I completed my PhD in 2013 under the supervision of Madalin Guta in the School of Mathematical Sciences at the University of Nottingham, resulting in my PhD thesis, Large Deviations and Dynamical Phase Transitions for Quantum Markov Processes. From 2014 to 2016 I was a postdoctoral researcher in the condensed matter theory group in the School of Physics and Astronomy at the University of Nottingham.

My research interests revolve around the study and classification of dynamical features of open quantum systems, from low-dimensional to complex many-body systems. In particular, I am interested in the mathematical description of such systems, which typically involves quantum probability and quantum dynamical semigroups; my goal is to use these theoretical frameworks in the study of concrete open quantum systems in order to explain and classify critical features such as dynamical heterogeneity and dynamical phase transitions.

(The above background image shows sample quantum jump trajectories in a particular many-body open quantum system.)

Merlijn van Horssen

LinkedIn

merlijn
@gmail.com

*van Horssen, M., Levi, E., Garrahan, J.P.*•Phys. Rev. B 92, 100305(R)•*2015*

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We study the real-time dynamics of a translationally invariant quantum spin chain, based on the East kinetically constrained glass model, in search for evidence of many-body localization in the absence of disorder. Numerical simulations indicate a change, controlled by a coupling parameter, from a regime of fast relaxation-corresponding to thermalization-to a regime of very slow relaxation. This slowly relaxing regime is characterized by dynamical features usually associated with nonergodicity and many-body localization (MBL): memory of initial conditions, logarithmic growth of entanglement entropy, and nonexponential decay of time correlators. We show that slow relaxation is a consequence of sensitivity to spatial fluctuations in the initial state. While numerical results and physical considerations indicate that relaxation time scales grow markedly with size, our finite size results are consistent both with an MBL transition, expected to only occur in disordered systems, and with a pronounced quasi-MBL crossover.

*van Horssen, M., Garrahan, J.P.*•Phys. Rev. E 91, 032132•*2015*

We consider an extension of classical stochastic reaction-diffusion (RD) dynamics to open quantum systems. We study a class of models of hard core particles on a one-dimensional lattice whose dynamics is generated by a quantum master operator and where particle hopping is coherent while reactions, such as pair annihilation or pair coalescence, are dissipative. These are quantum open generalisations of the A+A→∅ and A+A→A classical RD models. We characterise the relaxation of the state towards the stationary regime via a decomposition of the system Hilbert space into transient and recurrent subspaces. We provide a complete classification of the structure of the recurrent subspace (and the non-equilibrium steady states) in terms of the dark states associated to the quantum master operator and its general spectral properties. We also show that, in one dimension, relaxation towards these absorbing dark states is slower than that predicted by a mean-field analysis due to fluctuation effects, in analogy with what occurs in classical RD systems. Numerical simulations of small systems suggest that the decay of the density in one dimension, in both the open quantum A+A→∅ and A+A→A cases, may go asymptotically as t^(-b) with 1/2 < b < 1.

*van Horssen, M., Guta, M.*•J. Math. Phys. 56, 022109•*2015*

In this paper we consider the statistics of repeated measurements on the output of a quantum Markov chain. We establish a large deviations result analogous to Sanov's theorem for the empirical measure associated to finite sequences of consecutive outcomes of a classical stochastic process. Our result relies on the construction of an extended quantum transition operator (which keeps track of previous outcomes) in terms of which we compute moment generating functions, and whose spectral radius is related to the large deviations rate function. As a corollary to this we obtain a central limit theorem for the empirical measure. Such higher level statistics may be used to uncover critical behaviour such as dynamical phase transitions, which are not captured by lower level statistics such as the sample mean. As a step in this direction we give an example of a finite system whose level-one rate function is independent of a model parameter while the level-two rate is not.

*van Horssen, M., Guta, M.*•arXiv preprint, arXiv:1206.4956•*2013*

The theory of quantum jump trajectories provides a new framework for understanding dynamical phase transitions in open systems. A candidate for such transitions is the atom maser, which for certain parameters exhibits strong intermittency in the atom detection counts, and has a bistable stationary state. Although previous numerical results suggested that the "free energy" may not be a smooth function, we show that the atom detection counts satisfy a large deviations principle, and therefore we deal with a phase crossover rather than a genuine phase transition. We argue however that the latter occurs in the limit of infinite pumping rate. As a corollary, we obtain the Central Limit Theorem for the counting process.

The proof relies on the analysis of a certain deformed generator whose spectral bound is the limiting cumulant generating function. The latter is shown to be smooth, so that a large deviations principle holds by the Gärtner-Ellis Theorem. One of the main ingredients is the Krein-Rutman Theorem which extends the Perron-Frobenius theory to a general class of positive compact semigroups.

*Lesanovsky, I., van Horssen, M., Guta, M., Garrahan, J.P.*•Phys. Rev. Lett. 110, 150401•*2013*

We describe how to characterize dynamical phase transitions in open quantum systems from a purely dynamical perspective, namely, through the statistical behavior of quantum jump trajectories. This approach goes beyond considering only properties of the steady state. While in small quantum systems dynamical transitions can only occur trivially at limiting values of the controlling parameters, in many-body systems they arise as collective phenomena and within this perspective they are reminiscent of thermodynamic phase transitions. We illustrate this in open models of increasing complexity: a three-level system, a dissipative version of the quantum Ising model, and the micromaser. In these examples dynamical transitions are accompanied by clear changes in static behavior. This is however not always the case, and in general dynamical phases need to be uncovered by observables which are strictly dynamical, e.g. dynamical counting fields. We demonstrate this via the example of a class of models of dissipative quantum glasses, whose dynamics can vary widely despite having identical (and trivial) stationary states.

*Catana, C., van Horssen, M., Guta, M.*•Phil. Trans. R. Soc. A, Vol. 370, No. 1979, pp. 5308–5323•*2012*

System identification is closely related to control theory and plays an increasing role in quantum engineering. In the quantum set-up, system identification is usually equated to process tomography, i.e. estimating a channel by probing it repeatedly with different input states. However, for quantum dynamical systems such as quantum Markov processes, it is more natural to consider the estimation based on continuous measurements of the output, with a given input that may be stationary. We address this problem using asymptotic statistics tools, for the specific example of estimating the Rabi frequency of an atom maser. We compute the Fisher information of different measurement processes as well as the quantum Fisher information of the atom maser, and establish the local asymptotic normality of these statistical models. The statistical notions can be expressed in terms of spectral properties of certain deformed Markov generators, and the connection to large deviations is briefly discussed.

*van Horssen, M.*•PhD thesis, The University of Nottingham•Download (PDF)•*2014*

Supervisor: Dr. Madalin Guta

University of Nottingham, Nottingham, United Kingdom • * 1 October, 2015*

University of Nottingham, Nottingham, United Kingdom • * 22-23 June, 2015*

King's College, London, United Kingdom• * 18-19 December, 2014*

University of Nottingham, Nottingham, United Kingdom • *29-30 May, 2014*

University of Nottingham, Nottingham, United Kingdom • *4-5 July 2013*

University of Nottingham, Nottingham, United Kingdom • *7-9 November 2012*

University of Nottingham, Nottingham, United Kingdom • *1-3 July 2012.*

University of Nottingham, Nottingham, United Kingdom • *25-26 June 2012*

Nicolaus Copernicus University, Torun, Poland • *20-24 June 2012*

Aberystwyth University, Aberystwyth, United Kingdom• *24-25 May 2012*

Imperial College London, London, United Kingdom• *24-27 January 2012*

International Centre for Mathematical Sciences, Edinburgh, United Kingdom• *16-18 January 2012*

The Kavli Royal Society International Centre, Chicheley, United Kingdom • *12-13 December 2011*

Aberystwyth University, Aberystwyth, United Kingdom • * 25-26 January 2011*

Ladek Zdroj, Poland• *8-13 February 2010*

Third Nottingham Workshop on Quantum Non-equilibrium Dynamics• University of Nottingham, Nottingham, United Kingdom• *22June 2015*

Mathematical physics of non-equilibrium quantum systems• King's College, London, United Kingdom• *18 December 2014*

Quantum information seminar• University of Nottingham, Nottingham, United Kingdom• *12 February 2014*

Quantum information seminar• University of Nottingham, Nottingham, United Kingdom• *18 March 2013*

Student conference on mathematical and theoretical aspects of Quantum Mechanics• University of Nottingham, Nottingham, United Kingdom• *7 November 2012*

QuaCS Quantum Correlations Students Workshop• University of Nottingham, Nottingham, United Kingdom• *3 July 2012*

Workshop on New Directions in Quantum Statistics• University of Nottingham, Nottingham, United Kingdom• *25 June 2012*

44 Symposium on Mathematical Physics• Nicolaus Copernicus University, Torun , Poland• *22 June 2012*

Invited speaker at Analysis of Quantum Control and Open Systems• Aberystwyth University, Aberystwyth, United Kingdom• *24 May 2012*

Postgraduate probability seminar• University of Nottingham, Nottingham, United Kingdom• *14 March 2012*

Merlijn van Horssen

LinkedIn

merlijn
@gmail.com