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MS10: Quantum chaos and semi-classical dynamics

Organisers: Eva-Maria Graefe, E-mail: e.m.graefe@imperial.ac.uk; Brian Winn, E-mail: B.Winn@lboro.ac.uk.

All talks are in room CC021 James France Building. For the abstracts, please click on the titles.

Sub-session 1

Monday 3rd September 2018, 10.30am--12.30pm
  • Juan-Diego Urbina, Universität Regensburg, Germany
  • Henning Schomerus, Lancaster, UK
  • Igor Lesanovsky, Nottingham, UK
  • Charlie Johnson, Bristol, UK

Sub-session 2

Monday 3rd September 2018, 3.30pm--5.30pm
  • Jens Bolte, Royal Holloway, UK
  • Benjamin Geiger, Universität Regensburg, Germany
  • Sebastian Müller, Bristol, UK

Sub-session 3

Tuesday 4th September 2018, 10.30am--12.30pm
  • Steve Mudute-Ndumbe, Imperial College, UK
  • Karol Życzkowski, Jagiellonian University, Poland
  • MS Santhanam, Indian Institute of Science Education and Research, India
  • Mark Everitt, Loughborough, UK

Sub-session 4

Tuesday 4th September 2018, 3.30pm--5.30pm
  • Julius Kullig, Universität Magdeburg, Germany
  • Anna Maltsev, Queen Mary, UK
  • Roman Schubert, Bristol, UK


  • 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 square-well potential that violates the assumptions of the Kolomogorov-Arnold-Moser theorem and we report on the quantum manifestations of non-KAM chaotic dynamics in this system, ranging from barrier power-law localization to non-integral h-bar scaling of break-times. Secondly, another variant of the noisy kicked rotor model using an atom-optics set-up 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 non-exponential, effectively slowing down the progression of the localised state from quantum to classical.

  • Julius Kullig, Universität Magdeburg
    Title: Exceptional points with whispering-gallery modes in optical microdisk cavities

    A striking signature of the non-Hermitian 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 long-lived whispering-gallery modes can be achieved via an refractive index adjustment and small boundary deformations which can be explained by a semiclassical approach based on resonance-assisted tunneling. Additionally, we cover exceptional point of higher order where more than two involved modes coalesce.

  • Juan-Diego Urbina, Universität Regensburg
    Title: Interfering mean-field paths in Fock space and the breaking of quantum-classical correspondence in many-body systems: from out-of-time-order correlators to many-body spin echo

    The emergence of quantum technologies requires both a deep understanding and simulation power of many-body 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 large-N version of many-body quantum mechanics that provides a correct and tractable description of quantum interference beyond the characteristic time-scale where quantum dynamics gets interference-dominated and thus departs drastically from its classical description. This many-body semiclassics, based on approximating path integrals in Fock space by coherent sums over interfering mean-field solutions, accounts in an comprehensible way for several many-body interference effects, ranging from many-body spin echo to the saturation of out-of-time-order correlators.

  • Henning Schomerus, Lancaster University
    Title: Zero modes in complex systems - robust features against a random background

    I explore applications of random-matrix 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 zero-modes H. Schomerus, M. Marciani, C. W. J. Beenakker Phys. Rev. Lett. 114, 166803 (2015)
    • [2] Zero-voltage 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 wave-packets 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 non-Hermitian Hamiltonian. This allows us to use previous results on non-Hermitian propagation of wave-packets 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 two-dimensional toroidal phase space whose quantisation can be seen as a realisation of the Berry-Keating model for the asymptotic density of the Riemann zeros. Another example is a magnetic Laplacian on a four-dimensional 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 many-body 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 quasi-chaotic behavior we studied a momentum-truncated version of the attractive Lieb-Liniger 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 short-time behavior of certain out-of-time-ordered 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 anti-unitary (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 complex-conjugation type, or broken time-reversal symmetry. We introduce a method to build systems with arbitrary time-reversal-type symmetries, and demonstrate that they indeed follow their RME predictions.

  • Steve Mudute-Ndumbe, Imperial College
    Title: A PT-symmetric version of a kicked top

    PT-symmetric 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 PT-symmetry 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 PT-symmetric. We will explore both the classical and the quantum PT-symmetric 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 PT-symmetry.

  • 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 well-known 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 Frobenius-Perron 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 M-dimensional 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 non-equilibrium phase. This phase is characterised by limit cycles corresponding to high-dimensional 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 pattern-retrieval 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, well-established 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 high-energy 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 phase-space 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 phase-space 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 phase-space 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.