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

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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....................................................................................................................................9

Limitations......................................................................................................................................10

Error Metrics...................................................................................................................................11

Patterns of interest...........................................................................................................................12

Conclusion..........................................................................................................................................14

Bibliography......................................................................................................................................15

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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.

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Figure 1: A Bose-Einstein Condensate [10]

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 spectrum 2, 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 Hamiltonian2 Red-detuned meaning that the laser has a lower frequency than the resonant frequency of the atom.

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Figure 2: Optical lattice of cold atoms [11]

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.

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].

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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]

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, 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

4 Can also use a diffraction grating but this is difficult and expensive to manufacture [16]

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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].

pixel can therefore produce a phase retardation between 0 and 2π, or one full rotation of the long axis of the liquid crystals.

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 E out 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 separate into two functions for amplitude and phase which can be minimised separately, so this adds another 256 x 256 dimensions to the problem.

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Figure 6a. Steepest Descent method [18] 6b. Conjugate Gradient descent method [18]

CG minimisation is much more powerful and applicable, though less easy to visualise. It

minimises a quadratic function6 f (x)=12

xT Ax+xT b+c, where x is an N-dimensional vector,

b and care constants, and A is a matrix whose significance will become apparent. A is defined to be positive-definite such that xT 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 thatf (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.

FFT shift=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

6 Meet the cost function!

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Figure 7 Output intensity for Gaussian line with incorrect and correct Fourier Transform

mix functions operating in different scales. Most importantly, numerically there is a large difference between 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=∑

nm∈MR

❑

I out nm

∑nm

❑

I out nm

η =√ 1N MR

∑nm∈MR

❑ ( I out nm−I target nm )2

I target nm2

ϵ¿∑

nm∈MR

❑

|Φ target nm−Φout nm|

∑nm∈MR

❑

|Φ target nm|

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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=|Enm|

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.

Cost functions

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 Enm

❑ contains all the phase and amplitude information.

F=|∑nm

❑

E target nm¿ Eout nm

❑ W nm❑ |

2

CF=(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 Anm. The phase difference is corrected by the cyclic term Φcyclicnm explained in the diagram above. It performs much better than CF, as demonstrated in Fig 9 on the next page.

Overlap= 1N ∑

nm

❑

Aout nm A target nm W nm cos (Φout nm+Φcyclic nm−Φ target nm )

Co=(1−Overlap )2

N=√∑nm

❑

( Aout nmW nm )2❑∑nm

❑

( A target nmW nm )2

Co’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 Coby 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.

7 David Bowman’s idea

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Cost function Iteration Light efficiency Г

Fidelity F Phase error ϵ RMS error η

CF 59 0.08 0.951083 2.68349 1.7891109∗CF 800 0.12 0.999998 0.672920 0.00296

Co 440 0.19 0.997845 0.032919 0.05741109∗Co 800 0.28 0.999998 0.000275 0.00028

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 W nmis 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.

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Figure 10 Increasing iterations reduces optical vortices [16].

Figure 9a. Target intensity 9b. Output intensity with Co9c. Phase output with Co 9d. Output Intensity with CF

9e. Output phase with CF. Images by Valentin Chardonnet.

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 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.

8 This means 0 and 2π on neighbouring pixels; this is discontinuity in the phase.

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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.

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.

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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.

Figure 12 LG00, LG01 and LG02 [19].

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.

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