Abstract Since the Hilbert space for state vectors describing Nature is, of course, of higher dimension than two, generalization of the state vector evolution equation given in the previous chapter is needed. It is explained why the Brownian motion function needs to be replaced by its time derivative, that is, white noise w(t). The collapse-generating operator A is allowed to be arbitrary. As an example, the case of a free particle moving in one dimension, where A=X (the particle’s position operator), is completely solved. Then, further generalizing, the single collapse-generating operator A is replaced by a family Ai, a complete commuting set of operators, one of whose joint eigenstates is the end result of collapse. Each operator is linked to its own white noise function wi(t). The chapter then discussed time-ordering, which comes into play in the generalized evolution equation. The chapter also covers the associated density matrix. The chapter concludes with a final generalization where the discrete index i is replaced by the continuous index x, so the operators A(x) (x is a point in space) and the white noise field w(x,t) appear in the evolution equation.
Abstract Since the Hilbert space for state vectors describing Nature is, of course, of higher dimension than two, generalization of the state vector evolution equation given in the previous chapter is needed. It is explained why the Brownian motion function needs to be replaced by its time derivative, that is, white noise w(t). The collapse-generating operator A is allowed to be arbitrary. As an example, the case of a free particle moving in one dimension, where A=X (the particle’s position operator), is completely solved. Then, further generalizing, the single collapse-generating operator A is replaced by a family Ai, a complete commuting set of operators, one of whose joint eigenstates is the end result of collapse. Each operator is linked to its own white noise function wi(t). The chapter then discussed time-ordering, which comes into play in the generalized evolution equation. The chapter also covers the associated density matrix. The chapter concludes with a final generalization where the discrete index i is replaced by the continuous index x, so the operators A(x) (x is a point in space) and the white noise field w(x,t) appear in the evolution equation.
Abstract This chapter introduces the two equations that completely define CSL, for the simplest case, that is, a state vector in a two-dimensional Hilbert space. One equation is the modified evolution equation for the state vector which depends upon a Brownian motion function of time, B(t), a Hermitian operator A called the “collapse-generating operator,” and a collapse rate parameter λ. The state vector evolution is linear, but not unitary, so its norm changes with time. The second equation is called the probability rule. It says the probability a particular B(t) occurs in Nature is proportional to the squared norm of the state vector, that is, high norm state vectors are most likely to occur. The chapter shows how an initial state vector that is a superposition of the eigenstates of A evolves to one eigenstate, with the Born probability, as predicted by standard quantum theory. Finally, it discusses the density matrix description of the ensemble of evolutions under all possible B(t).
Abstract CSL so far is a general framework to describe dynamical collapse of a statevector to one or another joint eigenstate of a set of mutually commuting collapse-generating operators. This chapter makes the final selection of the collapse generating operators. First, the choice of mass-density operators, A(x)= D(x), is discussed. This leads to an infinite rate of energy increase for particles, which is ameliorated by a slightly different choice, “smeared” mass-density operators, A(x)=∫dx’f(x-x’)D(x’), where f is a Gaussian of width a, f(z) exp-z2/2a2. This introduces a second parameter to the theory (in addition to the collapse rate λ), the width parameter a. This completes construction of non-relativistic CSL, the theory proposed to describe Nature. The chapter goes on to discuss the density matrix, followed by two important calculations, the first consequences of the theory. First, for an initial state that is an object in a superposition of two different places, is the rate to collapse to one or another place, and second, the rate of energy increase of an object due to the wave function narrowing inherent in collapse.
Quantum mechanics exhibits a broad collection of theoretical results in complete agreement with experimental evidence. Besides showing unquestionable success in the description of isolated systems, it can be also successfully used to characterize non-isolated quantum system. In such a case, phenomena like dissipation, diffusion or decoherence. The theory of open quantum systems provides the framework where such features can be conveniently explained. \nThis thesis is about decoherence and collapse models. Albeit conceptually they are very far from each other, they both belong the framework of open quantum systems. Indeed, they consider systems interacting with an external entity: an environment for decoherence models, a noise for collapse models. Although the external influence has different origin, they can be described by similar dynamical equations and, to confirm or falsify them, the same experimental tests can be performed. \nThe quantum Brownian motion model can be considered the paradigm of decoherence models. 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. Our derivation allows to compute the time evolution of physically relevant quantities in a much easier way than previous formulations. Moreover, we are not bound to compute the time evolution of the state of the system, which in general is a complicated task. The explicit dependence on the initial state appears only in the initial expectation values and not in the dynamics. This makes possible the derivation of expectation values also for nontrivial states. \nAnother decoherence model we considered is a recently proposed model, based on the mass-energy equivalence. The model describes a decoherence source acting universally on every system whose superposition is extended on positions experiencing different gravitational potentials. We studied the conditions under which this mechanism becomes the dominant decoherence effect in typical interferometric experiments. We show that current experiments are off by several orders of magnitude. New ideas are needed to achieve the necessary requirements. \nThe second part of this thesis concerns collapse models, in particular the Continuous Spontaneous Localization (CSL) model. We focus on experimental tests that can probe it. In this respect, experiments can be grouped in two classes: interferometric tests and non-interferometric ones. The first class includes those experiments, which directly create and detect quantum superpositions of the center of mass of massive systems. The strongest bounds on the CSL parameters come from the second class of non-interferometric experiments, which are sensitive to small position displacements and detect CSL-induced diffusion in position. \nWe investigate how we can benefit from the non-interferometric perspective given by optomechanical setups, which have reached high sensitivities as force and position sensors. Three examples are considered.\nFirst, we compute the upper bounds on the CSL parameters, which can be inferred by the gravitational wave detectors LIGO, LISA Pathfinder and AURIGA.\nSecond, we report new results from an experiment based on a high-quality cantilever cooled to millikelvin temperature. High accuracy measurements of the cantilever thermal fluctuations reveal a nonthermal force noise of unknown origin. This excess noise is compatible with the CSL heating predicted by Adler. Several physical mechanisms able to explain the observed noise have been ruled out.\nThird, we propose an unattempted non-interferometric test aimed to investigate the still unexplored region of the CSL parameter space. Our proposal relies on torsional degrees of freedom rather than the usual vibrational ones. We believe that the test that has been put forward here, will eventually probe the unexplored CSL parameter space.
Abstract Chapter 2 considered the case where the collapse-generating operator, to one of whose eigenstates collapse takes place, has a discrete spectrum. This Supplement considers an example where the spectrum is continuous, specifically, the case where the collapse-generating operator is the position operator. This chapter is a supplement to Chapter 2. Whereas Chapter 2 considers the case where the collapse–generating has a discrete spectrum, Chapter 9 considers an example where the spectrum is continuous. Specifically, it covers the case where the collapse–generating operator is the position operator. It also explains how the Born probability rule is obeyed in this example.
We continue the analysis of models of spontaneous wave function collapse with stochastic dynamics driven by non-white Gaussian noise. We specialize to a model in which a classical "noise" field, with specified autocorrelator, is coupled to a local nonrelativistic particle density. We derive general results in this model for the rates of density matrix diagonalization and of state vector reduction, and show that (in the absence of decoherence) both processes are governed by essentially the same rate parameters. As an alternative route to our reduction results, we also derive the Fokker-Planck equations that correspond to the initial stochastic Schrödinger equation. For specific models of the noise autocorrelator, including ones motivated by the structure of thermal Green's functions, we discuss the qualitative and qantitative dependence on model parameters, with particular emphasis on possible cosmological sources of the noise field.
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description