Abstract We discuss the confinement of an electron in a Penning trap as the realisation of a one-dimensional quantum-mechanical oscillator. In such a situation, the non-zero ground-state energy of the electron in the confining electric and magnetic fields of the trap (as compared to the classical zero of the energy of a free electron) plays a role in certain kinds of investigations, and can be directly measured with high precision in photo-detachment spectroscopy of confined atomic anions such as H − . We detail a concrete implementation of such a measurement and discuss its implications.
Abstract We discuss the confinement of an electron in a Penning trap as the realisation of a one-dimensional quantum-mechanical oscillator. In such a situation, the non-zero ground-state energy of the electron in the confining electric and magnetic fields of the trap (as compared to the classical zero of the energy of a free electron) plays a role in certain kinds of investigations, and can be directly measured with high precision in photo-detachment spectroscopy of confined atomic anions such as H − . We detail a concrete implementation of such a measurement and discuss its implications.
Zero-point confinement in a suitable trap is briefly discussed as a quantum-mechanical equivalent of the classical single particle at rest in free space. So far, such confinement has been realized only for the 150 GHz cyclotron motion in geonium, a single electron permanently confined in a Penning trap. The most important result of mono-ion spectroscopy is a new, 10 000 times smaller, radius for the electron. This result was obtained by analyzing our g-factor data, g/2 = 1.001 159 652 193 (4), on the basis of a near-Dirac particle model. RF spectroscopy in geonium relies on the Continuous Stem Gerlach Effect in which a spin flip is detected as a small change in the ~ 60 MHz axial oscillation frequency of the electron in the trap. The Kaufmann Effect or relativistic mass shift may become a superior alternative : operating a geonium apparatus as a frequency selective mini-synchro-cyclotron has produced an easily detectable shift in the axial frequency. Like in Habann's 1926 split-anode magnetron, energy is quickly transferred from the oscillating electron to a resonant circuit, but by us for the purpose of detection, damping and cooling of the oscillatory motion. Conversely, localizing the electron to ~ 60 μm in the node of a standing wave in the trap cavity, where this cavity looks like a short, when it is resonant with the cyclotron motion, has made it possible to decrease the natural line width ten-fold. By driving the axial frequency not on resonance but on a side band higher by the ~ 12 kHz magnetron frequency, it has been possible to force the magnetron motion to absorb the excess in the photon energy and thereby shrink the magnetron radius to ~ 15 μm. By an analogous laser spectroscopic procedure the oscillation amplitude of a Ba+ ion in a different trap has been reduced to ~ 120 nm or less. Like in Nuclear Magnetic Resonance, confinement is now much smaller than the wave length and no motional side bands appear in the optical spectrum. Such a mono-ion oscillator, but using Tl+ or a similar ion, may make an optical frequency standard with a 1 000-day reproducibility of 10 -18 possible and thereby allow one to watch the expansion of the universe from one's laboratory.
Abstract Coupled harmonic oscillators are ubiquitous in physics and play a prominent role in quantum science. They are a cornerstone of quantum mechanics and quantum field theory, where second quantization relies on harmonic oscillator operators to create and annihilate particles. Descriptions of quantum tunneling, beamsplitters, coupled potential wells, ``hopping terms'', decoherence, and many other phenomena rely on coupled harmonic oscillators. However, the ability to couple separate quantum harmonic oscillators directly, with coupling rates that substantially exceed decoherence rates, has remained elusive. Here, we realize high-fidelity coherent exchange of single motional quanta between harmonic oscillators -- in this case, spectrally separated harmonic modes of motion of a trapped ion crystal -- where the timing, strength, and phase of the coupling are controlled through an oscillating electric potential with suitable spatial variation. The coupling rate can be made much larger than the decoherence rates, enabling demonstrations of high-fidelity quantum state transfer, entanglement of motional modes, and Hong-Ou-Mandel-type interference. We also project a harmonic oscillator into its ground state by measurement and preserve that state during repetitions of the projective measurement, an important prerequisite for non-destructive syndrome measurement in continuous-variable quantum error correction. Controllable coupling between harmonic oscillators has potential applications in quantum information processing with continuous variables, quantum simulation, and precision measurements. It can also enable cooling and quantum logic spectroscopy involving motional modes of trapped ions that are not directly accessible.
We present a detailed account of the design choices required for planar Penning traps that feature a highly harmonic confining potential. High harmonicity is indispensable for a number of applications, particularly for confinement of single electrons as qubits in quantum-information processing. The present work extends previous studies [D. Goldmann and G. Gabrielse, Phys. Rev. A 81, 052335 (2010)] by a fully analytic treatment of finite electrode gaps and their relevance for small traps on the millimetre size scale and below, when relative gap sizes are non-negligible. We derive the overall trap potential with a particular focus on different models of finite-gap potentials and show how to find the optimum trap geometry and electrode voltages to minimize anharmonicities. The analytic calculations are compared with detailed finite-element simulations.
Zero-point level confinement1 in a trap is the quantum-mechanical equivalent of the classical single particle at rest in free space. Such confinement has been demonstrated—by the continuous Stern-Gerlach effect—only for the 150-GHz cyclotron motion in geonium, a single electron permanently confined in a cold Penning trap. Localizing the electron to ≈ 60 μm in the node of a standing wave in the trap cavity approximated free space, but decreased spontaneous emission tenfold. Driving the 60-MHz axial motion on a sideband higher by the ≈ 12-kHz magnetron frequency forced the magnetron motion to absorb the excess photon energy, shrinking its radius to ~ 15μm. Analogous laser cooling has reduced the oscillation amplitude of a Ba+ ion in a Paul rf trap to < 120 nm. As in nuclear magnetic resonance, confinement is now much smaller than the wavelength and sidebands disappear. Quantum jumps to and from an electronic metastable level of Ba+ have been demonstrated. Using In+ or a similar ion, this may make a monoion oscillator optical frequency standard with a 1000-day reproducibility of 10-18 possible.
A trapped-ion harmonic oscillator represents a rich and well-controlled platform for tests of quantum physics, with applications to quantum simulations, quantum sensing, and quantum information processing. Owing to its net charge, the ion's motion is affected by changes in the electric potential. Consequently, electric field and potential curvature noise will limit the coherence of the ion's harmonic oscillation. While in some respects, this serves as a limitation in the performance of the trapped ion system (for example, in the fidelity of two-qubit gates or quantum simulations with motional quanta), it also opens an opportunity to use the ion as a precise field sensor and to use engineered quantum states of motion to enhance the sensitivity to these fields beyond the standard quantum limit (SQL). This thesis describes the implementation of such applications on a single beryllium ion confined in a Paul trap. In the thesis, I present the theory of an ideal harmonic oscillator and the implementation of this system using the motion of a single trapped ion, including producing special quantum states of motion; these states include the energy eigenstates of the harmonic oscillator, called number states, up to n=100, superpositions of number states of the form (1/√2)(|0> + |n>), with n up to 18, and coherently displaced number states with an average occupation up to approximately 300. I describe how we use these quantum states of motion for precise sensing of fluctuations of the harmonic oscillator frequency. The sensitivity of the (1/√2)(|0> + |n>) state ideally follows the 1/n Heisenberg scaling for frequency sensitivity. Additionally, I describe investigations of the spectrum of motional frequency noise using a series of coherent displacements of the motion of the ion, with features similar to Ramsey and dynamical decoupling sequences for two-level systems. While these displaced states don't improve sensitivity over the SQL, they can be simply and rapidly implemented with trapped ions. I discuss extensions to all of these experiments, including to multiple ions in a new trap designed to confine 3 or 4 ions in individually tunable potential wells.
Zero‐point level confinement in a trap is the quantum‐mechanical equivalent of the classical single particle at rest in free space. Such confinement has been demonstrated—by the Continuous Stern‐Gerlach Effect—for only the 150 GHz cyclotron motion in geonium, a single electron permanently confined in a cold Penning trap. This confinement made possible a new determination of the radius of the electron, some 10 000 times smaller than the accepted upper limit of ≊10−16 cm. Localizing the electron to ≊60 μm in the node of a standing wave in the trap cavity approximated free space, however decreasing spontaneous emission ten‐fold. Driving the 60 MHz axial motion on a side band higher by the ≊12 kHz magnetron frequency, forced the magnetron motion to absorb the excess photon energy, and shrank its radius to ≊15 μm. Analogous laser cooling has reduced the oscillation amplitude of a Ba+ ion in a Paul rf trap to <120 nm. As in Nuclear Magnetic Resonance, confinement here is much smaller than the wave length and side bands disappear. Quantum jumps to and from an electronic metastable level of Ba+ have been demonstrated. Using In+ or similar, these techniques may make a mono‐ion oscillator optical frequency standard with a 1000‐day reproducibility of 10−18 possible. Further experiments are proposed.