Background

Results

Conclusions

Quantum optics is the field of research in which the interaction of light and matter at the atomic scale is investigated.

Theory

Transport Dynamics in Quantum

Gas Microscopes

Major experimental and

computational advances

have been made recently

which have allowed

many new, radical

experiments to be

undertaken (Figure [1])

and new theories to be

tested.

Figure [1]

Aims

Laser beam

Optical lattice

Particles `hopping’ Figure [2]

at the university of strathclyde

department of

physics Malcolm Jardine | Supervisors: Andrew J. Daley, Anton Buyskikh

Special thanks: Jorge Yago

Funded by: Interns@Strathclyde

Any further questions please contact: malcolm.jardine.2013@uni.strath.ac.uk

Quantum optics experiments can be undertaken by creating an optical

lattice using counter-propagating laser beams. This is an artificial crystal

of light that physicists can insert particles into, where there is single

particle control and resolution.

Goal of project was to simulate the dynamics of a large number of

atoms in an optical lattice.

Allows us to emulate electron transport in materials such as high

temperature superconductors and address fundamental ideas about

matter.

Method

Particles interacting

In response to the inability of the Hubbard model to be calculated, the

technique of using Matrix Product States (MPS) [2] was developed. This

represents the system in a different way: instead of all possible

combinations, only the most relevant ones are used. This information is

then broken into small pieces that are affordable to compute.

Quantum mechanics has to consider all possible

combinations of particles on lattice.

Lattice of 300 particles on 100 lattice sites has more configurations

than there are particles in the universe.

Conventional methods on computers cannot solve worthwhile sized

Hubbard model systems.

Log 1

0(E

xact

—M

PS)

Time

Site occupation

allowed

Difference in hopping between exact and MPS calculations

Figure [3]

The Hubbard model was solved numerically using standard methods but

could only be solved for small systems, which is not useful for

experiments. Therefore MPS was needed to go to bigger systems.

The Hubbard Model provides an effective mathematical description of

an optical lattice

This model accounts for atoms `hopping’ between neighbouring sites,

or interacting if they are on the same site

Problem arises trying to solve this computationally

Hubbard Model Challenges

Single atoms in an optical lattice

in a quantum gas microscope

We computationally model these many-particle

interacting systems to study matter at a

fundamental level where quantum physics

dominates, accessing properties such as

movement of atoms along the lattice.

Figure 3 shows the difference in the magnitude of hopping that was

predicted by the MPS and exact calculations. The two solutions should

converge, and this is seen up to certain degree of accuracy.

Using these results it is now possible to start going beyond what is

possible via standard techniques. Therefore more realistic and useful

systems can be simulated.

MPS technique can be used on much larger quantum optics systems than exact calculations are able to do.

Investigated this by showing MPS techniques can replicate the result of a Hubbard model simulation which used conventional methods.

The future steps will be to take the MPS code and apply it to more interesting systems and geometries, with the possibility that these can be

replicated with experiments.

[1] Haller, E., Hudson, J., Kelly, A., Cotta, D.A., Peaudecerf, B., Bruce, G.D. and Kuhr, S. (2015) ‘Single-atom imaging of fermions in a quan-

tum-gas microscope’, Nature Physics, 11(9), pp. 738–742. doi: 10.1038/nphys3403

[2] Schollwöck, U. (2011) ‘The density-matrix renormalization group in the age of matrix product states’, Annals of Physics, 326(1)

i1

i2

iN

. . . i1

i2

A1 A2 . . .

iN

AN MPS

construction

What information is kept? The information that is most important for

the low energy dynamics.