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John Gough

Publications and source records attributed to John Gough.

36 records · Page 2Linked to original sources
arXiv2012

Asymptotic Stochastic Transformations for Nonlinear Quantum Dynamical Systems

The Ito and Stratonovich approaches are carried over to quantum stochastic systems. Here the white noise representation is shown to be the most appropriate as here the two approaches appear as Wick and Weyl orderings, respectively. This introduces for the first time the Stratonovich form for SDEs driven by Poisson processes or quantum SDEs including the conservation process. The relation of the nonlinear Heisenberg ODES to asymptotic quantum SDEs is established extending previous work on linear (Schrodinger) equations. This is shown to generalize the classical integral transformations between the various forms of stochastic calculi and to extend the Khasminskii theorem to the quantum setting.

math-ph↗
arXiv2009

Quantum Dissipative Systems and Feedback Control Design by Interconnection

The purpose of this paper is to extend J.C. Willems' theory of dissipative systems to the quantum domain. This general theory, which combines perspectives from the quantum physics and control engineering communities, provides useful methods for analysis and design of dissipative quantum systems. We describe the interaction of the plant and a class of exosystems in general quantum feedback network terms. Our results include an infinitesimal characterization of the dissipation property, which generalizes the well-known Positive Real and Bounded Real Lemmas, and is used to study some properties of quantum dissipative systems. We also show how to formulate control design problems using quantum network models, which implements Willems' `control by interconnection' for open quantum systems. This control design formulation includes, for example, standard problems of stabilization, regulation, and robust control.

quant-ph↗
arXiv2007

Filtering of Wide Sense Stationary Quantum Stochastic Processes

We introduce a concept of a quantum wide sense stationary process taking values in a C*-algebra and expected in a sub-algebra. The power spectrum of such a process is defined, in analogy to classical theory, as a positive measure on frequency space taking values in the expected algebra. The notion of linear quantum filters is introduced as some simple examples mentioned.

math-ph↗
arXiv2006

Quantum Flows as Markovian Limit of Emission, Absorption and Scattering Interactions

We consider a Markovian approximation, of weak coupling type, to an open system perturbation involving emission, absorption and scattering by reservoir quanta. The result is the general form for a quantum stochastic flow driven by creation, annihilation and gauge processes. A weak matrix limit is established for the convergence of the interaction-picture unitary to a unitary, adapted quantum stochastic process and of the Heisenberg dynamics to the corresponding quantum stochastic flow: the convergence strategy is similar to the quantum functional central limits introduced by Accardi, Frigerio and Lu$^{[ 1]}$. The principal terms in the Dyson series expansions are identified and re-summed after the limit to obtain explicit quantum stochastic differential equations with renormalized coefficients. An extension of the Pul\UNICODE{0xe9} inequalities$^{[ 2]}$ allows uniform estimates for the Dyson series expansion for both the unitary operator and the Heisenberg evolution to be obtained.

quant-ph↗
arXiv2006

Singular perturbation of quantum stochastic differential equations with coupling through an oscillator mode

We consider a physical system which is coupled indirectly to a Markovian resevoir through an oscillator mode. This is the case, for example, in the usual model of an atomic sample in a leaky optical cavity which is ubiquitous in quantum optics. In the strong coupling limit the oscillator can be eliminated entirely from the model, leaving an effective direct coupling between the system and the resevoir. Here we provide a mathematically rigorous treatment of this limit as a weak limit of the time evolution and observables on a suitably chosen exponential domain in Fock space. The resulting effective model may contain emission and absorption as well as scattering interactions.

math-ph↗
arXiv2006

Quantum Stratonovich Stochastic Calculus and the Quantum Wong-Zakai Theorem

We extend the Ito -to- Stratonovich analysis or quantum stochastic differential equations, introduced by Gardiner and Collett for emission (creation), absorption (annihilation) processes, to include scattering (conservation) processes. Working within the framework of quantum stochastic calculus, we define Stratonovich calculus as an algebraic modification of the Ito one and give conditions for the existence of Stratonovich time-ordered exponentials. We show that conversion formula for the coefficients has a striking resemblance to Green's function formulae from standard perturbation theory. We show that the calculus conveniently describes the Markov limit of regular open quantum dynamical systemsin much the same way as in the Wong-Zakai approximation theorems of classical stochastic analysis. We extend previous limit results to multiple-dimensions with a proof that makes use of diagrammatic conventions.

math-ph↗
arXiv2005

Quantum Stochastic Generators

We discuss stochastic derivations, stochastic Hamiltonians and the flows that they generate, algebraic fluctuaion-dissipation theorems, etc., in a language common to both classical and quantum algebras. It is convenient to define distinct notions of time-ordered exponentials to take account of the breakdown of the Leibniz rule in the Ito calculus. We introduce a notion of quantum Stratonovich calculus and show how it relates to Stratonovich-Dyson time ordered exponentials. We then use it to demonstrate a natural way to add stochastic derivations.

quant-ph↗
arXiv2005

Optimal Quantum Feedback Control for Canonical Observables

We show that the stochastic Schrodinger equation for the filtered state of a system, with linear free dynamics, undergoing continual non-demolition measurement or either position or momentum, or both together, can be solved explicitly within a class of Gaussian states which we call extended coherent states. The asymptotic limit yields a class of relaxed states which we describe explicitly. Bellman's principle is then applied directly to optimal feedback control of such dynamical systems and the Hamilton Jacobi Bellman equation for the minimum cost is derived. The situation of quadratic performance criteria is treated as the important special case and solved exactly for the class of relaxed states.

quant-ph↗
arXiv2005

Quantum Statistical Field Theory and Combinatorics

This is a set of review notes on combinatorial aspects of Bosonic quantum field theory. We collect together several related issues concerning moments of distributions, moments of stochastic processes and Ito's formula, and Green's functions and cumulant moments in quantum field theory.

quant-ph↗
arXiv2005

Stochastic Schrodinger equations as limit of discrete filtering

We consider an open model possessing a Markovian quantum stochastic limit and derive the limit stochastic Schrodinger equations for the wave function conditioned on indirect observations using only the von Neumann projection postulate. We show that the diffusion (Gaussian) situation is universal as a result of the central limit theorem with the quantum jump (Poissonian) situation being an exceptional case. It is shown that, starting from the correponding limiting open systems dynamics, the theory of quantum filtering leads to the same equations, therefore establishing consistency of the quantum stochastic approach for limiting Markovian models.

quant-ph↗
arXiv2004

Feynman Diagrams and the Quantum Stochastic Calculus

We present quantum stochastic calculus in terms of diagrams taking weights in the algebra of observables of some quantum system. In particular, we note the absence of non-time-consecutive Goldstien diagrams. We review recent results in Markovian limits in these terms.

quant-ph↗
arXiv2004

The Stochastic Sector of Interacting-Free Quantum Field Theory

The Quantum Stochastic Limit of a quantum mechanical particle coupled to a quantum field without the neglect of the response details of the interaction (i.e. not making the dipole approximation) is made following the treatment of Accardi and Lu [6] and the corresponding Quantum Stochastic Structure is derived. The stochastic sector for the noise is constructed and is shown to be of a qualitatively new type. We also include a physical discussion on the limit noise which obeys Interacting-Free statistics and include a new shorter proof of the noise convergence and also a new construction of Interacting-Free Fock Space.

quant-ph↗
arXiv2004

Continuous measurement of canonical observables and limit Stochastic Schrodinger equations

We derive the stochastic Schrodinger equation for the limit of continuous weak measurement where the observables monitored are canonical position and momentum. To this end we extend an argument due to Smolianov and Truman from the von Neumann model of indirect measurement of position to the Arthurs and Kelly model for simultaneous measurement of position and momentum. We only require unbiasedness of the detector states and an integrability condition sufficient to ensure a central limit effect. Despite taking a weak interaction, as opposed to weak measurement limit, the resulting stochastic wave equation is of the same form as that derived in a recent paper by Scott and Milburn for the specific case of joint Gaussian states.

quant-ph↗
arXiv2004

Boson-Fermion unification implemented by Wick calculus

We construct a transformation between Bose Fock space and Fermi Fock space that is super-symmetric in the sense that it converts Boson fields into Fermi fields over a fixed one-particle space. The transformation involves the spectral splitting of the one-particle space into a continuous direct integral of internal spaces. We present a theory of integration on the Fock spaces over the square intergable functions taking values in these internal spaces which we refer to as a Wick calculus: this is a natural generalization of the theory of quantum stochastic calculus which used a single fixed internal space.

math-ph↗
arXiv2003

DUCT: An Interactive Define-Use Chain Navigation Tool for Relative Debugging

This paper describes an interactive tool that facilitates following define-use chains in large codes. The motivation for the work is to support relative debugging, where it is necessary to iteratively refine a set of asser-tions between different versions of a program. DUCT is novel because it exploits the Microsoft Intermediate Language (MSIL) that underpins the .NET Framework. Accordingly, it works on a wide range of programming languages without any modification. The paper describes the design and implementation of DUCT, and then illustrates its use with a small case study.

cs.SE↗
arXiv2003

Holevo-ordering and the continuous-time limit for open Floquet dynamics

We consider an atomic beam reservoir as a source of quantum noise. The atoms are modelled as two-state systems and interact one-at-a-time with the system. The Floquet operators are described in terms of the Fermionic creation, annihilation and number operators associated with the two-state atom. In the limit where the time between interactions goes to zero and the interaction is suitably scaled, we show that we may obtain a causal (that is, adapted) quantum stochastic differential equation of Hudson-Parthasarathy type, driven by creation, annihilation and conservation processes. The effect of the Floquet operators in the continuous limit is exactly captured by the Holevo ordered form for the stochastic evolution.

quant-ph↗
arXiv2003

Quantum Markovian Approximations for Fermionic Reservoirs

We establish a quantum functional central limit for the dynamics of a system coupled to a Fermionic bath with a general interaction linear in the creation, annihilation and scattering of the bath reservoir. Following a quantum Markovian limit, we realize the open dynamical evolution of the system as an adapted quantum stochastic process driven by Fermionic Noise.

quant-ph↗