We introduce a comprehensive analytical benchmark, relying on Fokker-Planck formalism, to study m... more We introduce a comprehensive analytical benchmark, relying on Fokker-Planck formalism, to study microbial dynamics in presence of both biotic and abiotic forces. In equilibrium, we observe a balance between the two kinds of forces, leading to no correlations between species abundances. This implies that real microbiomes, where correlations have been observed, operate out of equilibrium. Therefore, we analyze non-equilibrium dynamics, presenting an ansatz for an approximate solution that embodies the complex interplay of forces in the system. This solution is consistent with Taylor's law as a coarse-grained approximation of the relation between species abundance and variance, but implies subtler effects, predicting unobserved structure beyond Taylor's law. Motivated by this theoretical prediction, we refine the analysis of existing metagenomic data, unveiling a novel universal macroecological pattern. Finally, we speculate on the physical origin of Taylor's law: building upon an analogy with Brownian motion theory, we propose that Taylor's law emerges as a Fluctuation-Growth relation resulting from equipartition of environmental resources among microbial species.
The theory of quantum Brownian motion describes the properties of a large class of open quantum s... more 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 s... more 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 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.
Objectivity constitutes one of the main features of the macroscopic classical world. An important... more Objectivity constitutes one of the main features of the macroscopic classical world. An important aspect of the quantum-to-classical transition issue is to explain how such a property arises from the microscopic quantum theory. Recently, within the framework of open quantum systems, there has been proposed such a mechanism in terms of the, so-called, spectrum broadcast structures. These are multipartite quantum states of the system of interest and a part of its environment, assumed to be under an observation. This approach requires a departure from the standard open quantum systems methods, as the environment cannot be completely neglected. In the present paper we study the emergence of such a state structure in one of the canonical models of the condensedmatter theory: the spin-boson model, describing the dynamics of a two-level system coupled to an environment made up by a large number of harmonic oscillators. We pay much attention to the behavior of the model in the non-Markovian regime, in order to provide a testbed to analyze how the non-Markovian nature of the evolution affects the surfacing of a spectrum broadcast structure.
Brownian motion is the random movement of a particle suspended in a fluid. This phenomenon played... more Brownian motion is the random movement of a particle suspended in a fluid. This phenomenon played a very important role in the history of science because it leaded to the idea that matter is made up by atoms. In this chapter we briefly present the fundamental results concerning classical Brownian motion, focusing on a description of the original observations and of the main theoretical attempts to study it.
In the previous chapter we described the main theoretical attempts to treat the grains motion det... more In the previous chapter we described the main theoretical attempts to treat the grains motion detected by Robert Brown. These theories are purely classical and rely on phenomenological equations, i.e. equations that are not derived in a Hamiltonian framework, but are proposed starting from experimental results that one aims to interpret. In this chapter we move our analysis to the quantum domain. The standard procedures of quantization are based on the existence of Hamiltonians (or equivalently Lagrangians) for the system in which one is interested. So, the first step to approach the study of Brownian motion in the quantum regime is to look for a Hamiltonian description of the phenomenon observed by Robert Brown. Precisely, one has to write a Hamiltonian leading to the phenomenological equations, such as those of Einstein [Eq. (2.2)] and Langevin [Eq. (2.18)]. Then, by replacing functional variables with operator ones it is possible to obtain a quantum Hamiltonian for Brownian motion. This is the point of view adopted, for instance, by Caldeira and Leggett (1983a). In Sect. 3.1 we introduce the Hamiltonian of QBM. It describes a quantum particle, usually trapped in a harmonic potential, coupled to a set of non-interacting harmonic oscillators. The Hamiltonian encodes all the information to study the physics of QBM. There exist several tools to do this. In the current chapter we consider the master equation formalism, i.e. we use an equation for the reduced density operator of the central Brownian particle. Such a topic belongs to standard textbook material, but we present the re-examination published in the paper of Massignan et al. (2015). Here we focused on the evaluation of the stationary state of the system, and the analysis of its geometrical configuration in the phase-space as the system parameters, such as temperature and interaction strength, vary. This kind of analysis leads to the detection of special effects for the impurity, such as squeezing and cooling. We discuss the regime of validity of the method adopted.
The previous chapter was devoted to the discussion of the QBM model. We treated such a model by m... more The previous chapter was devoted to the discussion of the QBM model. We treated such a model by means of a Born-Markov master equation and we studied its stationary solution. All the theory we developed in Chap. 3 refers to a Hamiltonian model where the interaction depends linearly on the position of the central particle. This is the conventional case, and we will refer to it in the following also as linear case.
We study the dynamics of a quantum impurity immersed in a Bose-Einstein condensate as an open qua... more We study the dynamics of a quantum impurity immersed in a Bose-Einstein condensate as an open quantum system in the framework of the quantum Brownian motion model. We derive a generalized Langevin equation for the position of the impurity. The Langevin equation is an integro-differential equation that contains a memory kernel and is driven by a colored noise. These result from considering the environment as given by the degrees of freedom of the quantum gas, and thus depend on its parameters, e.g. interaction strength between the bosons, temperature, etc. We study the role of the memory on the dynamics of the impurity. When the impurity is untrapped, we find that it exhibits a super-diffusive behavior at long times. We find that back-flow in energy between the environment and the impurity occurs during evolution. When the particle is trapped, we calculate the variance of the position and momentum to determine how they compare with the Heisenberg limit. One important result of this paper is that we find position squeezing for the trapped impurity at long times. We determine the regime of validity of our model and the parameters in which these effects can be observed in realistic experiments.
My master thesis regards "Marshal et.al. optomechanical system" in a quantum-classical ... more My master thesis regards "Marshal et.al. optomechanical system" in a quantum-classical hybrid approach
We revisit decoherence process of a qubit register interacting with a thermal bosonic bath. We ge... more We revisit decoherence process of a qubit register interacting with a thermal bosonic bath. We generalize the previous studies by considering not only the register's behavior but also of a part of its environment. In particular, we are interested in information flow from the register to the environment, which we describe using recently introduced multipartite quantum state structures called Spectrum Broadcast Structures. Working in two specific cases of: i) two-qubit register and ii) collective decoherence, we identify the regimes where the environment acquires almost complete information about the register state. We also study in more detail the interesting causal aspects, related to the finite propagation time of the field disturbances between the qubits. Finally, we describe quantum state structures which appear due to the presence of protected spaces.
Perhaps the largest debate in network Ecology, the emergence of structural patterns stands out as... more Perhaps the largest debate in network Ecology, the emergence of structural patterns stands out as a multifaceted problem. To the methodological challenges –pattern identification, statistical significance– one has to add the relationship between candidate architectures and dynamical performance. In the case of mutualistic communities, the debate revolves mostly around two structural arrangements (nestedness and modularity) and two requirements for persistence, namely feasibility and stability. So far, it is clear that the former is strongly related to nestedness, while the latter is enhanced in modular systems. Adding to this, it has recently become clear that nestedness and modularity are antagonistic patterns –or, at the very least, their coexistence in a single system is problematic. In this context, this work addresses the role of the interaction architecture in the emergence and maintenance of both properties, introducing the idea of hybrid architectural configurations. Specifi...
During the last decades, the study of cities has been transformed by new approaches combining eng... more During the last decades, the study of cities has been transformed by new approaches combining engineering and complexity sciences. Network theory is playing a central role, facilitating the quantitative analysis of crucial urban dynamics, such as mobility, city growth or urban planning. In this framework, betweenness, a well-known centrality measure, represents a fundamental tool, standing out as a proxy of traffic density and congestion. In this work, we focus on the spatial aspects of congestion. Analyzing a large amount of real city networks, we show that most cities present a set of congestion regimes, separated by abrupt transitions. To help unveiling this spatial dependence of the betweenness, we introduce a simple model composed of a grid connected to a set of tree graphs. This structure, coined as the GT-model, allows us to analytically describe in terms of betweenness, how and why congestion emerges in particular geographical areas of monocentric cities. The model predicts ...
In the previous chapter we have proposed a Lindblad model for QBM, exploring both the cases of li... more In the previous chapter we have proposed a Lindblad model for QBM, exploring both the cases of linear and non-linear coupling. Lindblad equations differ from the original Born-Markov ones just for a few extra-terms (only one term in the linear case), curing the forbidden area detected in Chaps. 3 and 4.
The previous chapter was devoted to the discussion of the QBM model. We treated such a model by m... more The previous chapter was devoted to the discussion of the QBM model. We treated such a model by means of a Born-Markov master equation and we studied its stationary solution. All the theory we developed in Chap. 3 refers to a Hamiltonian model where the interaction depends linearly on the position of the central particle. This is the conventional case, and we will refer to it in the following also as linear case.
We introduce a comprehensive analytical benchmark, relying on Fokker-Planck formalism, to study m... more We introduce a comprehensive analytical benchmark, relying on Fokker-Planck formalism, to study microbial dynamics in presence of both biotic and abiotic forces. In equilibrium, we observe a balance between the two kinds of forces, leading to no correlations between species abundances. This implies that real microbiomes, where correlations have been observed, operate out of equilibrium. Therefore, we analyze non-equilibrium dynamics, presenting an ansatz for an approximate solution that embodies the complex interplay of forces in the system. This solution is consistent with Taylor's law as a coarse-grained approximation of the relation between species abundance and variance, but implies subtler effects, predicting unobserved structure beyond Taylor's law. Motivated by this theoretical prediction, we refine the analysis of existing metagenomic data, unveiling a novel universal macroecological pattern. Finally, we speculate on the physical origin of Taylor's law: building upon an analogy with Brownian motion theory, we propose that Taylor's law emerges as a Fluctuation-Growth relation resulting from equipartition of environmental resources among microbial species.
The theory of quantum Brownian motion describes the properties of a large class of open quantum s... more 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 s... more 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 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.
Objectivity constitutes one of the main features of the macroscopic classical world. An important... more Objectivity constitutes one of the main features of the macroscopic classical world. An important aspect of the quantum-to-classical transition issue is to explain how such a property arises from the microscopic quantum theory. Recently, within the framework of open quantum systems, there has been proposed such a mechanism in terms of the, so-called, spectrum broadcast structures. These are multipartite quantum states of the system of interest and a part of its environment, assumed to be under an observation. This approach requires a departure from the standard open quantum systems methods, as the environment cannot be completely neglected. In the present paper we study the emergence of such a state structure in one of the canonical models of the condensedmatter theory: the spin-boson model, describing the dynamics of a two-level system coupled to an environment made up by a large number of harmonic oscillators. We pay much attention to the behavior of the model in the non-Markovian regime, in order to provide a testbed to analyze how the non-Markovian nature of the evolution affects the surfacing of a spectrum broadcast structure.
Brownian motion is the random movement of a particle suspended in a fluid. This phenomenon played... more Brownian motion is the random movement of a particle suspended in a fluid. This phenomenon played a very important role in the history of science because it leaded to the idea that matter is made up by atoms. In this chapter we briefly present the fundamental results concerning classical Brownian motion, focusing on a description of the original observations and of the main theoretical attempts to study it.
In the previous chapter we described the main theoretical attempts to treat the grains motion det... more In the previous chapter we described the main theoretical attempts to treat the grains motion detected by Robert Brown. These theories are purely classical and rely on phenomenological equations, i.e. equations that are not derived in a Hamiltonian framework, but are proposed starting from experimental results that one aims to interpret. In this chapter we move our analysis to the quantum domain. The standard procedures of quantization are based on the existence of Hamiltonians (or equivalently Lagrangians) for the system in which one is interested. So, the first step to approach the study of Brownian motion in the quantum regime is to look for a Hamiltonian description of the phenomenon observed by Robert Brown. Precisely, one has to write a Hamiltonian leading to the phenomenological equations, such as those of Einstein [Eq. (2.2)] and Langevin [Eq. (2.18)]. Then, by replacing functional variables with operator ones it is possible to obtain a quantum Hamiltonian for Brownian motion. This is the point of view adopted, for instance, by Caldeira and Leggett (1983a). In Sect. 3.1 we introduce the Hamiltonian of QBM. It describes a quantum particle, usually trapped in a harmonic potential, coupled to a set of non-interacting harmonic oscillators. The Hamiltonian encodes all the information to study the physics of QBM. There exist several tools to do this. In the current chapter we consider the master equation formalism, i.e. we use an equation for the reduced density operator of the central Brownian particle. Such a topic belongs to standard textbook material, but we present the re-examination published in the paper of Massignan et al. (2015). Here we focused on the evaluation of the stationary state of the system, and the analysis of its geometrical configuration in the phase-space as the system parameters, such as temperature and interaction strength, vary. This kind of analysis leads to the detection of special effects for the impurity, such as squeezing and cooling. We discuss the regime of validity of the method adopted.
The previous chapter was devoted to the discussion of the QBM model. We treated such a model by m... more The previous chapter was devoted to the discussion of the QBM model. We treated such a model by means of a Born-Markov master equation and we studied its stationary solution. All the theory we developed in Chap. 3 refers to a Hamiltonian model where the interaction depends linearly on the position of the central particle. This is the conventional case, and we will refer to it in the following also as linear case.
We study the dynamics of a quantum impurity immersed in a Bose-Einstein condensate as an open qua... more We study the dynamics of a quantum impurity immersed in a Bose-Einstein condensate as an open quantum system in the framework of the quantum Brownian motion model. We derive a generalized Langevin equation for the position of the impurity. The Langevin equation is an integro-differential equation that contains a memory kernel and is driven by a colored noise. These result from considering the environment as given by the degrees of freedom of the quantum gas, and thus depend on its parameters, e.g. interaction strength between the bosons, temperature, etc. We study the role of the memory on the dynamics of the impurity. When the impurity is untrapped, we find that it exhibits a super-diffusive behavior at long times. We find that back-flow in energy between the environment and the impurity occurs during evolution. When the particle is trapped, we calculate the variance of the position and momentum to determine how they compare with the Heisenberg limit. One important result of this paper is that we find position squeezing for the trapped impurity at long times. We determine the regime of validity of our model and the parameters in which these effects can be observed in realistic experiments.
My master thesis regards "Marshal et.al. optomechanical system" in a quantum-classical ... more My master thesis regards "Marshal et.al. optomechanical system" in a quantum-classical hybrid approach
We revisit decoherence process of a qubit register interacting with a thermal bosonic bath. We ge... more We revisit decoherence process of a qubit register interacting with a thermal bosonic bath. We generalize the previous studies by considering not only the register's behavior but also of a part of its environment. In particular, we are interested in information flow from the register to the environment, which we describe using recently introduced multipartite quantum state structures called Spectrum Broadcast Structures. Working in two specific cases of: i) two-qubit register and ii) collective decoherence, we identify the regimes where the environment acquires almost complete information about the register state. We also study in more detail the interesting causal aspects, related to the finite propagation time of the field disturbances between the qubits. Finally, we describe quantum state structures which appear due to the presence of protected spaces.
Perhaps the largest debate in network Ecology, the emergence of structural patterns stands out as... more Perhaps the largest debate in network Ecology, the emergence of structural patterns stands out as a multifaceted problem. To the methodological challenges –pattern identification, statistical significance– one has to add the relationship between candidate architectures and dynamical performance. In the case of mutualistic communities, the debate revolves mostly around two structural arrangements (nestedness and modularity) and two requirements for persistence, namely feasibility and stability. So far, it is clear that the former is strongly related to nestedness, while the latter is enhanced in modular systems. Adding to this, it has recently become clear that nestedness and modularity are antagonistic patterns –or, at the very least, their coexistence in a single system is problematic. In this context, this work addresses the role of the interaction architecture in the emergence and maintenance of both properties, introducing the idea of hybrid architectural configurations. Specifi...
During the last decades, the study of cities has been transformed by new approaches combining eng... more During the last decades, the study of cities has been transformed by new approaches combining engineering and complexity sciences. Network theory is playing a central role, facilitating the quantitative analysis of crucial urban dynamics, such as mobility, city growth or urban planning. In this framework, betweenness, a well-known centrality measure, represents a fundamental tool, standing out as a proxy of traffic density and congestion. In this work, we focus on the spatial aspects of congestion. Analyzing a large amount of real city networks, we show that most cities present a set of congestion regimes, separated by abrupt transitions. To help unveiling this spatial dependence of the betweenness, we introduce a simple model composed of a grid connected to a set of tree graphs. This structure, coined as the GT-model, allows us to analytically describe in terms of betweenness, how and why congestion emerges in particular geographical areas of monocentric cities. The model predicts ...
In the previous chapter we have proposed a Lindblad model for QBM, exploring both the cases of li... more In the previous chapter we have proposed a Lindblad model for QBM, exploring both the cases of linear and non-linear coupling. Lindblad equations differ from the original Born-Markov ones just for a few extra-terms (only one term in the linear case), curing the forbidden area detected in Chaps. 3 and 4.
The previous chapter was devoted to the discussion of the QBM model. We treated such a model by m... more The previous chapter was devoted to the discussion of the QBM model. We treated such a model by means of a Born-Markov master equation and we studied its stationary solution. All the theory we developed in Chap. 3 refers to a Hamiltonian model where the interaction depends linearly on the position of the central particle. This is the conventional case, and we will refer to it in the following also as linear case.
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