Quasi Exactly Solvable Difference Equations

Several explicit examples of quasi exactly solvable `discrete' quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogues of the well-known quasi exactly solvable systems, the harmonic oscillator (with/without the centrifugal potential) deformed by a sextic potential and the 1/sin^2x potential deformed by a cos2x potential. They have a finite number of exactly calculable eigenvalues and eigenfunctions.Comment: LaTeX with amsfonts, no figure, 17 pages, a few typos corrected, a reference renewed, 3/2 pages comments on hermiticity adde

Infinitely many shape invariant potentials and cubic identities of the Laguerre and Jacobi polynomials

We provide analytic proofs for the shape invariance of the recently discovered (Odake and Sasaki, Phys. Lett. B679 (2009) 414-417) two families of infinitely many exactly solvable one-dimensional quantum mechanical potentials. These potentials are obtained by deforming the well-known radial oscillator potential or the Darboux-P\"oschl-Teller potential by a degree \ell (\ell=1,2,...) eigenpolynomial. The shape invariance conditions are attributed to new polynomial identities of degree 3\ell involving cubic products of the Laguerre or Jacobi polynomials. These identities are proved elementarily by combining simple identities.Comment: 13 page

Unified Theory of Annihilation-Creation Operators for Solvable (`Discrete') Quantum Mechanics

The annihilation-creation operators $a^{(\pm)}$ are defined as the positive/negative frequency parts of the exact Heisenberg operator solution for the `sinusoidal coordinate'. Thus $a^{(\pm)}$ are hermitian conjugate to each other and the relative weights of various terms in them are solely determined by the energy spectrum. This unified method applies to most of the solvable quantum mechanics of single degree of freedom including those belonging to the `discrete' quantum mechanics.Comment: 43 pages, no figures, LaTeX2e, with amsmath, amssym

Generalised Calogero-Moser models and universal Lax pair operators

Calogero-Moser models can be generalised for all of the finite reflection groups. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H_3, H_4, and the dihedral group I_2(m), besides the well-known ones based on crystallographic root systems, namely those associated with Lie algebras. Universal Lax pair operators for all of the generalised Calogero-Moser models and for any choices of the potentials are constructed as linear combinations of the reflection operators. The consistency conditions are reduced to functional equations for the coefficient functions of the reflection operators in the Lax pair. There are only four types of such functional equations corresponding to the two-dimensional sub-root systems, A_2, B_2, G_2, and I_2(m). The root type and the minimal type Lax pairs, derived in our previous papers, are given as the simplest representations. The spectral parameter dependence plays an important role in the Lax pair operators, which bear a strong resemblance to the Dunkl operators, a powerful tool for solving quantum Calogero-Moser models.Comment: 37 pages, LaTeX2e, no macro, no figur

Equilibrium Positions, Shape Invariance and Askey-Wilson Polynomials

We show that the equilibrium positions of the Ruijsenaars-Schneider-van Diejen systems with the trigonometric potential are given by the zeros of the Askey-Wilson polynomials with five parameters. The corresponding single particle quantum version, which is a typical example of "discrete" quantum mechanical systems with a q-shift type kinetic term, is shape invariant and the eigenfunctions are the Askey-Wilson polynomials. This is an extension of our previous study [1,2], which established the "discrete analogue" of the well-known fact; The equilibrium positions of the Calogero systems are described by the Hermite and Laguerre polynomials, whereas the corresponding single particle quantum versions are shape invariant and the eigenfunctions are the Hermite and Laguerre polynomials.Comment: 14 pages, 1 figure. The outline of derivation of the result in section 2 is adde

Comment on "Memory Effects in an Interacting Magnetic Nanoparticle System"

In Phys. Rev. Lett. 91 167206 (2003), Sun et al. study memory effects in an interacting nanoparticle system with specific temperature and field protocols. The authors claim that the observed memory effects originate from spin-glass dynamics and that the results are consistent with the hierarchical picture of the spin-glass phase. In this comment, we argue their claims premature by demonstrating that all their experimental curves can be reproduced qualitatively using only a simplified model of isolated nanoparticles with a temperature dependent distribution of relaxation times.Comment: 1 page, 2 figures, slightly changed content, the parameters involved in Figs. 1 and 2 are changed a little for a semi-quantitative comparision with experimental result

A new family of shape invariantly deformed Darboux-P\"oschl-Teller potentials with continuous \ell

We present a new family of shape invariant potentials which could be called a ``continuous \ell version" of the potentials corresponding to the exceptional (X_{\ell}) J1 Jacobi polynomials constructed recently by the present authors. In a certain limit, it reduces to a continuous \ell family of shape invariant potentials related to the exceptional (X_{\ell}) L1 Laguerre polynomials. The latter was known as one example of the `conditionally exactly solvable potentials' on a half line.Comment: 19 pages. Sec.5(Summary and Comments): one sentence added in the first paragraph, several sentences modified in the last paragraph. References: one reference ([25]) adde

Experimental demonstration of entanglement assisted coding using a two-mode squeezed vacuum state

We have experimentally realized the scheme initially proposed as quantum dense coding with continuous variables [Ban, J. Opt. B \textbf{1}, L9 (1999), and Braunstein and Kimble, \pra\textbf{61}, 042302 (2000)]. In our experiment, a pair of EPR (Einstein-Podolski-Rosen) beams is generated from two independent squeezed vacua. After adding two-quadrature signal to one of the EPR beams, two squeezed beams that contain the signal were recovered. Although our squeezing level is not sufficient to demonstrate the channel capacity gain over the Holevo limit of a single-mode channel without entanglement, our channel is superior to conventional channels such as coherent and squeezing channels. In addition, optical addition and subtraction processes demonstrated are elementary operations of universal quantum information processing on continuous variables.Comment: 4 pages, 4 figures, submitted to Phys. Rev.