A stochastic approach to open quantum systems

Stochastic methods are ubiquitous to a variety of fields, ranging from Physics to Economy and Mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in contact with a somehow bigger system, an environment, that is considered in thermal equilibrium. Any small fluctuation of the environment has some random effect on the system. In Physics, stochastic methods have been applied to the investigation of phase transitions, thermal and electrical noise, thermal relaxation, quantum information, Brownian motion etc. In this review, we will focus on the so-called stochastic Schr\"odinger equation. This is useful as a starting point to investigate the dynamics of open quantum systems capable of exchanging energy and momentum with an external environment. We discuss in some details the general derivation of a stochastic Schr\"odinger equation and some of its recent applications to spin thermal transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly discuss the advantages of this formalism with respect to the more common approach in terms of the reduced density matrix. The applications discussed here constitute only a few examples of a much wider range of applicability.Comment: 43 pages, 9 figures, iopart style, published in Journal of Physics: Condensed Matte

Application of a time-convolutionless stochastic Schrödinger equation to energy transport and thermal relaxation

arXiv:1203.3785v3Quantum stochastic methods based on effective wave functions form a framework for investigating the generally non-Markovian dynamics of a quantum-mechanical system coupled to a bath. They promise to be computationally superior to the master-equation approach, which is numerically expensive for large dimensions of the Hilbert space. Here, we numerically investigate the suitability of a known stochastic Schrödinger equation that is local in time to give a description of thermal relaxation and energy transport. This stochastic Schrödinger equation can be solved with a moderate numerical cost, indeed comparable to that of a Markovian system, and reproduces the dynamics of a system evolving according to a general non-Markovian master equation. After verifying that it describes thermal relaxation correctly, we apply it for the first time to the energy transport in a spin chain. We also discuss a portable algorithm for the generation of the coloured noise associated with the numerical solution of the non-Markovian dynamics.RB and RDA acknowledge support from MICINN (FIS2010-21282-C02-01 and PIB2010US-00652), the Grupos Consolidados UPV/EHU del Gobierno Vasco (IT-319-07) and ACI-Promociona (ACI2009-1036), and the financial support of CONSOLIDER-INGENIO 2010 NanoTherm (CSD2010-00044). RB acknowledges financial support from IKERBASQUE, Basque Foundation for Science and the Ministerio de Educación, Cultura y Deporte (FPU12/01576). CT acknowledges financial support from Deutsche Forschungsgemeinschaft, in part through Research Unit FOR 1154 Towards Molecular Spintronics. RD'A acknowledges support from Diputacion Foral de Gipuzkoa via grant number Q4818001B.Peer Reviewe

Modelling thermo-electric transport and excited states in low dimensional systems

117 p.The interaction of radiation with matter at the nanoscale has an inexhaustible range of applicationsin electronics, biotechnology and medicine. At the nanoscale, the length scale wherethe classical and quantum worlds meet, quantum effects dominate the light¿matter interactionand unique phenomena arise. This work addresses fundamental questions on the overlap ofquantum theory, non-equilibrium thermodynamics and material science.As the exact description of these quantum phenomena is not feasible, we discuss how the openquantum system approach can be used to study thermal relaxation and thermo-electric transportat the nanoscale. The basic concepts of thermal relaxation are studied from first principles. Asthe conditions for relaxation are connected with the non-Markovian nature of the equation ofmotion, we discuss a time-local stochastic Schr¿odinger equation. Remarkably, this equationcan describe thermal relaxation and transport dynamics correctly. Furthermore, this thesis introducesa thermal transport theory where the temperature field is established by radiation ofclassical blackbodies. The combination of this theory with the techniques of time-dependentcurrent density functional theory provides an ab initio tool to study thermal transport in manybodysystems. This approach is general and can be adapted to describe both electron andphonon dynamics. In this way, combined with the time-dependent current DFT, it provides aunified way to investigate ab initio electrical and thermal transport beyond linear response. Theobservation of thermo-electric transport in macroscopic bodies does not disturb the system orchange the flow of energy. However, when moving towards the nanoscale, measurements mayinfluence the system and has to be considered. We demonstrate that the choice of location ofthese local measurements provides control of the direction of the energy flow and of the particlecurrents separately. These results seem to violate the second law of thermodynamics. Bytreating decoherence as a thermodynamic bath we resolve this contradiction. In order to furtheradvance the applications of light¿matter interactions for realisable materials, the electronic andoptical properties of 2D layered semiconductors are studied. 2D materials have establishedtheir place as candidates for the next generation of opto-electronic devices. Specifically, theelectronic and optical properties of TiS3 and In2Se3 are theoretically investigated within DFTand many-body perturbation theory. This work constitutes a first step towards exploiting thetrichalcogenide family in 2D opto-electronical applications, such as chemical sensors, passiveoptical polarisers, fast photodetectors, and battery technologies

Titanium trisulfide (TiS3): a 2D semiconductor with quasi-1D optical and electronic properties

We present characterizations of few-layer titanium trisulfide (TiS3) flakes which, due to their reduced in-plane structural symmetry, display strong anisotropy in their electrical and optical properties. Exfoliated few-layer flakes show marked anisotropy of their in-plane mobilities reaching ratios as high as 7.6 at low temperatures. Based on the preferential growth axis of TiS3 nanoribbons, we develop a simple method to identify the in-plane crystalline axes of exfoliated few-layer flakes through angle resolved polarization Raman spectroscopy. Optical transmission measurements show that TiS3 flakes display strong linear dichroism with a magnitude (transmission ratios up to 30) much greater than that observed for other anisotropic two-dimensional (2D) materials. Finally, we calculate the absorption and transmittance spectra of TiS3 in the random-phase-approximation (RPA) and find that the calculations are in good agreement with the observed experimental optical transmittance.Comment: 18 pages, 4 figures, including Supporting Information (6 pages, 6 figures

Hierarchies of Hofstadter butterflies in 2D covalent-organic frameworks

The Hofstadter butterfly is one of the first and most fascinating examples of the fractal and self-similar quantum nature of free electrons in a lattice pierced by a perpendicular magnetic field. However, the direct experimental verification of this effect on single-layer materials is still missing as very strong and inaccessible magnetic fields are necessary. For this reason, its indirect experimental verification has only been realized in artificial periodic 2D systems, like moir\'e lattices. The only recently synthesized 2D covalent-organic frameworks might circumvent this limitation: Due to their large pore structures, magnetic fields needed to detect most features of the Hofstadter butterfly are indeed accessible with today's technology. This work opens the door to making this exotic and theoretical issue from the 70s measurable and might solve the quest for the experimental verification of the Hofstadter butterfly in single-layer materials. Moreover, the intrinsic hierarchy of different pore sizes in a 2D covalent-organic framework adds additional complexity and beauty to the original butterflies and leads to a directly accessible playground for new physical observations

Hierarchies of Hofstadter butterflies in 2D covalent organic frameworks

The Hofstadter butterfly is one of the first and most fascinating examples of the fractal and self-similar quantum nature of free electrons in a lattice pierced by a perpendicular magnetic field. However, the direct experimental verification of this effect on single-layer materials is still missing as very strong and inaccessible magnetic fields are necessary. For this reason, its indirect experimental verification has only been realized in artificial periodic 2D systems, like moiré lattices. The only recently synthesized 2D covalent organic frameworks might circumvent this limitation: Due to their large pore structures, magnetic fields needed to detect most features of the Hofstadter butterfly are indeed accessible with today technology. This work opens the door to make this exotic and theoretical issue from the 70s measurable and might solve the quest for the experimental verification of the Hofstadter butterfly in single-layer materials. Moreover, the intrinsic hierarchy of different pore sizes in 2D covalent organic framework adds additional complexity and beauty to the original butterflies and leads to a direct accessible playground for new physical observations

Modelling thermo-electric transport and excited states in low dimensional systems

A thesis submitted in partial fulfillment for the degree of Doctor of Philosophy in the Faculty of Physics, Chemistry and Materials Science, Department of Materials Physics.The interaction of radiation with matter at the nanoscale has an inexhaustible range of applications in electronics, biotechnology and medicine. At the nanoscale, the length scale where the classical and quantum worlds meet, quantum effects dominate the light–matter interaction and unique phenomena arise. This work addresses fundamental questions on the overlap of quantum theory, non-equilibrium thermodynamics and material science. As the exact description of these quantum phenomena is not feasible, we discuss how the open quantum system approach can be used to study thermal relaxation and thermo-electric transport at the nanoscale. The basic concepts of thermal relaxation are studied from first principles. As the conditions for relaxation are connected with the non-Markovian nature of the equation of motion, we discuss a time-local stochastic Schrödinger equation. Remarkably, this equation can describe thermal relaxation and transport dynamics correctly. Furthermore, this thesis introduces a thermal transport theory where the temperature field is established by radiation of classical blackbodies. The combination of this theory with the techniques of time-dependent current density functional theory provides an ab initio tool to study thermal transport in manybody systems. This approach is general and can be adapted to describe both electron and phonon dynamics. In this way, combined with the time-dependent current DFT, it provides a unified way to investigate ab initio electrical and thermal transport beyond linear response. The observation of thermo-electric transport in macroscopic bodies does not disturb the system or change the flow of energy. However, when moving towards the nanoscale, measurements may influence the system and has to be considered. We demonstrate that the choice of location of these local measurements provides control of the direction of the energy flow and of the particle currents separately. These results seem to violate the second law of thermodynamics. By treating decoherence as a thermodynamic bath we resolve this contradiction. In order to further advance the applications of light–matter interactions for realisable materials, the electronic and optical properties of 2D layered semiconductors are studied. 2D materials have established their place as candidates for the next generation of opto-electronic devices. Specifically, the electronic and optical properties of TiS3 and In2Se3 are theoretically investigated within DFT and many-body perturbation theory. This work constitutes a first step towards exploiting the trichalcogenide family in 2D opto-electronical applications, such as chemical sensors, passive optical polarisers, fast photodetectors, and battery technologies.Peer reviewe