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Physica Scripta

Nonlinear harmonic boson oscillator

Bogdan M. Mihalcea

Published Sep 1, 2010• 10 citations
Publisher

Abstract

This paper deals with the study of the quantum Hamiltonian for a boson confined in a nonlinear ion trap. The expected value of the quantum Hamiltonian on coherent states is obtained. Then, the equation of motion for the boson results using the time-dependent variational principle (TDVP) and the Hamilton equations of motion. The resulting equation of motion is equivalent to the one describing a perturbed classical oscillator.

DOI
10.1088/0031-8949/2010/t140/014056

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This paper deals with the study of the quantum Hamiltonian for a boson confined in a nonlinear ion trap. The expected value of the quantum Hamiltonian on coherent states is obtained. Then, the equation of motion for the boson results using the time-dependent variational principle (TDVP) and the Hamilton equations of motion. The resulting equation of motion is equivalent to the one describing a perturbed classical oscillator.
Spectral properties of the Hamiltonian function which characterizes a trapped ion are investigated. In order to study semiclassical dynamics of trapped ions, coherent state orbits are introduced as sub-manifolds of the quantum state space, with the Kähler structure induced by the transition probability. The time dependent variational principle (TDVP) is applied on coherent states' orbits. The Hamilton equations of motion on Kähler manifolds of the type of classical phase spaces naturally arise. The associated classical Hamiltonian is obtained from the expected values on symplectic coherent states of the quantum Hamiltonian. Spectral information is thus coded within the phase portrait. We deal with the bosonic realization of the Lie algebra of the SU(1,1) group, which we particularize for the case of an ion confined in a combined, Paul and Penning trap. This formalism can be applied to Hamiltonians which are nonlinear in the infinitesimal generators of a dynamical symmetry group, such as the case of ions confined in electrodynamic traps. Discrete quasienergy spectra are obtained and the corresponding quasienergy states are explicitly realized as coherent states parameterized by the stable solutions of the corresponding classical equations of motion. A correspondence between quantum and classical stability domains is thus established, using the Husimi representation.
We investigate the spectral properties of the Hamiltonian function that describes an ion confined within a nonlinear trap. The Hamiltonian is then particularized for the case of dynamic traps and we introduce algebraic models with the aim to characterize the associated dynamics in such traps. The coherent states formalism for dynamic groups and the time-dependent variational principle are applied in order to study the semiclassical behaviour of the confined ion. We deal with the bosonic realization of the Lie algebra of the SU(1,1) group, which we particularize for the case of an ion confined in a combined (Paul and Penning) trap. We infer the equations of motion for the ion in a semiclassical approach. We suggest an algorithm by means of which we can associate a classical Hamiltonian with the quantum Hamiltonian that describes the ion. The classical Hamiltonian implicitly contains spectral information on the quantum system, which means we can dequantify the system.
"We investigate quantum dynamics for an ion confined within an oscillating quadrupole field, starting from two well known and elegant approaches. It is established that the Hamilton equations of motion, in both Schrodinger and Heisenberg representations, are equivalent to the Hill equation. One searches for a linear independent solution associated to a harmonic oscillator (HO). An adiabatic invariant, which is also a constant of motion, is introduced based on the Heisenberg representation. Thus, the state of the non-autonomous system can be determined at any subsequent moment of time. The quantum states for trapped ions are demonstrated to be Fock (number) states, while the exact solutions of the Schrodinger equation for a trapped ion are exactly the quasienergy states. Semiclassical dynamics is also investigated for many-body systems of trapped ions, where the wavefunction associated to the Schrodinger equation is prepared as a Gauss package multiplied by a Hermite polynomial. We also discuss time evolution for the system under investigation and supply the propagator."
Quasiclassical dynamics of trapped ions is described by applying the time dependent variational principle (TDVP) on coherent state orbits, introduced as sub-manifolds of the quantum state space. The classical Hamilton functions associated to the system are obtained as the expected values of the quantum Hamiltonian on symplectic coherent states. Such formalism can be applied to Hamilton functions which are nonlinear in the infinitesimal generators of a dynamical symmetry group. We give the explicit expression of the quantum Hamilton function in case of nonlinear electromagnetic traps. Discrete quasienergy spectra result, and the corresponding quasienergy states are explicitly realized as coherent states parameterized by the stable solutions of the corresponding classical equations of motion. A dequantization algorithm is also suggested. The method is then applied to combined (Paul and Penning) and ideal Paul traps. We obtain the explicit expressions of the classical Hamilton functions for such particular traps, as well as the classical Hamilton equations of motion in complex coordinates for an octupole trap. In the pseudopotential approximation, the points of minimum of the Hamiton function define equilibrium configurations for trapped ions, of interest for quantum logic.
Following the paper by Combescure [Ann. Phys. (N.Y.) 204, 113 (1990)], we apply the quantum singular time-dependent oscillator model to describe the relative one-dimensional motion of two ions in a trap. We argue that the model can be justified for low-energy excited states with the quantum numbers $n\ensuremath{\ll}{n}_{\mathrm{max}}\ensuremath{\sim}100$, provided the dimensionless constant characterizing the strength of the repulsive potential is large enough ${g}_{*}\ensuremath{\sim}{10}^{5}$. Time-dependent Gaussian-like wave packets generalizing odd coherent states of the harmonic oscillator and excitation number eigenstates are constructed. We show that the relative motion of the ions, in contradistinction to its center-of-mass counterpart, is extremely sensitive to the time dependence of the binding harmonic potential since the large value of g∗{g}_{*}g∗​ results in a significant amplification of the transition probabilities between energy eigenstate even for slow time variations of the frequency.
A quantum-mechanical model of a damped harmonic oscillator (both with time-independent and time-dependent parameters) is studied in the framework of the linear Schr\"odinger equation with a Hermitian nonstationary Hamiltonian. Integrals of the motion of this equation and their eigenstates, including coherent states, are constructed. The influence of an external harmonic force to the time evolution of various average values calculated over coherent states is considered, including the resonance case. The specific symmetry of the Hamiltonian leading to the new concept of loss-energy states is discussed.