A quantum mechanical model of Rabi oscillations between two interacting harmonic oscillator modes and the interconversion of modes in a Penning trap

Journal Article · Fri Jan 15 00:00:00 EST 1999 · AIP Conference Proceedings

DOI:https://doi.org/10.1063/1.57446· OSTI ID:21207932

Kretzschmar, Martin ^[1]

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1. Institut fuer Physik, Johannes-Gutenberg-Universitaet, 55099 Mainz (Germany)

When a Penning trap is operated with an additional quadrupole driving field with a frequency that equals a suitable combination (sum or difference) of the frequencies of the fundamental modes of motion (modified cyclotron, magnetron and axial frequency), then a periodic conversion of the participating modes into each other is observed, strongly resembling the Rabi oscillations in a 2-level atom driven by a laser field tuned to the transition frequency. This investigation attempts to understand on a fundamental level how and why the motion of a classical particle in a macroscopic apparatus can be truely analogous to the oscillations of states of quantum mechanical 2-level systems (2-level atom or magnetic resonance). Ion motion in a Penning trap with an additional quadrupole driving field is described in a quantum mechanical frame work. The Heisenberg equations of motion for the creation and annihilation operators of the interacting oscillators have been explicitly solved, the time development operator of the Schroedinger picture has been determined. The driving field provides for two types of intermode interaction: Type I preserves the total number of excitation quanta present in the two interacting modes, the system oscillates between the modes with a frequency corresponding to the Rabi frequency in two-level systems. Type II preserves the difference of the numbers of excitation quanta present in the two interacting modes, it causes the ion motion to become unbounded. The two types of interaction are associated in a natural way with a SU(2) and a SU(1,1) Lie algebra. The three generators of these algebras form a vector operator that we denote as the Bloch vector operator. The Hilbert space decomposes in a natural way into invariant subspaces, finite dimensional in the case of type I interaction (SU(2)-algebra) and infinite dimensional in the case of type II interaction (SU(1,1)-algebra). The physics of the 2-level atom in the laser field can be described in the 2-dimensional (the lowest nontrivial) sector of the Hilbert space associated with the type I (SU(2)-algebra) interaction. The Bloch vector, well known from quantum optics, is the expectation value of our Bloch operator. On the other hand, the description of ion motion in the Penning trap requires the whole infinite dimensional Hilbert space of our model. Classical ion trajectories are obtained by calculating for the observables corresponding to position and momentum the expectation values with respect to minimum uncertainty coherent oscillator states.

• OSTI ID: 21207932
• Journal Information: AIP Conference Proceedings, Vol. 457, Issue 1; Conference: International conference on trapped charged particles and fundamental physics, Asilomar, CA (United States), 31 Aug - 4 Sep 1998; Other Information: DOI: 10.1063/1.57446; (c) 1999 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0094-243X
• Country of Publication: United States
• Language: English

Related Subjects

74 ATOMIC AND MOLECULAR PHYSICS
ANNIHILATION OPERATORS
ATOMS
EQUATIONS OF MOTION
EXCITATION
EXPECTATION VALUE
HARMONIC OSCILLATOR MODELS
HILBERT SPACE
IONS
LASER RADIATION
LIE GROUPS
MAGNETIC RESONANCE
OSCILLATIONS
PHOTON-ATOM COLLISIONS
QUANTUM MECHANICS
SCHROEDINGER PICTURE
TRAPS
TWO-DIMENSIONAL CALCULATIONS
WAVE FUNCTIONS