Abstract All quantum systems are subject to noise and imperfections due to stray fields, inhomogeneities or drifting experimental controls. An understanding of the effects of noise and decoherence is critical to the progress towards fully functional quantum devices. In this perspective, we focus on noise in quantum systems which are modelled by a dynamic stochastic parameter in the Hamiltonian. We will outline exact evolution equations describing the ensemble average dynamics for a variety of common noise types and their connections. We will also highlight an approximate evolution equation valid in the weak noise limit for an arbitrary classical stochastic field. This framework should serve as a starting point for identifying signatures of differing noise types and optimisation of robust control protocols.
Abstract All quantum systems are subject to noise and imperfections due to stray fields, inhomogeneities or drifting experimental controls. An understanding of the effects of noise and decoherence is critical to the progress towards fully functional quantum devices. In this perspective, we focus on noise in quantum systems which are modelled by a dynamic stochastic parameter in the Hamiltonian. We will outline exact evolution equations describing the ensemble average dynamics for a variety of common noise types and their connections. We will also highlight an approximate evolution equation valid in the weak noise limit for an arbitrary classical stochastic field. This framework should serve as a starting point for identifying signatures of differing noise types and optimisation of robust control protocols.
We develop a formally exact master equation to describe the evolution of the average density matrix of a closed quantum system driven by a stochastic Hamiltonian. The average over stochastic processes generally results in decoherence effects in closed system dynamics, in addition to the unitary evolution. We then show that, for an important class of problems in which the Hamiltonian is proportional to a Gaussian random process, the 2nd-order master equation yields exact dynamics. The general formalism is applied to study the examples of a two-level system, two atoms in a stochastic magnetic field and the heating of a trapped ion, where we find phenomena such as decoherence-induced disentanglement.
This thesis studies randomness in quantum dissipative and chaotic dynamics. We first focus on a detailed study of the effect of noise in the Hamiltonian of a quantum system, going beyond the noise-average and characterizing higher moments. The main quantity we introduce is the Stochastic Operator Variance (SOV), which is an observable characterizing the spread of trajectories around the average evolution. Interestingly, this quantity fulfills different types of uncertainty relations and is related to quantum information scrambling through out-of-time-order correlators (OTOC). We illustrate the SOV-OTOC connection in a stochastic version of the Lipkin-Meshkov-Glick (LMG) model, which shows a positive Lyapunov exponent from an unstable saddle point. We find that under the action of the noise, this Lyapunov exponent can change sign, thus stabilizing the unstable phase of the model while destabilizing the stable one. We then study the interplay between noise and decay in a non-Hermitian Hamiltonian. The noise-average evolution follows an antidephasing master equation beyond Lindblad form. We characterize the purity dynamics and the steady states of this master equation, and study this new evolution in a stochastically driven version of the Dissipative Qubit. By characterizing its spectral and steady state properties, we find that there are three phases: the PT unbroken, PT broken, and a novel Noise Induced (NI) phase where the qubit converges to the lossy state. We further investigate the validity of our model to explain experimental data, such as the residual damping rate of the PT unbroken phase. In the last Chapter, we study randomness as a model for chaotic dynamics. In particular, leveraging a Wigner-like surmise for the k-th neighbor level spacing distribution, we compute analytically the k-th neighbor Spectral Form Factor (knSFF), which characterizes the contribution of the k-th neighbor spacings to the Spectral Form Factor (SFF). We study the properties of the individual knSFF and characterize their role in building the universal ramp of the SFF. Interestingly, we find that the very short-range and very long-range spacings are the ones that contribute the most to the extent of the ramp. We finish by discussing our results and possible pathways for further research.
The ensemble-averaged dynamics of open quantum systems are typically irreversible. We show that this irreversibility need not hold at the level of individually monitored quantum trajectories. Our main results are analytical stochastic Schrödinger equations for quantum reverse diffusion, along with corresponding stochastic master equations. These equations describe the exact and approximate stochastic reverse processes for continuously monitored Pauli channels, including time-dependent depolarizing noise. We show that the reverse processes generalize the forward dynamics by combining the noise effects of the forward processes with an additional stochastic drift that dynamically steers a quantum state back to its initial configuration. Consequently, the exact reverse stochastic Schrödinger equations admit closed-form solutions that can be implemented in real-time without the need for variational techniques. Our findings establish an analytical framework for quantum state recovery, noise-resilient quantum gates, quantum generative modelling, quantum tomography via forward-reverse cycles, and potential paradigms for quantum error correction based on reverse diffusion.
ABSTRACT Stochastic evolution underpins several approaches to the dynamics of open quantum systems, such as random modulation of Hamiltonian parameters, the stochastic Schrödinger equation (SSE), and the stochastic Liouville equation (SLE). These approaches replace the explicit system–environment coupling with an effective system‐only dynamics, where dissipative behavior emerges from ensemble averaging. Stochastic Hamiltonians, in particular, have long served as phenomenological tools in physical chemistry to include environmental effects without recourse to an explicit microscopic derivation. In this work, we aim at a self‐contained and accessible presentation of these approaches to further elaborate on their common roots in essential concepts of stochastic calculus and to delineate the conditions under which they are equivalent. We also discuss how different formulations naturally lead to different numerical time‐integration schemes, better suited for either classical simulation platforms, based on finite‐difference approximations, or quantum algorithms, that employ random unitary maps. Our analysis aims at providing a unified perspective and actionable recipes for classical and quantum implementations of stochastic evolution in the simulation of open quantum systems.
We develop a new master equation as a unified description of the effects of both quantum noise (system-bath interaction) and classical noise on a system's dynamics, using a two-dimensional series expansion method. When quantum and classical noises are both present, their combined effect on a system's dynamics is not necessarily a simple sum of the two individual effects. Thus previous master equations for open systems and those for classical noise, even when jointly used, may not capture the full physics. Our formalism can determine whether there is interference between quantum and classical noises and will be able to capture and describe such interference if there is any (in a perturbative manner). We find that, interestingly, second-order interference between quantum and classical noises vanishes identically. This work thus also serves to justify simple additive treatments of quantum and classical noises, especially in the weak coupling regime. For a Zeeman-splitted atom in a stochastic magnetic field interacting with an optical cavity, we use the formalism developed herein to find the overall decoherence rate between the atom's energy levels.
We study the quantum dynamics generated by a non-Hermitian Hamiltonian subject to stochastic perturbations in its anti-Hermitian part, describing fluctuating gains and losses. The dynamics averaged over the noise is described by an "antidephasing" master equation. We characterize the resulting state evolution and analyze its purity. The properties of such dynamics are illustrated in a stochastic dissipative qubit. Our analytical results show that adding noise allows for a rich control of the dynamics, stabilizing the lossy state and making state purification possible to a greater variety of steady states.