We study the localization problem in quantum stochastic mechanics. We start from the Edwards model for a particle in a bath of scattering centers and prove static localization of the ground state wavefunction of the particle in a one dimensional square well coupled to Dirac delta like scattering centers in arbitrary but fixed positions. We see how the localization increases for increasing coupling $g$. Then we choose the scattering centers positions as pseudo random numbers with a uniform probability distribution and observe an increase in the localization of the average of the ground state over the many positions realizations. We discuss how this averaging procedure is consistent with a picture of a particle in a Bose-Einstein condensate of of non interacting boson scattering centers interacting with the particle with Dirac delta functions pair potential. We then study the dynamics of the ground state wave function. We conclude with a discussion of the affine quantization version of the Lax model which reduces to a system of contiguous square wells with walls in arbitrary positions independently of the coupling constant $g$.
We study the localization problem in quantum stochastic mechanics. We start from the Edwards model for a particle in a bath of scattering centers and prove static localization of the ground state wavefunction of the particle in a one dimensional square well coupled to Dirac delta like scattering centers in arbitrary but fixed positions. We see how the localization increases for increasing coupling g. Then we choose the scattering centers positions as pseudo random numbers with a uniform probability distribution and observe an increase in the localization of the average of the ground state over the many positions realizations. We discuss how this averaging procedure is consistent with a picture of a particle in a Bose-Einstein condensate of of non interacting boson scattering centers interacting with the particle with Dirac delta functions pair potential. We then study the dynamics of the ground state wave function. We conclude with a discussion of the affine quantization version of the Lax model which reduces to a system of contiguous square wells with walls in arbitrary positions independently of the coupling constant g.
We present a mapping between the Edwards model of disorder describing the motion of a single particle subject to randomly positioned static scatterers and the Bose polaron problem of a light quantum impurity interacting with a Bose-Einstein condensate (BEC) of heavy atoms. The mapping offers an experimental setting to investigate the physics of Anderson localization where, by exploiting the quantum nature of the BEC, the time evolution of the quantum impurity emulates the disorder-averaged dynamics of the Edwards model. Valid in any space dimension, the mapping can be extended to include interacting particles, arbitrary disorder, or confinement and can be generalized to study many-body localization. Moreover, the corresponding exactly solvable disorder model offers means to benchmark variational approaches used to study polaron physics. Here we illustrate the mapping by focusing on the case of an impurity interacting with a one-dimensional BEC through a contact interaction. While a simple wave function based on the expansion in the number of bath excitations misses the localization physics entirely, a coherent state Ansatz combined with a canonical transformation captures the physics of disorder and Anderson localization.
We study the ground state of a large bosonic system trapped in a symmetric double-well potential, letting the distance between the two wells increase to infinity with the number of particles. In this context, one should expect an interaction-driven transition between a delocalized state (particles are independent and all live in both wells) and a localized state (particles are correlated, half of them live in each well). We start from the full many-body Schr{ö}dinger Hamiltonian in a large-filling situation where the on-site interaction and kinetic energies are comparable. When tunneling is negligible against interaction energy, we prove a localization estimate showing that the particle number fluctuations in each well are strongly suppressed. The modes in which the particles condense are minimizers of nonlinear Schr{ö}dinger-type functionals.
We study the localization of a quantum particle in a one-dimensional disordered system with long-range hopping amplitudes t(r)∝r−a. In contrast to the standard one-dimensional Anderson model (a→∞), in which all states are localized and the localization length is minimal at the band edge, the long-range model with $1
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 thermometer El 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.
In this work we give a positive answer to the following question: does Stochastic Mechanics uniquely define a three-dimensional stochastic process which describes the motion of a particle in a Bose–Einstein condensate? To this extent we study a system of N trapped bosons with pair interaction at zero temperature under the Gross–Pitaevskii scaling, which allows to give a theoretical proof of Bose–Einstein condensation for interacting trapped gases in the limit of N going to infinity. We show that under the assumption of strictly positivity and continuous differentiability of the many-body ground state wave function it is possible to rigorously define a one-particle stochastic process, unique in law, which describes the motion of a single particle in the gas and we show that, in the scaling limit, the one-particle process continuously remains outside a time dependent random “interaction-set” with probability one. Moreover, we prove that its stopped version converges, in a relative entropy sense, toward a Markov diffusion whose drift is uniquely determined by the order parameter, that is the wave function of the condensate.
We formulate a quantum Monte Carlo (QMC) method for calculating the ground state of many-boson systems. The method is based on a field-theoretical approach, and is closely related to existing fermion auxiliary-field QMC methods which are applied in several fields of physics. The ground-state projection is implemented as a branching random walk in the space of permanents consisting of identical single-particle orbitals. Any single-particle basis can be used, and the method is in principle exact. We apply this method to an atomic Bose gas, where the atoms interact via an attractive or repulsive contact two-body potential parametrized by the s-wave scattering length. We choose as the single-particle basis a real-space grid. We compare with exact results in small systems, and arbitrarily-sized systems of untrapped bosons with attractive interactions in one dimension, where analytical solutions exist. Our method provides a way to systematically improve upon the mean-field Gross-Pitaevskii (GP) method while using the same framework, capturing interaction and correlation effects with a stochastic, coherent ensemble of non-interacting solutions. to study the role of many-body correlations in the ground state, we examine the properties of the gas, such as the energetics, condensate fraction, and the density and momentum distributions as a function of the number of particles and the scattering length, both in the homogenous and trapped gases. Results are presented for systems with up to 1000 bosons. Comparing our results to the mean-field GP results, we find significant departure from mean field at large positive scattering lengths. The many-body correlations tend to increase the kinetic energy and reduce the interaction energy compared to GP. In the trapped gases, this results in a qualitatively different behavior as a function of the scattering length. Possible experimental observation is discussed.