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Physical Review Letters

Quantum motion in a Paul trap

Lowell S. Brown

Published Feb 4, 1991• 230 citations
Publisher

Abstract

Quasienergy eigenstates are constructed for a quantum harmonic oscillator with a periodic, time-dependent ``spring constant.'' This is done by a sequence of canonical transformations. The wave function in the new variables is that of an ordinary oscillator.
DOI
10.1103/physrevlett.66.527

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Operators conjugate to the Hamiltonian are constructed explicitly for the quantum harmonic oscillator by two approaches in the space spanned by the eigenstates of qqq and the eigenstates of ppp. The operators are quantum analogs of a classical angle variable divided by the oscillator frequency. Matrix elements have been evaluated in the coherent state representation. Either conjugate operator can be used to construct an explicitly time-dependent operator invariant. It can be used as the starting point in a new perturbative procedure for constructing invariant operators for nonlinear, nonautonomous Hamiltonians.
By using Husimi’s test-function method, we construct n exact solutions of a single Paul trapped ion modelled by a quantum harmonic oscillator with a time-dependent ‘spring constant’. Any n th solution is a combination of the Fock state and coherent state, which describes a wavepacket train consisting of n + 1 packets. Its centre oscillates like the classical harmonic oscillator with variable frequency, while the width and height of each packet change simultaneously. When the deformation is small, it behaves like a propagated and non-deformed train, and the large deformation is identified with breathing of the wavepacket train.