Supersymmetric Transformations Research Articles

Topological state engineering via supersymmetric transformations

The quest to explore new techniques for the manipulation of topological states simultaneously promotes a deeper understanding of topological physics and is essential in identifying new ways to harness their unique features. Here, we examine the potential of supersymmetric transformations to systematically address, alter and reconfigure the topological properties of a system. To this end, we theoretically and experimentally study the changes that topologically protected states in photonic lattices undergo as supersymmetric transformations are applied to their host system. In particular, we show how supersymmetry-induced phase transitions can selectively suspend and re-establish the topological protection of specific states. Furthermore, we reveal how understanding the interplay between internal symmetries and the symmetry constraints of supersymmetric transformations provides a roadmap to directly access the desirable topological properties of a system. Our findings pave the way for establishing supersymmetry-inspired techniques as a powerful and versatile tool for topological state engineering.

Unified supersymmetric transformations for the harmonic oscillator and its rational extension

We discussed transformations of supersymmetric quantum mechanics by considering a general six-parameter function, from which the superpotential and the supersymmetric partner potentials V−(r) and V+(r) were constructed. By specific choice of the parameters, V−(r) reproduced the radial harmonic oscillator, or its rationally extended version, with solutions containing X1-type Laguerre polynomials. The choice of parameters also determined which of the four possible SUSY transformations Ti, i = 1, ...4 were employed. Within this formalism, the relation between the radial harmonic oscillator and its rational extension can be discussed in a straightforward way, furthermore, several aspects of SUSYQM and exactly solvable potentials can be demonstrated in a unified framework on a level suitable for basic quantum mechanics courses.

Higher-order Darboux transformations and Wronskian representations for Schrödinger equations with quadratically energy-dependent potentials

We construct higher-order Darboux transformations for Schrödinger equations with quadratically energy-dependent potentials. Similar to the case of conventional Darboux (also known as supersymmetric) transformations, both the solutions and the potentials of transformed equations are expressed through Wronskians. We discuss properties of our Darboux transformations and provide an application.

Effective Interactions: Resonances and Off-Shell Characteristics

The influence of resonances on the analytical properties and off-shell characteristics of effective interactions has been investigated. This requires, among others, the knowledge of the Jost function in regions of physical interest on the complex kplane when the potentials are given in a tabular form. The latter are encountered in inverse scattering and supersymmetric transformations. To investigate the effects of resonances on the analytical properties of the potential, we employed the Marchenko inverse scattering method to construct, phase and bound state equivalent local potentials but with different resonance spectra. It is shown that the inclusion of resonances changes the shape, strength, and range of the potential which in turn would modify the bound and scattering wave functions in the interior region. This could have important consequences in calculations of transition amplitudes in nuclear reactions, which strongly depend on the behaviour of the wave functions at short distances. Finally, an exact method to obtain the Jost solutions and the Jost functions for a repulsive singular potential is presented. The effectiveness of the method is demonstrated using the Lennard-Jones (12,6) potential.

Mode-sorter design using continuous supersymmetric transformation.

We propose to use a continuous supersymmetric (SUSY) transformation of a dielectric permittivity profile in order to design a photonic mode sorter. The iso-spectrality of the SUSY transformation ensures that modes of the waveguide preserve their propagation constants while being spatially separated. This global matching of the propagation constants, in conjunction with the adiabatic modification of the refractive index landscape along the propagation direction, results in the negligible modal cross-talk and low scattering losses in the sorter. We show that a properly optimized SUSY mode sorter outperforms a standard asymmetric Y-splitter by reducing the cross-talk by at least two orders of magnitude. Moreover, the SUSY sorter is capable of sorting either transverse-electric or transverse-magnetic polarized modes and operates in a broad range of wavelengths. At the telecommunication wavelength, the 300-μm-long SUSY sorter provides the cross-talk of -35 dB and a broad operation bandwidth. The design proposed here paves the way toward efficient signal manipulation in integrated photonic devices.

Ladder operators for the BenDaniel-Duke Hamiltonians and their SUSY partners

Position dependent mass systems can be described by a class of operators which include the Ben Daniel-Duke Hamiltonians. The usual methods to solve this kind of problems are, in general, either numerical or those looking for a connection with constant mass problems. In this paper we impose the existence of first-order ladder operators to fix our initial system. Then, we perform the first and second order supersymmetric transformations to generate families of Hamiltonians whose eigenfunctions are known analytically for a given mass profile.

Time-dependent Darboux (supersymmetric) transformations for non-Hermitian quantum systems

We propose time-dependent Darboux (supersymmetric) transformations that provide a scheme for the calculation of explicitly time-dependent solvable non-Hermitian partner Hamiltonians. Together with two Hermitian Hamilitonians the latter form a quadruple of Hamiltonians that are related by two time-dependent Dyson equations and two intertwining relations in form of a commutative diagram. Our construction is extended to the entire hierarchy of Hamiltonians obtained from time-dependent Darboux–Crum transformations. As an alternative approach we also discuss the intertwining relations for Lewis–Riesenfeld invariants for Hermitian as well as non-Hermitian Hamiltonians that reduce the time-dependent equations to auxiliary eigenvalue equations. The working of our propsals is discussed for a hierarchy of explicitly time-dependent rational, hyperbolic, Airy function and nonlocal potentials.

Prepotentials for linearized supergravity

Linearized supergravity in arbitrary dimension is reformulated into a first order formalism which treats the graviton and its dual on the same footing at the level of the action. This generalizes previous work by other authors in two directions: (1) we work in arbitrary space-time dimension, and (2) the gravitino field and supersymmetry are also considered. This requires the construction of conformally invariant curvatures (the Cotton fields) for a family of mixed symmetry tensors and tensor-spinors, whose properties we prove (invariance; completeness; conformal Poincaré lemma). We use these geometric tools to solve the Hamiltonian constraints of the graviton and gravitino through the introduction of prepotentials (two tensor fields for the graviton, one tensor-spinor for the gravitino) enjoying (linearized) conformal invariance. The action and equations of motion take a geometrically simple form in terms of the Cotton tensor(-spinors) of the prepotentials. In particular, the equations of motion of the graviton are equivalent to twisted self-duality conditions. We express the supersymmetric transformations of the graviton and gravitino into each other in terms of the prepotentials. We also reproduce the dimensional reduction of supergravity within the prepotential formalism. Finally, our formulas in dimension five are recovered from the dimensional reduction of the already known prepotential formulation of the 6D maximally supersymmetric theory.

Influence of orthogonalization procedure on astrophysical S-factor for the direct α + d →6Li + γ capture process in a three-body model

The astrophysical S-factor for the direct [Formula: see text] capture reaction is calculated in a three-body model based on the hyperspherical Lagrange-mesh method. A sensitivity of the E1 and E2 astrophysical S-factors to the orthogonalization method of Pauli forbidden states in the three-body system is studied. It is found that the method of orthogonalising pseudopotentials (OPP) yields larger isotriplet ([Formula: see text]) components than the supersymmetric transformation (SUSY) procedure. The E1 astrophysical S-factor shows the same energy dependence in both cases, but strongly different absolute values. At the same time, the E2 S-factor does not depend on the orthogonalization procedure. As a result, the OPP method yields a very good description of the direct data of the LUNA collaboration at low energies, while the SUSY transformation strongly underestimates the LUNA data.

Integrated photonic devices based on adiabatic transitions between supersymmetric structures.

We introduce adiabatic transitions connecting two supersymmetric partner profiles by smoothly modifying the transverse refractive index profile along the propagation direction. With this transformation, one of the transverse electric modes evolves adapting its shape and propagation constant without being coupled to other guided or radiated modes while the rest of the modes are radiated. This technique offers a systematic way to manipulate the modal content in systems of optical waveguides and engineer efficient and robust photonic devices such as tapered waveguides, single-waveguide mode filters, beam splitters and interferometers. Numerical simulations show that very high fidelities and transmitted powers are obtained for a broad range of devices lengths and light's wavelengths.

Analyzing supersymmetric transformed α-nucleus potentials with electric-multipole transitions

Alpha(4He or α)-cluster models have often been used to describe light nuclei. Towards the application to multi-cluster systems involving heavy clusters, we study the relative wave functions of the α+16O and α+40Ca systems generated from phase-shift-equivalent potentials. In general, a potential between clusters is deep accommodating several redundant bound states which should be removed in an appropriate way. To avoid such a complicated computation, we generate a shallow-singular potential by using supersymmetric transformations from the original deep potential. Changes in the relative wave functions by the transformations are quantified with electric-multipole transitions which give a different radial sensitivity to the wave function depending on their multipolarity. Despite the fact that the original and transformed potentials give exactly the same phase shift, some observables are unfavorably modified. A possible way to obtain a desired supersymmetric potential is proposed.

Supersymmetric Transformations in Optical Fibers

We thank Sergio Lechago for his valuable help with the numerical simulations. This work is supported by Spanish National Plan projects [No. MINECO/FEDER UE XCORE TEC2015-70858-C2-1-R, No. PHUTURE TEC2015-73581-JIN (AEI/FEDER, UE), and No. HIDRASENSE RTC-2014-2232-3]. A.M.'s work is supported by F.P.I. Grant No. BES-2013-062952.

Controlling Random Waves with Digital Building Blocks Based on Supersymmetry

Harnessing multimode waves allows high information capacity through modal expansions. Although passive multimode devices including waveguides, couplers, and multiplexers have been demonstrated for broadband responses in momentum or frequency domains, collective switching of multimodes remains a challenge, due to the difficulty in imposing consistent dynamics on all eigenmodes. Here we overcome this limitation by realizing digital switching of spatially random waves, based on supersymmetric pairs of multimode potentials. We reveal that supersymmetric transformations of any parity-symmetric potential derive the parity reversal of all eigenmodes, which allows the complete isolation of random waves at the 'off' state. Building blocks for binary and many-valued logics are then demonstrated for random waves: a harmonic pair for binary switching of arbitrary wavefronts and a P\"oschl-Teller pair for multi-level switching which implements the fuzzy membership function. Our results establishing global phase matching conditions for multimode dynamics will lay the foundation of multi-channel digital photonics.

Boundary effects in super-Yang\u2013Mills theory

In this paper, we shall analyze a three dimensional supersymmetry theory with mathcal {N} = 2 supersymmetry. We will analyze the quantization of this theory, in the presence of a boundary. The effective Lagrangian used in the path integral quantization of this theory, will be given by the sum of the gauge fixing term and the ghost term with the original classical Lagrangian. Even though the supersymmetry of this effective Lagrangian will also be broken due to the presence of a boundary, it will be demonstrated that half of the supersymmetry of this theory can be preserved by adding a boundary Lagrangian to the effective bulk Lagrangian. The supersymmetric transformation of this new boundary Lagrangian will exactly cancel the boundary term generated from the supersymmetric transformation of the effective bulk Lagrangian. We will analyze the Slavnov–Taylor identity for this mathcal {N} = 2 Yang–Mills theory with a boundary.

Implications of Lorentz symmetry violation on a 5D supersymmetric model

Field models with [Formula: see text] extra spatial dimensions have a larger [Formula: see text] Lorentz symmetry which is broken down to the standard [Formula: see text] four-dimensional one by the compactification process. By considering Lorentz violating operators in a 5D supersymmetric Wess–Zumino model, which otherwise conserve the standard four-dimensional Poincaré invariance, we show that supersymmetry (SUSY) can be restored upon a simple deformation of the supersymmetric transformations. However, SUSY is not preserved in the effective 4D theory that arises after compactification when the 5D Lorentz violating operators do not preserve [Formula: see text] bulk parity. Our mechanism unveils a possible connection among Lorentz violation and the Scherk–Schwarz mechanism. We also show that parity preserving models, on the other hand, do provide well defined supersymmetric KK models.

Dirac matrices as elements of a superalgebraic matrix algebra

The paper considers a Clifford extension of the Grassmann algebra, in which operators are built from Grassmann variables and by the derivatives with respect to them. It is shown that a subalgebra which is isomorphic to the usual matrix algebra exists in this algebra, the Clifford exten-sion of the Grassmann algebra is a generalization of the matrix algebra and contains superalgebraic operators expanding matrix algebra and produces supersymmetric transformations.

Trigonometric potentials arising from the spheroidal equation: Supersymmetric partners and integral formulas

We construct supersymmetric partners of a quantum system featuring a class of trigonometric potentials that emerge from the spheroidal equation. Examples of both standard and confluent supersymmetric transformations are presented. Furthermore, we use integral formulas arising from the confluent supersymmetric formalism to derive new representations for single and multiple integrals of spheroidal functions.

Optical revivals in nonuniform supersymmetric photonic arrays

We investigate the problem of wavepacket revivals in coupled nonuniform linear optical structures. Starting from the photonic Bloch lattices and J(x) arrays, whose propagators are fully periodic, we use cascaded discrete supersymmetric transformations to generate a family of nonuniform isospectral lattices. These new structures exhibit perfect imaging for any initial condition despite the apparent lack of order in their physical parameters. We note, however. that the SUSY-induced disordered coefficients are not random but, rather, inherit some of the features associated with the original array.

N=1super Feynman rules for any superspin: Noncanonical SUSY

Super Feynman rules for any superspin are given for massive $ \mathcal{N}=1 $ supersymmetric theories, including momentum superspace on-shell legs. This is done by extending, from space to superspace, Weinberg's perturbative approach to quantum field theory. Superfields work just as a device that allow one to write super Poincare-covariant superamplitudes for interacting theories, relying neither in path integral nor canonical formulations. Explicit transformation laws for particle states under finite supersymmetric transformations are offered. $ \mathit{C}, \mathit{P}, \mathit{T}, $ and $ \mathcal{R} $ transformations are also worked out. A key feature of this formalism is that it does not require the introduction of auxiliary fields, and when introduced, their purpose is just to render supersymmetric invariant the time-ordered products in the Dyson series. The formalism is tested for the cubic scalar superpotential. It is found that when a superparticle is its own antisuperparticle the lowest-order correction of time-ordered products, together with its covariant part, corresponds to the Wess-Zumino model potential.

Four dimensional supersymmetric theories in presence of a boundary

In this paper, we study N=1 supersymmetric theories in four dimensions in presence of a boundary. We demonstrate that it is possible to preserve half the supersymmetry of the original theory by suitably modifying it in presence of a boundary. This is done by adding new boundary terms to the original action, such that the supersymmetric variation of the new terms exactly cancels the boundary terms generated by the supersymmetric transformation of the original bulk action. We also analyze the boundary projections of such supercharges used in such a theory. We study super-Yang–Mills theories in presence of a boundary using these results. Finally, we study the Born–Infeld action in presence of a boundary. We analyze the boundary effects for the Born–Infeld action coupled to a background dilaton and an axion field. We also analyze the boundary effects for a non-abelian Born–Infeld action. We explicitly construct the actions for these systems in presence of a boundary. This action preserves half of the original supersymmetry.