Biorthonormal Basis Research Articles

Theory of quantum fluctuations of optical dissipative structures and its application to the squeezing properties of bright cavity solitons

We present a method for the study of quantum fluctuations of dissipative structures forming in nonlinear optical cavities, which we illustrate in the case of a degenerate, type I optical parametric oscillator. The method consists in (i) taking into account explicitly, through a collective variable description, the drift of the dissipative structure caused by the quantum noise, and (ii) expanding the remaining--internal--fluctuations in the biorthonormal basis associated to the linear operator governing the evolution of fluctuations in the linearized Langevin equations. We obtain general expressions for the squeezing and intensity fluctuations spectra. Then we theoretically study the squeezing properties of a special dissipative structure, namely, the bright cavity soliton. After reviewing our previous result that in the linear approximation there is a perfectly squeezed mode irrespective of the values of the system parameters, we study which are the better squeezed modes at the different bifurcations that the cavity soliton can suffer, as well as the squeezing level attainable when the homodyne detection is made with a plane-wave local oscillator field. In this last case, the squeezing level is quite large in most of the existence domain of the cavity soliton. We then take into account the effect of the detectormore » size and find that, in general, squeezing degrades as the detector is made smaller, except for detector sizes around the cavity soliton size, in which case there is an interesting deviation from the general trend.« less

Non-Hermitian polarizers: a biorthogonal analysis

The non-Hermitian operators of the ideal nonorthogonal multilayer optical polarizers are spectrally analyzed in the framework of skew-angular biorthonormal vector bases. It is shown that these polarizers correspond to skew projectors and their operators are generated by skew projectors, exactly as the canonical ideal polarizers correspond to Hermitian projectors. Thus the common feature of all the polarizers (Hermitian and non-Hermitian) is that their "nuclei" are (orthogonal or skew) projectors--the generating projectors. It is shown that if these nonorthogonal polarizers are looked upon as variable devices, two kinds of degeneracy may occur for suitable values of the inner parameter of the device: The corresponding operators may become normal (more precisely, Hermitian) or, on the contrary, very pathological--defective and singular. In the first case their eigenvectors and biorthogonal conjugate eigenvectors collapse into a unique pair of eigenvectors; in the second case their eigenvectors (as well as their biorthogonal conjugates) collapse into a single vector.

On existence of a biorthonormal basis composed of eigenvectors of non-Hermitian operators

We present a set of necessary conditions for the existence of a biorthonormal basis composed of eigenvectors of non-Hermitian operators. As an illustration, we examine these conditions in the case of normal operators. We also provide a generalization of the conditions which is applicable to non-diagonalizable operators by considering not only eigenvectors but also all root vectors.

Block second-order Krylov subspace methods for large-scale quadratic eigenvalue problems

In this paper, we first introduce a block second-order Krylov subspace G m 1 ( A , B ; Q 1 ) based on a pair of square matrices A and B and an orthonormal matrix Q 1. Then we present a block second-order Arnoldi procedure for generating an orthonormal basis of G m 1 ( A , B ; Q 1 ) and a block second-order biorthogonalization procedure for generating biorthonormal basis of G m 1 ( A , B ; Q 1 ) and G m 1 ( A T , B T ; P 1 ) . By applying the projection techniques, we derive two block second-order Krylov subspace methods for solving a large-scale quadratic eigenvalue problem (QEP). These methods are applied to the QEP directly. Hence they preserve essential structures and properties of the QEP. Some theoretical results are given. Numerical experiments report the effectiveness of these methods.

The second-order biorthogonalization procedure and its application to quadratic eigenvalue problems

Given a pair of matrices and starting vectors, we present a procedure to generate the biorthonormal basis of the second-order right and left Krylov subspaces. The application is to solve the large-scale quadratic eigenvalue problems via oblique projection technique. This method can take full advantage of the sparseness of large-scale system as well as the superior convergence behavior of Krylov subspace based methods by implicit linearization, which makes the solution acceptable in terms of both cost and time.

Rabi oscillations in a two-level atomic system with a pseudo-Hermitian Hamiltonian

Rabi oscillations of a two-level atomic system with damping effects are described by the use of a pseudo-Hermitian Hamiltonian. The Rabi frequency and the amplitudes for being in the upper and lower level depend in the present system on both the Hermitian resonant interaction between the em field and the atomic dipole and on damping effects. The time development of the two-level system is studied by using the metric of the pseudo-Hermitian Hamiltonian and its bi-orthonormal basis of states. The special characteristics of the present system are related to PT (parity-time reversal) and C (generalized conjugation operator) invariant properties of the pseudo-Hermitian Hamiltonian.

Vector analysis of dispersion for the fundamental cladding mode in photonic crystal fibers

The dispersions of the fundamental cladding mode in photonic crystal fibers(PCFs)are computed with a vectorial effective index approach in this paper. By comparing with the results obtained from biorthonormal-basis method or scalar effective method,it is found that the results of this paper are in accordance with that obtained from the biorthonormalbasis method, and that the scalar effective method is a good approximation at a lower air-filling fraction or higher normalized wave-number A/λ. For high accuracy at larger air-filling fraction or lower normalized wave-number A/λ,it is necessary to consider the vectorial method. The relation between effective index of cladding and microstructure of PCFs is discussed. The computation and analysis in this paper can provide a theoretic reference for designing of PCFs.

The Gram Matrix of a PT-Symmetric Quantum System

The eigenstates of a diagonalizable PT-symmetric Hamiltonian satisfy unconventional completeness and orthonormality relations. These relations reflect the properties of a pair of bi-orthonormal bases associated with non-hermitean diagonalizable operators. In a similar vein, such a dual pair of bases is shown to possess, in the presence of PT symmetry, a Gram matrix of a particular structure: its inverse is obtained by simply swapping the signs of some its matrix elements.

Degeneracy of Resonances, Jordan Blocks, and Gamow–Jordan Eigenfunctions

We show that a degeneracy of resonances is associated with a second rank pole in the scattering matrix and a Jordan chain of generalized eigenfunctions of the radial Schrodinger equation. The generalized Gamow-Jordan eigenfunctions are basis elements of an expansion in complex resonance energy eigenfunctions. In this biorthonormal basis, any operator f(H r \(\overline B\) which is a regular function of the Hamiltonian is represented by a nondiagonal complex matrix with a Jordan block of rank 2.

Jordan blocks and Gamow-Jordan eigenfunctions associated with a degeneracy of unbound states

An accidental degeneracy of unbound states gives rise to a double pole in the scattering matrix, a double zero in the Jost function, and a Jordan chain of length 2 of generalized Gamow-Jordan eigenfunctions of the radial Schr\"odinger equation. The generalized Gamow-Jordan eigenfunctions are basis elements of an expansion in bound- and resonant-state energy eigenfunctions plus a continuum of scattering wave functions of a complex wave number. In this biorthonormal basis, any operator ${f(H}_{r}^{(\mathcal{l})})$ which is a regular function of the Hamiltonian is represented by a complex matrix that is diagonal except for a Jordan block of rank 2. The occurrence of a double pole in the Green's function, as well as the non exponential time evolution of the Gamow-Jordan generalized eigenfunctions are associated with the Jordan block in the complex energy representation.

A Multi-Channel Multi-Sampling Rate Theorem

The Mobius function is utilised to propose a class of multi-channel, multi-sampling rate theorems. Our construction facilitates a parallel decomposition of the Fourier transform into an array of general transform operators. A practical example is discussed that employs a complex square wave biorthonormal basis.

Spectral line shapes in dissipative systems: Molecules adsorbed on metal surfaces

The temporal evolution of rates and probabilities of transitions, induced by light between states of a molecule interacting with a dissipative medium, are derived for a Hamiltonian containing an anti-Hermitian term which describes dissipation. Our theoretical treatment introduces a biorthonormal basis set of decaying eigenstates to obtain the rates and probabilities from perturbation theory. The formalism is developed for excitation by continuous and pulsed light. As an example, it is applied to the excitation of molecules adsorbed on solid surfaces, taking into account the dissipation of energy into the solid as the molecule relaxes or is desorbed.

Generalized Localized Matter Wavepackets

Studying the system of the non-Hermitian (NH) generalized parametric oscillator, we use the Wei–Norman factorization method, as well as the biorthonormal basis formalism, in order to initiate a complex probability density. Generalizing the usual Yuen procedure for the calculation of the variances Δq, Δp in terms of the complex probability density, we obtain a reduction in the position and an increase in the momentum uncertainty, in discrete time intervals. In the framework of the macroscopic quantum tunneling (MQT) effect, such a suppression of the matter wavepacket spreading provides a mechanism to localize a quantum particle, and is responsible for its behavior as a macroscopic state, inhibiting the tunneling process. Applications of squeezed matter wavepackets can be found in the control of excited-state chemical reactions, in MQT for Josephson junctions, and in the tunneling of trapped atoms in optical crystals.

Eigenmodes of decay and fragmentation processes.

Linear rate equations are used to describe the cascading decay of an initial heavy cluster into fragments. This representation is based upon a triangular matrix of transition rates. We expand the state vector of mass multiplicities, which describes the process, into the biorthonormal basis of eigenmodes provided by the triangular matrix. When the transition rates have a scaling property in terms of mass ratios at binary fragmentation vertices, we obtain solvable models with explicit mathematical properties for the eigenmodes. A suitable continuous limit provides an interpolation between the solvable models. It gives a general relationship between the decay products and the elementary transition rates.

M-ary wavelet transform and formulation for perfect reconstruction in M-band filter bank

The binary wavelet transform is generalized and extended to the M-ary biorthonormal case. The computational equivalence between the discrete wavelet analysis and the M-band multirate signal filtering is indicated. The equivalence allows the perfect reconstruction requirement in a filter bank to be investigated from the vector space decomposition/reconstruction in wavelet analysis. From the construction of the biorthonormal wavelet bases, the necessary and sufficient condition for the filters in a perfect reconstruction filter bank is formulated. Under this formulation, an additional optimization procedure is then used to model the frequency domain requirement in filter bank design.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Periodic sampling basis and associated biorthonormal basis of signal spaces composed of spline functions

AbstractThis paper derives a Fourier‐like expansion formula in the signal space composed of spline functions, with the sampled value sequence of the signal as the expansion coefficients. Here, the domain is a finite closed interval and the boundary condition is periodic. The signal space composed of spline functions is one of the generalizations of the band‐limited signal space with the continuous differentiability as the parameter. Consequently, the Fourier‐like expansion formula derived in this paper is a generalization of Cahn's sampling theorem in the band‐limited signal space.

Calculation of transition density matrices by nonunitary orbital transformations

AbstractAn efficient scheme for calculating one‐ and two‐electron transition density matrices for two wave functions is described. The method applies to CAS (complete active space) wave functions and certain multireference CI expansions. The orbital sets of the two wave functions are not assumed to be equal. They are transformed to a biorthonormal basis, and the corresponding transformation of the CI coefficients is carried out directly, using the one‐electron coupling coefficients.

Physical boson basis states in the boson expansion theories.

The use of physical boson basis states is stressed for the calculations in the boson space. The explicit form of physical boson basis states in terms of bosons is derived for the nonunitary boson mapping of Dyson. The ambiguity in the normalization introduced due to the use of bi-orthonormal basis states is satisfactorily resolved, resulting in a Hermitian matrix. This Hermitian matrix is found to coincide with the Hamiltonian matrix in the fermion space. The model cases where the use of a boson basis is justified are shown to be consistent with our view of using physical boson basis states.

Biorthonormal electronic basis sets in the theory of electron transfer reactions

We have used the quantum-chemical concept of biorthonormal electronic basis functions to derive the two-centre electron exchange integrals in non-diabatic electron transfer reactions. It is shown that only non-diagonal parts induce the transfer corresponding to inclusion of the diagonal parts in the zero-order hamiltonians. This result coincides with the result derived previously by means of scattering theory.

An effective (exchange) Hamiltonian for many-electron systems with degenerate and nonorthogonal localized orbitals

Using biorthonormal basis sets, the orthonormal formulation of magnetic interactions between ions in an insulator, previously given by Fuchikami and Tanabe, is extended to nonorthogonal function spaces. The nonorthogonality is entirely absorbed in the coefficients of each operator appearing in the effective Hamiltonian. To make a numerical comparison, both methods are applied to a few simple and exactly solvable problems. We discuss the effect of nondiagonal matrix elements between spatially equivalent excited configurations in connection with the perturbational expansion applied.