The Ramsey method in high-precision mass spectrometry with Penning traps: Theoretical foundations

This paper presents in a quantum mechanical framework a theoretical description of the interconversion of the magnetron and modified cyclotron motional modes of ions in a Penning trap due to excitation by external rf-quadrupole fields with a frequency near the true cyclotron frequency. The work aims at a correct description of the resonance line shapes that are observed in connection with more complicated excitation schemes using several excitation pulses, such as Ramsey’s method of separated oscillating fields. Quantum mechanical arguments together with the “rotating wave approximation” suggest a model Hamiltonian that permits a rigorous solution of the corresponding Heisenberg equations of motion. We show that the ion motion in a Penning trap with an additional rf-quadrupole field is dynamically analogous to nuclear-magnetic-resonance. This is done by introducing the concept of the “Bloch vector operator”, which is a vector operator obeying the commutation rules of an angular momentum operator and which is the analogue of a nuclear spin. During the quadrupole excitation the expectation value of the Bloch vector operator, which is an ordinary real three-vector, performs a precessional and nutational motion similar to the spin in nuclear-magnetic-resonance experiments. The frequency of the interconversion of the magnetron and modified cyclotron motional modes of the ions is the analogue of the Rabi frequency. Hence the applicability of Ramsey’s ideas to Penning trap physics becomes understandable. Resonance line shapes are deduced from the general solution of the dynamical problem for arbitrary values of the excitation time, pulse structure and detuning of the quadrupole radiation. Results are given for excitation schemes with up to five pulses. A comparison of the theoretical results to experimental data is found in the accompanying paper by George et al. [S. George, et al., IJMS, this issue].