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SIAM Journal on Control and Optimization

Spatial Control Model and Analysis of Quantum Fields in One-Dimensional Waveguides

Wen-Bin Dong, Re-Bing Wu, Wei Zhang, Chunwen Li, Tzyh‐Jong Tarn

Published Jan 1, 2016• 9 citations
Publisher

Abstract

In this paper, we explore quantum control systems tuned by spatially varying functions, which are different from the broadly studied quantum control systems with time-dependent controls. Such control problems exist in the design of quantum waveguides and materials (e.g., fibers, photonic crystals, and metamaterials) and have not been considered from a control theoretic point of view. The spatial quantum control model is built from quantized Maxwell equations for dielectric waveguides, where the spatially varying dielectric parameters are taken as controls. The model can be applied to both passive and active dielectrics or their mixtures such as metamaterials. We derive the iterative equation for slab structures, which is analogous to a discrete-time control system. For homogeneous waveguides, the model is parallel with the well-known Langevin equation in the time domain. However, for inhomogeneous and dispersive waveguides, the control model becomes bilinear even for linear dielectric materials and hence is more complicated. The resulting input-output relation is derived in terms of irrational transfer functions, and the statistical properties of quantum noises are analyzed. This work provides a new perspective on modeling and analyzing quantum control systems, on the basis of which further design and optimization with spatial parameters can be explored.

DOI
10.1137/140999190

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In this paper, we explore quantum control systems tuned by spatially varying functions, which are different from the broadly studied quantum control systems with time-dependent controls. Such control problems exist in the design of quantum waveguides and materials (e.g., fibers, photonic crystals, and metamaterials) and have not been considered from a control theoretic point of view. The spatial quantum control model is built from quantized Maxwell equations for dielectric waveguides, where the spatially varying dielectric parameters are taken as controls. The model can be applied to both passive and active dielectrics or their mixtures such as metamaterials. We derive the iterative equation for slab structures, which is analogous to a discrete-time control system. For homogeneous waveguides, the model is parallel with the well-known Langevin equation in the time domain. However, for inhomogeneous and dispersive waveguides, the control model becomes bilinear even for linear dielectric materials and hence is more complicated. The resulting input-output relation is derived in terms of irrational transfer functions, and the statistical properties of quantum noises are analyzed. This work provides a new perspective on modeling and analyzing quantum control systems, on the basis of which further design and optimization with spatial parameters can be explored.
The design of many quantum optical devices is attributed to the determination of spatially distributed parameters (e.g., refraction index in photonic crystal), which calls for a theoretical description for the spatial propagation of quantum fields. This paper discusses the spatial evolution of quantized electromagnetic fields in waveguide with spatially variant dielectric parameters, which is in parallel with the time evolution in the standard input-output formalism [C. W. Gardiner and M. J. Collett, Phys.Rev.A 31, 3761 (1985)]. This formalism is non-trivial to quantum optics in nonlinear and dispersive medium, and provides a new perspective of modeling and analyzing the control of quantum fields by matter.
This thesis is concerned with the control and filtering problems for quantum systems. Two kinds of control problems are considered for a class of quantum systems: optimal control and trajectory tracking control. Filtering of a class of linear quantum systems is also investigated. \n \nFor the optimal control of quantum systems, the system in question evolves on a manifold inR^3 and is modeled as a bilinear control form whose states are represented as coherence vectors. An associated matrix Lie group system with state space S O(3) is introduced in order to facilitate solving the given problem. The controllability as well as the reachable set of the system is analyzed in detail. The maximum principle is applied to the optimal control for systems evolving on the Lie group of special orthogonal matrices of dimension 3, with cost that is quadratic in the control input. Controlling the evolutions of coherence vector is equivalent to controlling the states of an associated matrix Lie group system. Thus, the optimal control problem for single spin-1/2 systems can be transformed to steer state of the matrix Lie group system from the identity matrix to a final matrix corresponding to a target state of single spin-1/2 systems in an optimal fashion. The obtained result is employed to perform a reversible logic quantum operation NOT on single spin-1/2 systems. Explicit expressions for the optimal control are given which are linked to the initial state of the system. The main contribution of the result is to extend the existing result to a larger class of systems, generalizing existing wave-function-based results to single spin-1/2 systems whose states are represented by density matrices such that it allows the inclusion of mixed states. \n \nFor the tracking control problem of closed single spin-1/2 quantum ensembles, based on the Liouville-von Neumann equation, mathematical model of dynamics for single spin-1/2 quantum ensembles are formulated as a bilinear form with states evolving on Bloch sphere. For a typical setting in NMR, one in general would not cancel the applied electromagnetic field along a specified direction, thus leading to the dynamics in a plane perpendicular to that direction on the Bloch sphere. One natural question arises that how can one design a feedback controller such that given an initial condition for quantum ensembles, the system evolutions can be steered to the pre-determined trajectory. An associated matrix Lie group differential equation describing transfers from the initial state being an identity matrix is naturally introduced. The tracking problem for the system of interest is transformed to a stabilization problem such that the control techniques are able to be used. A distance function as well as a distance vector defined on the Lie group SO(3) are introduced, which can be used to construct a Lyapunov function to deal with the tracking problem. In order to deal with the tracking problem, the Lyapunov-based feedback design method is employed to obtain the feedback control such that the dynamics of system can converge to the reference trajectory. \n \nFor filtering of quantum systems, the filter problem of a class of linear quantum stochastic systems with quantum noise is investigated. The quantum system model is first analyzed, then the necessary conditions of nondemolition are obtained by applying the observation principle. The relations between the input and output noise are necessary for the nondemolition measurement of the quantum output channels. Given prescribed quantum system, the filter of the system is derived using those necessary conditions.
Using quantum theory, we study the propagation of an optical field in an inhomogeneous dielectric, and apply this scheme to traveling optical fields in a waveguide. We introduce a field-atom interaction Hamiltonian and derive the refractive index using quantum optics. We show that the transmission and reflection of optical fields at an interface between different materials can be described with normalized Fresnel coefficients and that this representation is related to the beam splitter operator. We then study the propagation properties of the optical fields for two types of slab waveguides: step-index and graded-index. The waveguides are divided into multiple layers to represent the spatial dependence of the optical field. We can evaluate the number of photons in an arbitrary volume in the waveguide using this procedure. Using the present method, the quantum properties of weak optical fields in a waveguide are revealed, while coherent states with higher amplitudes reduces to representation of classical waveguide optics.
A quantization scheme is developed for the radiation and higher order electromagnetic fields in one dimensional periodic, dispersive and absorbing dielectric medium. For this structure, the Green function is solved based on the plane wave expansion method, thus the photon operators, commutation relations and quantum Langevin equations are given and studied based on the Green function approach, moreover, the input-output relations are also derived. It is proved that this quantum theory can be reduced back to that of the predecessors' study on the homogenous dielectric. Based on this method, we find that the transformation of the photon state through the lossy grating is non-unitary and that the notable non-unitary transformation can be obtained by tuning the imaginary part of the permittivity, we also discussed the excellent quantum optical properties for the grating which are similar to the classical optical phenomena. We believe our work is very beneficial for the control and regulation of the quantum light based on gratings.
These notes, intended to be self contained and tutorial, present a direct, macroscopic approach to quantizing light inside a linear-response dielectric material when both spectral dispersion and spatial nonuniformity are present, but the spectral region of interest is optically transparent so that explicit treatment of the underlying physics of the medium is not needed. The approach taken is based on the macroscopic Maxwell equations and a corresponding Hamiltonian, without the use of Lagrangians or any dynamical model for the medium, and uses a standard mode-based quantization method. The treatment covers: energy density and flux in a dispersive dielectric; a summary of the inverse permittivity formalism; a new derivation of the mode normalization condition; a direct proof of the nonorthogonality of the modes; examples of quantized field expressions for the general case and various special cases; the relationship between group velocity and energy flux; the band approximation and the continuum limit; and quantum optical treatment of waveguide modes.
An investigation is made of the coherence of the electromagnetic field in a plane dielectric waveguide excited by an external field with specific coherence properties. The quantum theory of an electromagnetic field in one-dimensionally inhomogeneous media is employed to derive an expression which relates the correlation functions of the field inside the waveguide to the external field. This expression is used to study the variation of the coherence of the radiation at various distances z from the area where the radiation is coupled into the waveguide. If the external field has a finite line width, the spatial coherence is a periodic function of z when z ≪ z cr and tends to a constant value when z ≫ z cr . When z ≪ z cr , the temporal coherence is the same as for the external source, whilst when z ≫ z cr , the temporal coherence as a function of τ has maxima which move away from the point τ = 0 with increasing z . It is found that the coherence tunes τ 1 and τ 2 determined following Wolf and Mandel behave differently when z ≫ z cr : τ 1 2 increases quadratically with increasing z , whereas τ 2 tends to a constant value. The form of the radiation modulation spectrum is deduced. Curves are plotted for the spatial and temporal coherence by way of a specific example.