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arXiv:quant-ph/9902047

One-Parameter Gaussian State of Anharmonic Oscillator: Nonlinear Realization of Bogoliubov Transformation

Jung Kon Kim, Sang Pyo Kim, Kwang-Gi Park

Published Feb 11, 1999• 0 citations
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Abstract

We find a one-parameter Gaussian state for an anharmonic oscillator with quadratic and quartic terms, which depends on the energy expectation value. For the weak coupling constant, the Gaussian state is a squeezed state of the vacuum state. However, for the strong coupling constant, the Gaussian state represents a different kind of condensation of bosonic particles through a nonlinear Bogoliubov transformation of the vacuum state.

DOI
10.48550/arxiv.quant-ph/9902047

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We find a one-parameter Gaussian state for an anharmonic oscillator with quadratic and quartic terms, which depends on the energy expectation value. For the weak coupling constant, the Gaussian state is a squeezed state of the vacuum state. However, for the strong coupling constant, the Gaussian state represents a different kind of condensation of bosonic particles through a nonlinear Bogoliubov transformation of the vacuum state.
By using the invariant method we find one-parameter squeezed Gaussian states for both time-independent and time-dependent oscillators. The squeezing parameter is expressed in terms of the energy expectation value for the time-independent case and represents the degree of mixing of positive and negative frequency solutions for the time-dependent case. A minimum uncertainty proposal is advanced to select, uniquely, vacuum states at each moment of time. We show that the Gaussian states with minimum uncertainty coincide with the true vacuum state for the time-independent oscillator and the Bunch-Davies vacuum for a massive scalar field in a de Sitter spacetime.
We show that the nonlinear auxiliary equation for the invariant introduced by Lewis and Riesenfeld for a time-dependent harmonic oscillator is satisfied by the amplitude of a complex soltuion to the classical equation of motion. One-parameter squeezed Gaussian states are found whose parameter denotes the mixing of positive and negative frequency solutions. The minimization of the energy expectation value is shown to provide a criterion to select the vacuum state. PACS number(s): 42.50.D; 03.65.G; 03.65.-w Typeset using REVTEX 1 Harmonic oscillators have played many important roles in quantum physics, partly because they are exactly solvable quantum mechanically and partly because any system around an equilibrium can be approximated as a harmonic oscillator system. As a non-stationary system, a time-dependent quantum harmonic oscillator can also be exactly solved. One can find typical time-dependent harmonic oscillators in a system of harmonic oscillators interacting
We have further extended the two-step approach developed by Chung and Chew [N. N. Chung and L. Y. Chew, Phys. Rev. A 76, 032113 (2007)] to the solution of the quantum dynamics of general systems of NNN-coupled anharmonic oscillators. The idea is to employ an optimized basis set to represent the dynamical quantum states of these oscillator systems. The set is generated via the action of the optimized Bogoliubov transformed bosonic operators on the optimal squeezed vacuum product state. The procedure requires (i) applying the two-step approach to the eigendecomposition of the time evolution operator and (ii) transforming the representation of the initial state from the original to the optimal bases. We have applied the formalism to examine the dynamics of squeezing and entanglement of several anharmonic oscillator systems.
In this article, for pedagogical purposes we have discussed the application of nondegenerate perturbation theory up to the third order to compute energy eigenvalues and wave functions for the quantum anharmonic oscillator. Energy levels of a single quartic oscillator for 𝜆 values in range of 0.1-1 are given. Perturbed and non-perturbed wave functions of the levels up to the fourth excited level are compared. Ground, first and second excited energy levels are also calculated by applying finite differences method and, results are compared with the ones obtained via perturbation theory. It is found that perturbation theory gives comparable results only for a small 𝜆 parameter and for the ground state. The quartic term in the Hamiltonian of the anharmonic oscillator leads to a more effective confinement of the particle which is deduced from the plots of wavefunctions and probability distributions. Meanwhile, the number of zero crossing nodes of the wavefunctions increases as the energy level increases, which is an expected result for both the harmonic and anharmonic oscillator.
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.