Supersymmetric Transformations Research Articles

ON THE SUPER FIELD REALIZATION OF SUPER-CASIMIR ${\mathcal W} {\mathcal A} _n$ ALGEBRAS

We give an explicit quantum super field construction of the [Formula: see text] super-Casimir [Formula: see text] algebras, which is obtained from supersymmetric Miura transformation for the Lie super-algebra [Formula: see text]. And also we give an extension of this algebras including a super-vertex operator which depends on simple root system of [Formula: see text].

Solving Multiphoton Jaynes-Cummings Model with Field Nonlinearity by Supersymmetric Unitary Transformation

We introduce a supersymmetric unitary transformation, to diagonalize the multiphoton Jaynes–Cummings model Hamiltonian based on supersymmetric quantum mechanics theory, that includes any forms of intensity-dependent coupling and field nonlinearity. On doing so, we obtain its eigenvalue and eigenstates, and the time evolution of state vector.

Symmetries of fermions in the lowest Landau level

In the first paper we presented a general theory of conformal fields on a complex plane, and in the second paper we wrote down, for the first time, a supersymmetric transformation of the conformal fields. I think I was expected to present a historical talk at the conference. But, since I had already written an essay on the reminiscences of my entire research activities in [3] and furthermore since a part of the article has been printed again in [4], I chose not to talk about my recollections on this matter. Instead, under the title Symmetries of Fermions in the Lowest Landau Level, I discussed the subjects, which may or may not be suited to the theme of this conference, namely (a) the dynamic symmetries of edge excitations and (b) the method of collective coordinates for planar fermions and the bosonization. Since I believe the viewpoint I took on the latter subject is unconventional, in this written report I expanded the discussion and have written it in detail.

A GENERALIZED MULTIPHOTON JAYNES–CUMMINGS MODEL FOR TWO COLLECTIVELY RADIATING ATOMS SOLVED VIA SUPERSYMMETRIC UNITARY TRANSFORMATION

The multiphoton Jaynes–Cummings model is theoretically generalized to the simplest case of collectively radiating atoms. For this Jaynes–Cummings model, we find its supersymmetric structure. Based on this, we introduce a supersymmetric unitary transformation to diagonalize the Hamiltonian, and in so doing, its eigenvalues and eigenfunctions are obtained. Some transition processes are calculated.

Solving a coupled-channel cavity QED model using supersymmetric unitary transformation

In this paper, we use supersymmetric unitary transformation theory to solve a coupled-channel cavity QED model which includes the Stark term and frequency detuning. We give the eigenvalue, eigenstates and time evolution of the state vector of the system.

Forbidden states and the three-body bound state collapse

The appearance of bound states with large binding energies of several hundred MeV in the three-body system, known as bound state collapse, is investigated. For this purpose three classes of two-body potentials are employed; local potentials equivalent to nonlocal interactions possessing a continuum bound state, in addition to the usual negative-energy bound state; local potentials with a strong attractive well sustaining a forbidden state; and supersymmetric transformation potentials. It is first shown that local potentials equivalent to the above nonlocal ones have a strong attractive well in the interior region which supports, in addition to the physical deuteron state, a second bound state (usually called a pseudobound state) with a large binding energy, which is responsible for the bound state collapse in the three-body (and in general to the N-body) system. Second, it is shown that local potentials with a forbidden state also generate a three-body bound state collapse, implying that the role played by the forbidden state is similar to the one played by the pseudobound state. Finally, it is shown that the removal of the forbidden state via supersymmetric transformations also results in the disappearance of the collapse. Thus one can safely argue that the presence of unphysical bound states with large binding energies in the two-body system is responsible for the bound state collapse in the three-body system.

Ghost components in the Jost function and a new class of phase-equivalent potentials

We show that the introduction of components, in the Jost function, that create new bound states while leaving the S-matrix unchanged, generates potentials behaving as r-2 at large distances. We demonstrate that the modified Jost functions can be obtained by applying two successive supersymmetric transformations to the original potential. We further show that transparent potentials, with Sl(k)≡1, can also be obtained by successive supersymmetric transformations. They are characterized by the property that their SUSY-2 partners resemble centrifugal barriers. Finally, the relation of these transformations to the asymptotic normalization constants of the inverse scattering problem is discussed. We show that the two supersymmetric transformations that remove a bound state provide a potential which is the same as that obtained via the Marchenko inverse scattering procedure, when the asymptotic normalization constant is set to zero.

Supersymmetric transformations in coupled-channel systems

A transformation of supersymmetric quantum mechanics for N coupled channels is presented, which allows the introduction of up to N degenerate bound states without altering the remaining spectrum of the Hamiltonian. Phase equivalence of the Hamiltonian can be restored by two successive supersymmetric transformations at the same energy. The method is successfully applied to the 3S1-3D1 coupled channels of the nucleon-nucleon system and a set of Moscow-type potentials is thus generated.

Deep l-independent potentials from supersymmetric inversion

Central potentials, independent of both the energy and the angular momentum l, are constructed from elastic phase shifts. The ambiguities of a fixed- l inversion are exploited to fit the phase shifts of other partial waves. The fitted quantities are the energy and normalization constant of forbidden bound states, and the small-distance behavior of the potential. The method is based on supersymmetric transformations of the radial Schrodinger equation. It is applied to the alpha+alpha collision in nuclear physics, and to the e(-)+Ne collision in atomic physics.

Multiphoton Jaynes-Cummings model solved via supersymmetric unitary transformation

For multiphoton Jaynes-Cummings(JC) model Hamiltonian, we find its supersymmetric structure. Based on this, we introduce a supersymmetric unitary transformation to diagonalize the JC Hamiltonian, in so doing, its eigenvalues and eigenfunctions are obtained. The transition process is also calculated.

Solving a Generalized Jaynes-Cummings Model by Supersymmetric Unitary Transformation

For the generalized Jaynes-Cummings model Hamiltonian which can describe two collectively radiation atoms, we find its supersymmetric structure. Based on supersymmetric quantum mechanics theory, we introduce a supersymmetric unitary transformation, in which the supersymmetric unitary transformation operator can be constructed by supersymmetric generators of the super-Lie algebra, to diagonalize the Hamiltonian. On doing so, its eigenvalue and eigenstates are obtained.

Exact solution for Super-Jaynes-Cummings model

Based on the linear quantum transformation theory, we further propose a supersymmetric unitary transformation, whose operator can be constructed by the supersymmetric generators of Lie superalgebra. Therefore, by making use of the supersymmetric unitary transformation, we can obtain the eigenspectrum, the eigenstates and the super-partition function for the Super-Jaynes-Cummings model when the coupling constants are Grassmann valued.

PT-symmetric sextic potentials

The family of complex PT-symmetric sextic potentials is studied to show that for various cases the system is essentially quasi-solvable and possesses real, discrete energy eigenvalues. For a particular choice of parameters, we find that under supersymmetric transformations the underlying potential picks up a reflectionless part.

RESONANCE FLUORESCENCE THEORY ANALYZED BY SUPERSYMMETRIC TRANSFORMATION

In this paper, use is made of supersymmetric transformation to discuss resonance fluorescence of two-level atom driven by a strong laser field and that of three-level atom driven by two single-mode laser fields. The supersymmetric derivation makes the analysis of the physical processes more accurate, clear and efficient. As a result, some new physical processes are revealed by this method.

Phase-equivalent energy-dependent potentials

Supersymmetric transformations, i.e., factorizations of the radial Schr\"odinger equation, are generalized to potentials with a linear dependence on energy. These transformations are equivalent to generalized Darboux transformations and do not affect the energy dependence of the potential. A pair of transformations allows the removal of a bound state without modification of the phase shift. The method is applied to the removal of the forbidden bound states of a deep energy-dependent $\ensuremath{\alpha}{+}^{16}\mathrm{O}$ potential generating a phase-equivalent shallow potential with an ${r}^{\ensuremath{-}2}$ repulsive core at the origin and the same energy dependence as the initial potential.

Representations of Superfields. II

D. Using these operators, we construct the projection operators which decompose a linear vector superfields into irreducible representations. The forms of the irreducible representations are explicitly described, and the equivalence of the representations under supersymmetric transformations is discussed. Finally, we construct spin operators for superfields that satisfy the same commutation relation as the angular momentum operators, [si ,s j ]= ieijksk.

Conditionally exactly solvable potentials and supersymmetric transformations

A general procedure is presented to construct conditionally exactly solvable (CES) potentials using the techniques of supersymmetric quantum mechanics. The method is illustrated with potentials related to the harmonic oscillator problem. Besides recovering known results, new CES potentials are also obtained within the framework of this general approach. The conditions under which this method leads to CES potentials are also discussed.

Resonances and off-shell characteristics of effective interactions

The importance of including experimental resonances in constructing effective intercluster interactions has been investigated. For this, we first address the question of how to obtain the analytical properties of the Jost function in regions of physical interest on the complex k-plane when the potential is given in a tabular form. We then employ the Marchenko inverse scattering method to construct, numerically, phase equivalent local potentials supporting the same bound state(s) but having different resonance spectra which affect the off-shell characteristics of the corresponding scattering amplitudes. This implies that the inclusion of the experimental resonances in constructing a potential would change its shape, strength, and range which in turn would modify the bound and scattering wave functions in the interior region. This is expected to have important consequences in calculations of transition amplitudes in nuclear reaction theories, which strongly depend on the behaviour of the wave functions at short distances. The influence of Supersymmetric Transformations on the position and movement of resonances has also been investigated.

Supersymmetry in a three-body model of halo nuclei

Supersymmetric transformations allow eliminating forbidden bound states from a potential without affecting its scattering properties. They are applied to the core-neutron potential in a three-body model of the 6 He , 11 Li and 14 Be halo nuclei and compared with the pseudo-potential method. The supersymmetric approach provides larger binding energies and smaller radii for these three nuclei. When the core-neutron interaction is readjusted to fit the experimental binding energy, the radii of the halo nuclei obtained with the two methods become very close. The 14 Be wave function qualitatively differs in both approaches.

The KW theorem for the SKP hierarchy

The supersymmetric Kadomtsev-Petviashvili (SKP) hierarchy is first introduced by Manin and Radul. In this letter, by the factorization L n = L n L n−1 ⋯ L 1 with L j = D+ u j,0 + u j,−1 D −1+ u j,−2 D −2+⋯, j=1,2⋯ n, being the independent super-pseudodifferential operators (S ΨDOs), we construct the supersymmetric Miura transformation for the SKP hierarchy, which leads to decomposition of the second Poisson brackets based on L n to a direct sum. Each term in the sum contains the second brackets for L j .