# Determination of hyperfine splittings and Landé $g_{J}$ factors of
$5s~{}^{2}S_{1/2}$ and $5p~{}^{2}P_{1/2,3/2}$ states of 111,113Cd+ for a
microwave frequency standard

J. Z. Han  These authors contributed equally to this work. State Key
Laboratory of Precision Measurement Technology and Instruments, Key Laboratory
of Photon Measurement and Control Technology of Ministry of Education,
Department of Precision Instrument, Tsinghua University, Beijing 100084, China
R. Si  These authors contributed equally to this work. Shanghai EBIT Lab, Key
Laboratory of Nuclear Physics and Ion-beam Application, Institute of Modern
Physics, Department of Nuclear Science and Technology, Fudan University,
Shanghai 200433, China  H. R. Qin  Department of Physics, Tsinghua University,
Beijing 100084, China  N. C. Xin  State Key Laboratory of Precision
Measurement Technology and Instruments, Key Laboratory of Photon Measurement
and Control Technology of Ministry of Education, Department of Precision
Instrument, Tsinghua University, Beijing 100084, China  Y. T. Chen  State Key
Laboratory of Precision Measurement Technology and Instruments, Key Laboratory
of Photon Measurement and Control Technology of Ministry of Education,
Department of Precision Instrument, Tsinghua University, Beijing 100084, China
S. N. Miao  State Key Laboratory of Precision Measurement Technology and
Instruments, Key Laboratory of Photon Measurement and Control Technology of
Ministry of Education, Department of Precision Instrument, Tsinghua
University, Beijing 100084, China  C. Y. Chen<EMAIL_ADDRESS>Shanghai
EBIT Lab, Key Laboratory of Nuclear Physics and Ion-beam Application,
Institute of Modern Physics, Department of Nuclear Science and Technology,
Fudan University, Shanghai 200433, China  J. W. Zhang<EMAIL_ADDRESS>State Key Laboratory of Precision Measurement Technology and Instruments, Key
Laboratory of Photon Measurement and Control Technology of Ministry of
Education, Department of Precision Instrument, Tsinghua University, Beijing
100084, China  L. J. Wang<EMAIL_ADDRESS>State Key Laboratory of
Precision Measurement Technology and Instruments, Key Laboratory of Photon
Measurement and Control Technology of Ministry of Education, Department of
Precision Instrument, Tsinghua University, Beijing 100084, China  Department
of Physics, Tsinghua University, Beijing 100084, China

###### Abstract

Regarding trapped-ion microwave-frequency standards, we report on the
determination of hyperfine splittings and Landé $g_{J}$ factors of 111,113Cd+.
The hyperfine splittings of the $5p~{}^{2}P_{3/2}$ state of 111,113Cd+ ions
were measured using laser-induced fluorescence spectroscopy. The Cd+ ions were
confined in a linear Paul trap and sympathetically cooled by Ca+ ions.
Furthermore, the hyperfine splittings and Landé $g_{J}$ factors of the
$5s~{}^{2}S_{1/2}$ and $5p~{}^{2}P_{1/2,3/2}$ levels of 111,113Cd+ were
calculated with greater accuracy using the multiconfiguration
Dirac–Hartree–Fock scheme. The measured hyperfine splittings and the
Dirac–Hartree–Fock calculation values were cross-checked, thereby further
guaranteeing the reliability of our results. The results provided in this work
can improve the signal-to-noise ratio of the clock transition and the accuracy
of the second-order Zeeman shift correction, and subsequently the stability
and accuracy of the microwave frequency standard based on trapped Cd+ ions.

##  I Introduction

With their improvements in accuracy over time, atomic clocks have played an
important role in practical applications Hinkley  _et al._ (2013); Burt  _et
al._ (2021) and testing the fundamental physics Dzuba  _et al._ (2016); Wcisło
_et al._ (2016); Safronova  _et al._ (2018). Indeed, the microwave-frequency
atomic clock plays a vital role in satellite navigation Bandi  _et al._
(2011), deep space exploration Prestage and Weaver (2007); Burt  _et al._
(2016), and timekeeping Diddams  _et al._ (2004). Among the many clock
proposals, trapped-ion microwave-frequency clocks have attracted wide
attention from researchers because the ions are well isolated from the
external environment in an ion trap. The setup is conducive to improvements in
the transportability of atomic clocksSchwindt  _et al._ (2016); Mulholland
_et al._ (2019a, b); Hoang  _et al._ (2021). Such clocks are also considered
the next generation of practical microwave clocks Schmittberger and Scherer
(2020).

Cadmium ions (Cd+) benefit from a simple and distinct electronic structure,
which is easily controlled, manipulated, and measurable with high precision.
The microwave-frequency standard based on laser-cooled 113Cd+ has achieved an
accuracy of $1.8\times 10^{-14}$ and a short-term stability of $4.2\times
10^{-13}/\sqrt{\tau}$ Miao  _et al._ (2021). The high performance and
potential for miniaturization make this frequency standard suitable in
establishing a ground-based transportable frequency reference for navigation
systems and for comparing atomic clocks between remote sites Zhang  _et al._
(2012); Wang S. G (2013); Miao  _et al._ (2015, 2021). Moreover, it has been
proposed as a means to achieve an ultra-high level of accuracy down to
$10^{-15}$ Han  _et al._ (2021), highlighting the importance of accurately
evaluating systematic frequency shifts.

Optical pumping is a fundamental process in operating a trapped-ion microwave
frequency standard. The optical pumping efficiency determines directly the
signal-to-noise ratio of the “clock signal,” which affects the short-term
stability and measurement accuracy of the ground-state hyperfine splitting
(HFS) for such frequency standards. Realizing optical pumping for the 113Cd+
microwave-frequency standard requires a blueshift in the laser frequency of
the Doppler-cooling transition
$5s~{}^{2}S_{1/2}~{}F=1,~{}m_{F}=1$–$5p~{}^{2}P_{3/2}~{}F=2,~{}m_{F}=2$ to
reach the $5p~{}^{2}P_{3/2}~{}F=1$ hyperfine level. However, there are no
precise measurements available of the HFSs for other excited states Li  _et
al._ (2018). A preliminary measurement of the HFS for the $5p~{}^{2}P_{3/2}$
level of the 113Cd+ ion is approximately 800 MHz Tanaka  _et al._ (1996).
Therefore, to improve the optical pumping efficiency and hence the performance
of the 113Cd+ microwave frequency standard, the HFSs of the $5p~{}^{2}P_{3/2}$
level of the 113Cd+ ion need to be determined with greater accuracy. From the
perspective of atomic structure calculations, the high-precision measurements
of the HFSs for the $5p~{}^{2}P_{3/2}$ level of 111,113Cd+ can also be used
for testing and developing calculation models of the atomic structure.

In a trapped-ion microwave frequency standard, an external magnetic field is
applied to provide the quantization axis to break the degeneracy of the
ground-state magnetic level. Among all the systematic frequency shifts of a
frequency standard, one dominant shift is the second-order Zeeman shift (SOZS)
induced by the external magnetic field Berkeland  _et al._ (1998); Phoonthong
_et al._ (2014); Miao  _et al._ (2021). The precise estimation of this SOZS
and the calibration of the external magnetic field require accurate knowledge
of the ground-state Landé $g_{j}$ factor Han  _et al._ (2019). The external
magnetic field in our latest laser-cooled microwave-frequency standard based
on trapped 113Cd+ ions is approximately $8000$ nT Miao  _et al._ (2021).
However, only two theoretical studies have provided a value of the ground-
state Landé $g_{J}$ factor of Cd+, one giving $2.00286(53)$, calculated using
the relativistic-coupled-cluster (RCC) theory Han  _et al._ (2019), and the
other giving $2.002291(4)$, calculated by the $\Lambda$-approach RCC
($\Lambda$-RCC) theory Yu  _et al._ (2020a). The two Landé $g_{J}$ factors
have a difference of $0.0006$ that generates a relative frequency shift of
$6.6\times 10^{-14}$. This large systematic shift obviously falls short
inaccuracy of our latest 113Cd+ microwave-frequency standard ($1.8\times
10^{-14}$) Miao  _et al._ (2021). Therefore, re-determining the ground-state
$g_{J}$ factor of 113Cd+ is imperative if further improvements in accuracy for
this microwave-frequency standard are to be attained.

In this work, the HFSs of the $5p~{}^{2}P_{3/2}$ level of the 113Cd+ ion is
measured using the laser-induced fluorescence (LIF) technique. To maintain a
low-temperature environment, the 113Cd+ ions are sympathetic-cooled by laser-
cooled 40Ca+ ions, a technique that improves the accuracy of measurements.
Furthermore, the HFSs and Landé $g_{J}$ factors of both the $5s~{}^{2}S_{1/2}$
and $5p~{}^{2}P_{1/2,3/2}$ levels were calculated using the multiconfiguration
Dirac–Hartree–Fock (MCDHF) method. Electron correlation effects are carefully
investigated and taken into account. Off-diagonal terms are also included to
improve the calculation accuracy of the HFSs for the $5p~{}^{2}P_{1/2,3/2}$
level in Cd+. Cross-checking the measured and calculated HFS results ensures
the reliability and accuracy of the results provided in this work. Our results
are of great importance for further improving the performance of the Cd+
microwave-frequency standard.

##  II Experiment

To obtain the HFSs of the $5p~{}^{2}P_{3/2}$ level for 111,113Cd+, we first
measure the frequency shifts from the
$5s~{}^{2}S_{1/2}~{}F=1$–$5p~{}^{2}P_{3/2}~{}F=2$ transition of 111,113Cd+ to
the $5s~{}^{2}S_{1/2}$–$5p~{}^{2}P_{3/2}$ transition of 114Cd+. Briefly, for
the experimental setup (see Ref. Han  _et al._ (2021) for details), we
prepared crystals of two ion species consisting of approximately $10^{5}$ Ca+
and Cd+ ions in a linear Paul trap. The Ca+ and Cd+ ions are produced by
selected-photoionization using laser beams of wavelength 423-nm (Ca
$4s^{2}~{}^{1}S_{0}$–$4s4p~{}^{1}P_{1}$) / 374-nm (Ca $4s4p~{}^{1}P_{1}$–
Continuum), and 228-nm (Cd $5s^{2}~{}^{1}S_{0}$–$5s5p~{}^{1}P_{1}$). The Ca+
are used as coolant ions that are Doppler-cooled using lasers beams of
wavelength 397-nm (Ca+ $4s~{}^{2}S_{1/2}$–$4p~{}^{2}P_{1/2}$) and 866-nm (Ca+
$3d~{}^{2}D_{3/2}$–$4p~{}^{2}P_{1/2}$). The Cd+ ions are sympathetically-
cooled to less than 0.5 K through Coulomb interactions with the Ca+ ions. The
frequency shifts of the $5s~{}^{2}S_{1/2}~{}(F=1)$–$5p~{}^{2}P_{3/2}~{}(F=2)$
transition of 111,113Cd+ and the $5s~{}^{2}S_{1/2}$–$5p~{}^{2}P_{3/2}$
transitions of 114Cd+ were measured using scanning frequencies in a weak
214.5-nm probe laser beam. The
$5s~{}^{2}S_{1/2}~{}(F=1,m_{F}=1)$–$5p~{}^{2}P_{3/2}~{}(F=2,m_{F}=2)$
transition is a cycling transition that was used to cool and detect the
111,113Cd+ ions. Although the circularly polarized cooling laser beam excites
a cycling transition, ions may, as a result of the polarization impurity,
still, leak to the $5s~{}^{2}S_{1/2}~{}F=0$ state via $5p~{}^{2}P_{3/2}~{}F=1$
state. To increase detection efficiency, 20-dBm microwave radiation resonant
with the ground-state hyperfine transition (15.2-GHz for 113Cd+ and 14.5-GHz
for 111Cd+) is applied during LIF detection. The frequency of each laser beam
is measured using a high-precision wavemeter (HighFinesse WS8-2).

Figure 1:  Schematic of the experiment setup for the 111,113Cd+ HFS
measurements. Sympathetic cooling technology is used to maintain the Cd+ cloud
at low temperatures and improve measurement accuracy. The energy level scheme
(not to scale) for 111,113Cd+ is also presented.

In obtaining the measured LIF spectrum (Fig. 2), the beam intensity is
maintained below 5 $\mu$W/mm2 (saturation parameter 0.0006) to reduce the
cooling and heating effects of the probe beam. The fitted curve is the Voigt
profile Zuo  _et al._ (2019); Han  _et al._ (2021), expressed as

$\displaystyle F$ $\displaystyle=$ $\displaystyle F_{0}+(F_{L}\ast
F_{G})(\nu),$ $\displaystyle F_{L}(\nu)$ $\displaystyle=$
$\displaystyle\frac{2A}{\pi}\frac{\nu_{L}}{4(\nu-\nu_{c})^{2}+\nu_{L}^{2}},$
$\displaystyle F_{G}(\nu)$ $\displaystyle=$ $\displaystyle\sqrt{\frac{4\ln
2}{\pi}}\frac{e^{-\frac{4\ln 2}{\nu^{2}_{G}}\nu^{2}}}{\nu_{G}},$ (1)

where $F_{0}$ is the offset, $\nu$ is the laser beam frequency, $\nu_{c}$ is
the ion resonance frequency, $A$ is the area, $\nu_{L}$ is the Lorentzian
width, $\nu_{G}$ is the Gaussian width of Doppler broadening. The line profile
is slightly asymmetrically because of the heating and cooling effects of the
probe beam, which lead to a slight increase in the uncertainty of the
estimated transition frequency.

Figure 2: Typical LIF spectrum of Cd+, using 113Cd+ as an example. The
frequency of the probe laser beam (214 nm wavelength) is scanned around the
resonance frequency over a range of 600 MHz. The measured line profiles are
fitted with a Voigt function. The inset is an image of the two-species ion
cloud of Cd+ and Ca+ captured by an EMCCD camera; aberration has blurred the
image.

Measurements present three sources of uncertainty:

  * i)

Statistical uncertainties. For the Cd+ $5s~{}^{2}S_{1/2}$–$5p~{}^{2}P_{3/2}$
transition, $\nu_{L}$ is 60.13 MHz which represent the natural width, the
fitted $\nu_{G}$ is approximately 30 MHz, and the ion temperature is estimated
to be approximately 100 mK. The statistical uncertainty associated with the
transition frequencies of 114Cd+ and the 111,113Cd+ is approximately 1 MHz,
and thus the statistical uncertainty in their differences is approximately 1.4
MHz;

  * ii)

Instrument uncertainties. The uncertainty arising from the drift in the
wavemeter is less than 0.5 MHz in a laboratory environment Liu  _et al._
(2018);

  * iii)

Systematic uncertainties. Most systematic shifts are common to the
$5s~{}^{2}S_{1/2}$–$5p~{}^{2}P_{3/2}$ transitions of both 114Cd+ and
111/113Cd+ and thus cancel each other out. Because the
$5s~{}^{2}S_{1/2}~{}(F=1)$–$5p~{}^{2}P_{3/2}~{}(F=2)$ transition of Cd+ is
sensitive to magnetic fields, the Zeeman shift becomes the dominant
contributor to systematic uncertainties.

In a weak field ($\mu_{B}B\sim 0.14$ MHz $\ll$ hyperfine constant $A$), the
Zeeman shift for a specific energy level is expressed as

$E_{Zeeman}=g_{F}M_{F}\mu_{B}\cdot B,$ (2)

where $g_{F}$ is given by

$g_{F}=\frac{F(F+1)+J(J+1)-I(I+1)}{2F(F+1)}g_{J},$ (3)

where $J=L+S$ the total electron angular momentum ($S$ and $L$ the spin and
orbital angular momenta), and $F=I+J$ the total angular momentum with $I$
denoting the nuclear spin. In typical conditions of our experiment, $B\sim
8000$ nT Miao  _et al._ (2015, 2021). By introducing values $g_{J}=2.002257$
and $1.334056$ for the levels $5s~{}^{2}S_{1/2}$ and $5p~{}^{2}P_{3/2}$
calculated in this work (see text below), the Zeeman shift for the
$5s~{}^{2}S_{1/2}~{}(F=1)$–$5p~{}^{2}P_{3/2}~{}(F=2)$ transition of Cd+ is
estimated to be 0.11 MHz. Therefore, the total systematic shifts for the
$5s~{}^{2}S_{1/2}~{}(F=1)$–$5p~{}^{2}P_{3/2}~{}(F=2)$ transitions of Cd+ are
estimated to be below 0.5 MHz.

The final frequencies for the
$5s~{}^{2}S_{1/2}~{}(F=1)$–$5p~{}^{2}P_{3/2}~{}(F=2)$ transitions of
111,113Cd+ and that for $5s~{}^{2}S_{1/2}$–$5p~{}^{2}P_{3/2}$ of 114Cd+ are
determined to be 4649.0(1.6) MHz and 4041.8(1.6) MHz, respectively.

In LS-coupling, the energy shifts after the hyperfine interaction are
expressed as Foot  _et al._ (2005)

$E_{HFS}=A\langle I\cdot J\rangle=\frac{A}{2}[F(F+1)-I(I+1)-J(J+1)].$ (4)

Therefore, for 111,113Cd+, we have

$\nu_{s,F=1\rightarrow p,F=2}^{111/113}=\nu_{s\rightarrow
p}^{111/113}+\frac{1}{4}\nu_{HFS,s}-\frac{3}{8}\nu_{HFS,p},$ (5)

where $\nu_{s,F=1\rightarrow p,F=2}^{111/113}$ is the transition frequency of
$5s~{}^{2}S_{1/2}~{}(F=1)$–$5p~{}^{2}P_{3/2}~{}(F=2)$ of 111,113Cd+;
$\nu_{s\rightarrow p}^{111/113}$ is the transition frequency of
$5s~{}^{2}S_{1/2}$–$5p~{}^{2}P_{3/2}$; $\nu_{HFS,s}$ is the HFS of
$5s~{}^{2}S_{1/2}$; and $\nu_{HFS,p}$ is the HFS of $5p~{}^{2}P_{3/2}$. In
reference to 114Cd+, through a linear transformation, Eq. (5) may be expressed
as

$\displaystyle\Delta\nu_{s,F=1\rightarrow p,F=2}^{111/113,114}$
$\displaystyle=$ $\displaystyle\Delta\nu_{s\rightarrow
p}^{111/113,114}+\frac{1}{4}\nu_{HFS,s}^{111/113}$ $\displaystyle-$
$\displaystyle\frac{3}{8}\nu_{HFS,p}^{111/113},$

where $\Delta\nu_{s,F=1\rightarrow p,F=2}^{111/113,114}=\nu_{s,F=1\rightarrow
p,F=2}^{111/113}-\nu_{s\rightarrow p}^{114}$ and $\Delta\nu_{s\rightarrow
p}^{111/113,114}=\nu_{s\rightarrow p}^{111/113}-\nu_{s\rightarrow p}^{114}$.
With our measurements, $\Delta\nu_{s,F=1\rightarrow p,F=2}^{111/113,114}$ are
respectively 4649.0(1.6) MHz and 4041.8(1.6) MHz, whereas
$\Delta\nu_{s\rightarrow p}^{111/113,114}$ are 1314.3(22)[023] MHz and
555.2(23)[008] MHz Hammen  _et al._ (2018). From our previous measurements
obtained through double-resonance microwave laser spectroscopy, the
$\nu_{HFS,s}^{111/113}$ were accurately measured to be 14530507349.9(1.1) Hz
Zhang  _et al._ (2012) and 15199862855.02799(27) Hz Miao  _et al._ (2021),
from which we derived $\nu_{HFS,p}^{111/113}$ to be 794.6(3.6) MHz and
835.5(2.9) MHz.

##  III Theory

###  III.1 Multiconfiguration Dirac–Hartree–Fock approach

The MCDHF method C. Froese Fischer  _et al._ (2016), as implemented in the
Grasp computer package Jönsson  _et al._ (2013); C. Froese Fischer  _et al._
(2019), is employed to obtain wave functions referred to as atomic state
functions. Specifically, they are approximate eigenfunctions of the Dirac
Hamiltonian describing a Coulombic system given by

$H_{\rm
DC}=\sum_{i=1}^{N}(c~{}\bm{\alpha_{i}}\cdot\bm{p_{i}}+(\beta_{i}-1)c^{2}+V_{i})+\sum_{i<j}^{N}\frac{1}{r_{ij}},$
(7)

where $V_{i}$ denotes the monopole part of the electron–nucleus interaction
for a finite nucleus and $r_{ij}$ the distance between electrons $i$ and $j$;
$\bm{\alpha}_{i}$ and $\beta_{i}$ are the Dirac matrices for electron $i$.

Electron correlations are included by expanding $|\Gamma J\rangle$, an atomic
state function, over a linear combination of configuration state functions
(CSFs) $|\gamma J\rangle$,

$|\Gamma J\rangle=\sum_{\gamma}c_{\gamma}|\gamma J\rangle,$ (8)

where $\gamma$ represents the parity and all the coupling tree quantum numbers
needed to define the CSF uniquely. The CSFs are four-component spin-angular
coupled, antisymmetric products of Dirac orbitals of the form

$\phi({\bf r})=\frac{1}{r}\left(\begin{array}[]{c}P_{n\kappa}(r)\chi_{\kappa
m}(\theta,\phi)\\\ iQ_{n\kappa}(r)\chi_{-\kappa
m}(\theta,\phi)\end{array}\right).$ (9)

The radial parts of the one-electron orbitals and the expansion coefficients
$c_{\gamma}$ of the CSFs are obtained by the self-consistent relativistic
field (RSCF) procedure. In the following calculations of the relativistic
configuration interaction (RCI), the Dirac orbitals are kept fixed, and only
the expansion coefficients of the CSFs are determined for selected eigenvalues
and eigenvectors of the complete interaction matrix. This procedure includes
the Breit interaction and the leading quantum electrodynamic (QED) effects
(vacuum polarization and self-energy).

The restricted active-set method is used in obtaining the CSF expansions by
allowing single and double (SD) substitutions from a selected set of reference
configurations to an active set (AS) of given orbitals. The configuration
space is increased step by step by increasing the number of layers,
specifically, a set of virtual orbitals. These virtual orbitals are optimized
in the RSCF procedure while all orbitals of the inner layers are fixed.

The interaction between the electrons and the electromagnetic multipole
moments of the nucleus splits each fine structure level into multiple
hyperfine levels. The interaction couples the nuclear spin $\bm{I}$ with the
total electronic angular momentum $\bm{J}$ to obtain total angular momentum
$\bm{F=I+J}$.

The hyperfine contribution to the Hamiltonian is represented by a multipole
expansion

$H_{\rm HFS}=\sum_{k\geq 1}\bf{T^{(k)}}\cdot\bf{M^{(k)}},$ (10)

where $\bf{T^{(k)}}$ and $\bf{M^{(k)}}$ are spherical tensor operators of rank
$k$ in the electronic and nuclear spaces, respectively Lindgren and Rosén
(1975). The $k=1$ term represents the magnetic dipole interaction, and the
$k=2$ term the electric quadrupole interaction. Higher-order terms are minor
and can often be neglected.

For the scheme considered in this work (111,113Cd+ with $I=1/2$), only the
magnetic dipole interaction is non-zero. To first-order, the fine-structure
level $\gamma J$ is then split according to

$\displaystyle\langle\Gamma IJFM_{F}|\bm{T^{(1)}\cdot M^{(1)}}|\Gamma
IJFM_{F}\rangle$ (11) $\displaystyle=$
$\displaystyle(-1)^{I+J+F}\left\\{\begin{array}[]{ccc}I&J&F\\\
J&I&1\end{array}\right\\}\langle\Gamma J||T^{(1)}||\Gamma
J\rangle\langle\Gamma I||M^{(1)}||\Gamma I\rangle,$

where the coefficient in curly brackets in the 6j symbol of the rotation
group. The reduced matrix elements of the nuclear tensor operators are related
to the conventional nuclear magnetic dipole moment,

$\langle\Gamma I||M^{(1)}||\Gamma
I\rangle=\mu_{I}\sqrt{\frac{(2I+1)(I+1)}{I}}.$ (12)

The hyperfine interaction energy contribution to a specified hyperfine level
is then given by

$\langle\Gamma IJFM_{F}|\bm{T^{(1)}\cdot M^{(1)}}|\Gamma
IJFM_{F}\rangle=\frac{1}{2}A_{J}C,$ (13)

where $A_{J}$ is the magnetic dipole hyperfine constants

$A_{J}=\frac{\mu_{I}}{I}\frac{1}{\sqrt{J(J+1)}}\langle\gamma
J||T^{(1)}||\gamma J\rangle,$ (14)

and $C=F(F+1)-J(J+1)-I(I+1)$.

However, this method discards the off-diagonal hyperfine interaction. This is
not sufficient for ${}^{2}P_{1/2,3/2}$ because the hyperfine interaction
between the two $F=1$ hyperfine levels is non-negligible owing to their minor
splitting. To account for the off-diagonal hyperfine effects, we consider the
second-order hyperfine interaction between ${}^{2}P_{1/2,3/2}$. The
contribution associated with sublevel labelled $\gamma IJFM_{F}$ may be
expressed as

$\frac{|\langle\gamma
IJFM_{F}|H_{hfs}|\gamma^{\prime}IJ^{\prime}FM_{F}\rangle|^{2}}{E_{J}-E_{J^{\prime}}}.$
(15)

In the relativistic theory, choosing the direction of the magnetic field as
the $z$-direction of the interaction and neglecting all diamagnetic
contributions, the interaction between the magnetic moment of the atom and an
external field may be written as

$H_{M}=(N_{0}^{(1)}+\Delta N_{0}^{1})B,$ (16)

where the last term is the Schwinger QED correction Cheng and Childs (1985).
To first order, the fine-structure splitting in the energy level is

$\displaystyle\langle\Gamma JM_{J}|N_{0}^{(1)}+\Delta N_{0}^{1}|\Gamma
JM_{J}\rangle B$ (17) $\displaystyle=$
$\displaystyle\frac{M_{J}}{\sqrt{J(J+1)}}\langle\Gamma J||\bm{N^{(1)}+\Delta
N^{(1)}}||\Gamma J\rangle B.$

Factoring out the dependence on the $M_{J}$ quantum number, the energies are
expressed in terms of the Landé $g_{J}$ factor

$g_{J}=2\frac{\langle\Gamma J||\bm{N^{(1)}+\Delta N^{(1)}}||\Gamma
J\rangle}{\sqrt{J(J+1)}}.$ (18)

The energy splittings are then given by

$g_{J}M_{J}\frac{B}{2}.$ (19)

Table 1: Calculated hyperfine splitting (HFS) (in MHz) and Landé $g_{J}$ factors for $4d^{10}5s\ ^{2}S_{1/2}$ of 111,113Cd+ obtained through the MCDHF approach. | MCDHF2+RCI | MCDHF3+RCI
---|---|---
|  111HFS |  113HFS | $g_{J}$ |  111HFS |  113HFS | $g_{J}$
5 | 11841 | 12386 | 2.002243 | 12925 | 13521 | 2.002242
6 | 13421 | 14040 | 2.002245 | 14096 | 14746 | 2.002247
7 | 13671 | 14301 | 2.002254 | 14414 | 15079 | 2.002250
8 | 13879 | 14518 | 2.002260 | 14476 | 15143 | 2.002256
9 | 13884 | 14524 | 2.002262 | 14515 | 15184 | 2.002257
10 | 13925 | 14567 | 2.002263 | 14541 | 15211 | 2.002257
11 | 13919 | 14561 | 2.002263 | 14532 | 15202 | 2.002257
12 | 13921 | 14562 | 2.002262 | 14537 | 15207 | 2.002257
Final | 13921(7) | 14563(7) | 2.002262(2) | 14536(9) | 15206(9) | 2.002257(1)

###  III.2 Computation Strategy

Initially, the MCDHF calculation was performed using the extended optimal-
level scheme for the states of the $4d^{10}5s$ and $4d^{10}5p$ configurations,
and these occupied orbitals were determined simultaneously and maintained
throughout subsequent calculations. Because the $4f$ subshell in both the
$4d^{10}5s$ and $4d^{10}5p$ configurations is vacant, imaging the strong pair
correlations between $4d^{2}$ and $4f^{2}$, and between $4d^{2}$ and $5d^{2}$
is easy. The strong core-core (CC) correlation of the $4d$ electrons and the
single valence $5s/5p$ electron do not allow us to include only the valence
correlation; hence, our MCDHF calculation starts from the CC4d mode, in which
the $5s/5p$ and $4d$ electrons can be SD-excited to the $n\leq 8,l\leq 6$
level (AS8, where ‘8’ labels the maximum principal quantum number in the
corresponding AS). To investigate the correlation effects of the inner core
electrons, we performed a few RCI calculations using the MCDHF wavefunctions
derived from the CC4d calculation. This calculation method is labelled
MCDHF1+RCI in this paper. From the plots of the $A_{J}$ and $g_{J}$ factors
for the ${}^{2}S_{1/2}$ and ${}^{2}P_{1/2,3/2}$ levels of 113Cd+ from this
calculation method (Fig. 3), we see that they are sensitive to the core
correlations, and even the core–valence (CV) correlation of the $1s$ orbital.
To describe the wavefunctions better, we performed a second calculation that
includes the strong CV and CC correlations in the RSCF procedure rather than
only including the CV/CC correlations in the RCI procedure.

Figure 3: Hyperfine structure constants $A_{J}$ and Landé $g_{J}$ factors of the $4d^{10}5s\ ^{2}S_{1/2}$ and $4d^{10}5p\ ^{2}P_{1/2,3/2}$ levels for 113Cd+, from the various correlation models used in our MCDHF1+RCI calculation. Table 2: Hyperfine splitting (HFS) (in MHz) and Landé $g_{J}$ factors for the $4d^{10}5p\ ^{2}P_{1/2,3/2}$ level of 111,113Cd+. HFSn.o indicate that no off-diagonal contributions were included, whereas HFSw.o refers to HFS calculations with off-diagonal contributions. | MCDHF2+RCI | MCDHF3+RCI
---|---|---
|  111HFSn.o |  113HFSn.o | $g_{J}$ |  111HFSn.o |  113HFSn.o |  111HFSw.o |  113HFSw.o | $g_{J}$
$4d^{10}5p\ ^{2}P_{1/2}$
5 | 1816 | 1900 | 0.665833 | 2061 | 2156 | 2067 | 2150 | 0.665829
6 | 2099 | 2196 | 0.665825 | 2267 | 2371 | 2270 | 2368 | 0.665821
7 | 2054 | 2148 | 0.665825 | 2255 | 2359 | 2259 | 2355 | 0.665820
8 | 2142 | 2241 | 0.665820 | 2310 | 2416 | 2314 | 2413 | 0.665814
9 | 2123 | 2220 | 0.665822 | 2300 | 2406 | 2304 | 2403 | 0.665818
10 | 2146 | 2245 | 0.665821 | 2324 | 2432 | 2328 | 2428 | 0.665818
11 | 2144 | 2243 | 0.665820 | 2316 | 2422 | 2319 | 2419 | 0.665816
12 | 2145 | 2244 | 0.665820 | 2319 | 2425 | 2322 | 2422 | 0.665816
Final | 2140(24) | 2239(25) | 0.665821(2) | 2314(25) | 2420(26) | 2317(25) | 2417(26) | 0.665817(4)
$4d^{10}5p\ ^{2}P_{3/2}$
5 | 546 | 571 | 1.334062 | 644 | 674 | 650 | 680 | 1.334056
6 | 699 | 732 | 1.334057 | 787 | 824 | 791 | 827 | 1.334052
7 | 699 | 731 | 1.334060 | 768 | 803 | 772 | 807 | 1.334056
8 | 727 | 761 | 1.334060 | 794 | 830 | 797 | 834 | 1.334056
9 | 722 | 755 | 1.334062 | 786 | 822 | 789 | 826 | 1.334058
10 | 729 | 763 | 1.334060 | 789 | 828 | 793 | 832 | 1.334055
11 | 728 | 762 | 1.334059 | 789 | 824 | 792 | 828 | 1.334057
12 | 729 | 762 | 1.334060 | 789 | 825 | 793 | 829 | 1.334056
Final | 727(8) | 760(8) | 1.334061(2) | 789(8) | 823(9) | 792(8) | 830(9) | 1.334056(3)

In our second calculation, based on the above investigations, we included CV
and CC correlations for the $4d$, $4p$, $4s$, $3d$, and $3p$ electrons, as
well as the CV correlation down to the $1s$ subshell by allowing restricted SD
excitations to $n\leq 12,l\leq 6$ (AS12), in the RSCF calculation. RCI
calculations were also performed to include the Breit and QED effects. Note
that this calculation also started from the MCDHF calculation for the
$4d^{10}5s$ and $4d^{10}5p$ configurations and hence is labelled MCDHF2+RCI.

In analyzing the wavefunction compositions from the MCDHF2+RCI calculation, we
noticed strong correlations between $4d^{10}5s$, $4d^{8}4f^{2}5s$, and
$4d^{9}4f5p$ configurations, and between $4d^{10}5p$, $4d^{8}4f^{2}5p$, and
$4d^{8}5p5d^{2}$ configurations. Therefore, instead of starting from the DHF
calculation where only $4d^{10}5s$ and $4d^{10}5p$ were included in the CSF
list, we allowed the $4d$ and $5s/5p$ electrons to be SD-excited to
{$5s,5p,5d,4f$} to generate the CSF list as a starting point of our third
calculation approach. In this way, the spectroscopic orbitals together with
the $5d$ and $4f$ orbitals are optimized together, and the correlation effect
between the essential CSFs is included in the beginning. The CV and CC
correlation effects are included by systematically increasing the virtual
excitations to AS12; this calculation method is labelled MCDHF3+RCI.

For $4d^{10}5s\ ^{2}S_{1/2}$, because there are no other levels with which to
have strong hyperfine interactions, we only included the diagonal
contributions to its HFS. The calculated HFSs and $g_{J}$ factors with an
increasing AS size from MCDHF2+RCI and MCDHF3+RCI calculations are listed in
Table 1. We find some fluctuations in our calculated HFSs with increasing AS
size, but the values from the last few ASs generally tend to some specific
value. We, therefore, took the average of the last three values (AS10, AS11,
and AS12) as our final calculated result, with the maximum difference between
them taken as the calculation error. Although the final splitting for 113Cd+
from the MCDHF2+RCI calculation (i.e., 14563(7) MHz) is much smaller than the
experimental measurement (i.e., 15199 MHz), the MCDHF3+RCI calculation,
(15206(9) MHz) shows a significant improvement with the experimental value
being within the estimated uncertainty of the latter calculation. Following a
similar method, the $g_{J}$ factors of $2S_{1/2}$ from MCDHF2+RCI and
MCDHF3+RCI calculations were 2.002262(2) and 2.002257(1), respectively. The
HFSs and $g_{J}$ factors for $4d^{10}5s\ ^{2}P_{1/2,3/2}$ with an increasing
AS are listed in Table 2. Following the same method as used in determining our
final calculation results and their uncertainties, the HFSs for the
${}^{2}P_{3/2}$ level of 111Cd+/113Cd+ from MCDHF2+RCI and MCDHF3+RCI
calculations when not including the off-diagonal contributions are
727(8)/760(8) MHz and 789(8)/823(9) MHz, respectively. With off-diagonal
contributions included, the MCDHF3+RCI results increase to 792(8)/830(9) MHz.

##  IV Results and discussions

The measured HFSs for the $5p~{}^{2}P_{3/2}$ level and the calculated HFSs and
Landé $g_{J}$ factors for the $5s~{}^{2}S_{1/2}$ and $5p~{}^{2}P_{1/3,3/2}$
levels in this work are listed in Table 3; other experimental and calculated
results are also listed for comparison. For the HFSs, our group’s previous
high-accuracy measurements for the 111,113Cd+ ground state provided an
excellent benchmark for the atomic structure calculation of Cd+. The present
HFSs for the $5s~{}^{2}S_{1/2}$ state calculated using the MCDHF method show
stronger agreement with our previous experimental results than those of
previous theoretical results Dixit  _et al._ (2008); Li  _et al._ (2018). The
present measured HFSs for the $5p~{}^{2}P_{3/2}$ level is also in agreement
with the present theoretical results. The cross-checking between experiment
and theory ensures the reliability of the Cd+ $5p~{}^{2}P_{3/2}$ HFSs
determined in this work. We recommend the adoption of 794.6(3.6) and
835.5(2.9) as the blue-shifted frequencies for optical pumping in the
microwave-frequency standard based on 111/113Cd+.

Regarding the Landé $g_{J}$ factors, there are no experimental results for
Cd+. Accurate calculations of Landé $g_{J}$ factors has proven complicated
even for alkali atoms and alkali-like ions because they are sensitive to
electron correlations. Those calculated in this work using the MCDHF method
show strong deviations from previous RCC results Han  _et al._ (2019). The
ground state Landé $g_{J}$ factor calculated in this work ($2.002257(1)$)
agrees with the recent result calculated using the $\Lambda$-RCC theory
($2.002291(4)$) Yu  _et al._ (2020b) to the fourth decimal place, although
there is no overlap within their margins of uncertainty. To our knowledge,
there also exists a significant difference in results between the
$\Lambda$-RCC calculations with the configuration interaction and the many-
body perturbation (CI+MBPT) calculations in Yb+ ground-state Landé $g_{J}$
factor Gossel  _et al._ (2013); Yu  _et al._ (2020b). Comparing the results of
the same physical quantity from different calculation methods is also of great
significance for developing atomic structure calculation models and
understanding the role of electronic correlation effects. Therefore, we
encourage more experimental and theoretical research on the Landé $g_{J}$
factors of Cd+.

For precaution, we recommend the value 2.00226(4) for the Cd+ ground state
Landé $g_{J}$ factor in the evaluation of the SOZS of the microwave frequency
standard of trapped Cd+ ions. The SOZS can be estimated using the Breit–Rabi
formula,

$\Delta\nu^{(2)}_{Zeem}(B)=-\frac{[g_{j}-g_{I}]^{2}\mu_{B}^{2}B^{2}}{2h^{2}A_{hf}},$
(20)

for which $B\sim 8000$ nT for the Cd+ microwave frequency standard during
actual operations. Thus, the fractional frequency shifts incurred when using
the value of $g_{J}=2.00226(4)$ is $4.4\times 10^{-15}$. The fractional
frequency shifts produced by this $g_{J}$ factor for the Cd+ ground-state can
meet current accuracy requirements for the best Cd+ microwave frequency
standard ($1.8\times 10^{-14}$). However, for further developments of this
standard, the ground state $g_{J}$ factor of Cd+ also needs to be determined
more accurately.

Table 3: Measured HFSs for $5p~{}^{2}P_{3/2}$ and the calculated HFSs and Landé $g_{J}$ factors of $5s~{}^{2}S_{1/2}$ and $5p~{}^{2}P_{1/3,3/2}$ of this work. Results of the Cd+ HFSs and Landé $g_{J}$ factors from other works are also listed for comparison. 111HFS |  113HFS | $g_{J}$ | Method |
---|---|---|---|---
$5s~{}^{2}S_{1/2}$ |
14530.507 | 15199.863 |  | Exp. Zhang  _et al._ (2012) |
14536(9) | 15206(9) | 2.002257(1) | MCDHF (This work) |
14478(175) | 15146(183) |  | RCC Li  _et al._ (2018) |
| 15280 |  | RCC Dixit  _et al._ (2008) |
|  | 2.00286(53) | RCC Han  _et al._ (2019) |
|  | 2.002291(4) |  $\Lambda$-RCC Yu  _et al._ (2020a) |
$5p~{}^{2}P_{3/2}$ |
794.6(3.6) | 835.5(2.9) |  | Exp. (This work) |
| 800 |  | Exp. Tanaka  _et al._ (1996) |
792(8) | 830(9) | 1.334056(3) | MCDHF (This work) |
794(12) | 832(12) |  | RCC Li  _et al._ (2018) |
| 812.04 |  | RCC Dixit  _et al._ (2008) |
|  | 1.33515(43) | RCC Han  _et al._ (2019) |
$5p~{}^{2}P_{1/2}$ |
2317(25) | 2417(26) | 0.665817(4) | MCDHF (This work) |
2333(31) | 2441(33) |  | RCC Li  _et al._ (2018) |
| 2430 |  | RCC Dixit  _et al._ (2008) |
|  | 0.66747(83) | RCC Han  _et al._ (2019) |

##  V Conclusion

We reported on the determination of HFSs and Landé $g_{J}$ factors for the
$5s~{}^{2}S_{1/2}$ and $5p~{}^{2}P_{1/2,3/2}$ levels of 111,113Cd+. The HFSs
of the $5p~{}^{2}P_{3/2}$ level was measured using the laser-induced-
fluorescence technique. The Cd+ ions were co-trapped with Ca+ ions in the same
linear ion trap and sympathetically cooled through the Coulomb interaction
with laser-cooled Ca+ ions. Furthermore, the HFSs and Landé $g_{J}$ factors
for both levels of interest were calculated using the MCDHF calculation. Three
computational strategies were followed to account for the electronic
correlation effects more comprehensively. The final calculated HFSs were in
perfect agreement with the measured HFSs of this work and our previous work,
which from cross-checks, demonstrated the reliability of the calculations and
the experiments. The HFSs and Landé $g_{J}$ factors determined in this work
can further improve the efficiency of the optical pumping procedure and the
accuracy of the second-order Zeeman correction, and the stability and accuracy
of the microwave frequency standard based on trapped Cd+ ions.

## Acknowledgements

We thank Z. M. Tang for the helpful discussions. This work is supported by the
National Key R&D Program of China (No. 2021YFA1400243), National Natural
Science Foundation of China (Nos. 91436210, 12074081, 12104095).

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# Composite search of active particles in three-dimensional space
based on non-directional cues

Justus A. Kromer  Stanford University, Stanford, California, United States of
America  Andrea Auconi  TU Dresden, Dresden, Germany  Benjamin M. Friedrich
<EMAIL_ADDRESS>TU Dresden, Dresden, Germany

###### Abstract

We theoretically address minimal search strategies of active, self-propelled
particles towards hidden targets in three-dimensional space. The particles can
sense if a target is close, e.g., by detecting signaling molecules released by
a target, but they cannot deduce any directional cues. We focus on composite
search strategies, where particles switch between extensive outer search and
intensive inner search; inner search is started when the proximity of a target
is detected and ends again when a certain inner search time has elapsed. In
the simplest strategy, active particles move ballistically during outer
search, and transiently reduce their directional persistence during inner
search. In a second, adaptive strategy, particles exploit a dynamic scattering
effect by reducing directional persistence only outside a well-defined target
zone. These two search strategies require only minimal information processing
capabilities and a single binary or tertiary internal state, respectively, yet
increases the rate of target encounter substantially. The optimal inner search
time scales as a power-law with exponent $-2/3$ with target density,
reflecting a trade-off between exploration and exploitation.

active Brownian particle, intermittent search, non-directional cues,
chemokinesis, persistent random walk, correlated random walk

Turning active colloids into micro-scale robots will enable envisioned
biomedical or environmental applications Li et al. (2017); Jurado-Sánchez and
Wang (2018). Yet, the information processing capabilities of these self-
propelled agents remain limited. This motivates theoretical analysis of
minimal strategies that are simulataneously simple and effective. Here, we
address optimal random search for hidden targets Mijalkov et al. (2016); Nava
et al. (2018). We propose a minimal model of an Active Brownian Particle (ABP)
that regulates its directional persistence in response to local cues. Previous
work showed that for ABP without internal states or memory, such a
chemokinesis strategy does not provide any advantage compared to simple
ballistic motion Kromer et al. (2020). Yet, a single binary or tertiary
internal state allows these agents to increase their rate of target encounter
substantially.

Our question is thus: How to continue a random search for hidden targets if
information is received that a target must be close, yet no directional
information is given? This situation frequently arises when search agents
detect dilute chemical signals released by targets. Examples include
chemokinetic navigation of motile cells, foraging of animals Fraenkel and Gunn
(1961), and odor-sniffing robots Webb (2002); Settles (2005). For micrometer-
sized agents, the signal-to-noise ratio of sensing chemical gradients is low
at sub-nanomolar concentrations, preventing effective directed chemotaxis,
nonwithstanding the fact that the chemical signal itself is above detection
threshold, and thus indicates the proximity of a target releasing signaling
molecules.

If search agents can only detect the presence of a nearby target within a
given threshold distance, we can decompose the search problem into an outer
and an inner search problem, far and close to a target, respectively. While
the main purpose of outer search is the exploration of search space (extensive
search Benhamou (2014)), the agent may choose a different exploitation
strategy in the vicinity of a target for inner search (intensive search).
Composite search strategies allow agents to switch between different search
modes Benhamou (1992); Plank and James (2008); Bénichou et al. (2011);
Benhamou (2014); Bartumeus et al. (2014); Nolting et al. (2015); Palyulin et
al. (2016), e.g., for outer and inner search.

The performance of search strategies strongly depends on the dimension of
search space. According to Pólya’s theorem Pólya (1921), a diffusive particle
will eventually find a target in one- and two-dimensional space almost surely,
while such a particle may never find the target in three dimensional space. A
colloquial version of Pólya’s theorem states: A drunk man will find his way
home, but a drunk bird may get lost forever Kak . This marks an important
qualitative difference between search problems in two and three space
dimensions. Previous research often investigated search problems in two-
dimensional space, largely motivated by animal foraging. Among others, this
includes theoretical and computational studies of composite search strategies
for hidden targets Knoppien and Reddingius (1985); Benhamou (2007); Reynolds
(2009); Nolting et al. (2015); Schwarz et al. (2016), the narrow escape
problem Schuss et al. (2007); Bénichou and Voituriez (2008); Schwarz et al.
(2016); Mangeat and Rieger (2019), or local search around a home base Noetel
et al. (2018a, b). Yet, microswimmers must usually search a three-dimensional
environment. Therefore, in the present manuscript, we focus on the less-
studied problem of search in three-dimensional space.

The optimal strategy for outer search depends on the distribution of target
sites: while Lévy walks are optimal for certain fractal distributions of
targets (or if targets can be revisited), simple ballistic motion is optimal
in infinite domains with uncorrelated random target positions, where each
target can be visited only once Viswanathan et al. (1999); Bartumeus et al.
(2002); Tejedor et al. (2012). This latter case exactly corresponds to a well-
stirred suspension of targets (which are consumed upon encounter), and we will
therefore assume ballistic motion for outer search.

Directional persistence is a key kinetic parameter of self-propelled
microswimmers, which is set by a competition between swimming speed and
directional fluctuations; directional fluctuations can be of thermal origin or
originate from active fluctuations of the propulsion mechanism itself. A
microswimmer may thus control its directional persistence by different means:
(i) transiently changing its speed Vuijk et al. (2018), for instance in
response to the local concentration of a chemical Peng et al. (2015) or
external fields Zhang et al. (2009), (ii) transiently changing its effective
size $L$ (since the thermal rotational diffusion coefficient changes as $\sim
1/L^{3}$ Einstein (1906)), or (iii) up- or down-regulating any stochastic part
of active propulsion Berg and Brown (1972); Ma et al. (2014). Despite these
different possible implementations, we will consider a generic model, assuming
agents can effectively regulate their directional persistence.

In this short communication, we consider a minimal model of an active particle
that employs composite search in three-dimensional space. The particle
switches between ballistic motion in outer search and intensive inner search.
Our model generalizes Active Brownian Particles (ABP), frequently used as
minimal model for cell motility, e.g. of biological or artificial
microswimmers Romanczuk et al. (2012); Bechinger et al. (2016) (also called
correlated or persistent random walkers Gillis (1955); Masoliver et al.
(1989); Friedrich (2008)). We find that a single binary internal state allows
these agents to substantially increase their rate of target encounter. The
optimal inner search time (‘giving-up time’) decreases with target density
$\rho$ as $\sim\rho^{-2/3}$, reflecting a exploration-exploitation trade-off
Reddy et al. (2016) between continuing inner search to eventually find the
nearest target, or searching elsewhere for other targets.

Figure 1:  Composite search. (a) We consider an Active Brownian Particle (ABP)
with velocity $v$ and rotational diffusion coefficient $D_{\mathrm{rot}}$ as a
generic model of a self-propelled microswimmer. (b) The ABP searches for
randomly distributed hidden targets (red) in three-dimensional space,
alternating between an outer search (blue, here corresponding to ballistic
motion), and inner search for a restricted time $\tau$, (red, here
corresponding to low-persistence motion). Inner search is triggered whenever
the ABP senses the vicinity of a target (rosé), and ends once a certain inner
search time $\tau$ has elapsed. This defines a composite search strategy based
on non-directional cues.

#### Model: Active Brownian Particles.

We consider an ABP moving along a trajectory $\textbf{R}(t)$ in three-
dimensional space with speed $v$ and rotational diffusion coefficient
$D_{\mathrm{rot}}$. Rotational diffusion causes its tangent
$\mathbf{t}=\dot{\mathbf{R}}/v$ to decorrelate on a time-scale
$\tau_{p}=l_{p}/v$ set by the persistence length
$l_{p}=v/(2D_{\mathrm{rot}})$, i.e.,
$\langle\mathbf{t}(t_{0})\cdot\mathbf{t}(t_{0}+t)\rangle=\exp(-|t|/\tau_{p})$
Daniels (1952).

We first consider the case of a single spherical target of radius $R_{0}$
located at $\mathbf{R}{=}\mathbf{0}$. Due to spherical symmetry, the time-
dependent distance $R(t)=|\mathbf{R}(t)|$ of the agent to the origin, and the
time-dependent angle $\psi(t)$ enclosed by the tangent $\mathbf{t}$ and the
radial direction $\mathbf{e}_{R}=-\mathbf{R}/R$, decouple from other
coordinates Friedrich and Jülicher (2007, 2009); Kromer et al. (2020)

$\displaystyle\dot{R}$ $\displaystyle=$ $\displaystyle-v\cos\psi,$ (1)
$\displaystyle\dot{\psi}$ $\displaystyle=$
$\displaystyle\frac{v}{R}\,\sin\psi+\sqrt{2D_{\mathrm{rot}}}\,\xi(t)+D_{\mathrm{rot}}\,\mathrm{cot}\,\psi.$
(2)

Here, $\xi(t)$ is Gaussian white noise with $\langle\xi(t)\rangle=0$ and
$\langle\xi(t)\xi(t^{\prime})\rangle=\delta(t-t^{\prime})$.

Intriguingly, the case of a position-dependent speed $v(\mathbf{x})$
(orthokinesis Fraenkel and Gunn (1961)), and the case of a position-dependent
rotational diffusion coefficient $D_{\mathrm{rot}}(\mathbf{x})$ (klinokinesis)
can be mapped onto each other using a local re-parameterization of time Kromer
et al. (2020): if we formally re-scale local time by a factor
$\Phi=v_{0}/v(\mathbf{x})$, then ABPs apparently move with constant speed
$v_{0}=v(\mathbf{x})\Phi$, while the new rotational diffusion coefficient
reads $D_{\mathrm{rot}}\Phi$. We emphasize that the shapes of trajectories do
not change under such time re-parametrization. For ABP with constant speed, it
is known that the steady-state probability distribution $p^{\ast}(\mathbf{x})$
is spatially homogeneous, even if the rotational diffusion coefficient
$D_{\mathrm{rot}}(\mathbf{x})$ depends on position Blanco and Fournier (2003);
Bénichou et al. (2005); Kromer et al. (2020).

In the following, we assume a minimal model with constant speed $v$ and
switchable rotational diffusion coefficient $D_{\mathrm{rot}}$, as detailed
next.

#### Inner and outer search.

We now turn to the full search problem for multiple, non-revisitable targets
(Poisson distributed with density $\rho$), combining inner and outer search,
see Fig. 1. In outer search mode, the ABP moves ballistically
($D_{\mathrm{rot}}=0$), i.e., it uses the optimal strategy to find non-
revisitable targets Viswanathan et al. (1999); Bartumeus et al. (2002);
Tejedor et al. (2012). Close to a target, the ABP uses time-restricted inner
search with high rotational diffusion coefficient $D_{\mathrm{rot}}$ (low
directional persistence), see Fig. 1. This simple strategy only requires the
agent to detect that a target must be within a detection radius $R_{1}$.

Whenever the ABP enters one of the spheres of radius $R_{1}$ surrounding the
targets, it switches to one of the following inner search strategies for a
limited inner search time $\tau$:

  1. (i-a)

Inner search with fixed inner search time: The ABP transiently increases its
rotational diffusion coefficient $D_{\mathrm{rot}}$ (i.e., reduces its
directional persistence) upon detecting a target zone for a total duration
$\tau$ (red curves in Fig. 2).

  2. (i-b)

Stochastic timer: Same as strategy (i-a), but for random inner search times
$\tau$ drawn from a distribution $q(\tau)$ with mean ${\overline{\tau}}$ (blue
curves in Fig. 2).

  3. (ii)

Adaptive inner search: During inner search, the ABP uses a position-dependent
rotational diffusion coefficient $D_{\mathrm{rot}}(\mathbf{x})$ that depends
on its distance $R$ from the nearest target: the agent uses ballistic motion
with $D_{\mathrm{rot}}=D_{1}=0$ if the target is within the detection radius
with $R\leq R_{1}$, while the agent moves with reduced directional persistence
with $D_{\mathrm{rot}}=D_{2}>0$ for $R>R_{1}$. This strategy exploits a
generic scattering effect that provides ABPs with multiple search attempts
Kromer et al. (2020). Like in strategy (i-a), inner search is terminated after
an inner search time $\tau$, after which the agent resumes outer search with
only ballistic motion until proximity of the next target is detected . Below
we show that the rate of target encounter is increased for adaptive inner
search (black curves in Fig. 2).

#### (i-a) Fixed inner search time.

We introduce a strict timer for inner search: The ABP automatically resets its
internal state back to outer search after a maximal inner search time $\tau$
(except in the unlikely event that the agent finds itself again in a target
zone defined by $R\leq R_{1}$). This defines a composite search strategy
Nolting et al. (2015).

What is the optimal time $\tau_{\mathrm{opt}}$ the ABP should spend in ‘inner
search mode’ before switching back to ballistic motion (‘giving-up time’
Nolting et al. (2015)), in order to maximize the rate $k$ of target
encounters? The answer depends on the density of targets $\rho$ and the
probability $p_{\mathrm{in}}(\tau)$ of successful inner search within time
$\tau$. The derivative $\mathcal{F}(\tau)=dp_{\mathrm{in}}(\tau)/d\tau$
denotes the (non-normalized) distribution of first-passage times for inner
search.

For composite search consisting of inner and outer search, we have
$k=k_{\mathrm{out}}\,p_{\mathrm{in}}(\tau)$, where $k_{\mathrm{out}}$ denotes
the rate at which the ABP enters the target zone around some target. For
ballistic motion, $k_{\mathrm{out}}$ equals the product of $\rho$ and the
effective search volume per unit time $\sigma=\pi R_{1}^{2}\,v$ Rothschild and
Swann (1949). Thus,
$k_{\mathrm{out}}\approx\rho\,\sigma(1-k_{\mathrm{out}}\tau)$, where we take
into account the time fraction $k_{\mathrm{out}}\tau$ spent in inner search.
We neglect the small probability that the ABP is again inside a target zone at
the end of inner search. This expression can be rewritten as
$k_{\mathrm{out}}=[(\rho\,\sigma)^{-1}+\tau]^{-1}$. The optimal duration
$\tau_{\mathrm{opt}}$ of inner search that maximizes $k(\tau)$ must satisfy
$0=\partial k/\partial\tau_{|\tau=\tau_{\mathrm{opt}}}$, which we can re-
arrange as

$\frac{\partial}{\partial\tau}\ln
p_{\mathrm{in}}(\tau)_{|\tau=\tau_{\mathrm{opt}}}=[(\rho\,\pi
R_{1}^{2}\,v)^{-1}+\tau_{\mathrm{opt}}]^{-1}\quad.$ (3)

Eq. (3) embodies the trade-off between continuation of inner search and
decampment to new targets in the spirit of a Marginal Value Theorem Charnov
(1976). If targets are denser, less time should be spent in inner search, see
Fig. 2(b), red curves. For long times $\tau D_{\mathrm{rot}}\gg 1$, persistent
random walks can be approximated by diffusion. This implies an asymptotic
Sparre-Anderson scaling $dp_{\mathrm{in}}/d\tau\sim\tau^{-3/2}$ Feller (2008).
In the limit of sparse targets, where the volume fraction occupied by the
target zones of all targets is much smaller than one, $\rho V_{1}\ll 1$, a
straight-forward but lengthy calculation gives (see Supplemental Material (SM)
text)

$\tau_{\mathrm{opt}}\sim\rho^{-2/3}\quad.$ (4)

The ABP will spend considerably more time in outer search compared to inner
search, $k_{\mathrm{out}}\tau_{\mathrm{opt}}\ll 1$. Together with
$p_{\mathrm{in}}(\tau_{\mathrm{opt}})\lesssim\lim_{\tau\rightarrow\infty}p_{\mathrm{in}}(\tau)\approx
R_{0}/R_{1}$, we conclude for the rate of target encounter for composite
search, $k\approx\rho\,\pi R_{0}R_{1}\,v$. We may re-state this result
equivalently as follows: composite search with fixed inner search time
according to strategy (i-a) increases the effective cross-sectional area of
target encounter to $k/(\rho v)\lesssim\pi R_{0}R_{1}$, compared to $k/(\rho
v)=\pi R_{0}^{2}$ for pure ballistic motion. A detailed mathematical
derivation can be found in SM text.

#### (i-b) Stochastic timer.

Above we considered the case of a fixed inner search time $\tau$ for sake of
simplicity, yet real artificial microswimmers will likely not be able to
control inner search time precisely. Instead, we expect that the microswimmer
can only realize a distribution $q(\tau)$ of inner search times that vary
around a mean value ${\overline{\tau}}$. In the possibly simplest example,
switching back to outer search mode could be controlled by a single-rate
process with rate constant $1/{\overline{\tau}}$; in this case, inner search
times $\tau$ follow an exponential distribution with mean ${\overline{\tau}}$

$q_{\overline{\tau}}(\tau)={\overline{\tau}}^{-1}\exp(-\tau/{\overline{\tau}})\quad.$
(5)

All results derived above hold also in this case (and in fact even for
$q_{\overline{\tau}}(\tau)={\overline{\tau}}^{-1}f(\tau/{\overline{\tau}})$
with some arbitrary function $f$), provided the probability
$p_{\mathrm{in}}(\tau)$ of successful inner search is replaced by an effective
probability distribution
${\overline{p}}_{\mathrm{in}}({\overline{\tau}})=\int_{0}^{\infty}\\!d\tau\,p_{\mathrm{in}}(\tau)q_{\overline{\tau}}(\tau)$.
In particular, we have the asymptotic scaling
$d{\overline{p}}_{\mathrm{in}}/d{\overline{\tau}}\sim{\overline{\tau}}^{-3/2}$,
and Eq. (4) holds analogously for the optimal mean inner search time
${\overline{\tau}}$, see Fig. 2, blue curves.

Figure 2:  Optimal inner search time decreases with target density. (a):
Simulated probability density $\mathcal{F}(\tau)=dp_{\mathrm{in}}(\tau)/d\tau$
of first-passage times $\tau$ to find a spherical target of radius $R_{0}$ for
search agents starting at a distance $R_{1}$ from the target for different
variants of inner search. Red curves: agents with constant rotational
diffusion coefficient $D_{\mathrm{rot}}$ in inner search [strategy (i-a)].
Black: agents employing adaptive inner search [strategy (ii)] with ballistic
motion inside the target zone $R\leq R_{1}$ and constant rotational diffusion
coefficient $D_{\mathrm{rot}}$ outside, as in Kromer et al. (2020). Blue
curves: probability density
${\overline{\mathcal{F}}}({\overline{\tau}})=d{\overline{p}}_{\mathrm{in}}({\overline{\tau}})/d{\overline{\tau}}$
to find the target for agents with stochastic timer, for which maximal inner
search time follows an exponential distribution with mean ${\overline{\tau}}$,
see Eq. (5) [strategy (i-b)]. All curves: dashed:
$D_{\mathrm{rot}}{=}v/(4R_{0})$, solid: $D_{\mathrm{rot}}{=}10\,v/(4R_{0})$,
time-scale $\tau_{0}=R_{0}/v$. The probability densities are not normalized,
as the total probability to eventually find the target is less than one even
for infinite search times. (b): Optimal time limit $\tau_{\mathrm{opt}}$ for
inner search decreases for increasing target density [cases as in panel (a)].
Parameters: $R_{1}/R_{0}=20$, volume of target zone $V_{1}{=}4\pi
R_{1}^{3}/3$.

#### (ii) Adaptive inner search.

The probability for successful inner search $p_{\text{in}}(\tau)$ can be
further increased by choosing an even more efficient inner search strategy
Kromer et al. (2020). Instead of using a constant rotational diffusion
coefficient $D_{\mathrm{rot}}$ during inner search, the agent may employ a
position-dependent rotational diffusion coefficient
$D_{\mathrm{rot}}(\mathbf{x})$ during inner search and switch its rotational
diffusion coefficient between two values $D_{1}$ and $D_{2}$ inside and
outside a target zone of radius $R_{1}$ around the target, respectively. If
$D_{1}\ll D_{2}$, this adaptive inner search strategy increases the
probability to eventually find the target substantially by exploiting a
dynamic scattering effect Kromer et al. (2020). In short, ABP that head away
from the target will switch to motion with low directional persistence as soon
as they move beyond distance $R_{1}$, which results in a high probability to
reverse direction and enter the target zone again. There, the ABP resumes
ballistic motion, providing the ABP with another attempt to hit the target. In
the following, we set $D_{1}=0$ for the rotational diffusion coefficient
$D_{\mathrm{rot}}$ inside the target zone.

An analytical calculation shows that the normalized rate of target encounter,
$k/(v\rho)$, which represents an effective cross-sectional area of targets,
equals $\pi R_{1}^{2}$ for adaptive search in the limit of sparse targets
($\rho V_{1}\ll 1$) and large rotational diffusion coefficient ($D_{2}\gg
v/R_{0}$), see SM text for details. For comparison, we had found lower
effective cross-sectional areas $k/(v\rho)\lesssim\pi R_{0}R_{1}$ for inner
search with constant $D_{\mathrm{rot}}\gg v/R_{0}$, and $k/(v\rho)=\pi
R_{0}^{2}$ for pure ballistic motion above, see also Fig. 3.

#### Numerical results confirm scaling law.

Fig. 2(a) shows simulation results for the first passage time density
$\mathcal{F(\tau)}$ to find the target exactly after time $\tau$ if the ABP
starts at distance $R_{1}$ (with random initial direction as in Kromer et al.
(2020)). For moderate values of the rotational diffusion coefficient
$D_{\mathrm{rot}}$, all three inner search strategies introduced above, i.e.,
inner search with fixed timer (red curves), inner search with stochastic timer
(blue curves), and adaptive inner search (black curves), give rise to similar
distributions $\mathcal{F(\tau)}$. For a high rotational diffusion
coefficient, however, the distribution $\mathcal{F(\tau)}$ is substantially
higher for adaptive inner search. In all cases, numerical results confirm the
asymptotic scaling $\mathcal{F}(\tau)\sim\tau^{-3/2}$.

Fig. 2(b) displays the corresponding optimal inner search time
$\tau_{\mathrm{opt}}$, which was numerically determined from Eq. (3).
Generally, strategies for which the density of first passage times is
comparably higher for short inner search times $\tau$ display shorter optimal
inner search times $\tau_{\mathrm{opt}}$. Intuitively, several short inner
search episodes are more efficient than fewer longer ones. All cases confirm
the asymptotic scaling $\tau_{\mathrm{opt}}\sim\rho^{-2/3}$.

Figure 3:  Comparison of random search strategies. Shown are schematics of the
different search strategies. In addition to the two strategies studied in this
manuscript, composite search with timer for inner search, and adaptive inner
search studied in this manuscript, pure ballistic motion (with
$D_{\mathrm{rot}}=0$) and pure diffusive motion (with $D_{\mathrm{rot}}\gg
v/R_{0}$), as well as instantaneous chemokinesis, and inverted chemokinesis
are shown. For instantaneous chemokinesis, the ABP changes its rotational
diffusion coefficient instantaneously as function of target distance, with
$D_{\mathrm{rot}}=D_{1}$ inside the target zone (rosé, target distance $R\leq
R_{1}$, green trajectory), and $D_{\mathrm{rot}}=D_{2}$ outside the target
zone ($R>R_{1}$, blue trajectory). The case $D_{1}=0$ and $D_{2}\gg v/R_{0}$
labeled chemokinesis exploits a beneficial inward scattering effect, whereas
the case $D_{1}\gg v/R_{0}$ and $D_{2}=0$ labeled inverted chemokinesis
suffers from disadvantageous outward scattering Kromer et al. (2020). The
different strategies are positioned according to a qualitative assessment of
their control complexity and search performance. SM text contains quantitative
results for the limit of low target density and high rotational diffusion
coefficient.

#### Discussion.

We studied optimal search strategies for hidden targets, whose presence can be
sensed at a distance. Using a minimal model of a chemokinetic agent that can
switch between different rotational diffusion coefficients, we explain how
search agents employing composite search find targets more efficiently if they
posses memory in the form of an internal state (‘outer search mode’ versus
‘inner search mode’).

Intuitively, the improved search success can be explained as follows: if
targets are randomly distributed in an infinite search domain (and can only be
visited once), the optimal search strategy that maximizes the rate of target
encounter is ballistic motion, i.e., moving simply along a straight path
Viswanathan et al. (1999); Bartumeus et al. (2002); Tejedor et al. (2012). If
a search agent, however, receives the information that a target must be close,
it is advantageous to transiently switch to less persistent motion to search
the local neighborhood more thoroughly. Specifically, if a target of radius
$R_{0}$ is known to be at most a distance $R_{1}$ away, the probability to
eventually find this target using Brownian motion is
$p^{\ast}(R_{0}|R_{1})=R_{0}/R_{1}$ if search time is unlimited Berg (1993).
Mathematically, the conditional mean-first passage time to find the target
diverges in this case, but this is not a practical limitation for search
agents that use a finite inner search time. Indeed, a small fraction of
eventually succesful realizations will require a long time to reach the target
(power-law tail of the distribution of conditional first-passage times). Yet,
most of those realizations that reach the target will have done so already
within a time-scale $\sim R_{1}^{2}/D_{\mathrm{eff}}$, where
$D_{\mathrm{eff}}=v_{0}^{2}/(6D_{\mathrm{rot}})$ is the effective
translational diffusion coefficient of the search agent. Thus, the search
agent faces a trade-off choice: shall it continue the inner search for a
particular target that is known to have been close, to get closer to the best
achievable probability $\lim_{\tau\rightarrow\infty}p_{\text{in}}(\tau)$ to
find this particular target? Or rather give up inner search, wander off for
good, and try to find other targets, to ultimately maximize its rate $k$ of
target encounters? This trade-off choice defines an optimal time
$\tau_{\mathrm{opt}}$ for inner search as function of target density: if the
density $\rho$ of targets is higher, it pays off to abandon inner search
earlier, $\tau_{\mathrm{opt}}\sim\rho^{-2/3}$.

Our generic framework allows to investigate different combinations of outer
and inner search strategies, see also Fig. 3 and the SM. In general,
strategies with ballistic motion during outer search outperform strategies
with effective diffusive motion. For ballistic outer search, different inner
search strategies differ in search performance (quantified in terms of an
effective cross-sectional area of targets in the SM), with adaptive inner
search performing best. Future work will address the effect of measurement
noise in target zone detection. More complex strategies can be realized if
agents respond in a gradual manner to the distance from the nearest target,
e.g., in the form of distance-dependent swimming speed Vuijk et al. (2018),
distance-dependent rotational diffusion coefficient Kromer et al. (2020), or
for dimers of elastically coupled active and passive particles in a spatial
activity field Vuijk et al. (2020).

Search strategies, such as the adaptive inner search strategy presented here,
where agents increase their directional persistence when they move towards the
target, are conceptually related to the principle of run-and-tumble
chemotaxis, where changes in a temporal concentration signal regulate the
strength of directional fluctuations Berg and Brown (1972). Yet our minimal
implementation is much simpler: in contrast to chemotaxis, it suffices if the
agent compares local concentration of signaling molecules diffusing from a
target to a fixed threshold.

Chemotactic navigation could additionally increase the probability to find the
target when the search agent is already very close to the target and chemical
gradients can be measured with sufficient precision Berg and Purcell (1977);
Bialek and Setayeshgar (2005); Mora and Nemenman (2019); Novak and Friedrich
(in press); Lange and Friedrich (in press). We speculate that composite random
search as described here might have been an evolutionary precursor of
chemotaxis strategies that additionally require sensory adaption. Composite
random search strategies without sensory adaptation may represent a promising
route for the design of future active microswimmers that need to find hidden
targets.

###### Acknowledgements.

AA acknowledges support by the DFG (FR3429/3-1); BMF acknnowleges support
through the Heisenberg program of the DFG (FR3429/4-1); JAK, AA, and BMF were
supported through the Excellence Initiative by the German Federal and State
Governments (Clusters of Excellence cfaed EXC-1056 and PoL EXC-2068). We would
like to dedicate this manuscript to the memory of Lutz Schimansky-Geier.

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##  Appendix A Supplemental Material

Justus A. Kromer, Andrea Auconi, Benjamin M. Friedrich:  Composite search of
active particles in three-dimensional space based on non-directional cues

###  A.1 Numerical methods

For numeric integration of Eqs. (1), (2) for the inner search problem, we used
an explicit Euler-Maruyama method with integration time step $\Delta t=2\times
10^{-3}\,R_{0}/v$. ABPs were initially positioned at $R_{1}$ with initial
direction angle $\psi$ distributed according to $p(\psi,t=0)=\sin(2\psi)$ for
$\psi\in[0,\pi/2]$ Kromer et al. (2020). Simulations were stopped after a
maximum search time of $2\times 10^{4}\,R_{0}/v$ (corresponding to a maximum
trajectory length of $10^{3}\,R_{1}$). Consistent results were obtained in
preliminary simulations of ABP trajectories in three-dimensional space using
an Euler-Heun scheme with matrix exponentials for propagation of the Frenet-
Serret frame.

To compute the effective probability distribution
${\overline{p}}_{\mathrm{in}}({\overline{\tau}})=\int_{0}^{\infty}\\!d\tau\,p_{\mathrm{in}}(\tau)q_{\overline{\tau}}(\tau)$
for search agents with stochastic timer [blue curves in Fig. 2], we used
simulation results for $p_{\mathrm{in}}(\tau)$ for $0\leq\tau\leq
t_{\mathrm{max}}$ with $t_{\mathrm{max}}=5\times 10^{4}\,\tau_{0}$, and
extrapolated $\mathcal{F}(\tau)=c\,\tau^{-3/2}$ for $\tau>t_{\mathrm{max}}$
with prefactor $c$ determined by a fit.

###  A.2 Analytic derivation of scaling law

We can mathematically prove the observed power law scaling of the optimal
inner search time $\tau_{\mathrm{opt}}$ in the limit of low directional
persistence, $D_{\mathrm{rot}}\gg v/R_{0}$. In this limit, we can approximate
the motion of the agent for inner search by diffusive motion with effective
translational diffusion coefficient $D_{\mathrm{eff}}=l_{p}v/3=(2/3)\,vR_{0}$
(corresponding to the example of an effective translational diffusion
coefficient of an ABP with rotational diffusion coefficient
$D_{\mathrm{rot}}=v/(4R_{0})$ and persistence length
$l_{p}=v/(2D_{\mathrm{rot}})=2R_{0}$ as used in Fig. 2). For diffusive motion,
the probability to find a spherical target of radius $R_{0}$ within time
$\tau$ if starting at $R=R_{1}$, is given by

$p_{\mathrm{in}}(\tau)=\frac{R_{0}}{R_{1}}\,\left[1-\mathrm{Erf}\left(\frac{R_{1}-R_{0}}{\sqrt{4D_{\mathrm{eff}}\tau}}\right)\right]\quad.$
(S1)

We include the derivation of Eq. (S1) for completeness: we consider the
diffusion equation in spherical coordinates for a radially symmetric
probability density $p(R,t)$ for position in three-dimensional space (with
units of an inverse volume)

$\frac{\partial}{\partial
t}{p}=D_{\mathrm{eff}}\,R^{-1}\,\frac{\partial^{2}}{\partial
R^{2}}\,\left(R\,p\right)\quad,$ (S2)

with initial condition $p(R,0)=(4\pi R_{1}^{2})^{-1}\,\delta(R-R_{1})$, and
absorbing boundary conditions at $R=R_{0}$, $p(R_{0},t)=0$, and use the method
of images to find the time-dependent solution

$\displaystyle p(R,t)=\frac{1}{4\pi R_{1}^{2}\,\sqrt{4\pi
D_{\mathrm{eff}}t}}\frac{R_{1}}{R}\Big{[}$
$\displaystyle\exp\left(-\frac{(R-R_{1})^{2}}{4D_{\mathrm{eff}}t}\right)$
$\displaystyle-$
$\displaystyle\exp\left(-\frac{(R+R_{1}-2R_{0})^{2}}{4D_{\mathrm{eff}}t}\right)\Big{]}\quad.$

This solution satisfies the absorbing boundary condition $p(R_{0},t)=0$ at
$R=R_{0}$. Now, $p_{\mathrm{in}}(\tau)=\int_{0}^{\tau}dt\,\mathcal{F}(t)$,
where the probability current $J(t)$ at time $t$ to the absorbing sphere of
radius $R_{0}$ at the origin is given by
$\mathcal{F}(t)=4\pi\,R_{0}^{2}\,D_{\mathrm{eff}}\,\partial p(R,t)/\partial
R_{|R=R_{0}}$.

Eq. (S1) becomes exact in the limit $l_{p}\ll R_{0}$. Note that a similar
result, yet with additional prefactor $R_{0}v/(4D_{\mathrm{eff}})$, was
derived for a case of intermediate persistence length $l_{p}$ with $R_{0}\ll
l_{p}\ll R_{1}$ (or $R_{0}v\ll D_{\mathrm{eff}}\ll R_{1}v$) Friedrich (2008)

$p_{\mathrm{in}}(\tau)=\frac{R_{0}v}{4D_{\mathrm{eff}}}\,\frac{R_{0}}{R_{1}}\,\left[1-\mathrm{Erf}\left(\frac{R_{1}}{\sqrt{4D_{\mathrm{eff}}\tau}}\right)\right]\quad.$
(S3)

Here, $D_{\mathrm{eff}}=l_{p}v/3$ denotes the effective translational
diffusion coefficient of a persistent random walk in three-dimensional space
with general persistence length $l_{p}$.

For the rate $k_{\mathrm{out}}$ at which the ABP enters the target zone around
some target, we have

$\displaystyle k_{\mathrm{out}}=\rho v\,\pi
R_{1}^{2}(1-k_{\mathrm{out}}\tau)=\frac{\rho v\,\pi R_{1}^{2}}{1+\rho v\,\pi
R_{1}^{2}\,\tau}\quad.$ (S4)

By combining Eq. (S1) and Eq. (S4), we obtain for the rate $k$ of target
encounter

$\displaystyle k$ $\displaystyle=k_{\mathrm{out}}\,p_{\mathrm{in}}=\frac{\rho
v\,\pi R_{0}R_{1}}{1+\rho v\,\pi
R_{1}^{2}\,\tau}\,\left[1-\mathrm{Erf}\left(\frac{R_{1}-R_{0}}{\sqrt{4D_{\mathrm{eff}}\tau}}\right)\right]\quad.$
(S5)

In the limit of small targets $R_{0}\ll R_{1}$, we can rewrite this expression
as

$\displaystyle k$
$\displaystyle=\rho\,\pi\,R_{0}R_{1}\,v\,\frac{\beta^{2}}{\beta^{2}+\Lambda}\left[1-\mathrm{Erf}(\beta)\right]\quad,$
(S6)

where we introduced short-hand

$\Lambda=\frac{3\pi}{8}\,\frac{R_{1}}{R_{0}}\,\rho
R_{1}^{3},\quad\beta^{2}=\frac{3}{8}\,\frac{R_{1}}{v\tau}\,\frac{R_{1}}{R_{0}}=\frac{R_{1}^{2}}{4D_{\mathrm{eff}}\tau}\quad.$
(S7)

In the limit $\Lambda\ll 1$, corresponding to $\rho V_{1}\ll R_{0}/R_{1}$, we
find that $k$ becomes maximal for
$\beta_{\mathrm{opt}}=\pi^{1/6}\,\Lambda^{1/3}$. Thus, the optimal inner
search time $\tau_{\mathrm{opt}}$ reads

$\tau_{\mathrm{opt}}=\frac{3^{1/3}}{2\pi}\,\frac{R_{0}}{v}\,(\rho\,R_{0}^{2}R_{1})^{-2/3}\quad.$
(S8)

Hence, $\tau_{\mathrm{opt}}$ decreases if $R_{0}$ or $R_{1}$ become larger,
and increases if targets become sparser.

In the limit $\rho V_{1}\ll R_{0}/R_{1}$ with $\beta\ll 1$, the success
probability of inner search approaches the limit value for time-unrestricted
diffusive search, $\lim_{\tau\rightarrow\infty}p_{\mathrm{in}}=R_{0}/R_{1}$.
The time fraction spent in inner search mode is negligible in this limit,
$k\tau\approx(R_{0}^{2}R_{1}\rho)^{1/3}\ll 1$. For the rate $k$ of target
encounter, we find

$k\approx\rho\,\pi\,R_{0}R_{1}\,v\quad.$ (S9)

This result corresponds to an effective cross-sectional area of the target
$k/(v\rho)\approx\pi R_{0}R_{1}$ for the case of composite search with fixed
inner search considered here (in the appropriate limit case), see also Table
S1.

###  A.3 Reference case: Global diffusive search

As a reference case, we additionally consider a case of effective diffusive
search without time-restriction, and a high rotational diffusion coefficient
$D_{\mathrm{rot}}\gg v/R_{0}$ that is constant in time and space. As always,
we consider search in three-dimensional space for randomly distributed non-
revisitable targets. A calculation analogous to the derivation of Eq. (S9)
yields the steady-state target encounter rate $k_{\mathrm{diff}}$ for this
case (time-unrestricted diffusive search for non-revisitable targets in three-
dimensional space). This rate is well defined only in dimension $d\geq 3$,
which follows from Pólya’s classical result on time-discrete random walks on a
$d$-dimensional lattices Dvoretzky and Erdös (1951).

We first consider a purely diffusive search agent with effective translational
diffusion coefficient $D_{\mathrm{eff}}$ in three-dimensional space, as well
as Poisson distributed spherical targets with radii $R_{0}$. We switch to a
coordinate frame that is co-moving with the diffusive agent. In this frame,
the centers of the targets are diffusing and become absorbed once they reach a
spherical shell of radius $R_{0}$ concentric with the agent. We make the
simplifying assumption that the diffusive motion of the target centers can be
considered independent. The probability density $p_{\mathrm{target}}(R,t)$ of
those target centers that have not yet been absorbed is radially symmetric
(with units of an inverse volume) and obeys the diffusion equation Eq. (S2).
The steady-state probability density reads

$p_{\mathrm{target}}^{\ast}(R)=\rho\,\left(1-\frac{R_{0}}{R}\right)\quad,$
(S10)

satisfying the boundary conditions $p_{\mathrm{target}}^{\ast}(R_{0})=0$ and
$\lim_{R\rightarrow\infty}p_{\mathrm{target}}^{\ast}(R)=\rho$. Thus, we find
for the steady-state current of target centers to the absorbing shell

$\displaystyle k_{\mathrm{diff}}$ $\displaystyle=4\pi
R_{0}^{2}\,D_{\mathrm{eff}}\,\frac{\partial}{\partial
R}p_{\mathrm{target}}^{\ast}(R)_{|R=R_{0}}$ (S11) $\displaystyle=\rho\,4\pi
R_{0}\,D_{\mathrm{eff}}\quad.$ (S12)

An ABP with high rotational diffusion coefficient $D_{\mathrm{rot}}\gg
v/R_{0}$ can be approximately described as a diffusive particle with effective
translational diffusion coefficient $D_{\mathrm{eff}}=l_{p}v/3$, where
$l_{p}=v/(2D_{\mathrm{rot}})$ denotes the persistence length of the ABP. Thus,

$\frac{k_{\mathrm{diff}}}{\rho v}\approx\frac{4\pi}{3}\,R_{0}l_{p}\quad.$
(S13)

###  A.4 Limit values of target encounter rate

We report limit values of the target encounter rate $k$ for different inner
search strategies in the limit of large rotational diffusion coefficients, see
table S1. Target encounter rates are reported as an effective cross-sectional
area of the targets, $k/(v\rho)$. For the case of fixed inner search time with
effective diffusive motion in inner search (first column), the target encouter
rate equals $k=k_{\mathrm{out}}p_{\mathrm{in}}$, where
$k_{\mathrm{out}}/(v\rho)=\pi R_{1}^{2}$ is the normalized rate of target
encounter for ballistic outer search, and $p_{\mathrm{in}}\approx R_{0}/R_{1}$
the success probability of inner search. For the case of adaptive inner search
[with ballistic motion for $R\leq R_{1}$ and effective diffusive motion for
$R>R_{1}$, strategy (ii)] (second column), inward scattering of outgoing
trajectories implies $p_{\mathrm{in}}\lesssim 1$ Kromer et al. (2020). For
comparison, we also report the case of purely ballistic search, where the ABP
employs $D_{\mathrm{rot}}=0$ both for outer and inner search (third column);
in this case, $k/(v\rho)=\pi R_{0}^{2}$. Finally, we consider a case of
inverted adaptive search, where the ABP employs a high rotational diffusion
coefficient $D_{\mathrm{rot}}=D_{1}\gg v/R_{0}$ inside the target zone defined
by $R\leq R_{1}$, but ballistic motion outside, $D_{\mathrm{rot}}=D_{2}=0$
(fourth column). In this case, $p_{\mathrm{in}}\approx 0$ as consequence of an
outward scattering effect of ingoing trajectories Kromer et al. (2020).

Strategy for inner search (outer search: ballistic)
---
Fixed inner search time |  Adaptive inner search |  Ballistic inner search |  Inverted adaptive inner search
$\pi R_{0}R_{1}$ | $\pi R_{1}^{2}$ | $\pi R_{0}^{2}$ | $0$
Table S1:  Normalized rate of target encounter for different inner search
strategies reported as $k/(v\rho)$ with units of an effective cross-sectional
area of targets. We assume ballistic motion with $D_{\mathrm{rot}}=0$ in outer
search throughout. First column: we consider diffusive motion with constant
rotational diffusion coefficient $D_{\mathrm{rot}}\gg v/R_{0}$, corresponding
to the strategy (i-a) of fixed inner search time considered in the main text.
Analogous results are found for the strategy (i-b) with stochastic timer with
random inner search times. Second column: we consider adaptive inner search,
strategy (ii), with position-dependent rotational diffusion coefficient
$D_{\mathrm{rot}}(\mathbf{x})$ ($D_{1}=0$ and $D_{2}\gg v/R_{0}$) as
considered in the main text. Third column: additionally, we state the result
for an ABP that moves always ballistically, i.e., for which outer and inner
search are indistinguishable. Fourth column: finally, we consider an inverted
adaptive inner search strategy, where the position-dependent rotational
diffusion coefficient $D_{\mathrm{rot}}(\mathbf{x})$ equals $D_{1}\gg v/R_{0}$
for $R\leq R_{1}$ and $D_{2}=0$ for $R>R_{1}$.

Lastly, one may ask about composite search strategies that employ effective
diffusive motion in outer search, with high rotational diffusion coefficient
$D_{\mathrm{rot}}$. In this case, a composite search with constant rotational
diffusion coefficient during inner search (e.g., fixed inner search time) does
not provide any considerable benefit. Only adaptive inner search considerably
increases the rate of target encouter, yielding
$k/(v\rho)\approx(4\pi/3)\,R_{1}l_{p}$, where $D_{2}=v/(2l_{p})\gg v/R_{0}$ is
the rotational diffusion coefficient used during inner search outside the
target zone. (Here, we used Eq. (S13) for $k_{\mathrm{out}}$, replacing
$R_{0}$ by $R_{1}$, and $p_{\mathrm{in}}\approx 1$.) However, this rate is
still smaller than the corresponding rate for adaptive inner search and
ballistic outer search, see Table S1.