We consider the propagation of the electromagnetic field within an optical fibre. The field consists of a classical component and a quantum component. The classical component is large and it interacts with the matter within the fibre comprising of atoms that are vibrating, ie, phonons. This interaction causes scattering of the classical field component that interacts with the quantum component of the field, thereby altering the state of the quantum field. After constructing this model, we propose two methods for reducing this "classical-photon-phonon interaction noise". The first method is based on an optimal control algorithm wherein we generate a "control potential" in such a way that after incorporating this potential in the Schrodinger dynamics of the state the state tracks as closely as possible a desired "noiseless state". The second approach is based on Belavkin’s quantum filtering theory wherein we model the classical photons as a quantum Bosonic white noise process and by taking non-demolition measurements on the system, we generate on a real-time basis, an estimate of the evolving state which by application of a control potential is made to track a desired state. Analysis of the spectrum of the transmitted quantum electromagnetic field is carried out by using a quantum stochastic differential model for the noise based on the Hudson-Parthasarathy quantum stochastic calculus. Further, the basic ideas of Cq channel capacities in quantum information theory are used to analyze the rate of information transmission through this optical fibre quantum mechanical channel. Transmission of other kinds of particles like non-Abelian matter and gauge particles are also discussed and the role of superstring theory in building a foundation for corrections to the Yang-Mills action is also discussed. We also highlight the use of the quantum effective action of a field when it interacts with other fields and with random current sources. The quantum effective action, obtained using Feynman’s path integral methods for fields provides a firm foundation for corrections to the classical action by quantum effects and enables one to describe the quantum corrections to the field equations accurately.
We consider the propagation of the electromagnetic field within an optical fibre. The field consists of a classical component and a quantum component. The classical component is large and it interacts with the matter within the fibre comprising of atoms that are vibrating, ie, phonons. This interaction causes scattering of the classical field component that interacts with the quantum component of the field, thereby altering the state of the quantum field. After constructing this model, we propose two methods for reducing this "classical-photon-phonon interaction noise". The first method is based on an optimal control algorithm wherein we generate a "control potential" in such a way that after incorporating this potential in the Schrodinger dynamics of the state the state tracks as closely as possible a desired "noiseless state". The second approach is based on Belavkin’s quantum filtering theory wherein we model the classical photons as a quantum Bosonic white noise process and by taking non-demolition measurements on the system, we generate on a real-time basis, an estimate of the evolving state which by application of a control potential is made to track a desired state. Analysis of the spectrum of the transmitted quantum electromagnetic field is carried out by using a quantum stochastic differential model for the noise based on the Hudson-Parthasarathy quantum stochastic calculus. Further, the basic ideas of Cq channel capacities in quantum information theory are used to analyze the rate of information transmission through this optical fibre quantum mechanical channel. Transmission of other kinds of particles like non-Abelian matter and gauge particles are also discussed and the role of superstring theory in building a foundation for corrections to the Yang-Mills action is also discussed. We also highlight the use of the quantum effective action of a field when it interacts with other fields and with random current sources. The quantum effective action, obtained using Feynman’s path integral methods for fields provides a firm foundation for corrections to the classical action by quantum effects and enables one to describe the quantum corrections to the field equations accurately.
Scalability in quantum networks requires the co-propagation of quantum and classical signals within shared optical fibres. However, classical fields interact with thermally induced atomic vibrations (phonons), generating scattering and noise that degrade quantum information. We model this interaction using a random Hamiltonian and the Lindblad master equation, accounting for both deterministic and stochastic noise components. To mitigate these effects, we propose a novel control method that introduces a counter-potential to cancel the average interaction noise. Simulations indicate a significant reduction in the noise-to-signal energy ratio using this approach. Additionally, we analyse a fully quantized scenario involving quantum phonon lattices and noise. We outline a method for generating counter-TPCP (Trace-Preserving Completely Positive) maps to neutralize these interactions. Finally, we provide a Monte Carlo algorithm, based on the LucBouten method, to construct these counter maps using a random potential. This research fosters robust, high-efficiency quantum technology development.
In this paper, we formulate the basic Maxwell-Dirac equations for the electromagnetic field of photons interacting with the Dirac field of electrons and positrons taking into account noisy components of the electromagnetic field modeled in the context of the Hudson-Parthasarathy quantum stochastic calculus. These equations are formulated within a waveguide and result in Helmholtz differential equations for the z component of the em fields with Dirac current source. We demonstrate how quantum noise perturbs these fields and how one can calculate the moments of the resulting perturbed quantum fields due to interaction and noise in a coherent state of the photons and Fermions. We also discuss perturbations to these fields produced by quantum gravitational fields using the covariant Maxwell equations of general relativity. Specifically, using elementary perturbation theory, we derive Helmholtz equations for the z components of the perturbed electromagnetic fields with current sources being functions of the unperturbed electromagnetic and Dirac fields and also the gravitational metric perturbations. We indicate how these calculations can be used in quantum guide parameter estimation problems. We also indicate how by expressing the Hamiltonian of the Maxwell-Dirac field in terms of creation and annihilation operators and creation and annihilation quantum stochastic processes for the noise, we can derive the Hudson-Parthasarathy noisy Schrodinger equation for the unitary evolution in second quantized field space and how by measuring non-demolition noise, we can estimate the evolving state of the Maxwell-Dirac field on a real time basis. The paper provides a self-contained discussion of quantum noise in field theory. Some aspects of supersymmetry, string gauge field theory and non-Abelian gauge field theory in the presence of noise are also discussed here. The paper provides a self-contained account of how to treat quantum Bosonic and Fermionic noisy fields in the context of quantum field theory by highlighting the fact that quantum noisy fields have an additional time-component of quantum randomness as compared with that of conventional quantum fields.
This paper starts with a basic formulation of a distributed parameter transmission line with memristor per unit length in series and also memristor per unit length in parallel apart from the other standard distributed parameters. The result of having memristors distributed along the line is a nonlinear effect in the form of coupling between charge and current and coupling between charge and voltage. These nonlinear effects result in the generation of an infinite number of higher harmonic components along the line even when the input voltage is of a fixed frequency or a finite superposition of voltages of different frequencies. Taking into account random white noise line loading effects, we formulate the resulting line equations based on KCL and KVL in the form of an infinite dimensional nonlinear stochastic differential equation. We study the problem of quantizing the transmission line dynamics using the well known Hudson-Parthasarathy theory of quantum stochastic differential equations derived from the Hudson-Parthasarathy noisy Schrodinger equation constructed from an appropriate transmission line Hamiltonian and Lindblad operators. We then conclude by discussing the effect of quantum stochastic noise and quantum stochastic supersymmetric noise on the action functional of a field theory and how such a model can be used to obtain corrections to the quantum effective action functional or more precisely, to the TPCP map that evolves an initially mixed state of the field to another mixed state after a finite duration. Some remarks on training and testing of quantum neural networks based on the Belavkin quantum filter are presented. This involves constructing a quantum evolution of a mixed state in such a way that the evolving pdf of a system observable will be obtained from the conditional expectation of the noisy quantum state given non-demolition measurements that are strongly correlated with actually measured signals.
We consider the interaction between a classical time-varying electric field and the atomic electrical dipole moment operator of a finite state atom and calculate approximately the mixed state evolution of the atom with such an interaction Hamiltonian up to quadratic orders in the external classical electric field using Schrodinger’s equation for mixed state evolution. From this approximate expression for the state evolution, we evaluate the average atomic electric dipole moment as a linear-quadratic function of the electric field. Later on, we also take into account Brownian motion bath noise that couples to the atomic dynamics, so that the Schrodinger equation dynamics gets modified to the Lindblad dynamics of an open quantum system, i.e., the master equation. From the average atomic dipole moment, we obtain an expression for the average polarization field of atoms constituting the medium through which the classical electromagnetic wave propagates. This average polarization is a linear-quadratic causal functional of the electric field. The resulting Maxwell equations for wave propagation, taking into account this polarization current density, thus acquire a quadratic nonlinearity which can be used to explain higher harmonic frequencies at the output of an optical fiber when the input is excited by the electric field coming from a monochromatic laser. We are then also able to calculate the mean square fluctuations in the polarization field using the evolving atomic state, and hence the electric field propagating through the medium of atoms acquires quantum fluctuations which can be used to explain time-varying random shifts in the spectral lines. As mentioned above, we include in this article an extension of this problem to the case when there is white Gaussian noise corrupting the state evolution dynamics of the atom and also the case when there is a pairwise interaction between the spin dipoles of atoms at two different locations. In the former case, we derive an approximate formula for the polarization field as well as its temporal statistical correlations, and we show how this computation can be extended to the latter case too. When there is a pairwise interaction between the atomic spins, we use a version of the approximate quantum Boltzmann equation to calculate the first and second order marginals of the atomic states, which is then used to compute the average polarization as well as its spatio-temporal correlations. Temporal correlations using second marginals of the atomic states cannot be computed using classical probabilistic methods owing to Heisenberg uncertainty, so we propose a sequential measurement-based strategy involving computation of the joint probabilities of the atomic spin components at two different times by first measuring one component of the spin, allowing for state collapse, and then following it up with nonlinear time evolution from the collapsed two-particle state using the quantum Boltzmann equation, followed further by measurement of another component of the spin at the second time.
Optical fibers are the backbone of the world's communication systems. They also represent the simplest example of a photonic device, where photons in waveguides replace electrons in wires. More sophisticated devices than this can be made, in which all the optical components – including multiple waveguided modes – are integrated on a chip. It is important to understand the quantum noise properties of these systems. While in some respects these systems can be treated as one-dimensional, there are additional features. Dispersion is due not just to material properties, but also to the waveguide geometry. The nonlinearity comes from a combination of material and geometric properties as well. There is, of course, a quantum aspect to these photonic devices, in which quantum noise occurs due to random effects caused by nonlinearity and gain. Optical devices based on waveguides are, of course, not just simple fibers. One can have multiple waveguides with various types of linear coupling. The intrinsic nonlinearity of the dielectric leads to four-wave mixing, which can be an important source of correlated photons, both for fundamental tests of quantum mechanics, and for applications in quantum information. It is often essential to include effects due to anisotropies of various types, as well as material dispersion. In dielectrics, another effect that should be included is the coupling of vibrational modes of the (solid) dielectric to the propagating field, which gives rise to Raman and Brillouin scattering effects. This allows us to obtain a more detailed and correct theory of quantum noise than the simplified case treated in Chapter 3.
This manuscript presents an advanced generalization of the Belavkin quantum filter, applying it to Boson and Fermion noise via the Hudson-Parthasarathy quantum stochastic calculus. It focuses on estimating quantum system observables and states in noisy conditions, highlighting the transition from non-commutative to commutative observables post-filtering. The work introduces an algorithm for constructing quantum filter coefficients by solving an infinite series of linear algebraic equations, enabling real-time filtering in the observable domain. This development offers significant insights into quantum noise analysis and state estimation, with potential applications in quantum computing and information processing. The manuscript also explores the integration of classical noisy measurements with quantum non-demolition measurements, creating a hybrid estimation framework that enhances quantum filtering capabilities. The work has potential applications in estimating the time varying state of a quantum gravitational wave from measurements on electromagnetic noise fields with which the wave interacts.