* [50] T. Ikuta and A. Munemasa, Complex Hadamard matrices attached to a 3-class nonsymmetric association scheme, _Graphs Combin._ 35 (2019), 1293–1304, doi:10.1007/s00373-019-02044-z, https://doi.org/10.1007/s00373-019-02044-z.
  * [51] T. Ito, Designs in a coset geometry: Delsarte theory revisited, _European J. Combin._ 25 (2004), 229–238, doi:10.1016/S0195-6698(03)00102-1, https://doi.org/10.1016/S0195-6698(03)00102-1.
  * [52] T. Ito, Terwilliger algebras and the Weisfeiler-Leman stabilization, in: _Algebraic combinatorics and the Monster group_ , Cambridge Univ. Press, Cambridge, volume 487 of _London Math. Soc. Lecture Note Ser._ , pp. 492–517, 2024.
  * [53] G. A. Jones, Paley and the Paley graphs, in: _Isomorphisms, symmetry and computations in algebraic graph theory_ , Springer, Cham, volume 305 of _Springer Proc. Math. Stat._ , pp. 155–183, [2020] ©2020, doi:10.1007/978-3-030-32808-5“˙5, https://doi.org/10.1007/978-3-030-32808-5_5.
  * [54] H. Kajiura, M. Matsumoto and T. Okuda, Non-existence and construction of pre-difference sets, and equi-distributed subsets in association schemes, _Graphs Combin._ 37 (2021), 1531–1544, doi:10.1007/s00373-021-02279-9, https://doi.org/10.1007/s00373-021-02279-9.
  * [55] H. Kharaghani, T. Pender and S. Suda, Quasi-balanced weighing matrices, signed strongly regular graphs and association schemes, _Finite Fields Appl._ 83 (2022), Paper No. 102065, 25, doi:10.1016/j.ffa.2022.102065, https://doi.org/10.1016/j.ffa.2022.102065.
  * [56] H. Kharaghani, S. Sasani and S. Suda, A strongly regular decomposition of the complete graph and its association scheme, _Finite Fields Appl._ 48 (2017), 356–370, doi:10.1016/j.ffa.2017.08.012, https://doi.org/10.1016/j.ffa.2017.08.012.
  * [57] B. G. Kodalen and W. J. Martin, On the connectivity of graphs in association schemes, _Electron. J. Combin._ 24 (2017), Paper No. 4.39, 17.
  * [58] H. Kurihara, Character tables of association schemes based on attenuated spaces, _Ann. Comb._ 17 (2013), 525–541, doi:10.1007/s00026-013-0194-5, https://doi.org/10.1007/s00026-013-0194-5.
  * [59] Z. Li, S. Gao and H. Guo, Commutative weakly distance-regular digraphs of circle with fixed length, _Ars Combin._ 101 (2011), 301–307.
  * [60] W. Liu and J. Lin, Walk regular digraphs, _Ars Combin._ 95 (2010), 97–102.
  * [61] G. Luo and X. Cao, A construction of linear codes and strongly regular graphs from $q$-polynomials, _Discrete Math._ 340 (2017), 2262–2274, doi:10.1016/j.disc.2017.04.016, https://doi.org/10.1016/j.disc.2017.04.016.
  * [62] D. S. Lyubshin and S. V. Savchenko, Cayley digraphs with normal adjacency matrices, _Discrete Math._ 309 (2009), 4343–4348, doi:10.1016/j.disc.2007.03.086, https://doi.org/10.1016/j.disc.2007.03.086.
  * [63] W. J. Martin and H. Tanaka, Commutative association schemes, _European J. Combin._ 30 (2009), 1497–1525, doi:10.1016/j.ejc.2008.11.001, https://doi.org/10.1016/j.ejc.2008.11.001.
  * [64] L. Martínez, M. Asunción García, L. Legarreta and I. Malaina, Cyclotomic association schemes of broad classes and applications to the construction of combinatorial structures, _Art Discrete Appl. Math._ 5 (2022), Paper No. 3.02, 14, doi:10.26493/2590-9770.1436.0f3, https://doi.org/10.26493/2590-9770.1436.0f3.
  * [65] C. Meyer, _Matrix analysis and applied linear algebra_ , Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000, doi:10.1137/1.9780898719512, with 1 CD-ROM (Windows, Macintosh and UNIX) and a solutions manual (iv+171 pp.), https://doi.org/10.1137/1.9780898719512.
  * [66] G. Monzillo and A. Siciliano, Reconstructing a generalized quadrangle from the Penttila-Williford $4$-class association scheme, _European J. Combin._ 99 (2022), Paper No. 103425, 11, doi:10.1016/j.ejc.2021.103425, https://doi.org/10.1016/j.ejc.2021.103425.
  * [67] H. Nozaki, Polynomial properties on large symmetric association schemes, _Ann. Comb._ 20 (2016), 379–386, doi:10.1007/s00026-016-0300-6, https://doi.org/10.1007/s00026-016-0300-6.
  * [68] G. R. Omidi, A spectral excess theorem for normal digraphs, _J. Algebraic Combin._ 42 (2015), 537–554, doi:10.1007/s10801-015-0590-5, https://doi.org/10.1007/s10801-015-0590-5.
  * [69] S. Penjić, _Algebraic characterizations of distance-regular graphs_ , Master’s thesis, University of Sarajevo, 2013.
  * [70] A. Pott, Y. Tan, T. Feng and S. Ling, Association schemes arising from bent functions, _Des. Codes Cryptogr._ 59 (2011), 319–331, doi:10.1007/s10623-010-9463-z, https://doi.org/10.1007/s10623-010-9463-z.
  * [71] J. Qu and J. Lei, Three-class association schemes from partial geometric designs, _Discrete Math._ 344 (2021), Paper No. 112290, 12, doi:10.1016/j.disc.2021.112290, https://doi.org/10.1016/j.disc.2021.112290.
  * [72] R. J. Riebeek, _Computations in association schemes_ , Thesis Publishers, Amsterdam, 1998, dissertation, Technische Universiteit Eindhoven, Eindhoven, 1998\.
  * [73] J. J. Seidel, Strongly regular graphs, in: _Surveys in combinatorics (Proc. Seventh British Combinatorial Conf., Cambridge, 1979)_ , Cambridge Univ. Press, Cambridge-New York, volume 38 of _London Math. Soc. Lecture Note Ser._ , 1979 pp. 157–180.
  * [74] A. Sinak, Construction of minimal linear codes with few weights from weakly regular plateaued functions, _Turkish J. Math._ 46 (2022), 953–972, doi:10.55730/1300-0098.3135, https://doi.org/10.55730/1300-0098.3135.
  * [75] S. Suda, A two-fold cover of strongly regular graphs with spreads and association schemes of class five, _Des. Codes Cryptogr._ 78 (2016), 463–471, doi:10.1007/s10623-014-0012-z, https://doi.org/10.1007/s10623-014-0012-z.
  * [76] H. Suzuki, Thin weakly distance-regular digraphs, _J. Combin. Theory Ser. B_ 92 (2004), 69–83, doi:10.1016/j.jctb.2004.04.004, https://doi.org/10.1016/j.jctb.2004.04.004.
  * [77] H. Tanaka, A four-class subscheme of the association scheme coming from the action of ${\rm PGL}(2,4^{f})$, _European J. Combin._ 23 (2002), 121–129, doi:10.1006/eujc.2001.0511, https://doi.org/10.1006/eujc.2001.0511.
  * [78] L. Y. Tsiovkina, Covers of complete graphs and related association schemes, _J. Combin. Theory Ser. A_ 191 (2022), Paper No. 105646, 33, doi:10.1016/j.jcta.2022.105646, https://doi.org/10.1016/j.jcta.2022.105646.
  * [79] E. R. van Dam, Three-class association schemes, _J. Algebraic Combin._ 10 (1999), 69–107, doi:10.1023/A:1018628204156, https://doi.org/10.1023/A:1018628204156.
  * [80] E. R. van Dam and M. Muzychuk, Some implications on amorphic association schemes, _J. Combin. Theory Ser. A_ 117 (2010), 111–127, doi:10.1016/j.jcta.2009.03.018, https://doi.org/10.1016/j.jcta.2009.03.018.
  * [81] E. R. van Dam and G. R. Omidi, Directed strongly walk-regular graphs, _J. Algebraic Combin._ 47 (2018), 623–639, doi:10.1007/s10801-017-0789-8, https://doi.org/10.1007/s10801-017-0789-8.
  * [82] J. Vidali, Computing distance-regular graph and association scheme parameters in SageMath with sage-drg, _Sém. Lothar. Combin._ 82B (2020), Art. 105, 8.
  * [83] W. D. Wallis, A. P. Street and J. S. Wallis, _Combinatorics: Room squares, sum-free sets, Hadamard matrices_ , Lecture Notes in Mathematics, Vol. 292, Springer-Verlag, Berlin-New York, 1972.
  * [84] K. Wang, Commutative weakly distance-regular digraphs of girth 2, _European J. Combin._ 25 (2004), 363–375, doi:10.1016/j.ejc.2003.09.009, https://doi.org/10.1016/j.ejc.2003.09.009.
  * [85] K. Wang, A family of weakly distance-regular digraphs of girth 2, _Discrete Math._ 306 (2006), 2811–2815, doi:10.1016/j.disc.2006.05.030, https://doi.org/10.1016/j.disc.2006.05.030.
  * [86] K. Wang and Y.-q. Feng, Deza digraphs, _European J. Combin._ 27 (2006), 995–1004, doi:10.1016/j.ejc.2005.04.001, https://doi.org/10.1016/j.ejc.2005.04.001.
  * [87] K. Wang and H. Suzuki, Weakly distance-regular digraphs, volume 264, pp. 225–236, 2003, doi:10.1016/S0012-365X(02)00562-9, the 2000 $\rm Com^{2}MaC$ Conference on Association Schemes, Codes and Designs (Pohang), https://doi.org/10.1016/S0012-365X(02)00562-9.
  * [88] Y. Wang and R. Tao, Constructions of linear codes with small hulls from association schemes, _Adv. Math. Commun._ 16 (2022), 349–364, doi:10.3934/amc.2020114, https://doi.org/10.3934/amc.2020114.
  * [89] Y. Wu, J. Y. Hyun and Y. Lee, Characterization of $p$-ary functions in terms of association schemes and its applications, _J. Combin. Theory Ser. A_ 187 (2022), Paper No. 105576, 22, doi:10.1016/j.jcta.2021.105576, https://doi.org/10.1016/j.jcta.2021.105576.
  * [90] T.-T. Xia, Y.-Y. Tan, X. Liang and J. H. Koolen, On association schemes generated by a relation or an idempotent, _Linear Algebra Appl._ 670 (2023), 1–18, doi:10.1016/j.laa.2023.03.029, https://doi.org/10.1016/j.laa.2023.03.029.
  * [91] B. Xu, Characters of table algebras and applications to association schemes, _J. Combin. Theory Ser. A_ 115 (2008), 1358–1373, doi:10.1016/j.jcta.2008.02.005, https://doi.org/10.1016/j.jcta.2008.02.005.
  * [92] B. Xu, Partial geometric designs arising from association schemes, _European J. Combin._ 109 (2023), Paper No. 103658, 16, doi:10.1016/j.ejc.2022.103658, https://doi.org/10.1016/j.ejc.2022.103658.
  * [93] Y. Yang, B. Lv and K. Wang, Weakly distance-regular digraphs of valency three, I, _Electron. J. Combin._ 23 (2016), Paper 2.12, 14, doi:10.37236/5413, https://doi.org/10.37236/5413.
  * [94] Y. Yang, B. Lv and K. Wang, Weakly distance-regular digraphs of valency three, II, _J. Combin. Theory Ser. A_ 160 (2018), 288–315, doi:10.1016/j.jcta.2018.07.001, https://doi.org/10.1016/j.jcta.2018.07.001.
  * [95] Y. Yang, B. Lv and K. Wang, Quasi-thin weakly distance-regular digraphs, _J. Algebraic Combin._ 51 (2020), 19–50, doi:10.1007/s10801-018-0866-7, https://doi.org/10.1007/s10801-018-0866-7.
  * [96] Y. Yang and K. Wang, Thick weakly distance-regular digraphs, _Graphs Combin._ 38 (2022), Paper No. 37, 42, doi:10.1007/s00373-021-02450-2, https://doi.org/10.1007/s00373-021-02450-2.
  * [97] M. Yoshikawa, On association schemes of finite exponent, _European J. Combin._ 51 (2016), 433–442, doi:10.1016/j.ejc.2015.07.019, https://doi.org/10.1016/j.ejc.2015.07.019.
  * [98] Q. Zeng, Y. Yang and K. Wang, $P$-polynomial weakly distance-regular digraphs, _Electron. J. Combin._ 30 (2023), Paper No. 3.3, 13, doi:10.37236/11798, https://doi.org/10.37236/11798.
  * [99] P.-H. Zieschang, _An algebraic approach to association schemes_ , volume 1628 of _Lecture Notes in Mathematics_ , Springer-Verlag, Berlin, 1996, doi:10.1007/BFb0097032, https://doi.org/10.1007/BFb0097032.


# Trapped electrons and ions as particle detectors

Daniel Carney  Physics Division, Lawrence Berkeley National Laboratory,
Berkeley, CA  Hartmut Häffner  Department of Physics, University of
California, Berkeley, CA  David C. Moore  Wright Laboratory, Yale University,
New Haven, CT  Jacob M. Taylor  Joint Quantum Institute, National Institute of
Standards and Technology, Gaithersburg, MD  Joint Center for Quantum
Information and Computer Science, University of Maryland, College Park, MD

###### Abstract

Electrons and ions trapped with electromagnetic fields have long served as
important high-precision metrological instruments, and more recently have also
been proposed as a platform for quantum information processing. Here we point
out that these systems can also be used as highly sensitive detectors of
passing charged particles, due to the combination of their extreme charge-to-
mass ratio and low-noise quantum readout and control. In particular, these
systems can be used to detect energy depositions many orders of magnitude
below typical ionization scales. As illustrations, we suggest some
applications in particle physics. We outline a non-destructive time-of-flight
measurement capable of sub-eV energy resolution for slowly moving, collimated
particles. We also show that current devices can be used to provide
competitive sensitivity to models where ambient dark matter particles carry
small electric millicharges $\ll e$. Our calculations may also be useful in
the characterization of noise in quantum computers coming from backgrounds of
charged particles.

Detection of moving electrically charged particles is a ubiquitous problem in
physics. Most charged particle detectors are based on ionization or
scintillation of a detector medium, leading to detection thresholds on the
order of eV. In the past few decades, semiconducting and superconducting
devices based on lower energy excitations have been proposed with thresholds
down to the meV range Hertel  _et al._ (2019); Griffin  _et al._ (2018);
Hochberg  _et al._ (2019, 2016). However, for particles that interact via
long-range forces like the Coulomb force, the predominant signal may occur in
the center-of-mass or other collective mode of an object, which can have
extremely low energies. As an example, several of us have recently proposed
Carney  _et al._ (2020) and demonstrated Monteiro  _et al._ (2020) the direct
monitoring of impulses delivered to the motion of a macroscopic mass as a
pathway toward ultra-low threshold detection in optomechanical systems Carney
_et al._ (2021).

Here we point out that analogous techniques can be applied to electrical
interactions of charged particles with single trapped ions and electrons
Wineland  _et al._ (1973); Paul (1990); Dehmelt (1990). The achievable
detection thresholds can be below the $\mu$eV scale. Since the electric force
is long-ranged, even a single electron can offer substantial interaction
cross-sections at sufficiently low detector thresholds, leading to remarkable
detection reach. We focus primarily on electrons, since their extreme charge-
to-mass ratio (at least $10^{3}$ higher than any ion) means that they set a
fundamental sensitivity floor achievable with the known particles of nature.
However, the basic results carry over to ions by simply rescaling the electron
mass $m_{e}\to m_{\rm ion}$. We also note that the same results would apply to
any other long-range forces coupling to these systems, not just
electromagnetism.

Electrons confined in Penning traps Wineland  _et al._ (1973); Dehmelt (1990)
have a long history in metrology, notably in precision measurement of the
electron magnetic moment Hanneke  _et al._ (2008, 2011). Electrons in both
Penning and radio-frequency (rf) Paul traps have also been proposed as qubit
systems which could enable very fast gate times in quantum computing
Ciaramicoli  _et al._ (2003); Marzoli  _et al._ (2009); Daniilidis  _et al._
(2013), with single-electron trapping recently achieved experimentally in a
room-temperature rf Paul trap Matthiesen  _et al._ (2021). This offers the
exciting possibility of utilizing quantum information techniques to enhance
the sensitivity of the device Giovannetti  _et al._ (2004); Riedel (2013).

As an example highlighting the extreme sensitivity of these devices, we
consider the application of a single-electron sensor to searches for dark
matter candidates with small electric charge $q\ll e$ McDermott  _et al._
(2011); Dolgov  _et al._ (2013); Boddy  _et al._ (2018); Barkana  _et al._
(2018); Emken  _et al._ (2019a). Such “millicharged” dark matter (mDM) can
arise in models with weak mixing between the photon and hidden-sector forces
Holdom (1986), and has received significant recent interest to possibly
explain Barkana (2018); Berlin  _et al._ (2018); Barkana  _et al._ (2018) an
anomalous measurement in the 21 cm emission of cosmological hydrogen by the
EDGES collaboration Bowman  _et al._ (2018). We show that a single electron in
a typical trap configuration, utilizing only ground-state cooling and single-
quantum (“single-phonon”) readout, would be capable of searching novel
parameter space in viable scenarios in which $\sim 1\%$ of dark matter
consists particles with charges $\sim 1\%$ of the electron charge.

_Trapped electrons as impulse detectors._ Both Penning and Paul traps produce
electron (or ion) motion which is well-described by harmonic oscillations.
Penning traps use a static, axial magnetic field combined with a static
electric quadrupole potential to trap the electrons. Paul traps instead do not
use a magnetic field but rather a combination of a static electric quadrupole
potential and rapidly varying drive field. Time-averaging over the drive
produces dynamical stability. Electron motion in the Penning trap is
characterized by the axial, cyclotron, and magnetron frequencies
($\omega_{z},\omega_{c},\omega_{m}$); Paul traps are characterized by three
frequencies $\omega_{z},\omega_{x}\approx\omega_{y}$ as well as the
micromotion drive frequency $\Omega$. In either case, the frequencies are
tunable; in Penning traps $\omega_{m}<\omega_{z}<\omega_{c}$.

We propose to exploit quantum control and readout of the electron motion as a
sensitive impulse detector. Consider cooling the motion on an axis of our
choice, with frequency $\omega$, to near the ground state $\ket{0}$. Suppose a
rapid impulse is then delivered to the electron, imparting energy $\Delta E$.
If $\Delta E\gtrsim\omega$, energy levels above the ground state will be
populated. If some event, for example a particle collision, transfers momentum
$\Delta p$ to the electron, this will impart energy $\Delta E=\Delta
p^{2}/2m_{e}$. Thus, the fundamental detection threshold of a single-electron
detector is of order

$\Delta p_{\rm SQL}=\sqrt{2\hbar m_{e}\omega}.$ (1)

This is precisely the “standard quantum limit” (SQL) for momentum detection
using a harmonic oscillator Mozyrsky  _et al._ (2004); Clerk (2004); Carney
_et al._ (2021). In terms of energy transfer, (1) translates to a detection
threshold simply given by the trap frequency. For electrons trapped in the 10
MHz-10 GHz range, this corresponds to threshold energies around neV-$\mu$eV,
far below ionization levels. With ions, these frequencies can be down to the
10s of kHz (for example Poulsen and Drewsen (2012)), which would produce
similar momentum thresholds (1) as the increased mass competes with the
decreased frequency.

The essential duty cycle of a detector of this nature would proceed in three
steps. First, we prepare the motion near the ground state $\ket{0}$. The
system then evolves for some time $\Delta t$. After $\Delta t$ has passed, we
estimate the change in momentum. A detector “click” occurs if this momentum
change is greater than the threshold (1) 111We note that a detector of this
type can be calibrated by a variety of methods. The most straightforward would
be to use laser pulses of known shape Monteiro  _et al._ (2020); another would
be to use a charged particle source (possibly with retarding potential to
produce a slow beam).

We remark that the fundamental assumption we make in what follows is simply
that we can measure changes in momentum at or near the SQL in a bandwidth
similar to $\omega$. Thus, both the preparation and measurement elements can
be relaxed to include higher temperature initial states and/or less effective
measurement, so long as near SQL-limited momentum measurement is achieved.
This has been demonstrated, for example, using both non-demolition Hanneke
_et al._ (2011) and projective Johnson  _et al._ (2015) measurement of the
oscillator level. In addition to achieving measurement errors at SQL levels,
the primary experimental requirement is that the heating rate (in phonons per
second $\Gamma_{Q}$) should be small. Defining the quantum quality factor
$Q=\omega/\Gamma_{Q}$, our duty cycle is bounded by $\Delta t\lesssim
Q/\omega$, so that we will not obtain spurious clicks due to heating.

We substantiate these generic requirements by examining the cryogenic electron
Penning trap in Hanneke  _et al._ (2011), in which the $150~{}{\rm GHz}$
cyclotron mode was monitored via quantum non-demolition measurements. This
device satisfies all of the above requirements, and has $Q$ sufficiently high
such that the effective “duty cycle” defined above can be on the order of
days. Scaling this setup to lower-frequency modes (to lower the detection
threshold (1)) should be feasible. In a typical cryogenic ion trap with a
$d\sim 50~{}{\rm\mu m}$ spacing between the ion and the electrodes, the phonon
heating rates for modes around $1~{}{\rm MHz}$ are of order a few phonons per
second Chiaverini and Sage (2014); Sedlacek  _et al._ (2018), with a
corresponding quantum $Q\gtrsim 10^{5}$. These rates should scale roughly as
$1/d^{4}$ Brown  _et al._ (2021) and linearly in $m_{e}/m_{\rm ion}$, so to
achieve a rate of a few phonons per day would require building a trap with
$d=\mathcal{O}({\rm cm})$. Typical dimensions of Penning traps approach this
size and indeed the BASE collaboration finds heating rates of 2 quanta/day for
the 18 MHz cyclotron motion of an anti-proton stored in a cryogenic Penning
trap with $d=1.8$ mm Borchert  _et al._ (2019).

_Charged particle detection_. We now specialize to the case of a passing
electrically charged particle $\chi$ of mass $m_{\chi}$ and charge $q_{\chi}$.
The Coulomb force between this and the electron has magnitude
$F=\lambda/r^{2}$. We will use “natural” units $\hbar=c=e=1$, so
$\lambda=\alpha q_{\chi}$ in terms of the fine structure constant $\alpha$.

Let $b$ denote the impact parameter between $\chi$ and the electron and $v$
the relative initial velocity (see Figure 1). We can define a “fly-by time”
$\tau=b/v$, during which the majority of the impulse is transferred. The
simplest case, which we will assume here, is that $\tau\ll\omega_{i}^{-1}$ for
all the trap frequencies (including, in the Paul trap, the micromotion drive
frequency $\Omega$ 222When $\tau$ is larger than $\Omega^{-1}$ but smaller
than the inverse trapping frequencies $\omega_{i}^{-1}$ there may be important
corrections to scattering due to the micromotion of the system, but we will
assume Mathieu parameter $q_{i}\ll 1$, consistent with current experiments,
and neglect this correction in what follows. We also note that axial detection
will be insensitive to micromotion.). With the typical velocities and impact
parameters of interest here, this should be valid for modes with frequency
below around $100~{}{\rm GHz}$. In this limit, the electron is essentially a
free particle during the collision, and we have simple Rutherford scattering.

Figure 1: Kinematics of a scattering event in a Paul trap. The charged
particle $\chi$ impinges on the trapped electron with impact parameter $b$ and
velocity $v$.

The events with the lowest momentum transfer will be at large impact
parameter. This will produce small-angle scattering, for which the momentum
transfer is

$\Delta p=\frac{\lambda}{bv}.$ (2)

We note that (2) is true even for relativistic velocities $v\approx c$ 333The
1st edition of Jackson’s _Electrodynamics_ had a nice discussion of this (see
his equation 13.1), but this material seems to have been removed in subsequent
editions., although for sufficiently low velocity it breaks down because the
incoming particle will not have enough kinetic energy.

We can combine (1) and (2) to estimate an effective cross-sectional area,
$\sigma_{\rm eff}$, through which a passing particle will be detectable above
threshold. Setting these equal and solving for the impact parameter gives

$\displaystyle\begin{split}\sigma_{\rm eff}&=4\pi b^{2}=4\pi
x_{0}^{2}\frac{\lambda^{2}}{v^{2}}\\\ &\approx 40~{}{\rm
nm}^{2}\times\frac{q_{\chi}^{2}}{v^{2}}\times\left(\frac{1~{}{\rm
GHz}}{\omega}\right),\end{split}$ (3)

where $x_{0}^{2}=1/2m_{\rm e}\omega$ is the size of the ground-state
wavefunction of the electron. These cross-sections are sufficiently large to
allow even a single trapped ion or electron to detect interesting fluxes of
charged particles.

These estimates assume that the momentum transfer due to scattering is
entirely delivered to the mode of interest (say, the axial mode). For the
Coulomb interaction, this will be the case only if the particle $\chi$ is
incoming with velocity transverse to this axis. Thus if we know the direction
of the incoming particles, this alignment is the optimum configuration. If we
are instead trying to detect particles with unknown or random incoming
directions, we will be sensitive only to a fraction of incoming particles

$f_{A}(\Delta p)=\sqrt{1-\Delta p_{\rm th}^{2}/\Delta p^{2}}.$ (4)

This natural (and tunable) anisotropy enables directional sensitivity of the
detector and is thus a feature in applications like dark matter searches.

As an illustration, Figure 2 shows the expected sensitivity for a trapped
electron or ion to a given flux of background electrons. These could could
arise for example from cosmogenic or radiogenic particles (e.g. the $\sim
10^{3}~{}{\rm cm}^{-2}{\rm day}^{-1}$ astrophysical muon flux) hitting the
trap and causing showers of slow electrons. Alternatively, local sources of
radioactivity may produce secondary low-energy particles at non-negligible
fluxes, including secondary electrons or slow-moving recoiling daughter ions
from decays on electrode surfaces. While such radioactivity has been
considered as a possible source of errors in superconducting qubits
Vepsäläinen  _et al._ (2020); McEwen  _et al._ (2021), here we see that this
may also eventually be a consideration for trapped electron or ion based
systems.

One possible application of such a detector could be to time-of-flight
measurements of slowly moving charges in order to measure their kinetic energy
Jerkins  _et al._ (2010); Steinbrink  _et al._ (2013). For example, one could
use a retarding potential to slow electrons produced in nuclear reactions into
a collimated beam. To resolve the energy of particles in the beam, one could
use a coincidence measurement with a pair of trapped-charged devices spaced at
distance $L$. The single-shot measurement time to resolve an SQL fluctuation
is a single trap period $\Delta T=2\pi/\omega$ Clerk (2004). A coincidence
measurement would then be able to determine $v$ up to an error of order
$\Delta v=L\Delta T/T^{2}=v^{2}\Delta T/L$. Resolving a $\chi$ particle’s
energy to precision $\Delta E=m_{\chi}v\Delta v$ would thus require a baseline
of order $L=(2E)^{3/2}\Delta T/m^{1/2}\Delta E$. For example, determining the
energy of an $E\sim 1~{}{\rm eV}$ electron at a resolution of $\Delta
E=1~{}{\rm eV}$ could be achieved with a pair of $100~{}{\rm MHz}$ traps
spaced at 10 mm.

Figure 2: Detectable levels of flux of ambient charged particles (here taken
to be electrons), as a function of the ambient particle’s kinetic energy. Each
curve represents a trap with an electron or ion (here we use beryllium Biercuk
_et al._ (2010) to illustrate) with various frequencies. A background of
electrons with the flux shown will produce impulses above threshold (1) at a
rate of 1 event/day.

_Sensitivity to millicharged dark matter (mDM)_. As a possible application of
these ideas, we now consider detection of an ambient background of dark matter
particles with small charges $q_{\chi}\ll 1$. Other DM-electron couplings—for
example, coupling to electron number $L_{e}$ Arvanitaki  _et al._ (2018), or
via a massive mediator Emken  _et al._ (2019b)—would have similar detection
reach, as long as the range of the force is longer than the cross section (3).

Our detection target is thus an approximately homogeneous distribution of mDM
particles, with positive and negative charges in arbitrary proportion. We
assume these all have the same mass $m_{\chi}$. Astrophysical observations
independent of the detailed nature of dark matter indicate that the average
dark matter mass density should be around $0.3~{}{\rm GeV}/{\rm cm}^{3}$ Bovy
and Tremaine (2012), so the number density of mDM particles is

$n_{\chi}=\frac{0.3}{{\rm cm}^{3}}\times f_{\rm q}\times\left(\frac{1~{}{\rm
GeV}}{m_{\chi}}\right).$ (5)

Here $f_{q}\lesssim 4\times 10^{-3}$ Barkana  _et al._ (2018) is the fraction
of DM which is charged. These charges will deliver random impulses on our
detector. Let $f(\mbox{\boldmath$v$})$ denote their velocity distribution. The
number of events, per unit time and momentum transfer, is given by the
Rutherford formula

$\frac{dR}{d\Delta p}=n_{\chi}\frac{2\pi\lambda^{2}}{\Delta p^{3}}\eta(\Delta
p).$ (6)

Here $\eta(\Delta p)=\int_{v_{\rm
min}}^{\infty}d\mbox{\boldmath$v$}f(\mbox{\boldmath$v$})v^{-1}$ parametrizes
effects from the velocity distribution, and $v_{\rm min}=\Delta p/m_{e}$ is
the minimum incoming velocity kinematically necessary to produce a kick
$\Delta p$ to the electron Lin (2019). Integrating (6), multiplied by (4) for
single-axis monitoring, from $\Delta p=\Delta p_{\rm th}$ upwards will produce
the total number of events, per unit time, above the detection threshold
$\Delta p_{\rm th}$.

Figure 3: Projected millicharged dark matter sensitivity with various trap
frequencies and integration times. We consider detection along only a single
trap axis (4). Top plot: high-charge DM thermalized at room temperature [300
K] (or at liquid He temperature [4 K]). Bottom: low-charge DM virialized to
the galaxy. Currently excluded parameter space is shaded: bounds in light blue
come from stellar astrophysics Boehm  _et al._ (2013); Chang  _et al._ (2018),
dark brown from cosmology (BBN and CMB $N_{\rm eff}$ Boehm  _et al._ (2013);
Vogel and Redondo (2014)), dark blue from direct detection experiments Emken
_et al._ (2019b), and light brown from a variety of collider experiments Prinz
_et al._ (1998); Magill  _et al._ (2019); Acciarri  _et al._ (2020); Ball  _et
al._ (2020).

For electrically charged DM, the details of the distribution present at a
terrestrial experiment are poorly understood, since these charges interact
with astrophysical magnetic fields, the atmosphere, etc. Chuzhoy and Kolb
(2009); McDermott  _et al._ (2011); Barkana  _et al._ (2018). In particular,
at sufficiently large charge, mDM may lose substantial energy when transiting
the atmosphere and earth en route to an experiment Emken  _et al._ (2019a);
Pospelov and Ramani (2020); Afek  _et al._ (2020). Such energy loss explains
the lack of direct detection limits above $q_{\chi}\gtrsim 10^{-4}$ Emken  _et
al._ (2019b), because the subsequently slowed mDM will have kinetic energy
below existing dark matter detection thresholds based on ionization Angle  _et
al._ (2011); Aprile  _et al._ (2016); Agnes  _et al._ (2018) or excitation of
electron-hole pairs in Si Abramoff  _et al._ (2019); Aguilar-Arevalo  _et al._
(2019); Amaral  _et al._ (2020). In contrast, the substantially lower
thresholds possible with trapped electrons, as well as the the $v^{-2}$
enhancement in the cross section (3), could enable detection in the currently
unconstrained region of parameter space above the direct detection constraints
around $10^{-5}\lesssim q_{\chi}\lesssim 10^{-1}$.

In Figure 3 we present a pair of example sensitivities in these two different
charge regions. For charges $q\gtrsim 10^{-5}$ at relatively heavy mass
$m_{\chi}\gtrsim 1~{}{\rm GeV}$, DM is expected to be captured and thermalized
into the Earth Pospelov and Ramani (2020). For smaller charges, in contrast,
the DM should be virialized with the galaxy. In both plots we use the highly
conservative number density (5) with $f_{q}=0.4\%$; in particular, the number
density for captured and thermalized mCPs may be orders of magnitude larger
Pospelov and Ramani (2020). Even with these conservative assumptions, we see
that a single electron or ion monitored for a day has substantial detection
reach in novel parameter space.

Substantial improvements in cross-section could be reached with either
multiple charges in a single trap, multiple traps, or a combination, as in
quantum computing. For example, current ion traps can support crystals of
$\gtrsim 10^{2}$ charges in a single trap spaced by their mutual Coulomb
repulsion at $\gtrsim\mu$m distances Jordan  _et al._ (2019). To illustrate
the requirements to achieve sensitivity below the lower edge of cosmological
constraints at $q_{\chi}\lesssim 10^{-10}$, we also show the sensitivity with
$10^{6}$ electrons 444While this may seem like a large number, note that this
is an order of magnitude fewer qubits than required to factor a number
relevant to RSA encryption with Shor’s algorithm gidney2021factor..

We note two key advantages of this type of detection. One is the inherent
directional sensitivity. As described above, the trap has tunable anisotropy
between the different axes, so directional sensitivity is possible for single
events (by measuring multiple axes) or statistically with many events (by
looking for temporal variations in the event rate in a single axis). Second, a
number of traditional backgrounds are suppressed by the effective velocity
discrimination in (3).

Such background sources include the atmospheric muon flux described above,
charged particles originating from plateout of Rn daughters (including from
long-lived species such as 210Pb) on detector surfaces, and natural
radioactivity such as $\beta$s from 40K in detector materials. These
background sources have been studied in detail by rare-event searches for dark
matter and $0\nu\beta\beta$ decay Akerib  _et al._ (2020); Abgrall  _et al._
(2016); Leonard  _et al._ (2017), and can be mitigated by material selection
and handling to produce primary event rates sub-dominant to atmospheric muons.
As described above in the context of qubit errors, production of low energy
secondary charged particles remains an open question. Such secondaries have
been identified in precision experiments sensitive to sources of low-energy
$e^{-}$ such as KATRIN Fränkle  _et al._ (2011, 2020); Aker  _et al._ (2021).
Nonetheless, given the small trap geometries, it is expected that it is
feasible to reject a substantial fraction of such backgrounds by surrounding
the traps with traditional charged particle detectors and vetoing coincidences
with high-energy interactions. Furthermore, the use of multiple electrons and
searching for “tracks” would enable robust rejection of many backgrounds
Carney  _et al._ (2020). For these reasons, and the very low-event data from
existing traps Hanneke  _et al._ (2011); Borchert  _et al._ (2019), we assumed
a background-free search is possible in producing Figure 3, although more
detailed study would be necessary in an experimental realization.

In a very practical sense, our approach is to look for a sudden change in
momentum due to an “event”, when compared to stochastic and quantum noise
backgrounds. The SQL is a reasonable starting point for the limits of this
type of detection. We mention however that advanced techniques (e.g., using
motional Fock states Meekhof  _et al._ (1996), squeezing Burd  _et al._
(2019), or spin entanglement Gilmore  _et al._ (2017)) can push measurements
beyond this limit.

Conclusions. Single electrons have the highest charge-to-mass ratio known in
nature. Like atoms, electrons are universal—each has exactly the same
properties—and can be controlled and read out at the single-phonon level. Thus
single-electron systems provide a fundamental platform for detection of tiny
impulses, well below typical ionization energies. As a simple example, we
showed that just one to a few trapped electrons can be used to perform novel
searches for millicharged dark matter. Beyond the millicharged scenario, these
systems can enable ultra low-threshold detection to any putative new long-
range force coupling to the electron or ion. With trapped ions and electrons
emerging as a platform for quantum information processing, these systems (with
either one or many charges) also provide an opportunity to test quantum-
enhanced protocols for sensing.

###### Acknowledgements.

_Acknowledgements._ We thank Nikita Blinov, Gordan Krnjaic, Zhen Liu, Sam
McDermott, Nadav Outmezguine, Ryan Plestid, Trey Porto, and Steve Rolston for
discussions. DC is supported by the US Department of Energy under contract DE-
AC02-05CH11231 and Quantum Information Science Enabled Discovery (QuantISED)
for High Energy Physics grant KA2401032. HH acknowledges support from AFOSR
through grant FA9550-20-1-0162, the NSF QLCI program through grant number
OMA-2016245. DCM is supported, in part, by NSF Grant PHY-1653232 and the
Heising-Simons Foundation.

## References

  * Hertel  _et al._ (2019) S. A. Hertel, A. Biekert, J. Lin, V. Velan, and D. N. McKinsey, “Direct detection of sub-gev dark matter using a superfluid ${}^{4}\mathrm{He}$ target,” Phys. Rev. D 100, 092007 (2019).
  * Griffin  _et al._ (2018) S. Griffin, S. Knapen, T. Lin, and K. M. Zurek, “Directional Detection of Light Dark Matter with Polar Materials,” Phys. Rev. D 98, 115034 (2018), arXiv:1807.10291 [hep-ph] .
  * Hochberg  _et al._ (2019) Y. Hochberg, I. Charaev, S.-W. Nam, V. Verma, M. Colangelo, and K. K. Berggren, “Detecting sub-gev dark matter with superconducting nanowires,” Phys. Rev. Lett. 123, 151802 (2019).
  * Hochberg  _et al._ (2016) Y. Hochberg, Y. Zhao, and K. M. Zurek, “Superconducting detectors for superlight dark matter,” Phys. Rev. Lett. 116, 011301 (2016).
  * Carney  _et al._ (2020) D. Carney, S. Ghosh, G. Krnjaic, and J. M. Taylor, “Proposal for gravitational direct detection of dark matter,” Phys. Rev. D 102, 072003 (2020), arXiv:1903.00492 [hep-ph] .
  * Monteiro  _et al._ (2020) F. Monteiro, G. Afek, D. Carney, G. Krnjaic, J. Wang, and D. C. Moore, “Search for composite dark matter with optically levitated sensors,” Phys. Rev. Lett. 125, 181102 (2020), arXiv:2007.12067 [hep-ex] .
  * Carney  _et al._ (2021) D. Carney, G. Krnjaic, D. C. Moore, C. Regal, _et al._ , “Mechanical Quantum Sensing in the Search for Dark Matter,” Quantum Sci. Technol. 6, 024002 (2021), arXiv:2008.06074 [physics.ins-det] .
  * Wineland  _et al._ (1973) D. Wineland, P. Ekstrom, and H. Dehmelt, “Monoelectron oscillator,” Phys. Rev. Lett. 31, 1279 (1973).
  * Paul (1990) W. Paul, “Electromagnetic traps for charged and neutral particles,” Rev. Mod. Phys. 62, 531 (1990).
  * Dehmelt (1990) H. Dehmelt, “Experiments on the structure of an individual elementary particle,” Science 247, 539–545 (1990).
  * Hanneke  _et al._ (2008) D. Hanneke, S. Fogwell, and G. Gabrielse, “New measurement of the electron magnetic moment and the fine structure constant,” Phys. Rev. Lett. 100, 120801 (2008).
  * Hanneke  _et al._ (2011) D. Hanneke, S. F. Hoogerheide, and G. Gabrielse, “Cavity control of a single-electron quantum cyclotron: Measuring the electron magnetic moment,” Phys. Rev. A 83, 052122 (2011).
  * Ciaramicoli  _et al._ (2003) G. Ciaramicoli, I. Marzoli, and P. Tombesi, “Scalable quantum processor with trapped electrons,” Phys. Rev. Lett. 91, 017901 (2003).
  * Marzoli  _et al._ (2009) I. Marzoli _et al._ , “Experimental and theoretical challenges for the trapped electron quantum computer,” Journal of Physics B: Atomic, Molecular and Optical Physics 42, 154010 (2009).
  * Daniilidis  _et al._ (2013) N. Daniilidis, D. J. Gorman, L. Tian, and H. Häffner, “Quantum information processing with trapped electrons and superconducting electronics,” New Journal of Physics 15, 073017 (2013).
  * Matthiesen  _et al._ (2021) C. Matthiesen, Q. Yu, J. Guo, A. M. Alonso, and H. Häffner, “Trapping electrons in a room-temperature microwave paul trap,” Phys. Rev. X 11, 011019 (2021).
  * Giovannetti  _et al._ (2004) V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
  * Riedel (2013) C. J. Riedel, “Direct detection of classically undetectable dark matter through quantum decoherence,” Phys. Rev. D 88, 116005 (2013).
  * McDermott  _et al._ (2011) S. D. McDermott, H.-B. Yu, and K. M. Zurek, “Turning off the lights: how dark is dark matter?” Phys. Rev. D 83, 063509 (2011).
  * Dolgov  _et al._ (2013) A. D. Dolgov, S. L. Dubovsky, G. I. Rubtsov, and I. I. Tkachev, “Constraints on millicharged particles from planck data,” Phys. Rev. D 88, 117701 (2013).
  * Boddy  _et al._ (2018) K. K. Boddy, V. Gluscevic, V. Poulin, E. D. Kovetz, M. Kamionkowski, and R. Barkana, “Critical assessment of cmb limits on dark matter-baryon scattering: New treatment of the relative bulk velocity,” Phys. Rev. D 98, 123506 (2018).
  * Barkana  _et al._ (2018) R. Barkana, N. J. Outmezguine, D. Redigolo, and T. Volansky, “Strong constraints on light dark matter interpretation of the edges signal,” Phys. Rev. D 98, 103005 (2018).
  * Emken  _et al._ (2019a) T. Emken, R. Essig, C. Kouvaris, and M. Sholapurkar, “Direct Detection of Strongly Interacting Sub-GeV Dark Matter via Electron Recoils,” JCAP 09, 070 (2019a), arXiv:1905.06348 [hep-ph] .
  * Holdom (1986) B. Holdom, “Two u(1)’s and $\epsilon$ charge shifts,” Phys. Lett. B 166, 196 – 198 (1986).
  * Barkana (2018) R. Barkana, “Possible interaction between baryons and dark-matter particles revealed by the first stars,” Nature 555, 71–74 (2018).
  * Berlin  _et al._ (2018) A. Berlin, D. Hooper, G. Krnjaic, and S. D. McDermott, “Severely constraining dark-matter interpretations of the 21-cm anomaly,” Phys. Rev. Lett. 121, 011102 (2018).
  * Bowman  _et al._ (2018) J. D. Bowman, A. E. E. Rogers, R. A. Monsalve, T. J. Mozdzen, and N. Mahesh, “An absorption profile centred at 78 megahertz in the sky-averaged spectrum,” Nature 555, 67–70 (2018).
  * Mozyrsky  _et al._ (2004) D. Mozyrsky, I. Martin, and M. B. Hastings, “Quantum-limited sensitivity of single-electron-transistor-based displacement detectors,” Phys. Rev. Lett. 92, 018303 (2004).
  * Clerk (2004) A. A. Clerk, “Quantum-limited position detection and amplification: A linear response perspective,” Phys. Rev. B 70, 245306 (2004).
  * Poulsen and Drewsen (2012) G. Poulsen and M. Drewsen, “Adiabatic cooling of a single trapped ion,” (2012), arXiv:1210.4309 .
  * Note (1) We note that a detector of this type can be calibrated by a variety of methods. The most straightforward would be to use laser pulses of known shape Monteiro  _et al._ (2020); another would be to use a charged particle source (possibly with retarding potential to produce a slow beam).
  * Johnson  _et al._ (2015) K. G. Johnson, B. Neyenhuis, J. Mizrahi, J. D. Wong-Campos, and C. Monroe, “Sensing atomic motion from the zero point to room temperature with ultrafast atom interferometry,” Phys. Rev. Lett. 115, 213001 (2015).
  * Chiaverini and Sage (2014) J. Chiaverini and J. M. Sage, “Insensitivity of the rate of ion motional heating to trap-electrode material over a large temperature range,” Phys. Rev. A 89, 012318 (2014), arXiv:1310.4385 .
  * Sedlacek  _et al._ (2018) J. A. Sedlacek, J. Stuart, D. H. Slichter, C. D. Bruzewicz, R. McConnell, J. M. Sage, and J. Chiaverini, “Evidence for multiple mechanisms underlying surface electric-field noise in ion traps,” Phys. Rev. A 98, 63430 (2018), arXiv:1809.07761 .
  * Brown  _et al._ (2021) K. R. Brown, J. Chiaverini, J. M. Sage, and H. Häffner, “Materials challenges for trapped-ion quantum computers,” Nature Reviews Materials , 1–14 (2021).
  * Borchert  _et al._ (2019) M. J. Borchert, P. E. Blessing, J. A. Devlin, J. A. Harrington, T. Higuchi, J. Morgner, C. Smorra, E. Wursten, M. Bohman, M. Wiesinger, A. Mooser, K. Blaum, Y. Matsuda, C. Ospelkaus, W. Quint, J. Walz, Y. Yamazaki, and S. Ulmer, “Measurement of ultralow heating rates of a single antiproton in a cryogenic penning trap,” Phys. Rev. Lett. 122, 043201 (2019).
  * Note (2) When $\tau$ is larger than $\Omega^{-1}$ but smaller than the inverse trapping frequencies $\omega_{i}^{-1}$ there may be important corrections to scattering due to the micromotion of the system, but we will assume Mathieu parameter $q_{i}\ll 1$, consistent with current experiments, and neglect this correction in what follows. We also note that axial detection will be insensitive to micromotion.
  * Note (3) The 1st edition of Jackson’s _Electrodynamics_ had a nice discussion of this (see his equation 13.1), but this material seems to have been removed in subsequent editions.
  * Vepsäläinen  _et al._ (2020) A. Vepsäläinen _et al._ , “Impact of ionizing radiation on superconducting qubit coherence,” Nature 584, 551–556 (2020), arXiv:2001.09190 [quant-ph] .
  * McEwen  _et al._ (2021) M. McEwen _et al._ , “Resolving catastrophic error bursts from cosmic rays in large arrays of superconducting qubits,” (2021), arXiv:2104.05219 [quant-ph] .
  * Jerkins  _et al._ (2010) M. Jerkins, J. R. Klein, J. H. Majors, F. Robicheaux, and M. G. Raizen, “Using cold atoms to measure neutrino mass,” New Journal of Physics 12, 043022 (2010).
  * Steinbrink  _et al._ (2013) N. Steinbrink, V. Hannen, E. L. Martin, R. G. H. Robertson, M. Zacher, and C. Weinheimer, “Neutrino mass sensitivity by mac-e-filter based time-of-flight spectroscopy with the example of katrin,” New Journal of Physics 15, 113020 (2013).
  * Biercuk  _et al._ (2010) M. J. Biercuk, H. Uys, J. W. Britton, A. P. VanDevender, and J. J. Bollinger, “Ultrasensitive detection of force and displacement using trapped ions,” Nature Nanotechnology 5, 646–650 (2010).
  * Arvanitaki  _et al._ (2018) A. Arvanitaki, P. W. Graham, J. M. Hogan, S. Rajendran, and K. Van Tilburg, “Search for light scalar dark matter with atomic gravitational wave detectors,” Phys. Rev. D 97, 075020 (2018), arXiv:1606.04541 [hep-ph] .
  * Emken  _et al._ (2019b) T. Emken, R. Essig, C. Kouvaris, and M. Sholapurkar, “Direct detection of strongly interacting sub-gev dark matter via electron recoils,” Journal of Cosmology and Astroparticle Physics 2019, 070 (2019b).
  * Bovy and Tremaine (2012) J. Bovy and S. Tremaine, “On the local dark matter density,” The Astrophysical Journal 756, 89 (2012).
  * Lin (2019) T. Lin, “Tasi lectures on dark matter models and direct detection,” (2019), arXiv:1904.07915 [hep-ph] .
  * Boehm  _et al._ (2013) C. Boehm, M. J. Dolan, and C. McCabe, “A lower bound on the mass of cold thermal dark matter from planck,” JCAP 2013, 041 (2013).
  * Chang  _et al._ (2018) J. H. Chang, R. Essig, and S. D. McDermott, “Supernova 1987a constraints on sub-gev dark sectors, millicharged particles, the qcd axion, and an axion-like particle,” Journal of High Energy Physics 2018, 1–45 (2018).
  * Vogel and Redondo (2014) H. Vogel and J. Redondo, “Dark radiation constraints on minicharged particles in models with a hidden photon,” Journal of Cosmology and Astroparticle Physics 2014, 029 (2014).
  * Prinz  _et al._ (1998) A. Prinz _et al._ , “Search for millicharged particles at slac,” Phys. Rev. Lett. 81, 1175 (1998).
  * Magill  _et al._ (2019) G. Magill, R. Plestid, M. Pospelov, and Y.-D. Tsai, “Millicharged particles in neutrino experiments,” Phys. Rev. Lett. 122, 071801 (2019).
  * Acciarri  _et al._ (2020) R. Acciarri _et al._ , “Improved limits on millicharged particles using the argoneut experiment at fermilab,” Phys. Rev. Lett. 124, 131801 (2020).
  * Ball  _et al._ (2020) A. Ball _et al._ , “Search for millicharged particles in proton-proton collisions at s= 13 tev,” Phys. Rev. D 102, 032002 (2020).
  * Chuzhoy and Kolb (2009) L. Chuzhoy and E. W. Kolb, “Reopening the window on charged dark matter,” JCAP 2009, 014 (2009).
  * Pospelov and Ramani (2020) M. Pospelov and H. Ramani, “Earth-bound Milli-charge Relics,” (2020), arXiv:2012.03957 [hep-ph] .
  * Afek  _et al._ (2020) G. Afek, F. Monteiro, J. Wang, B. Siegel, S. Ghosh, and D. C. Moore, “Limits on the abundance of millicharged particles bound to matter,” (2020), arXiv:2012.08169 [hep-ex] .
  * Angle  _et al._ (2011) J. Angle _et al._ (XENON10), “A search for light dark matter in XENON10 data,” Phys. Rev. Lett. 107, 051301 (2011), [Erratum: Phys.Rev.Lett. 110, 249901 (2013)], arXiv:1104.3088 [astro-ph.CO] .
  * Aprile  _et al._ (2016) E. Aprile _et al._ (XENON), “Low-mass dark matter search using ionization signals in XENON100,” Phys. Rev. D 94, 092001 (2016), [Erratum: Phys.Rev.D 95, 059901 (2017)], arXiv:1605.06262 [astro-ph.CO] .
  * Agnes  _et al._ (2018) P. Agnes _et al._ (DarkSide), “Constraints on Sub-GeV Dark-Matter–Electron Scattering from the DarkSide-50 Experiment,” Phys. Rev. Lett. 121, 111303 (2018), arXiv:1802.06998 [astro-ph.CO] .
  * Abramoff  _et al._ (2019) O. Abramoff _et al._ (SENSEI), “SENSEI: Direct-Detection Constraints on Sub-GeV Dark Matter from a Shallow Underground Run Using a Prototype Skipper-CCD,” Phys. Rev. Lett. 122, 161801 (2019), arXiv:1901.10478 [hep-ex] .
  * Aguilar-Arevalo  _et al._ (2019) A. Aguilar-Arevalo _et al._ (DAMIC Collaboration), “Constraints on light dark matter particles interacting with electrons from damic at snolab,” Phys. Rev. Lett. 123, 181802 (2019).
  * Amaral  _et al._ (2020) D. W. Amaral _et al._ (SuperCDMS), “Constraints on low-mass, relic dark matter candidates from a surface-operated SuperCDMS single-charge sensitive detector,” Phys. Rev. D 102, 091101 (2020), arXiv:2005.14067 [hep-ex] .
  * Jordan  _et al._ (2019) E. Jordan, K. A. Gilmore, A. Shankar, A. Safavi-Naini, J. G. Bohnet, M. J. Holland, and J. J. Bollinger, “Near ground-state cooling of two-dimensional trapped-ion crystals with more than 100 ions,” Phys. Rev. Lett. 122, 053603 (2019).
  * Note (4) While this may seem like a large number, note that this is an order of magnitude fewer qubits than required to factor a number relevant to RSA encryption with Shor’s algorithm gidney2021factor.
  * Akerib  _et al._ (2020) D. S. Akerib _et al._ (LZ), “The LUX-ZEPLIN (LZ) radioactivity and cleanliness control programs,” Eur. Phys. J. C 80, 1044 (2020), arXiv:2006.02506 [physics.ins-det] .
  * Abgrall  _et al._ (2016) N. Abgrall _et al._ , “The Majorana Demonstrator radioassay program,” Nucl. Instrum. Meth. A 828, 22–36 (2016), arXiv:1601.03779 [physics.ins-det] .
  * Leonard  _et al._ (2017) D. S. Leonard _et al._ , “Trace radioactive impurities in final construction materials for EXO-200,” Nucl. Instrum. Meth. A 871, 169–179 (2017), arXiv:1703.10799 [physics.ins-det] .
  * Fränkle  _et al._ (2011) F. Fränkle _et al._ , “Radon induced background processes in the KATRIN pre-spectrometer,” Astropart. Phys. 35, 128–134 (2011), arXiv:1103.6238 [physics.ins-det] .
  * Fränkle  _et al._ (2020) F. Fränkle _et al._ , “KATRIN background due to surface radioimpurities,” (2020), arXiv:2011.05107 [physics.ins-det] .
  * Aker  _et al._ (2021) M. Aker _et al._ , “First direct neutrino-mass measurement with sub-eV sensitivity,” (2021), arXiv:2105.08533 [hep-ex] .
  * Meekhof  _et al._ (1996) D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, “Generation of nonclassical motional states of a trapped atom,” Phys. Rev. Lett. 76, 1796 (1996).
  * Burd  _et al._ (2019) S. C. Burd, R. Srinivas, J. J. Bollinger, A. C. Wilson, D. J. Wineland, D. Leibfried, D. H. Slichter, and D. T. C. Allcock, “Quantum amplification of mechanical oscillator motion,” Science 364, 1163–1165 (2019).
  * Gilmore  _et al._ (2017) K. A. Gilmore, J. G. Bohnet, B. C. Sawyer, J. W. Britton, and J. J. Bollinger, “Amplitude sensing below the zero-point fluctuations with a two-dimensional trapped-ion mechanical oscillator,” Phys. Rev. Lett. 118, 263602 (2017).