Phd Research:
Long-Range Interacting Quantum Spin Systems
Why do we focus so much on nearest neighbour or exponentially decreasing interacting systems? The answer to this question can be traced back to the pioneers of statistical physics, Boltzmann and Gibbs. These two founding fathers were interested in describing the different states of matter where the predominant interaction is of an electromagnetic nature. In the system they considered the presence of both positive and negative charges caused a screening effect which reduced the interactions to ones which are effectively short-ranged or exponentially decaying. Using short-range interacting systems we have made some big leaps forward in understanding different quantum phenomena such as quantum phase transitions and superconductivity to name but a few. However, the story does not end here... Recent developments in the experimental control of (ultra) cold atoms and trapped ions in optical lattices have opened the door for the development of quantum simulators predicted in 1982 by Richard Feynmann in his famous lecture.
Hexagonal patch of trapper Beryllium ions.
Great progress has been made by the group of John Bollinger at NIST in Boulder Colorado. They are able to trap order 300 Beryllium ions in a hexagonal patch where the Coulombic repulsion between the ions forces a triangular internal structure. By changing the frequencies of the laser light shone on the sample they are able to engineer Ising models with tunable long-range Interactions. If we want to use these recent experimental breakthroughs to develop quantum simulators and eventually fully functional quantum computers we have to understand the effects of long-range interaction in quantum systems.
In the astrophysical context researchers have been familiar with the effects of long-range interactions for some time. Here Lynden-Bell and Wood showed that gravitating gas spheres have negative heat capacities (entropy is NOT concave) i.e. adding heat to the system reduces the temperature. This was later confirmed by Thirring who explaining it by showing that systems with long-range interactions do not satisfy the equivalence of ensembles like their short-ranged counterparts do. So when taking the cumulative effect of long-range interactions into account you should expect to find some new and interesting physics.
In the astrophysical context researchers have been familiar with the effects of long-range interactions for some time. Here Lynden-Bell and Wood showed that gravitating gas spheres have negative heat capacities (entropy is NOT concave) i.e. adding heat to the system reduces the temperature. This was later confirmed by Thirring who explaining it by showing that systems with long-range interactions do not satisfy the equivalence of ensembles like their short-ranged counterparts do. So when taking the cumulative effect of long-range interactions into account you should expect to find some new and interesting physics.
Analytic calculations performed on long-range interacting Ising models showed that when the interactions are sufficiently long ranged a long-lived quasi-stationary state emerges. Depending on the type of initial conditions utilized it is possible to determine any of the correlation functions of this long-ranged Ising model. Hence, we are able to fully describe the behaviour of any observable in this systems. Remarkably the current experimental setup should have long enough decoherence times such that the formation of the long-lived quasi-stationary state should be seen. Our analytic results will be used to benchmark upcoming experimental setups which is one of the first steps in building a working quantum simulator!
Collaborators:
- Michael Kastner
- John Bollinger
- Brian Sawyer
- Jens Eisert
- Salvatorre Manmanna
- Kaden Hazzard
- Ana Maria Rey
- Michael Foss-Feig
- Emanuele Dalla Torre
- Tilman Pfau
Publications:
2014
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Kaden R. A. Hazzard, Mauritz van den Worm, Michael Foss-Feig, Salvatore R.
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2013
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Jens Eisert, Mauritz van den Worm, Salvatore R Manmana and Michael Kastner, Breakdown of quasi-locality in long-range quantum lattice models, Phys. Rev. Lett. 111, 260401 (2013)
We are all familiar with the notion of a light-cone emerging in relativistic physics where the speed of light is the maximum speed with which any information can propagate. Naively we might think that in a non-relativistic setting the absence of a finite maximum speed would imply that if we perform a local perturbation in some spatially extended lattice system then this perturbation can be "felt" arbitrarily far away for arbitrarily short times. However, what we see is the emergence of an effective sound cone, outside of which correlations decay extremely fast. These effective sound cones are given by Lieb-Robinson bounds. Hastings extended the original LR-bounds to systems with long-range power-law decaying interactions with the requirement that the exponent in the power-law be greater than the dimensionality of the system. By deriving lower bounds on the classical information capacity we show that this form of the exponent is not a mathematical artefact but reflects the true physical behaviour of long-range interacting systems where, in general, a causal region only appear for exponents greater than the dimensionality of the system. Surprisingly, when using multi-particle entangled GHZ states as initial states the causal region already appears for exponents greater than half the dimensionality. These results show that if we have some spatially extended lattice system with long-range interactions we might be able to use it to send information arbitrarily far in arbitrarily short times. However, when trying to measure this you will ultimately by confined by the laws of relativistic physics in which case the speed of propagation of information is once again bound by the speed of light.
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Mauritz van den Worm, Brian C. Sawyer, John J. Bollinger, Michael Kastner,
Relaxation timescales and decay of correlations in a long-range interacting quantum simulator, 2013 New J. Phys. 15 083007 In 1982 Richard Feynman famously proposed a hypothetical universal quantum simulator to simulate quantum systems. What made the idea so insightful is that a normal classical Turing machine experiences exponential slowdown when simulating quantum phenomena while his universal quantum simulator would not. Due to the vast improvements in both ultra-cold atomic and optical lattice experiments these ideas of Feynman are finally within grasp. During April 2012 Joe Britton and co-workers in the group of John Bollinger at NIST in Boulder Colorado published a breakthrough paper in Nature in which they show how they are able to engineer a long-range interacting Ising type Hamiltonian on hexagonal patches consisting of hundreds of beryllium ions arranged in a triangular lattice structure. In this work we show how to calculate exact analytic expressions for correlations functions of this system. Remarkably, for sufficiently long-range interactions a prethermalization plateau or long lived quasi-stationary state forms. These analytic results will be used to benchmark there future experiments.
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M.Sc Research
My Master's research was done under the supervision of Dr. Rocco Duvenhage at the University of Pretoria. We studied the application of techniques used in the theory of C*-Algebras to construct a noncommutative sigma model. This work was mostly based on an article by Vargese Mathai and Jonathan Rosenberg with the same title. A copy of my dissertation can be found below.
| A Noncommutative Sigma Model |
