Geometric inequalities from phase space translations

We establish a quantum version of the classical isoperimetric inequality relating the Fisher information and the entropy power of a quantum state. The key tool is a Fisher information inequality for a state which results from a certain convolution operation: the latter maps a classical probability distribution on phase space and a quantum state to a quantum state. We show that this inequality also gives rise to several related inequalities whose counterparts are well-known in the classical setting: in particular, it implies an entropy power inequality for the mentioned convolution operation as well as the isoperimetric inequality, and establishes concavity of the entropy power along trajectories of the quantum heat diffusion semigroup. As an application, we derive a Log-Sobolev inequality for the quantum Ornstein-Uhlenbeck semigroup, and argue that it implies fast convergence towards the fixed point for a large class of initial states.Comment: 37 pages; updated to match published versio

Nested Integrals and Rationalizing Transformations

A brief overview of some computer algebra methods for computations with nested integrals is given. The focus is on nested integrals over integrands involving square roots. Rewrite rules for conversion to and from associated nested sums are discussed. We also include a short discussion comparing the holonomic systems approach and the differential field approach. For simplification to rational integrands, we give a comprehensive list of univariate rationalizing transformations, including transformations tuned to map the interval $[0,1]$ bijectively to itself.Comment: manuscript of 25 February 2021, in "Anti-Differentiation and the Calculation of Feynman Amplitudes", Springe

Non-Markovian Dynamics and Entanglement of Two-level Atoms in a Common Field

We derive the stochastic equations and consider the non-Markovian dynamics of a system of multiple two-level atoms in a common quantum field. We make only the dipole approximation for the atoms and assume weak atom-field interactions. From these assumptions we use a combination of non-secular open- and closed-system perturbation theory, and we abstain from any additional approximation schemes. These more accurate solutions are necessary to explore several regimes: in particular, near-resonance dynamics and low-temperature behavior. In detuned atomic systems, small variations in the system energy levels engender timescales which, in general, cannot be safely ignored, as would be the case in the rotating-wave approximation (RWA). More problematic are the second-order solutions, which, as has been recently pointed out, cannot be accurately calculated using any second-order perturbative master equation, whether RWA, Born-Markov, Redfield, etc.. This latter problem, which applies to all perturbative open-system master equations, has a profound effect upon calculation of entanglement at low temperatures. We find that even at zero temperature all initial states will undergo finite-time disentanglement (sometimes termed "sudden death"), in contrast to previous work. We also use our solution, without invoking RWA, to characterize the necessary conditions for Dickie subradiance at finite temperature. We find that the subradiant states fall into two categories at finite temperature: one that is temperature independent and one that acquires temperature dependence. With the RWA there is no temperature dependence in any case.Comment: 17 pages, 13 figures, v2 updated references, v3 clarified results and corrected renormalization, v4 further clarified results and new Fig. 8-1

Amino Acid Classification in 2D NMR Spectra via Acoustic Signal Embeddings

Nuclear Magnetic Resonance (NMR) is used in structural biology to experimentally determine the structure of proteins, which is used in many areas of biology and is an important part of drug development. Unfortunately, NMR data can cost thousands of dollars per sample to collect and it can take a specialist weeks to assign the observed resonances to specific chemical groups. There has thus been growing interest in the NMR community to use deep learning to automate NMR data annotation. Due to similarities between NMR and audio data, we propose that methods used in acoustic signal processing can be applied to NMR as well. Using a simulated amino acid dataset, we show that by swapping out filter banks with a trainable convolutional encoder, acoustic signal embeddings from speaker verification models can be used for amino acid classification in 2D NMR spectra by treating each amino acid as a unique speaker. On an NMR dataset comparable in size with of 46 hours of audio, we achieve a classification performance of 97.7% on a 20-class problem. We also achieve a 23% relative improvement by using an acoustic embedding model compared to an existing NMR-based model

Multiple-Point and Multiple-Time Correlations Functions in a Hard-Sphere Fluid

A recent mode coupling theory of higher-order correlation functions is tested on a simple hard-sphere fluid system at intermediate densities. Multi-point and multi-time correlation functions of the densities of conserved variables are calculated in the hydrodynamic limit and compared to results obtained from event-based molecular dynamics simulations. It is demonstrated that the mode coupling theory results are in excellent agreement with the simulation results provided that dissipative couplings are included in the vertices appearing in the theory. In contrast, simplified mode coupling theories in which the densities obey Gaussian statistics neglect important contributions to both the multi-point and multi-time correlation functions on all time scales.Comment: Second one in a sequence of two (in the first, the formalism was developed). 12 pages REVTeX. 5 figures (eps). Submitted to Phys.Rev.

On thermal ionization for open quantum systems

This thesis addresses the phenomenon of ionization of an idealized atom by a surrounding infinitely extended quantized electromagnetic field at positive temperature. According to Planck's law one expects photons with arbitrary high energy, which eventually exceed the ionization threshold of the atom. Mathematically, this can be interpreted as the absence of time-invariant normal states in a suitable dynamical system. Such problems can be converted into a spectral-theoretical question by means of the self-adjoint generator of the time evolution - the Liouvillian L. In this setting it suffices to show that zero is not an eigenvalue of L. The goal of this thesis is the proof of thermal ionization for more concrete models with less restrictions than in previous works, including a QED-like coupling term with a spatial decay and an idealized atom, given as Schrödinger operator. With respect to the atom it will be differentiated between two cases: first, potentials with infinitely many bound states, but only finitely many coupled to the field, and second, compactly supported smooth potentials. For the latter there are, apart from the spatial decay, no further artificial restrictions required in the coupling. The proof is based on positive commutators. On the field it uses the generator of translations, and for the atom the generator of dilations (first case) or the generator of dilations in the space of scattering functions (second case). By means of an approximated version of Fermi's Golden Rule one obtains a uniform result in every bounded temperature range

Inference of transport phenomena in quantum devices

This thesis is concerned with charge transport in electrostatically defined quantum dot devices. Such devices display a wide range of transport phenomena in both open and closed configurations. The transport regime can be tuned experimentally by controlling the voltages applied to gate electrodes, but the precise electrostatic landscape which determines the transport regime is unknown. This uncertainty is given by variations in device fabrication, material defects, and sources of electrostatic disorder. The research chapters of this thesis consider a range of transport regimes in quantum dot devices, and infer properties of the device using both experimental and theoretical techniques. The first research chapter considers the detection of single charge transport events through a double quantum dot. By fitting an open quantum systems model to the sub-attoampere currents measured, tunnel rates are inferred. The second results chapter considers an electrostatic simulation of a quantum dot device and how it can be accelerated using deep learning. This accelerated model is then used in the third results chapter, along with experimental measurements of the transport regime, to inform a Bayesian inference algorithm and produce a set of disorder potentials to narrow the gap between simulation and reality. The final results chapter develops a differentiable quantum master equation solver which is used for parameter estimation in a theoretical study of transport in single and double quantum dots