Photon-added coherent states for shape invariant systems

Abstract

Since the introduction of the concept of canonical coherent states (CS), associated with the one dimensional harmonic oscillator by Schrödinger in 1926 1 , followed decades in which this concept has reached great investigations 2 -5 . CS are a useful mathematical framework for dealing with the connection between classical and quantum mechanics 5-7 . These states can globally be constructed in three equivalent ways: (i) by defining them as eigenstates of the lowering operator (called CS of the Barut-Girardello 8 type), (ii) by applying a unitary displacement operator on a ground state (Klauder-Perelomov CS 9 or CS of the Gazeau-Klauder type), and (iii) by considering them as quantum states with a minimum uncertainty relationship 10,11 .The CS for shape-invariant potentials (SIP) performed in this work 12,13 , belong to the Barut-Girardello type. They are built using algebraic approach based on the supersymmetric quantum mechanics (SUSY-QM) 14,15 . SUSY QM deals with the study of partner Hamiltonians which are isospectral, that is, they have almost the same energy eigenvalues. A number of such partner Hamiltonians satisfy an integrability condition called shape invariance 16 -18 .However, not all exactly solvable systems are shape-invariant.Recently, a considerable attention is devoted to photon-added CS (PACS) 19 -22 , first introduced by Agarwal and Tara 23 . The PACS represent interesting states generalizing both the Fock states and CS. Indeed, they are obtained by repeatedly operating the photon creation operator on an ordinary CS. In some previous works, the PACS were assimilated to nonlinear CS. Their various generalizations were also performed 24,25 . They evidence some nonclassical effects, for e.g, amplitude squeezing, sub-Poissonian behaviour, nonclassical quasi-probability distribution. In one of our previous papers 26 , a family of photon added as well as photon depleted CS related to the inverse of ladder operators acting on hypergeometric CS was introduced. Their squeezing and antibunching properties were investigated in both standard (nondeformed) and deformed quantum optics. Recently 27 , new generalized PACS were formulated by excitations on a family of generalized CS. Their non-classical features and quantum statistical properties were compared with those obtained by Agarwal. Besides, in another paper 28 , photon-subtracted generalized CS, which are reminiscent of the PACS, were introduced; their nonclassical features were also discussed. PACS find many applications in physics. In 29 , a generating Schrödinger-cat-like states of a single-mode

Shape Invariant Potential Formalism for Photon-Added Coherent State Construction

“…In this section, a construction of PA-SIPCS [36], and their physical and mathematical properties are presented.…”

“…One can check that for a PA-SIPCS (2.4) the expectation values of the operator N := B + B − [36] are:…”

“…In a recent work [36], we constructed photon-added CS for SIP and investigated different cases following the Infeld-Hull [21] classification.…”

Gazeau-Klauder coherent states in position-deformed Heisenberg algebra

“…We show that these states satisfy the Klauder's mathematical requirement to build coherent states. We also explore the statistical properties [29][30][31][32][33][34][35][36] of these states, such as the photon distribution, the photon mean number, the intensity correlation and the Mandel parameter. We find that these states are sub-Poissonian in nature.…”

Photon-added BarutGirardello like coherent states of time-dependent Landau problem

“…Various generalizations of these states taking into account their statistical properties were also performed [29,30,31,32]. Recently, one of us addressed conjonctly the study of photon added states into the generalized associated hypergeometric coherent states [33] and into a full characterization of shape invariant potentials using algebraic approach based on the supersymmetric quantum mechanics [34]. The original construction of photon added coherent states (PACSs) [24,25] obtained from the conventionnal coherent states based on the Weyl-Heisenberg group has been extended to a number of Lie groups with square integrable representations.…”