Laser excitation of the 1s-hyperfine transition in muonic hydrogen

P. Amaro1$\star$, A. Adamczak2, M. Abdou Ahmed3, L. Affolter4, F. D. Amaro5,
P. Carvalho1, T.-L. Chen6, L. M. P. Fernandes5, M. Ferro1, D. Goeldi4, T.
Graf3, M. Guerra1, T. W. Hänsch7,8,

C. A. O. Henriques5, Y.-C. Huang6, P. Indelicato9, O. Kara4, K. Kirch4,10, A.
Knecht10,

F. Kottmann4,10, Y.-W. Liu6, J. Machado1, M. Marszalek10, R. D. P. Mano5, C.
M. B. Monteiro5, F. Nez9, J. Nuber4,10, A. Ouf11, N. Paul9, R. Pohl11, 12, E.
Rapisarda10, J. M. F. dos Santos5,

J. P. Santos1, P. A. O. C. Silva5, L. Sinkunaite10, J.-T. Shy6, K. Schuhmann4,
S. Rajamohanan11, A. Soter4, L. Sustelo1, D. Taqqu4,10, L.-B. Wang6, F.
Wauters12, P. Yzombard9, M. Zeyen4, A. Antognini4,10$\dagger$,

1 Laboratory of Instrumentation, Biomedical Engineering and Radiation Physics
(LIBPhys-UNL), Department of Physics, NOVA School of Science and Technology,
NOVA University Lisbon, 2829-516 Caparica, Portugal

2 Institute of Nuclear Physics, Polish Academy of Sciences, PL-31342 Krakow,
Poland

3 Institut für Strahlwerkzeuge, Universität Stuttgart, 70569 Stuttgart,
Germany

4 Institute for Particle Physics and Astrophysics, ETH Zurich, 8093 Zurich,
Switzerland

5 LIBPhys-UC, Department of Physics, University of Coimbra, P-3004-516
Coimbra, Portugal

6 Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan

7 Ludwig-Maximilians-Universität, Fakultät für Physik, 80799 Munich, Germany

8 Max Planck Institute of Quantum Optics, 85748 Garching, Germany

9 Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-Université PSL,
Collège de France, 75005 Paris, France

10 Paul Scherrer Institute, 5232 Villigen–PSI, Switzerland

11 Institut für Physik, QUANTUM, Johannes Gutenberg-Universität Mainz, 55099
Mainz, Germany

12 PRISMA+ Cluster of Excellence and Institute of Nuclear Physics, Johannes
Gutenberg-Universität Mainz, 55099 Mainz, Germany

<EMAIL_ADDRESS><EMAIL_ADDRESS>
## Abstract

The CREMA collaboration is pursuing a measurement of the ground-state
hyperfine splitting (HFS) in muonic hydrogen ($\mu$p) with 1 ppm accuracy by
means of pulsed laser spectroscopy to determine the two-photon-exchange
contribution with $2\times 10^{-4}$ relative accuracy. In the proposed
experiment, the $\mu$p atom undergoes a laser excitation from the singlet
hyperfine state to the triplet hyperfine state, then is quenched back to the
singlet state by an inelastic collision with a H2 molecule. The resulting
increase of kinetic energy after the collisional deexcitation is used as a
signature of a successful laser transition between hyperfine states. In this
paper, we calculate the combined probability that a $\mu$p atom initially in
the singlet hyperfine state undergoes a laser excitation to the triplet state
followed by a collisional-induced deexcitation back to the singlet state. This
combined probability has been computed using the optical Bloch equations
including the inelastic and elastic collisions. Omitting the decoherence
effects caused by the laser bandwidth and collisions would overestimate the
transition probability by more than a factor of two in the experimental
conditions. Moreover, we also account for Doppler effects and provide the
matrix element, the saturation fluence, the elastic and inelastic collision
rates for the singlet and triplet states, and the resonance linewidth. This
calculation thus quantifies one of the key unknowns of the HFS experiment,
leading to a precise definition of the requirements for the laser system and
to an optimization of the hydrogen gas target where $\mu$p is formed and the
laser spectroscopy will occur.

###### Contents

  1. 1 Introduction
  2. 2 The experimental scheme
  3. 3 Theoretical framework
    1. 3.1 Bloch equations
    2. 3.2 Collisional rates
    3. 3.3 Rabi frequencies and matrix elements
    4. 3.4 Doppler broadening
  4. 4 Analytical expressions for two limiting regimes
    1. 4.1 Fermi-golden-rule regime
    2. 4.2 Rabi-oscillation regime: Two-level Bloch equations
  5. 5 Results and discussion
    1. 5.1 $\bar{\rho}_{33}$ at the experimental conditions
    2. 5.2 Resonance profile at the experimental conditions
  6. 6 Conclusion
  7. A Hyperfine dipole matrix elements
  8. B Results for a statistical distribution and $\tau=10$ ns

##  1 Introduction

Highly accurate measurements of atomic transitions can be used as precise
probes of low-energy properties of nuclei. As the Bohr radius of hydrogen-like
atoms decreases with increasing orbiting particle mass, muonic atoms (atoms
formed by a negative muon and a nucleus) have an enhanced sensitivity to
nuclear structure effects [1, 2, 3, 4, 5, 6, 7]. The Charge Radius Experiments
with Muonic Atoms (CREMA) collaboration in recent years has performed laser
spectroscopy of the $2s-2p$ (Lamb shift) transitions in muonic hydrogen
($\mu$p) [4, 6], muonic deuterium ($\mu$d) [8] and muonic helium
($\mu^{4}$He+) [9] and extracted the corresponding nuclear charge radii with
an unprecedented accuracy. The impact of the $\mu$p measurements on beyond-
standard-model searches, on precision atomic physics, and on the proton
structure can be found in recent reviews [10, 11, 12, 7]. Along this line of
research, the CREMA collaboration is presently aiming at the measurement of
the ground-state hyperfine splitting (HFS) in $\mu$p with 1 ppm relative
accuracy by means of pulsed laser spectroscopy.

From the measurement of the HFS, precise information about the magnetic
structure of the proton can be extracted [13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23]. Specifically, by comparing the measured HFS transition frequency with
the corresponding theoretical prediction based on bound-state QED calculations
[13, 19, 20, 5], the two-photon-exchange contribution can be extracted with
approximately $2\times 10^{-4}$ relative accuracy. Because the two-photon-
exchange contribution can be expressed as the sum of a finite-size (static,
elastic) part proportional to the Zemach radius ($R_{Z}$) and a polarizability
part (dynamic, virtual excitation), its determination can be used to extract
separately the two parts: the Zemach radius when the polarizability
contribution is assumed from theory [21, 16, 22, 24, 25, 13, 15, 18, 23], and
the polarizability contribution when taking $R_{Z}$ from electron-proton
scattering or hydrogen spectroscopy [26, 19, 27, 28].

Figure 1: (a) Hyperfine structure of the 1s-state in $\mu$p divided into the
triplet ($F=1$) and the singlet ($F=0$) states depending on the total angular
momentum of the muon-proton system. (b) The three-level system used in the
Bloch equations to model the laser excitation followed by collisional
deexcitation with an increase of kinetic energy ($E_{\text{kin}}$). Initially
all $\mu$p atoms are thermalized (average of $E_{\text{kin}}\approx 5$ meV) to
the singlet state with population $\rho_{11}$. The laser pulse drives the HFS
transition, exciting the $\mu$p atoms into the triplet state with population
$\rho_{22}$. An inelastic collision then deexcites the triplet state back to
the singlet state converting the transition energy into kinetic energy. This
singlet state with additional kinetic energy is the third level in the optical
Bloch equations with population $\rho_{33}$ and $E_{\text{kin}}\approx 100$
meV.

In this paper, we calculate the laser transition probability between singlet
and triplet sublevels of the ground state hyperfine-splitting in $\mu$p (see
Figure 1), accounting for the actual detection scheme used in the experiments
and considering collisional and Doppler effects. This transition probability
is one of the key quantities needed to evaluate the feasibility of the CREMA
hyperfine-splitting experiment, and to define the requirements for the
experimental setup.

Two other collaborations, the FAMU collaboration at RAL (UK) [29, 20, 30, 31]
and another one at J-PARC (Japan) [32, 33, 34], are also aiming at a precision
measurement of the HFS in $\mu$p. For all three experimental collaborations,
the main challenge is posed by the small laser-induced transition probability
resulting from the small matrix element (magnetic dipole-type transition,
$M1$), in conjunction with a transition wavelength at 6.8 $\mu$m where no
adequate (sufficiently powerful) laser technologies are available, and the
large volume where $\mu$p atoms are formed.

Prior to this work there were three other publications addressing the laser-
induced excitation probability between the hyperfine states: the FAMU
collaboration performed the calculation assuming the Fermi-golden rule with
Doppler convolution while neglecting collisional effects as well as laser
bandwidth [35]. This calculation was revised in Ref. [36] that corrected a
mistake in the matrix element. The J-PARC collaboration calculated the laser-
induced transition probability using the optical Bloch equations [37], also
omitting collisional effects given their low target density [34]. Laser
bandwidth was also omitted in this calculation. In contrast, decoherence
effects due to collisions and laser bandwidth were considered in this work, as
they reduce the transition probability by almost a factor of two at the
optimal experimental conditions. Therefore, collisional rates between $\mu$p
and H2 were calculated in this work for the CREMA experimental conditions
following the theory of Ref. [38].

The paper is organized as follows: in Sec. 2 we summarize the CREMA
experimental scheme needed to understand the experimental conditions in which
the laser excitation takes place. Section 3 introduces our theoretical
framework based on the optical Bloch equations. We also summarize there the
collisional effects and the transition matrix elements. In Sec. 4, starting
from the Bloch equations we derive well-known analytical expressions valid
only in certain regimes which serve to better understand the dynamics during
the laser excitation and the numerical results of Sec. 5. The latter section
presents the results obtained by integrating numerically the Bloch equations
at various experimental conditions and discusses in detail the impact of
collisions, Doppler-effects, and laser parameters.

##  2 The experimental scheme

In this section we briefly present the experimental scheme for the measurement
of the HFS pursued by the CREMA collaboration, to precisely define the goal
and framework of the calculation.

Figure 2: (Color online) Schematic (not to scale) of the setup for measuring
the HFS in $\mu$p. A negative muon beam is stopped in a H2 gas target at
cryogenic temperatures, pressures of about 1 bar, and a thickness of about 1
mm. The formed $\mu$p atoms are excited by the laser pulse whose intensity is
enhanced in the multi-pass cavity. A successful laser excitation of the $\mu$p
atoms followed by a collisional deexcitation leads to a $\mu$p atom with extra
kinetic energy. With this extra kinetic energy, the $\mu$p atom diffuses to
the target walls where x-rays are produced.

A low-energy muon beam of about 11 MeV/c momentum is stopped in a hydrogen gas
target ($\sim 1$ mm thickness, a pressure ranging from $0.5$ to 1 bar, and at
a temperature ranging from 20 to 50 K), wherein a $\mu$p atom is formed in a
highly excited state. The formed muonic atom quickly deexcites to the singlet
($F=0$) state of the ground state (see Figure 1) while thermalizing within
about 1 $\mu$s to the hydrogen gas temperature. When thermalized, the $\mu$p
atom is illuminated by a laser pulse at a wavelength of 6.8 $\mu$m (equivalent
to a frequency of 44 THz, or 0.18 eV energy) to drive the hyperfine
transitions. To enhance the laser transition probability, the laser pulse with
a duration of about 20 ns and 5 mJ of energy is coupled through a 0.5 mm wide
slit into a multi-pass cavity. The multiple reflections occurring in this
cavity, whose average number depends mainly on the mirror reflectivity and the
losses at the coupling slit, enhance the laser fluence at the position of the
$\mu$p atom and increase the effective pulse length. On-resonance laser light
thus excites the muonic atom from the singlet $F=0$ to the triplet $F=1$
sublevels. A subsequent inelastic collision with a hydrogen molecule deexcites
the $\mu$p atom from the triplet back to the singlet sublevels. In this
process the HFS transition energy of 0.18 eV is converted into kinetic energy:
on average 0.1 eV of kinetic energy is imparted to the $\mu$p. This kinetic
energy, which is much larger than the thermal energy, causes the $\mu$p atom
to diffuse away from the laser-illuminated volume, reaching one of the gold-
coated target walls in a time window between 100 and 300 ns after the laser
excitation. When the $\mu$p atom reaches the wall, the muon is transferred to
a gold atom forming muonic gold ($\mu$Au∗) in highly excited states. The
various x-rays of MeV energy produced in the subsequent deexcitation of
$\mu$Au∗ are used as signature of a successful laser excitation, so that the
HFS resonance can be determined by counting the number of $\mu$Au x-rays as a
function of the laser frequency.

In this paper, we present the calculation of the probability that a $\mu$p
atom initially in the singlet state and thermalized at the temperature of the
hydrogen gas will undergo the above described sequence of laser excitation and
collisional deexcitation, acquiring the extra $\sim 0.1$ eV of kinetic energy
needed to provide the observable signal.

##  3 Theoretical framework

###  3.1 Bloch equations

The laser excitation and the population dynamics between hyperfine states is
investigated in the framework of the density-matrix formalism, using optical
Bloch equations expressed as [39],

$\displaystyle\frac{d\rho_{11}}{dt}(t)$ $\displaystyle=$
$\displaystyle-\mbox{Im}\left(\Omega\rho_{12}e^{i\Delta
t}\right)+\Gamma_{\mathrm{sp}}\rho_{22}~{},$ (1)
$\displaystyle\frac{d\rho_{22}}{dt}(t)$ $\displaystyle=$
$\displaystyle\phantom{-}\mbox{Im}\left(\Omega\rho_{12}e^{i\Delta
t}\right)-(\Gamma_{i}+\Gamma_{\mathrm{sp}})\rho_{22}~{},$ (2)
$\displaystyle\frac{d\rho_{12}}{dt}(t)$ $\displaystyle=$
$\displaystyle\frac{i\Omega^{*}}{2}(\rho_{11}-\rho_{22})e^{-i\Delta
t}-\frac{\Gamma_{c}}{2}\rho_{12}~{},$ (3)
$\displaystyle\frac{d\rho_{33}}{dt}(t)$ $\displaystyle=$
$\displaystyle\Gamma_{i}\rho_{22}~{},$ (4)

where the detuning $\Delta=\omega_{r}-\omega$ is the departure of the laser
angular frequency $\omega$ from the atomic resonance angular frequency
$\omega_{r}$, $\Omega$ is the Rabi frequency, $\Gamma_{i}$ is the triplet
deexcitation rate caused by the inelastic collisions with the H2 gas leading
the $\mu$p atoms to gain extra kinetic energy, $\Gamma_{\mathrm{sp}}$ is the
spontaneous radiative decay rate from the triplet state (with negligible
change of the kinetic energy), and $\Gamma_{c}$ is the decay rate of the
coherence, referred as ”decoherence rate”. All deexcitation and decoherence
sources in equations (1)-(4) are in units of Hertz. Within our experimental
conditions, $\Gamma_{\mathrm{sp}}\ll\Gamma_{i}$ so that throughout this paper
$\Gamma_{\mathrm{sp}}$ can be neglected. Note that in equations (1)-(4) we
also neglect the muon decay, since at our experimental conditions the muon
decay rate is much smaller than the decoherence rate. The overall decay of all
populations due to muon decay ($\tau_{\mu}=2.2$ $\mu$s) can be treated
separately simply by multiplying our results by $\exp{(-t/\tau_{\mu})}$.

The diagonal density terms $\rho_{11}$ and $\rho_{22}$ are the populations of
the thermalized singlet and (excited) triplet states, respectively, while
$\rho_{12}$ is the coherence between these two states.

The third level with population $\rho_{33}$ is used to track the fraction of
$\rho_{22}$ that undergoes a collisionally-induced deexcitation from the
triplet to the singlet states, with consequent increase of the kinetic energy.
This third level is modeled as a dark state decoupled from the laser because
the additional 0.1 eV of kinetic energy (corresponding to $\sim$1000 K),
acquired by the $\mu$p in the inelastic collision, gives rise to such a large
Doppler shift that the atomic resonance becomes detuned from the exciting
laser frequency. Actually, within several hundred nanoseconds, the $\mu$p
atoms can thermalize again through elastic $\mu$p–H2 collisions. However,
$\mu$p atoms with the extra kinetic energy are very likely to exit the laser-
excitation volume within tens of nanoseconds. Hence, multiple cycles of laser-
excitation followed by collisional-quenching are negligible at our
experimental conditions, and thus modeling the third level as a dark state is
appropriate.

As initial conditions for the time $t=0$, we assume $\rho_{11}=1$ and
$\rho_{22}=\rho_{33}=0$ since in the experiment we choose the arrival time of
the laser pulse so that all $\mu$p atoms are deexcited to the singlet state
and thermalized to the hydrogen gas temperature. As a consequence of
$\rho_{22}=0$, for time $t=0$ the coherence $\rho_{12}=0$ is also zero. In
this paper, we assume that the laser excitation occurs from time $t=0$ until
$t=\tau$ at constant laser intensity $\mathcal{I}$, resulting in a laser
fluence of $\mathcal{F}=\mathcal{I}\tau$ [J/cm2]. The impact of the laser
fluence $\mathcal{F}$ and exposure time $\tau$ on the population dynamics is
evaluated throughout this paper. Exposure times of 10 ns and 100 ns are
considered. Note that the actual time (and spatial) distribution of the laser
intensity within the multi-pass cavity is complicated by the superpositions of
the folded laser beam spreading in the cavity, however at late times it
follows approximatively an exponential function with a lifetime ranging from
50 ns to 150 ns depending on the performance and geometry of the cavity.

In this work, the temporal evolution of the populations in the three levels is
computed by numerically integrating the optical Bloch equations for various
target conditions and laser performances. When combined with simulations of
the diffusion process, muon beam and detection system, this allows
optimization of the experimental setup to maximize statistical significance
(signal/$\sqrt{\text{background}}$). The parameters $\Gamma_{i}$ and
$\Gamma_{c}$ allow a phenomenological inclusion in the Bloch equations of the
laser bandwidth and of the $\mu$p–H2 elastic and inelastic collisions. The
Doppler effect is accounted for in a second step by folding the results from
the optical Bloch equation with a Doppler (Gaussian) profile for the (initial)
thermalized $\mu$p atoms.

###  3.2 Collisional rates

The decoherence rate is given by

$\Gamma_{c}={2\pi\Delta_{l}}+\Gamma_{e}^{F=0}+\Gamma_{e}^{F=1}+\Gamma_{i}+\Gamma_{\mathrm{sp}}~{},$
(5)

where $\Delta_{l}$ is the laser (FWHM) bandwidth, which contains a $2\pi$
factor to convert a bandwidth into a ”rate” (see [39]). $\Gamma_{i}$ is the
inelastic $\mu$p–H2 collision rate for the excited state,
$\Gamma_{e}^{\mathrm{F=1}}$ is the elastic collision rate for the triplet, and
$\Gamma_{e}^{\mathrm{F=0}}$ is the elastic collision rate for the singlet
state:

$\displaystyle\Gamma_{i}:$
$\displaystyle\quad\mu\mathrm{p}^{\mathrm{F=1}}+\mathrm{H}_{2}\rightarrow\mu\mathrm{p}^{\mathrm{F=0}}+\mathrm{H}_{2}^{*}$
(6) $\displaystyle\Gamma_{e}^{\mathrm{F=1}}:$
$\displaystyle\quad\mu\mathrm{p}^{\mathrm{F=1}}+\mathrm{H}_{2}\rightarrow\mu\mathrm{p}^{\mathrm{F=1}}+\mathrm{H}_{2}^{*}$
(7) $\displaystyle\Gamma_{e}^{\mathrm{F=0}}:$
$\displaystyle\quad\mu\mathrm{p}^{\mathrm{F=0}}+\mathrm{H}_{2}\rightarrow\mu\mathrm{p}^{\mathrm{F=0}}+\mathrm{H}_{2}^{*}\,.$
(8)

Here the ∗ indicates possible rotational excitations of the hydrogen molecule
after the collision. Note that these collisional cross-sections do not depend
on the magnetic sub-states of the $\mu$p atom [40, 41].

All the collisional rates ($\Gamma_{i}$, $\Gamma_{e}^{F=1}$ and
$\Gamma_{e}^{F=0}$) were obtained using

$\Gamma=\overline{\upsilon_{r}\sigma(\upsilon_{r})}\rho_{\mbox{\tiny
H}_{2}}\;,$ (9)

where $\upsilon_{r}$ is the $\mu$p–H2 relative velocity and $\rho_{\mbox{\tiny
H}_{2}}$ is the number density of H2 molecules, which was calculated assuming
that H2 behaves as an ideal gas. The term
$\overline{\upsilon_{r}\sigma(\upsilon_{r})}$ represents the folding of the
relative $\mu$p–H2 velocity ($\upsilon_{r}$) distribution for thermalized
$\mu$p atoms and H2 molecules with the velocity-dependent cross sections
$\sigma(\upsilon_{r})$ for the considered scattering process [38].

These rates were calculated following Ref. [38], which uses the velocity- and
spin-dependent cross sections between $\mu$p atoms and hydrogen molecules
summed over the final rotational states of the H2 molecules and averaged over
the distribution of the initial rotational states. The spin alignment of the
two nuclei of the molecule is taken into consideration when computing the
rotational states and their initial populations. Two distributions for the
initial rotational states were considered: statistical and Boltzmann
distributions. At room temperature, the Boltzmann distribution is similar to
the statistical distribution given by 75% ortho-hydrogen (odd rotational
number $K$) and 25% para-hydrogen (even $K$). For low temperatures and at
equilibrium, the Boltzmann distribution differs considerably from the
statistical distribution, e.g. at 22 K, basically all molecules have $K=0$.
However, if the gas is quickly cooled down, the ratio ortho- to para-hydrogen
existing at high temperature is retained so that 75% of the molecules have
$K=1$ and 25% have $K=0$. The Boltzmann distribution is only reached after a
long time because the conversion from $K=1$ to $K=0$ rotational states is a
very slow process having a rate of about 2% per week at the normal temperature
and pressure [42]. The actual rotational distribution in the experimental
conditions is therefore bounded between these two extreme cases. For this
reason, we give in Table 1 values of the collision rates for statistical and
Boltzmann distributions.

Table 1: $\mu$p–H2 collision rates in MHz for various H2 target pressures ($p$) and temperatures ($T$). Two distributions of ortho- and para-H2 are considered: statistical and Boltzmann.  $p=0.5$ bar |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
|  $T=22$ K |  |  $T=30$ K |  |  $T=50$ K
| Stat. | Boltz. |  | Stat. | Boltz. |  | Stat. | Boltz.
$\Gamma_{e}^{F=0}$ | 20 | 20 |  | 15 | 15 |  | 9 | 9
$\Gamma_{e}^{F=1}$ | 52 | 29 |  | 41 | 24 |  | 28 | 18
$\Gamma_{i}$ | 82 | 93 |  | 59 | 66 |  | 34 | 37
$p=1$ bar |  |  |  |  |  |  |  |
$\Gamma_{e}^{F=0}$ | 40 | 39 |  | 30 | 30 |  | 19 | 19
$\Gamma_{e}^{F=1}$ | 104 | 59 |  | 83 | 47 |  | 55 | 37
$\Gamma_{i}$ | 164 | 187 |  | 118 | 133 |  | 68 | 74
$p=2$ bar |  |  |  |  |  |  |  |
$\Gamma_{e}^{F=0}$ | 79 | 79 |  | 61 | 61 |  | 38 | 37
$\Gamma_{e}^{F=1}$ | 208 | 118 |  | 165 | 94 |  | 110 | 74
$\Gamma_{i}$ | 328 | 374 |  | 235 | 265 |  | 137 | 148
Table 2: Values of ${\mathcal{M}}$ and $\Omega/\sqrt{\mathcal{I}}$ for Lamb
shift and HFS transitions in $\mu$p and $\mu^{3}$He+. The analytical
expressions for the $2s-2p$ matrix elements agree with [43]. $a_{\mu}$ is the
muonic Bohr radius.

Atom | Transition |  ${\mathcal{M}}$ [m] |  $\frac{\Omega}{\sqrt{\mathcal{I}}}$ [$\mbox{m}/\sqrt{\mbox{Js}}$]
---|---|---|---
$\mu$p | $2s^{F=1}\rightarrow 2p_{3/2}^{F=2}$ | $\sqrt{5}a_{\mu}=6.367\times 10^{-13}$ | 2.65 $\times 10^{4}$
$\mu^{3}$He+ | $2s^{F=1}\rightarrow 2p_{3/2}^{F=2}$ | $\frac{\sqrt{5}}{2}a_{\mu}=2.969\times 10^{-13}$ | 1.24 $\times 10^{4}$
$\mu$p | $1s^{F=0}\rightarrow 1s^{F=1}$ | $\frac{\hbar}{4m_{\mu}c}\left(g_{\mu}+\frac{m_{\mu}}{m_{p}}g_{p}\right)$ | 5.12 $\times 10^{1}$ a
|  | $=1.228\times 10^{-15}$ |
$\mu^{3}$He+ | $1s^{F=1}\rightarrow 1s^{F=0}$ | $\frac{\hbar}{4\sqrt{3}m_{\mu}c}\left(g_{\mu}+\frac{m_{\mu}}{m_{\tiny\mbox{He}}}g_{\tiny\mbox{He}}\right)$ | 2.07 $\times 10^{1}$
|  | $=4.965\times 10^{-16}$ |

a 1.77$\times 10^{1}$ $\mbox{m}/\sqrt{\mbox{Js}}$ according to Ref. [37]

###  3.3 Rabi frequencies and matrix elements

The Rabi frequency
$\Omega=\sqrt{\frac{8\pi\alpha\mathcal{I}}{\hbar}}{\mathcal{M}}$ included in
the Bloch equations quantifies the coupling strength between the laser with
intensity $\mathcal{I}$, and the atomic transition. Here, $\alpha$ is the fine
structure constant and $\hbar$ the reduced Planck constant. We evaluated the
matrix element ${\mathcal{M}}$ for the M1 HFS transitions of $\mu$p and
$\mu^{3}$He+ in the ground state. To verify the calculations, we also
evaluated the matrix elements for the already measured $2s-2p$ transitions in
$\mu$p and $\mu^{3}$He+, where literature values of saturation fluences are
available [43].

Because the magnetic substates of the initial state can be assumed to be
statistically populated, the matrix element is averaged over the initial
magnetic sub-states $m$:

${\mathcal{M}}^{2}=\frac{1}{2F_{i}+1}\sum_{m,m^{\prime}}\left|\mathcal{M}^{(m,m^{\prime})}\right|^{2}.$
(10)

It is also summed over all possible final magnetic sub-states $m^{\prime}$
since the final magnetic sub-states are not detected (resolved) in our
experimental scheme. The matrix elements $\mathcal{M}^{(m,m^{\prime})}$ for
the $2s-2p$ (E1-type) and $1s$-HFS (M1-type) transitions are given by [44, 35]

$\displaystyle\mathcal{M}_{\mbox{E1}}^{(m,m^{\prime})}=\left\langle
2p\,{F^{\prime}}\,m^{\prime}\left|\bm{r}\cdot\bm{\hat{\varepsilon}}\right|2s\,{F}\,m\right\rangle,$
$\displaystyle\mathcal{M}_{\mbox{M1}}^{(0,m^{\prime})}=$ (11)
$\displaystyle\frac{1}{2m_{\mu}c}\left\langle
F^{\prime}=1\,m^{\prime}\left|(g_{\mu}\bm{S}+g_{p}\frac{m_{\mu}}{m_{p}}\bm{I})\cdot\bm{\hat{k}\times\hat{\varepsilon}}\right|F=0\,0\right\rangle,$

where $m_{\mu}$ is the muon mass, $m_{p}$ the proton mass, $\bm{\hat{k}}$ the
laser wavevector, $\bm{\hat{\varepsilon}}$ the laser polarization, $\bm{S}$
the spin operator of the muon, $\bm{I}$ the spin operator of the proton, and
$g_{\mu}$ and $g_{p}$ the $g$-factor of the muon ($\simeq 2.00$) and proton
($\simeq 5.58$), respectively. See Appendix A for more details about the
evaluation of these matrix elements.

Table 2 summarizes the numerical values of the (analytical) matrix elements
${\mathcal{M}}$ and $\Omega/\sqrt{\mathcal{I}}$ for the HFS and the most
intense Lamb shift transitions for $\mu$p and $\mu^{3}$He+. While the matrix
elements for the $2s-2p$ transitions are in agreement with published values
[43], the $\Omega/\sqrt{\mathcal{I}}$ value obtained for the $\mu$p HFS of
$5.12\times 10^{1}\;\mbox{m}/\sqrt{\mbox{Js}}$ is in disagreement with the
value of $1.77\times 10^{1}\;\mbox{m}/\sqrt{\mbox{Js}}$ published in Ref.
[37]. The difference of
$\sqrt{\frac{3}{2}}\left(\frac{g_{\mu}m_{p}+g_{p}m_{\mu}}{m_{p}+m_{\mu}}\right)\approx
2.9$ is traced back to a miscalculation of the matrix elements in Ref. [37]
(see Appendix A), and by not considering the proton spin-flip.

###  3.4 Doppler broadening

While the collisional effects and the laser bandwidth are already accounted
for in the Bloch equations, and eventually give rise to a Lorentzian profile
(see Sec. 4.1), the Doppler broadening needs to be treated separately.

Table 3: Saturation fluences for laser bandwidths of $\Delta_{l}=10$ MHz
($\mathcal{F}_{\mbox{sat}}^{\Delta_{l}\rightarrow 10}$) and $\Delta_{l}=100$
MHz ($\mathcal{F}_{\mbox{sat}}^{\Delta_{l}\rightarrow 100}$) for three
transitions in muonic atoms, accounting for collisional effects. For
comparison we also give the saturation fluence neglecting decoherence effects
($\mathcal{F}^{\Gamma_{c}\rightarrow 0}_{\mbox{sat}}$). The resonance
frequencies $\nu_{r}=\omega_{r}/2\pi$ for the three transitions have been
taken from Refs. [45, 5, 46]. The spontaneous decay rates $\Gamma_{\mbox{sp}}$
of the $2p$ states were taken from [47] and [48], while for the ground state
triplet in $\mu$p it is calculated here. The decoherence rates $\Gamma_{c}$
have been obtained using equation (5) with the values from Table 1 assuming a
Boltzmann distribution between ortho- and para-H2 (for a statistical
distribution see Appendix B). $\sigma_{D}$ is the Doppler standard deviation
(14).

Atom | Transition | $T$ | $p$ | $\nu_{r}$ | $\Gamma_{\text{sp}}$ | $\Gamma_{c}^{\Delta_{l}\rightarrow 10}$ | $\Gamma_{c}^{\Delta_{l}\rightarrow 100}$ | $\sigma_{D}$ | $\mathcal{F}^{\Gamma_{c}\rightarrow 0}_{\mbox{sat}}$ | $\mathcal{F}_{\mbox{sat}}^{\Delta_{l}\rightarrow 10}$ | $\mathcal{F}_{\mbox{sat}}^{\Delta_{l}\rightarrow 100}$
---|---|---|---|---|---|---|---|---|---|---|---
|  | [K] | [bar] | [THz] | [MHz] | [MHz] | [MHz] | [MHz] | [J/cm2] | [J/cm2] | [J/cm2]
$\mu$p | $2s^{F=1}\rightarrow 2p_{3/2}^{F=2}$ | 300 | 0.001 | 49.9 | 1.16$\times 10^{5}$ |  | 1.16$\times 10^{5}$ | 2.48$\times 10^{2}$ |  |  | 0.0165
$\mu^{3}$He+ | $2s^{F=1}\rightarrow 2p_{3/2}^{F=2}$ | 300 | 0.004 | 379 | 2.00$\times 10^{6}$ |  | 2.00$\times 10^{6}$ | 1.13$\times 10^{3}$ |  |  | 1.304
$\mu$p | $1s^{F=0}\rightarrow 1s^{F=1}$ |  |  | 44.2 | 1.23$\times 10^{-11}$ |  |  |  |  |  |
|  | 22 | 0.5 |  |  | 205 | 770 | 60 | 23 | 28 | 44
|  | 22 | 1 |  |  | 348 | 913 | 60 | 23 | 32 | 49
|  | 22 | 2 |  |  | 633 | 1198 | 60 | 23 | 40 | 58
|  | 30 | 0.5 |  |  | 168 | 733 | 70 | 27 | 31 | 47
|  | 30 | 1 |  |  | 273 | 838 | 70 | 27 | 34 | 50
|  | 50 | 0.5 |  |  | 128 | 693 | 90 | 35 | 37 | 53
|  | 50 | 1 |  |  | 192 | 757 | 90 | 35 | 39 | 55

We include the Doppler effect by convoluting the population
${\rho}_{33}(\omega)$ as obtained from the Bloch equations with a Gaussian
distribution describing the Doppler profile:

$\bar{\rho}_{33}(\omega)=\int_{-\infty}^{\infty}\rho_{33}(\omega^{\prime})\frac{1}{\sqrt{2\pi}\gamma_{D}}\exp\left(-\frac{(\omega-\omega^{\prime})^{2}}{2\gamma_{D}^{2}}\right)d\omega^{\prime}~{},$
(12)

with $\gamma_{D}$ being given by

$\displaystyle\gamma_{D}=\omega_{r}\sqrt{\frac{kT}{(m_{\mu}+m_{p})c^{2}}}$
$\displaystyle\simeq$ $\displaystyle 7.98\times
10^{7}\sqrt{T}~{}\text{[rad/s]},$ (13)
$\displaystyle\sigma_{D}=\frac{\gamma_{D}}{2\pi}$ $\displaystyle\simeq$
$\displaystyle 12.7\sqrt{T}\,~{}\text{[MHz]},$ (14)

so that the Doppler standard deviation (MHz) is $\sigma_{D}$. $k$ is the
Boltzmann constant, $c$ the speed of light, and $T$ the temperature in Kelvin
of the thermalized $\mu$p atom in the singlet state.

##  4 Analytical expressions for two limiting regimes

Before integrating numerically the three-level Bloch equations and
investigating how collisional and Doppler effects impact the $\rho_{33}$
population, it is interesting to derive analytical expressions from our
formalism to reproduce well-known results valid for low and high Rabi
frequencies, where low and high are relative to the other frequencies involved
in the problem.

###  4.1 Fermi-golden-rule regime

The laser-induced rate of population transfer $R(t,\omega)$ from $\rho_{11}$
to $\rho_{22}$ is given by the first term of the second Bloch equation (2),

$R(t,\omega)=\mbox{Im}\left(\Omega\rho_{12}e^{i\Delta t}\right)~{}.$ (15)

In the limit of low transition probability ($\Omega\ll\Gamma_{c}$), and
sufficiently long time ($t\gg 1/\Gamma_{c}$) to obtain stable $\rho_{22}$
population, this rate converts to the Lorentzian form of the Fermi-golden rule
[39],

$R_{\text{F}}^{\text{(L)}}(\omega)=\frac{|\Omega|^{2}}{4}\frac{\Gamma_{c}}{\left(\Delta^{2}+\left(\frac{\Gamma_{c}}{2}\right)^{2}+\frac{|\Omega|^{2}\Gamma_{c}}{2\Gamma_{i}}\right)}~{}.$
(16)

This result is obtained using the analytical solutions of $\rho_{22}$ and
$\rho_{12}$ of Refs. [49, 50, 51]. Note that the Fermi-golden-rule approach
does not contain population dynamics; it is a static approach.

The Doppler effect can be included by performing a convolution similar to
equation (12)

$\bar{R}_{\text{F}}^{\text{(L)}}(\omega)=\int_{-\infty}^{\infty}R_{\text{F}}^{\text{(L)}}(\omega^{\prime})\frac{1}{\sqrt{2\pi}\gamma_{D}}\exp\left(-\frac{(\omega-\omega^{\prime})^{2}}{2\gamma_{D}^{2}}\right)d\omega^{\prime}~{},$
(17)

resulting in a combined laser-excitation and collisional deexcitation
probability of

$\bar{\rho}_{33}(\omega)\approx\hat{\rho}_{33}(\omega)=\int_{0}^{\tau}\bar{R}_{\text{F}}^{\text{(L)}}(\omega)dt=\frac{\mathcal{F}}{\mathcal{F}_{\small\mbox{sat}}(\omega)}~{},$
(18)

where $\mathcal{F}_{\small\mbox{sat}}$ is the saturation fluence and
$\bar{\rho}_{33}(\omega)$ is the obtained Voigt profile that accounts for
Lorentzian (homogeneous) and Gaussian (inhomogeneous) broadenings. The hat-
symbol on the third level population $\hat{\rho}_{33}$ is used to denote that
it was computed in the Fermi-golden-rule approximation and including Doppler
effects.

Since the Voigt function at resonance has an analytical solution, at resonance
we can find an analytical expression for
$\mathcal{F}_{\mbox{sat}}(\omega_{r})$:

$\displaystyle\mathcal{F}_{\mbox{sat}}(\omega_{r})=\frac{\hbar\gamma_{D}^{2}}{\pi^{3/2}\alpha\Gamma_{c}\mathcal{M}^{2}}\frac{\varpi
e^{-\varpi^{2}}}{\mbox{Erfc}(\varpi)}~{},$ (19)

where $\varpi=\frac{\sqrt{2}\Gamma_{c}}{4\gamma_{D}}$ and Erfc is the
complementary error function. For $\Gamma_{c}\ll\sigma_{D}$ the saturation
fluence becomes

$\displaystyle\mathcal{F}^{\Gamma_{c}\rightarrow
0}_{\mbox{sat}}(\omega_{r})=\frac{\hbar\sigma_{D}}{(2\pi)^{3/2}\alpha\mathcal{M}^{2}}\approx\frac{\sqrt{T}}{0.2}\;~{}{[\text{J/cm}^{2}]}.$
(20)

In this limit, the laser transition probability becomes

$\displaystyle\hat{\rho}_{33}^{\tiny{\Gamma_{c}\rightarrow
0}}\approx\frac{0.2}{\sqrt{T}}\mathcal{F}\;,$ (21)

where $T$ and $\mathcal{F}$ are values of temperature and fluence in units of
K and J/cm2, respectively. This result is in agreement with the excitation
probability given in Ref. [36].

Table 3 summarizes the saturation fluences $\mathcal{F}_{\small\mbox{sat}}$
calculated from equation (19), for Lamb shift transitions in $\mu$p and
$\mu$He+, as well as for the HFS transition in $\mu$p for possible
experimental conditions. Obtained values of $\mathcal{F}_{\small\mbox{sat}}$
for Lamb shift transitions are in agreement with Ref. [43]. Four orders of
magnitude separate the $\mathcal{F}_{\small\mbox{sat}}$ for the Lamb shift and
the HFS transitions in $\mu$p, emphasizing the laser technology leap needed to
accomplish the hyperfine experiment. Values of
$\mathcal{F}^{\Gamma_{c}\rightarrow 0}_{\mbox{sat}}$ given by equation (20)
are also shown together with $\mathcal{F}_{\mbox{sat}}^{\Delta_{l}\rightarrow
10}$ and $\mathcal{F}_{\mbox{sat}}^{\Delta_{l}\rightarrow 100}$, which are the
saturation fluences for $\Delta_{l}=10$ MHz and 100 MHz, respectively. The
comparison of $\mathcal{F}_{\mbox{sat}}^{\Delta_{l}\rightarrow 10}$ to
$\mathcal{F}_{\mbox{sat}}^{\Delta_{l}\rightarrow 100}$ highlights the impact
of the laser bandwidth, while comparison of
$\mathcal{F}_{\mbox{sat}}^{\Delta_{l}\rightarrow 10}$ to
$\mathcal{F}^{\Gamma_{c}\rightarrow 0}_{\mbox{sat}}$ allows to appreciate the
impact of the collisional effects.

Table 5 in Appendix B shows that within our experimental conditions, the
decoherence rates and the saturation fluences are only marginally affected by
the assumed ortho- to para-H2 distributions. Therefore, throughout this study,
we assumed a Boltzmann distribution.

###  4.2 Rabi-oscillation regime: Two-level Bloch equations

For $\Gamma_{c}<\Omega$, the approximation of equation (18) is no longer
valid. In this limit, the $\rho_{22}$ population saturates and oscillates
according to

$\rho_{22}=\frac{|\Omega|^{2}}{|\Omega|^{2}+\Delta^{2}}\sin^{2}\left(\frac{\tau}{2}\sqrt{|\Omega|^{2}+\Delta^{2}}\right)\;,$
(22)

which on resonance simplifies to

$\displaystyle{\rho}_{22}=\sin^{2}\left(\sqrt{\frac{8\pi\alpha\mathcal{F}\tau}{\hbar}}\mathcal{M}\right)\;.$
(23)

These are the well-known analytical solutions of the two-level Bloch equations
valid in the Rabi-oscillation regime when broadening sources can be neglected
[39], i.e., when the Rabi frequency dominates over the broadening rates
$\Omega\gg\Gamma_{c}$ and $\Omega\gg\sigma_{D}$.

Reference [37] calculates the transition probability between hyperfine states
in $\mu$p using equation (22), hence without decoherence (collisional or laser
bandwidth) effects. Doppler broadening was included in a second step in their
calculations as a not-well-specified numerical average.

##  5 Results and discussion

###  5.1 $\bar{\rho}_{33}$ at the experimental conditions

Simulating the time-evolution of state populations using Bloch equations has
improved laser spectroscopy of molecules [52], muonium [53] and highly charged
ions [54]. Similarly, the findings of population dynamics obtained from Eq.
(1)-(4) can be used to optimize the target conditions ($T$ and $p$) of the HFS
experiment.

We present here the $\bar{\rho}_{33}$ populations for laser parameters and
hydrogen target conditions in the current region of interest for the CREMA HFS
experiment.

Figure 3:  (Color online) $\bar{\rho}_{33}$ population versus laser fluence
$\mathcal{F}$ for various H2 target conditions, two laser bandwidths
($\Delta_{l}=10$ MHz and $\Delta_{l}=100$ MHz), and two exposure times
($\tau=10$ ns and $\tau=100$ ns) in the range of interest for the CREMA HFS
experiment. We use the collision rates of Table 1 assuming Boltzmann
distribution between ortho- and para-H2. The solid lines represent the results
from the analytical expression of equation (18) that neglects population
dynamics. The saturation fluences $\mathcal{F}_{\text{sat}}$ are taken from
Table 3.

These populations obtained from the numerical integration of the Bloch
equations are shown by the dashed curves in Figure 3 and compared with the
approximations of equation (18), shown as solid lines. Already for fluences
larger than 5 J/cm2 the full-numerical calculations need to be used for
accurate predictions. At the prospected optimal conditions of $T=22$ K and
$p=0.5$ bar (based on preliminary simulations of the diffusion of the $\mu$p
atom in the H2 gas target), for $\Delta_{l}=10$ MHz, the analytical prediction
overestimate the transition probability by about 30% and 46% for fluences of
10 J/cm2 and 20 J/cm2, respectively. For $\Delta_{l}=100$ MHz, this difference
is reduced by 17% and 26% for the same fluences, respectively. Note that
fluences up to 20 J/cm2 are achievable locally in our multi-pass cavity. The
importance of the dynamics is also highlighted by the small but non-negligible
dependence of $\bar{\rho}_{33}$ on the exposure time $\tau$.

###  5.2 Resonance profile at the experimental conditions

The HFS resonance has to be searched for in a region of about 40 GHz ($\pm
3\sigma$) given by the uncertainty of the theoretical predictions arising
mainly from the uncertainty of the two-photon exchange contribution [13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23]. The resonance is scanned by counting the
number of x-rays for a few hours at a given laser frequency. Then the laser
frequency is shifted by a fixed amount and the x-ray counting is performed
anew. This is repeated until a statistically significant deviation of the
number of x-ray counts from the background level has been observed. Given the
large search range, the long time (few hours) per frequency point necessary to
observe a statistically significant deviation above background, and the
limited access at the PSI accelerator, it is important to optimize the
frequency step used to search for the resonance. For this reason, the
linewidth of the targeted resonance has to be known precisely.

Figure 4: (Color online) $\bar{\rho}_{33}$ versus the laser angular frequency
$\omega$ (line-shape) calculated by numerically integrating the Bloch equation
for $\tau=100$ ns, at various target conditions assuming Boltzmann
distributions between ortho- and para-H2: (a) $\mathcal{F}=10$ J/cm2 and
$\Delta_{l}=10$ MHz; (b) $\mathcal{F}=20$ J/cm2 and $\Delta_{l}=10$ MHz; (c)
$\mathcal{F}=10$ J/cm2 and $\Delta_{l}=100$ MHz; (d) $\mathcal{F}=20$ J/cm2
and $\Delta_{l}=100$ MHz. Values of $\bar{\rho}_{33}$ on resonance and FWHM
linewidth can be found in Table 4.

Figure 4 shows some line profiles obtained from numerical integration at
conditions relevant for the CREMA experiment. The figure clearly shows that
the reduction of the on-resonance excitation probability due to collisional
effects is correlated with a broadening of the line-shape.

For fluences $\mathcal{F}\lesssim 20$ J/cm2, the Rabi frequency is
sufficiently small so that this power broadening can be neglected. In this
regime the line shape is thus a Lorentzian profile with a FWHM of
approximately [39]

${\Gamma}_{L}^{2}=\Gamma_{c}^{2}+2|\Omega|^{2}\frac{\Gamma_{c}}{\Gamma_{i}}\;.$
(24)

When including the Doppler effect, a Voigt profile is obtained with a FWHM-
linewidth $\hat{\Gamma}_{V}$ given approximately by [55]:

$\hat{\Gamma}_{V}=0.53\Gamma_{L}+\sqrt{0.22\Gamma_{L}^{2}+5.54{\gamma_{D}^{2}}}\;.$
(25)

The $\hat{\Gamma}_{V}$, calculated using the analytical expressions of Eqs.
(24) and (25) are compared in Table 4 to the FWHM linewidth obtained by
numerically integrating the optical Bloch equations. For completeness, Table 4
also summarizes the $\hat{\rho}_{33}(\omega_{r})$ and
$\bar{\rho}_{33}(\omega_{r})$ as obtained from the analytical expression of
equation (18) and from the full numerical calculation, respectively. Here, a
Boltzmann distribution between ortho- and para-hydrogen was assumed while the
respective values for a statistical distribution are listed in Appendix B.
Their comparison shows a negligible dependence on the type of distribution
assumed. Tables 4, 6 and 7 provide the complete information needed to quantify
the combined probability of laser-excitation followed by collisional
deexcitation for the HFS experiment of the CREMA collaboration.

Table 4: $\hat{\Gamma}_{V}$ from the analytical expression of equation (25),
FWHM line-width $\Gamma_{V}$ extracted from fitting the line-profile obtained
from numerical integration of the Bloch equations for various $\omega$,
$\hat{\rho}_{33}(\omega_{r})$ from the analytical expression of equation (18)
and $\bar{\rho}_{33}(\omega_{r})$ from the numerical integration of the Bloch
equations. $\tau=100$ ns and Boltzmann distribution between ortho- and para-H2
are assumed (for $\tau=10$ ns and a statistical distribution see Appendix B).
The symbol “-“ indicates that $\hat{\rho}_{33}>1$, which is non-physical.

|  |  |  |  |  |  ((($\mathcal{F}=10$ J/cm2 |  |  (($\mathcal{F}=20$ J/cm2 |  |  (($\mathcal{F}=50$ J/cm2
---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  | ($\Omega=8$ MHz) |  | ($\Omega=11$ MHz) |  | ($\Omega=18$ MHz)
$p$ | $T$ | $\Delta_{l}$ | $\Gamma_{c}$ | $\sigma_{D}$ |  | $\hat{\Gamma}_{V}$ | $\Gamma_{V}$ | $\hat{\rho}_{33}$ | $\bar{\rho}_{33}$ |  | $\hat{\Gamma}_{V}$ | $\Gamma_{V}$ | $\hat{\rho}_{33}$ | $\bar{\rho}_{33}$ |  | $\hat{\Gamma}_{V}$ | $\Gamma_{V}$ | $\hat{\rho}_{33}$ | $\bar{\rho}_{33}$
$[\mathrm{bar}]$ | [K] | [MHz] | [MHz] | [MHz] |  | [MHz] | [MHz] |  |  |  | [MHz] | [MHz] |  |  |  | [MHz] | [MHz] |  |
0.5 | 22 | 100 | 770 | 60 |  | 220 | 228 | 0.23 | 0.19 |  | 223 | 232 | 0.45 | 0.33 |  | 231 | 267 | - | 0.59
1 | 22 | 100 | 913 | 60 |  | 235 | 242 | 0.21 | 0.18 |  | 236 | 256 | 0.41 | 0.32 |  | 241 | 283 | - | 0.60
0.5 | 22 | 10 | 205 | 60 |  | 161 | 167 | 0.36 | 0.25 |  | 163 | 176 | 0.71 | 0.38 |  | 169 | 190 | - | 0.57
1 | 22 | 10 | 348 | 60 |  | 174 | 183 | 0.31 | 0.25 |  | 175 | 187 | 0.63 | 0.40 |  | 178 | 208 | - | 0.64
0.5 | 30 | 100 | 733 | 70 |  | 239 | 247 | 0.21 | 0.17 |  | 243 | 251 | 0.43 | 0.30 |  | 254 | 277 | - | 0.54
1 | 30 | 100 | 838 | 70 |  | 249 | 256 | 0.20 | 0.17 |  | 251 | 269 | 0.40 | 0.31 |  | 257 | 287 | - | 0.57
0.5 | 30 | 10 | 168 | 70 |  | 182 | 189 | 0.32 | 0.21 |  | 185 | 194 | 0.65 | 0.32 |  | 191 | 208 | - | 0.48
1 | 30 | 10 | 273 | 70 |  | 190 | 199 | 0.30 | 0.22 |  | 192 | 204 | 0.60 | 0.36 |  | 196 | 226 | - | 0.56
0.5 | 50 | 100 | 693 | 90 |  | 283 | 288 | 0.19 | 0.15 |  | 290 | 295 | 0.38 | 0.28 |  | 308 | 316 | 0.95 | 0.42
1 | 50 | 100 | 757 | 90 |  | 287 | 294 | 0.18 | 0.15 |  | 290 | 299 | 0.37 | 0.27 |  | 300 | 328 | 0.91 | 0.49
0.5 | 50 | 10 | 127 | 90 |  | 228 | 230 | 0.27 | 0.16 |  | 233 | 236 | 0.53 | 0.23 |  | 240 | 249 | - | 0.35
1 | 50 | 10 | 192 | 90 |  | 231 | 236 | 0.26 | 0.18 |  | 231 | 240 | 0.51 | 0.27 |  | 240 | 256 | - | 0.43

##  6 Conclusion

In view of the upcoming HFS experiment of the CREMA collaboration, we
calculated the combined probability that a thermalized $\mu$p atom undergoes
laser excitation from the singlet to the triplet states followed by a
collisional deexcitation, where it acquires on average about 0.1 eV extra
kinetic energy. This calculation was performed accounting for collisional and
Doppler effects. The collisional effects together with the laser properties
such as bandwidth, exposure time, and fluence have been accounted for directly
in the three-level Bloch equations, while the Doppler broadening has been
accounted for in a second step by convoluting the results from the Bloch
equations with the Doppler profile.

We derived simple analytical expressions valid in limiting regimes (Fermi-
golden rule and Rabi-oscillations). Given the small Rabi frequency relative to
the homogeneous broadening, the proposed HFS experiment lies more closely to
the Fermi-golden-rule regime.

The collisional effects play an important role: while the inelastic collisions
trigger a successful laser transition, the elastic collisions decrease the
laser excitation probability to the triplet state. In the experimental
conditions of the CREMA HFS experiment, (around 0.5 bar, 22 K,
$\Delta_{l}=100$ MHz) the decoherence effects decrease the transition
probability for small flueces by about a factor of two (see Table 3).

For fluences reached in the optical cavity (up to 20 J/cm2 locally), the
dynamical effects beyond the Fermi approximation captured by the numerical
integration of the Bloch equations are significant, as can be seen by
comparing the solid and dashed lines in Figure 3 and $\hat{\rho}_{33}$ and
$\bar{\rho}_{33}$ in Table 4. We also demonstrate that $\bar{\rho}_{33}$ is
not significantly affected by the assumed distribution of the hydrogen
rotational states.

For the a fluence of 10 J/cm2, 100 MHz laser bandwidth and 100 ns laser
exposure, 19% of the $\mu$p atoms exposed to the laser light acquires the
extra 0.1 eV kinetic energy for a hydrogen gas target at 22 K temperature and
0.5 bar pressure.

This paper is the first in a set of studies dedicated to the $\mu$p HFS
experiment of the CREMA collaboration. In conjunction with studies of the
$\mu$p diffusion in the target gas, the optical multi-pass cavity and the
detection system which will be published elsewhere, this allows to optimize
the hydrogen target and define the specifications for the laser system.

## Acknowledgements

We acknowledge the support of the following grants: Fundação para a Ciência e
a Tecnologia (FCT), Portugal, and FEDER through COMPETE in the framework of
project numbers PTDC/FIS-AQM/29611/2017 and UID/04559/2020 (LIBPhys), and
contracts No.
SFRH/BPD/92329/2013 and No. PD/BD/128324/2017; Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Initiative EXC
1098 PRISMA (194673446), Excellence Strategy EXC PRISMA+ (390831469) and
DFG/ANR Project LASIMUS (DFG Grant Agreement 407008443); The French National
Research Agency with project ANR-18-CE92-0030-02; The European Research
Council (ERC) through CoG. #725039, and the Swiss National Science Foundation
through the projects SNF 200021_165854 and SNF
200020_197052.

##  Appendix A Hyperfine dipole matrix elements

Evaluation of the matrix elements follows similar methods as done in recent
works [56, 48, 57, 58]. The hyperfine matrix elements of equation (11) are
evaluated with standard angular reduction methods, which start by expanding
the electric dipole $\bm{r}\cdot\bm{\hat{\varepsilon}}$, and the magnetic
dipole $\bm{S}\cdot\bm{\hat{k}\times\hat{\varepsilon}}$ operators (similarly
with $\bm{I}\cdot\bm{\hat{k}\times\hat{\varepsilon}}$) in a spherical basis.
After considering the overall atomic state being the product coupling of the
nucleus and electron angular momenta, and using the Wigner-Eckart theorem, the
matrix element for the irreducible rank-one tensor $T_{\lambda}$ (components
$\lambda=\pm 1$) of the spherical expansion is given by [44, 59]

$\displaystyle\left\langle\beta^{\prime}F^{\prime}m^{\prime}J^{\prime}\left|T_{\lambda}\right|\beta
FmJ\right\rangle$ $\displaystyle=$
$\displaystyle(-1)^{F^{\prime}+I+F+1+J^{\prime}-m^{\prime}}\sqrt{[F,F^{\prime}]}\left(\begin{array}[]{ccc}F^{\prime}&1&F\\\
-m^{\prime}&\lambda&m\end{array}\right)$ (27) $\displaystyle\times$
$\displaystyle\left\\{\begin{array}[]{ccc}J&I&F\\\
F^{\prime}&1&J^{\prime}\end{array}\right\\}\left\langle\beta^{\prime}J^{\prime}\|T\|\beta
J\right\rangle~{}.$ (30)

Here, $T_{\lambda}=r_{\lambda}$ and $T_{\lambda}=S_{\lambda}$ are for the
$2s-2p$ and HFS cases, respectively. Both final and initial states are defined
by
$\left|f\right\rangle=\left|\beta^{\prime}F^{\prime}m^{\prime}J^{\prime}\right\rangle$
and $\left|i\right\rangle=\left|\beta FmJ\right\rangle$, where $F$ is the
total angular momentum , $m$ is its projection, and $J$ is the total angular
momentum of $\mu$. $\beta$ represents further quantum numbers, such as $\ell$
and $n$. The notation $[j_{1},j_{2},...]$ stands for $(2j_{1}+1)(2j_{2}+1)$…
The reduced matrix elements for E1 and M1 are given by

$\displaystyle\left\langle 2pJ^{\prime}\|r\|2s,J=1/2\right\rangle$
$\displaystyle=$
$\displaystyle\sqrt{\left[1/2,J^{\prime}\right]}\left(\begin{array}[]{ccc}J^{\prime}&1&1/2\\\
1/2&0&-1/2\end{array}\right)\int P_{2p}^{\prime}\,P_{2s}\,rdr~{},$ (33)

and

$\displaystyle\left\langle\beta^{\prime}J^{\prime}|S|\beta
J\right\rangle=\frac{\sqrt{6}}{2}\delta(\ell,\ell^{\prime})(-1)^{J^{\prime}-1/2}\sqrt{[J,J^{\prime}]}\left\\{\begin{array}[]{ccc}1/2&\ell&J\\\
J^{\prime}&1&1/2\end{array}\right\\}\hbar~{},$ (36)

for the $\bm{r}$ and $\bm{S}$ operators, respectively. The functions
$P_{2p}^{\prime}$ and $P_{2s}$ in equation (33) are hydrogenic radial
wavefunctions.

After performing the averaging over initial magnetic substates $m$ and
summation over final substates $m^{\prime}$, the matrix element are given by

$\displaystyle{\mathcal{M}}_{\mbox{E1}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\sqrt{[F^{\prime},J,J^{\prime}]}\left\\{\begin{array}[]{ccc}J&I&F\\\
F^{\prime}&1&J^{\prime}\end{array}\right\\}\left(\begin{array}[]{ccc}J^{\prime}&1&J\\\
1/2&0&-1/2\end{array}\right)\int P^{\prime}_{2p}\,P_{2s}\,rdr~{},$ (41)
$\displaystyle{\mathcal{M}}_{\mbox{M1}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{8}}\sqrt{[F^{\prime},F]}\left\\{\begin{array}[]{ccc}1/2&I&F\\\
F^{\prime}&1&1/2\end{array}\right\\}\left(\frac{g_{\mu}\hbar}{m_{\mu}c}+\frac{g_{p}\hbar}{m_{p}c}\right)~{},$
(44)

for the Lamb shift and HFS, respectively. While evaluating the matrix
elements, the overall contribution of light polarization is reduced to
$|\hat{\varepsilon}|^{2}=1$, thus suppressing any dependence on the laser
polarization (circular or linear) used. This is expected since the initial
state is not polarized, and the final state polarization is not observed.
Equation (44) is equivalent to equation (2) of Ref. [37] with $g_{\mu}=2$,
$g_{p}=0$ and $1/c\rightarrow e$ (different units), however we obtain a value
of $e\hbar/{(2mc)}$ instead of $e\hbar/(2\sqrt{6}mc)$ after its evaluation.

##  Appendix B Results for a statistical distribution and $\tau=10$ ns

Table 5 compares $\mathcal{F}_{\text{sat}}$ and $\Gamma_{c}$ for statistical
and Boltzmann distributions between ortho- and para-hydrogen, while Table 6
lists the transition probabilities $\bar{\rho}_{33}(\omega_{r})$ and
linewidths, similar to Table 4 but for a statistical distribution. Differences
of less than 5% are observed for both the saturation fluences and
$\bar{\rho}_{33}(\omega_{r})$ at the listed experimental conditions.

Table 7 contains the transition probabilities $\bar{\rho}_{33}(\omega_{r})$
and linewidths for $\tau=10$ ns. As observed, differences in
$\bar{\rho}_{33}(\omega_{r})$ between 10 ns and 100 ns are more pronounced for
$\Delta_{l}=100$ MHz, having differences around 26%, while for $\Delta_{l}=10$
MHz are negligible.

Table 5:  Decoherence rates and saturation fluences for statistical and Boltzmann distributions between ortho- and para-hydrogen, and for $\Delta_{l}=100$ MHz.  $T$ | $p$ | $\Gamma_{c}^{\text{Boltz.}}$ | $\Gamma_{c}^{\text{Stat.}}$ | $\mathcal{F}^{\text{Boltz.}}_{\mbox{sat}}$ | $\mathcal{F}_{\mbox{sat}}^{\text{Stat.}}$
---|---|---|---|---|---
[K] | [bar] | [MHz] | [MHz] | [J/cm2] | [J/cm2]
22 | 0.5 | 771 | 782 | 44 | 45
22 | 1 | 913 | 936 | 49 | 49
22 | 2 | 1198 | 1244 | 58 | 59
30 | 0.5 | 733 | 743 | 47 | 47
30 | 1 | 838 | 859 | 50 | 50
50 | 0.5 | 693 | 699 | 53 | 53
50 | 1 | 757 | 770 | 55 | 55
Table 6:  Same as Table 4 but for a statistical ortho- to para-distribution of
the initial rotational states of H2 molecules.

|  |  |  |  |  |  ((($\mathcal{F}=10$ J/cm2 |  |  (($\mathcal{F}=20$ J/cm2 |  |  (($\mathcal{F}=50$ J/cm2
---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  | ($\Omega=8$ MHz) |  | ($\Omega=11$ MHz) |  | ($\Omega=18$ MHz)
$p$ | $T$ | $\Delta_{l}$ | $\Gamma_{c}$ | $\sigma_{D}$ |  | $\hat{\Gamma}_{V}$ | $\Gamma_{V}$ | $\hat{\rho}_{33}$ | $\bar{\rho}_{33}$ |  | $\hat{\Gamma}_{V}$ | $\Gamma_{V}$ | $\hat{\rho}_{33}$ | $\bar{\rho}_{33}$ |  | $\hat{\Gamma}_{V}$ | $\Gamma_{V}$ | $\hat{\rho}_{33}$ | $\bar{\rho}_{33}$
$[\mathrm{bar}]$ | [K] | [MHz] | [MHz] | [MHz] |  | [MHz] | [MHz] |  |  |  | [MHz] | [MHz] |  |  |  | [MHz] | [MHz] |  |
0.5 | 22 | 100 | 782 | 60 |  | 222 | 231 | 0.22 | 0.18 |  | 225 | 234 | 0.45 | 0.32 |  | 235 | 269 | - | 0.58
1 | 22 | 100 | 936 | 60 |  | 238 | 246 | 0.20 | 0.18 |  | 240 | 259 | 0.40 | 0.32 |  | 245 | 287 | - | 0.59
0.5 | 22 | 10 | 217 | 60 |  | 162 | 169 | 0.35 | 0.25 |  | 165 | 177 | 0.71 | 0.38 |  | 171 | 193 | - | 0.57
1 | 22 | 10 | 371 | 60 |  | 176 | 182 | 0.31 | 0.24 |  | 177 | 191 | 0.62 | 0.40 |  | 182 | 213 | - | 0.64
0.5 | 30 | 100 | 744 | 70 |  | 241 | 249 | 0.21 | 0.17 |  | 245 | 253 | 0.43 | 0.29 |  | 258 | 279 | - | 0.53
1 | 30 | 100 | 859 | 70 |  | 251 | 259 | 0.20 | 0.17 |  | 254 | 270 | 0.40 | 0.30 |  | 261 | 290 | 0.99 | 0.56
0.5 | 30 | 10 | 178 | 70 |  | 183 | 190 | 0.32 | 0.21 |  | 186 | 196 | 0.64 | 0.32 |  | 194 | 211 | - | 0.48
1 | 30 | 10 | 293 | 70 |  | 192 | 197 | 0.29 | 0.22 |  | 194 | 208 | 0.59 | 0.36 |  | 199 | 226 | - | 0.56
0.5 | 50 | 100 | 699 | 90 |  | 285 | 290 | 0.19 | 0.13 |  | 292 | 295 | 0.38 | 0.23 |  | 310 | 317 | 0.95 | 0.41
1 | 50 | 100 | 770 | 90 |  | 288 | 295 | 0.18 | 0.15 |  | 292 | 301 | 0.36 | 0.26 |  | 303 | 331 | 0.91 | 0.48
0.5 | 50 | 10 | 134 | 90 |  | 229 | 233 | 0.27 | 0.16 |  | 233 | 234 | 0.53 | 0.23 |  | 242 | 251 | - | 0.35
1 | 50 | 10 | 205 | 90 |  | 233 | 239 | 0.25 | 0.18 |  | 235 | 243 | 0.51 | 0.28 |  | 242 | 260 | - | 0.43

Table 7: Same as Table 4 but for a laser exposure time of $\tau=10$ ns.

|  |  |  |  |  |  ((($\mathcal{F}=10$ J/cm2 |  |  (($\mathcal{F}=20$ J/cm2 |  |  (($\mathcal{F}=50$ J/cm2
---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  | ($\Omega=26$ MHz) |  | ($\Omega=36$ MHz) |  | ($\Omega=58$ MHz)
$p$ | $T$ | $\Delta_{l}$ | $\Gamma_{c}$ | $\sigma_{D}$ |  | $\hat{\Gamma}_{V}$ | $\Gamma_{V}$ | $\hat{\rho}_{33}$ | $\bar{\rho}_{33}$ |  | $\hat{\Gamma}_{V}$ | $\Gamma_{V}$ | $\hat{\rho}_{33}$ | $\bar{\rho}_{33}$ |  | $\hat{\Gamma}_{V}$ | $\Gamma_{V}$ | $\hat{\rho}_{33}$ | $\bar{\rho}_{33}$
$[\mathrm{bar}]$ | [K] | [MHz] | [MHz] | [MHz] |  | [MHz] | [MHz] |  |  |  | [MHz] | [MHz] |  |  |  | [MHz] | [MHz] |  |
0.5 | 22 | 100 | 770 | 60 |  | 245 | 257 | 0.23 | 0.14 |  | 269 | 258 | 0.45 | 0.24 |  | 328 | 291 | - | 0.42
1 | 22 | 100 | 913 | 60 |  | 249 | 271 | 0.21 | 0.15 |  | 263 | 282 | 0.41 | 0.27 |  | 301 | 304 | - | 0.49
0.5 | 22 | 10 | 205 | 60 |  | 177 | 189 | 0.36 | 0.24 |  | 190 | 190 | 0.72 | 0.39 |  | 219 | 208 | - | 0.56
1 | 22 | 10 | 348 | 60 |  | 184 | 205 | 0.31 | 0.22 |  | 194 | 210 | 0.63 | 0.38 |  | 218 | 227 | - | 0.60
0.5 | 30 | 100 | 733 | 70 |  | 271 | 276 | 0.21 | 0.11 |  | 301 | 278 | 0.43 | 0.20 |  | 371 | 312 | - | 0.34
1 | 30 | 100 | 838 | 70 |  | 267 | 285 | 0.20 | 0.14 |  | 285 | 285 | 0.40 | 0.25 |  | 330 | 319 | - | 0.44
0.5 | 30 | 10 | 168 | 70 |  | 201 | 207 | 0.33 | 0.22 |  | 214 | 208 | 0.65 | 0.36 |  | 246 | 226 | - | 0.49
1 | 30 | 10 | 273 | 70 |  | 203 | 215 | 0.30 | 0.21 |  | 214 | 221 | 0.60 | 0.35 |  | 240 | 238 | - | 0.54
0.5 | 50 | 100 | 693 | 90 |  | 333 | 314 | 0.19 | 0.08 |  | 374 | 315 | 0.38 | 0.13 |  | 470 | 351 | 0.95 | 0.22
1 | 50 | 100 | 757 | 90 |  | 315 | 315 | 0.18 | 0.10 |  | 341 | 327 | 0.37 | 0.19 |  | 404 | 350 | 0.91 | 0.33
0.5 | 50 | 10 | 127 | 90 |  | 252 | 246 | 0.27 | 0.18 |  | 269 | 253 | 0.53 | 0.29 |  | 305 | 272 | - | 0.38
1 | 50 | 10 | 192 | 90 |  | 249 | 252 | 0.26 | 0.18 |  | 263 | 258 | 0.51 | 0.30 |  | 293 | 285 | - | 0.43

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11institutetext: NeuralMind, Brazil
11email<EMAIL_ADDRESS>
22institutetext: Centro de Informática, Universidade Federal de Pernambuco,
Brazil
22email<EMAIL_ADDRESS>
#  Visconde: Multi-document QA with GPT-3 and Neural Reranking

Jayr Pereira 1122 0000-0001-5478-438X Robson Fidalgo 22 0000-0002-4714-2933
Roberto Lotufo 11 0000-0002-5652-0852 Rodrigo Nogueira 11 0000-0002-2600-6035

###### Abstract

This paper proposes a question-answering system that can answer questions
whose supporting evidence is spread over multiple (potentially long)
documents. The system, called Visconde, uses a three-step pipeline to perform
the task: decompose, retrieve, and aggregate. The first step decomposes the
question into simpler questions using a few-shot large language model (LLM).
Then, a state-of-the-art search engine is used to retrieve candidate passages
from a large collection for each decomposed question. In the final step, we
use the LLM in a few-shot setting to aggregate the contents of the passages
into the final answer. The system is evaluated on three datasets: IIRC,
Qasper, and StrategyQA. Results suggest that current retrievers are the main
bottleneck and that readers are already performing at the human level as long
as relevant passages are provided. The system is also shown to be more
effective when the model is induced to give explanations before answering a
question. Code is available at https://github.com/neuralmind-ai/visconde.

##  1 Introduction

In recent years, question-answering (QA) tasks that use relatively short
contexts (e.g., a paragraph) have seen remarkable progress in multiple domains
[14, 15]. However, in many cases, the necessary information to answer a
question is spread over multiple documents or long ones [36, 23, 7, 9]. To
solve this task, QA models are based on a pipeline comprised of a retriever
and a reader component [13, 8, 30, 28]. Most of these approaches rely on fine-
tuning large language models on supervised datasets, which may be available
for a variaty of domains. Other approaches use Transformers for long sequences
like LongT5 [10] to process the context document and the question at once [29,
33], which might not scale to longer sequences (e.g., documents with hundreds
of pages).

The few-shot capability of LLMs may reduce the costs for solving QA tasks, as
it allows one to implement QA systems for different domains without needing a
specific annotated dataset. In addition, recent studies showed that adding a
chain-of-thought (CoT) reasoning step before answering significantly improves
LLMs’ zero or few-shot effectiveness on diverse QA benchmarks [16]. In this
work, we propose Visconde,111The name is a homage to Visconde de Sabugosa a
fictional character invented by Monteiro Lobato that is a corn cob doll whose
wisdom comes from reading books. a QA system that combines a state-of-the-art
retriever and a few-shot (CoT) approach to induce an LLM to generate the
answer as a generative reader. The retriever is a multi-stage pipeline that
uses BM25 [27] to select candidate documents followed by a monoT5 reranker
[22]. The reader uses GPT-3 [2] in a few-shot setting that reason over the
retrieved records to produce an answer. We induce CoT by asking the model to
explain how the evidence documents can answer the question. Our system rivals
state-of-the-art supervised models in three datasets: IIRC, Qasper, and
StrategyQA.

Our main contribution is to show that current multi-document QA systems are
close to human-level performance as long as ground truth contexts are provided
as input to the reader. When a SOTA retriever selects the context, we observe
a significant drop in effectiveness. Thus, we argue future work on multi-
document QA should focus on improving retrievers.

##  2 Related Work

Most approaches for multi-document QA are typically based on a retriever
followed by a reader component [37]. The retriever aims to select relevant
documents for given a question, while the reader seeks to infer the final
answer from them. Recent studies used dense retrievers [30, 7] or commercial
search engines [17] for this task. For the reader component, some studies used
sequence-to-sequence models to generate natural language answers [18, 34, 12],
numerical reasoning models adapted to reason also over text [8, 30, 7], or
LLMs to aggregate information from multiple documents [21, 17].

Recent work enriches this pipeline by adding components to perform query
decomposition [24, 4, 6, 1, 11] or evidence retrieved by a web search engine
[21, 24, 17]. Our work is similar to these, but we focus on evaluating the
limitations of this method and found that the retrieval component needs more
work.

##  3 Our Method: Visconde

Visconde is a multi-document QA system that has three main steps: Question
decomposition, Document Retrieval, and Aggregation. As illustrated in Figure
1, the system first decomposes the user question when necessary and searches
for relevant documents to answer the subquestions. The retrieved documents are
the basis for generating an explanation and a final answer using an LLM.

Figure 1: Visconde QA flow.

Question Decomposition: We use GPT-3 (text-davinci-002) with five in-context
examples for question decomposition. In Figure 1 we show an example of a
question that needs to be decomposed (Q) extracted from the IIRC dataset [7]
and the subquestions generated by the model (Q1 and Q2). The five examples
used as few-shot in the prompt were randomly selected from the training set of
StrategyQA dataset [9], which has questions and decomposed subquestions.

Document Retrieval: For document retrieval, we used a strategy divided into
three main steps: 1) document indexing – we create an inverted index using
Pyserini [19]; 2) candidates retrieval – we use the Pyserini implementation of
the BM25 algorithm to retrieve candidate documents. 3) document reranking – we
rerank the top-1000 documents retrieved using a sequence-to-sequence model
designed for reranking, the monoT5 model [22], an adaptation of the T5 model
[25].222We used the 3 billion parameters version, whose checkpoint is
available at https://huggingface.co/castorini/monot5-3b-msmarco-10k monoT5
receives as input a sequence with the document and the query, and a softmax
function is applied only on the logits calculated by T5 to the tokens true and
false. The log probability of the token true is used as the document relevance
score given the question. The output is a list of documents ranked by the
relevance scores.

Aggregation: We use GPT-3 (text-davinci-002) as a few-shot learner for the
aggregation step. Different studies have shown that the effectiveness of LLM
can be improved by inducing it to first generate reasoning steps before
answering a question [32, 31, 16, 3]. We use CoT to induce the LLM to reason
over multiple documents and answer a question, as shown in the example in
Figure 2. In our prompt, each example has a list of context documents (e.g.,
[Document 1]), a question, an evidence paragraph, and the answer. The context
documents of the target example are the top-k documents from the retriever
step. When the question is decomposed, we use the top-k documents from each
subquestion. For the target example, an evidence paragraph is not provided,
and the LLM must generate it, as well as a final answer.

Figure 2: Reasoning prompt example. The bold text is the model’s output.

– Few-shot examples omitted due to space –

Example 5:

[Document 1]: Title: Giovanni Messe. Content: […] became aide-de-camp to King
Victor Emmanuel III, holding this post from 1923 to 1927.

– Documents omitted due to space –

[Document 5]: Title: World War I. Content: […] Germany signed an armistice on
11 November 1918, effectively ending the war.

Question: How long had the First World War been over when Messe was named
aide-de-camp?

Evidence: According to [Document 5], the First World War ended on November
11th, 1918. According to [Document 1], Messe was named aide-de-camp on 1923.
Therefore, the First World War had been over for 5 years when Messe was named
aide-de-camp.

Answer: 5 years

We tested two approaches for prompt construction: 1) using static prompts with
a pre-defined list of examples; and 2) using dynamic prompts by selecting in-
context examples that are similar to the test example. For dynamic prompts, we
encode the questions from the training dataset using SentenceTransformers [26]
and apply a KNN algorithm to find the $k$ most similar to the test question,
as Liu et al. [20] did for other tasks.333We used this model as our sentence
encoder: sentence-transformers/msmarco-bert-base-dot-v5.

##  4 Experiments

###  4.1 IIRC

The Incomplete Information Reading Comprehension (IIRC) dataset [7] consists
of information-seeking questions that require retrieving the necessary
information missing from the original context. Each original context is a
paragraph from the English Wikipedia, which comes with a set of links to other
Wikipedia pages.

Pre-processing: we used the dynamic prompt described in Section 3. For this,
we automatically generated reasoning paragraphs for 10% of the IIRC training
set (1340 questions) using GPT-3. In addition, we processed the context
articles provided by the dataset to create a searchable index.

Procedure: following the framework depicted in Figure 1, we first decomposed
the questions from the IIRC test set. We performed document retrieval on a
database of Wikipedia documents provided in the dataset. In the aggregation
step, we applied four methods: 1) using GPT-3 without CoT and providing the
links and the ground truth contexts, i.e., skipping the document retrieval
step; 2) using the reasoning step with the links and ground truth contexts; 3)
using reasoning step over the intersection of retrieved documents and the
documents cited by the main context; and 4) reasoning over the documents
retrieved from the entire Wikipedia subset provided by the dataset.

###  4.2 Qasper

Qasper [5] is an information-seeking QA dataset over academic research papers.
The task consists of retrieving the most relevant evidence paragraph for each
question and answering the question.

Procedure: we did not apply query decomposition because the questions in this
dataset are closed-ended, i.e., they do not require decomposition as they are
grounded in a single paper of interest [5]. For example, the question How is
the text segmented? only makes sense concerning its grounded paper. Besides,
we skipped the BM25 step for document retrieval as the monoT5 reranker can
score each paragraph in the paper in a reasonable time. The document retrieval
step consists of reranking the paper’s paragraphs based on the question and
choosing the top five as context documents. We did not notice any advantage in
using a dynamic prompt in this dataset.

###  4.3 StrategyQA

StrategyQA [9] is a dataset focused on open-domain questions that require
reasoning steps. This dataset has three tasks: 1) question decomposition,
measured using a metric called SARI, generally used to evaluate automatic text
simplification systems [35]; 2) evidence paragraph retrieval, measured as the
recall of the top ten retrieved results; and 3) question answering, measured
in terms of accuracy.

Pre-processing: we did not generate reasoning paragraphs for the training
examples since the context comprises long paragraphs that exceed the model
input size limit (4000 tokens). We processed the context articles provided by
the dataset to create a searchable index, by splitting the articles into
windows of three sentences each.

Procedure: we applied question decomposition and performed retrieval using the
approach described in Section 3. We used the top five retrieved documents for
each decomposed question as the context in the reading step.

###  4.4 Results

Table 1: Visconde and similar methods results on IIRC, Qasper and StrategyQA.

IIRC
---
Model | F1 | EM
Human | 88.4 | 85.7
Finetuned |  |
Ferguson et al. [7] | 31.1 | 27.7
Ferguson et al. [7] Linked pages | 32.5 | 29.0
Ferguson et al. [7] Gold Ctx | 70.3 | 65.6
PReasM (pretrain + finetuning) [30] Gold Ctx | - | 73.3
PReasM (pretrain + finetuning) [30] | - | 47.2
SupA+QA (supervised) [8] | 51.6 | -
Few-shot |  |
Visconde (4-shot dynamic prompt) Gold Ctx and CoT | 84.2 | 74.7
Visconde (4-shot dynamic prompt) Gold Ctx | 80.3 | 70.0
Visconde (4-shot static prompt) Gold Ctx | 74.3 | 62.7
Visconde (4-shot dynamic prompt) Linked pages | 48.2 | 40.7
Visconde (4-shot dynamic prompt) CoT | 47.9 | 40.0

Qasper
---
Model | Extractive | Abstractive | Boolean | Unanswerable | Evidence F1 | Answer F1
Human | 58.9 | 39.7 | 79.0 | 69.4 | 71.6 | 60.9
LED-base | 30.0 | 15.0 | 68.9 | 45.0 | 29.9 | 33.6
SOTA sup. [33] | - | - | - | - | - | 53.1
Visconde | 52.3 | 21.7 | 86.2 | 48.3 | 38.5 | 49.1

StrategyQA
---
Model | Acc | Recall@10 | SARI
Human | 87.00 | 0.586 | -
Baseline | 63.60 | 0.195 | -
Leaderboard’s SOTA | 69.80 | 0.537 | 0.555
GOPHER-280B OB${}^{PoE}_{Google}$ [17] | 66.20 | - | -
Visconde (1-shot, static prompt, gold evidences) | 73.80 | - | -
Visconde (1-shot, static prompt) | 69.43 | 0.331 | 0.570

In Table 1, we present the results of our experiments. First, we show the
results obtained in the IIRC dataset. Our approach outperforms the baselines
(i.e., Ferguson et al. [7]’s) in different settings: 1) Using the gold context
searched by humans (Gold Ctx); 2) Searching for context in the links the
dataset provides (Linked pages); and 3) Searching for contexts in the entire
dataset. We report Visconde’s results with and without CoT and using a static
prompt instead of a dynamic one. The dynamic prompt leads to better
performance in this dataset. When using the gold contexts, Visconde approaches
human performance in terms of F1. However, when the system has to search for
context, performance decreases. Also, the system performs better when using
CoT. With CoT, Visconde tends to perform better in questions requiring basic
arithmetic operations, which is consistent with the literature [16, 32]. By
inspecting the model output in relation to the expected answer, we noticed
that in some cases the system answers questions marked as unanswerable by the
human annotators. This may occur 1) because the retriever found a relevant
document containing the answer or 2) because GPT-3 answers the question even
when the necessary information is not in the context. Different ways to write
numerals may also affect the results. For example, the model might answer five
years, while 5 years is expected.

For Qasper, we present the results in terms of F1 for the answer and the
evidence. The LED-base model is the baseline [5]. The SOTA model is the model
proposed by Xiong et al. [33]. Visconde outperformed the baseline but did not
surpass the SOTA model. Xiong et al. [33]’s model is a long Transformer fine-
tuned on the task. Regarding evidence F1, our system outperforms the baseline,
but there is still a gap between our performance and human performance.
Visconde had a high performance on the boolean questions but a low score on
the abstractive ones. The table shows that even the human result is lower for
the abstractive question than other types.

For StrategyQA, we present the results in terms of answer accuracy, evidence
recall@10, and SARI for question decomposition. Automated methods are still
far from human performance. Our approach outperforms the baselines presented
in the paper [9] in terms of answer accuracy and evidence recall@10. We also
outperform the leaderboard’s SOTA
model444https://leaderboard.allenai.org/strategyqa/submissions/public.
Accessed on July 20 2022. in the quality of the questions decomposition
measured with SARI. However, we did not surpass SOTA’s recall@10 in retrieving
the appropriate evidence paragraphs and coming close in the answer accuracy.
We also outperform Lazaridou et al. ’s approach [17], which also uses a few-
shot LLM.

##  5 Conclusion

This paper describes a system for multi-document question answering that uses
a passage reranker to retrieve documents and large language models to reason
over them and compose an answer. Our system rivals state-of-the-art supervised
models in three datasets: IIRC, Qasper, and StrategyQA. Our results suggest
that using GPT-3 as a reader is close to human-level performance as long as
relevant passages are provided, while current retrievers are the main
bottleneck. We also show that inducing the model to give explanations before
answering a question improves effectiveness.

## Aknowledgements

This research was partially supported by Fundação de Amparo à Pesquisa do
Estado de São Paulo (FAPESP) (project id 2022/01640-2) and by Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior (CAPES) (Grant code:
88887.481522/2020-00). We also thank Centro Nacional de Processamento de Alto
Desempenho (CENAPAD-SP) and Google Cloud for computing credits.

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