The theory of quantum Brownian motion describes the properties of a large class of open quantum systems. Nonetheless, its description in terms of a Born-Markov master equation, widely used in the literature, is known to violate the positivity of the density operator at very low temperatures. We study an extension of existing models, leading to an equation in the Lindblad form, which is free of this problem. We study the dynamics of the model, including the detailed properties of its stationary solution, for both constant and position-dependent coupling of the Brownian particle to the bath, focusing in particular on the correlations and the squeezing of the probability distribution induced by the environment.
The theory of quantum Brownian motion describes the properties of a large class of open quantum systems. Nonetheless, its description in terms of a Born-Markov master equation, widely used in the literature, is known to violate the positivity of the density operator at very low temperatures. We study an extension of existing models, leading to an equation in the Lindblad form, which is free of this problem. We study the dynamics of the model, including the detailed properties of its stationary solution, for both constant and position-dependent coupling of the Brownian particle to the bath, focusing in particular on the correlations and the squeezing of the probability distribution induced by the environment.
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.
Quantum Brownian motion is a fundamental model for a proper understanding of open quantum systems in different contexts such as chemistry, condensed-matter physics, biophysics, and optomechanics. In this paper we propose a different approach to describe this model. We provide an exact and analytic equation for the time evolution of the operators and we show that the corresponding equation for the states is equivalent to well-known results in the literature. The dynamics is expressed in terms of the spectral density, regardless of the strength of the coupling between the system and the bath. Our derivation allows to compute the time evolution of physically relevant quantities in a much easier way than previous formulations. An example is explicitly studied.
An open quantum system is a quantum system that interacts with some environment whose degrees of freedom have been coarse grained away. This model describes non-equilibrium processes more general than scattering-matrix formulations. Furthermore, the microscopically-derived environment provides a model of noise, dissipation and decoherence far more general than Markovian (white noise) models. The latter are fully characterized by Lindblad equations and can be motivated phenomenologically. Non-Markovian processes consistently account for backreaction with the environment and can incorporate effects such as finite temperature and spatial correlations. We consider linear systems with bilinear coupling to the environment, or quantum Brownian motion, and nonlinear systems with weak coupling to the environment. For linear systems we provide exact solutions with analytical results for a variety of spectral densities. Furthermore, we point out an important mathematical subtlety which led to incorrect master-equation coefficients in earlier derivations, given nonlocal dissipation. For nonlinear systems we provide perturbative solutions by translating the formalism of canonical perturbation theory into the context of master equations. It is shown that unavoidable degeneracy causes an unfortunate reduction in accuracy between perturbative master equations and their solutions. We also extend the famous theorem of Lindblad, Gorini, Kossakowski and Sudarshan on completely positivity to non-Markovian master equations. Our application is primarily to model atoms interacting via a common electromagnetic field. The electromagnetic field contains correlations in both space and time, which are related to its relativistic (photon-mediated) nature. As such, atoms residing in the same field experience different environmental effects depending upon their relative position and orientation. Our more accurate solutions were necessary to assess sudden death of entanglement at zero temperature. In contrast to previous claims, we found that all initial states of two-level atoms undergo finite-time disentanglement. We were also able to access regimes which cannot be described by Lindblad equations and other simpler methods, such as near resonance. Finally we revisit the infamous Abraham-Lorentz force, wherein a single particle in motion experiences backreaction from the electromagnetic field. This leads to a number of well-known problems including pre-acceleration and runaway solutions. We found a more a more-suitable open-system treatment of the nonrelativistic particle to be perfectly causal and dissipative without any extraneous requirements for finite size of the particle, weak coupling to the field, etc..
We analyze the microscopic model of quantum Brownian motion, describing a Brownian particle interacting with a bosonic bath through a coupling which is linear in the creation and annihilation operators of the bath, but may be a nonlinear function of the position of the particle. Physically, this corresponds to a configuration in which damping and diffusion are spatially inhomogeneous. We derive systematically the quantum master equation for the Brownian particle in the Born-Markov approximation and we discuss the appearance of additional terms, for various polynomials forms of the coupling. We discuss the cases of linear and quadratic coupling in great detail and we derive, using Wigner function techniques, the stationary solutions of the master equation for a Brownian particle in a harmonic trapping potential. We predict quite generally Gaussian stationary states, and we compute the aspect ratio and the spread of the distributions. In particular, we find that these solutions may be squeezed (superlocalized) with respect to the position of the Brownian particle. We analyze various restrictions to the validity of our theory posed by non-Markovian effects and by the Heisenberg principle. We further study the dynamical stability of the system, by applying a Gaussian approximation to the time-dependent Wigner function, and we compute the decoherence rates of coherent quantum superpositions in position space. Finally, we propose a possible experimental realization of the physics discussed here, by considering an impurity particle embedded in a degenerate quantum gas.
The Lindblad theory of open quantum systems has been successfully applied in various fields ranging from quantum optics, condensed matter physics, and quantum thermodynamics to quantum chemistry. Nevertheless, there are situations where we can compare its predictions to those of exact methods or numerically exact results and find discrepancies between them. One example is the case where a single harmonic oscillator is coupled to a thermal bath of independent harmonic oscillators. In this case, for certain parameters, the exact master equation describes equilibration of the system to a thermal state with a temperature different from the one predicted by the Lindblad master equation. The Lindblad theory, based on a weak coupling between the system and environment as well as the Born-Markov approximation, predicts that the system will evolve to a thermal state whose temperature is equal to the initial temperature of the bath. This result is a consequence of the Born approximation, which supposes that throughout the evolution, the bath only fluctuates around its initial equilibrium state. In this master thesis we study the validity of the Born approximation as well as its consistency with other approximations. By starting from a Hamiltonian for the total system we show that, if the the Born approximation is not invoked, the dynamics of the system is determined by a hierarchy of equations for matrices which can be used to reconstruct the reduced density matrix of the system. This hierarchy approach is fundamentally different from the standard approach, which describes the system dynamics with a weak-coupling master equation, because the hierarchy does not restrict the amount of entanglement possible between the system and the environment. Furthermore, we show how invoking the Born approximation reduces to the usual result and how a generalised ansatz, which we refer to as a generalised Born approximation, reduces the hierarchy to a master equation where the bath temperature is an additional degree of freedom. Finally, we analyse the conservation of the energy of the total system and its relationship to the Markov and secular approximations, and discuss whether imposing it as a condition can fix the time-dependent temperature.
We develop a Markovian master equation in the Lindblad form that enables the efficient study of a wide range of open quantum many-body systems that would be inaccessible with existing methods. The validity of the master equation is based entirely on properties of the bath and the system-bath coupling, without any requirements on the level structure within the system itself. The master equation is derived using a Markov approximation that is distinct from that used in earlier approaches. We provide a rigorous bound for the error induced by this Markov approximation; the error is controlled by a dimensionless combination of intrinsic correlation and relaxation timescales of the bath. Our master equation is accurate on the same level of approximation as the Bloch-Redfield equation. In contrast to the Bloch-Redfield approach, our approach ensures preservation of the positivity of the density matrix. As a result, our method is robust, and can be solved efficiently using stochastic evolution of pure states (rather than density matrices). We discuss how our method can be applied to static or driven quantum many-body systems, and illustrate its power through numerical simulation of a spin chain that would be challenging to treat by existing methods.