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arXiv:1501.04046

Interaction as stochastic noise

Roberto D’Agosta

Published Jan 16, 2015• 0 citations
arXivHTMLPDF

Abstract

Interaction is so ubiquitous that imaging a world free from it is a difficult fantasy exercise. At the same time, in understanding any complex physical system, our ability of accounting for the mutual interaction of its constituents is often insufficient when not the restraining factor. Many strategies have been devised to control particle-particle interaction and explore the diverse regimes, from weak to strong interaction. Beautiful examples of these achievements are the experiments on Bose condensates, or the recent experiments on the dynamics of spin chains. Here I introduce another possibility, namely replacing the particle-particle interaction with an external stochastic field, and once again reducing the dynamics of a many-body system to the dynamics of single-particle systems. The theory is exact, in the sense that no approximations are introduced in decoupling the many-body system in its non-interacting sub-parts. Moreover, the equations of motion are linear, and no unknown external potential is inserted.

DOI
10.48550/arxiv.1501.04046

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Interaction is so ubiquitous that imaging a world free from it is a difficult fantasy exercise. At the same time, in understanding any complex physical system, our ability of accounting for the mutual interaction of its constituents is often insufficient when not the restraining factor. Many strategies have been devised to control particle-particle interaction and explore the diverse regimes, from weak to strong interaction. Beautiful examples of these achievements are the experiments on Bose condensates, or the recent experiments on the dynamics of spin chains. Here I introduce another possibility, namely replacing the particle-particle interaction with an external stochastic field, and once again reducing the dynamics of a many-body system to the dynamics of single-particle systems. The theory is exact, in the sense that no approximations are introduced in decoupling the many-body system in its non-interacting sub-parts. Moreover, the equations of motion are linear, and no unknown external potential is inserted.
Stochastic methods are ubiquitous to a variety of fields, ranging from physics to economics 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 somewhat bigger system, an environment with which it 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 and so on. In this review, we will focus on the so-called stochastic Schrödinger 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 detail the general derivation of a stochastic Schrödinger 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.
The study of open quantum systems forms one of the main problems of modern physics. This thesis aims at contributing to this field, and focuses on the investigation of non-Markovian dynamics of open quantum systems. Here, as a special case, quantum many-body systems provide for a rich field of research. With the ability to control them being of essential importance for the development of quantum technologies, the characterization of their dynamical properties has also drawn a lot of interest lately. One of the main methodological tasks in the study of open quantum systems is to reduce the infinite degrees of freedom of the total system. In order to achieve this, the thesis makes use of the tensor network method matrix product states and combines it with the picture of the quantum stochastic Schrödinger equation. However, when computing the dynamics of many-body systems of system size N, even the reduced system dynamics scale with 2N, making larger system sizes difficult to access. Therefore, especially when computing non-Markovian time-dynamics where the information backflow from the reservoir has to be taken into account, efficiently constructed algorithms are required. This thesis proposes these type of algorithms and demonstrates their application. They enable an access to larger system sizes, thus making it possible to investigate non-equilibrium steady-states of quantum many-body systems in a non-Markovian interaction for system sizes up to N=30. This thesis consists of three parts. The first part provides an introduction to the theory of open quantum systems and to the methodology for modeling them. The second part examines many-body system-reservoir interaction in the non-Markovian regime by investigating the case of fully coherent self-feedback. It uses the one-dimensional Heisenberg spin-1/2 chain as a paradigmatic and recently intensely studied many-body system. It is demonstrated that this gives rise to new conditions for the feedback phase and to highly non-trivial states within the system as well as to the possibility of partially characterizing the system non-invasively and storing excitation within it. Next, a novel approach to the boundary-driven Heisenberg chain is presented, where the chain is exposed to a coherent driving field, thus enabling a comparison to the well-studied incoherently driven model. The third part demonstrates the use of an external pump to control the feedback phase of a Lambda-type three-level system, where the external pump pulse is shown to be a new control parameter. The thesis concludes with the proposition to enhance the indistinguishability of two photons by feedback control.
This work investigates the theoretical and experimental decoherence of phonons in weakly and strongly-interacting Bose-Einstein Condensates (BECs). The theoretical analysis treats phonons as open quantum systems where the environment comprises all other quasi-particle modes of the BEC. The phonons are assumed to be Gaussian states and the time in which they decohere is estimated from the evolution of their purity and nonclassical depth in the dissipative channel. The calculations are performed for various BEC systems and it is found that the excited phonon states will always decohere much more rapidly than the rate at which they relax back to equilibrium with the environment. Part II of this work considers how the decoherence of the phonons can be measured experimentally. The experiment that is currently being investigated uses a strongly-interacting [superscript]6Li BEC where the scattering length can be varied with an external magnetic field. In a strongly-interacting Bose gas the mutual interaction between the condensed and noncondensed components plays a greater role than in weakly-interacting gases and results in distinct absorption images. Understanding the effects from this mutual interaction is vital to model the in-situ absorption images of a strongly-interacting BEC in order to extract accurate information such as that which will facilitate the experimental measurement of decoherence. Three theoretical models that could be used to fit in-situ absorption images of a strongly-interacting gas are analysed. These are the bi-modal, semi-ideal and Hartree-Fock models, which will be fit against the absorption images of the [superscript]6Li BEC for various scattering lengths. The validity of these models is also investigated to determine when beyond mean-field effects may be observed in the Bose gas. Controlling decoherence is essential to the operation and physical realisation of many quantum information tasks and quantum technologies. Recently, new technologies have emerged from relativistic quantum information science that, in principle, are more precise than their non-relativistic counterparts. The practical setup of these devices utilizes phonons of BECs but the decoherence of the phonons has not been considered. In this work, one of the BEC systems used to estimate the decoherence time of the phonons has been chosen to be based on these devices. This calculation is expected to inform the practical realization of these devices and inspire future related studies. Analogue gravity investigations based on BECs also utilize phonons, for example, in the analogue of Hawking radiation. The quantum properties of these states is of particular interest in these studies and understanding how they decohere, and at what rate, could potentially inform the theory of black hole physics as well as dictate what is possible to measure experimentally.
Understanding the non-equilibrium quantum dynamics of many-body systems is one of the most challenging problems in modern theoretical physics. While numerous approximate and exact solutions exist for systems in equilibrium, examples of non-equilibrium dynamics of many-body systems that allow reliable theoretical analysis are few and far between. In this paper, we discuss a broad class of time-dependent interacting systems subject to external linear and parabolic potentials, for which the many-body Schrödinger equation can be solved using a scaling transformation. We demonstrate that scaling solutions exist for both local and non-local interactions, and derive appropriate self-consistency equations. We apply this approach to several specific experimentally relevant examples of interacting bosons in one and two dimensions. As an intriguing result, we find that weakly and strongly interacting Bose gases expanding from a parabolic trap can exhibit very similar dynamics.