Gentle Perturbations of the Free Bose Gas I

It is demonstrated that the thermal structure of the noncritical free Bose Gas is completely described by certain periodic generalized Gaussian stochastic process or equivalently by certain periodic generalized Gaussian random field. Elementary properties of this Gaussian stochastic thermal structure have been established. Gentle perturbations of several types of the free thermal stochastic structure are studied. In particular new models of non-Gaussian thermal structures have been constructed and a new functional integral representation of the corresponding euclidean-time Green functions have been obtained rigorously.Comment: 51 pages, LaTeX fil

Quantum Brownian motion in Bose-Einstein condensates

Quantum Brownian motion is one of the most prominent examples of an open quantum system, a system which cannot be treated in isolation from its environment. The simplest method to study the dynamics of a system undergoing such a type of motion, that satisfies Heisenberg Uncertainty principle is the approach of Quantum Generalized Langevin Equations (QGLE), which was used throughout this thesis. A Quantum Brownian motion approach is used in this work to study the Bose polaron problem. In this case, one transforms the original problem into one where the impurities are treated as quantum Brownian particles interacting with a bath composed of the Bogoliubov modes of the condensate. Then by deriving the relevant QGLE, it was shown that the dynamics of the Bose polaron exhibit memory effects. This was studied for both a free Bose-Einstein condensate (BEC) and a harmonically trapped one, in both cases for experimentally relevant parameters. Taking advantage of this recent theoretical development, we study a number of phenomena that can be examined under this prism and show how various microdevices can be constructed and controlled. In the first project, we study the creation of entanglement and squeezing of two uncoupled impurities that are immersed in a single common (BEC) bath. We treat these impurities as two quantum Brownian particles. We study two scenarios:(i) In the absence of an external potential, we observe sudden death of entanglement;(ii) In the presence of an external harmonic potential, where entanglement survives even at the asymptotic time limit. In our second work, we studied the diffusive behavior of a Bose Polaron immersed in a coherently coupled two-component BEC. The particle superdiffuses if it couples in the same manner to both components, i.e. if it couples either attractively or repulsively to both of them. This is the same behavior of an impurity immersed in a single BEC. Conversely, we find that it exhibits a transient nontrivial subdiffusive behavior if it couples attractively to one of the components and repulsively with the other. We show how the magnitude of the anomalous exponent reached and the duration of the subdiffusive interval can be controlled with the Rabi frequency of the coherent coupling between the two components and the coupling strength of the impurity to the BEC. Then we proceeded with the construction of two microdevices, a quantum sub-nk thermometer and a heat diode. In the first project, we introduced a novel minimally disturbing method for sub-nK thermometry in a BEC. In this case, the impurity acted as a thermometer, where one detects temperature fluctuations from measurements of the position and momentum of the impurity. Crucially, these cause minimal backaction on the BEC and hence, realize a nondemolition temperature measurement. Following the paradigm of the emerging field of quantum thermometry, we combine tools from quantum parameter estimation and the theory of open quantum systems to solve the problem in full generality. We thus avoid any simplification, such as demanding thermalization of the impurity atoms. In our final work, we investigated the heat transport and the control of heat current among two spatially separated trapped BECs, each of them at a different temperature. To allow for heat transport among the two independent BECs we consider a link made of two dipole-dipole interacting harmonically trapped impurities, each of them interacting with one of the BECs. We address the dependence of heat current and current-current correlations on the physical parameters of the system. Interestingly, we show that heat rectification, can occur in our system, when a periodic driving on the trapping frequencies of the impurities is considered. Therefore, our system is a possible setup for the implementation of a phononic circuit, and hence contributes in the general framework of using BECs as platforms for quantum information processing.El movimiento Browniano, es un ejemplo de un sistema abierto, es decir un sistema que no se puede tratar en aislamiento. El método más simple para estudiar la dinámica de dicho sistema, que cumple el principio de la incertidumbre de Heisenberg es el de Quantum Generalized Langevin Equations (QGLE), que es el método que se usa en esta tesis. La perspectiva de Quantum Brownian motion se ha usado para estudiar muchos sistemas, entre ellos el problema de Bose polaron. En este caso, uno pasa el problema original a uno donde las impurezas se tratan como partículas Brownianas quanticas interactuando con un baño compuesto de modos de Bogoliubov del condensado. Después de derivar la QGLE relevante, se puede demostrar que la dinámica del Bose polaron muestra efectos de memoria. Esto se ha estudiado tanto en un Bose Einstein Condensate (BEC) libre como en uno atrapado en un trapo harmónico, para parámetros relevantes en experimentos. Aprovechando de este reciente desarrollo, estudiamos muchos fenómenos que se pueden investigar bajo este prisma y mostramos cómo se pueden construir y controlar varios microdispositivos. En el primer proyecto, estudiamos la creación de enlazamiento y squeezing de dos impurezas no acopladas, inmersas en un único BEC baño común. Estudiamos dos senarios: (i) en la ausencia de un potencial externo, donde observamos la muerte repentina del enlazamiento (ii) en la presencia de un trapo externo harmónico, donde el enlazamiento sobrevive incluso en el límite asintótico de largos tiempos. En nuestro segundo trabajo, estudiamos el comportamiento difusivo de un Bose polaron inmerso en un BEC de dos componentes que están acopladas coherentemente. La partícula es superdffusa si se acopla en la misma manera a los dos componentes, i.e. atractivamente o repulsivamente, como en el caso de un unico BEC. En el caso contrario, encontramos que la partícula muestra un comportamiento transitorio non-trivial. Mostramos como la magnitud del exponente anómalo y la duración del periodo transitorio se pueden controlar a través de la frecuencia Rabi del acoplamiento coherente entre los dos componentes y la fuerza del acoplamiento de la impureza a los dos componentes del BEC. En seguida, procedemos con la construcción de dos microdispositivos, un termómetro quántico y un diodo térmico. En el primer proyecto, hemos introducido un nuevo método de mínimo disturbio, que sirve para termometría en temperaturas sub-nK en un BEC. Nuestra técnica está basada otra vez en el modelo de Bose polaron, donde esta vez la impureza inmersa en un BEC sirve como un termómetro. La propuesta es detectar fluctuaciones de la temperatura de las medidas de la posición y el impulso de la impureza. Crucialmente, estas causan una reacción mínima en el BEC y, por lo tanto, realizan una medida de la temperatura no demoledora. En nuestro trabajo, evitamos cualquiera simplificación, como la imposición de la termalización de la impureza, o del acoplamiento débil de la impureza con el BEC. En el último trabajo, investigamos el transporte de calor y el control de corrientes de calor entre dos BECs espacialmente separados y atrapados harmónicamente, en temperaturas distintas. El flujo de calor entre los dos BECs, esta facilitado a través de dos impurezas harmónicamente atrapadas, cada una interactuando con su propio BEC. Las impurezas estan acopladas a traves de interacciones de dipolo-dipolo. Examinamos la dependencia del corriente de calor y sus correlaciones en los parámetros físicos del sistema. Mostramos que la rectificación del corriente del calor, i.e. el flujo de calor unidireccional puede ocurrir en el sistema, cuando aplicamos una conducción periódica en las frecuencias de los trapos de las impurezas. Por lo tanto, nuestro sistema es una posible configuración para la implementación de un circuito fononico.Postprint (published version

Quantum Brownian motion revisited : extensions and applications

Quantum Brownian motion represents a paradigmatic model of open quantum system, namely a system which cannot be treated as an isolated one, because of the unavoidable interaction with the surrounding environment. In this case the central system is constituted by a quantum particle, while the bath is made up by a large set of uncoupled harmonic oscillators. In the original model, the interaction between the system and the environment shows a linear dependence on the particle position. Such a particular form corresponds to a homogeneous environment, inducing a damping and diffusion which depends on the state. This is not the most general situation: often the environment shows an inhomogeneous character given by a space-dependent density, involving a non-linearity in the coupling with the central system. One of the main motivations of the thesis is the study of quantum Brownian motion in presence of this non-linear coupling. In particular we focus on the case in which the bath-particle interaction depends quadratically on the position of the latter. There exist several techniques aimed to treat the physics of the model. For instance one could consider the master equation, namely an equation ruling the temporal evolution of the state of the Brownian particle, here represented by its reduced density matrix. We derive such an equation in the Born-Markov regime and look into its stationary solution, studying its configuration in the phase space. For a non-linear quadratic coupling the stationary state may be approximated by means of a Gaussian Wigner function, that experiences genuine position squeezing (i.e. the position variance of the particle takes a value smaller than that associated to the Heisenberg principle, although this is fulfilled) at low temperature and as the coupling with the bath grows. However, the Born-Markov master equation is not the most appropriate tool to investigate the regime in which squeezing occurs, since the underlying hypothesis in general fail at strong coupling and low temperature, leading to violations of the Heisenberg principle. To overcome this problem we recall alternative methods, such as a Lindblad equation, namely a master equation constructed to preserve the positivity of the state at any time, and Heisenberg equations. In particular we employ the Heisenberg equation formalism to explore the behavior of the Bose polaron, i.e. an impurity embedded in a Bose-Einstein condensate. This experimentally feasible system attracted a lot of attention in the last years. We derive the equation of motion of the impurity position showing that it shows the same form of the famous equation derived by Langevin in 1909 in the context of classical Brownian motion. The main difference lies in the fact that the impurity Langevin-like equation for the impurity carries a certain amount of memory effects, while the original one was purely Markovian. An important part of the work is devoted to the solution of the motion equation for the impurity, in order to calculate the position variance that can be measured in experiments. For this goal we distinguish the case in which the impurity is trapped in a harmonic potential and that where it is free of any trap. In the latter case the impurity the position variance exhibits a quadratic dependence on time (i.e. ballistic diffusion), as a consequence of memory effects. When the impurity is trapped in a harmonic potential it approaches an equilibrium state localized in average in the middle of the trap. Here, at low temperature and for certain values of the coupling strength we detect genuine position squeezing. When we consider a gas with a Thomas-Fermi profile we find that such an effect is improved if we make the gas trap tighter. Genuine squeezing plays an important role in the context of quantum metrology and opens a wide range of possibility to design new protocols, such as the quantum thermometerEl movimiento Browniano cuántico es uno de los principales modelos de sistema abierto, es decir un sistema cuyo comportamiento no se puede tratar de manera separada de su entorno. Este modelo describe la física de una partícula acoplada a un entorno de osciladores. En la versión original del modelo la interacción entre la partícula y el entorno manifiesta una dependencia lineal de la posición de ambos los sistemas. Esta forma analítica del acoplamiento corresponde a un entorno homogeneo, asociado a una fricción y una difusión que dependen del estado del sistema. En todo caso, esta no es la situación más general: a menudo el enorno es inhomogeneo, ya que la densidad no es constante, y esto produce una interacción cuya dependencia de la posición de la partícula no es lineal. Una de las motivaciones principales de esta tesis es el estudio del movimiento Browniano cuántico en presencia de acoplamiento non-lineal. En particular, estudiamos el caso de dependencia cuadrática en la posición de la partícula. Existen muchas técnicas para abordar el modelo. Por ejemplo, se puede emplear la master equation, o sea un ecuación que gobierna la evolución en el tiempo del estado de la partícula, representado por el operador densidad reducido. Derivamos esta ecuación en el régimen de Born-Markov, y estudiamos la forma del estado estacionario en el espacio de las fases. Cuando el acoplamiento es cuadrático, este estado se puede aproximar por medio de una función de Wigner de forma Gausiana, cuya peculiaridad es la emergencia de genuine position squeezing (la varianza de la posición adquiere un valor más bajo de el asociado a la cota de Heisenberg) a temperaturas bajas y cuando el acoplamiento crece. Sin embargo, la ecuación de Born-Markov no es la herramienta más adecuada para tratar el régimen en el que detectamos squeezing, porque las hipótesis subyacentes en general no valen a temperaturas bajas e interacción fuerte, llevando a violaciones del principio de Heisenberg. Para superar este obstáculo es posible emplear métodos alternativos, por ejemplo la ecuación de Lindblad, es decir una ecuación cuya forma sirve para preservar la positividad del estado en cualquier instante, y las ecuaciones de Heisenberg. En particular, aplicamos el formalismo de las ecuaciones de Heisenberg para investigar el comportamiento del Bose polaron, o sea una impureza en un condensado de Bose-Einstein. Es un sistema realista experimentalmente que ha atraido mucha atención recientemente. Derivamos la ecuación del movimiento de la impureza y mostramos que su forma analítica es la misma que la de la ecuación de Langevin para el movimiento Browniano clásico. La diferencia principal es que en este caso la dinámica acarrea efectos de memoria. Una parte importante del trabajo consiste en solucionar esta ecuación del movimiento para evaluar la varianza de la posición, que se puede medir en experimentos. Aquí diferenciamos dos casos: cuando la impureza está atrapada en un potencial armónico, y cuando no hay trampa armónica. En el segundo caso la varianza es proporcional al cuadrado del tiempo (difusión balística), como consecuencia de los efectos de memoria. Cuando la impureza está atrapada alcanza un estado de equilibrio localizado en el medio de la trampa. En este estado, bajando la temperatura y considerando valores del coupling más fuertes detectamos otra vez squeezing. Si consideramos un gas con una densidad de Thomas-Fermi se puede comprobar que este efecto se puede optimizar aprietando la trampa del gas. El estudio del squeezing es muy importante en el marco de la metrología cuántica porque permite el desarrollo de nuevo protocolos como el termometro cuántico.Postprint (published version

Quantum Brownian motion revisited : extensions and applications

Quantum Brownian motion represents a paradigmatic model of open quantum system, namely a system which cannot be treated as an isolated one, because of the unavoidable interaction with the surrounding environment. In this case the central system is constituted by a quantum particle, while the bath is made up by a large set of uncoupled harmonic oscillators. In the original model, the interaction between the system and the environment shows a linear dependence on the particle position. Such a particular form corresponds to a homogeneous environment, inducing a damping and diffusion which depends on the state. This is not the most general situation: often the environment shows an inhomogeneous character given by a space-dependent density, involving a non-linearity in the coupling with the central system. One of the main motivations of the thesis is the study of quantum Brownian motion in presence of this non-linear coupling. In particular we focus on the case in which the bath-particle interaction depends quadratically on the position of the latter. There exist several techniques aimed to treat the physics of the model. For instance one could consider the master equation, namely an equation ruling the temporal evolution of the state of the Brownian particle, here represented by its reduced density matrix. We derive such an equation in the Born-Markov regime and look into its stationary solution, studying its configuration in the phase space. For a non-linear quadratic coupling the stationary state may be approximated by means of a Gaussian Wigner function, that experiences genuine position squeezing (i.e. the position variance of the particle takes a value smaller than that associated to the Heisenberg principle, although this is fulfilled) at low temperature and as the coupling with the bath grows. However, the Born-Markov master equation is not the most appropriate tool to investigate the regime in which squeezing occurs, since the underlying hypothesis in general fail at strong coupling and low temperature, leading to violations of the Heisenberg principle. To overcome this problem we recall alternative methods, such as a Lindblad equation, namely a master equation constructed to preserve the positivity of the state at any time, and Heisenberg equations. In particular we employ the Heisenberg equation formalism to explore the behavior of the Bose polaron, i.e. an impurity embedded in a Bose-Einstein condensate. This experimentally feasible system attracted a lot of attention in the last years. We derive the equation of motion of the impurity position showing that it shows the same form of the famous equation derived by Langevin in 1909 in the context of classical Brownian motion. The main difference lies in the fact that the impurity Langevin-like equation for the impurity carries a certain amount of memory effects, while the original one was purely Markovian. An important part of the work is devoted to the solution of the motion equation for the impurity, in order to calculate the position variance that can be measured in experiments. For this goal we distinguish the case in which the impurity is trapped in a harmonic potential and that where it is free of any trap. In the latter case the impurity the position variance exhibits a quadratic dependence on time (i.e. ballistic diffusion), as a consequence of memory effects. When the impurity is trapped in a harmonic potential it approaches an equilibrium state localized in average in the middle of the trap. Here, at low temperature and for certain values of the coupling strength we detect genuine position squeezing. When we consider a gas with a Thomas-Fermi profile we find that such an effect is improved if we make the gas trap tighter. Genuine squeezing plays an important role in the context of quantum metrology and opens a wide range of possibility to design new protocols, such as the quantum thermometerEl movimiento Browniano cuántico es uno de los principales modelos de sistema abierto, es decir un sistema cuyo comportamiento no se puede tratar de manera separada de su entorno. Este modelo describe la física de una partícula acoplada a un entorno de osciladores. En la versión original del modelo la interacción entre la partícula y el entorno manifiesta una dependencia lineal de la posición de ambos los sistemas. Esta forma analítica del acoplamiento corresponde a un entorno homogeneo, asociado a una fricción y una difusión que dependen del estado del sistema. En todo caso, esta no es la situación más general: a menudo el enorno es inhomogeneo, ya que la densidad no es constante, y esto produce una interacción cuya dependencia de la posición de la partícula no es lineal. Una de las motivaciones principales de esta tesis es el estudio del movimiento Browniano cuántico en presencia de acoplamiento non-lineal. En particular, estudiamos el caso de dependencia cuadrática en la posición de la partícula. Existen muchas técnicas para abordar el modelo. Por ejemplo, se puede emplear la master equation, o sea un ecuación que gobierna la evolución en el tiempo del estado de la partícula, representado por el operador densidad reducido. Derivamos esta ecuación en el régimen de Born-Markov, y estudiamos la forma del estado estacionario en el espacio de las fases. Cuando el acoplamiento es cuadrático, este estado se puede aproximar por medio de una función de Wigner de forma Gausiana, cuya peculiaridad es la emergencia de genuine position squeezing (la varianza de la posición adquiere un valor más bajo de el asociado a la cota de Heisenberg) a temperaturas bajas y cuando el acoplamiento crece. Sin embargo, la ecuación de Born-Markov no es la herramienta más adecuada para tratar el régimen en el que detectamos squeezing, porque las hipótesis subyacentes en general no valen a temperaturas bajas e interacción fuerte, llevando a violaciones del principio de Heisenberg. Para superar este obstáculo es posible emplear métodos alternativos, por ejemplo la ecuación de Lindblad, es decir una ecuación cuya forma sirve para preservar la positividad del estado en cualquier instante, y las ecuaciones de Heisenberg. En particular, aplicamos el formalismo de las ecuaciones de Heisenberg para investigar el comportamiento del Bose polaron, o sea una impureza en un condensado de Bose-Einstein. Es un sistema realista experimentalmente que ha atraido mucha atención recientemente. Derivamos la ecuación del movimiento de la impureza y mostramos que su forma analítica es la misma que la de la ecuación de Langevin para el movimiento Browniano clásico. La diferencia principal es que en este caso la dinámica acarrea efectos de memoria. Una parte importante del trabajo consiste en solucionar esta ecuación del movimiento para evaluar la varianza de la posición, que se puede medir en experimentos. Aquí diferenciamos dos casos: cuando la impureza está atrapada en un potencial armónico, y cuando no hay trampa armónica. En el segundo caso la varianza es proporcional al cuadrado del tiempo (difusión balística), como consecuencia de los efectos de memoria. Cuando la impureza está atrapada alcanza un estado de equilibrio localizado en el medio de la trampa. En este estado, bajando la temperatura y considerando valores del coupling más fuertes detectamos otra vez squeezing. Si consideramos un gas con una densidad de Thomas-Fermi se puede comprobar que este efecto se puede optimizar aprietando la trampa del gas. El estudio del squeezing es muy importante en el marco de la metrología cuántica porque permite el desarrollo de nuevo protocolos como el termometro cuántico

Progress in Group Field Theory and Related Quantum Gravity Formalisms

Following the fundamental insights from quantum mechanics and general relativity, geometry itself should have a quantum description; the search for a complete understanding of this description is what drives the field of quantum gravity. Group field theory is an ambitious framework in which theories of quantum geometry are formulated, incorporating successful ideas from the fields of matrix models, ten-sor models, spin foam models and loop quantum gravity, as well as from the broader areas of quantum field theory and mathematical physics. This special issue collects recent work in group field theory and these related approaches, as well as other neighbouring fields (e.g., cosmology, quantum information and quantum foundations, statistical physics) to the extent that these are directly relevant to quantum gravity research

The Kardar-Parisi-Zhang equation and universality class

Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new universality class has emerged to describe a host of important physical and probabilistic models (including one dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the Kardar-Parisi-Zhang (KPZ) universality class and underlying it is, again, a continuum object -- a non-linear stochastic partial differential equation -- known as the KPZ equation. The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with {\it narrow wedge} initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact one-point distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media. As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.Comment: 57 pages, survey article, 7 figures, addition physics ref. added and typo's correcte

Effect of electron-phonon interaction in nanostructures and ultracold quantum gases

The subject of this thesis is the effect of electron-phonon interaction in two classes of mesoscopic systems. The first class includes molecular quantum dots. They are believed to be good candidates for future realizations of transistors on the nanoscale. Using the concept of full counting statistics (FCS), the charge transfer for several models is characterized. On the one hand, the main focus of this work lies on systems with rather strong electron-phonon interactions, on the other hand, it lies on models with strongly correlated electrodes described by Tomonaga-Luttinger liquids. Based on a generalized Keldysh formalism, perturbative and non-perturbative methods have been provided to calculate the FCS. Using double quantum dot models, the analogy with multi-level systems is discussed. The second class contains the BEC polaron problem. The BEC polaron is based on the analogy of immersed quantum gases with electrons in crystal lattices. Using imaginary-time path integral Monte Carlo methods, variational principles and perturbation theory, the effective Fröhlich model is investigated. The similarity to the emission of Cherenkov radiation is discussed

Nonequilibrium Quantum Field Theory

Bringing together the key ideas from nonequilibrium statistical mechanics and powerful methodology from quantum field theory, this 2008 book captures the essence of nonequilibrium quantum field theory. Beginning with the foundational aspects of the theory, the book presents important concepts and useful techniques, discusses issues of basic interest, and shows how thermal field, linear response, kinetic theories and hydrodynamics emerge. It also illustrates how these concepts are applied to research topics including nonequilibrium phase transitions, thermalization in relativistic heavy ion collisions, the nonequilibrium dynamics of Bose-Einstein condensation, and the generation of structures from quantum fluctuations in the early Universe. This self-contained book is a valuable reference for graduate students and researchers in particle physics, gravitation, cosmology, atomic-optical and condensed matter physics. It has been reissued as an Open Access publication