We apply quantum mechanical sum rules to pairs of one-dimensional systems defined by potential energy functions related by parity. Specifically, we consider symmetric potentials, $V(x) = V(-x)$, and their parity-restricted partners, ones with $V(x)$, but defined only on the positive half-line. We extend recent discussions of sum rules for the quantum bouncer by considering the parity-extended version of this problem, defined by the symmetric linear potential, $V(z) = F|z|$ and find new classes of constraints on the zeros of the Airy function, $Ai(z)$, and its derivative $Ai'(z)$. We also consider the parity-restricted version of the harmonic oscillator and find completely new classes of mathematical relations, unrleated to those of the ordinary oscillator problem. These two soluble quantum-mechanical systems defined by power-law potentials provide examples of how the form of the potential (both parity and continuity properties) affects the convergence of quantum-mechanical sum rules. We also discuss semi-classical predictions for expectation values and the Stark effect for these systems.
Mechanical resonators are emerging as an important new platform for quantum science and technologies. A large number of proposals for using them to store, process, and transduce quantum information motivates the development of increasingly sophisticated techniques for controlling mechanical motion in the quantum regime. By interfacing mechanical resonators with superconducting circuits, circuit quantum acoustodynamics (cQAD) can make a variety of important tools available for manipulating and measuring motional quantum states. Here we demonstrate direct measurements of the phonon number distri…
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-Pöschl-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…
Dirac's ket-bra formalism is the "language" of quantum mechanics and quantum field theory. In Refs.(Fan et al, Ann. Phys. 321 (2006) 480; 323 (2008) 500) we have reviewed how to apply Newton-Leibniz integration rules to Dirac's ket-bra projectors. In this work by alternately using the technique of integration within normal, antinormal, and Weyl ordering of operators we not only derive some new operator ordering identities, but also deduce some useful integration formulas regarding to Laguerre and Hermite polynomials. This opens a new route of deriving mathematical integration formulas by virtu…
Solvability of the ubiquitous quantum harmonic oscillator relies on a spectrum generating osp(1|2) superconformal symmetry. We study the problem of constructing all quantum mechanical models with a hidden osp(1|2) symmetry on a given space of states. This problem stems from interacting higher spin models coupled to gravity. In one dimension, we show that the solution to this problem is the Plyushchay family of quantum mechanical models with hidden superconformal symmetry obtained by viewing the harmonic oscillator as a one dimensional Dirac system, so that Grassmann parity equals wavefunction …
The coherence properties of mechanical resonators are often limited by multiple unavoidable forms of loss -- including phonon-phonon and phonon-defect scattering -- which result in the scattering of sound into other resonant modes and into the phonon bath. Dynamic suppression of this scattering loss can lift constraints on device structure and can improve tolerance to defects in the material, even after fabrication. Inspired by recent experiments, here we introduce a model of phonon losses resulting from disorder in a whispering gallery mode resonator with acousto-optical coupling between opti…
We propose to realize mechanical parity-time PT symmetry in two coupled optomechanical systems. To provide gain to one mechanical resonator and the same amount of damping to the other, the two optical cavities should be driven by blue- and red-detuned laser fields, respectively. After adiabatically eliminating the degrees of freedom of the cavity modes, we derive a formula to describe the PT symmetry of two coupled mechanical resonators. Mechanical PT-symmetric phase transition is demonstrated by the dynamical behavior of the mechanical resonators. Moreover, we study the effect of the quantum …
We develop a rigorous theoretical framework for interaction-induced phenomena in the waveguide quantum electrodynamics (QED) driven by mechanical oscillations of the qubits. Specifically, we predict that the simplest set-up of two qubits, harmonically trapped over an optical waveguide, enables the ultrastrong coupling regime of the quantum optomechanical interaction. Moreover, the combination of the inherent open nature of the system and the strong optomechanical coupling leads to emerging parity-time (\PT) symmetry, quite unexpected for a purely quantum system without artificially engineered …
Photon number states are assigned a parity of if their photon number is even and a parity of if odd. The parity operator, which is minus one to the power of the photon number operator, is a Hermitian operator and thus a quantum mechanical observable though it has no classical analog, the concept being meaningless in the context of classical light waves. In this paper we review work on the application of the parity operator to the problem of quantum metrology for the detection of small phase shifts with quantum optical interferometry using highly entangled field states such as the so-called N00…
Particles of spin 1/2 and 1 in external Abelian monopole field are considered. P-inversion like operators N-s commuting with the respective Hamiltonians are constructed: N(bisp.) is diagonalized onto the relevant wave functions, whereas N(vect.) does not. Such a paradox is rationalized through noting that both these operators are not self-conjugate. It is shown that any N-parity selection rules cannot be produced. Non-Abelian problems for doublets of spin 1/2 and 1 particles are briefly discussed; the statement is given of that corresponding discrete operators are self-conjugate and selection …
We consider the phenomenon of quantum mechanical contextuality, and specifically parity-based proofs thereof. Mermin's square and star are representative examples. Part of the information invoked in such contextuality proofs is the commutativity structure among the pertaining observables. We investigate to which extent this commutativity structure alone determines the viability of a parity-based contextuality proof. We establish a topological criterion for this, generalizing an earlier result by Arkhipov.
Localized electronic and nuclear spin qubits in the solid state constitute a promising platform for storage and manipulation of quantum information, even at room temperature. However, the development of scalable systems requires the ability to entangle distant spins, which remains a challenge today. We propose and analyze an efficient, heralded scheme that employs a parity measurement in a decoherence free subspace to enable fast and robust entanglement generation between distant spin qubits mediated by a hot mechanical oscillator. We find that high-fidelity entanglement at cryogenic and even …
We investigate the energy spectrum for hybrid mechanical systems described by non-parity-symmetric quantum Rabi models. A set of analytical solutions in terms of the confluent Heun functions and their analytical energy spectrum are obtained. The analytical energy spectrum includes regular and exceptional parts, which are both confirmed by direct numerical simulation. The regular part is determined by the zeros of the Wronskian for a pair of analytical solutions. The exceptional part is relevant to the isolated exact solutions and its energy eigenvalues are obtained by analyzing the truncation …
A review of fundamentals and physical applications of fractional quantum mechanics has been presented. Fundamentals cover fractional Schrödinger equation, quantum Riesz fractional derivative, path integral approach to fractional quantum mechanics, hermiticity of the Hamilton operator, parity conservation law and the current density. Applications of fractional quantum mechanics cover dynamics of a free particle, new representation for a free particle quantum mechanical kernel, infinite potential well, bound state in δ-potential well, linear potential, fractional Bohr atom and fractional oscilla…
For eleven examples of one-dimensional quantum mechanics with shape-invariant potentials, the Darboux-Crum transformations in terms of multiple pseudo virtual state wavefunctions are shown to be equivalent to Krein-Adler transformations deleting multiple eigenstates with shifted parameters. These are based upon infinitely many polynomial Wronskian identities of classical orthogonal polynomials, i.e. the Hermite, Laguerre and Jacobi polynomials, which constitute the main part of the eigenfunctions of various quantum mechanical systems with shape-invariant potentials.
We study the bound states of a quantum mechanical system consisting of a simple harmonic oscillator with an inverse square interaction, whose interaction strength is governed by a constant $α$. The singular form of this potential has doubly-degenerate bound states for $-1/4\leqα<0$ and $α>0$; since the potential is symmetric, these consist of even and odd-parity states. In addition we consider a regularized form of this potential with a constant cutoff near the origin. For this regularized potential, there are also even and odd-parity eigenfunctions for $α\geq-1/4$. For attractive potentials w…
The paper concerns a problem of the Dirac fermion doublet in the external monopole potential obtained by embedding the Abelian monopole solution in the non-Abelian scheme. In this case, the doublet-monopole Hamiltonian is invariant under operations consisting of a complex and one parametric Abelian subgroup in S0(3.C). This symmetry results in a certain freedom in choosing a discrete operator N(A) (A is a complex number) entering the complete set of quantum variables. The same complex number A represents an additional parameter at the basis functions. The generalized inversion like operator N(…
When independent Bose-Einstein condensates (BEC), described quantum mechanically by Fock (number) states, are sent into interferometers, the measurement of the output port at which the particles are detected provides a binary measurement, with two possible results $\pm1$. With two interferometers and two BEC's, the parity (product of all results obtained at each interferometer) has all the features of an Einstein-Podolsky-Rosen quantity, with perfect correlations predicted by quantum mechanics when the settings (phase shifts of the interferometers) are the same. When they are different, signif…
We consider a two-parameter non hermitean quantum-mechanical hamiltonian that is invariant under the combined effects of parity and time reversal transformation. Numerical investigation shows that for some values of the potential parameters the hamiltonian operator supports real eigenvalues and localized eigenfunctions. In contrast with other PT symmetric models, which require special integration paths in the complex plane, our model is integrable along a line parallel to the real axis.
Two-dimensional PT-symmetric quantum-mechanical systems with the complex cubic potential V_{12}=x^2+y^2+igxy^2 and the complex Henon-Heiles potential V_{HH}=x^2+y^2+ig(xy^2-x^3/3) are investigated. Using numerical and perturbative methods, energy spectra are obtained to high levels. Although both potentials respect the PT symmetry, the complex energy eigenvalues appear when level crossing happens between same parity eigenstates.
The ability of matter to be superposed at two different locations while being intrinsically connected by a quantum phase is among the most counterintuitive predictions of quantum physics. While such superpositions have been created for a variety of systems, the in-situ observation of the phase coherence has remained out of reach. Using a heralding measurement on a spin-oscillator entangled state, we project a mechanical trapped-ion oscillator into a superposition of two spatially separated states, a situation analogous to Schrödinger's cat. Quantum interference is clearly observed by extractin…
