# Measurement of infrared magic wavelength for an all-optical trapping of
40Ca+ ion clock

Yao Huang  State Key Laboratory of Magnetic Resonance and Atomic and Molecular
Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of
Sciences, Wuhan 430071, China  Key Laboratory of Atomic Frequency Standards,
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan
430071, China  Hua Guan  State Key Laboratory of Magnetic Resonance and Atomic
and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese
Academy of Sciences, Wuhan 430071, China  Key Laboratory of Atomic Frequency
Standards, Wuhan Institute of Physics and Mathematics, Chinese Academy of
Sciences, Wuhan 430071, China  Chengbin Li  State Key Laboratory of Magnetic
Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and
Mathematics, Chinese Academy of Sciences, Wuhan 430071, China  Key Laboratory
of Atomic Frequency Standards, Wuhan Institute of Physics and Mathematics,
Chinese Academy of Sciences, Wuhan 430071, China  Huaqing Zhang  State Key
Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan
Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan
430071, China  Key Laboratory of Atomic Frequency Standards, Wuhan Institute
of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
University of Chinese Academy of Sciences, Beijing 100049, China  Baolin Zhang
State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan
430071, China  Key Laboratory of Atomic Frequency Standards, Wuhan Institute
of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
University of Chinese Academy of Sciences, Beijing 100049, China  Miao Wang
State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan
430071, China  Key Laboratory of Atomic Frequency Standards, Wuhan Institute
of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
University of Chinese Academy of Sciences, Beijing 100049, China  Liyan Tang
State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan
430071, China  Key Laboratory of Atomic Frequency Standards, Wuhan Institute
of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
Tingyun Shi  State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy
of Sciences, Wuhan 430071, China  Key Laboratory of Atomic Frequency
Standards, Wuhan Institute of Physics and Mathematics, Chinese Academy of
Sciences, Wuhan 430071, China  K. Gao<EMAIL_ADDRESS>State Key Laboratory
of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of
Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China  Key
Laboratory of Atomic Frequency Standards, Wuhan Institute of Physics and
Mathematics, Chinese Academy of Sciences, Wuhan 430071, China  Center for Cold
Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China

(August 27, 2024)

###### Abstract

For the first time, we experimentally determine the infrared magic wavelength
for the 40Ca+ $4s\,^{2}\\!S_{1/2}\rightarrow 3d\,^{2}\\!D_{5/2}$ electric
quadrupole transition by observation of the light shift canceling in 40Ca+
optical clock. A ”magic” magnetic field direction is chosen to make the magic
wavelength insensitive to both the linear polarization purity and the
polarization direction of the laser. The determined magic wavelength for this
transition is 1056.37(9) nm, which is not only in good agreement with
theoretical predictions but also more precise by a factor of about 300. Using
this measured magic wavelength we also derive the differential static
polarizability to be $-44.32(32)$ a.u., which will be an important input for
the evaluation of the blackbody radiation shift at room temperatures. Our work
paves a way for all-optical-trapping of 40Ca+ optical clock.

###### pacs:

32.10.Dk, 06.20.F, 06.30.Ft, 37.10.Ty

With rapid development of laser technology, state-of-the-art optical clocks
have now reached an accuracy or frequency stability at the level of 10-18 or
higher Ushijima15 ; Huntemann16 ; McGrew18 ; Brewer19 ; Bothwell19 , which is
two orders of magnitude better than the state-of-the-art microwave atomic
clocks. At this level of accuracy, optical clocks can play a critical role in
redefining the second Targat13 , in searching for variation of fundamental
constants Huntemann14 ; Safronova18 , and in chronometric leveling Grotti18 .
For many neutral-atom optical lattice clocks, the ac-Stark shift due to black
body radiation (BBR) or lattice lasers McGrew18 ; Bothwell19  can be a
limiting factor for achieving such high accuracy McGrew18 ; Bothwell19 ; for
ion-based clocks, on the other hand, micromotion shifts Huntemann16 ; Brewer19
may limit the accuracy of some clocks. One way to reduce the micromotion
shifts is to apply the all-optical trapping technique Schneider10 ; Huber14 ;
Lambrecht17 , where the micromotion shift will be gone when the rf field is
switched off. Since the laser used for all-optical trapping can be chosen at a
magic wavelength LeBlanc07 ; Arora11 ; Herold12 ; Holmgren12 , the energy
shift in the relevant transition will be zero and thus the trapping potential
will introduce no shift in the clock transition frequency. Therefore, for a
magic-wavelength optical-trapped ion, both the micromotion and ac-Stark shift
can be greatly suppressed. In addition to the accuracy of a clock, the
frequency stability is also a very important issue when evaluating a clock.
Comparing to neutral-atom lattice clocks, the stability of a single ion clock
is limited by the signal to noise ratio. Recently, the optical trapping of
Coulomb ion crystals has been demonstrated Schmidt18 , which sheds a light on
the development of all-optical trapping ion clocks using multiple ions to
achieve a better frequency stability

Precision measurements of magic wavelengths in atoms are also very important
in fundamental studies of atomic structure. For example, a measurement of line
strength ratio by magic wavelength can bring a new perspective for determining
accurate transition matrix elements, which are important in testing atomic
computational methods and in interpreting atomic parity non-conservation
Derevianko00 ; Sahoo06 ; Porsev09 . Precision measurements of magic
wavelengths in ions can be used to derive relevant oscillator strengths and
polarizabilities for clock states Liu15 , which is essential for evaluating
the BBR shift at the 10-18 level at room temperatures.

The magic wavelengths of Ca+ have recently been studied both theoretically
Tang13 ; Kaur15 ; Jiang17  and experimentally Liu15 . Two magic wavelengths
for the $4s_{1/2}\rightarrow 3d_{5/2}$ ($m_{J}$ = 1/2, 3/2) clock transitions
near 395.79 nm have been measured to high accuracy, which are in well
agreement with all existing theoretical predictions. However, these magic
wavelengths are very close to the $4s_{1/2}\rightarrow 4p_{3/2}$ and
$4s_{1/2}\rightarrow 4p_{1/2}$ resonant transitions. The near resonant light
has high spontaneous photon scattering rates that can result in a high heating
process Haycock97 . Thus, these magic wavelengths are not ideal choices for
the optical trapping of the ions. Therefore, in order to do optical trapping
of ions, it is important to search for magic wavelengths far off any resonant
transitions; for 40Ca+ in particular, magic wavelengths in the infrared region
are desirable.

In this Letter, we will report the experimental measurement of an infrared
magic wavelength by observation of the light shift canceling in 40Ca+ optical
clock. The clock has an uncertainty of 2.2 $\times$ 10-17 and a 10-16 level
stability at a few seconds Huang19 . The clock is suitable for making a
differential measurement, the clock uncertainty would only introduce a
negligible measurement uncertainty of $<$ 0.001 nm. We will present a method
to extract a reduced transition matrix element using our measured magic
wavelength. We will also determine a static differential polarizability that
is an important parameter in evaluating the BBR shift at room temperatures.

Calculating or measuring an infrared magic wavelength is very different from
measuring a near-resonance magic wavelength Liu15 . Briefly speaking,in
theoretical calculation, the predicted magic wavelengths have much larger
uncertainty compared to the near-resonance magic wavelengths; in the
experiments, for a near-resonance magic wavelength, it is much less sensitive
to magnetic field direction, laser propagation direction, and laser
polarization direction. For measuring a far-off-resonance magic wavelength,
however, one needs to carefully control the laser and magnetic field
conditions and carefully evaluate systematic shifts. To setup the experiment,
first of all, a single 40Ca+ ion is trapped in a miniature ring Paul trap and
the temperature of ion is laser cooled to a few mK. To measure the magic
wavelength, the clock laser is locked to the Zeeman components of clock
transition and the light shift on the clock transition can be observed by
switching on and off the laser with wavelength around 1050 nm (named Lm laser
for short in the following sections). To keep the Lm laser linearly polarized
during the measurement, a polarizer (Glan-Tyler Prism) is placed in the light
path before ion-light interaction takes place. In doing so, the linear
polarization purity can reach $>$ 99%, which can be derived by analyzing the
incident and transmission lights of Lm laser. The wavelength of the Lm laser
used in the experiment is measured with a precision of 100 MHz by a wavemeter
(WS-7, HighFinesse GmbH). The power of Lm laser is measured using a commercial
power meter (S120VC, Thorlabs Inc.) with a power variation within 5%. To
increase the measurement accuracy, a ”magic” magnetic field direction is
chosen to make the magic wavelength insensitive to both the linear
polarization purity and the polarization direction of the laser.

The ac Stark shift caused by a laser can be written in the form

$\begin{split}\Delta
E_{i}=&-\frac{F^{2}}{2}\bigg{[}\alpha_{i}^{S}(\omega)+A\cos\theta_{k}\frac{m_{J}}{2J}\alpha_{i}^{V}(\omega)\\\
&+\frac{3\cos^{2}\theta_{p}-1}{2}\cdot\frac{3m_{J}^{2}-J(J+1)}{J(2J-1)}\alpha_{i}^{T}(\omega)\bigg{]},\end{split}$
(1)

where $F$ is the strength of the ac electromagnetic field,
$\alpha_{i}^{S}(\omega)$, $\alpha_{i}^{V}(\omega)$, and
$\alpha_{i}^{T}(\omega)$ are, respectively, the scalar, the vector, and the
tensor polarizabilities for quantum state $i$ at frequency $\omega$, and the
tensor component will be taken into account only when $J>1/2$. Also in Eq.
(1), the laser polarization $A$, the angle $\theta_{k}$ between the laser
propagation direction $\hat{k}$ and the magnetic field direction $\hat{B}$,
the angle $\theta_{p}$ between the laser polarization direction and $\hat{B}$
are all important parameters affecting the ac Stark shift. In previous
theoretical calculations Tang13 ; Kaur15 ; Jiang17 , $A=0$ and
$\cos\theta_{p}=1$ were chosen when calculating the polarizabilities and
extracting the magic wavelengths under a linearly polarized laser field.

We first consider the case where $A=0$ and $\cos\theta_{p}=1$ in our
experiment. Unlike the 395 nm magic wavelength measurement, it is found that
the magic wavelength here is very sensitive to the parameters $A$,
$\theta_{k}$, and $\theta_{p}$. Thus, we have to make sure that these
parameters are very stable and precise. The parameter $A$ is measured to be
0.005(5) that corresponds to an almost linear polarization, but the
$A\cos\theta_{k}$ term still affects the measurement because the ac Stark
shifts to the sublevels $m_{J}=-3/2$ and $m_{J}=3/2$ are found to be
different. Setting $\cos\theta_{k}$ to be 0 will lower the effect caused by
the polarization impurity.

In the experimental setup, the Lm laser polarization and propagation
directions are kept unchanged. In the beginning of our measurement, the
background magnetic field of the ion is compensated to 0 by adjusting the
currents in the three pairs of Helmholtz coils. The magnetic field amplitude
can be measured by observing the clock transition Zeeman components. By
adjusting the currents in the coils, the relationship between the current in
each pair of coils and the magnetic field it produces is measured. By changing
the currents in the coils, one can produce the magnetic field of any direction
while keeping the amplitude constant. In the end of our measurement, the
compensated background magnetic field is measured again so that the background
magnetic field drift amplitude can be evaluated.

To measure the magic wavelength $\lambda_{m}$, we studied ac Stark shift
within a few nanometers around $\lambda_{m}$. We measured the ac Stark shifts
at six wavelengths of Lm laser, each being measured for about 2000 s. Then the
six points were fitted linearly and the magic wavelength was obtained.
Evaluation of systematic shifts is of great importance in the measurement of
the infrared magic wavelength since it is sensitive to the above-mentioned
parameters. The systematic shifts caused by the uncertainties in $\theta_{k}$
and $\theta_{p}$, by the laser power, by the broadband laser spectrum, and by
the background magnetic field drift were also evaluated.

For estimating the systematic shift due to $\theta_{p}$, we scanned
$\theta_{p}$ from $-30^{\circ}$ to $30^{\circ}$. We found that the measured
magic wavelength became longer when $\theta_{p}$ was near 0, as observed in
Ref. Jiang17 . Experimentally we can change $\theta_{p}$ until the measured
magic wavelength becomes the longest. According to the precision of
$\theta_{p}$ that we can experimentally have, $\theta_{p}$ could cause a
measurement uncertainty of 0.03 nm. For estimating the systematic shift due to
$\theta_{k}$ and $A$, we let the laser pass through a polarizer before it
enters into the vacuum chamber. However, for some reasons, such as the
viewports that would change the polarization slightly, we can still see strong
effects caused by $A$. Technically, the magnetic field direction can be
adjusted to make $\cos\theta_{k}=0$. By scanning $\theta_{k}$, it was found
that the measured magic wavelength was longer when $\theta_{k}$ was closer to
90∘. When measuring the magic wavelength difference between $m_{J}=3/2$ and
$m_{J}=-3/2$, we found that this difference came to 0 when
$\theta_{k}=90^{\circ}$, indicating that the $A\cos\theta_{k}$ term no longer
contributed to the systematic shift. Experimentally we can change $\theta_{k}$
until the measured magic wavelength difference between $m_{J}=3/2$ and
$m_{J}=-3/2$ becomes 0. The experimental precision of $\theta_{k}$ would cause
a measurement uncertainty of 0.01 nm.

Figure 1: The magic wavelength as a function of $\theta_{p}$. $\theta_{p}$
represents the angle between magnetic field direction and the laser
polarization. Each data point shows the average of an experiment lasts for 1-4
hours. The error bars only include the statistical errors, yet the systematic
errors caused by the magnetic field drifting, the laser power drifting, and
the laser pointing drifting are not included. The fitted solid curve is a
polynomial fit of the data set to the 4th order.

The background magnetic field may be changing during the measurement. Since
the measurement was found to be sensitive to the magnetic field direction, the
effects of magnetic field change should be considered. By measuring the
compensated magnetic field amplitude (which should be about 0) every few
hours, the background magnetic field would only be changed by less than 30 nT
during the whole experiment. Since the applied magnetic field amplitude is
3800 nT, we estimated that both $\theta_{p}$ and $\theta_{k}$ would gain an
uncertainty of less than $0.5^{\circ}$ due to the background magnetic field
change. According to the relationship between the magic wavelength and those
parameters, magnetic field change during the whole experiment would cause a
magic wavelength measurement uncertainty of 0.08 nm.

Table 1 lists the systematic error budget. Details about the systematic shift
evaluation can be found in the Supplementary Materials.

Table 1: Uncertainty budget for the infrared magic wavelength measurement. Effects with both shift and uncertainty smaller than 0.001 nm are not listed. Units are in nm. Source | Shift | Uncertainty
---|---|---
Statistical | - | 0.02
$\theta_{p}$ | 0 | 0.03
$\theta_{k}$ | 0 | 0.01
Laser power | $-0.03$ | 0.03
Broadband laser spectrum | 0.005 | 0.005
Background magnetic field shift | 0 | 0.08
Total uncertainty | $-0.04$ | 0.09
Magic wavelength |  |
with correction |  | 1056.37(9)

With the corrections shown in Table 1, the infrared magic wavelength for
$|m_{J}|=3/2$ is determined as 1056.37(9) nm. To date, there are a few
theoretical calculations on this wavelength Tang13 ; Kaur15 ; Jiang17 , as
listed in Table 2. One can see that our result is in fairly good agreement
with these calculations but with much smaller uncertainty.

Theoretically, using the perturbation theory, the dynamic electric dipole
polarizabilities of a given atomic state can be expressed as

$\displaystyle\alpha_{i}^{S}$
$\displaystyle(\omega)=\frac{2}{3(2J_{i}+1)}\sum_{k}\frac{\Delta
E_{ki}|\langle\Psi_{i}||D||\Psi_{k}\rangle|^{2}}{\Delta
E_{ki}^{2}-\omega^{2}}$ (2) $\displaystyle\alpha_{i}^{V}$
$\displaystyle(\omega)=\sqrt{\frac{24J_{i}}{(J_{i}+1)(2J_{i}+1)}}$
$\displaystyle\times\sum_{k}(-1)^{(J_{i}+J_{k}+1)}\begin{Bmatrix}J_{i}&1&J_{i}\\\
1&J_{k}&1\end{Bmatrix}\frac{\omega|\langle\Psi_{i}||D||\Psi_{k}\rangle|^{2}}{\Delta
E_{ki}^{2}-\omega^{2}}$ $\displaystyle\alpha_{i}^{T}$
$\displaystyle(\omega)=\sqrt{\frac{40J_{i}(2J_{i}-1)}{3(J_{i}+1)(2J_{i}+1)(2J_{i}+3)}}$
$\displaystyle\times\sum_{k}(-1)^{(J_{i}+J_{k})}\begin{Bmatrix}J_{i}&2&J_{i}\\\
1&J_{k}&1\end{Bmatrix}\frac{\Delta
E_{ki}|\langle\Psi_{i}||D||\Psi_{k}\rangle|^{2}}{\Delta
E_{ki}^{2}-\omega^{2}}$

where $D$ is the electric dipole transition operator. It is noted that, when
$\omega=0$, $\alpha_{i}^{S}(\omega)$, $\alpha_{i}^{V}(\omega)$, and
$\alpha_{i}^{T}(\omega)$ are referred, respectively, as the static scalar,
vector, and tensor polarizabilities for state $i$. The uncertainties of the
polarizabilities are governed by the uncertainties of the reduced transition
matrix elements. Under our experimental conditions, the ac Stark shift at the
magic wavelength includes the contributions from $\alpha_{4s}^{S}(\omega)$,
$\alpha_{3d_{5/2}}^{S}(\omega)$, and $\alpha_{3d_{5/2}}^{T}(\omega)$, and the
contribution from $\alpha^{V}(\omega)$ can be neglected.

Since the ac Stark shift of the clock transition at the magic wavelength is
zero, the dynamic polarizabilities are the same for both $4s_{1/2}$ and
$3d_{5/2}$ states. Theoretical works Tang13 ; Safronova11  show that the
contributions from the $4s_{1/2}\rightarrow 4p_{1/2}$ and $4s_{1/2}\rightarrow
4p_{3/2}$ transitions dominate the polarizability of the $4s_{1/2}$ state, and
the contributions to the polarizability of the $3d_{5/2}$ state are dominated
by the $3d_{5/2}\rightarrow 4p_{3/2}$ transition that constitutes over 80% of
the polarizability. Based upon the magic wavelength measured here, the energy
levels of atomic states in Ca+ given by NIST Kramida18 , the experimentally
obtained high precision matrix elements for the $4s_{1/2}\rightarrow 4p_{1/2}$
and $4s_{1/2}\rightarrow 4p_{3/2}$ transitions Liu15 , and other reduced
matrix elements from RCC Safronova11 ; Kaur17  and DFCP calculations, the
matrix element $|\langle 3d_{5/2}||D||4p_{3/2}\rangle|$ is extracted to be
3.295(15) a.u..

The BBR shift to the $4s_{1/2}\rightarrow 5d_{5/2}$ clock transition frequency
can be evaluated according to

$\displaystyle\Delta_{\rm BBR}(4s_{1/2}\rightarrow 5d_{5/2})=$
$\displaystyle-\frac{1}{2}(\alpha_{0,4s_{1/2}}-\alpha_{0,3d_{5/2}})$ (3)
$\displaystyle\times(831.9{\rm V}/{\rm m})^{2}\bigg{(}\frac{T(\rm
K)}{300}\bigg{)}^{4}$

where $\alpha_{0}$ is the static electric-dipole polarizability. Combining the
matrix element $\left|\langle 3d_{5/2}\left\|D\right\|4p_{3/2}\rangle\right|$
obtained above and other matrix elements from both experiment and theoretical
calculations, the differential static polarizability between the $4s_{1/2}$
and $3d_{5/2}$ states is determined to be $-44.32(32)$ a.u.. The corresponding
BBR shift at 300 K is 0.3816(28) Hz. Comparing to the existing theoretical
values, as listed in Table 2, the present value agrees with and slightly
better than the best previous theoretical calculation of Ref. Safronova11 .
The fractional uncertainty of BBR shift can now be updated to be 6.8$\times
10^{-18}$. The uncertainty due to the knowledge of the dynamic
polarizabilities can be further reduced with the method in Ref. Barrett19 .

Table 2: Comparison of the infrared magic wavelength (nm) and the Ca+ blackbody radiation shift (Hz) at 300 K. | Present | Theory
---|---|---
|  | All-order | DFCP
|  | method | method
Magic wavelength | 1056.37(9) | 1052.26 Kaur15  | 1074(26) Tang13
|  |  | 1074(32) Jiang17
BBR shift | 0.3816(28) | 0.3811(44) Safronova11  | 0.380(14) Arora07
|  | 0.31(1) Sahoo09  | 0.368 Mitroy08

In summary, we have performed an experimental determination of the infrared
magic wavelength in Ca+ with uncertainty less than $0.1$ nm. Our result agrees
well with theoretical values but with 1-2 orders of magnitude improvement. By
using our measured result, the differential static scalar polarizability has
been determined as $-44.32(32)$ a.u., also in agreement with the previous
theoretical values but with higher accuracy. The blackbody radiation shift at
300 K has then evaluated as 0.3816(28) Hz, which is also in good agreement
with our recent measurement Huang19 . It is noted that the infrared magic
wavelength for the $4s_{1/2}\rightarrow 5d_{5/2}$ transition ($m_{J}=1/2$) was
also predicted theoretically in Ref. Tang13 . The matrix element of
$3d_{5/2}\rightarrow 4f_{7/2}$ transition, whose theoretical uncertainty is
1.1% using relativistic all-order method, could be extracted and improved from
further measurement on this magic wavelength, which can help reduce the BBR
shift uncertainty further. Although the differential static scalar
polarizability can be experimentally obtained with a better accuracy by
measuring the magic rf field Huang19  that could result in the BBR shift with
lower uncertainty, it requires that the differential static polarizability of
the clock transition is negative Huang19 ; Dube14 . However, many optical
clock candidates, such as Yb+, In+, Sr, and Yb, do not satisfy this criterion.
The scheme in this work, which uses the magic wavelength to extract the
transition matrix elements, can be an alternative and more general way to
determine the differential static polarizability.

Furthermore, the determination of the infrared magic wavelength is also a very
important step for building an all-optical trapping ion optical clock in the
near future. Long-time all-optical trapping of the ions has already been
achieved recently by Schatz’s group Lambrecht17 . It is found that one can
trap an ion with optical dipole trap only if the trap potential is higher than
the ion kinetic motion energy, and the heating rate of the dipole trap will be
higher with relatively near resonance wavelength. The ion lifetime in dipole
trap would be a few ms with a few hundreds of GHz red detuning lasers
Schneider10 ; Huber14 ; yet the lifetime can be extended to a few second with
a few hundred THz far-off-resonance lasers. To realize an ion-based optical
clock with all-optical trapping scheme, lifetime of at least 100 ms is
required and the heating rate should be maintained as low as possible in order
to lower the Doppler and Stark shifts. Building a clock with infrared lasers
of hundreds THz of red detuning is a better choice comparing to the 395 nm
laser. Besides, one can easily obtain a fiber laser with higher power ($>$ 60
W) at the Ca+ infrared magic wavelength in the range of $1000\sim 1100$ nm.
The all-optical trapping ion optical clock scheme can be used to trap multiple
ions Schmidt18 , which will potentially increase clock stability. However, The
magic wavelength in sensitive to the alignment of the beam and its
polarization relative to the magnetic field orientation, in our case, these
effect would limit the precision of the magic wavelength to the 0.1 nm level,
this would limit the accuracy of the optical clocks. In the practical point of
view, building a high accuracy all optical ion clock would require techniques
to make the laser pointing and magnetic field more stable.

We thank Jun Jiang, Yongbo Tang, Fangfei Wu, V. Yudin, A. Taichenachev,
Zongchao Yan, B. Sahoo, and J. Ye for help and fruitful discussions. This work
is supported by the National Key R&D Program of China (Grant Nos.
2018YFA0307500, 2017YFA0304401, 2017YFA0304404, 2017YFF0212003), the Natural
Science Foundation of China (Grant Nos. 11634013, 11622434, 91736310,
11774388), the Strategic Priority Research Program of the Chinese Academy of
Sciences (Grant No. XDB21030100), CAS Youth Innovation Promotion Association
(Grant Nos. 2015274, 2018364), and Hubei Province Science Fund for
Distinguished Young Scholars (Grant No. 2017CFA040).

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# The effect of wave dark matter on equal mass black hole mergers

Josu C. Aurrekoetxea<EMAIL_ADDRESS>Astrophysics,
University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH,
United Kingdom  Katy Clough  Geometry, Analysis and Gravitation, School of
Mathematical Sciences, Queen Mary University of London, Mile End Road, London
E1 4NS, United Kingdom  Jamie Bamber  Departments of Physics and Astronomy,
University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA  Pedro G.
Ferreira  Astrophysics, University of Oxford, Denys Wilkinson Building, Keble
Road, Oxford OX1 3RH, United Kingdom

###### Abstract

For dark matter to be detectable with gravitational waves from binary black
holes, it must reach higher than average densities in their vicinity. In the
case of light (wave-like) dark matter, the density of dark matter between the
binary can be significantly enhanced by accretion from the surrounding
environment. Here we show that the resulting dephasing effect on the last ten
orbits of an equal mass binary is maximized when the Compton wavelength of the
scalar particle is comparable to the orbital separation, $2\pi/\mu\sim d$. The
phenomenology of the effect is different to the channels that are usually
discussed, where dynamical friction (along the orbital path) and radiation of
energy and angular momentum drive the dephasing, and is rather dominated by
the radial force (the spacetime curvature in the radial direction) towards the
overdensity between the black holes. Whilst our numerical studies limit us to
scales of the same order, this effect may persist at larger separations and/or
particle masses, playing a significant role in the merger history of binaries.

Introduction.— Gravitational-wave observations provide a unique window that
can be used not only to infer the astrophysical properties of black holes
(BHs), but also to gather information about the environments they live in. The
presence of matter around BHs during a binary merger event results in
modifications to the trajectories, which in turn changes the gravitational-
wave signal in a characteristic way [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19]. Environments may arise from standard matter, such
as accretion discs, or from dark matter (DM). In this work we focus on the
latter case.

Figure 1: Dephasing in the coalescence time $\Delta t_{\mathrm{c}}$ for a
10-orbit binary for different scalar masses $\mu$ and initial densities
$\rho_{0}$. Here $\bar{t}_{\mathrm{c}}$ is the merger (coalescence) time in
the absence of a dark matter cloud. The effect is maximized for $\mu\approx
0.45M^{-1}$, corresponding to a Compton wavelength of the dark matter particle
that is comparable to the initial separation of the orbit
$\lambda_{c}=2\pi/\mu\sim d_{0}\approx 12M$. Whilst we focus on the regime of
small separations for numerical reasons, larger ones may support effects from
smaller mass DM candidates. We also note that the effect persists even at
larger masses that would normally already be showing a more particle-like
behaviour. The effect could therefore be more generic than our study suggests.

The DM energy densities required to give significant effects on the signal are
high relative to the expected average galactic values, with the latter
determined by large scale observations [20, 21, 22, 23, 24]. Therefore the
impact of such effects may be expected to be small [6]. However, average
galactic densities describe DM on large scales only, and its distribution on
small scales (in particular the parsec and below scales relevant for
astrophysical BHs) is not well constrained. There exist several mechanisms
that could create DM overdensities around isolated BH. One well known
possibility is the superradiant instability, in which a bosonic field extracts
energy and angular momentum from a highly spinning black hole via repeated
scattering in the ergoregion [25, 26, 27, 28, 29, 30] (see [31] for a review).
Another more prosaic effect is simply the accretion of dark matter in the
potential well around BHs, which results in the formation of “dark matter
spikes” [32] (a combination of both superradiance and accretion may lead to
even higher densities [33]). Such spikes were originally proposed in the
context of WIMP-like dark matter, but in general their profile is a power law
with an exponent that depends on the effective equation of state of the dark
matter [34, 35, 36, 37, 38, 39, 40]. However, they also occur for low mass,
wave-like DM candidates, with a form that is dependent on the relative Compton
wavelength of the DM particle and the black hole horizon [41, 42, 43, 44, 45,
46]. In both cases, the DM density near the BHs depends on the asymptotic dark
matter environment and on the particle properties.

However, a key question is whether these overdensities around isolated objects
persist during a binary merger event. In the case of heavy (particle-like) DM
[47], N-body simulations have shown that they disperse for equal mass mergers,
meaning that objects close to merger or with a violent merger history are
likely to have lost their DM environment [48, 49, 50]. Dark matter spikes
nevertheless remain relevant for intermediate and extreme mass ratio inspirals
(IMRIs and EMRIs) or primordial black holes, with signatures potentially
detectable in next generation space and ground based observations [51, 52, 53,
54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65]. For light or wave-like DM
[66] (see [67, 68, 69] for reviews), much work has focused on the impact of
black holes moving in galactic DM halos [70, 71, 72, 73, 74, 75, 76, 77, 78]
or with superradiant clouds [79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90,
91, 92, 93, 94, 95, 96, 97, 98, 99]. Some of this work has suggested that the
cloud is not completely lost. In a previous publication [100], we demonstrated
that overdensities around equal mass binaries grew into a quasi-stationary
profile that persisted up until the merger (see also [101, 102, 103, 104]).

In this paper, we build on our study to better understand how generic such an
effect is, and to properly quantify the impact that the DM has on the binary
close to merger. We focus on the effect of wave DM on equal mass BH mergers,
and in particular its dependence on the mass of the scalar particle. We
simulate a 10-orbit binary black hole in an initially homogeneous dark matter
environment starting from initial conditions satisfying the Hamiltonian and
momentum constraints. We identify the decay of the orbit (and, as a
consequence, dephasing of the gravitational wave signal) as being a direct
result of the scalar cloud. Our key results are illustrated in Fig. 1, where
we show the dephasing is maximized when the mass of the scalar particle is
such that its Compton wavelength is comparable to the initial separation of
the orbit $\lambda_{c}=2\pi/\mu\sim d_{0}$. In addition, we are able to
quantify the different channels that contribute to the dephasing in our
scenario, finding the dominant effect to be driven not by radiation or
dynamical friction drag forces, as are often discussed, but rather the
attraction of the binary to the central overdensity.

Key background and physical setup.— We consider a minimally coupled massive
complex scalar field $\Phi$ described by the action

$S=\int\differential^{4}x\sqrt{-g}\left(\frac{R}{16\pi
G}-\frac{1}{2}\left(\nabla_{\mu}\Phi\right)^{*}\left(\nabla^{\mu}\Phi\right)-V(\Phi)\right),$
(1)

with a quadratic potential

$V(\Phi,\Phi^{*})=\frac{1}{2}\mu^{2}\Phi^{*}\Phi.$ (2)

where $\mu$ is a parameter related to the scalar field mass111The parameter
$\mu$ is the inverse length scale $\mu=2\pi/\lambda_{c}=m_{s}c/\hbar$
associated with the scalar field mass $m_{s}$. In Planck units $\mu=m_{s}$, so
it is common to refer to $\mu$ simply as “the scalar mass”.. The dynamics of
the scalar field is given by the Klein-Gordon equation coupled to gravity

$\left[\nabla^{\alpha}\nabla_{\alpha}-\mu^{2}\right]\Phi=0\,.$ (3)

Figure 2: Cloud density for two values of $\mu$: one large with
$\lambda_{c}\approx d_{0}$ (right) in which case the binary obtains an
enhanced density cloud, and one smaller $\lambda_{c}\gg d_{0}$ (left) in which
case the pressure coming from the long wavelength of the collective
excitations of the field prevents a high density cloud from forming. We refer
to these as the “cloud” and “no cloud” cases respectively. Simulation movie in
[105].

In the case of a single BH immersed in a reservoir of such scalar DM, the
stationary solution near the black hole is described by the Heun functions
[42, 106, 107, 108], with a power law envelope and characteristic oscillations
in the spatial profile on length scales set by the scalar wavelength. In the
case of a binary no analytic form for a stationary state is known, but
simulations [100] using the numerical codes grchombo [109] and grdzhadzha
[110] have indicated that for a range of initial configurations and within a
few orbits, the scalar matter evolves into a persistent quasi-stationary
profile with density spikes near the black holes and an overdensity between
them.

Ideally we would set this “natural” quasi-stationary DM configuration as an
initial condition, and study the impact the cloud has on the binary merger
using general relativity. However, even if an analytic form was known, a
consistent solution of the GR constraints would lead to changes to the initial
effective masses and momenta of the black holes for different densities and
profiles, making comparisons of the subsequent evolutions difficult to
interpret. In particular, it is hard to know if the additional dephasing is
arising due to matter effects or due to the increased initial eccentricity of
the orbits. One can mitigate this by applying eccentricity reduction schemes
to the initial data, but the fact that the clouds can be very dense near the
horizon makes this challenging as the eccentricity is extremely sensitive to
small changes.

In this paper we take a simpler approach. Given the short relaxation timescale
of the cloud ($\sim 2$ orbits), compared to the timescale of the merger we are
simulating ($\sim 10$ orbits), we start all simulations from a homogeneous
configuration with fixed initial density

$\rho_{0}=\mu^{2}\phi_{0}^{2}\,,$ (4)

and allow the cloud to build up dynamically during the simulation. To do so,
we choose homogeneous initial conditions for the real and imaginary components
of the scalar field, $\Phi=(\phi_{0},0)$ and
$\partial_{t}\Phi=(0,\mu\phi_{0})$. As we vary $\mu$ (which gives us different
cloud configurations, Fig. 2), we adjust $\phi_{0}$ so as to we keep the
initial density $\rho_{0}$ unchanged, thus the initial trajectories of the
binary as a result of solving the initial data will be the same in all cases,
which allows comparison between different masses (and different scalar cloud
profiles) in a more controlled way. We still need to be conscious of the
effect of the transient phase, but since this can be clearly identified in the
evolution during the first few orbits, it becomes easier to separate out.

We decompose the line element in the ADM form [111]

$ds^{2}=-\alpha^{2}\mathrm{d}t^{2}+\gamma_{ij}(\mathrm{d}x^{i}+\beta^{i}\mathrm{d}t)(\mathrm{d}x^{j}+\beta^{j}\mathrm{d}t),$
(5)

where $\gamma_{ij}$ is the three-dimensional spatial metric that we decompose
into a conformal factor and a conformally related metric
$\gamma_{ij}=\bar{\gamma}_{ij}/\chi$. The lapse and shift gauge functions
$\alpha$ and $\beta^{i}$ determine the choice of spatial hyperslicings and
their coordinates, which in numerical relativity are dynamically determined.
The extrinsic curvature tensor
$K_{ij}=(2D_{(i}\beta_{j)}-\partial_{t}\gamma_{ij})/2\alpha$ is decomposed
into a trace $K$ and a traceless part $A_{ij}$, i.e.
$K_{ij}=A_{ij}+(1/3)K\gamma_{ij}$.

We solve the Hamiltonian constraint using Bowen-York initial data using the
CTTK hybrid method [112]. In the homogeneous limit, this reduces to choosing
the trace of the extrinsic curvature tensor $K^{2}=24\pi G\rho$ and solving
for a correction of the conformal factor $\chi$ sourced by the traceless
components $A_{ij}$. We use the open-source numerical relativity code grchombo
[109, 113] with adaptive mesh refinement [114] to solve the full Einstein
equations using the CCZ4 formalism [115] with the moving puncture gauge [116,
117, 118, 119, 120]. We use a simulation box length $L=512M$ and $9$ levels of
mesh refinement and impose reflecting boundary conditions at $z=0$, while for
the other boundaries we apply zeroth order extrapolating boundaries to the
scalar field and Sommerfeld boundary conditions to the metric. Further
technical details and the parameters for each case are given in the
Supplemental Material.

Figure 3: Dephasing of the gravitational wave signal due to the accretion,
dynamical friction and emission of wave dark matter around a binary black hole
merger. Top panel is the real part of the $\psi_{22}$ mode, whilst mid and
bottom panels are its modulus and phase, respectively. The black solid line
corresponds to $(\mu,\phi_{0})=(68,50)\times 10^{-4}$, which we refer to as
“no cloud”, as the Compton wavelength of the scalar field is much larger and
we do not efficiency excite a DM cloud. The blue solid line corresponds to
$(\mu,\phi_{0})=(4300,0.79)\times 10^{-4}$, and causes a $\Delta
t_{\mathrm{c}}/\bar{t}_{\mathrm{c}}\approx 10\%$ dephasing of the merger time.
The initial densities for both these cases is $\rho_{0}\approx 10^{-9}M^{-2}$.
Movie in [105].

Dephasing of the binary.— We study the tensor gravitational-wave modes
$\psi_{lm}$ emitted by the binary black hole extracting the Newman-Penrose
scalar $\Psi_{4}$ with tetrads proposed by [121], projected into spin-weight
$-2$ spherical harmonics ${}_{-2}Y^{lm}$

$r_{\text{ex}}\psi_{lm}=\oint_{S^{2}}r_{\mathrm{ex}}\Psi_{4}|_{r=r_{\mathrm{ex}}}\left[{}_{-2}\bar{Y}^{lm}\right]\,\mathrm{d}\Omega\,,$
(6)

where $\mathrm{d}\Omega=\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi$ is
the area element on the $S^{2}$ unit sphere222Due to the non zero flux of
matter at the boundary, we are doing the extraction of gravitational waves in
a region that is not strictly asymptotically flat, but in each constant
density case the small non zero value of $K$ is the same, meaning that our
comparisons are still meaningful.. The merger or coalescence time for our
10-orbit binary in vacuum is $\bar{t}_{\mathrm{c}}\approx 2000\,M$, defined as
when $|\psi_{22}|$ peaks, which is the dominant mode. This is also the case
for small initial densities $\rho_{0}$ since there is less backreaction of the
matter on the binary metric. For a given density, a smaller effect is also
seen for masses $\mu\ll M^{-1}$, as the Compton wavelength $\lambda_{c}\gg d$
and the cloud is not efficiently excited, see Fig. 2. We refer to the case of
small $\mu$ as the “no cloud” configuration (it has the same initial, non zero
density, but no structure forms around the binary), and the higher $\mu$ case
as “with cloud”.

A typical result can be seen in Fig. 3. As expected, the presence of wave dark
matter around the binary results in a dephasing of the gravitational-wave
signal $\Delta t_{\mathrm{c}}\equiv t_{\mathrm{c}}-\bar{t}_{\mathrm{c}}$,
which is caused by effects like accretion and dynamical friction from the
cloud. We will give explanations and order of magnitude estimates for the
various effects in the following section.

In Fig. 1 we compare the dephasing for different DM masses
$\mu\in\\{0.0068,\,0.86\\}M^{-1}$, corresponding to wavelengths
$\lambda_{c}\in\\{924,\,7\\}M$ that span a range above and below the initial
binary separation $d_{0}\approx 12\,M$. We find that the dephasing is
maximized for $\mu\approx 0.45M^{-1}$, corresponding to $\lambda_{c}\approx
14M\approx d_{0}$. If the mass is smaller $\mu<0.45M^{-1}$, the cloud quickly
becomes suppressed and the dephasing becomes negligible, but we note that
larger separations earlier in the lifetime of the binary may support clouds at
smaller masses. If the mass is larger $\mu>0.45M^{-1}$, the dephasing is
smaller but remains significant, and we still find an efficient excitation of
the cloud. One the one hand, this is not so surprising – even at our highest
mass, we are still in a regime where $\mu\approx M^{-1}$, and so as the merger
radiates gravitational waves and inspirals in, it eventually approaches an
orbital separation comparable to $\lambda_{c}$. On the other hand, in other
studies (see e.g. [80]) one often finds that the behaviour at this limit is
already reasonably well described by the particle limit, and so we might have
expected to see a greater dissipation of the cloud and suppression of the
effect. The fact that this is not the case implies that the mass does not need
to be very finely tuned for the effects to be significant, and motivates a
more detailed study to find the boundary between the wave and particle
regimes.

We also vary the asymptotic energy density to find the value at which the
dephasing is detectable in our simulations during the last 10 orbits, which
gives an indication of the value required for effects to be significant at
merger. See the conclusion section for these values in physical units. Smaller
values may still give detectable effects if observations can happen over a
longer time frame (e.g. by combined LISA and LVK observations), but here we
aim to consider the simplest case where the dephasing is significant in the
merger part of the signal alone.

Quantification of the causes of the dephasing.— To quantify the origins of the
dephasing we want to identify the changes in the energy, angular and radial
momentum of the binary that relate to the presence of the cloud of matter. In
the Newtonian picture, these would include the effects of accretion of the
matter and gravitational forces coming from the uneven distribution of the
surrounding cloud (e.g. the effect of dynamical friction where the object
builds up an overdense tail of matter). In the GR picture these are curvature
effects not forces, but we can still quantify them, up to an unavoidable
slicing dependence333Here we say slicing dependence rather than gauge
dependence because the defined quantities are scalars and in principle do not
depend on the gauge. In practise, however, their definition is with respect to
the coordinate observers of the dynamical evolution and so choosing a
different slicing will result in physically different scalars, with
consequently different values..

We follow the approach of [122, 123, 124, 80], and define a current
$J^{\mu}=\xi^{\nu}T^{\mu}_{\nu}$ in the direction $\xi^{\nu}$ and associated
charge and flux

$Q=-n_{\mu}J^{\mu}\qquad F=\alpha N_{i}J^{i}\,,$ (7)

where $N_{i}$ is the outward normal direction to the surface that bounds the
volume. If $\xi^{\nu}$ is a Killing vector then $\nabla_{\mu}J^{\mu}=0$ and
the change in charge is balanced by a flux through a surface. When this is not
the case the conservation laws require an additional “source” term

$S=\alpha T^{\mu}_{\nu}\nabla_{\mu}\xi^{\nu}\,,$ (8)

describing the exchange of the charge between matter and curvature. It is this
quantity that corresponds to gravitational forces in the Newtonian limit, and
that quantifies the way in which momentum is extracted from the binary by the
matter444In our plots we also include in this quantity the accretion of the
matter charge into spheres around the BHs, since the volume integral is not
defined at the singularity. The split between accretion and the source term
depends strongly on the location of the spheres and the gauge and so is not
very meaningful (see [80, 124] for a more detailed discussion). The total
quantity remains slicing dependent but to a significantly lesser extent. The
conservation law is then written as

$\partial_{t}\left(\int Q\mathrm{d}V\right)=\int S\mathrm{d}V-\int
F\mathrm{d}A\,,$ (9)

where these correspond to the change in charge in the cloud, the exchange
between matter from/to curvature, and the flux to/from infinity. In this work
we will focus on extracting the charges and fluxes related to the energy,
angular and radial momentum, via $\xi_{t}^{\nu}=(1,0,0,0)$,
$\xi_{\phi}^{\nu}=(0,-y,x,0)$ and $\xi_{r}^{\nu}=-(0,x,y,z)/r$. We write the
explicit expressions in terms of the ADM variables in the Supplemental
Material. In Fig. 4 we plot the time integration of each of these quantities.
In each case the black line should equal the sum of the blue and red lines and
provides a check on the error. It is the red line that quantifies the
extraction of the relevant charge from the binary by the matter, and therefore
this is what drives the dephasing. The initial period before $t-r_{ex}\approx
40M$ will contain transient effects related to the growth of the cloud from an
unphysical initial state, but after this the effects are representative of the
quasi-stationary state. We discuss below their evolution and roughly quantify
their effect on the dephasing.

Figure 4: Conservation law for the energy, angular (corotating) and radial
inward momentum currents of the matter in a sphere of radius $40M$ around the
binary, that as we can see in Fig. 2, contains the main cloud overdensity. The
black line shows the change in the total integrated charge of the cloud: $\int
Q\mathrm{d}V$. The red line describes the time integration of the exchange
between matter and curvature $\int\left(\int S\mathrm{d}V\right)\mathrm{d}t$.
The blue line is (minus) the total flux across the outer bounding surface
$\int\left(\int-F\mathrm{d}A\right)\mathrm{d}t$, positive and negative
representing ingoing and outgoing flow, respectively. We see from this the
transfers of energy and momentum from the curvature to the matter and can
infer the energy accretion onto the binary, the extraction and radiation of
angular momentum and the inwards radial force due to momentum accretion and
the central overdensity.

In the top panel of Fig 4, we see that the energy of the matter within the
volume increases, due to the flux of matter energy across the outer surface –
this is simply reflecting the fact that the central cloud density grows over
time due to accretion from the environment. The increase is partially offset
by a negative source term, which is mainly driven by the accretion of energy
into the BHs, increasing their masses by approximately 1% over the course of
the merger (we can check that this agrees to the change in their measured
masses from the AH finder). Outside of the BHs there is very little exchange
in the energy between the curvature and the matter cloud since the spacetime
settles into a quasi-stationary state within the first $\approx 250M$ (it
would be exactly zero for a spacetime with a time-like Killing vector). After
the merger, the energy in the cloud around the remnant decreases slightly as
some is accreted onto the remnant, but it does not completely dissipate.

In the second panel, the angular momentum held in the matter cloud initially
increases as the curvature of the binary “stirs up” the cloud during the
transient phase, then reaches a reasonably steady state during which the
amount of extraction of angular momentum from the spacetime curvature (the
stirring) is balanced by its flux out of the outer surface. The result of this
is that the angular momentum of the spacetime of the binary is decreased, and
carried away by scalar waves. After merger there is an increased flux of
radiation from the outer surface which carries away all the angular momentum
built up in the cloud during the inspiral. We can view the source from/to
curvature as a dynamical friction effect – the extraction of the angular
momentum of the spacetime of the binary by the matter. Can this loss account
for the dephasing observed? A very approximate expression for the dephasing
over one period $\Delta T$ as a result of the loss of angular momentum $\Delta
J$ is

$\frac{\Delta T}{T}\sim\frac{\Delta J}{J}$ (10)

This uses the Newtonian expression $L=mr^{2}\omega$ and assumes a roughly
circular orbit, so that over the period
$|\dot{r}\dot{\theta}|\ll|r\ddot{\theta}|$ and $\ddot{r}$ can be neglected.
Assuming a constant rate of dephasing, $\Delta J/J^{\mathrm{bbh}}_{0}$ of the
system should be around 10% to account for the dephasing observed in this
case. We see that this is far from being the case, with the loss only of order
$0.2\%$. Something else is required to explain the dephasing.

In the third panel, the radial momentum held in the matter cloud is tracked.
Here, we see that the matter cloud initially gains some inward radial
momentum. However, the accretion of inward radial momentum from the outer
surface is roughly balanced by its loss into curvature - partly as a result of
the binary accreting the ingoing momentum, and partly as a result of it being
attracted to the central overdensity. This gives the binary an inward pull
that accelerates during the final plunge. Quantifying the effect is difficult
as it is happening in an extremely non linear regime. However, roughly
speaking the inward radial force on each BH is
$-\mathrm{d}P_{r}/(2\mathrm{d}t)$, i.e. minus the slope of the red line in the
plot. Treating this as a constant in time force:

$\frac{\Delta r}{r_{o}}\sim\frac{\Delta P_{r}\Delta t}{Md_{0}}$ (11)

Putting in the numbers for the time from $t_{\mathrm{ret}}\approx 500M$ to
$t_{\mathrm{ret}}\approx 1500M$ gives $\Delta r/r_{0}\approx 20\%$. To an
order of magnitude estimate, this explains well the dephasing that is
observed. After merger, the flux of radial momentum from the outer surface is
balanced by the accretion into the BH remnant and so the total inward momentum
of the matter remains (approximately) constant, as expected for the final
stationary state.

Conclusion.— Using general relativistic simulations of a binary accreting dark
matter, we have shown that the dephasing in the gravitational-wave signal of
an equal mass black hole merger is maximized when the Compton wavelength of
the dark matter particle is comparable to the orbital distance of the binary,
$2\pi/\mu\sim d$. We need the mass of the scalar to be sufficiently large for
a central overdensity to build up – low mass scalars suppress structure on
smaller scales than their Compton wavelength. Converting into physical units,
the optimal scalar mass to induce dephasing in the last 10 orbits of an equal
mass binary with total mass $M$ is then

$\mu\approx 5\times
10^{-17}\,\left(\frac{M}{10^{6}M_{\odot}}\right)^{-1}\,\mathrm{eV}\,,$ (12)

which can result in a $10\%$ dephasing during the last 10 orbits of the binary
(taking the blue line in Fig. 1) for asymptotic densities around the BH of

$\rho_{0}\approx
10^{20}\,\left(\frac{M}{10^{6}M_{\odot}}\right)^{-2}\,\frac{M_{\odot}}{\mathrm{pc}^{3}}\,.$
(13)

This is high relative to the average DM density, but the dephasing that we
obtain is only over a short period of the binary’s lifetime ($\sim 10$
orbits), and has a cumulative effect. Therefore smaller densities could give
sufficient dephasing to be detectable assuming the effect is triggered at
larger separations (which would also allow lower mass candidates to contribute
to the effect). In particular, multi-band observations between the LISA and
LVK detectors should be able to probe the whole range of frequencies the
binary explores, accumulating effects and thus probing smaller densities than
the ones discussed here. There is also potential for direct detection of the
cloud for models with standard model couplings [125].

The simulations in this work demonstrate that accumulation of wave-like dark
matter between the binary could have a significant effect on the merger
history of binaries, unlike in particle cases where the dark matter tends to
disperse. As recently suggested in [70], it could even go some way to
explaining the final parsec problem. In particular, we highlight the
importance of considering the radial force arising from any central
overdensity that forms, in addition to the radiation of waves carrying angular
momentum and energy away from the binary. As noted above, the effects remain
significant even at the higher end of the masses that we can probe in our
simulations, at which $\mu M\approx 1$. Further investigations should be made
to determine the point at which particle-like behaviour takes effect, and to
study the importance of the relativistic features in our simulations such as
the presence of black hole horizons.

Acknowledgements.— We would like to thank Jean Alexandre, Emanuele Berti,
Gianfranco Bertone, Robin Croft, Giuseppe Ficarra, Thomas Helfer, Charlie Hoy,
Lam Hui, Macarena Lagos, Eugene Lim, Miren Radia, Dina Traykova, Rodrigo
Vicente, Sebastian von Hausegger and Helvi Witek for helpful conversations. We
thank the GRChombo collaboration (www.grchombo.org) for their support and code
development work. JCA acknowledges funding from the Beecroft Trust and The
Queen’s College via an extraordinary Junior Research Fellowship (eJRF). KC
acknowledges funding from the UKRI Ernest Rutherford Fellowship (grant number
ST/V003240/1). JB acknowledges funding from a Science and Technology
Facilities Council (STFC) PhD studentship and funding from National Science
Foundation (NSF) Grant PHY-2006066. PGF acknowledges support from STFC and the
Beecroft Trust.

This work was performed using the DiRAC@Durham facility managed by the
Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility
(www.dirac.ac.uk) under DiRAC RAC15 Grant ACTP316. The equipment was funded by
BEIS capital funding via STFC capital grants ST/P002293/1, ST/R002371/1 and
ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. Part
of this work was performed using the DiRAC Data Intensive service at
Leicester, operated by the University of Leicester IT Services, which forms
part of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was
funded by BEIS capital funding via STFC capital grants ST/K000373/1 and
ST/R002363/1 and STFC DiRAC Operations Grant ST/R001014/1. This work also used
the DiRAC@Durham facility managed by the Institute for Computational Cosmology
on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was
funded by BEIS capital funding via STFC Capital Grants ST/P002293/1,
ST/R002371/1 and ST/S002502/1, Durham University and STFC Operations Grant
ST/R000832/1. DiRAC is part of the National e-Infrastructure.

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## Numerical implementation, diagnostics and convergence tests

We evolve the gravity sector solving the Einstein field equations for a line
element that we decompose in the usual ADM form [111]

$ds^{2}=-\alpha^{2}\mathrm{d}t^{2}+\gamma_{ij}(\mathrm{d}x^{i}+\beta^{i}\mathrm{d}t)(\mathrm{d}x^{j}+\beta^{j}\mathrm{d}t),$
(14)

where $\gamma_{ij}$ is the three-dimensional spatial metric that we decompose
into a conformal factor and a conformally related metric
$\gamma_{ij}=\bar{\gamma}_{ij}/\chi$. The lapse and shift gauge functions
$\alpha$ and $\beta^{i}$ determine the choice of spatial hyperslicings and
their coordinates, which in numerical relativity are dynamically determined.
The extrinsic curvature tensor
$K_{ij}=(2D_{(i}\beta_{j)}-\partial_{t}\gamma_{ij})/2\alpha$ is decomposed
into a trace $K$ and a traceless part $A_{ij}$, i.e.
$K_{ij}=A_{ij}+(1/3)K\gamma_{ij}$. We evolve use the CCZ4 formulation [115]
and the moving puncture gauge [116, 117, 118, 120] with grchombo [109, 114,
113].

We solve the Hamiltonian constraint using Bowen-York initial data using the
CTTK hybrid method [112] (see table 1 for the binary parameters). In the
homogeneous limit, this reduces to choosing the trace of the extrinsic
curvature tensor $K^{2}=24\pi G\rho$ and solving for a correction of the
conformal factor $\chi$ sourced by the traceless components $A_{ij}$. We
choose $K<0$ so that the scalar field is initially decaying, which chooses the
more conservative impact on the merger. The value is small and the effect of
either choice on the overall trends observed is not significant.

$d/M$ | $12.21358$ | $|p_{x}|/M$ | $5.10846\times 10^{-4}$
---|---|---|---
$M_{\mathrm{BH}}/M$ | $0.48847892320123$ | $|p_{y}|/M$ | $8.41746\times 10^{-2}$
$T/M$ | $271.34$ | $|p_{z}|/M$ | $0$
Table 1: Black hole binary initial parameters. The black holes are initially
aligned along the $x$ axis in the $z=0$ plane, with initial momenta
$\vec{p}_{1}=(-|p_{x}|,+|p_{y}|,0)$ for the BH with initial position
$\vec{r}_{1}=(d/2,0,0)$ and $\vec{p}_{2}=(+|p_{x}|,-|p_{y}|,0)$ for the one at
$\vec{r}_{2}=(-d/2,0,0)$.

We use a simulation box length $L=512M$ and $9$ levels of mesh refinement (See
Fig. 5 for convergence tests). We impose reflecting boundary conditions at
$z=0$, while for the other boundaries we impose either zeroth order
extrapolating boundary conditions (matching the values on the exterior ghost
cells to the value (radially directed) inside the simulation grid) or
Sommerfeld boundary conditions.

Figure 5: Convergence testing of the gravitational-wave phase evolution for
the largest dephasing case: $(\mu,\phi_{0})=(4300,2.5)\times 10^{-3}$, so that
the initial density is $\rho_{0}\approx 10^{-8}M^{-2}$. We use three different
resolutions with $N^{3}$ number of grid points on the coarsest level. The
decrease in the error is consistent with $1^{\mathrm{st}}$ order.

Following the approach of [122, 123], the energy momentum tensor is decomposed
into the energy density, momentum density and stress-energy density measured
by normal observers as

$T^{\mu\nu}=\rho n^{\mu}n^{\nu}+S^{\mu}n^{\nu}+S^{\nu}n^{\mu}+S^{\mu\nu}.$
(15)

The energy and angular momentum currents are related to the time-like, angular
and radial directions $J^{\mu}_{t}=T^{\mu}_{\nu}\xi^{\nu}_{t}$,
$J^{\mu}_{\phi}=T^{\mu}_{\nu}\xi^{\nu}_{\phi}$ and
$J^{\mu}_{r}=T^{\mu}_{\nu}\xi^{\nu}_{r}$, where $\xi_{t}^{\nu}=(1,0,0,0)$,
$\xi_{\phi}^{\nu}=(0,-y,x,0)$ and $\xi_{r}^{\nu}=-(0,x,y,z)/r$. The respective
charges $Q$ and fluxes $F$ in terms of ADM quantities are then

$\displaystyle Q_{t}$ $\displaystyle=-\alpha\rho+\beta_{k}S^{k}$ (16)
$\displaystyle F_{t}$
$\displaystyle=N_{i}\left(\beta^{i}(\alpha\rho-\beta^{j}S_{j})+\alpha(\beta^{k}S_{k}^{i}-\alpha
S^{i}\right)$ (17) $\displaystyle Q_{\\{{\phi,r\\}}}$
$\displaystyle=S_{i}\xi^{i}_{\\{{\phi,r\\}}}$ (18) $\displaystyle
F_{\\{{\phi,r\\}}}$
$\displaystyle=-N_{i}\beta^{i}S_{j}\xi^{j}_{\\{{\phi,r\\}}}+\alpha
N_{i}S^{i}_{j}\xi^{j}_{\\{{\phi,r\\}}}\,,$ (19)

where $N_{i}=(x,y,z)/r$ is the normalised radial unit vector, with
$s_{i}=(x,y,z)/r$ and $N_{i}=s_{i}/\sqrt{(\gamma^{jk}s_{j}s_{k})}$. The source
terms

$\displaystyle S_{t}=$
$\displaystyle-\rho\partial_{t}\alpha+S_{i}\partial_{t}\beta^{i}+\frac{\alpha}{2}S^{ij}\partial_{t}\gamma_{ij}$
(20) $\displaystyle S_{\\{{\phi,r\\}}}=$ $\displaystyle\alpha
S^{\mu}_{\nu}\partial_{\mu}\xi^{\nu}_{\\{{\phi,r\\}}}+\alpha
S^{\mu}_{\nu}{}^{(3)}\Gamma^{\nu}_{\mu\sigma}\xi^{\sigma}_{\\{{\phi,r\\}}}$
$\displaystyle-
S_{\nu}\beta^{i}\partial_{i}\xi^{\nu}_{\\{{\phi,r\\}}}+S_{\nu}\xi^{\mu}_{\\{{\phi,r\\}}}\partial_{\mu}\beta^{\nu}-\rho\xi^{\mu}_{\\{{\phi,r\\}}}\partial_{\mu}\alpha\,$
(21)

where $\partial_{t}\gamma_{ij}=-2\alpha K_{ij}+D_{i}\beta_{j}+D_{j}\beta_{i}$,
and both $\partial_{t}\alpha$ and $\partial_{t}\beta^{i}$ are given by our
moving puncture gauge conditions. The quantities $\partial_{\mu}\xi^{\nu}$ are
all zero except $\partial_{x}\xi^{y}=1$ and $\partial_{y}\xi^{x}=-1$

We also track the flux through inner surfaces that move together with the
black holes, which introduces additional advection terms to the flux

$F^{\mathrm{BH}}=\alpha
N_{i}^{\mathrm{BH}}J^{i}-N_{i}^{\mathrm{BH}}\beta^{i}\left(Q-S/2\right)\,,$
(22)

where $N_{i}^{\mathrm{BH}}$ is defined above.