Entangled states in supersymmetric quantum mechanics

We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integrals of motion. We construct the most general cubic algebra and we present specific realizations. We use them to calculate the energy spectrum. All classical and quantum superintegrable potentials separable in Cartesian coordinates with a third order integral are known. The general formalism is applied to quantum reducible and irreducible rational potentials separable in Cartesian coordinates in E2. We also discuss these potentials from the point of view of supersymmetric and PT-symmetric quantum mechanics.

We revisit the localization computation of the expectation values of ’t Hooft operators in mathcal{N} = 2* SU(N) theory on ℝ3 × S1. We show that the part of the answer arising from “monopole bubbling” on ℝ3 can be understood as an equivariant integral over a Kronheimer-Nakajima moduli space of instantons on an orbifold of ℂ2. It can also be described as a Witten index of a certain supersymmetric quiver quantum mechanics with mathcal{N} = (4, 4) supersymmetry. The map between the defect data and the quiver quantum mechanics is worked out for all values of N. For the SU(2) theory, we compute several examples of these line defect expectation values using the Witten index formula and confirm that the expressions agree with the formula derived by Okuda, Ito and Taki [16]. In addition, we present a Type IIB construction — involving D1-D3-NS5-branes — for monopole bubbling in mathcal{N} = 2* SU(N) SYM and demonstrate how the quiver quantum mechanics arises in this brane picture.

We study the model of a mass enhancement in the supersymmetric quantum mechanics. This model is so simple that it may be implemented as a quantum simulation of the mass enhancement taking place when supersymmetry (SUSY) is spontaneously broken. It is evolved from a prototype based on the quantum Rabi model. Here, our quantum simulation means the realization of the target quantum phenomenon as a physical entity with some quantum-information devices. The original prototype is given as a mathematical model, and has the transition from the N=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {N}}=2$$\\end{document} SUSY to its spontaneous breaking, though it has no mass enhancement. It is recently reported that the transition is observed for this prototype in a trapped-ion experiment with devising a method experimentally to realize the transition. The model proposed in this paper describes how the mass enhancement takes place in the fermionic states by the excitation of the 1-mode light boson in the SUSY breaking. In this model, the bosonic and fermionic states are graded by qubits. Although our model’s interaction does not have the Higgs potential, the mass enhancement in the fermionic states is caused by the radiative enhancement with the help of the X-gate’s swap between the fermionic and bosonic states.

As a direct generalization of the model of supersymmetric quantum mechanics· by Witten, which describes the motion of a spin one·half particle in the one·dimensional space, we construct a model of the supersymmetric quantum mechanics in the three·dimensional space, which describes the motion of a spin one· half particle in central and spin· orbit potentials in the context of the nonrelativistic quantum mechanics. With the simplest choice of the (super) potential, this model is shown to reduce to the model of the harmonic oscillator plus constant spin· orbit potential of unit strength of both positive and negative signs, which was studied in detail in our recent paper in connection with accidental degeneracy as well as the graded groups. This simplest model is discussed in some detail as an example of the three­ dimensional supersymmetric quantum mechanical system, where the supersymmetry is an exact symmetry of the system. More general choice of a polynomial superpotential is also discussed. It is shown that the supersymmetry cannot be spohtaneously broken for any polynomial superpotential in our three-dimen· sional model; this result is contrasted to the corresponding one in the one·dimensional model.

The spin-weighted spheroidal equation in the case of s = 1/2 is thoroughly studied by using the perturbation method from the supersymmetric quantum mechanics. The first-five terms of the superpotential in the series of parameter β are given. The general form for the n-th term of the superpotential is also obtained, which could also be derived from the previous terms Wk, k < n. From these results, it is easy to obtain the ground eigenfunction of the equation. Furthermore, the shape-invariance property in the series of parameter β is investigated and is proven to be kept. This nice property guarantees that the excited eigenfunctions in the series form can be obtained from the ground eigenfunction by using the method from the supersymmetric quantum mechanics. We show the perturbation method in supersymmetric quantum mechanics could completely solve the spin-weight spheroidal wave equations in the series form of the small parameter β.

Bulk Witten indices and the number of normalizable ground states in supersymmetric quantum mechanics of orthogonal, symplectic and exceptional groups

Theoretical physics seems to be in a kind of schizophrenic state. Many phenomena in the observable macroscopic world obey nonlinear evolution equations, whereas the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. I claim that linearity in quantum mechanics is not as essential as it apparently seems since quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown where complex Riccati equations appear in time-dependent quantum mechanics and how they can be treated and compared with similar space-dependent Riccati equations in supersymmetric quantum mechanics. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation. Finally, it will be shown that (real and complex) Riccati equations also appear in many other fields of physics, like statistical thermodynamics and cosmology.

We review basic aspects of the mathematical theory of abstract supersymmetric quantum mechanics, which can be applied to supersymmetric quantum mechanics both with finite degrees of freedom and with infinite degrees of freedom, including supersymmetric quantum field theory.

We construct a counterexample to show that if supersymmetry is broken in supersymmetric quantum mechanics then the ground state and also the excited states are nondegenerate.Received 22 May 1985DOI:https://doi.org/10.1103/PhysRevD.32.1597©1985 American Physical Society

We constructN=2 supersymmetric quantum Hamiltonians with meromorphic superpotentials on compact Riemann surfaces and investigate the topological properties of these Hamiltonians.L 2-cohomology groups for supercharge (a deformed\(\bar \partial \) operator) are considered and the Witten index for the supersymmetric Hamiltonian with meromorphic superpotential is calculated in terms of Euler characteristic of the Riemann surface and the degree of a divisor of poles for the differential of the superpotential.

It is shown how symmetries of Dirac equations can be used to obtain constants of motion for nonrelativistic supersymmetric quantum Hamiltonians. In particular, conserved supercharges are found for a spin-1/2 particle in the field of a dyon which yield under anticommutation the generalized Runge-Lenz symmetry of the system.

In this work, supersymmetric quantum mechanics will be used to obtain complex\nsolutions to Painleve IV equation with real parameters. We will also focus on\nthe properties of the associated Hamiltonians, i.e. the algebraic structure,\nthe eigenfunctions and the energy spectra.\n

We present a case demonstrating the connection between supersymmetric quantum mechanics (SUSYQM), reflectionless scattering, and soliton solutions of integrable partial differential equations. We show that the members of a class of reflectionless Hamiltonians, namely, Akulin's Hamiltonians, are connected via supersymmetric chains to a potential-free Hamiltonian, explaining their reflectionless nature. While the reflectionless property in question has been mentioned in the literature for over two decades, the enabling algebraic mechanism was previously unknown. Our results indicate that the multisoliton solutions of the sine-Gordon and nonlinear Schrödinger equations can be systematically generated via the supersymmetric chains connecting Akulin's Hamiltonians. Our findings also explain a well-known but little-understood effect in laser physics: when a two-level atom, initially in the ground state, is subjected to a laser pulse of the form V(t) = (nh/τ)/cosh(t/τ), with n being an integer and τ being the pulse duration, it remains in the ground state after the pulse has been applied, for any choice of the laser detuning.

Supersymmetric quantum mechanics provides a simple way to understand the scattering of a particle from a potential that contains a bound state that is Pauli blocked to the projectile. The singularity of the resulting potential found empirically in studies of nucleon-alpha scattering emerges naturally in supersymmetric quantum mechanics as does a direct and unique way to calculate the complete potential.

For a general class of boson–fermion Hamiltonians H acting in the tensor product Hilbert space L2(ℝn) ⊗ ∧(ℂr) of L2(ℝn) and the fermion Fock space ∧(ℂr) over ℂr(n, r ∈ ℕ), we establish, in terms of an n-dimensional conditional oscillator measure, a functional integral representation for the trace Tr (F ⊗ zN f e-tH)(F ∈ L∞(ℝn), z ∈ ℂ∖{0}, t &gt; 0), where N f is the fermion number operator on ∧(ℂr). We prove a Golden–Thompson type inequality for | Tr (F ⊗ zN f e-tH)|. Also we discuss applications to a model in supersymmetric quantum mechanics and present an improved version of the Golden–Thompson inequality in supersymmetric quantum mechanics given by Klimek and Lesniewski ([Lett. Math. Phys.21 (1991) 237–244]). An upper bound for the number of the supersymmetric states is given as well as a sufficient condition for the spontaneous supersymmetry breaking. Moreover, we derive a functional integral representation for the analytical index of a Dirac type operator on ℝn (Witten index) associated with the supersymmetric quantum mechanical model.