The quantum theory of the Penning trap

Figures & data

Figure 1. The classical radial motion of a charged particle in the circular Penning trap traces out an epicyclic curve [1] in the x-y plane. The cyclotron motion of frequency ${\mathit{\omega }}_{+}$ is superposed onto a slow magnetron drift orbit with frequency ${\mathit{\omega }}_{-}$. The relative size of the orbits is not drawn to scale.

Figure 2. Left: The expectation values of the coupled Hamiltonian in the Fock states $|n_x,n_z⟩$, as given in (89(89) $$\begin{array}{c}\hfill ⟨{\widehat{\mathcal{H}}}_d⟩=\frac{ħ{\mathit{\omega }}_0}{2}(N+1)+\frac{ħ}{2}\mathit{\delta }l.\end{array}$$(89) ). These are the so-called ‘bare’ states of the system. Right: Expectation values of the coupled Hamiltonian in the ‘dressed’ states $|n_{\mathit{\alpha }}{n}_{\mathit{\beta }}⟩$. The effects of the dressing are the formation of an avoided crossing between the $l=n_z-n_x$ sub-levels of the system at the point $\mathit{\delta }=0$. The size of the splitting is dependent on the electric coupling field strength in (67(67) $$\begin{array}{c}\hfill E_p(t)=\text{Re}\left({\mathit{ϵ}}_p{e}^{i{\mathit{\omega }}_{p}t}\right)(x{\widehat{e}}_z+z{\widehat{e}}_x)\end{array}$$(67) ), where the renormalized strength $\mathit{\xi }$ is defined in Equation (76(76) $$\begin{array}{c}\hfill \mathit{\xi }=\frac{e}{4m}\frac{1}{\sqrt{{\mathit{\omega }}_1{\mathit{\omega }}_z}}{\mathit{ϵ}}_p.\end{array}$$(76) ) (not shown to scale).

Brown, L.S.; Gabrielse, G. Geonium Theory: Physics of a Single Electron or Ion in a Penning Trap. Rev. Modern Phys. 1986, 58, 233–311.

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