# Observation of Dynamic Nuclear Polarization Echoes

Nino Wili<EMAIL_ADDRESS>Anders B. Nielsen  José P. Carvalho  Niels Chr.
Nielsen<EMAIL_ADDRESS>Interdisciplinary Nanoscience Center (iNANO) and
Department of Chemistry, Aarhus University, Gustav Wieds Vej 14, DK-8000
Aarhus C, Denmark

###### Abstract

It is demonstrated that the time evolution of the electron-nuclear
polarization transfer process during pulsed dynamic nuclear polarization (DNP)
can be reversed on a microsecond timescale, leading to the observation of DNP
echoes. The DNP echoes are induced by consecutive application of two pulse
trains that produce effective Hamiltonians that differ only in the sign of the
effective hyperfine coupling. The experiments have been performed on a frozen
solution of trityl radicals in water/glycerol on a home-built X-band EPR/DNP
spectrometer at $80\text{\,}\mathrm{K}$. We envisage that DNP echoes will play
an important role in future development of pulsed DNP for sensitivity-enhanced
NMR, hyperfine spectroscopy, and quantum sensing.

##  I Introduction

Echo phenomena are a cornerstone of coherent spectroscopy. They are central to
magnetic resonance, including nuclear magnetic resonance (NMR) [1, 2] and
electron paramagnetic resonance (EPR) [3, 4], but echoes have also been
observed in Ruby lasers [5] and in microwave [6] and infrared spectroscopy
[7]. In magnetic resonance, echoes are formed by refocusing an interaction in
the spin Hamiltonian that leads to an apparent signal decay. For example, the
pioneering Hahn echo [8] refocuses heteronuclear spin-spin couplings and
inhomogeneities in the external field. Another important example is the magic
echo and variants thereof, which have been shown to invert the evolution in a
homonuclear system of strongly coupled spins [9, 10]. Remarkably, this
observation violates the spin temperature hypothesis and highlights that many
“relaxation” phenomena in magnetic resonance are in fact coherent in nature
and accordingly may be dealt with through pulsed operation, and refocused at
least in principle. This phenomenon has been referred to as “time-reversal”
experiments [11], as the system evolves as if time was reversed.

Besides spin echoes, also polarization and coherence transfer elements are
fundamental both in NMR and EPR [2, 4]. In NMR of solids, the most prominent
polarization transfer scheme is cross-polarization (CP) [12], where two
dipolar coupled heteronuclei are spin locked by radio-frequency (rf) fields
with the same nutation frequency, known as Hartmann-Hahn matching [13]. In the
context of echoes, it has been shown that it is possible to invert the sign of
the effective heteronuclear coupling Hamiltonian by strategically choosing the
phases and offsets of both spin locks, leading to so-called cross-polarization
echoes [14].

In spin systems containing electron spins and nuclear spins, the electron-
nuclear polarization transfer is known as dynamic nuclear polarization (DNP)
[15]. Historically, most DNP experiments have been conducted with continuous
wave (CW) microwave (mw) irradiation. This is especially the case for high-
field, high-resolution DNP-enhanced NMR aimed at investigating chemical and
biological systems with orders of magnitude higher sensitivity than reachable
by conventional NMR [16, 17]. Pulsed DNP was introduced in 1987 with the
Nuclear spin Orientation via Electron spin Locking (NOVEL) pulse sequence [18,
19]. This experiment is the analogon of the CP experiment, but because of the
large difference between gyromagnetic ratios of electron and nuclear spins,
polarization transfer through the pseudosecular hyperfine coupling is in this
case driven by matching the spin lock Rabi frequency of the electrons to the
Larmor frequency of the nuclei. In this particular case no irradiation of the
nuclei is required. In recent years, many other pulsed DNP schemes have been
developed [20, 21, 22, 23, 24], and it was recognized that the description of
pulsed DNP sequences in many respects resembles that of magic-angle spinning
(MAS) recoupling sequences [21]. In the latter case, the pulse sequences
interfere with the physical sample rotation. In pulsed DNP, the pulse sequence
interferes with the rotation in spin space due to the nuclear Zeeman
interaction.

In this work, we show that it is possible to invert the effective hyperfine
couplings in pulsed DNP, which leads to the formation of dynamic nuclear
polarization echoes. Such echoes may have important applications in further
development of pulsed DNP for atomic structure analysis and emerging quantum
technologies.

##  II Overall idea

We consider a system of one electron spin and $N_{I}$ nuclear spins of the
same isotope. The Hamiltonian of this system, in the electron spin rotating
frame and employing the high-field approximation for the electron spin, is
given in angular frequencies by

$\displaystyle\mathcal{H}$
$\displaystyle=\Delta\omega_{S}S_{z}+\mathcal{H}_{\text{mw}}$
$\displaystyle+\sum_{i}^{N_{I}}\omega_{I}I_{iz}+S_{z}\left(A_{x}^{i}I_{ix}+A_{y}^{i}I_{iy}+A_{z}^{i}I_{iz}\right)$
$\displaystyle+\mathcal{H}_{\text{nn}},$ (1)

where $\Delta\omega_{S}$ is the electron spin offset,
$\mathcal{H}_{\text{mw}}$ describes the microwave irradiation,
$\omega_{I}=-\gamma_{I}B_{0}$ is the nuclear Zeeman frequency, $A^{i}$ is the
anisotropic hyperfine interaction between the electron and nucleus $i$, and
$\mathcal{H}_{\text{nn}}$ corresponds to the nuclear-nuclear couplings.

Our goal is to generate an effective Hamiltonian that leads to electron-
nuclear spin polarization transfer, and then to invert the sign of the
effective Hamiltonian to create echoes which can be observed on either of the
two spin species. This should be achieved by irradiating the electrons only.
We will restrict ourselves to periodic irradiation schemes, where
$\mathcal{H}_{\text{mw}}(t+\tau_{\text{m}})=\mathcal{H}_{\text{mw}}(t)$ with
$\tau_{\text{m}}$ being the period of the Hamiltonian. It was shown by single-
spin vector analysis and average Hamiltonian theory[25, 26, 23, 27] and
operator-based Floquet theory [21], that the first-order effective Hamiltonian
governing the polarization transfer in DNP can be written as

$\displaystyle\mathcal{H}_{\text{eff}}$
$\displaystyle=-\omega_{\text{eff}}^{(S)}\tilde{S}_{z}+\omega_{\text{eff}}^{(I)}I_{z}$
$\displaystyle+\sum_{i}^{N_{I}}\frac{B_{i}}{4}\left(\left(a_{\Delta}\tilde{S}^{+}I^{-}+a_{\Delta}^{*}\tilde{S}^{-}I^{+}\right)\right.$
$\displaystyle+\left.\left(a_{\Sigma}\tilde{S}^{+}I^{+}+a_{\Sigma}^{*}\tilde{S}^{-}I^{-}\right)\right)$
$\displaystyle+\tilde{\mathcal{H}}_{\text{nn}},$ (2)

with $B=\sqrt{A_{x}+A_{y}}$. The tilde indicates an effective frame, where the
$z$-axis points along the overall axis of rotation over one period of the
periodic pulse sequence. The magnitude of the effective fields is given by
$\omega_{\text{eff}}=\beta_{\text{eff}}/\tau_{\text{m}}$, where
$\beta_{\text{eff}}$ is the overall angle of rotation. We choose the
convention $|\beta_{\text{eff}}|\leq\pi$, such that the effective field is at
most half the modulation frequency, $|\omega_{\text{eff}}|\leq\
\omega_{\text{m}}/2=\pi/\tau_{\text{m}}$. If the nuclear spins are not
irradiated, then the nuclear effective field is
$\omega_{\text{eff}}^{(I)}=\omega_{I}-\text{round}\left(\omega_{I}/\omega_{\text{m}}\right)\omega_{\text{m}}$.
The strength of the effective hyperfine coupling is encoded in the scaling
factors $a_{\Delta}$ and $a_{\Sigma}$, which depend on the details of the
electron spin trajectory and are complex in general [23].

Depending on the size and signs of the electron or nuclear spin effective
fields, the polarization can be transferred in the zero-quantum (ZQ), or
double-quantum (DQ) subspaces. Usually it is desirable that only one of the
two takes place, which is the case either if
$|\omega_{\text{eff}}^{(S)}|\approx|\omega_{\text{eff}}^{(I)}|\gg|B_{i}|$ or
if $|\omega_{\text{eff}}^{(S)}|\approx|\omega_{\text{eff}}^{(I)}|\approx 0$
and either $a_{\Delta}=0$ or $a_{\Sigma}=0$ 111Another special case is
$|\omega_{\text{eff}}^{(S)}|\approx|\omega_{\text{eff}}^{(I)}|\approx\omega_{\text{m}}/2$.
In this case there are additional time-independent terms in the interaction
frame that have to be considered. However, in this case one could look at two
periods of the repeating element, which would again lead to
$|\omega_{\text{eff}}^{(S)}|\approx|\omega_{\text{eff}}^{(I)}|\approx 0$..

Let us assume that we choose a ZQ-resonance condition with a sizable effective
field and a negative Zeeman frequency of the nuclei. In this case, the DQ
terms in the effective Hamiltonian are truncated, and we are left with

$\displaystyle\mathcal{H}_{\text{eff}}$
$\displaystyle=-\omega_{\text{eff}}^{(S)}\tilde{S}_{z}-|\omega_{\text{eff}}^{(I)}|I_{z}$
$\displaystyle+\sum_{i}^{N_{I}}\frac{B_{i}}{4}\left(a_{\Delta}\tilde{S}^{+}I^{-}+a_{\Delta}^{*}\tilde{S}^{-}I^{+}\right)$
$\displaystyle+\tilde{\mathcal{H}}_{\text{nn}}.$ (3)

So far, the action of this type of Hamiltonians has essentially only been
considered for the $\tilde{S}_{z}\rightarrow I_{z}$ transfer corresponding to
an inversion ($\pi$ rotation) in ZQ space. Accordingly, if only these two
operators are of interest, then the phase of $a_{\Delta}$ does not matter.
However, for our goal, this phase is essential, since it allows for an
inversion of the coupling Hamiltonian. If the effective field of the electron
spins points along $z$, the phase of $a_{\Delta}$ can simply be controlled by
the overall phase of the mw pulse sequence. This phase change corresponds to a
$S_{z}$ rotation, and a phase change of $\pi$ leads to

$\displaystyle e^{i\pi
S_{z}}\sum_{i}^{N_{I}}\frac{B_{i}}{4}\left(a_{\Delta}{S}^{+}I^{-}+a_{\Delta}^{*}{S}^{-}I^{+}\right)e^{-i\pi
S_{z}}$
$\displaystyle=-\sum_{i}^{N_{I}}\frac{B_{i}}{4}\left(a_{\Delta}{S}^{+}I^{-}+a_{\Delta}^{*}{S}^{-}I^{+}\right).$
(4)

The effective fields and the nuclear coupling Hamiltonian are not affected by
this phase change. However, the DNP echo formation still occurs if 1) the
mismatch of the effective field is not too large, and 2) the timescale of the
electron-nuclear transfer is much shorter than any nuclear-nuclear spin
dynamics. Condition 1) should be mostly fulfilled for reasonable pulse
sequences and narrow EPR spectra, and condition 2) may be fulfilled by
choosing reasonably short DNP echo times. We should note that the approach of
inverting the coupling Hamiltonian is fundamentally different from inverting
the effective field. The latter approach changes the sequence from a ZQ to a
DQ one. The dynamics then occur in different subspaces, and no echo would be
formed. For example, simply inverting the phase of the spin lock in NOVEL does
not lead to the formation of a DNP echo.

##  III DNP pulse sequence

In principle, any periodic DNP pulse sequence that acts only on the electron
spins and generates an effective rotation around $z$ can be used. We chose a
conceptually simple building block consisting of two $\pi$ pulses that differ
in their phases by $\Delta\phi$, as illustrated in Fig. 1 A). It can be looked
at as a special case of TPPM DNP [24] with added delays. The delay $d$ between
the pulses is introduced to make the sequences less susceptible to pulse
transients. Overall, the modulation time of the sequence is given by
$\tau_{\text{m}}=2(t_{p}+d)$. During that time, the nuclear spins rotate
around $I_{z}$ by an angle $\beta_{\text{eff}}^{(I)}=\omega_{I}\cdot
2(t_{p}+d)$. For an on-resonance spin packet, the overall electron spin
rotation is given by $\beta_{\text{eff}}^{(S)}=2\Delta\phi$. The two angles
have to match up (modulo $2\pi$). We chose
$\Delta\phi=(2\pi-\omega_{I}\tau_{\text{m}})/2$. More concretely, with a
nuclear Zeeman frequency of $\omega_{I}/2\pi=$$-14.787\text{\,}\mathrm{MHz}$,
we used $t_{p}=$$12\text{\,}\mathrm{ns}$, $d=$$10\text{\,}\mathrm{ns}$, and
$\Delta\phi=1.098\approx$ $63\text{\,}\mathrm{\SIUnitSymbolDegree}$. The exact
optimal value for $\Delta\phi$ was set experimentally, and could deviate from
this prediction by a few degrees. The scaling factor for these values were
numerically calculated to be $|a_{\Delta}|=0.3371$.

Figure 1:  Pulse sequences used in this work. A) Basic DNP building block. Two
$\pi$ pulses of length $t_{p}$ with a phase difference of $\Delta\phi$,
separated by a delay $d$. The overall length of one element is
$\tau_{m}=2(t_{p}+d)$. The sequence leads to an overall rotation around
$S_{z}$ by an angle of $2\Delta\phi$. B) Generation of DNP echo and EPR
readout. The basic building block in A) is repeated for a time
$t_{\text{inv}}$, at which point the overall phase is inverted. The sequence
is then continued up to a total time $t$. At this point, the electron
polarization is detected via a Hahn echo. The DNP echo forms for
$t=2t_{\text{inv}}$. C) Formation of DNP echo and detection on the nuclear
spins. The sequence in B), excluding the EPR readout, is repeated $h$ times,
with a repetition time $t_{\text{rep}}$ on the the order of the electron
$T_{1,e}$. The polarization of the bulk protons is then detected with a solid
echo. The residual bulk proton polarization is destroyed by a saturation train
after each readout. Filled and open pulses in readout corresponds to $\pi/2$
and $\pi$ pulses, respectively.

##  IV Experimental

All experiments were conducted on a home-built X-band pulsed EPR/DNP
spectrometer (based on the design of Doll et al. [29]) with a sample of
$5\text{\,}\mathrm{mM}$ trityl (OX063) in a H2O:D2O:Glycerol-d5 solution
(1:3:6 by volume) at 80 K. This is a common system for pulsed DNP studies
[30], although usually with a slightly lower degree of solvent protonation. A
commercial MD4 electron-nuclear double resonance probe (Bruker BioSpin)
extended with an external tuning and matching circuit was used, and NMR
signals were detected with a Spincore iSpin-NMR console (SpinCore
Technologies, Inc., Gainesville, FL).

##  V Results

Figure 2:  Observation of DNP echoes via NMR and EPR read out. Top: NMR
signals Bottom: Corresponding EPR signals. A) Reference signals for
$t_{\text{inv}}=0$. B)-F) $t_{\text{inv}}=$$132\text{\,}\mathrm{ns}$,
$220\text{\,}\mathrm{ns}$, $440\text{\,}\mathrm{ns}$,
$880\text{\,}\mathrm{ns}$, and $2200\text{\,}\mathrm{ns}$. The expected echo
position at $t=2t_{\text{inv}}$ is marked by a dashed line. The reference
signal from A) is shown in gray in the other panels. The apparent stop of
small oscillations in the NMR signals after around
$2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ is due to a coarser sampling
interval at longer times.

DNP echo formation for trityl was observed both on the electron and the
nuclear spins. For the electron spins, we used the sequence shown in Fig. 1
B). The basic building block described above was repeated for a time
$t_{\text{inv}}$, at which point the overall phase of the building block was
changed by $\pi$. After a total time $t$, and a waiting time of
$10\text{\,}\mathrm{\SIUnitSymbolMicro s}$, which is long enough for all
remaining electron spin coherences to decay, but much shorter than the
electron $T_{1,e}$ (about $2\text{\,}\mathrm{ms}$), the electron polarization
was read out with a simple Hahn echo sequence.

For the nuclear polarization, it is not really possible to detect the protons
that are very close to the unpaired electron. This is due to the low
concentration of unpaired electrons, fast nuclear spin relaxation, and
hyperfine couplings that exceed the detection bandwidth. However, the
electron-nuclear spin dynamics can be imprinted onto the bulk proton
magnetization by making use of nuclear spin diffusion. For this to occur, one
has to repeat the DNP pulse sequence a large number of times, see Fig. 1 C)
pulse sequence. The mw pulse sequence in Fig. 1 B), excluding the Hahn echo,
was repeated 1000 times with a repetition time of $2\text{\,}\mathrm{ms}$.
During the waiting times, the nuclear spin polarization diffuses to the bulk
protons. This polarization is then read out on the nuclear spins by a solid-
echo sequence.

The NMR and EPR signals observed for different phase inversion times
$t_{\text{inv}}$ are shown in Fig. 2. Panel A) shows the polarization dynamics
in the absence of the overall phase change, while panels B) - F) were recorded
with increasing values for $t_{\text{inv}}$. One clearly observes the
formation of DNP echoes at a time $2t_{\text{inv}}$, indicated by dashed
lines. For short $t_{\text{inv}}$ times, i.e. $132\text{\,}\mathrm{ns}$, the
inversion is essentially perfect, and the nuclear polarization goes back to
zero at the expected echo position, while the the electron polarization goes
back to the initial intensity. At increasingly long $t_{\text{inv}}$ times,
the refocusing is no longer perfect, because the matching condition is not
perfectly fulfilled for all spin packets. Additionally, higher-order terms of
the effective coupling Hamiltonian can also lead to non-perfect refocusing.
While the nuclear-nuclear spin couplings might also play a role, we expect
this effect to be negligible on the presented time scale. Notwithstanding, the
echo formation is still clearly visible in all cases shown here.

One curiosity is that in some cases, the overall nuclear polarization obtained
with the phase inversion is higher than without it, most prominently in Fig. 2
D). While this was not the goal of this work, it does give some interesting
hints if one want to obtain as high a nuclear polarization as possible in DNP
experiments. These features are qualitatively reproduced in numerical
simulations (see Supplementary Information). In the presence of mismatches due
to electron offsets and mw inhomogeneity over the sample volume, the phase
inversion can partially correct some imperfections, as it essentially acts as
a composite pulse in the ZQ/DQ subspace.

##  VI Discussion and Outlook

We have shown that it is possible to generate and observe dynamic nuclear
polarization echoes. This is achieved by choosing a pulsed DNP sequence that
forms an effective $z$-rotation of the electron spin. Inverting the overall
phase of this sequence causes inversion of the electron-nuclear spin coupling
term in the effective Hamiltonian responsible for the polarization transfer.
This effectively lets the electron-nuclear spin system evolve backwards in
time. The DNP echo formation can be observed both on the electrons and the
nuclei. To supplement the observations in the preceding sections, we here add
different aspects that may put the results into perspective and suggest
avenues to further investigations and applications.

In principle, any pulse sequence leading to an overall $z$-rotation could be
employed. The recently introduced PulsePol sequence [20] should be a good
alternative candidate. We did not pursue this in this work, because in our
setup, the obtainable electron spin Rabi frequency is only about a factor of
2-3 larger than the nuclear Zeeman frequency. This implies that either the
pulses would fill a significant amount of the modulation time, or the
modulation time would get very long and thereby the scaling factor low. In the
former case, the PulsePol sequence generates both ZQ and DQ transfer. In
typical setups that investigate nitrogen-vacancy centers, the electron spin
Rabi frequency is much larger than the nuclear Zeeman frequency of 13C, which
alleviates this problem.

It might just as well be possible to observe DNP echoes with pulse sequences
where the effective field does not point along $z$. In this case, however, the
inversion of the coupling Hamiltonian is more complicated, since one has to
change the phase of the pulse sequence in the effective frame, in a time that
it short compared to the spin dynamics of the system. In the case of NOVEL, an
infinitely short $\pi$ pulse along the spin-lock axis would work in theory. In
practice, the spin dynamics during the pulse would need to be compensated
somehow.

The pulse scheme proposed here inverts the effective hyperfine coupling during
DNP, but it leaves the nuclear-nuclear couplings unaffected. In principle, it
might be possible to invert the sign of the complete Hamiltonian, although we
expect this to be rather challenging to realize in practice.

Serendipitously, we observed that a larger overall nuclear polarization can be
achieved with the phase inversion. This observation could be explored further,
i.e. by employing known composite pulse schemes in the ZQ/DQ subspaces.

We expect that the DNP echo could form the basis for pulse sequences that are
used to detect nuclear spins in the vicinity of an electron spin. In the
magnetic resonance community, this is known as hyperfine spectroscopy. Since
the DNP echo and its decay depend on all hyperfine and nuclear-nuclear spin-
spin couplings, it might also be possible to infer information about the
coupling network around electron spins. Of course, it should also be possible
to use DNP echoes in other quantum sensing approaches [31], including single-
spin readout experiments such as the ones employed in nitrogen-vacancy centers
[32].

###### Acknowledgements.

##  VII Acknowledgements

### Funding:

NW was supported by the Swiss National Science Foundation (Postdoc.Mobility
grant 206623). We acknowlege support from the Novo Nordisk Foundation (NERD
grant NNF22OC0076002) and the Aarhus University Research Foundation (AUFF,
grant AUFF-E-2021-9-22).

### Author contributions:

All authors developed and discussed the theory. N.W. built the spectrometer,
performed the measurements and wrote the initial draft. All authors helped
reviewing the manuscript.

### Competing Interests:

The authors declare that they have no competing interests.

### Data and materials availability:

All data needed to evaluate the conclusions in the paper are present in the
paper and/or the Supplementary Materials. Raw data and processing scripts will
be deposited on Zenodo upon acceptance of the manuscript.

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# Gravitational Collider Physics via Pulsar-Black Hole Binaries II:
Fine and Hyperfine Structures are Favored

Xi Tong1,2<EMAIL_ADDRESS>Yi Wang1,2<EMAIL_ADDRESS>Hui-Yu Zhu1,2
<EMAIL_ADDRESS>1Department of Physics, The Hong Kong University of
Science and Technology,
Clear Water Bay, Kowloon, Hong Kong, P.R.China  2The HKUST Jockey Club
Institute for Advanced Study, The Hong Kong University of Science and
Technology,
Clear Water Bay, Kowloon, Hong Kong, P.R.China

###### Abstract

A rotating black hole can be clouded by light bosons via superradiance, and
thus acquire an atom-like structure. If such a gravitational atom system is
companioned with a pulsar, the pulsar can trigger transitions between energy
levels of the gravitational atom, and these transitions can be detected by
pulsar timing. We show that in such pulsar-black hole systems, fine and
hyperfine structure transitions are more likely to be probed than the Bohr
transition. Also, the calculation of these fine and hyperfine structure
transitions are under better analytic control. Thus, these fine and hyperfine
structure transitions are more ideal probes in the search for gravitational
collider signals in pulsar-black hole systems.

##  1\. Introduction

Compact objects play an important role in astrophysics and particle cosmology.
With their astronomical mass squeezed into a compact region of spacetime,
these objects provide an ideal platform for high energy astrophysical
phenomena as well as a test ground for gravity (Shapiro and Teukolsky,, 1983;
Camenzind,, 2007).

Arguably, the most notable of the compact objects are pulsars and black holes.
Pulsars are usually rotating neutron stars that emit beams of electromagnetic
radiation from their magnetic poles. The beam sweeps across the earth
periodically due to the rotation. To a distant observer, despite its complex
internal dynamics, a pulsar can be simply viewed as a clock that ticks by
emitting periodic radio pulses. The precise periodicity of the pulsar makes it
an excellent timing tool. For instance, pulsar timing is used to measure the
orbital decay of binary systems (Hulse and Taylor,, 1975; Weisberg and
Taylor,, 2004), to detect low-frequency Gravitational Waves (GWs) (Hobbs et
al.,, 2010; Dewdney et al.,, 2009; Arzoumanian et al.,, 2020), to probe
gravity in the strong-field regime (Angélil et al.,, 2010), and even to serve
as autonomous space navigation beacons (Becker et al.,, 2013).

Black holes also play a central role in modern physics. Although isolated
stationary black holes are classically characterized only by three parameters,
they are known to carry more structures in the presence of perturbations
(Konoplya and Zhidenko,, 2011; Hawking et al.,, 2016). Not only does the
horizon emit Hawking radiation quantum mechanically (Hawking,, 1974), which
inspired numerous studies on the long-standing information paradox (Hawking,,
1975), a rotating black hole carries a dissipative ergoregion capable of
radiating particles on a classical level (Zel’Dovich,, 1971; Press and
Teukolsky,, 1972). This phenomenon known as superradiance has also been widely
studied for over half a century (see a comprehensive review given by Brito et
al., 2015b ). For bosonic particles with mass $\mu\lesssim(GM_{B})^{-1}$,
where $M_{B}$ is the mass of the black hole, superradiance triggers an
instability in the spectrum of the black hole bound states (Damour et al.,,
1976). This leads to the formation of a bosonic cloud of size
$r_{1}\sim\mathcal{O}(10^{1}\text{-}10^{3})GM_{B}$ around the black hole, with
an energy spectrum similar to that of the hydrogen atom. In isolation, such a
gravitational atom emits monochromatic GWs through pair annihilations as well
as spontaneous level transitions (Arvanitaki and Dubovsky,, 2011). When a
binary companion is introduced, the periodic orbital motion may hit the
resonance band of the gravitational atom and induce Landau-Zener transitions
(Landau,, 1932; Zener,, 1932) between different energy levels. The
backreaction effect produces floating or sinking/kicked orbits observable from
the GW signatures emitted by the binary. This recently proposed framework
aimed at probing ultralight bosons is known as Gravitational Collider Physics
(GCP) (Baumann et al., 2019a, ; Baumann et al., 2019b, ; Baumann et al.,,
2020).

However, GW (Ng et al.,, 2020) is not the only observation channel for GCP
resonances and ultralight bosons. The backreaction on the binary can be
naturally viewed as a change of orbital period derivative, $i.e.$, a timing
problem. Given the simple yet accurate time periodicity of the pulsar, it is
natural to consider the Pulsar-Black Hole (PSR-BH) binary as a viable probe of
the GCP resonances. This PSR-BH-radio observation channel has recently been
verified in (Ding et al.,, 2021) for Bohr transitions of the gravitational
atom111See other PSR-BH approaches to ultralight bosons in (Kavic et al.,,
2020; Seymour and Yagi,, 2020)..

Bohr transitions change the principal quantum number, thus due to the narrow
mass range of the pulsar, the corresponding resonance frequency is relatively
high, and the orbital period can be as short as $P_{r}\sim\mathcal{O}(1)$s.
Hence the pulsar timing accuracy proves to be always sufficient. However,
there are still several problems faced by the PSR-BH-radio channel for Bohr
transitions. (i) The short binary period during Bohr transitions suggests that
the binary is near the end of the inspiral process. Binaries with such a short
period are statistically disfavored. Given the fact that the number of
observable PSR-BH binaries in our Galaxy is limited (Faucher-Giguère and
Loeb,, 2011; Shao and Li,, 2018; Chattopadhyay et al.,, 2020), the event rate
for Bohr transitions in the PSR-BH binaries may be extremely low. (ii) During
a Bohr transition, the binary separation is comparable to the size of the
boson cloud. This may threaten the validity of the quadrupole approximation,
leading to the inadequacy of considering a narrow resonance with $\Delta m=2$
only. Additional effects such as dynamical friction (Zhang and Yang,, 2020),
upscattering effects (Wong,, 2020), and the emergence of additional molecular
states (Ikeda et al.,, 2021) may also dramatically change the prediction.

Compared to the Bohr transition, fine/hyperfine GCP transitions are
observationally more probable and theoretically cleaner to analyze. In this
work, we set out to analyze the fine/hyperfine GCP transitions of PSR-BH
binaries for ultralight scalar bosons. The advantages of probing GCP with
fine/hyperfine transition are: (i) The fine and hyperfine splittings are
suppressed by extra factors of $\alpha^{2}$ and $\alpha^{3}$ respectively,
where $\alpha\equiv GM_{B}\mu\ll 1$ is the gravitational fine structure
constant. Therefore, they have a much longer resonance period. In addition,
some fine/hyperfine transitions give floating orbits which enjoy an extremely
long duration. This dramatically increases the event rate. (ii) The increase
in resonance period is accompanied by the increase in the binary separation,
which is now much greater than the cloud radius. This ensures the validity of
the quadrupole approximation and the narrow resonance.

Apparently, a disadvantage of fine/hyperfine transitions is that the signal is
weaker than Bohr transitions. The long orbital period gives rise to an orbital
decay that may be too tiny to be detected by the first generation of space GW
detectors (Amaro-Seoane et al.,, 2017; Luo et al.,, 2020; Mei et al.,, 2021;
Ando et al.,, 2009). However, thanks to the well-established timing accuracy
of pulsars, a long-term observation is sufficient to capture the
fine/hyperfine resonances, as we will show below.

This paper is organized as follows. In Sect. 2, we review some technical
details of the gravitational atom and GCP transitions. In Sect. 3, we discuss
the problems faced by the Bohr transitions and motivate our study for
fine/hyperfine transitions. Then in Sect. 4, we focus on the major
fine/hyperfine transitions induced by the quadrupole moment of the orbiting
pulsar and analyze their observational feasibility. We conclude and give
future prospects in Sect. 5.

##  2\. The Gravitational Atom

In this section, we briefly review the basic aspects of the gravitational atom
and GCP following Baumann et al., 2019a  and Baumann et al., (2020). A Kerr
black hole is equipped with a dissipative ergosphere that can amplify incoming
waves (Penrose,, 1969; Zel’Dovich,, 1971). For a massive bosonic field, its
mass serves as a natural mirror that reflects back the amplified modes (Press
and Teukolsky,, 1972; Cardoso et al.,, 2004), from which an instability is
generated (Damour et al.,, 1976). This superradiance instability leads to the
growth of a bosonic cloud, whose behavior is governed by the Schrödinger
equation with corrections from Kerr spacetime,

$i\partial_{t}\psi(t,\vec{x})=\left(-\frac{1}{2\mu}\partial_{\vec{x}}^{2}-\frac{\alpha}{r}+\mathcal{O}(\alpha^{2})\right)\psi(t,\vec{x})\leavevmode\nobreak\
,$ (1)

where $\alpha\ll 1$ guarantees the validity of non-relativistic expansion. The
solutions of the Schrödinger equation with in-going boundary condition at the
horizon are atomic states $|nlm\rangle$ labeled by the principal, angular and
magnetic quantum numbers222Throughout this paper, we assume the ultralight
boson is a (pseudo)scalar.. The frequency of each eigenstate is in general
complex: $\omega_{nlm}=E_{nlm}+i\Gamma_{nlm}$, with

$\displaystyle E_{nlm}$
$\displaystyle=\mu\Bigg{(}1-\frac{\alpha^{2}}{2n^{2}}-\frac{\alpha^{4}}{8n^{4}}-\frac{(3n-2l-1)\alpha^{4}}{n^{4}(l+1/2)}$
(2) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\frac{2\tilde{a}m\alpha^{5}}{n^{3}l(l+1/2)(l+1)}+O(\alpha^{6})\Bigg{)},$

and

$\Gamma_{nlm}=2\tilde{r}_{+}C_{nlm}(m\Omega_{H}-\omega_{nlm})\alpha^{4l+5}\leavevmode\nobreak\
,$ (3)

where $\tilde{r}\equiv r/M_{B}$ and $\tilde{a}\equiv a/M_{B}\lesssim 1$ is the
dimensionless black hole spin. For simplicity, we will assume $\tilde{a}\simeq
1$ throughout this work, since astrophysical BHs in binaries generally have a
large spin (O’Shaughnessy et al.,, 2005; Nielsen,, 2016). $C_{nlm}$ can be
found in Baumann et al., 2019a . Notice that although (3) is derived under
Detweiler’s approximation (Detweiler,, 1980) with $\alpha\ll 1$, numerical
studies have confirmed it validity for $\alpha<0.5$ (Brito et al., 2015b, ).
For a positive $\Gamma_{nlm}$, the total mass of the state $M_{nlm}$ will grow
at a timescale $T^{\text{(growth)}}_{nlm}\equiv\Gamma_{nlm}^{-1}$ until
saturation, then it slowly depletes via the emission of GWs. The depletion
power of a highest helicity state with $l=m$ is

$\dot{M}_{nlm}=-B_{nl}\left(\frac{M_{nlm}}{M_{B}}\right)^{2}\alpha^{4l+10}\leavevmode\nobreak\
.$ (4)

The coefficient $B_{nl}$ is computed by Yoshino and Kodama, (2014) in the flat
spacetime limit as

$B_{nl}=\frac{16^{l+1}l(2l-1)\Gamma(2l-1)^{2}\Gamma(l+n+1)^{2}}{n^{4l+8}(l+1)\Gamma(l+1)^{4}\Gamma(4l+3)\Gamma(n-l)^{2}}.$
(5)

However, we caution the reader that numerical analysis shows that (5)
underestimates the depletion power by approximately one order of magnitude
(Yoshino and Kodama,, 2014; Brito et al., 2015a, ). The scaling powers of
$M_{nlm}$ and $\alpha$, in contrast, are robust for $\alpha\ll 1$.
Nevertheless, due to its analytical generality for all $n$ and $l$, we will
adopt (5) to estimate the depletion time scale,

$T_{nlm}^{\text{(deplete)}}\approx
B_{nl}^{-1}\frac{M_{B}^{2}}{M_{nlm,0}}\alpha^{-4l-10}\leavevmode\nobreak\ ,$
(6)

and warn the reader about the potential $\mathcal{O}(10)$ uncertainty due to
(5). Here $M_{nlm,0}$ is the initial mass of the cloud state, whose value at
saturation can be estimated using angular momentum conservation (see Table. 1
in Baumann et al., (2020)).

Now let us come to the binary system case. The motion of a binary companion
will generate a periodic tidal perturbation on the gravitational atom, leading
to level crossings in the atomic spectrum. At a large binary separation, the
tidal perturbation is solely dependent on the mass of the binary companion.
Thus it can be any astronomical object compact enough to fit into the orbit.
In particular, we consider a pulsar of mass $M_{P}$. GCP transitions happen
when the frequency of pulsar revolution matches the energy difference between
two atomic states. After expanding the tidal perturbation into multipole
moments and calculating its matrix elements between atomic states, one obtains
the following selection rule for a process
$|nlm\rangle\to|n^{\prime}l^{\prime}m^{\prime}\rangle$ (Baumann et al., 2019a,
):

$\displaystyle-m^{\prime}+m_{*}+m=0$ (7) $\displaystyle
l+l_{*}+l^{\prime}=2p,\text{ for }p\in\mathbb{Z}$
$\displaystyle|l-l^{\prime}|\leqslant l_{*}\leqslant
l+l^{\prime}\leavevmode\nobreak\ .$

For large circular equatorial orbits, the resonance period is

$P_{r}=2\pi\left|\frac{\Delta m}{\Delta E}\right|\leavevmode\nobreak\ ,$ (8)

with $\Delta E\equiv
E_{n^{\prime}l^{\prime}m^{\prime}}-E_{nlm},\leavevmode\nobreak\ \Delta m\equiv
m^{\prime}-m$. We see that a larger $\Delta E$ generically corresponds to
shorter orbital periods.

The atomic transition produces backreaction to the binary, resulting in
floating orbit and sinking orbit. We call transition with $\Delta E<0$
floating orbit, where the boson cloud loses its energy, delaying the orbital
decay. In contrast, $\Delta E>0$ gives sinking orbit, where the period of
binary system decreases faster with their energy given to the cloud. The total
transition time is $\Delta t_{\text{tot}}=\Delta t+\Delta t_{c}$, where
$\Delta t$ is the transition time without backreaction, and $\Delta t_{c}$ is
the extra time caused by the backreaction. Here $\Delta t_{c}$ is positive for
floating orbits and is negative for sinking orbits. The detailed expressions
of $\Delta t$ and $\Delta t_{c}$ can be found in (Baumann et al.,, 2020). A
GCP transition can sometimes turn a superradiant state into a non-superradiant
state with negative $\Gamma_{n^{\prime}l^{\prime}m^{\prime}}$, where the cloud
is absorbed into the BH. Interestingly, it is recently pointed out this may be
avoided under certain conditions (Takahashi and Tanaka,, 2021).

##  3\. Bohr transitions vs fine and hyperfine transitions

For Bohr transitions among the lowest a few states, the energy difference is
typically large, hence a short binary period. For instance, a Bohr transition
from $n=3$ to $n=2$ gives a resonance period

$P_{3\to 2}=\frac{288GM_{B}}{5\alpha^{3}}=(2.6\text{
s})\times\left(\frac{M_{B}}{5M_{\odot}}\right)\left(\frac{\alpha}{0.12}\right)^{-3}\leavevmode\nobreak\
.$ (9)

Denoting $q\equiv\frac{M_{P}}{M_{B}}$, the time left until the merger is then

$T^{(\text{merger})}_{3\to 2}=(1\text{
day})\times\frac{(1+q)^{1/3}}{q}\left(\frac{M_{B}}{5M_{\odot}}\right)\left(\frac{\alpha}{0.12}\right)^{-8}\leavevmode\nobreak\
,$ (10)

suggesting that the PSR-BH binary is near the end of the inspiral phase.
Although the orbital decay at this stage is significant for both pulsar timing
and GW detectors, the likelihood of encountering a binary at this stage is
smaller than that in the middle of the inspiral phase by at least several
orders of magnitude. Yet the total number of observable PSR-BH binaries in our
Galaxy is estimated to be $\mathcal{O}(10^{2}\text{-}10^{3})$ (Faucher-Giguère
and Loeb,, 2011; Shao and Li,, 2018; Chattopadhyay et al.,, 2020). Therefore,
the event rate for Bohr transitions may be extremely low.

Another problem for Bohr transitions comes from the validity of multipole
expansion. The binary separation for a typical $n=3$ to $n=2$ Bohr transition
is

$R_{3\to 2}=\left(4.8\times 10^{3}\text{
km}\right)\times(1+q)^{1/3}\left(\frac{M_{B}}{5M_{\odot}}\right)\left(\frac{\alpha}{0.12}\right)^{-2}\leavevmode\nobreak\
.$ (11)

The size of the bosonic cloud for the state with principal quantum number $n$
is $r_{n}=n^{2}r_{1}$, with $r_{1}=M_{B}\alpha^{-2}$ being the Bohr radius.
For $n=3$, we have

$r_{3}=9M_{B}\alpha^{-2}=\left(4.6\times 10^{3}\text{
km}\right)\left(\frac{M_{B}}{5M_{\odot}}\right)\left(\frac{\alpha}{0.12}\right)^{-2}\leavevmode\nobreak\
.$ (12)

Hence the multipole expansion, in particular, the narrow resonance
approximation of the $l_{*}=m_{*}=2$ quadrupole moment (Baumann et al.,, 2020)
may be questionable for $q\sim 1$. This is because when $R_{3\to 2}$ is close
to $r_{3}$, the pulsar is already moving inside the cloud, and higher
multipole moments that mediate other transitions with $|\Delta m|>2$ are non-
negligible. In addition, the dynamical friction of the cloud (Zhang and Yang,,
2020), upscattering effects (Wong,, 2020) and the formation of molecular
states (Ikeda et al.,, 2021) can also have important impacts on the
transition. As a result, an accurate account for Bohr transitions may require
a non-perturbative treatment.

For fine ($\Delta n=0,\Delta l\neq 0$) and hyperfine ($\Delta n=\Delta
l=0,\Delta m\neq 0$) transitions, however, both difficulties can be evaded.
From (2), we see that the energy difference in a fine (hyperfine) transition
is smaller by a factor of $\alpha^{2}$ ($\alpha^{3}$) than a Bohr transition,
leading to a much longer resonance period. This means they can happen for PSR-
BH binaries in the middle phase of inspiral, which is statistically more
favored. The binary separation is also enlarged by a factor of $\alpha^{-4/3}$
($\alpha^{-2}$) for a fine (hyperfine) transition, making the multipole
expansion well-defined and the $|\Delta m|=m_{*}=2$ approximation accurate.

Moreover, for $n\leqslant 3$, fine/hyperfine transitions always occur with
$\Delta E<0$. This is because the initial cloud state has maximal helicity and
the highest energy within the same $l$-multiplet, and the excitation to a
higher $l$-multiplet is forbidden for $n\leqslant 3$. This suggests that these
phenomenologically interesting fine/hyperfine transitions give rise to
floating orbits. Thus the time spent on the GCP resonance is extended by
$\Delta t_{c}$, further increasing the likelihood of detection. If $\Delta
t_{c}\gg\Delta t\gtrsim T^{(\text{deplete})}$, once the binary hits the
resonance band, the pulsar will be stuck on the floating orbit until the cloud
depletes, which typically takes $10^{8}$ yr. This greatly enhances the
detection likelihood.

Table 1A comparison between Bohr transitions and fine/hyperfine transitions | Transition |  $P_{r}$ (hr) |  $\Delta t$ (yr) |  $\Delta t_{c}$ (yr) | $r_{n}/R_{r}$ |  $T^{(\text{growth})}$ (yr) |  $T^{(\text{deplete})}$ (yr) |  $T^{(\text{merge})}$ (yr)
---|---|---|---|---|---|---|---|---
Bohr | $|322\rangle\to|200\rangle$ | $6.4\times 10^{-4}$ | $2.8\times 10^{-3}$ | $2.2\times 10^{-3}$ | $0.96$ | $9600$ | $10^{13}$ | $7.9\times 10^{-3}$
$|322\rangle\to|100\rangle$ | $9.9\times 10^{-5}$ | $1.9\times 10^{-5}$ | $2.8\times 10^{-5}$ | 3.3 | 9570 | $10^{13}$ | $5.5\times 10^{-5}$
$|311\rangle\to|21-1\rangle$ | $7.0\times 10^{-4}$ | $3.6\times 10^{-3}$ | $1.4\times 10^{-2}$ | 0.89 | $4.7\times 10^{-2}$ | $10^{5}$ | $1.0\times 10^{-2}$
$|211\rangle\to|31-1\rangle$ | $7.1\times 10^{-4}$ | $3.6\times 10^{-3}$ | $-1.4\times 10^{-2}$ | $0.89$ | $1.7\times 10^{-2}$ | $10^{5}$ | $1.0\times 10^{-2}$
Fine | $|322\rangle\to|300\rangle$ | $1.9\times 10^{-2}$ | $25$ | $6.3$ | $9.8\times 10^{-2}$ | $9600$ | $10^{13}$ | $72$
Hyperfine | $|322\rangle\to|320\rangle$ | 12 | $7.5\times 10^{8}$ | $2.2\times 10^{7}$ | $1.3\times 10^{-3}$ | $9600$ | $10^{13}$ | $2.1\times 10^{9}$
$|321\rangle\to|32-1\rangle$ | $6.4$ | $1.3\times 10^{8}$ | $2.4\times 10^{7}$ | $2.0\times 10^{-3}$ | $6.4\times 10^{5}$ |  $10^{5}$-$10^{13}$ | $3.8\times 10^{8}$
$|311\rangle\to|31-1\rangle$ | $1.3$ | $1.8\times 10^{6}$ | $5.6\times 10^{5}$ | $6.0\times 10^{-3}$ | $4.7\times 10^{-2}$ | $10^{5}$ | $5.1\times 10^{6}$
$|211\rangle\to|21-1\rangle$ | 0.38 | $7.0\times 10^{4}$ | $3.3\times 10^{4}$ | $6.0\times 10^{-3}$ | $1.7\times 10^{-2}$ | $10^{5}$ | $2.0\times 10^{5}$

A quantitative comparison of Bohr transitions and fine/hyperfine transitions
is shown in TABLE. 1, where the mass parameters are fixed to be
$\alpha=0.12,M_{B}=5M_{\odot}$ and $M_{P}=1.4M_{\odot}$. Here we have
enumerated all GCP transitions that involve states with $n,n^{\prime}\leqslant
3$, and that are mediated by the $l_{*}=m_{*}=2$ quadrupole moment. Because of
the uncertainty in the $T^{(\text{deplete})}$ formula (6), we have only kept
its order of magnitude. Also note that the depletion time for the state
$|321\rangle$ has not yet been computed in the literature to our best
knowledge. Therefore, we only give its possible range estimated by
$T_{322}^{(\text{deplete})}$ and $T_{311}^{(\text{deplete})}$. It is clear
from TABLE. 1 that fine/hyperfine transitions solve all issues aforementioned,
by having a much larger $P_{r}$, a much longer $\Delta t,T^{(\text{merger})}$,
and a much smaller ratio $r_{n}/R_{r}$.

Going beyond the lowest a few states, we can find more interesting structures
emerging. Given the $l_{*}=m_{*}=|\Delta m|=2$ constraint and the selection
rules (7), one can find all possible quadrupole-mediated GCP transitions with
$n,n^{\prime}\leqslant n_{\text{max}}$. The formula for the total number of
allowed transitions is

$N_{\text{tot},n_{\text{max}}}^{(B/F/HF)}=\frac{1}{2}\left(n_{\max}-2\right)n_{\max}\left(n_{\max}^{2}-2n_{\max}+3\right)\leavevmode\nobreak\
,$ (13)

where $n_{\text{max}}\geqslant 4$. Within the $n$-th energy level, the number
of allowed fine transitions and hyperfine transitions are given by

$\displaystyle N_{n}^{(F)}$ $\displaystyle=2n^{2}-10n+14$ $\displaystyle
N_{n}^{(HF)}$ $\displaystyle=n^{2}-3n+3\leavevmode\nobreak\
,\leavevmode\nobreak\ \text{with }n\geqslant 4\leavevmode\nobreak\ .$ (14)

We have enumerated all allowed transitions with $n_{\text{max}}=4$ and
$n_{\text{max}}=8$ in FIG. 4. The transition graph with
$n_{\text{max}}\geqslant 4$ neatly factorize into the direct product of four
connected subgraphs. This can be understood as the consequence of the
quadrupole approximation. Because $\Delta m=2$, states with odd $m$ cannot
jump to states with even $m^{\prime}$, vice versa. In addition, parity
conservation further divides these two sets into parity-odd (odd $l$ and
$l^{\prime}$) families and parity-even (even $l$ and $l^{\prime}$) families,
leading to the four disjoint sectors. Note that although higher multipole
moments ($l_{*}\geqslant 3$) are able to mediate Bohr transitions between
these sectors, they are in general too small to influence fine/hyperfine
transitions.

The advantage of fine/hyperfine transitions over Bohr transitions persists as
more GCP transitions are included. In FIG. 4, we have shown the time left
until merger as well as the ratio of cloud size and binary separation for all
transitions up to $n_{\text{max}}=10$. It is clear that the three types of
transitions occupy different regions in the parameter space. Most Bohr
transitions lie in the gray region where the multipole expansion breaks down,
while fine/hyperfine transitions are far safer. Fine/hyperfine transitions
also correspond to a much longer time before merger, hence a higher event
rate.

Figure 1.— The distribution of Bohr/fine/hyperfine transitions in the
parameter space spanned by the time $T^{\text{merger}}$ left until merger as
well as the ratio of cloud size and binary separation $r_{n}/R_{r}$. Here we
have considered all quadrupole-mediated GCP transitions up to
$n_{\text{max}}=10$. There are in total $N_{\text{tot},10}^{(B/F/HF)}=3320$
transitions. The parameters are again chosen as $\alpha=0.12$,
$M_{B}=5M_{\odot}$ and $M_{P}=1.4M_{\odot}$. The gray region is where the
multipole expansion and narrow resonance approximation break down. The dashed
line represents the age of the universe. A smaller ratio $r_{n}/R_{r}$ gives
better quadrupole approximation and a longer $T^{\text{merger}}$ enhances the
event rate444Notice that a $T^{\text{merger}}$ longer than the age of the
universe does not mean the transition cannot happen, nor the binary does not
exist. It just means that the binary is going to spend an extremely long time
in the inspiral phase.. Hence the advantage of fine/hyperfine transitions over
Bohr transitions is obvious.

##  4\. Uncovering fine and hyperfine structures: Pulsar timing accuracy

The direct observation of GCP transitions for a PSR-BH binary relies on an
accurate measurement of orbital motion, which is recorded as modulations in
the Rømer delay of pulsar time-of-arrivals. Unlike Bohr transitions,
fine/hyperfine transitions occur at a much longer orbital period, where the GW
emission is still weak. The resonance frequency lies in the range of the
space-based GW detectors such as LISA. At a low orbital frequency, the
corresponding orbital decay is much slower. Such a weak effect may require a
long-term observation that lasts more than a decade. As a comparison, LISA
only has a lifetime of 4-6 years (Amaro-Seoane et al.,, 2017). Therefore, it
is questionable whether the first generation of space GW detectors (Amaro-
Seoane et al.,, 2017; Luo et al.,, 2020; Mei et al.,, 2021; Ando et al.,,
2009) are precise enough to probe the GCP transitions in due time. In
contrast, radio telescopes are earth-based and can last many decades. For
instance, the Arecibo telescope built in 1963 had been functioning for 57
years before its tragic collapse in 2020. Thus a long-term observation of a
PSR-BH binary may reveal the tiny deviations of orbital decay and uncover the
fine/hyperfine structure of the gravitational atom.

The orbital decay of the PSR-BH binary, according to Hulse and Taylor, (1975)
and Weisberg and Taylor, (2004), can be observed by recording the periastron
time shift

$\Delta_{P}=t-P(0)\int^{t}_{0}\frac{1}{P(t)}{\rm{d}}t^{\prime}\leavevmode\nobreak\
.$ (15)

If we consider a small timescale with respect to that of a significant orbital
decay, we can linearize the period change by $P(t)\simeq P(0)+\dot{P}t$. The
periastron time shift then increases quadratically with observation time:

$\Delta_{P}=\frac{1}{2}\frac{\dot{P}}{P}t^{2}.$ (16)

Under the influence of isolated atomic transitions, the orbital decay due to
GW emission can be approximated as (Ding et al.,, 2021)

$\displaystyle\dot{P}$
$\displaystyle=-\frac{96}{5}(2\pi)^{8/3}(GM_{B})^{5/3}\frac{q}{(1+q)^{1/3}}P^{-5/3}$
(17) $\displaystyle\times\frac{1}{1\pm\frac{\Delta t_{c}}{\Delta
t}\times\Pi(\frac{P-P_{r}}{\Delta P_{r}})},$

where $\Pi$ is the Heaviside-Pi window function that characterize the
bandwidth of the GCP resonance, $i.e.$, $\Delta
P_{r}\simeq\frac{0.6q}{1+q}P_{r}$. The deviation of $P(t)$ and hence
$\Delta_{P}(t)$ from the general relativity prediction will be the signal that
we are after.

In order to successfully detect GCP resonances, we need to ensure that the
uncertainty $\sigma_{\Delta_{P}}$ is smaller than total Periastron time shift
$\Delta_{P}$. Notice that the periastron time shift is not measured from its
definition (15), which demands an accurate measurement of the orbital period
as a function of time. Rather, we measure the periastron time shift directly
by counting the accumulated orbital periods. We determine one orbital period
by counting the number of pulses between two pulses with equal Rømer delay and
equal time derivative of Rømer delay. In order to resolve an orbital period,
the maximal Rømer delay must be larger than the pulse width $w$, which is on a
similar order as the pulse period $\tau$. Therefore, we demand

$\frac{2R_{r}}{1+q}>\tau\leavevmode\nobreak\ ,$ (18)

which automatically implies $P_{r}>\tau$. The error during one continuous
observation window is also $w\sim\tau$. Suppose we can observe the pulsar for
$t_{obs}$ every day, which means we can measure $t_{obs}/P$ periods every day.
That is, for every single continuous measurement, the error can be estimated
by $\frac{\tau}{\text{min}(t_{obs},t)/P}$. If we observe for $0<t\leqslant
T_{obs}$, where $T_{obs}$ is the maximal observation time, then the number of
independent measurement is $\lceil t/1\text{ day}\rceil$, where $\lceil\rceil$
is ceil function. In summary, the uncertainty for Periastron time shift is

$\sigma_{\Delta_{P}}=\frac{1}{\sqrt{\lceil
t/1\text{day}\rceil}}\frac{\tau}{\text{min}(t_{obs},t)/P}\leavevmode\nobreak\
.$ (19)

We can detect the GCP transitions only if the difference between the
periastron time shift with transition backreaction and that without it is
greater than the observation uncertainty,

$\Big{|}\Delta_{P}|_{GCP}-\Delta_{P}|_{GR}\Big{|}>\sigma_{\Delta_{P}}\leavevmode\nobreak\
.$ (20)

In addition, there are two more constraints on the model parameters. Namely,
the superradiant time scale of boson cloud should be short enough to observe,
and the cloud should be stable on an astrophysical timescale (Baumann et al.,,
2020),

$T^{\text{(growth)}}\lesssim 10^{6}\text{ yrs, }T^{\text{(deplete)}}\gtrsim
10^{8}\text{ yrs}\leavevmode\nobreak\ .$ (21)

Figure 2.— The feasible regions for $\alpha$, $M_{B}$ and different pulsar
periods $\tau$. Here we have taken the mass of the pulsar to be
$M_{P}=1.4M_{\odot}$ and assumed an observation time $T_{obs}=1$ decade. The
black dashed line given by $\Delta t=\Delta t_{c}$ divides the parameter space
into the upper half with strong backreaction and the lower half with weak
backreaction. The timing accuracy edge acquires a turning point near this line
because the periastron time shift has different asymptotic behaviors across
the line.

Combining the constraints (18), (20) and (21), we obtain the feasible
parameter region for the fine/hyperfine transitions555Note that the transition
$|321\rangle\to|32-1\rangle$ is not shown due to our lack of information on
$T_{321}^{(\text{deplete})}$. shown in FIG. 2. Overall, transitions starting
with $|322\rangle$ give a wide range of parameter region that covers
$10^{-3}M_{\odot}<M_{B}<10^{8}M_{\odot}$ and $0.06<\alpha<0.5$. In contrast,
transitions starting with $|311\rangle$ and $|211\rangle$ allow a relatively
limited parameter space with significantly smaller $\alpha$. Notice that the
right edge of the parameter space is constrained by the depletion time, which
is subjected to an $\mathcal{O}(10)$ uncertainty. The lower edge is
constrained by the Rømer delay resolvability (18). The upper edge is
constrained by the timing accuracy of periastron time deviation (20). The left
edge is either constrained by the timing accuracy (for hyperfine transitions)
or the cloud growth time (for fine transitions).

Thus we see that timing accuracy plays an important role in probing hyperfine
transitions. Naturally, pulsars with shorter rotations period $\tau$ provide a
finer resolution of the orbital motion, thereby increasing the timing
accuracy. Alternatively, for a given PSR-BH system, one can also extend the
observation time $T_{\text{obs}}$ to increase the timing accuracy. This fact
is demonstrated in FIG. 3 for the hyperfine transition
$|322\rangle\to|320\rangle$. Some parameter choices may require decades of
observation for a clear detection, a task suitable only for ground-based
apparatus such as radio telescopes.

The fine/hyperfine transitions for states with $n\geqslant 4$ are
qualitatively similar. Since the superradiance growth rate and the GW
depletion rate is not very sensitive to the principal quantum number, the
feasible parameter regions for $|n22\rangle\to|n20\rangle$ and
$|n22\rangle\to|n00\rangle$ are similar to that plotted in FIG. 2. Transitions
starting with higher angular quantum number $l$ generally require a larger
$\alpha$, since their growth rate are further suppressed by powers of
$\alpha$.

Figure 3.— The feasible parameter space for $\alpha$, $M_{B}$ and different
observe time $T_{\text{obs}}$ for the hyperfine transition
$|322\rangle\to|320\rangle$. Here we have chosen $M_{P}=1.4M_{\odot}$ and
$\tau=0.1$s.

##  5\. Conclusion

In this paper, we focused on probing fine/hyperfine GCP transitions in a PSR-
BH binary with pulsar timing. Starting from a general review of the
gravitational atom and GCP, we pointed out the problems faced by Bohr
transitions. Namely, the Bohr transitions may be extremely rare because they
happen near the end of the inspiral phase. The quadrupole narrow resonance
approximation may also become invalid when the binary separation is comparable
to the cloud size. Then we showed that these problems can be evaded in the
fine/hyperfine transitions, which typically enjoy a longer resonance period
and a greater binary separation. All fine/hyperfine transitions with
$n\leqslant 3$ lead to floating orbits that delay the orbital decay, further
enhancing the likelihood of observation. The advantage persists for higher
energy levels, as the three types of transitions occupy distinct regions in
the parameter space. The subsequent analysis of pulsar timing accuracy
demonstrates the feasibility of detecting fine/hyperfine transitions in GCP.
In particular, we find that the fine transition $|n22\rangle\to|n00\rangle$
and hyperfine transitions $|n22\rangle\to|n20\rangle$ give wide parameter
regions that can be probed via pulsar timing. Increasing the total observation
time also increases the timing accuracy, making the detection of transitions
with large black hole mass possible. In the spirit of multi-messenger
astronomy, due to the long lifetime of radio telescopes and the stringent
accuracy requirement, the observation of fine/hyperfine transitions via this
PSR-BH-radio channel should serve as a complement to observing Bohr
transitions via the BH-BH-GW channel.

However, there are still many questions left unanswered in the current work,
and we hope to address them in the future. We conclude this paper by
mentioning a few of them. First, our estimation of boson cloud depletion time
is based on an approximate formula with a considerable amount of error,
especially at large $\alpha$ and when $l\neq m$. We hope to improve our
constraints in the future using more accurate numerical solutions of cloud
depletion. Second, we have argued that the event rate of fine/hyperfine
transitions is greatly enhanced by their long orbital periods and floating
orbits, yet we did not give any explicit estimation of the likelihood of
observation. To perform such an analysis, one needs to consider the initial
distribution of BHs and pulsars in the Galaxy as well its evolution history.
Third, in addition to pulsar timing, the measurement of the Doppler effect and
the ellipsoidal modulation of a White Dwarf (WD) in a binary is nowadays
accurate enough for detecting the orbital decay (Hermes et al.,, 2012; Burdge
et al., 2019a, ; Burdge et al., 2019b, ). Since there are more WDs than
pulsars in the Galaxy, it is interesting to consider probing GCP resonances in
WD-BH binaries, whose event rate can be further enhanced. We can also consider
the Doppler effect of a generic Neutron Star (NS) which may not necessarily be
a pulsar. This possibility becomes more exciting after the recent LIGO-Virgo
discovery of two NS-BH mergers by Abbott et al., (2021). The question is, of
course, whether the accuracy can reach the requirement of detecting the
deviations in the orbital decay due to GCP resonances. We leave a detailed
analysis to future works.

(a) Allowed GCP transitions up to $n_{\text{max}}=4$

(b) Allowed GCP transitions up to $n_{\text{max}}=8$

Figure 4.— The transition graph for states with $n,n^{\prime}\leqslant
n_{\text{max}}$. The purple, blue and orange arrows represent Bohr
transitions, fine transitions and hyperfine transitions, respectively. The
size of the vertices indicates the superradiance growth rate, $i.e.$, a large
vertex corresponds to a fast-growing cloud.

## Acknowledgment

We would like to thank Qianhang Ding and Chon Man Sou for helpful discussions.
This work was supported in part by the CRF grant C6017-20GF by the RGC of Hong
Kong SAR, and the NSFC Excellent Young Scientist (EYS) Scheme (Hong Kong and
Macau) Grant No. 12022516.

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