# Single Microwave Photon Detection with a Trapped Electron

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## Abstract

**:**

## 1. Introduction

## 2. Basics of the Geonium Chip Planar Penning Trap

#### 2.1. Overview of the Experimental Setup

#### 2.2. Detection of the Trapped Electron’s Axial Motion

- The detection is non destructive: the axial frequency can be monitored continuously, without losing the trapped particle [19].
- The measurement of ${\omega}_{z}$ occurs with the axial motion being in thermal equilibrium with the LC resonator, typically at 4 K or eventually even at higher temperatures [23].
- The detection of the axial motion leaves the electron’s cyclotron quantum state unaffected [5]. Thus, while the axial motion is in thermal equilibrium with the LC resonator at 4 K, the cyclotron quantum state can be in thermal equilibrium with the 80 mK temperature provided by the ADR.
- While ${\omega}_{z}$ and the LC circuit must be resonantly coupled (see Figure 2b), the value of the trapping magnetic field, and therefore of ${\omega}_{p}$, can be freely chosen without affecting the detection of the axial motion.
- The frequency ${\omega}_{z}$ can be measured with an accuracy of 1 Hz in around 5 s. This is the lapse required to measure an FFT spectrum of 1 Hz resolution (∼1 s each) and average it for a good signal-to-noise ratio. As explained in Section 5, such long measurement time would not allow for high quantum efficiency in MW photon detection. However, frequency variations $\mathrm{\Delta}{\omega}_{z}$ can be determined substantially faster, by recording the phase evolution of the axial motion but without waiting for a full $2\pi $ oscillation [26]. This will permit very rapid MW photon detection, as detailed in Section 5.

## 3. Detection of Microwave Photons with a Trapped Electron

#### 3.1. Interaction of a Cyclotron Quantum Harmonic Oscillator with an Itinerant Microwave

#### 3.2. Dynamics of Two Coupled Quantum Harmonic Oscillators

#### 3.3. Principle of Microwave Photon Detection: The Continuous Stern-Gerlach Effect

#### 3.4. Generation of a Magnetic Bottle in the Geonium Chip

#### 3.5. Quantum Non Demolition Photon Detection

## 4. Quantum Efficiency of MW Photon Detection by a Single Trapped Electron

#### 4.1. Electric Field Strength of a Single Itinerant Photon Propagating along a CPW Transmission-Line

#### 4.2. Probability of MW Photon Absorption by the Quantum Cyclotron Oscillator

#### Characteristic Admittance of the Trapped Electron

#### 4.3. Internal Quantum Efficiency of Microwave Photon Detection

## 5. Speed of Quantum Jump Detection versus Cyclotron Radiative Lifetime

#### 5.1. Quantum Cyclotron Radiative Lifetime in a CPW Transmission-Line

#### 5.2. Measurement Time of $\mathrm{\Delta}{\omega}_{z}$

## 6. Comparison with Other Single Microwave Photon Detectors

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

ADR | Adiabatic Demagnetisation Refrigerator |

QND | Quantum Non Demolition |

MW | Microwave |

EM | Electromagnetic |

CPW | Coplanar-Waveguide |

FFT | Fast Fourier Transform |

EDM | Electric Dipole Moment |

CPB | Cooper Pair Box |

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**Figure 1.**(

**a**) Sketch of the geonium chip. The electrons are captured at some height ${y}_{0}$ above the central electrode, the so-called ring [11]. DC-voltages applied to the electrodes generate the electrostatic potential well around which the particles are captured and oscillate; (

**b**) Motion of a trapped electron.

**Figure 2.**(

**a**) Basic cryogenic setup with the 4 K pulse-tube cryocooler hosting one geonium chip and an RF superconducting coil for the detection of the electron’s axial motion; (

**b**) Detection of the axial motion. The particle induces some charge upon the chip’s surface which generates the detection signal. The latter is amplified and measured with a FFT spectrum analyser.

**Figure 3.**(

**a**) Possible introduction of the microwave signal to be measured into the geonium chip with the trapped electron. The picture shows a sketch of the cryogenic vacuum chamber which encloses the electron and builds also a 3D microwave cavity. This cavity is used to inhibit cyclotron spontaneous emission into free space (see text); (

**b**) The continuous Stern-Gerlach effect. The upper diagram shows the variation of the axial frequency caused by the absorption of one MW photon and the subsequent spontaneous emission after a cyclotron radiative lifetime of ${\tau}_{s}$. The lower graph shows the variation of the axial trapping potential experienced by the electron as a function of the cyclotron quantum number (a negative magnetic curvature ${B}_{2}<0$ is assumed).

**Figure 4.**(

**a**) Input port for the MW signal and input admittance ${Y}_{in}$ of the combined system, electron + CPW transmission-line terminated by a load admittance ${Y}_{L}$; (

**b**) Dimensions of the CPW- line which guides the microwave signal towards the trapped electron.

**Figure 5.**Characteristic admittance of the quantum cyclotron oscillator. Computed with Equations (7) and (8), truncating the series at $n=\mathrm{60,000}$. Both upper graphs assume $S=0.7$ mm, while the lower ones $S=0.1$ mm. All assume a 40 mm chip width on a quartz substrate of $0.675$ mm thickness, 200 nm gold layer and with $0.02$ S characteristic admittance of the CPW.

**Figure 6.**Radiative lifetime of the quantum cyclotron oscillator coupled to a CPW. Calculated with the same CPW dimensions assumed in Figure 5, with $S=0.7$ mm in both graphs.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Cridland, A.; Lacy, J.H.; Pinder, J.; Verdú, J. Single Microwave Photon Detection with a Trapped Electron. *Photonics* **2016**, *3*, 59.
https://doi.org/10.3390/photonics3040059

**AMA Style**

Cridland A, Lacy JH, Pinder J, Verdú J. Single Microwave Photon Detection with a Trapped Electron. *Photonics*. 2016; 3(4):59.
https://doi.org/10.3390/photonics3040059

**Chicago/Turabian Style**

Cridland, April, John Henry Lacy, Jonathan Pinder, and José Verdú. 2016. "Single Microwave Photon Detection with a Trapped Electron" *Photonics* 3, no. 4: 59.
https://doi.org/10.3390/photonics3040059