Statistical Effects in Anyonic Quantum Walks

121 Pages·2013·1.44 MB·English

Statistical Effects in Anyonic
Quantum Walks
By
Lauri Lehman
A thesis submitted to Macquarie University
for the degree of
Doctor of Philosophy
Department of Physics and Astronomy
June 2013
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c Lauri Lehman, 2013.
(cid:13)
Typeset in LATEX2ε.
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Except where acknowledged in the customary manner,
the material presented in this thesis is, to the best of my
knowledge, original and has not been submitted in whole
or part for a degree in any university.
Lauri Lehman
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Acknowledgements
I would like to thank my parents Lasse and Anneli and my sister Sisko for their con-
tinuous support and understanding. They never complained about waking up at early
hours for our phone calls, and were always there for me when I had a chance to visit
back home during my thesis project. It has been a pleasure to host my parents during
their visits to Australia, and I will always remember our adventures in the Red Centre
and the Great Barrier Reef.
A special thanks goes also to my supervisor Gavin Brennen, who has been a sup-
portive and determined coordinator for this project. When I started my thesis project
back in 2009, I had no idea what kind of ride I was about to get into. Gavin’s wide
range of interests and his precision and creativity in research have served as an inspi-
ration for my own research, and I consider him as a role model as a physicist. I wish
him the best of luck on his career and hope that this project will not be the end of our
research efforts.
Jiannis Pachos was also an inspiring supervisor during my six-month visit to Leeds
in 2010. His vision and guidance were essential in delivering the results for this project.
Va´clav Zatloukal was a good buddy, colleague and housemate in Leeds, and many of
the results presented in this thesis are entirely credited to him.
I have met some great people during my stay in Australia. The student group at
Macquarie: Aharon, Andrew, Chris, Gerardo, Ingo, Ivan, Johann, Mauro, Michael,
Nora, Ressa, Thorn, Tommaso and Xavi have been of special importance, and our
numerous coffee breaks have been vital to my well-being here. Finally, I would like
to thank Andreas, Casper, Chris, Eleonora, Emma, Francesca, Georg, Helena, Hugo,
Jarkko, Joe, Joppe, Jussi, Katrina, Kayla, Kyle, Kush, Magda, Matti, Michail, Mika,
Mikko, Nadz, Oldoozie, Pekka, Pete, Samuli, Sara, Shira, Sia, Stephen, Thomas, Tony,
Ville, Wenjie and William for being good friends!
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List of Publications
L. Lehman, V. Zatloukal, J. Pachos, G. Brennen and Z. Wang. Quantum Walks
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withNon-AbelianAnyons. Phys. Rev. Lett. 106,230404(2011); arXiv: 1009.0813
L. Lehman, D. Ellinas and G. Brennen. Quantum Walks of SU(2) Anyons on
k
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a Ladder. Proceedings of the International Workshop on Theoretical Aspects of
the Discrete Time Quantum Walk. J. Comput. Theor. Nanosci. 10 (7), 1634
(2013); arXiv:1203.1999
V. Zatloukal, L. Lehman, S. Singh, J. Pachos and G. Brennen. Transport Prop-
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erties of Anyons in Random Topological Environments. Submitted. Preprint
arXiv:1207.5000
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Abstract
Anyonic quantum statistics is an exotic phenomenon of identical particles in quan-
tum mechanics. When particles are confined in two spatial dimensions, exchanges of
identical particles can induce phase factors in the wave function of Abelian anyons
or matrix valued transformations of the wave function of non-Abelian anyons. As a
result, systems of anyons may have richer properties than those of bosons and fermions
in three spatial dimensions. There is strong theoretical support for the existence of
anyons in some engineered two-dimensional systems such as 2D electron gases, strongly
correlated spin lattices and as edge modes of nanowires. In the future, anyons could
be used in topological quantum computation to perform highly efficient information
processing with very small error rates.
The phases of matter described by dynamically interacting anyons have recently
beenstudiedinchainswhereanyonsinteractviaHeisenberg-typeexchangeinteractions.
In this thesis, new kind of anyonic interactions are studied, induced purely by braiding
during free evolution. Such interactions are of topological origin, and the information
about the interactions is stored non-locally. A quantum walk model is used to study
the effects of these braiding interactions on the dynamical behaviour of anyons.
Theanyonicquantumwalkisaquasi-one-dimensionalgeneralizationofthediscrete-
time quantum walk which allows the simulation of anyonic dynamics analytically. The
moving anyon and a chain of stationary anyons interact via braiding statistics, and
the behaviour of the anyon is studied in three cases. Striking differences are found be-
tween particles with conventional boson or fermion statistics and non-Abelian anyonic
statistics.
The random walk is a dynamical model that describes the motion of a particle on
a lattice. In physics, it is used to describe Brownian motion of fluids and gases. In
such systems, energy transport is diffusive, and the order of the system approaches a
highly mixed state without any information content. The quantum version of the ran-
dom walk, the quantum walk, has nonintuitive properties. Generally, the information
content in a unitarily evolving system is fixed, which leads to unexpected transport
phenomena. An initially localized particle does not propagate diffusively, but escapes
the starting point with ballistic speed.
Most of our results for non-Abelian anyons use the Ising model anyon which is
most likely to be measured in experiments. First, when each anyonic site is occupied
by one Ising anyon, the propagation of the anyon becomes diffusive. More precisely, the
variance of the spatial probability distribution of the particle depends linearly on the
number of time steps. This is in stark contrast to bosons, fermions and Abelian anyons
which propagate ballistically, and the variance depends linearly on the square of the
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number of time steps. The essential reason for this slowdown is that the non-Abelian
anyons possess an extra degree of freedom called the fusion Hilbert space. This space
can be viewed as an environment for the normal degrees of freedom of the particle,
inducing decoherence in the quantum walk. This thesis opens the line for studies of
this novel kind of decoherence mechanism in quantum walks.
In the second case the study is extended to more general anyon models by losing
some of the information about the history of the evolution. In this case the system is
subject to decoherence, and the total system evolution is not unitary. The behaviour of
the particle in this model is found to be diffusive for all the non-Abelian anyon models
studied, while Abelian anyons behave ballistically also in this model.
One peculiarity of quantum systems is their behaviour under disorder. Quantum
mechanical particles moving in random local potentials are known to freeze and not
move at all. Such a phenomenon is known as Anderson localization. Studies have
shown that Anderson localization happens also in quantum walks with random spatial
fluctuations in the coin parameters. In the third case, the transport properties of
anyons are studied under topological randomness, allowing the occupations of anyons
change between experiments. Bosons and fermions would propagate ballistically under
such randomness, and Abelian anyons are shown to localize. The results show that
non-Abelian anyons behave diffusively at short time scales, and it is argued that they
do so in the long time limit as well.
In all cases, non-Abelian anyons are shown to have very different dynamical prop-
erties than bosons, fermions and Abelian anyons. A possible simulation of the anyonic
quantum walk in Fractional Quantum Hall systems is also discussed.