Dipole induced transitions in an anharmonic oscillator: A dynamical mean field model M. Berrondo a,⇑ , J. Récamier b a Department of Physics and Astronomy, Brigham Young University, Provo, UT 84602, USA b Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Apdo. Postal 48-3, Cuernavaca, Morelos 62251, Mexico article info Article history: Received 3 September 2010 In final form 24 December 2010 Available online 31 December 2010 abstract We calculate transition probabilities between discrete states of an anharmonic oscillator induced by a time dependent dipole. Our prototype oscillator includes a quadratic term n 2 in the quantum number n. The corresponding unperturbed Hamiltonian is expressed in terms of deformed ladder operators. To consider the perturbation due to the dipole interaction we introduce a dynamical mean field model Ham- iltonian for the oscillator part. The resulting system is a parametric harmonic oscillator with a time dependent frequency determined self-consistently. We present results for the time dependence of tran- sition probabilities for different pulses, as well as the expectation values of position and momentum. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction The interaction of a quantum charge with a time dependent dipole is an ubiquitous problem appearing in many instances. The specific case of a harmonic oscillator is indeed a favorite example of time dependent perturbation theory in text books [1]. On the other hand, the nonlinear response of a harmonic oscillator inter- acting with a time and space dependent classical electric field was considered using an algebraic method with a nonperturbative approach [2]. In this work we consider the response of a charge q in a one dimensional anharmonic oscillator induced by an external time dependent electric dipole. Our calculations are performed nonperturbatively exploiting the algebraic properties of the system. In particular, we consider a model time dependent Hamiltonian where both the oscillator and the dipole interaction contain time dependent coefficients. The operator part of the Hamiltonian is written in terms of elements of a Lie algebra. Given that the result- ing algebra is closed under commutation, we can express the time evolution operator U as a product of exponentials each correspond- ing to a single generator of the algebra [3]. Within this scheme the time dependence is concentrated in the c-number coefficients fa i g appearing in the exponents, as will become evident in what follows. The differential equation fulfilled by the evolution operator U is re- placed by a set of coupled ordinary differential equations for the a’s that can be solved numerically. Armed with this U we can calculate the time dependence of (a) transition probabilities between two of the oscillator states induced by the dipole, (b) the expectation value of the Heisenberg position and momentum operators, and (c) a plot of the corresponding phase space trajectory. It has been customary to proceed to the interaction representation as the first step in this kind of problems [4]. Although this step seems to be appropriate in perturbation theory, it can be obviated in our instance. In practice, the difference between these two approaches is an unobservable phase factor and the time dependence of the resulting model unperturbed Hamiltonian makes the interaction representation awkward. For the present work the unperturbed oscillator system differs from the harmonic case by the presence of a term of the form N 2 in the Hamiltonian, where N is the number operator. Depending on the sign of this term, the resulting anharmonic energy spectrum is expanded or contracted with respect to the uniformly spaced harmonic spectrum. A typical example of contracted spectrum appears in the case of a Morse potential [5–7] which has become the workhorse of molecular vibrations due to its dissociative char- acter at large energies. In contrast the Pöschl–Teller trigonometric potential [6,8] presents an expanding spectrum with the distance between consecutive energy levels increasing with the quantum number. The degree of anharmonicity will be defined by a dimen- sionless parameter v, positive for ‘expanded’ spectra, negative for ‘contracted’ spectra and zero for the harmonic case. A very intuitive way to include anharmonicities in the Hamilto- nian is to ‘deform’ the harmonic oscillator ladder operators so as to preserve the step-up and step-down properties with respect to the number operator N [4]. As a result both the commutator and the anti-commutator are now functions of N. An optimal situation results when the commutator is a linear function of N thus yielding a simple closed algebra of order three, while the anti-commutator (proportional to the Hamiltonian) includes an N 2 anharmonicity term in the energy spectrum [4]. 0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.12.081 ⇑ Corresponding author. E-mail addresses: berrondo@byu.edu (M. Berrondo), pepe@fis.unam.mx (J. Récamier). Chemical Physics Letters 503 (2011) 180–184 Contents lists available at ScienceDirect Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett