Measurement of ultra-low heating rates of a single antiproton in a cryogenic Penning trap

TL;DR

This study measures extremely low motional heating rates of a single antiproton in a cryogenic Penning trap, demonstrating noise levels significantly lower than previous ion trap experiments, advancing precision in antimatter research.

Contribution

First detailed measurement of motional heating of a single antiproton in a cryogenic Penning trap with ultra-low noise levels, surpassing prior ion trap results.

Findings

Cyclotron quantum transition rate of 6(1) quanta/h

Electric field noise spectral density below 7.5(3.4)×10^{-20} V^2/m^2/Hz

Noise levels over two orders of magnitude lower than previous experiments

Abstract

We report on the first detailed study of motional heating in a cryogenic Penning trap using a single antiproton. Employing the continuous Stern-Gerlach effect we observe cyclotron quantum transition rates of 6(1) quanta/h and an electric field noise spectral density below $7.5(3.4)\times 10^{-20}\,\text{V}^{2}\text{m}^{-2} \text{Hz}^{-1}$, which corresponds to a scaled noise spectral density below $8.8(4.0)\times 10^{-12}\,\text{V}^{2}\text{m}^{-2}$, results which are more than two orders of magnitude smaller than those reported by other ion trap experiments.

Figures

Tables

Table 1. Table 1: Parasitic voltage fluctuation and heating rate contributions.

Observed $S_V$	$225\left(54\right)\text{pV}{\text{Hz}}^{-1/2}$
Axial detection system	$1.5\text{pV}{\text{Hz}}^{-1/2}$
RC filter stages
Electrode Johnson noise	$\sim 3\times {10}^{-3}\text{pV}{\text{Hz}}^{-1/2}$
Blackbody radiation	${\omega }_{+}\times S_E\left({\omega }_{+}\right)\sim 6\times {10}^{-14}{\text{V}}^2{\text{m}}^{-2}$
Background pressure

Equations

$$\begin{array}{rl}& \Delta {\nu }_z(n_{+},n_{-},m_s)\\ & =\frac{h{\nu }_{+}}{4{\pi }^2{m}_{\bar{p}}{\nu }_z}\frac{B_2}{B_0}\left[\left(n_{+}+\frac{1}{2}\right)+\frac{{\nu }_{-}}{{\nu }_{+}}\left(n_{-}+\frac{1}{2}\right)+\frac{g_{\bar{p}}{m}_s}{2}\right].\end{array}$$

$${\sigma }_{{\nu }_z}(\tau )\propto \sqrt{{\sigma }_v(\tau {)}^2+{\sigma }_{\text{FFT}}(\tau {)}^2+\tau \left(\Delta {\nu }_{z,+}^2{\zeta }_{+}\right)}.$$

$${\zeta }_{+}=\frac{q^2{n}_{+}}{2m_{\bar{p}}\hslash {\omega }_{+}}{S}_E\left({\omega }_{+}\right),$$

$$S_E\left({\omega }_{+}\right)={\Lambda }^2\left(\rho ,z\right)S_V\left({\omega }_{+}\right),$$

$${\Lambda }^2\left(\rho ,z\right)=\sum _{n=1}^5{\left(\frac{\partial V_n}{\partial \rho }\right)}^2\propto {\rho }^2,$$

Full text

Quantum control techniques applied to trapped charged particles, well-isolated from environmental influences, have very versatile applications in metrology and quantum information processing. For example, elegant experiments on co-trapped laser cooled ions in Paul traps have provided highly precise state-of-the-art quantum logic clocks schmidt2005 , enabled the development of exquisite atomic precision sensors kitching2011 and the implementation of quantum information algorithms applied with highly entangled ion-crystals blatt2008 . Decoherence effects from noise driven quantum transitions, commonly referred to as anomalous heating turchette2000 ; brownnutt2015 , affect the scalability of multi-ion systems, which would enable even more powerful algorithms. Trapped particles are also highly sensitive probes to test fundamental symmetries, and to search for physics beyond the standard model kosteleckyPenning ; safronova2018 . The most precise values of the mass of the electron sturm2014 and the most stringent tests of bound-state quantum electrodynamics sturm2011 are based on precise frequency measurements on highly-charged ions in Penning traps. Measurements of the properties of trapped electrons hanneke2008 and positrons vandyck1987 provide the most sensitive tests of quantum electrodynamics and of the fundamental charge-parity-time (CPT) invariance in the lepton sector cpt ; dehmelt1999 .

Our experiments smorra2015epj make high-precision comparisons of the fundamental properties of protons and antiprotons, and provide stringent tests of CPT invariance in the baryon sector. We recently reported on an improved determination of the proton magnetic moment with a fractional precision of 300 parts in a trillion schneider2017science and the first high-precision determination of the antiproton magnetic moment with a fractional precision of 1.5 parts in a billion smorra2017nature . This measurement, based on a newly invented multi-trap method, improves the fractional precision achieved in previous studies Jack2012Proton ; Jack2013Antiproton by more than a factor of 3000. These multi-trap based high-precision magnetic moment measurements on protons and antiprotons require low-noise conditions much more demanding than in any other ion trap experiment. Compared to experiments on electrons and positrons hanneke2008 ; vandyck1987 , the 660-fold smaller proton/antiproton magnetic moment makes it much more challenging to apply high-fidelity single particle spin-quantum spectroscopy techniques smorra2017 . Our experiments become possible only in cryogenic ultra-low-noise Penning-trap instruments, which provide energy stabilities of the particle motion on the peV/s range, effectively corresponding to a parasitic transition rate acceptance limit of, at most, two motional quanta over several minutes of measurement time.

In this Letter we report on the characterization of the electric field fluctuations in a cryogenic Penning trap by explicit measurements of cyclotron quantum transition rates of a single antiproton using the continuous Stern-Gerlach effect dehmelt1973 .

The observed electric field spectral noise density is more than two orders of magnitude lower than in room temperature Penning traps goodwin2016 and more than 1000 times smaller than observed in cryogenic Paul trap experiments brownnutt2015 . Based on heating rate measurements at various particle orbits we identify fluctuations in the trapping field caused by residual voltage noise as the dominant heating mechanism. Anomalous heating is not observed within our measurement resolution.

The measurements are conducted in the cryogenic spin-state analysis trap of the BASE apparatus at CERN smorra2015epj , which is shown in Fig. 1. The Penning trap is realized using a superconducting magnet at $1.945$ T combined with a quadrupolar electrostatic potential provided from a set of five carefully designed cylindrical electrodes with an inner diameter of 3.6 mm CCRodegheri2012 . The central ring electrode is made out of a Co/Fe alloy, which distorts the nearly homogeneous axial magnetic field to ${B_{z}=B_{0}+B_{2}\left(z^{2}-\rho^{2}/2\right)}$, deliberately generating a magnetic inhomogeneity of $B_{2}~{}=~{}272\,\text{kT}\text{m}^{-1}$. The trajectory of a single antiproton stored in a Penning trap is composed of three harmonic oscillator modes. The modified cyclotron motion at $\nu_{+}$ and the magnetron motion at $\nu_{-}$ are perpendicular to the magnetic field, while the particle oscillates along the magnetic field lines with axial frequency $\nu_{z}$. For the BASE analysis trap, $\nu_{+}\approx 17.845\,$MHz, $\nu_{-}\approx 10\,$kHz and $\nu_{z}\approx 675\,$kHz.

The gold-plated OFHC electrodes are placed inside an indium-sealed vacuum chamber which is cooled to ${T\approx 6\,}$K. Cryo-pumping provides an ultra-high vacuum with pressures $<3\times 10^{-18}$ mbar which enables storage times $>10\,a$ sellner2017 . Radio frequency (rf) lines equipped with high order low-pass and band-pass filters as well as high-insulation switches are used for particle manipulation (Fig$.\,$1 c). The axial oscillation frequency $\nu_{z}$ is measured by an image current detection system nagahama2016 . The detector’s time transient is processed with a Fast Fourier Transform (FFT) spectrum analyzer. Once cooled to thermal equilibrium, the particle signature appears as a dip in the resulting frequency spectrum wineland1975 (see Fig$.\,$1 b). A least-squares fit of the recorded spectra yields the axial frequency $\nu_{z}$. In the measurements reported here, we apply active electronic feedback cooling (see Fig. 1 a) dehmelt1986 ; durso2003 , which enables measurements at low axial temperature ($T_{z}\approx 1{.}92(10)$ K) and high axial frequency stability smorra2017 .

For explicit measurements of modified cyclotron transition rates we utilize the continuous Stern-Gerlach effect dehmelt1973 . Here, the interaction of the particle’s magnetic moment $\mu_{z}=\mu_{+}+\mu_{-}+\mu_{s}$ with the strong magnetic inhomogeneity $B_{2}$ results in a magnetostatic axial energy $E_{B,z}=-\mu_{z}\times{B}_{z}(z)$, where $\mu_{+}$ and $\mu_{-}$ are the angular magnetic moments associated with the modified cyclotron and the magnetron mode, while $\mu_{s}$ is the spin magnetic moment. As a result, the antiproton’s axial frequency $\nu_{z}=\nu_{z,0}+\Delta\nu_{z}$ becomes a function of the radial quantum states:

[TABLE]

Transitions in the corresponding states $\left(m_{s},n_{+},n_{-}\right)$ lead to axial frequency shifts of $\Delta\nu_{z,\text{s}}~{}=~{}172(10)\,$mHz, ${\Delta\nu_{z,\text{+}}~{}=~{}}$62(4) mHz and ${\Delta\nu_{z,-}~{}=~{}}$40(3) $\mu$Hz, respectively.

To determine the transition rate $\zeta_{+}$ of the cyclotron motion we first prepare a particle at low radial energy with $n_{+}<200$ mooser2013 . Then, we record sequences of axial frequency measurements $\nu_{z,k}$ with an averaging time $\tau_{0}~{}=~{}50$ s. Subsequently, we evaluate the standard deviation $\sigma_{\nu_{z}}(\tau)=\sigma\left(\langle\nu_{z,j+1}\rangle\left(\tau\right)-\langle\nu_{z,j}\rangle\left(\tau\right)\right)$, where $\langle\nu_{z,j}\rangle(\tau)$ represents the mean values of a sub-series of axial frequency measurements with an averaging time $\tau=l\times\tau_{0}$. A result of such an overlapping differential Allan deviation $\sigma_{\nu_{z}}(\tau)$ allan1966 is shown in Fig. 2 as blue filled circles. Various measured and simulated contributions to $\sigma_{\nu_{z}}(\tau)$ are also plotted in Fig. 2. The contribution from voltage fluctuations $\sigma_{v}(\tau)$ (dark red triangles) is extracted from simultaneous measurement of the voltage supply stability, as shown in Fig 1. The contribution from white frequency measurement noise, $\sigma_{\text{FFT}}(\tau)\propto\delta\nu_{z}^{1/2}$ $\text{SNR}^{-1/4}$ (dark red squares) is calculated smorra2015epj , $\delta\nu_{z}$ being the line-width of the axial frequency dip and SNR the signal-to-noise ratio (see Fig. 1 b). At small averaging times ($\tau<100\,$s), these two contributions dominate. Meanwhile, with long averaging times ($\tau>250\,$s), $\sigma_{\nu_{z}}(\tau)$ is dominated by transition rates $\zeta_{+}$ in the modified cyclotron mode,

[TABLE]

By analyzing such data and comparing the Allan deviation to Monte-Carlo simulated noise-driven random walks, we extract an absolute cyclotron transition rate of $\zeta_{+}=6(1)\,\text{h}^{-1}$, see Fig. 2. Note that $\zeta_{+}$ describes a nearly undirected random walk.

The observed transition rates can be related to the noise spectral density of the radial electric field $S_{E}(\omega_{+})$ at the modified cyclotron frequency. Considering first order transitions in a noise-driven quantum mechanical oscillator savard1997 , cyclotron transitions rates are given by

[TABLE]

where $S_{E}(\omega_{+})$ is the spectral density of electric field noise acting on the particle’s cyclotron motion. The average increase of $n_{+}$ is given by the heating rate d$\overline{n}_{+}/\text{d}t~{}=~{}\zeta_{+}~{}\times~{}1/(2n_{+})$ for $n_{+}\gg 1$. Together with the determination of a lower limit for $n_{+}$ based on the continuous Stern-Gerlach effect ulmer2011prlobservation we obtain an upper limit for the electric field spectral density of $S_{E}(\omega_{+})\leq 7.5^{+3.4}_{-2.8}\times 10^{-20}\,\text{V}^{2}\text{m}^{-2}\text{Hz}^{-1}$. The absolute resolution of our axial frequency measurements is limited by environmental variations of temperature, cryoliquid levels, and pressure, which impose uncertainties on the determination of both the cyclotron quantum number $n_{+}$ as well as the transition rate $\zeta_{+}$.

Nevertheless, our upper limit for $S_{E}(\omega_{+})$ is far below the results reported by cryogenic Paul trap labaziewicz2008_1 ; labaziewicz2008_2 ; wang2010 ; chiaverini2014 ; niedermayr2014 ; bruzewicz2015 ; sedlacek2018 and room temperature Penning-trap experiments goodwin2016 ; sawyer2014 ; stutter2018 . The current best limits extracted from those experiments are $S_{E}(\omega)=2.4\times 10^{-15}$ $\text{V}^{2}\text{m}^{-2}\text{Hz}^{-1}$ niedermayr2014 ; brownnutt2015 and $S_{E}(\omega)=8\times 10^{-16}\,\text{V}^{2}\text{m}^{-2}\text{Hz}^{-1}$ goodwin2016 ; stutter2018 , respectively. Fig. 3 a) displays the commonly used scaled electric field noise $\omega S_{E}\left(\omega\right)$ which accounts for the $1/\omega-$dependence of the heating rate turchette2000 ; brownnutt2015 . Our result $\omega S_{E}\left(\omega\right)\leq 8.8^{+4.0}_{-3.2}\times 10^{-12}\,\text{V}^{2}\text{m}^{-2}$ sets an upper limit which is a factor of 1800 niedermayr2014 lower than the best reported Paul trap heating rates and a factor of 230 lower than the best Penning trap goodwin2016 . Fig. 3 b) plots the heating rate $\text{d}\overline{n}/\text{d}t$ for various experiments, which is in our case below $0.1\,\text{h}^{-1}$. The corresponding energy increase $\text{d}E/\text{d}t$, plotted in Fig. 3 c), is on the order of peV/s, demonstrating to our knowledge the highest energy stability of a particle in any ion trap experiment.

To further investigate the residual drive mechanism, we measure transition rates $\zeta_{+}(\rho_{-})$ as a function of the particle’s magnetron radius $\rho_{-}$, thereby changing the trapping field at the particle position. We excite the magnetron mode and record series of axial frequency sequences $\Omega_{k}(\nu_{z},\rho_{-})$ for in total 7 different magnetron radii, thereby tracing a radial range of $6\,\mu\text{m}\leq\rho_{-}\leq 65\,\mu$m.

The results of these measurements are displayed in Fig. 4. In Fig. 4 a) we show the measured axial frequency fluctuation $\sigma_{\nu_{z}}\left(\rho_{-},\tau=250\,\text{s}\right)$. For the data points displayed in Fig. 4 b), we analyze the transition rate $\zeta_{+}(\rho_{-})$ of each dataset $\Omega_{k}(\nu_{z},\rho_{-})$ and determine the spectral density $S_{V}\left(\omega_{+}\right)$ of an equivalent effective voltage noise source present on each trap electrode:

[TABLE]

where $\Lambda\left(\rho,z\right)$ describes the relation between the electric field at the particle position $\vec{x}=\left(\rho,z\right)$ and the potential $V_{n}$ created by the $n$-th electrode:

[TABLE]

for low cyclotron energies, $\rho\approx\rho_{-}$. The linear increase of $\sigma_{\nu_{z}}\left(\tau\right)\propto\rho_{-}$ observed in Fig. 4 a) reflects a quadratic increase of transition rates $\zeta_{+}\propto\rho_{-}^{2}$ (Eq. 2). This is expected from Eq. (3, 4, 5), assuming electrode voltage noise $S_{V}$ as the dominant source of electric field fluctuations. We obtain $S_{V}=225(54)\,\text{pV}\,\text{Hz}^{-1/2}$. Anomalous heating reported from Paul traps turchette2000 ; brownnutt2015 scales with $d^{-4}$, $d$ denoting the electrode-ion-distance. Since the variation of $d$ is small ($\Delta d/d=1/60$) for the considered magnetron radii, anomalous heating would result in a nearly constant electric field noise spectral density. Since a clear increase is observed in $\zeta_{+}$, anomalous heating is ruled out as the dominant heating mechanism. Its effect is constrained to be below $S_{E}\left(\omega_{+}\right)\leq 7.5(3.4)\times 10^{-20}\,\text{V}^{2}\text{m}^{-2}\text{Hz}^{-1}$.

In order to investigate contributions to $S_{V}$ we consider the experimental setup depicted in Fig. 1. The effective parallel resistance of the axial detection system at the cyclotron frequency contributes about $1.5\,\text{pV}\,\text{Hz}^{-1/2}$. The Johnson noise of the electrode RC-filters is below $1\,\text{pV}\,\text{Hz}^{-1/2}$, the electrode Johnson noise is on the order of $10^{-3}\,\text{pV}\,\text{Hz}^{-1/2}$. None of these mechanisms can explain the observed voltage fluctuations. Field fluctuations arising from blackbody radiation are estimated to be $\omega_{+}\times S_{E}^{(\text{BB})}\approx 6\times 10^{-14}\,\text{V}^{2}\text{m}^{-2}$ henkel1999 ; brownnutt2015 , which is two orders of magnitude lower than our limit of $\omega S_{E}\left(\omega\right)\leq 8.8^{+4.0}_{-3.2}\times 10^{-12}\,\text{V}^{2}\text{m}^{-2}$. A trapped ion polarizes neutral background gas atoms and thereby induces collisions described by the Langevin rate $\gamma$, which is proportional to the background gas density wineland1998 ; chiaverini2014 . From our antiproton lifetime measurement sellner2017 we derived upper limits for the partial pressure of hydrogen $p_{\text{upper},H}<1.2\times 10^{-18}\,$mbar and helium $p_{\text{upper},He}<2.7\times 10^{-18}$ mbar leading to $\zeta_{+}<4\times 10^{-9}\text{s}^{-1}$. Voltage supply (UM1-14) noise at $\nu_{+}$ is ruled out by independent measurements. Therefore we assume parasitic coupling of stray EMI noise onto the trap electrodes to be the dominant source of electric field fluctuations in our trap. A further improvement to achieve even lower heating rates which will enhance the sensitivity of our experiment will be subject of future experimental studies.

We acknowledge financial support of RIKEN Pioneering Project Funding, RIKEN FPR program, RIKEN JRA program, the Max-Planck Society, the CERN fellowship program, the EU (Marie Sklodowska-Curie grant agreement No. 721559) and the KAS/BMBF PhD fellowship program. We acknowledge support from CERN, in particular from the AD operation team.