We present a method for obtaining evolution operators for linear quantum trajectories. We apply this to a number of physical examples of varying mathematical complexity, in which the quantum trajectories describe the continuous projection measurement of physical observables. Using this method we calculate the average conditional uncertainty for the measured observables, being a central quantity of interest in these measurement processes.
We present a method for obtaining evolution operators for linear quantum trajectories. We apply this to a number of physical examples of varying mathematical complexity, in which the quantum trajectories describe the continuous projection measurement of physical observables. Using this method we calculate the average conditional uncertainty for the measured observables, being a central quantity of interest in these measurement processes.
In this thesis we consider primarily the dynamics of quantum systems subjected to continuous observation. In the Schrödinger picture the evolution of a continuously monitored quantum system, referred to as a `quantum trajectory', may be described by a stochastic equation for the state vector. We present a method of deriving explicit evolution operators for linear quantum trajectories, and apply this to a number of physical examples of varying mathematical complexity. In the Heisenberg picture evolution resulting from continuous observation may be described by quantum Langevin equations. We use this method to examine the noise spectrum that results from a continuous observation of the position of a moving mirror, and examine the possibility of detecting the noise resulting from the quantum back-action of the measurement. In addition to the work on continuous measurement theory, we also consider the problem of reconstructing the state of a quantum system from a set of measurements. We present a scheme for determining the state of a single cavity mode from the photon statistics measured both before and after an interaction with one or two two-level atoms.
Abstract The dynamics of a quantum system, undergoing unitary evolution and continuous monitoring, can be described in term of quantum trajectories. Although the averaged state fully characterizes expectation values, the entire ensemble of stochastic trajectories goes beyond simple linear observables, keeping a more attentive description of the entire dynamics. Here we go beyond the Lindblad dynamics and study the probability distribution of the expectation value of a given observable over the possible quantum trajectories. The measurements are applied to the entire system, having the effect of projecting the system into a product state. We develop an analytical tool to evaluate this probability distribution at any time t. We illustrate our approach by analyzing two paradigmatic examples: a single qubit subjected to magnetization measurements, and a free hopping particle subjected to position measurements.
Introduction A very general concept of a quantum trajectory would be the path taken by the state of a quantum system over time. This state could be conditioned upon measurement results, as we considered in Chapter 1. This is the sort of quantum trajectory we are most interested in, and it is generally stochastic in nature. In ordinary use, the word trajectory usually implies a path that is also continuous in time. This idea is not always applicable to quantum systems, but we can maintain its essence by defining a quantum trajectory as the path taken by the conditional state of a quantum system for which the unconditioned system state evolves continuously. As explained in Chapter 1, the unconditioned state is that obtained by averaging over the random measurement results which condition the system. With this motivation, we begin in Section 4.2 by deriving the simplest sort of quantum trajectory, which involves jumps (that is, discontinuous conditioned evolution). In the process we will reproduce Lindblad's general form for continuous Markovian quantum evolution as presented in Section 3.6. In Section 4.3 we relate these quantum jumps to photon-counting measurements on the bath for the model introduced in Section 3.11, and also derive correlation functions for these measurement records. In Section 4.4 we consider the addition of a coherent field (the ‘local oscillator’) to the output before detection. In the limit of a strong local oscillator this is called homodyne detection, and is described by a continuous (diffusive) quantum trajectory.
Quantum trajectories describe the stochastic evolution of an open quantum system conditioned on continuous monitoring of its output, such as by an ideal photodetector. In practice an experimenter has access to an output filtered through various electronic devices, rather than the microscopic states of the detector. This introduces several imperfections into the measurement process, of which only inefficiency has previously been incorporated into quantum trajectory theory. However, all electronic devices have finite bandwidths, and the consequent delay in conveying the output signal to the observer implies that the evolution of the conditional state of the quantum system must be non-Markovian. We present a general method of describing this evolution and apply it to avalanche photodiodes and to photoreceivers. We include the effects of efficiency, dead time, bandwidth, electronic noise and dark counts. The essential idea is to treat the quantum system and classical detector jointly, and to average over the latter to obtain the conditional quantum state. The significance of our theory is that quantum trajectories for realistic detection are necessary for sophisticated approaches to quantum feedback, and our approach could be applied in many areas of physics.
Quantum trajectories describe the evolution of quantum systems undergoing indirect measurement. In the discrete-time setting, the state of the system is updated by applying Kraus operators according to the measurement results. From an experimental perspective, these Kraus operators can depend on unknown physical parameters p. An interesting and powerful method has been proposed in [1] to estimate a parameter in a finite set; however, complete results of convergence were lacking. This article fills this gap by rigorously showing the consistency of the method, whereas there was only numerical evidence so far. When the parameter belongs to a continuous set, we propose an algorithm to approach its value and show simulation results.
This project introduces a computational framework for quantum trajectory reconstruction using weak measurements and probabilistic inference. In standard quantum mechanics, physical systems are described by a wavefunction, and observable quantities are inherently probabilistic. As a result, particle trajectories are not directly accessible. This work proposes an operational approach in which trajectories are treated as latent variables inferred from noisy measurement data. The framework combines three core components: A numerical solver for the time-dependent Schrödinger equation (using FFT-based split-operator methods) A weak measurement model that simulates continuous, noisy observations of the system Inference engines, including Sequential Monte Carlo (particle filtering) and recurrent neural networks, to reconstruct the most probable trajectory The central idea is to reformulate quantum dynamics as an inference problem over hidden states, where the goal is to estimate the trajectory that best explains the observed measurements. This approach does not violate the principles of quantum mechanics, including the Principe d'incertitude de Heisenberg, but instead provides a new computational lens through which quantum systems can be analyzed. Potential applications include: Quantum sensing and metrology Quantum control and error correction Hybrid AI-quantum systems Advanced simulation of open quantum systems The project is fully reproducible and open-source, with modular architecture designed for research extensibility.