Narrow Results

Based on the coherent state (S1) and the operator ${(x{a}^{†}+ya)}^m$, we induce other three quantum states (here we abbreviate them as S2, S3 and S4). S2 is obtained by operating the operator on S1 directly. S3 is an orthogonal state of S1 constructed from the orthogonalizer relevant with that operator. S4 is a continuous-variable (CV) qubit state superposed from S1 and S3. We study and compare the mathematical and physical properties of such four quantum states. We demonstrate some statistical properties for S1–S4, including the mean photon number (MPN), anti-bunching effect, quadrate squeezing, photon number distribution, Husimi Q-function and Wigner function. The numerical results show some interesting non-classical characters for such states. It is worthy to note that the photon-added coherent state introduced by Agarwal and Tara is only a special case of our considered states.

An orthogonal state of coherent state is produced by applying an orthogonalizer related with Hermite-excited superposition operator $H_m(x{a}^{†}+ya)$. Using some technique, we cleverly deal with the normalization and discuss the nonclassical and non-Gaussian characters of the orthogonal state. The analytical expressions for the Wigner functions of the orthogonal state are derived in detail. Numerical results show that the orthogonal state will exhibit its richly nonclassical and non-Gaussian character by changing the interaction parameters.

A new version of the generalized conjugate direction (GCD) method for nonsymmetric linear algebraic systems is proposed which is oriented on large and ill-conditioned sets of equations. In distinction from the known Krylov subspace methods for unsymmetrical matrices, the method uses explicitly computed A-conjugate (in generalized sense) vectors, along with an orthogonal set of residuals obtained in the Arnoldi orthogonalization process. Employing entire sequences of orthonormal basis vectors in the Krylov subspaces, similarly to GMRES and FOM, ensures high stability of the method. But instead of solution of a linear set of equations with a Hessenberg matrix in each iteration for determining the step we use A-conjugate vectors and some simple recurrence formulas. The performance of the proposed algorithm is illustrated by the results of extensive numerical experiments with large-scale ill-conditioned linear systems and by comparison with the known efficient algorithms.