Martin Horvat

Exponential complexity of an adiabatic algorithm for an NP-complete problem

We prove an analytical expression for the size of the gap between the ground and the first excited state of quantum adiabatic algorithm for the 3-satisfiability, where the initial Hamiltonian is a projector on the subspace complementary to the ground state. For large problem sizes the gap decreases exponentially and as a consequence the required running time is also exponential.

Transport in a disordered tight-binding chain with dephasing

We studied transport properties of a disordered tight-binding model (XX spin chain) in the presence of dephasing. Focusing on diffusive behaviour in the thermodynamic limit at high energies, we analytically derived the dependence of conductivity on dephasing and disorder strengths. As a function of dephasing, conductivity exhibits a single maximum at the optimal dephasing strength. The scaling of the position of this maximum with disorder strength is different for small and large disorders. In addition, we studied periodic disorder for which we found a resonance phenomenon, with conductivity having two maxima as a function of dephasing strength. If the disorder is non-zero only at a random fraction of all sites, conductivity is approximately the same as in the case of a disorder on all sites but with a rescaled disorder strength.

The discriminatory value of cardiorespiratory interactions in distinguishing awake from anaesthetised states: a randomised observational study

Depth of anaesthesia monitors usually analyse cerebral function with or without other physiological signals; non‐invasive monitoring of the measured cardiorespiratory signals alone would offer a simple, practical alternative. We aimed to investigate whether such signals, analysed with novel, non‐linear dynamic methods, would distinguish between the awake and anaesthetised states. We recorded ECG, respiration, skin temperature, pulse and skin conductivity before and during general anaesthesia in 27 subjects in good cardiovascular health, randomly allocated to receive propofol or sevoflurane. Mean values, variability and dynamic interactions were determined. Respiratory rate (p = 0.0002), skin conductivity (p = 0.03) and skin temperature (p = 0.00006) changed with sevoflurane, and skin temperature (p = 0.0005) with propofol. Pulse transit time increased by 17% with sevoflurane (p = 0.02) and 11% with propofol (p = 0.007). Sevoflurane reduced the wavelet energy of heart (p = 0.0004) and respiratory (p = 0.02) rate variability at all frequencies, whereas propofol decreased only the heart rate variability below 0.021 Hz (p < 0.05). The phase coherence was reduced by both agents at frequencies below 0.145 Hz (p < 0.05), whereas the cardiorespiratory synchronisation time was increased (p < 0.05). A classification analysis based on an optimal set of discriminatory parameters distinguished with 95% success between the awake and anaesthetised states. We suggest that these results can contribute to the design of new monitors of anaesthetic depth based on cardiovascular signals alone.

Uni-directional transport properties of a serpent billiard

We present a dynamical analysis of a classical billiard chain—a channel with parallel semi-circular walls, which can serve as a model for a bent optical fibre. An interesting feature of this model is the fact that the phase space separates into two disjoint invariant components corresponding to the left and right uni-directional motions. Dynamics is decomposed into the jump map, a Poincare map between the two ends of a basic cell, and the time function, travelling time across a basic cell of a point on a surface of section. The jump map has a mixed phase space where the relative sizes of the regular and chaotic components depend on the width of the channel. For a suitable value of this parameter, we can have almost fully chaotic phase space. We have studied numerically the Lyapunov exponents, time auto-correlation functions and diffusion of particles along the chain. As a result of the singularity of the time function, we obtain marginally normal diffusion after we subtract the average drift. The last result is also supported by some analytical arguments.

Wigner function statistics in classically chaotic systems

We have studied statistical properties of the values of the Wigner function W(x) of 1D quantum maps on compact 2D phase space of finite area V. For this purpose we have defined a Wigner function probability distribution P(w) = (1/V) ∫ δ(w − W(x))dx, which has, by definition, fixed first and second moments. In particular, we concentrate on relaxation of time-evolving quantum states in terms of W(x), starting from a coherent state. We have shown that for a classically chaotic quantum counterpart the distribution P(w) in the semiclassical limit becomes a Gaussian distribution that is fully determined by the first two moments. Numerical simulations have been performed for the quantum sawtooth map and the quantized kicked top. In a quantum system with Hilbert space dimension N(~1/) the transition of P(w) to a Gaussian distribution was observed at times t ∝ log N. In addition, it has been shown that the statistics of Wigner functions of propagator eigenstates is Gaussian as well in the classically fully chaotic regime. We have also studied the structure of the nodal cells of the Wigner function, in particular the distribution of intersection points between the zero manifold and arbitrary straight lines.

Railway switch transport model.

We propose a simple model of coupled heat and particle transport based on zero-dimensional classical deterministic dynamics, which is reminiscent of a railway switch whose action is a function only of the particles energy. It is shown that already in the minimal three-terminal model, where the second terminal is considered as a probe with zero net particle and heat currents, one can find extremely asymmetric Onsager matrices as a consequence of time-reversal symmetry breaking of the model. This minimalistic transport model provides a better understanding of thermoelectric heat engines in the presence of time-reversal symmetry breaking.

Relativistic Positioning System in perturbed spacetime

We present a variant of a Global Navigation Satellite System called a Relativistic Positioning System (RPS), which is based on emission coordinates. We modelled the RPS dynamics in a space-time around Earth, described by a perturbed Schwarzschild metric, where we included the perturbations due to Earth multipoles (up to the 6th), the Moon, the Sun, Venus, Jupiter, solid tide, ocean tide, and Kerr rotation effect. The exchange of signals between the satellites and a user was calculated using a ray-tracing method in the Schwarzschild space-time. We find that positioning in a perturbed space-time is feasible and is highly accurate already with standard numerical procedures: the positioning algorithms used to transform between the emission and the Schwarzschild coordinates of the user are very accurate and time efficient -- on a laptop it takes 0.04 s to determine the users spatial and time coordinates with a relative accuracy of

Basics of Numerical Analysis

10^{-28}-10^{-26}

The bends on a quantum waveguide and cross-products of Bessel functions

and

The ensemble of random Markov matrices

10^{-32}-10^{-30}