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Journal of Physics A Mathematical and Theoretical

Anharmonic oscillators and generalized squeezed states

Davood Afshar, Mojtaba Jafarpour

Published Jul 15, 2008• 0 citations
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Abstract

We present a new method to study the eigenvalues and eigenfunctions of anharmonic oscillators. It involves a new class of Bogoliubov transformations and leads to the introduction of k-photon coherent states. We consider the Hamiltonians for the simple harmonic and anharmonic oscillators as the two generators of a Lie algebra, whose other generators may be found exactly, or up to any desired order of the perturbation parameter involved. An element of this Lie group, turning out to be the multi-photon operator, transforms the anharmonic Hamiltonian to the harmonic one, thus facilitating the calculation of the eigenvalues and eigenfunctions of the former. The transformation of the ordinary annihilation and creation operators leads to generalized ones, corresponding to generalized oscillation modes, and also helps us out to introduce multi-photon coherent states. We specifically consider four-photon coherent states in detail and study time dependent position and momentum uncertainties in these states.

DOI
10.1088/1751-8113/41/30/304015

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We present a new method to study the eigenvalues and eigenfunctions of anharmonic oscillators. It involves a new class of Bogoliubov transformations and leads to the introduction of k-photon coherent states. We consider the Hamiltonians for the simple harmonic and anharmonic oscillators as the two generators of a Lie algebra, whose other generators may be found exactly, or up to any desired order of the perturbation parameter involved. An element of this Lie group, turning out to be the multi-photon operator, transforms the anharmonic Hamiltonian to the harmonic one, thus facilitating the calculation of the eigenvalues and eigenfunctions of the former. The transformation of the ordinary annihilation and creation operators leads to generalized ones, corresponding to generalized oscillation modes, and also helps us out to introduce multi-photon coherent states. We specifically consider four-photon coherent states in detail and study time dependent position and momentum uncertainties in these states.
Using Lie algebra methods, we find a new class of Bogoliubov transformations which generalize the notion of squeezed states. The Hamiltonians for the simple harmonic and anharmonic oscillators, turn out to be the generators of a Lie group, whose other generators may be found exactly, or up to any desired order of the perturbation parameter. An element of this Lie group, which is realized as the multi-photon operator, transforms the anharmonic Hamiltonian to the harmonic one. The transformation of the ordinary annihilation and creation operators under this unitary transformation leads to the introduction of multi-photon coherent states. We specifically consider four-photon coherent states in detail and study the time dependent position and momentum uncertainties in these states.
We present a new Lie algebraic method to study the eigenvalues and eigenfunctions of quantum anharmonic oscillators. We consider the Hamiltonians for the simple harmonic and anharmonic oscillator as the two generators of a Lie algebra, whose other generators may be found exactly or up to any order of the parameter involved. Speciflcally, the closed commutator algebra for the quartic anharmonic oscillator is established in a perturbation sense. An element of this Lie group, turning out to be the four-photon operator, transforms the quartic anharmonic oscillator Hamiltonian to the harmonic one in a perturbation sense; thus, facilitating the calculation of the eigenvalues and eigenfunctions of the former.
We introduce a linear, canonical transformation of the fundamental single-mode field operators a and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ that generalizes the linear Bogoliubov transformation familiar in the construction of the harmonic oscillator squeezed states. This generalization is obtained by adding a nonlinear function of any of the fundamental quadrature operators X1{X}_{1}X1​ and X2{X}_{2}X2​ to the linear transformation, thus making the original Bogoliubov transformation quadrature dependent. Remarkably, the conditions of canonicity do not impose any constraint on the form of the nonlinear function, and lead to a set of nontrivial algebraic relations between the c-number coefficients of the transformation. We examine in detail the structure and the properties of the quantum states defined as eigenvectors of the transformed annihilation operator b. These eigenvectors define a class of multiphoton squeezed states. The structure of the uncertainty products and of the quasiprobability distributions in phase space shows that besides coherence properties, these states exhibit a squeezing and a deformation (cooling) of the phase-space trajectories, both of which strongly depend on the form of the nonlinear function. The presence of the extra nonlinear term in the phase of the wave functions has also relevant consequences on photon statistics and correlation properties. The nonquadratic structure of the associated Hamiltonians suggests that these states be generated in connection with multiphoton processes in media with higher nonlinearities. We give a detailed description of the quadratic nonlinear transformation, which defines four-photon squeezed states. In particular, the behaviors of the second-order correlation function g(2)(0){g}^{(2)}(0)g(2)(0) and of the fourth-order correlation function g(4)(0){g}^{(4)}(0)g(4)(0) are studied. The former exhibits super-Poissonian statistics, while the latter indicates photon bunching in the four-photon emissions.
The squeezed harmonic oscillator Hamiltonian H = ωa†a + αa2 + α*a†2 is analysed, where a† and a are harmonic oscillator creation and annihilation operators, ω is real, and α is a time independent constant. For the case that ω2 − 4|α|2 > 0 it is known that the Hamiltonian possesses real and positive eigenvalues. The exact eigenstates are constructed by using a Bogoliubov transformation to express them in terms of the algebra and states of the harmonic oscillator. Known properties of these states are reviewed and new results are derived. Modified coherent states are constructed from the exact eigenstates and shown to yield a unit projection operator for the original harmonic oscillator Hilbert space via generalized Gaussian integrals. Transition elements between harmonic oscillator coherent states are then evaluated exactly by using the properties of the modified coherent state projection operator.
By applying higher powers of the Bogoliubov transformation operator bdagger = v*a + mu*adagger to the two-photon coherent states (or minimum uncertainty squeezed states) we construct a new type of quantum state which we call the generalized excited two-photon coherent states. Analytic expressions for the quantum statistical properties are derived, and through numerical computation the phase space quasi-probability distributions are found. These states can exhibit highly nonclassical behaviour depending on the degree of excitation m and other parameters. For particular values of two parameters lambda and rho, these generalized states reduce to other classes of coherent states formerly reported. Our theory thus presents a much broader approach to these types of quantum states.