
MS10:
Quantum chaos and semiclassical dynamics
All talks are in room CC021 James France Building. For the abstracts, please click on the
titles.
Subsession 1
Monday 3rd September 2018, 10.30am12.30pm
 JuanDiego Urbina, Universität Regensburg, Germany
 Henning Schomerus, Lancaster, UK
 Igor Lesanovsky, Nottingham, UK
 Charlie Johnson, Bristol, UK
Subsession 2
Monday 3rd September 2018, 3.30pm5.30pm
 Jens Bolte, Royal Holloway, UK
 Benjamin Geiger, Universität Regensburg, Germany
 Sebastian Müller, Bristol, UK
Subsession 3
Tuesday 4th September 2018, 10.30am12.30pm
 Steve MuduteNdumbe, Imperial College, UK
 Karol Życzkowski, Jagiellonian University, Poland
 MS Santhanam, Indian Institute of Science Education
and Research, India
 Mark Everitt, Loughborough, UK
Subsession 4
Tuesday 4th September 2018, 3.30pm5.30pm
 Julius Kullig, Universität Magdeburg, Germany
 Anna Maltsev, Queen Mary, UK
 Roman Schubert, Bristol, UK
Abstracts
 MS Santhanam, Indian Institute of Science Education
and Research
Title: Chaos, localisation and decoherence in kicked rotor variants
Kicked rotor is one of the simple models of Hamiltonian chaos and
and was thoroughly investigated, in theory and experiments, druing the
last 40 years. The temporal mean energy growth of the kicked rotor
displays distinct behaviour in its classical and quantum regimes. The
classical regime is associated with linear increase in energy while
the quantum regime display dynamical localisation. In this
presentation, we discuss two variants of kicked rotor model. Firstly,
we study a kicked rotor placed in a finite squarewell potential that
violates the assumptions of the KolomogorovArnoldMoser theorem and
we report on the quantum manifestations of nonKAM chaotic dynamics in
this system, ranging from barrier powerlaw localization to
nonintegral hbar scaling of breaktimes. Secondly, another variant
of the noisy kicked rotor model using an atomoptics setup is studied
in which the kicking sequence is manipulated. In this version, kicks
are skipped during certain intervals of time, taken from a Lévy
distribution. We show through theory, simulation and experiments that
the decoherence rate in this case becomes nonexponential, effectively
slowing down the progression of the localised state from quantum to
classical.
 Julius Kullig, Universität Magdeburg
Title: Exceptional points with whisperinggallery modes in optical
microdisk cavities
A striking signature of the nonHermitian physics in open systems is
the existence of exceptional points in parameter space. In contrast to
a conventional degeneracy, an exceptional point results in the
coalescence of the eigenvalues and simultaneously that of the
corresponding eigenstates. We study exceptional points in passive
optical microdisk cavities. In these systems exceptional points with
involved longlived whisperinggallery modes can be achieved via an
refractive index adjustment and small boundary deformations which can
be explained by a semiclassical approach based on resonanceassisted
tunneling. Additionally, we cover exceptional point of higher order
where more than two involved modes coalesce.
 JuanDiego Urbina, Universität Regensburg
Title: Interfering meanfield paths in Fock space and the breaking of
quantumclassical correspondence in manybody systems: from
outoftimeorder correlators to manybody spin echo
The emergence of quantum technologies requires both a deep
understanding and simulation power of manybody quantum
systems. Exploiting the key (and complicated) quantum coherent aspects
of complex scenarios gets further complicated by the implicit large
number N of particles involved. In this talk we report on our
progress towards a largeN version of manybody quantum mechanics
that provides a correct and tractable description of quantum
interference beyond the characteristic timescale where quantum
dynamics gets interferencedominated and thus departs drastically from
its classical description. This manybody semiclassics, based on
approximating path integrals in Fock space by coherent sums over
interfering meanfield solutions, accounts in an comprehensible way
for several manybody interference effects, ranging from manybody
spin echo to the saturation of outoftimeorder correlators.
 Henning Schomerus, Lancaster University
Title:
Zero modes in complex systems  robust features against a random background
I explore applications of randommatrix ensembles with topological
features, resulting from eigenstates of the Hamiltonian with
eigenvalue pinned to zero. These zero modes leave their mark in the
chaotic dynamics [1] and transport [2] through superconducting
systems, and also govern the topological mode competition in
semiclassical bosonic systems with loss and nonlinearities [3].
 [1] Effect of chiral symmetry on chaotic scattering from Majorana
zeromodes H. Schomerus, M. Marciani, C. W. J. Beenakker
Phys. Rev. Lett. 114, 166803 (2015)
 [2] Zerovoltage conductance peak from weak antilocalization in a
Majorana nanowire D. I. Pikulin, J. P. Dahlhaus, M. Wimmer,
H. Schomerus, and C. W. J. Beenakker New J. Phys. 14,
125011 (2012)
 [3] Topological dynamics and excitations in lasers and condensates
with saturable gain or loss S. Malzard, E. Cancellieri, and
H. Schomerus submitted (2018).
 Roman Schubert, University of Bristol
Title:
Dynamics of Gaussian wave packets and their superpositions in open
quantum systems
The dynamics of a quantum system which is coupled to an environment
can in many situations be described by a master equation of Lindblad
type. We extend the well known results on semiclassical propagation of
wavepackets from the Schroedinger equation to the Lindblad equation
and show how dissipation and diffusion induced by the environment
affect the evolution of the centre of the wave packet and of its
covariance. This extends previous results by Brodier and Ozorio de
Almeida. We furthermore recast the phase space representation of the
Lindblad equation as a Schroedinger equation for the Wignerfunction
with a nonHermitian Hamiltonian. This allows us to use previous
results on nonHermitian propagation of wavepackets and apply them to
the Lindblad dynamics of the superposition of Gaussian states and
their decoherence.
This is joint work with E. M. Graefe, B. Longstaff and T. Plastow.
 Jens Bolte, Royal Holloway, University of London
Title:
Weyl quantisation of Hamiltonian flows on toroidal phase spaces
Weyl quantisation on a compact classical phase space leads
to finite dimensional quantum Hilbert spaces, with the dimension of
the Hilbert space serving as a semiclassical parameter. The
kinematical aspects of this are well known from the quantisation of
torus maps. In this talk we consider Schrödinger equations (in
continuous time) and prove a Gutzwiller trace formula. As a first
application we discuss an inverted harmonic oscillator on a
twodimensional toroidal phase space whose quantisation can be seen as
a realisation of the BerryKeating model for the asymptotic density of
the Riemann zeros. Another example is a magnetic Laplacian on a
fourdimensional toroidal phase space.
The talk is based on joint work with Sebastian Egger, Stefan Keppeler,
and with Lewis Proctor.
 Benjamin Geiger, Universität Regensburg
Title:
Fast scrambling and quantum butterfly effect in an integrable system
The investigation of scrambling of information in interacting quantum
systems has recently attracted a lot of attention as a manifestation
of manybody quantum chaos. However, it has been demonstrated that
certain integrable systems that are subject to quantum phase
transitions allow for fast information scrambling if they are tuned
close to their critical point [1]. To investigate the origin of this
quasichaotic behavior we studied a momentumtruncated version of the
attractive LiebLiniger gas using established semiclassical methods
applicable for integrable systems. We find that for finite particle
numbers, the quantum critical behavior has its origin in the
appearance of a separatrix in the classcial phase space that renders
the classical dynamics locally unstable. This has various effects on
the underlying quantum system, one of them being a fast growth of
multiparticle entanglement leading to fast scrambling in reduced
density matrices. Furthermore, the instability of the classical system
is sufficient to reproduce the shorttime behavior of certain
outoftimeordered correlators predicted for chaotic systems, where
the classical stability exponent takes the role of the Lyapunov
exponent.
 [1] Dvali et al. Phys. Rev. D 88, 124041 (2013)
 Charlie Johnson, University of Bristol
Title:
Generalised time reversal symmetries on quantum graphs
The spectral statistics of a chaotic quantum system are given by a
Random Matrix Ensemble, of a type determined completely by the
symmetries of the quantum system  whether they be unitary (geometric)
or antiunitary (time reversal or spectral mirror
symmetries). Previous work has demonstrated the use of Quantum Graphs
in constructing chaotic quantum systems with arbitrary unitary
symmetries, but has been restricted to systems with either a basic
complexconjugation type, or broken timereversal symmetry. We
introduce a method to build systems with arbitrary timereversaltype
symmetries, and demonstrate that they indeed follow their RME
predictions.
 Steve MuduteNdumbe, Imperial College
Title: A PTsymmetric version of a kicked top
PTsymmetric Hamiltonians that are not Hermitian but can nevertheless
possess a purely real spectrum have been the focus of intense research
interests over the last decades. The interplay of PTsymmetry and
quantum chaos, however, is not well explored. Here we introduce a
modified version of the kicked top, where the Hamiltonian is no longer
Hermitian but is now PTsymmetric. We will explore both the classical
and the quantum PTsymmetric system, showing how the behaviour differs
from both the closed kicked top and a completely open kicked top. In
particular, we will see how the behaviour is drastically different
depending on the regime of PTsymmetry.
 Sebastian Müller, University of Bristol
Title: Discrete symmetries in quantum chaos and the tenfold way
Wigner and Dyson classified quantum systems into three symmetry
classes according to their behaviour under time reversal. For fully
chaotic systems these symmetry classes determine the statistics of the
energy levels. In the 1990s Altland and Zirnbauer extended this
classification to ten symmetry classes by including systems that have
a mirror symmetry in their spectra, arising e.g. for Andreev
billiards, topological insulators or in nuclear physics. It is
wellknown that additional discrete geometric symmetries can alter the
Altland/Zirnbauer classification. In the presence of these
symmetries, one has to consider instead the spectral properties of
subspectra of the system connected to representations of the symmetry
group. These subspectra can then be classified according to the
tenfold way.
We provide a simple and complete theory on how discrete geometric
symmetries affect the tenfold classification, and which symmetry class
describes the spectral properties of their subspectra. It is obtained
by an extension of Wigner's corepresentation theory and uses simple
indicators to determine the symmetry class. We illustrate our
theoretical results with examples.
 Karol Życzkowski, Jagiellonian University
Title: Quantum chaos and irreversible dynamics
We study classically chaotic systems and their quantum analogues. For
closed systems the quantum dynamics is reversible and can be described
by unitary evolution operators. In the case of open systems
interacting with an environment the dynamics becomes irreversible.
One uses then the notion of quantum operations  completely positive
trace preserving maps  which send the set of density matrices into
itself.
Spectral properties of quantum operations are analyzed and a quantum
analogue of the FrobeniusPerron theorem concerning stochastic
matrices is formulated. Under assumption of strong chaos in the
corresponding classical system and a strong decoherence (i.e. strong
coupling with an Mdimensional environment) the spectrum of a quantum
map Φ displays a universal behaviour: it contains the leading
eigenvalue λ_1 = 1 and displays a spectral gap: all other
eigenvalues are restricted to the disk of radius R = M^{1/2}.
Sequential action of the map Φ brings all pure states exponentially fast
to the unique invariant state, while the convergence rate
is determined by modulus of the subleading eigenvalue, R = λ_2.
 Igor Lesanovsky, University of Nottingham
Title: Open system quantum generalisations of Hopfield neural networks
We discuss ideas for developing an understanding of how quantum
effects may impact on the dynamics of neural networks. We implement
the dynamics of neural networks in terms of Markovian open quantum
systems, which allows us to treat thermal and quantum coherent
effects on the same footing. This leads quite naturally to an open
quantum generalisation of the Hopfield neural network, the simplest
toy model of associative memory. We determine its phase diagram and
show that quantum fluctuations may give rise to a qualitatively new
nonequilibrium phase. This phase is characterised by limit cycles
corresponding to highdimensional stationary manifolds that may be
regarded as a generalisation of storage patterns to the quantum
domain.
 [1] P. Rotondo, M. Marcuzzi, J. P. Garrahan, I. Lesanovsky and M.
Müller, Open quantum generalisation of Hopfield neural networks,
Journal of Physics A 51, 115301 (2018)
 [2] E. Fiorelli, P. Rotondo, M. Marcuzzi, J. P. Garrahan and I.
Lesanovsky, Quantum accelerated approach to the thermal state of
classical spin systems with applications to patternretrieval in
the Hopfield neural network,
arXiv:1806.02747 (2018)
 Anna Maltsev, Queen Mary University of London
Title: Localization and landscape functions on quantum graphs
I will discuss localization and other properties of eigenfunctions of
the Schrödinger operator on quantum graphs. The motivation is to
understand how graph structure impacts eigenfunction behavior. I will
present two estimates based on the Agmon method to show that a tree
structure aids the exponential decay at energies below the essential
spectrum. I will furthermore present adaptations of the landscape
function approach, wellestablished for R^n, to quantum
graphs and its limitations. In our context, a ``landscape function''
is a function that controls the localization properties of normalized
eigenfunctions through a pointwise inequality. The connectedness of a
graph can present a barrier to the existence of universal landscape
functions in the highenergy regime, as we demonstrate with simple
examples. However, at low and moderate energies landscape functions
can be made explicit. This talk is based on joint work with Evans
Harrell.
 Mark Everitt, Loughborough University
Title:
A general approach to quantum mechanics as a statistical theory
Since the very early days of quantum theory there have been numerous
attempts to interpret quantum mechanics as a statistical theory. This
is equivalent to describing quantum states and ensembles together with
their dynamics entirely in terms of phasespace distributions. Finite
dimensional systems have historically been an issue. In recent works
[Phys. Rev. Lett. 117, 180401 and Phys. Rev. A
96, 022117] we
presented a framework for representing any quantum state as a complete
continuous Wigner function. Here we extend this work to its partner
function  the Weyl function. In doing so we complete the phasespace
formulation of quantum mechanics  extending work by Wigner, Weyl,
Moyal, and others to any quantum system. This work is structured in
three parts. Firstly we provide a brief modernized discussion of the
general framework of phasespace quantum mechanics. We extend previous
work and show how this leads to a framework that can describe any
system in phase space  putting it for the first time on a truly
equal footing to Schrödinger's and Heisenberg's formulation of quantum
mechanics. Importantly, we do this in a way that respects the unifying
principles of "parity" and "displacement" in a natural broadening of
previously developed phase space concepts and methods. Secondly we
consider how this framework is realized for different quantum systems;
in particular we consider the proper construction of Weyl functions
for some example finite dimensional systems. Finally we relate the
Wigner and Weyl distributions to statistical properties of any quantum
system or set of systems.


