. MRAF,
on the other hand, doesn't directly minimise errors, but instead iterates between the SLM plane
and the output plane until errors stagnate [3].
It is important to note that although CG calculates a local minimum very effectively, it may
still not find the global minimum.
Amplitude and phase control
The CG algorithm for SLM phase calculation was developed by Tiffany Harte in St Andrews,
with the main program in Fortran which she also wrote later in Python; this was never properly
debugged. Debugging the python code took a significant amount of time during the project.
There were several immediate problems.
Firstly, it was not possible to control both phase and
amplitude simultaneously and the output patterns all had
a checkerboard effect, which is illustrated in Fig 7. By
recognising that a discrete Fourier transform was used
rather than a continuous transform, both these issues
could be solved. After a discrete Fourier transform, the
frequency ends of the spectrum are swapped so that the
high frequency components are in the centre; however, to
be used correctly it’s required that zero frequency components are central in the array. The
Fourier transform function in the program from SciPy didn’t have an inbuilt Fourier shift to
correct this. By adding two Fourier shifts to the original operation, as shown below, this was
fixed, which eliminated the checkerboard effect. It also allowed partial phase control.
𝐹𝐹𝑇𝑠ℎ𝑖𝑓𝑡 = 𝑓𝑓𝑡𝑠ℎ𝑖𝑓𝑡( 𝑓𝑓𝑡( 𝑓𝑓𝑡𝑠ℎ𝑖𝑓𝑡( ) ))
Normalisation is of crucial importance in computational physics, as it imposes conservation of
quantities in the given problem. Hence, a normalisation over number of pixels was added to
account for the discrete Fourier transform. The target intensity and input intensity were also
correctly normalised so that the sum over all pixels added to 1.
A second major insight into the algorithm was to notice Python’s treatment of radians. Python’s
functions can be made to run between 0 and 2π or –π and π, and it’s not correct to mix functions
operating in different scales. Most importantly, numerically there is a large difference between
6
Meet thecost function!
Figure 7 Output intensity for Gaussian line
with incorrect and correct Fourier Transform

9. 9
either ends of these scales, but geometrically they should be equivalent since angle is a periodic
quantity. Therefore, cyclic boundary conditions needed to be imposed in the computation,
whose principle is illustrated by the following diagram.
Figure 8 Cyclic phase conditions on output vs. target phase Φ implemented in the code. Diagram by V. Chardonnet
Another problem became apparent that using a random guess phase didn’t work as well as
using an average over outputs SLM phase. To improve this still further a random phase was
added on top of the average phase which reduced errors.
The final step in getting full amplitude and phase control was finding the best cost function,
which deserves its own section.
Error Metrics
Certain metrics were chosen to quantify the associated errors with both experiment and theory.
Three metrics are used define the quality of the patterns produced in the region of interest as
used by St Andrews cold atoms group [16]: the light efficiency Г, the root-mean-square (RMS)
fractional error η and the phase error ϵ.
Г =
∑ 𝐼𝑜𝑢𝑡 𝑛𝑚𝑛𝑚 ∈𝑀𝑅
∑ 𝐼𝑜𝑢𝑡 𝑛𝑚𝑛𝑚
η =√
1
𝑁 𝑀𝑅
∑
( 𝐼𝑜𝑢𝑡 𝑛𝑚−𝐼𝑡𝑎𝑟𝑔𝑒𝑡 𝑛𝑚 )
2
𝐼𝑡𝑎𝑟𝑔𝑒𝑡 𝑛𝑚
2𝑛𝑚 ∈𝑀𝑅
ϵ =
∑ | 𝛷 𝑡𝑎𝑟𝑔𝑒𝑡 𝑛𝑚−𝛷 𝑜𝑢𝑡 𝑛𝑚 |𝑛𝑚 ∈𝑀𝑅
∑ | 𝛷 𝑡𝑎𝑟𝑔𝑒𝑡 𝑛𝑚 |𝑛𝑚 ∈𝑀𝑅
MR is the measure region, where the value of the weighting is 1, 𝑁 𝑀𝑅 is the number of pixels
in the measure region, 𝐼 𝑛𝑚 = | 𝐸 𝑛𝑚 |2
. These metrics consider the values of each pixel and
compare the target with the current output. Setting a limit for the errors can be used as a
stagnation condition.
Different cost functions and patterns performed with very differently under the error metrics,
so optimising combinations of parameters, such as pairing the best cost function with a
particular pattern, was crucial.

10. 10
Costfunctions
The choice of cost function is of paramount importance in the success of the algorithm. Many
cost functions were tried, but there are two that gave the best results: the overlap function and
the fidelity function. Note that any cost function tried had to fulfil certain requirements, such
as being real-valued. Hence, only squares of the electric field could be used, but this meant
some information is lost.
The first cost function with success in full phase and amplitude control was a function based
on the fidelity F. The electric field 𝐸 𝑛𝑚 contains all the phase and amplitude information.
𝐹 = |∑ 𝐸𝑡𝑎𝑟𝑔𝑒𝑡 𝑛𝑚
∗
𝐸𝑜𝑢𝑡 𝑛𝑚
𝑛𝑚
𝑊𝑛𝑚 |
2
𝐶 𝐹 = (1 − 𝐹)2
The second, and superior, cost function tried was the overlap function7. The idea comes from
the theory of the scalar product, which quantifies the projection of one vector onto another. The
angle between the vectors is given by the phase difference at each pixel, and the length of each
vector is given by the amplitude of the electric field at the pixel 𝐴 𝑛𝑚. The phase difference is
corrected by the cyclic term 𝛷 𝑐𝑦𝑐𝑙𝑖 𝑐 𝑛𝑚 explained in the diagram above. It performs much better
than CF, as demonstrated in Fig 9 on the next page.
𝑂𝑣𝑒𝑟𝑙𝑎𝑝 =
1
𝑁
∑ 𝐴 𝑜𝑢𝑡 𝑛𝑚 𝐴𝑡𝑎𝑟𝑔𝑒𝑡 𝑛𝑚 𝑊𝑛𝑚 cos(𝛷 𝑜𝑢𝑡 𝑛𝑚 + 𝛷 𝑐𝑦𝑐𝑙𝑖𝑐 𝑛𝑚 − 𝛷𝑡𝑎𝑟𝑔𝑒𝑡 𝑛𝑚)
𝑛𝑚
𝐶 𝑜 = (1 − 𝑂𝑣𝑒𝑟𝑙𝑎𝑝)2
𝑁 = √∑( 𝐴 𝑜𝑢𝑡 𝑛𝑚 𝑊𝑛𝑚)2
𝑛𝑚
∑(𝐴𝑡𝑎𝑟𝑔𝑒𝑡 𝑛𝑚 𝑊𝑛𝑚)
2
𝑛𝑚
𝐶 𝑜’s landscape is very flat, partly due to the normalisation constant’s scaling, and mostly due
to the hyper-dimensionality of the space. The latter statement is what makes finding the best
cost function so difficult; there are many pixels, each of which contributes a phase and
amplitude, and these 256 x 256 = 65536 pixels are collectively being fitted to the target.
However, by multiplying 𝐶 𝑜 by a factor, the peaks and troughs of this complicated landscape
are amplified. The larger the factor, the easier it is for the algorithm to find a local minimum,
or perhaps even the global minimum. This improvement is illustrated in the diagram below for
the LG01 mode, halting iterations at 800.
Cost function Iteration Light efficiency Г Fidelity F Phase error ϵ RMS error η
𝐶 𝐹 59 0.08 0.951083 2.68349 1.7891
109
∗ 𝐶 𝐹 800 0.12 0.999998 0.672920 0.00296
𝐶 𝑜 440 0.19 0.997845 0.032919 0.05741
109
∗ 𝐶 𝑜 800 0.28 0.999998 0.000275 0.00028
7
David Bowman’s idea

11. 11
Limitations
Though my part in the project was primarily working on optimising the CG algorithm for
different patterns, experimental implementation was the focus for the project. There were some
experimental limitations with the optical set-up which are discussed in this section.
For example, light from the input scatters off the SLM pixels causing bright spots on the zeroth
order diffraction as well as horizontal and vertical bright lines through the centre which
interfere with the laser output; the solution was to have the output in a corner.
Another limitation is that the optics are diffraction limited, and the Fourier plane output pixels
may start to merge in experiment. To avoid this problem, an artificial 256 x 256 padding of
zeros is added to the 256 x 256 pixel array for the SLM and laser input. This produces a 512 x
512 output plane array which has better resolution experimentally, but has no effect on the
theoretical result.
Producing a good result for the entire output plane is computationally expensive, and
sometimes impossible. Therefore, instead of searching for a local minimum with CG over the
whole plane, an idea from MRAF is borrowed. The weighting array 𝑊𝑛𝑚 is used to define a
measure region, where the value of the array is 1, and otherwise where the value is 0. This
significantly improves results for all patterns.
Optical vortices are a prolific source of annoyance in CGHs that have been successfully
eradicated here due to phase control. They are singularities in the intensity output due to phase
winding8 occurring in the phase output, which, when phase is not controlled by the algorithm,
is free. While not controlling phase, vortices can’t be filled, but by increasing the number of
iterations, many can be pushed out of the centre of the measure region. This problem is mostly
resolved by controlling the phase, but for certain patterns such as the concentric rings pattern
8
This means 0 and 2π on neighbouring pixels; this is discontinuity in the phase.
Figure 10 Increasing iterations reduces optical vortices [16].
Figure 9a. Target intensity 9b. Output intensity with 𝐶 𝑜 9c. Phase output with 𝐶 𝑜 9d. Output Intensity with 𝐶 𝐹 9e.
Output phase with 𝐶 𝐹. Images by Valentin Chardonnet.

12. 12
where there is a discontinuity in phase along a line, this still creates the vortices. The solution
for this is yet to occur.
The more of the output plane the pattern takes up, the worse the CG performs due to the Fourier
transform. However, if the pattern is too thin, the CG may overlook it. To get better results,
these conflicting problems must be balanced, and about 5-10 pixels in width was found to be
the best compromise.
Patterns of interest
Several patterns have been sufficiently well made by the algorithm that they could be used in
experiment. As well as evaluating the errors associated over the whole measure region, the
smoothness and other qualities of the pattern should be assessed for the top 10 % of light, which
is where atoms would accumulate. Other patterns could be made by the algorithm as well, such
as a ring with a cut to allow tunnelling, flat top, and many others.
One pattern that appeared quite successful was graphene. Atoms are trapped in a hexagonal
graphene-like crystal structure. This can be used for gauge fields and to study atomic
diffraction. An image of the target amplitude and resulting intensity pattern are shown to the
right. The only problem with implementing this pattern in experiment is that the light efficiency
only went up to about 8 % which is insufficient for experimental realisation.
Another pattern that performed well in the CG algorithm was the pattern of concentric rings
with a phase gradient. This traps atoms in the red rings while allowing tunnelling between. The
phase target associated causes rotation. The patterns are shown on the next page.
Figure 11. 10a. Target Intensity 10b. Output intensity with optical vortex on the discontinuous line between the rings
10c.Output phase with singularity 10d. Output intensity with continuous phase 10e. Output phase with less
discontinuity. Image by Valentin Chardonnent

13. 13
Other patterns that were experimentally viable were the Laguerre-Gauss (LG) modes. They are
analytic light modes that have an intensity and phase associated, as shown in Figure 12 below.
It illustrates how the wave-fronts of light are effected by the phase.
Yet more patterns that performed well were the Gaussian line and a ring with a cut. This cut is
interesting because this pattern could be used to investigate the physics of atomic tunnelling.
Figure 12 LG00, LG01 and LG02 [19].

14. 14
Figure 13: The algorithm for SLM phase pattern calculation with conjugate-gradient
minimisation as done by Tiffany Harte [16]. The main cost function is now different.

15. 15
Conclusion
The Conjugate-gradient method has been shown to effectively control the phase and amplitude.
After debugging the code, in distinguishing a discrete Fast Fourier Transform from a
continuous one, the majority of project work was done on optimising the code for different
patterns. This was done by trying different cost functions, improving initial conditions for the
guess phase, choosing a particular region to focus calculation in, and adding a smoothing term.
For each pattern, different parameters and guess phases had to be used, and, although accurate,
the grid search method is very inefficient. However, once the SLM phase plate giving the best
result for error metrics for a particular pattern is calculated, it can be used repetitively.
Despite this method being an extremely powerful iterative minimisation procedure, it produces
significant errors. Even in theory this method doesn’t produce a light efficiency Г greater than
30%, and for some patterns produces a Г so low (about 5%) that it is unsuitable for experiment.
In practice, there are also many experimental barriers to obtaining a very good pattern. To
improve the errors from this method, a better optimisation procedure than grid search could be
tried, such as adding momentum.

16. 16
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