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The oscillation frequencies of charged particles in a Penning trap can serve as sensors for spectroscopy when additional field components are introduced to the magnetic and electric fields used for confinement. The presence of so-called “magnetic bottles” and specific electric anharmonicities creates calculable energy-dependences of the oscillation frequencies in the radiofrequency domain which may be used to detect the absorption or emission of photons both in the microwave and optical frequency domains. The precise electronic measurement of these oscillation frequencies therefore represents an optical sensor for spectroscopy. We discuss possible applications for precision laser and microwave spectroscopy and their role in the determination of magnetic moments and excited state life-times. Also, the trap-assisted measurement of radiative nuclear de-excitations in the X-ray domain is discussed. This way, the different applications range over more than 12 orders of magnitude in the detectable photon energies, from below

Penning traps usually serve as a mere means for ion confinement under well-defined conditions. For spectroscopic applications, localization of ions and the possibility to cool their motion to very low velocities are the main features. However, as will become evident later, also specific properties of the ion confinement and the interaction between the ions and the storing fields can be employed for advanced spectroscopy.

Spectroscopy in charged particle traps such as Penning traps has widely been used for precision spectroscopy of stored ions at low velocities, as has been detailed out in [

Laser and microwave sources for excitation of transitions are available within a broad range of frequencies [

It is therefore desirable to have alternative detection schemes at hand which are non-optical. Confined ions in a Penning trap, typically single ions, offer such a possibility. When the confining fields are chosen in such a way that the ion oscillation frequencies in the trap depend unambiguously on the energy of the ion motion, they may serve as an electronic detector for the absorption or emission of microwave, optical, or X-ray photons, as will be detailed out below.

The well-known continuous Stern-Gerlach effect [

The confinement of ions in a Penning trap [

In a so-called “ideal” Penning trap, _{0} is the electric trapping potential, _{0} and the axial trap size _{0}. The factor _{2} equals unity in the ideal Penning trap, imperfections will be explained below. The two superimposed radial oscillation frequencies are given by
_{+} is called the “modified cyclotron frequency” and _{−} is the “magnetron drift frequency”. _{c}

The presence of either a magnetic or electrostatic imperfection represents a coupling of the oscillatory motions such that the individual oscillation frequencies become dependent on the energies of all motions. This is due to the fact that in imperfect fields the effective field strength (which determines the oscillation frequencies) experienced by the ion depends on its position in the trap and deviates from the ideal value when the energy (amplitude) of a motion changes. Using the hierarchy _{−} ≪ _{z}_{+} of the frequencies [_{E}_{B}_{R}_{L}

Let the anharmonicity of the electrostatic potential near the trap centre be described by the expansion [_{k}_{2} and to account for electric imperfections characterized by _{4}. Higher-order contributions are considered as negligible. The next term _{6} is suppressed with respect to the term in _{4} by a factor of (^{2}, which typically is of order 10^{−4} or smaller. Odd terms vanish because of the point symmetry with respect to the trap centre. The coefficients including _{4} depend on applied voltages and can be written as [_{k}_{C}_{2} only represents a linear scaling of the trapping potential, a non-vanishing term _{4} ≠ 0 leads to frequency dependences described by
_{z}/ω_{+}. The bottom line of the matrix _{E}

Let the symmetry axis _{z}_{0} and the radial coordinate be _{0} is the homogeneous part of the field, _{1} describes a _{2} characterizes a so-called “magnetic bottle” and its dependence on both an axial and a radial coordinate. Higher-order terms are not of relevance for the present discussion. A magnetic bottle is therefore a magnetic field inhomogeneity of the kind
_{2} ≠ 0 superimposed on the magnetic trapping field _{0} with radial symmetry around the trap centre. The presence of _{2} ≠ 0 results in a dependence of the oscillation frequencies on the motional energies given by

Relativistic frequency shifts can be understood in terms of the relativistic mass shift with the kinetic energy in the respective motion and lead to

For the three ion oscillations, this is a straightforward transformation acting on the motional frequencies through the mass shift. Following the discussion in [^{2})^{−1/2}, the free ion cyclotron frequency is given by _{c}^{−15}, this effect will be neglected. Typical values for both the shift terms _{4}) := 6_{4}/(_{0}) and _{2}) := _{2}/(2_{−}_{+} _{0}) in ^{−5}/eV to 10^{−3}/eV in highly charged ions and from 10^{−3}/eV to 10^{−1}/eV in singly charged ions. The value of ^{2} can be approximated by
^{−5} to 10^{−3} in highly charged ions and from 10^{−3} to 10^{−1} in singly charged ions. The relativistic term 1/(^{2}) is of order 10^{−10}/eV for light ions and of order 10^{−11}/eV for heavy ions. A graphic representation of the typical magnitude of these shift terms is given in _{0} between 10V and 1000V, magnetic fields _{0} between 1 T and 10 T, and values of _{2} = 10mT/mm^{2} and _{4} = 0.5 have been assumed.

Out of the twelve dependences described by ^{2} or higher may be assumed too small in magnitude for a significant contribution to the coupling, since
^{−2} or smaller, see above.

Inserting relations (1) to (3) into _{−} ≪ _{z}_{+} ≈ _{c}_{C}_{0} in

We employ quantum mechanical first order perturbation theory to describe the energy shift of the confined ion due to the presence of the magnetic bottle by
_{+} and _{z}_{S}_{z}_{z}_{z}_{z}_{B}_{e}_{z}_{+}, _{−}, _{z}_{S}_{+}, _{−}_{z}_{S}_{+}, _{−}_{z}_{S}_{−} ≪ _{z}_{+} ≈ _{c}_{L}_{B}_{0}_{S}_{J}μ_{B}B_{0}_{S}_{e}_{J}_{s}

Looking at ^{2} or higher). They can be used for a detection of changes in the oscillation energy by observing the corresponding shifts in the oscillation frequencies. These terms are Δ_{+}(_{z}_{2}), Δ_{z}_{z}_{4}), Δ_{z}_{+}, _{2}) _{L}_{z}_{2}). Additionally, _{z}_{s}_{2}) and Δ_{+}(_{s}_{2}) ^{2} in

Terms in group A can be used to detect changes in the oscillation energy of confined ions, e.g., due to laser cooling or heating, and thus serve as an electronic detector for optical photons.

Terms in group B allow to determine a spin change of the system, e.g., of a single electron bound in an ion, and therefore can serve as an electronic detector for microwave photons which induce spin transitions. This is the basis also for the continuous Stern-Gerlach effect.

The relativistic terms in group C make oscillation frequencies dependent on kinetic energies even for ideal confining fields, but are generally too small in magnitude for spectroscopic purposes. However, the “direct” relativistic mass effect due to ^{2} allows to weigh internal excitation energies by the corresponding frequency shift, e.g., of nuclear isomeric states in ions.

Additionally, the dependence Δ_{L}_{z}_{+}) can be used for a manipulation of the Larmor frequency, which may be of use in spectroscopy as in group B. The dependences in group A can alternatively be used to measure magnetic bottle strengths by electronic means: for known trap geometry and confining fields, the electric anharmonicity _{4} can be chosen by variation of the voltage ratio _{C}_{0} such that it cancels the effect of the _{2} term, see e.g., _{C}_{0} such that the total energy-dependence of the oscillation frequency vanishes, yields the corresponding _{2}.

Any of these possible applications relies on a detection of the corresponding oscillation frequency shift. For confined ions, the typical axial and radial frequencies are roughly of order MHz and can be measured electronically with high accuracy, as discussed in detail in [^{−10} can be detected by application of a phase-sensitive detection scheme as outlined in [

Assume an ion stored in a magnetic bottle with _{2} ≠ 0 superimposed to the magnetic trapping field _{0}. The terms Δ_{+}(_{z}_{2}) and Δ_{z}_{+}, _{2}) then describe the dependence of the radial frequency _{+} on the axial energy _{z}_{z}_{+} of the ion. The respective energies can individually be set to well-defined values by application of initial cooling, e.g., by resistive cooling to the cryogenic ambience temperature using a resonance circuit [_{+}(_{z}_{z}_{+}). Scanning a narrow-band laser over the transition of interest, the resonance is found as a maximum shift of the corresponding ion oscillation frequency, which is detected electronically. The applicability and potential of such a scheme has been described in detail in [_{+} corresponding to the decrease of axial oscillation energy _{z}

In a situation as described above, there are two mechanisms which can be used to change the energy of the ion oscillation in a well-defined way. One is the cooling or heating by the detuned laser, the other is the cooling or heating by a resonance circuit as used for initial resistive cooling. A balance between any two opposing mechanisms, e.g., laser heating against resistive cooling, results in a zero oscillation frequency shift as a function of time and may be used to determine the desired rate (inverse lifetime) Γ of the used optical transition. The power transferred to the ion by the laser is given by
^{2}/Γ^{2} is the saturation parameter which is proportional to the square of the on-resonance Rabi frequency Ω. ^{2} ≪ 1, _{E}

Another possibility, independent from electronic power transfer, makes use of the fact that the rate at which the observed frequency _{+} in the above example shifts is directly proportional to the desired transition rate Γ, since

Alternatively, one can make use of the fact that the light pressure of laser cooling shifts the ion axially from the trap centre by an amount much larger than the motional amplitude. In the presence of a magnetic inhomogeneity, this results in a shift of the radial frequencies which can be measured. An axially asymmetric trapping potential may be used to restore the ion position and thus its radial frequencies, yielding the value of the shift. This directly determines the desired value of the transition rate Γ. The shift Δ_{L}_{E}_{A}_{1} is the first term in the expansion of the trapping potential as given by _{A}_{0} such that one is free to choose the axial frequency.

When an ion is confined in the presence of a magnetic bottle, the terms in group B (_{z}_{s}_{2}) and Δ_{+}(_{s}_{2})) provide that the ion oscillation frequencies depend on the spin orientation of an electron bound in the ion relative to the magnetic field. Especially for hydrogen-like ions, this so-called ‘continuous Stern-Gerlach effect’ offers a possibility to determine the spin orientation of the electron (which is an intrinsic ion property described by the magnetic spin quantum number _{s}_{0} of several Tesla strength, the Larmor frequency of electrons is in the microwave domain at typically 100 GHz, corresponding to photon energies of order meV. For protons, this number is still smaller by a factor of about 658, such that photon energies are of order

Alternatively, the term Δ_{L}_{z}_{+}) (

Using the terms in group B, the magnetic moment of the bound electron (and thus its _{J}^{12}C^{5+} [^{16}O^{7+} [

The relativistic mass effect provides a change of the oscillation frequencies due to the mass change of the confined ion when its energy content is changed. The relativistic mass shift Δ^{2} changes the oscillation frequencies, such that the absorption or emission of a photon with energy Δ^{−10}/eV for light ions is for optical spectroscopy at the limit of the current resolution, however, this does not restrict the principle idea. Absorption of a photon of several eV energy by a light ion may serve as a proof of principle. Given a sensitivity higher by an order of magnitude, the relativistic frequency shift could be a valuable tool in finding the famous low-lying nuclear transition in ^{229}Th [

In highly charged ions, electronic excitation energies are much higher (up to order 100 keV) and could thus be detected much more easily, however the corresponding lifetimes of the excited states are extremely short. In few-electron ions, the upper state lifetime scales with the nuclear charge number ^{−4} for electric dipole, as ^{−6} for magnetic dipole and as ^{−10} for electric quadrupole transitions, such that only for ^{3}S_{1} state in Li^{+} with a lifetime of about 50 seconds and a decay energy of about 60 eV [

In the study of nuclear de-excitations, however, both the upper state lifetime and the photon energies are potentially high. With photon energies in the keV to MeV region, the expected relative frequency shifts are of order 10^{−7} to several 10^{−5} and thus easily detectable. The corresponding ion recoil energy due to photon emission is given by ^{2}/(2

The radioactive decay of isotopes is followed by a discontinuous change of the mass-to-charge ratio of the ion (due to α or

The radiative decay of long-lived nuclear isomers, however, does not change the mass-to-charge ratio and can therefore only be seen by the relativistic mass shift corresponding to the emitted photon energy. Corresponding measurements of long-lived nuclear isomers have been performed, e.g., on ^{65m}Fe [^{68m}Cu [_{EC}^{26}Si has been determined by such a trap-assisted measurement with a relative accuracy of about 10^{−5} [_{g}^{−16}m^{2}. At typical cryogenic vacua, this lifetime is of order 1,000 seconds for highly charged ions. Correspondingly, ^{−5}. Assuming a frequency resolution of some 10^{−10}, the transition energies of these nuclei can be measured with a relative accuracy of about 10^{−5}, which is substantially more precise than any of the measurements shown in

We have discussed concepts for the detection of microwave, optical and X-ray photon absorption or emission by charged particles confined in a Penning trap. A common feature is the electronic and non-destructive measurement of oscillation frequency shifts in the radiofrequency domain following photon absorption or emission. Using specific inhomogeneities of the trap’s confining fields, the oscillation frequencies in the trap depend on the energy of the particles which is changed in photon absorption or emission. Thus, the particle oscillation serves as a sensor for photons and can be employed for spectroscopy. As examples, we have discussed the continuous Stern-Gerlach effect in the microwave domain, ’blind’ spectroscopy in the optical domain and the radiative de-excitation of nuclear isomers in the X-ray domain. These examples span over 12 orders of magnitude in the photon energy, from ^{−10} for the determination of magnetic moments using the Stern-Gerlach effect, potentially even beyond 10^{−10} for the determination of electronic transition energies using “blind” spectroscopy, and up to about 10^{−5} for X-ray spectroscopy of radiative de-excitation of nuclear isomers. The applications require transitions which can either be excited inside the trap or which are long-lived. On the other hand, only a single particle is needed and hence also rare species can be examined. The omission of direct photon detection makes the applications system-unspecific and reduces the experimental effort to already established electronic detection methods. The discussed applications serve for precision measurements of magnetic moments (

^{171}Yb

^{+}Ions

^{16}O

^{7+}

^{3}

_{1}Li

^{+}

^{65}Fe with Penning Trap Mass Spectrometry

_{EC}

^{26}Si

_{2}and He

Typical geometry of a cylindrical Penning trap with open endcaps and correction electrodes between the central ring and the endcap electrodes. The confinement of ions by electric and magnetic fields is indicated schematically. The trapping region is located around the centre of the arrangement, inside the hollow cylinder electrodes. Details are given in [

Typical values of _{4}), _{2}) and ^{2} both for singly charged (^{2}) is shown for masses between

Illustration of the “blind spectroscopy” concept: a single stored ion is axially laser cooled on the optical transition of interest and the corresponding radial frequency shift is measured electronically. The cooling laser is scanned over the transition of interest and the resonance is detected as a maximum frequency shift.

Metastable nuclear isomers with transition lifetimes between 1 and 1,000 seconds, for which either no reliable energy measurement exists and / or the transition energy is not known to better than 1 keV. Data taken from [^{−5}.