# Proposal of ensemble qubits with two-atom decay

Wei Qin  Theoretical Quantum Physics Laboratory, Cluster for Pioneering
Research, RIKEN, Wako-shi, Saitama 351-0198, Japan  Adam Miranowicz
Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research,
RIKEN, Wako-shi, Saitama 351-0198, Japan  Institute of Spintronics and Quantum
Information, Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań,
Poland  Franco Nori  Theoretical Quantum Physics Laboratory, Cluster for
Pioneering Research, RIKEN, Wako-shi, Saitama 351-0198, Japan  Center for
Quantum Computing, RIKEN, Wako-shi, Saitama 351-0198, Japan  Department of
Physics, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

###### Abstract

We propose and analyze a novel approach to implement ensemble qubits. The
required anharmonicity is provided by a simultaneous decay of two atoms (i.e.,
two-atom decay), which is achieved by fully quantum degenerate parametric
amplification. For an atomic ensemble, the two-atom decay generates and
stabilizes a 2D quantum manifold, which is spanned by the ground and single-
excited superradiant states. Moreover, this nonlinear decay process can
strongly suppress transitions to higher-excited superradiant states, and
convert residual transitions into an effective decay from the single-
excitation superradiant state to the ground state. Our method does not require
Rydberg dipole blockade and, thus, strong atom-atom interactions, compared to
previous work. This indicates that it can be applied to typical atomic or spin
ensembles in simple experimental setups. Remarkably, our idea is compatible
with the cavity protection mechanism, and therefore spin dephasing due to
inhomogeneous broadening can be strongly suppressed. The presented ensemble
qubit provides a new platform for quantum information processing, and also
extends the range of applications of atomic or spin ensembles.

##  I Introduction

Ensembles of atoms or spins have been attracting considerable interest,
because of their long coherence times (even at room temperatures) and their
ability to interface between the microwave and optical regimes. Moreover,
ensembles of atoms or spins have been used to, e.g., build quantum repeaters
for long distance quantum communication Duan  _et al._ (2001); Kuzmich  _et
al._ (2003); Hammerer  _et al._ (2010); Sangouard  _et al._ (2011), design
hybrid systems for powerful quantum computation Xiang  _et al._ (2013);
Kurizki  _et al._ (2015); Gu  _et al._ (2017); Clerk  _et al._ (2020), and
generate nonclassical states for high-precision quantum measurements Kitagawa
and Ueda (1993); Agarwal  _et al._ (1997); Hald  _et al._ (1999); Ma  _et al._
(2011); Pezzè  _et al._ (2018); Qin  _et al._ (2020); Groszkowski  _et al._
(2020); Qin  _et al._ (2021); Liu  _et al._ (2022). Despite such considerable
developments, exploiting an ensemble as a single qubit remains a challenge,
due to the requirement of a strong anharmonicity. Previous approaches, which
have been proposed to encode an ensemble qubit, usually rely on Rydberg dipole
blockade Lukin  _et al._ (2001); Brion  _et al._ (2007); Møller  _et al._
(2008); Yan  _et al._ (2008); Mei  _et al._ (2009); Zwierz and Kok (2010);
Saffman  _et al._ (2010); Ebert  _et al._ (2015); Saffman (2016); Spong  _et
al._ (2021), and require strong dipole-dipole interactions between highly
excited atoms. However, for typical ensembles, it is much easier to control
atoms in the ground state manifold, and in this case atom-atom interactions
are extremely weak. Thus, a new mechanism, which can generate a strong
anharmonicity in atomic ensembles, is desirable for the formation of ensemble
qubits.

In the present paper, we describe the use of the two-atom decay to provide a
strong anharmonicity. The two-atom decay, proposed in our recent work by
exploiting a fully quantum degenerate parametric amplifier (DPA) Qin  _et al._
(2021), refers to a simultaneous decay of two atoms in an ensemble. As usual,
we assume the atomic ensemble to be in the low-excitation regime, where the
number of excited atoms is much smaller than the total number of atoms. In
this regime, the spin-wave approximation can be applied, and the ensemble
behaves as a harmonic oscillator. However, the nonlinear two-atom decay
conserves the parity of the number of the excited atoms, and as a result the
ensemble states are restricted to a 2D quantum manifold, spanned by the ground
state and a symmetric collective state with only one excitation (i.e., a
single-excitation superradiant state). Moreover, due to the two-atom decay,
the transitions out of the 2D quantum subspace (e.g., driven by an external
driving) are strongly suppressed, and the residual transitions are converted
into an effective decay from the single-excitation superradiant state to the
ground state. This indicates that an ensemble qubit is formed.

We also show that our proposal is compatible with the cavity protection
technique. That is, a strong coupling between the ensemble and the signal mode
makes the superradiant spin-wave mode largely detuned and, thus, decoupled
from the subradiant spin-wave modes. Consequently, spin dephasing due to
inhomogeneous broadening, which is the main source of noise of atomic
ensembles, can be strongly suppressed. In contrast to previous work on
ensemble qubits, our proposal requires neither Rydberg dipole blockade nor
strong atom-atom interactions. Therefore, it could be applied to common atomic
or spin ensembles, thus enabling a significant simplification of the
experimental complexity.

The remaining of the paper is organized as follows. The physical system used
to form an ensemble qubit, i.e., an atomic ensemble coupled to the signal mode
of a fully quantum DPA, is introduced in Sec. II. Included are the system
Hamiltonian and the corresponding master equation. In Sec. III, we discuss a
parametric coupling between an atom pair and a pump-mode photon, obtain the
two-atom decay by adiabatically eliminating the pump mode, and then, with such
a nonlinear atomic decay, analyze the form of an ensemble qubit. Sec. IV
presents Rabi oscillations of the ensemble qubit subject to an external field,
and shows that the transitions out of the ensemble-qubit subspace, driven by
this external field, can be converted to an effective decay process of the
ensemble qubit. We demonstrate in Sec. V that our idea is compatible with the
cavity protection mechanism, and thus can strongly suppress spin dephasing due
to inhomogeneous broadening for the ensemble qubit. Conclusions are drawn in
Sec. VI. Finally, three Appendixes provide some detailed derivations.

Figure 1: Schematics of our setup used to form an ensemble qubit with a fully
quantum degenerate parametric amplifier. The pump and signal modes,
$\hat{a}_{p}$ and $\hat{a}_{s}$, coupled via a single-photon parametric
coupling of strength $J$, are represented by two cavities, for simplicity. An
ensemble consists of $N$ two-level atoms, each with a transition frequency
$\omega_{q}$, and it is coupled to the signal mode with a single-atom coupling
of strength $g$. Here, $\omega_{p}$, $\omega_{s}$ are the resonance
frequencies of the pump and signal modes, and $\kappa_{p}$, $\kappa_{s}$ are
their respective photon loss rates.

##  II physical model

The basic idea is schematically illustrated in Fig. 1. We consider a fully
quantum DPA, with a pump mode $\hat{a}_{p}$ of frequency $\omega_{p}$, and a
signal mode $\hat{a}_{s}$ of frequency $\omega_{s}$. These two modes are
coupled through a single-photon parametric coupling of strength $J$. We
further assume that an ensemble of $N$ two-level atoms is coupled to the
signal mode with a single-atom strength $g$, and that all the atoms have the
same transition frequency $\omega_{q}$. The system can thus be described by
the Hamiltonian,

$\displaystyle\hat{H}=\;$
$\displaystyle\omega_{s}\hat{a}_{s}^{\dagger}\hat{a}_{s}+\omega_{p}\hat{a}_{p}^{\dagger}\hat{a}_{p}+\omega_{q}\hat{S}_{z}$
$\displaystyle+g\left(\hat{a}_{s}^{\dagger}\hat{S}_{-}+\hat{a}_{s}\hat{S}_{+}\right)+J\left(\hat{a}_{p}\hat{a}_{s}^{\dagger
2}+\hat{a}_{p}^{\dagger}\hat{a}_{s}^{2}\right),$ (1)

where $\hat{S}_{z}=\frac{1}{2}\sum_{j=1}^{N}\sigma_{j}^{z}$ and
$\hat{S}_{\pm}=\sum_{j=1}^{N}\sigma_{j}^{\pm}$ are collective spin operators
of the ensemble. Here, $\sigma_{j}^{z}$ and $\sigma_{j}^{\pm}$ are the Pauli
matrices of the $j$th atom. Upon introducing the Holstein-Primakoff
transformation Holstein and Primakoff (1940), the collective spin can be
mapped to a bosonic mode $\hat{s}$, i.e.,

$\displaystyle\hat{S}_{-}=\;$
$\displaystyle\hat{s}\sqrt{N-\hat{s}^{\dagger}\hat{s}},$ (2)
$\displaystyle\hat{S}_{z}=\;$
$\displaystyle-\frac{N}{2}+\hat{s}^{\dagger}\hat{s}.$ (3)

We restrict our discussions to the case of $N\rightarrow\infty$, so that the
number of excited atoms is much smaller than the total number of atoms, i.e.,
$\mbox{$\langle\hat{s}^{\dagger}\hat{s}\rangle$}\ll N$. In this case, we can
apply the spin-wave approximation and, as a consequence, have

$\hat{S}_{-}\simeq\sqrt{N}\hat{s}.$ (4)

The Hamiltonian $\hat{H}$ is accordingly transformed to

$\displaystyle\hat{\mathcal{H}}=\;$
$\displaystyle\omega_{s}\hat{a}_{s}^{\dagger}\hat{a}_{s}+\omega_{p}\hat{a}_{p}^{\dagger}\hat{a}_{p}+\omega_{q}\hat{s}^{\dagger}\hat{s}$
$\displaystyle+g_{\rm
col}\left(\hat{a}_{s}^{\dagger}\hat{s}+\hat{a}_{s}\hat{s}^{\dagger}\right)+J\left(\hat{a}_{p}\hat{a}_{s}^{\dagger
2}+\hat{a}_{p}^{\dagger}\hat{a}_{s}^{2}\right),$ (5)

where $g_{\rm col}=g\sqrt{N}$ is the strength of the collective coupling of
the ensemble to the signal mode. Furthermore, we use the Lindblad dissipator
$\mathcal{L}\left(o\right)\rho=\left(2\hat{o}\hat{\rho}\hat{o}^{\dagger}-\hat{o}^{\dagger}\hat{o}\hat{\rho}-\hat{\rho}\hat{o}^{\dagger}\hat{o}\right)/2$
to describe photon losses. Therefore, the full dynamics of the system can be
determined by the following master equation:

$\dot{\hat{\rho}}=-i\left[\hat{\mathcal{H}},\hat{\rho}\right]+\kappa_{p}\mathcal{L}\left(\hat{a}_{p}\right)\hat{\rho}+\kappa_{s}\mathcal{L}\left(\hat{a}_{s}\right)\hat{\rho},$
(6)

where $\kappa_{p}$ and $\kappa_{s}$ are the photon loss rates of the pump and
signal modes, respectively.

In the general case, the single-photon coupling $J$ is very weak, so that a
strong pump-mode driving is needed to generate a strong parametric (i.e., two-
photon) driving for the signal mode. In this case, the pump mode is treated
classically and its quantum nature is neglected Scully and Zubairy (1997);
Drummond and Ficek (2004); Drummond and Hillery (2014). Such a type of DPA is
called semiclassical, and its intracavity squeezing has been extensively
studied for various applications Huang and Agarwal (2009); Lü  _et al._
(2015); Qin  _et al._ (2018); Leroux  _et al._ (2018); Ge  _et al._ (2019);
Qin  _et al._ (2019a, b); Chen  _et al._ (2019); Li  _et al._ (2020); Burd
_et al._ (2021); Tang  _et al._ (2022); Peano  _et al._ (2015); Zhou  _et al._
(2021); Chen  _et al._ (2021, 2022). For example, applying a semiclassical DPA
to the Jaynes-Cummings model can enhance the ability of cavity and circuit
quantum electrodynamics systems to process quantum information, which was
predicted in Refs. Qin  _et al._ (2018); Leroux  _et al._ (2018); Ge  _et al._
(2019) and demonstrated experimentally in Ref. Burd  _et al._ (2021).
Recently, with the rapid development of quantum devices, a strong coupling
$J$, which can range from several tens of kHz to several tens of MHz Leghtas
_et al._ (2015); Touzard  _et al._ (2018); Lescanne  _et al._ (2020); Chang
_et al._ (2020); Vrajitoarea  _et al._ (2020); Guo  _et al._ (2016); Bruch
_et al._ (2019); Wang  _et al._ (2021), has been achieved in experiments. This
strong single-photon coupling allows to exploit the quantum nature of the pump
mode, i.e., a fully quantum DPA, for modern quantum technologies Gerry and
Hach III (1993); Gilles  _et al._ (1994); Mirrahimi  _et al._ (2014); Guillaud
and Mirrahimi (2019); Puri  _et al._ (2019); Sun  _et al._ (2019); Qin  _et
al._ (2022); Cai  _et al._ (2021); Ma  _et al._ (2021); Zhou  _et al._ (2022).
As demonstrated below, a fully quantum DPA can be used to confine the state of
an atomic ensemble to a 2D quantum manifold spanned by the ground state and a
single-excitation superradiant state, forming an ensemble qubit.

##  III Ensemble qubit

Figure 2: (a) Parametric coupling between an atom pair and a real pump photon
mediated by a virtual signal-photon pair. Two excited atoms emit two virtual
signal photons, which then are up-converted into a real pump photon. (b)
Transition processes corresponding to the two-atom single-photon conversion
shown in (a). Figure 3: (a) Lowest energy levels of the system Hamiltonian in
Eq. (II) as a function of $\omega_{p}/\omega_{q}$. The two levels indicated by
the arrows corresponds to the states $|10\rangle$$|0\rangle$ and
$|00\rangle$$|2\rangle$, respectively. Here, we set $g_{\rm
col}=0.03\omega_{q}$, $J=3g_{\rm col}$, and $\Delta_{q}=20g_{\rm col}$. (b)
Enlarged view of the boxed region in (c), showing an avoided level crossing
around $\omega_{p}/\omega_{q}=2$. This avoided crossing originates from the
hybridization of the states $|00\rangle$$|2\rangle$ and
$|10\rangle$$|0\rangle$ due to the parametric coupling $\chi$ in Eq. (10).

In this section, we show how to form a two-level ensemble, i.e., an ensemble
qubit, by using a fully quantum DPA. To proceed, we assume that both couplings
of the signal mode to the ensemble and to the pump mode are largely detuned,
i.e.,

$\displaystyle\omega_{s}-\omega_{q}\equiv\;$ $\displaystyle\Delta_{q}\gg
g_{\rm col},$ (7) $\displaystyle 2\omega_{s}-\omega_{p}\equiv\;$
$\displaystyle\Delta_{p}\gg J.$ (8)

But at the same time, the pump mode frequency is approximately equal to the
doubled atomic transition frequency, i.e.,

$\omega_{p}\simeq 2\omega_{q}.$ (9)

Note that here, a slight detuning between $\omega_{p}$ and $2\omega_{q}$ has
been assumed to compensate some undesired resonance shifts of the atoms and
the pump mode (see below for details). Under these assumptions, we can predict
a parametric coupling between atom pairs and real pump photons, i.e., the
joint absorption and emission of a pump photon by two atoms Garziano  _et al._
(2016, 2020). This parametric coupling is mediated by a virtual signal-photon
pair, as shown in Fig. 2(a). Note that the virtual-photon mediated coupling
has been previously exploited for quantum nonlinear optics Kockum  _et al._
(2019, 2017); Stassi  _et al._ (2017).

To better understand the form of the above parametric coupling, we demonstrate
the virtual intermediate transitions between the states
$|00\rangle$$|2\rangle$ and $|10\rangle$$|0\rangle$ in Fig. 2(b). Here, the
first ket $|m_{p}m_{s}\rangle$ ($m_{p},m_{s}=0,1,2,\cdots$) in the pair
accounts for the DPA state with $m_{p}$ photons in the pump mode and $m_{s}$
photons in the signal mode, and the second ket $|n\rangle$ ($n=0,1,2,\cdots$)
accounts for the symmetric superradiant state, $|S=N/2,m_{z}=-N/2+n\rangle$,
of the collective spin, with $n$ excited atoms. The coupling $g$ drives two
virtual transitions in turn from the state $|00\rangle$$|2\rangle$, via the
state $|01\rangle$$|1\rangle$ with a detuning
$\Delta_{q}=\omega_{s}-\omega_{q}$, to the state $|02\rangle$$|0\rangle$
detuned by $2\Delta_{q}$. Subsequently, a virtual transition from
$|02\rangle$$|0\rangle$ to the real state $|10\rangle$$|0\rangle$ is driven by
the parametric coupling $J$.

Thus, the Hamiltonian $\mathcal{H}$ in Eq. (II), after time averaging in the
interaction picture (see Appendix A for details), becomes

$\hat{\mathcal{H}}_{\rm avg}=\chi\left(\hat{a}_{p}\hat{s}^{\dagger
2}+\hat{a}_{p}^{\dagger}\hat{s}^{2}\right),$ (10)

where

$\chi=\frac{g_{\rm col}^{2}J}{\Delta_{q}^{2}}$ (11)

is the strength of an effective parametric coupling between the ensemble and
the pump mode. We numerically diagonalize the system Hamiltonian
$\hat{\mathcal{H}}$ in Eq. (II), and plot the lowest energy levels in Fig.
3(a). It seems as though the levels of the states $|10\rangle$$|0\rangle$ and
$|00\rangle$$|2\rangle$ cross around $\omega_{p}/\omega_{q}=2$. However, in
fact, the predicted parametric coupling $\chi$ can hybridize the states
$|10\rangle$$|0\rangle$ and $|00\rangle$$|2\rangle$ and, as a result, induce
an avoided crossing, with two hybridized states

$\mbox{$|\varphi_{\pm}\rangle$}=\frac{1}{\sqrt{2}}\left(\mbox{$|10\rangle$}\mbox{$|0\rangle$}\pm\mbox{$|00\rangle$}\mbox{$|2\rangle$}\right).$
(12)

This avoided level crossing becomes apparent in the enlarged view shown in
Fig. 3(b).

Let us now consider how to implement a two-level atomic ensemble with the
time-averaged Hamiltonian $\hat{\mathcal{H}}_{\rm avg}$. We assume that the
photon loss of the pump mode is very strong, so that the pump photon emitted
jointly by two excited atoms is lost rather than reabsorbed by these two
atoms. This process leads to the desired two-atom decay. To be more explicit,
we adiabatically eliminate the pump mode $\hat{a}_{p}$ of the time-averaged
Hamiltonian $\hat{\mathcal{H}}_{\rm avg}$ (see Appendix B), and then obtain an
adiabatic master equation associated only with the degree of freedom of the
ensemble, i.e.,

$\dot{\hat{\rho}}_{\rm ens}=\kappa_{\rm
2at}\mathcal{L}\left(\hat{s}^{2}\right)\hat{\rho}_{\rm ens},$ (13)

where

$\kappa_{\rm 2at}=\frac{4\chi^{2}}{\kappa_{p}}$ (14)

is the rate of the two-atom decay. It is clear that the two-atom decay
conserves the parity of the number of excited atoms. Therefore, the state
space of the ensemble can be decomposed into even and odd subspaces, according
to the eigenvalues $\pm 1$ of the parity operator
$P=\exp\left(i\pi\hat{s}^{\dagger}\hat{s}\right)$. Under the master equation
in Eq. (13), the ensemble can be cooled to its ground state $|0\rangle$ for an
even-parity initial state, and to a single-excitation superradiant state
$|1\rangle$ for an odd-parity initial state. Here, the ket $|n\rangle$
($n=0,1,2,\dots$) refers to the collective spin state
$|S=N/2,m_{z}=-N/2+n\rangle$ as already mentioned above, such that

$\mbox{$|n\rangle$}\equiv\mbox{$|S=N/2,m_{z}=-N/2+n\rangle$}.$ (15)

This indicates that the steady state of the ensemble is restricted by the two-
atom decay into a 2D quantum manifold $\mathcal{C}$ spanned by $|0\rangle$ and
$|1\rangle$, and can be expressed as

$\displaystyle\hat{\rho}_{\rm ens}^{\rm ss}=\;$ $\displaystyle
c_{00}\mbox{$|0\rangle$}\mbox{$\langle
0|$}+c_{11}\mbox{$|1\rangle$}\mbox{$\langle 1|$}$
$\displaystyle+c_{01}\mbox{$|0\rangle$}\mbox{$\langle
1|$}+c_{10}\mbox{$|1\rangle$}\mbox{$\langle 0|$},$ (16)

where $c_{i,j}$ ($i,j=0,1$) are some coefficients, which are determined below.

In order to determine the coefficients $c_{ij}$, we define the following two
conserved quantities Simaan and Loudon (1978); Albert and Jiang (2014):

$\displaystyle\hat{\Pi}_{00}=\;$
$\displaystyle\sum_{n=0}\mbox{$|2n\rangle$}\mbox{$\langle 2n|$},$ (17)
$\displaystyle\hat{\Pi}_{01}=\;$
$\displaystyle\frac{\left(\hat{s}^{\dagger}\hat{s}-1\right)!!}{\hat{s}^{\dagger}\hat{s}!!}\hat{\Pi}_{00}\hat{s},$
(18)

where $n!!=n\times\left(n-2\right)!!$ is the double factorial. Under the
master equation in Eq. (13), the time evolution of an operator $\hat{\Pi}$ in
the Heisenberg picture is given by

$\dot{\hat{\Pi}}=\frac{1}{2}\kappa_{\rm 2at}\left(2\hat{s}^{\dagger
2}\hat{\Pi}\hat{s}^{2}-\hat{s}^{\dagger
2}\hat{s}^{2}\hat{\Pi}-\hat{\Pi}\hat{s}^{\dagger 2}\hat{s}^{2}\right).$ (19)

After a straightforward calculation, we obtain

$\dot{\hat{\Pi}}_{00}=\dot{\hat{\Pi}}_{01}=0,$ (20)

implying that these two quantities are indeed conserved.

According to Refs. Simaan and Loudon (1978); Albert and Jiang (2014), the
operators $\hat{\Pi}_{00}$ and $\hat{\Pi}_{01}$ determine the population,
$c_{00}$, of the ground state $|0\rangle$, and the complex coherence,
$c_{01}$, between $|0\rangle$ and the single-excitation superradiant state
$|1\rangle$. As a result, the coefficients $c_{ij}$ can be calculated as

$\displaystyle c_{00}=\;$ $\displaystyle{\rm
Tr}\left[\hat{\Pi}_{00}\hat{\rho}_{\rm ens}\left(0\right)\right],$ (21)
$\displaystyle c_{11}=\;$ $\displaystyle 1-c_{00},$ (22) $\displaystyle
c_{01}=\;$ $\displaystyle{\rm Tr}\left[\hat{\Pi}_{01}^{\dagger}\hat{\rho}_{\rm
ens}\left(0\right)\right],$ (23) $\displaystyle c_{10}=\;$ $\displaystyle
c_{01}^{*},$ (24)

where $\hat{\rho}_{\rm ens}\left(0\right)$ is the initial state of the
ensemble. Clearly, if the ensemble is initially in the even (or odd) parity
subspace, we have $c_{00}=1$ (or $c_{11}=1$), and all other coefficients
vanish.

Figure 4: Time evolution of the state error $\eta$, with the ensemble being
initially in different spin coherent states of coherent amplitudes $\alpha=1$,
$2$, and $3$. We assumed that $J=3g_{\rm col}$, $\Delta_{q}=20g_{\rm col}$,
$\kappa_{p}=5\chi$, $\kappa_{s}=0.3\kappa_{p}$, and that both the pump and
signal modes are initialized in the vacuum.

In order to further confirm the steady-state prediction given above, we
perform numerical simulations in Fig. 4. Specifically, we plot the time
evolution of the state error $\eta=1-\mathcal{F}$, where $\mathcal{F}$ is the
state fidelity. Initially, the ensemble is assumed to be in a spin coherent
state $\hat{\rho}_{\rm ens}\left(0\right)=\mbox{$|\theta,\phi\rangle$}$, which
is defined as

$\mbox{$|\theta,\phi\rangle$}=\hat{R}\left(\theta,\phi\right)\mbox{$|0\rangle$}.$
(25)

Here,

$\hat{R}\left(\theta,\phi\right)=\exp\left[\frac{1}{\sqrt{N}}\left(\alpha\hat{S}_{+}-\alpha^{*}\hat{S}_{-}\right)\right],$
(26)

where $\alpha=\sqrt{N}\theta\exp\left(i\phi\right)/2$, is a rotation operator
of the collective spin, which rotates the ground state $|0\rangle$ of the
ensemble by an angle $\theta$ about the axis
$\left(\sin\phi,-\cos\phi,0\right)$ of the collective Bloch sphere. Clearly,
under the spin-wave approximation (i.e., $\hat{S}_{-}\simeq\sqrt{N}\hat{s}$),
the rotation operator $\hat{R}\left(\theta,\phi\right)$ becomes a bosonic
displacement operator with a coherent amplitude $\alpha$, and accordingly, the
spin coherent state $|\theta,\phi\rangle$ becomes approximated by a bosonic
coherent state $|\alpha\rangle$. With such an initial state, a straightforward
calculation gives

$\displaystyle c_{00}=\;$
$\displaystyle\frac{1}{2}\left(1+e^{-2\left|\alpha\right|^{2}}\right),$ (27)
$\displaystyle c_{01}=\;$
$\displaystyle\alpha^{*}e^{-\left|\alpha\right|^{2}}I_{0}\left(\left|\alpha\right|^{2}\right),$
(28)

where
$I_{0}\left(\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\right)$
is the modified Bessel function of the first kind.

In Fig. 4, we numerically integrate the original master equation in Eq. (6)
for $\alpha=1$, $2$ and $3$ by using the QutiP toolbox Johansson  _et al._
(2012, 2013), and obtain the actual states of the system. Then, we calculate
the fidelity $\mathcal{F}$ between these actual states and the ideal states
analytically predicted by the coefficients in Eq. (27) and (28). We find from
Fig. 4 that for realistic parameters of $g_{\rm col}/2\pi=10$ MHz and
$J/2\pi=30$ MHz, an evolution time of $t=2/\chi\simeq 0.42$ $\mu$s can keep
the state error $\eta$ below $0.1$ even for an initial spin coherent state of
$\alpha=3$. Thus, as expected, the ensemble states are driven by the two-atom
decay into a stabilized 2D quantum manifold $\mathcal{C}$ of the states
$|0\rangle$ and $|1\rangle$, with a high fidelity.

##  IV Rabi oscillations

Having demonstrated the formation of a collective qubit, let us now discuss
how to manipulate this qubit. We assume that the ensemble is driven by an
external field of amplitude $\Omega_{d}$, phase $\theta_{d}$, and frequency
$\omega_{d}$. As shown in Eq. (4) in Appendix A, the atomic transition
frequency is shifted by an amount $\delta_{q}=g_{\rm col}^{2}/\Delta_{q}$, due
to the second-order perturbation of the largely detuned Hamiltonian
$\hat{\mathcal{H}}$ in Eq. (II). Thus, in order to make the ensemble driven
resonantly by the external field, we set

$\omega_{d}=\omega_{q}+\delta_{q}.$ (29)

This yields a driving Hamiltonian under the spin-wave approximation, i.e.,

$\hat{\mathcal{H}}_{d}=\Omega_{d}\left(e^{-i\theta_{d}}\hat{s}+e^{i\theta_{d}}\hat{s}^{\dagger}\right).$
(30)

Note that, here, $\hat{\mathcal{H}}_{d}$ has been transformed to the rotating
frame, which the time-averaged Hamiltonian $\hat{\mathcal{H}}_{\rm avg}$ in
Eq. (10) and the adiabatic master equation in Eq. (13) are in. Including the
driving Hamiltonian $\hat{\mathcal{H}}_{d}$, the master equation in Eq. (13)
can be rewritten as,

$\dot{\hat{\rho}}_{\rm ens}=-i\left[\hat{\mathcal{H}}_{d},\hat{\rho}_{\rm
ens}\right]+\kappa_{\rm 2at}\mathcal{L}\left(\hat{s}^{2}\right)\hat{\rho}_{\rm
ens}.$ (31)

Figure 5: Probabilities, $P_{0}$ and $P_{1}$, of the ensemble being in the
states $|0\rangle$ and $|1\rangle$, respectively, for
$\Omega_{d}=0.1\kappa_{\rm 2at}$ and $\theta_{d}=0$. The Rabi oscillations
between the states $|0\rangle$ and $|1\rangle$ are demonstrated. Symbols are
the exact results obtained by integrating the original master equation in Eq.
(6), with the driving Hamiltonian $\hat{\mathcal{H}}_{d}$ in Eq. (30)
transformed to the proper frame. Curves are the effective predictions given by
the master equation in Eq. (35). The parameters used here are the same as
those in Fig. 4.

Clearly, the driving $\Omega_{d}$ can cause transitions out of the desired
subspace $\mathcal{C}$. However, as long as $\Omega_{d}\ll\kappa_{\rm 2at}$,
the two-atom decay can bring the excited out-of-subspace states quickly back
to the subspace $\mathcal{C}$ and, as a result, the dominant dynamics of the
ensemble is still restricted into $\mathcal{C}$. To see this more clearly, we
express the driving Hamiltonian $\hat{\mathcal{H}}_{d}$ and the two-atom decay
operator $\hat{s}^{2}$ in terms of the states $|0\rangle$, $|1\rangle$, and
$|2\rangle$ as follows:

$\displaystyle\hat{\mathcal{H}}_{d}=\;$
$\displaystyle\Omega_{d}\left(e^{i\theta_{d}}\mbox{$|1\rangle$}\mbox{$\langle
0|$}+e^{i\theta_{d}}\sqrt{2}\mbox{$|2\rangle$}\mbox{$\langle 1|$}+{\rm
H.c.}\right),$ (32) $\displaystyle\hat{s}^{2}=\;$
$\displaystyle\sqrt{2}\mbox{$|0\rangle$}\mbox{$\langle 2|$}.$ (33)

It can be seen that the driving $\Omega_{d}$ drives the transitions
$\mbox{$|0\rangle$}\leftrightarrow\mbox{$|1\rangle$}\leftrightarrow\mbox{$|2\rangle$}$,
but then the state $|2\rangle$ decays to the state $|0\rangle$ via the two-
atom decay. This results in a coherent coupling between the states $|0\rangle$
and $|1\rangle$, and at the same time an effective decay from the state
$|1\rangle$ to the state $|0\rangle$. To further quantify these processes, we
now adiabatically eliminate the state $|2\rangle$ (see Appendix C), and obtain
the Hamiltonian

$\hat{\mathcal{H}}^{\rm
qubit}_{d}=\Omega_{d}\left(e^{-i\theta_{d}}\hat{\sigma}_{-}+e^{i\theta_{d}}\hat{\sigma}_{+}\right),$
(34)

with the corresponding master equation,

$\displaystyle\dot{\hat{\rho}}_{\rm ens}=-i\left[\hat{\mathcal{H}}^{\rm
qubit}_{d},\hat{\rho}_{\rm
ens}\right]+\gamma\mathcal{L}\left(\hat{\sigma}_{-}\right)\hat{\rho}_{\rm
ens},$ (35)

where $\gamma=4\Omega^{2}_{d}/\kappa_{\rm 2at}$ is the rate of the effective
decay of the ensemble qubit. Here, we have defined
$\hat{\sigma}_{-}=\mbox{$|0\rangle$}\mbox{$\langle 1|$}$ and
$\hat{\sigma}_{+}=\hat{\sigma}_{-}^{\dagger}$ as the lowering and raising
operators of the ensemble qubit, respectively. As expected,
$\hat{\mathcal{H}}^{\rm qubit}_{d}$ represents a coherent driving of the
ensemble qubit, which can arbitrarily rotate the state of this ensemble qubit
on the Bloch sphere.

In Fig. 5, we plot the time evolution of the probabilities, $P_{0}$ and
$P_{1}$, of finding the ensemble qubit being in the states $|0\rangle$ and
$|1\rangle$, respectively. We find that the exact results are in excellent
agreement with the effective predictions. The Rabi oscillations between the
states $|0\rangle$ and $|1\rangle$ are clear, implying that the driving
$\Omega_{d}$ can effectively manipulate the ensemble qubit. Furthermore, the
amplitude of the Rabi oscillations decreases mostly due to the ensemble decay
$\gamma$ as induced by the two-atom decay.

##  V Inhomogeneous broadening

So far, we have considered a model, where there is no spin dephasing. Now, we
consider inhomogeneous broadening, which is the main source of noise in
typical atomic ensembles. It is described by the following Hamiltonian

$\hat{H}_{\rm inh}=\frac{1}{2}\sum_{j=1}^{N}\delta_{j}\hat{\sigma}^{z}_{j},$
(36)

where $\delta_{j}=\omega_{j}-\omega_{q}$. Here, $\omega_{j}$ is the transition
frequency of the $j$th atom, and $\omega_{q}$ can be considered as the average
of transition frequencies of all the atoms, i.e.,
$\omega_{q}=\sum_{j=1}^{N}\omega_{j}/N$. The detrimental effects of spin
dephasing due to inhomogeneous broadening can, in principle, be completely
canceled using spin echo pulses Hahn (1950). Below, we show that our proposal
can inherently suppress these detrimental effects, without applying spin echo
techniques.

Figure 6: (a) Schematic illustration of suppressing spin dephasing induced by
inhomogeneous broadening. The superradiant mode $\hat{s}$ is coupled to the
end mode $\hat{d}$ of a chain of subradiant modes, with a strength
$\delta_{\rm inh}$ and a detuning $\delta_{q}$. The effect of inhomogeneous
broadening can be strongly suppressed, as long as $\delta_{\rm
inh}\ll\delta_{q}$. (b), (c) Time evolution of the coherence,
$\mbox{$\langle\hat{\sigma}_{-}\rangle$}=\left|\mbox{$\langle\hat{\sigma}_{-}\rangle$}\right|\exp(i\phi)$,
of the ensemble qubit, with and without the cavity protection. The modulus
$\left|\mbox{$\langle\hat{\sigma}_{-}\rangle$}\right|$ is plotted in (b), and
the phase $\phi$ is in (c). Here, we integrated the adiabatic master equation
in Eq. (13), with an additional inhomogeneous broadening, and the initial
state is assumed to be a spin coherent state of amplitude $\alpha=1$. The
horizontal dash-dotted line refers to the ideal value of
$\left|\mbox{$\langle\hat{\sigma}_{-}\rangle$}\right|$, given by Eq. (28). We
set $N=6$, $\delta_{\rm inh}=0.1\delta_{q}$, and the other parameters are the
same as those in Fig. 4.

In the low excitation regime, each two-level atom can be modelled by a bosonic
mode $b_{j}$ Kurucz  _et al._ (2009, 2011); Diniz  _et al._ (2011); Kubo  _et
al._ (2012), such that

$\displaystyle\hat{\sigma}_{-}^{j}\simeq\;$ $\displaystyle\hat{b}_{j},$ (37)
$\displaystyle\hat{\sigma}_{z}^{j}\simeq\;$
$\displaystyle-1+2\hat{b}^{\dagger}_{j}\hat{b}_{j}.$ (38)

Correspondingly, the Hamiltonian $\hat{H}_{\rm inh}$ becomes

$\hat{\mathcal{H}}_{\rm
inh}=\sum_{j=1}^{N}\delta_{j}\hat{b}^{\dagger}_{j}\hat{b}_{j}.$ (39)

According to the definition of the mode $\hat{s}$ given in Eq. (4), it is
clearly seen that

$\hat{s}=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\hat{b}_{j}.$ (40)

Since the mode $\hat{s}$ is coupled to the cavity photon, it is thus called
the superradiant spin-wave mode. In addition to this mode, there also exist
$(N-1)$ orthogonal dark modes, which are uncoupled from the cavity photon.
These dark modes are, thus, called subradiant. In the case of no inhomogeneous
broadening, i.e., $\delta_{j}=0$, the superradiant mode $\hat{s}$ is decoupled
from the subradiant modes, and it is not affected by them.

In the presence of inhomogeneous broadening, the Hamiltonian
$\hat{\mathcal{H}}_{\rm inh}$ can be modeled as a chain of coupled subradiant
modes Kurucz  _et al._ (2011), as shown in Fig. 6(a), and the superradiant
mode $\hat{s}$ is coupled to the subradiant mode

$\hat{d}=\frac{1}{\sqrt{N}\delta_{\rm
inh}}\sum_{j=1}^{N}\delta_{j}\hat{b}_{j}$ (41)

at the end of this chain. Here, $\delta_{\rm inh}$ is the variance of the
atomic transition frequencies, and is defined as

$\delta_{\rm inh}^{2}=\frac{1}{N}\sum_{j=1}^{N}\delta_{j}^{2}.$ (42)

The coupling strength between the modes $\hat{s}$ and $\hat{d}$ is found to be
$\delta_{\rm inh}$, according to the Heisenberg equation of motion for the
mode $\hat{s}$. However, as discussed above and in Appendix A, the resonance
of the superradiant mode $\hat{s}$ is shifted by an amount $\delta_{q}$, due
to its detuned coupling to the cavity mode $\hat{a}_{s}$. Under the assumption
that

$\delta_{\rm inh}\ll\delta_{q},$ (43)

the coupling between the superradiant mode $\hat{s}$ and the subradiant mode
$\hat{b}$ becomes far detuned, and therefore can be neglected. This is the so-
called cavity protection effect Kurucz  _et al._ (2009, 2011); Diniz  _et al._
(2011); Putz  _et al._ (2014).

In Figs. 6(b) and 6(c), we perform numerical simulations, with and without the
cavity protection. Specifically, we integrate the adiabatic master equation in
Eq. (13), with the inhomogeneous-broadening Hamiltonian
$\hat{\mathcal{H}}_{\rm inh}$ given in Eq. (39), and then calculate the
complex coherence,
$\mbox{$\langle\hat{\sigma}_{-}\rangle$}=\left|\mbox{$\langle\hat{\sigma}_{-}\rangle$}\right|\exp(i\phi)$,
of the ensemble qubit. Note that, here, $\hat{\mathcal{H}}_{\rm inh}$ needs to
be transformed, by a unitary operator
$\exp\left(i\delta_{q}\hat{s}^{\dagger}\hat{s}t\right)$, to the rotating frame
of the adiabatic master equation. In fact, this transformation reflects the
suppression effect of the detuning $\delta_{q}$ on the inhomogeneous-
broadening induced spin dephasing. The frequency shifts $\delta_{j}$ are
randomly chosen according to a Gaussian distribution of mean $0$ and variance
$\delta_{\rm inh}$. In Fig. 6(b), we see that the cavity protection stabilizes
the modulus $\left|\mbox{$\langle\hat{\sigma}_{-}\rangle$}\right|$ at almost
the ideal value given in Eq. (28). Furthermore, Fig. 6(c) shows that, with
cavity protection, the phase $\phi$ increases linearly with the evolution
time, and thus can be subtracted in a proper frame. In sharp contrast, the
coherence of the ensemble qubit (or quantum information) is easily destroyed
in the case of no cavity protection. This indicates that our proposal can
inherently and strongly suppress the effect of inhomogeneous broadening, as
long as $\delta_{\rm inh}\ll\delta_{q}$.

##  VI Conclusions

We have introduced a method of how to generate an ensemble qubit. The method
does not rely on Rydberg dipole blockade, and eliminates the need for the
strong atom-atom interactions. The anharmonicity required for the formation of
a qubit is provided by the nonlinear two-atom decay. We have demonstrated that
the two-atom decay can restrict the ensemble states into a 2D quantum
manifold, spanned by the ground state and a symmetric single-excitation
superradiant state. The transitions to higher excited superradiant states are
strongly suppressed by the two-atom decay, with the residual transitions being
converted into an effective decay between the single-excitation superradiant
state and the ground state. We have also shown that spin dephasing induced by
inhomogeneous broadening can be strongly suppressed, due to the cavity
protection effect, with which our proposal is compatible. This indicates that
our ensemble qubit can have a long lifetime. Finally, we expect that our
ensemble qubit could find and stimulate diverse applications in modern quantum
technologies.

##  VII acknowledgments

W.Q. was supported in part by the Incentive Research Project of RIKEN. A.M.
was supported by the Polish National Science Centre (NCN) under the Maestro
Grant No. DEC-2019/34/A/ST2/00081. F.N. is supported in part by: Nippon
Telegraph and Telephone Corporation (NTT) Research, the Japan Science and
Technology Agency (JST) [via the Quantum Leap Flagship Program (Q-LEAP), and
the Moonshot R&D Grant Number JPMJMS2061], the Japan Society for the Promotion
of Science (JSPS) [via the Grants-in-Aid for Scientific Research (KAKENHI)
Grant No. JP20H00134], the Asian Office of Aerospace Research and Development
(AOARD) (via Grant No. FA2386-20-1-4069), and the Foundational Questions
Institute Fund (FQXi) via Grant No. FQXi-IAF19-06.

##  Appendix A Derivation of the time-averaged Hamiltonian in Eq. (10)

In this Appendix, we show in detail how to derive the time-averaged
Hamiltonian $\hat{\mathcal{H}}_{\rm avg}$. To begin, we rewrite the
Hamiltonian in Eq. (II) in the interaction picture, yielding

$\displaystyle\hat{\mathcal{H}}_{\rm I}=\;$ $\displaystyle g_{\rm
col}\left(\hat{a}_{s}^{\dagger}\hat{s}e^{i\Delta_{q}t}+\hat{a}_{s}\hat{s}^{\dagger}e^{-i\Delta_{q}t}\right)$
$\displaystyle+J\left(\hat{a}_{p}\hat{a}_{s}^{\dagger
2}e^{i\Delta_{p}t}+\hat{a}_{p}^{\dagger}\hat{a}_{s}^{2}e^{-i\Delta_{p}t}\right),$
(1)

where $\Delta_{q}=\omega_{s}-\omega_{q}$ and
$\Delta_{p}=2\omega_{s}-\omega_{p}$. According to the formalism of Refs. Gamel
and James (2010); Shao  _et al._ (2017), the second-order and third-order
time-averaged Hamiltonians are given, respectively, by

$\displaystyle\hat{\mathcal{H}}_{\rm TA}^{\left(2\right)}=\;$
$\displaystyle\frac{g_{\rm
col}^{2}}{\Delta_{q}}\left(\hat{a}_{s}^{\dagger}\hat{a}_{s}-\hat{s}^{\dagger}\hat{s}\right)$
$\displaystyle-\frac{J^{2}}{\Delta_{p}}\left(2\hat{a}_{p}^{\dagger}\hat{a}_{p}+4\hat{a}_{p}^{\dagger}\hat{a}_{p}\hat{a}_{s}^{\dagger}a_{s}-\hat{a}_{s}^{\dagger}\hat{a}_{s}^{\dagger}\hat{a}_{s}a_{s}\right).$
(2)

and

$\hat{\mathcal{H}}_{\rm
TA}^{\left(3\right)}=\chi\left(\hat{a}_{p}\hat{s}^{\dagger 2}e^{-i\delta
t}+\hat{a}_{p}^{\dagger}s^{2}e^{i\delta t}\right).$ (3)

Here, $\delta=\omega_{p}-2\omega_{q}$ and $\chi=g_{\rm
col}^{2}J/\Delta_{q}^{2}$. Note that as demonstrated below, the detuning
$\delta$ is small but nonzero, so as to compensate some undesired shifts
induced by the second-order process.

Since the signal mode $\hat{a}_{s}$ is initialized in the vacuum state, the
second-order time-averaged Hamiltonian is reduced to

$\hat{\mathcal{H}}_{\rm TA}^{\left(2\right)}=-\frac{g_{\rm
col}^{2}}{\Delta_{q}}\hat{s}^{\dagger}\hat{s}-\frac{2J^{2}}{\Delta_{p}}\hat{a}_{p}^{\dagger}\hat{a}_{p}.$
(4)

It is seen that the second-order process induces a Lamb shift Scully and
Zubairy (1997) for atoms (i.e., the first term), and a resonance shift for the
pump mode (i.e., the second term). In order to compensate these two undesired
shifts, we set

$\delta=\frac{2J^{2}}{\Delta_{p}}-\frac{2g_{\rm col}^{2}}{\Delta_{q}}.$ (5)

Under this condition, the time-averaged Hamiltonian $\hat{\mathcal{H}}_{\rm
avg}$ can be achieved in a proper frame.

##  Appendix B Adiabatic elimination of the pump mode $\hat{a}_{p}$

In this appendix, we demonstrate how to adiabatically eliminate the pump mode
$\hat{a}_{p}$ of the time-averaged Hamiltonian $\hat{\mathcal{H}}_{\rm avg}$
in Eq. (10). We now consider the following master equation,

$\dot{\hat{\rho}}=-i\left[\hat{\mathcal{H}}_{\rm
avg},\hat{\rho}\right]+\kappa_{p}\mathcal{L}\left(\hat{a}_{p}\right)\hat{\rho}.$
(6)

Under the assumption of $\kappa_{p}\gg\chi$, the photon population of the pump
mode is very low. This allows us to only consider the vacuum state $|0\rangle$
and single-photon state $|1\rangle$ of the pump mode. Here, the number in the
ket refers to the number of photons of the pump mode. As a result, the density
matrix $\rho$ can be reexpressed as

$\hat{\rho}=\rho_{00}\mbox{$|0\rangle$}\mbox{$\langle
0|$}+\rho_{01}\mbox{$|0\rangle$}\mbox{$\langle
1|$}+\rho_{10}\mbox{$|1\rangle$}\mbox{$\langle
0|$}+\rho_{11}\mbox{$|1\rangle$}\mbox{$\langle 1|$}.$ (7)

It follows, upon substituting it into the master equation in Eq. (6), that

$\displaystyle\dot{\rho}_{00}=\;$ $\displaystyle
i\chi\left(\rho_{01}\hat{s}^{2}-\hat{s}^{\dagger
2}\rho_{10}\right)+\kappa_{p}\rho_{11},$ (8) $\displaystyle\dot{\rho}_{11}=\;$
$\displaystyle i\chi\left(\rho_{10}\hat{s}^{\dagger
2}-\hat{s}^{2}\rho_{01}\right)-\kappa_{p}\rho_{11},$ (9)
$\displaystyle\dot{\rho}_{01}=\;$ $\displaystyle
i\chi\left(\rho_{00}\hat{s}^{\dagger 2}-\hat{s}^{\dagger
2}\rho_{11}\right)-\frac{\kappa_{p}}{2}\rho_{01},$ (10)

and $\rho_{10}=\rho_{01}^{\dagger}$. Then setting $\dot{\rho}_{01}=0$ gives

$\rho_{01}=i\frac{2}{\kappa_{p}}\chi\left(\rho_{00}\hat{s}^{\dagger
2}-\hat{s}^{\dagger 2}\rho_{11}\right),$ (11)

and, in turn, the adiabatic master equation only with the degree of freedom of
the ensemble, as follows:

$\displaystyle\dot{\hat{\rho}}_{\rm
ens}=\dot{\rho}_{00}+\dot{\rho}_{11}=\kappa_{\rm
2at}\mathcal{L}\left(\hat{s}^{2}\right)\hat{\rho}_{\rm ens},$ (12)

where $\kappa_{\rm 2at}=4\chi^{2}/\kappa_{p}$. Note that in Eq. (12), we have
set $\rho_{11}=0$, due to the fact that the photon population of the pump mode
is very low.

##  Appendix C Adiabatic elimination of the superradiant state $|2\rangle$

We begin with the master equation in Eq. (31). In the limit
$\Omega_{d}\ll\kappa_{\rm 2at}$, the coupling of the superradiant states
$|1\rangle$ and $|2\rangle$, which is induced by the driving $\Omega_{d}$, is
considered as only a perturbation. Therefore, the state $|2\rangle$ can be
adiabatically eliminated. To do so, we define, according to Ref. Reiter and
Sørensen (2012), a non-Hermitian Hamiltonian

$\hat{\mathcal{H}}_{\rm NH}=-\frac{i}{2}\kappa_{\rm 2at}\hat{s}^{\dagger
2}\hat{s}^{2}=-i\kappa_{\rm 2at}\mbox{$|2\rangle$}\mbox{$\langle 2|$},$ (13)

and a perturbative coupling

$\hat{\mathcal{V}}=\sqrt{2}\Omega_{d}e^{i\theta_{d}}\mbox{$|2\rangle$}\mbox{$\langle
1|$}.$ (14)

With them, we can obtain an effective coupling between the states $|0\rangle$
and $|1\rangle$, described by

$\displaystyle\hat{\mathcal{H}}^{\rm qubit}_{d}=$
$\displaystyle-\frac{1}{2}\hat{\mathcal{V}}^{\dagger}\left[\hat{\mathcal{H}}_{\rm
NH}^{-1}+\left(\hat{\mathcal{H}}_{\rm
NH}^{-1}\right)^{\dagger}\right]\hat{\mathcal{V}}$
$\displaystyle+\Omega_{d}\left(e^{-i\theta_{d}}\hat{\sigma}_{-}+e^{i\theta_{d}}\hat{\sigma}_{+}\right),$
(15) $\displaystyle=\;$
$\displaystyle\Omega_{d}\left(e^{-i\theta_{d}}\hat{\sigma}_{-}+e^{i\theta_{d}}\hat{\sigma}_{+}\right).$
(16)

and an effective Lindblad operator

$2\kappa_{\rm 2at}\mathcal{L}\left(\mbox{$|0\rangle$}\mbox{$\langle
2|$}\hat{\mathcal{H}}_{\rm
NH}^{-1}\hat{\mathcal{V}}\right)=\gamma\mathcal{L}\left(\hat{\sigma}_{-}\right),$
(17)

where $\gamma=4\Omega^{2}_{d}/\kappa_{\rm 2at}$. Thus, the master equation in
Eq. (35) can be achieved.

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aainstitutetext: Instituut-Lorentz, Leiden University, Niels Bohrweg 2, 2333
CA Leiden, The Netherlandsbbinstitutetext: Institut für Astroteilchen Physik,
Karlsruher Institut für Technologie (KIT), Hermann-von-Helmholtz-Platz 1,
76344 Eggenstein-Leopoldshafen, Germany ccinstitutetext: Department of
Physics, Taras Shevchenko National University of Kyiv, 64 Volodymyrs’ka str.,
Kyiv 01601, Ukraineddinstitutetext:  IFPU, Institute for Fundamental Physics
of the Universe, via Beirut 2, I-34014 Trieste, Italyeeinstitutetext:  SISSA,
via Bonomea 265, I-34132 Trieste, Italy ffinstitutetext:  INFN, Sezione di
Trieste, SISSA, Via Bonomea 265, 34136, Trieste, Italy

# Sensitivity of the FACET experiment to Heavy Neutral Leptons and Dark
Scalars

Maksym Ovchynnikov c Viktor Kryshtal d,e,f Kyrylo Bondarenko
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We analyze the potential of the recently proposed experiment FACET (Forward-
Aperture CMS ExTension) to search for new physics. As an example, we consider
the models of Higgs-like scalars with cubic and quartic interactions and Heavy
Neutral Leptons. We compare the sensitivity of FACET with that of other
proposed “intensity frontier” experiments, including FASER2, SHiP, etc. and
demonstrate that FACET could probe an interesting parameter space between the
current constraints and the potential reach of the above mentioned proposals.

##  1 Introduction

Despite its success in describing accelerator data, the Standard model (SM)
fails to explain several observed phenomena constituting beyond the Standard
model problems: neutrino masses, dark matter, and the matter-antimatter
asymmetry. These problems may be resolved by extending the SM particle content
with some new particles. One class of extensions is where new particles
interact with SM via renormalizable operators suppressed by very small
couplings, the so-called portals. Depending on the spin of the mediator field
entering the portal operator, there are three types of portals – scalar,
vector, and fermion Alekhin:2015byh ; Agrawal:2021dbo .

To search for portal particles, many experiments have been proposed during the
last years. Examples include dedicated beam experiments such as SHiP, DUNE,
SHADOWS, NA62; LHC-based experiments, such as MATHUSLA, Codex-b, ANUBIS, AL3X.
There is a class of LHC-based experiments that have decay volume covering
large pseudorapidities, which is called the far-forward experiments. The
particles produced in the far-forward direction have large energies,
$E=\mathcal{O}(1\text{ TeV})$, which means that their lifetime is increased by
$\gamma\sim 10^{3}(1\text{ GeV}/m)$. Therefore, as compared with the off-axis
experiments located at the same distance, the far-forward experiments may
probe particles with shorter lifetimes.

The representatives of this class are already running FASER FASER:2018bac ;
FASER:2019aik , FASER$\nu$ FASER:2019dxq ; FASER:2020gpr  and SND@LHC
SHiP:2020sos  experiments. Their proposed upgrades, FASER2/FASER$\nu$2 and
AdvSND, would be installed at the Far Forward physics facility and work during
the High Luminosity phase of the LHC Feng:2022inv . Recently, a new far-
forward experiment FACET has been proposed Cerci:2021nlb . Apart from covering
$\simeq 4$ times larger solid volume and having longer decay volume, it would
be located in 100 meters downwards the CMS interaction point – $\simeq 5$
times closer than SND@LHC/FASER, and in this way allows to probe even shorter
lifetimes.

In this work, we estimate the sensitivity of the FACET experiment to models of
scalar and fermion portals, making a qualitative comparison of its sensitivity
with FASER2. The final results are shown in Fig. 1. We see that due to larger
decay volume and closer distance from the interaction point, FACET allows to
significantly extend the probed parameter space as compared to FASER2.

Figure 1: Sensitivity of FACET and FASER2 to the models of HNLs and Higgs-like
scalars. Left panel: Higgs-like scalars, assuming $\text{Br}(h\to SS)=0.05$.
The solid red line shows the sensitivity of FACET including the production of
scalars from $h$ and $B$, while the dashed line denotes the sensitivity to
scalars from $h$ only. Right panel: HNLs that mix predominantly with
$\nu_{e}$. For comparison, we also show the sensitivity of SHiP and MATHUSLA
experiments from Agrawal:2021dbo . The region excluded by the previous
experiments is given from Abdullahi:2022jlv  for HNLs and from Agrawal:2021dbo
for Higgs-like scalars.

The content of the paper is as follows. In Sec. 2, we briefly describe the
scalar and neutrino portals. In Sec. 3, we describe the FACET experiment. In
Sec. 4, we compare the reach of the FACET and FASER2 experiments based on
semi-analytic estimates, considering scalars and heavy neutral leptons
produced by decays of $B$ mesons and Higgs bosons. In Sec. 5, we discuss the
obtained results and make their comparison with the literature. Finally, in
Sec. 6, we make conclusions.

##  2 Portals

###  2.1 Scalar portal with the quartic coupling

The general form of the Lagrangian of the scalar portal below the electroweak
scale Alekhin:2015byh  is

$\mathcal{L}_{\text{scalar portal}}=\theta m_{h}^{2}hS+\frac{\alpha}{2}hSS,$
(1)

where $h$ is the Higgs boson field, $S$ is a new scalar particle (called also
the Higgs-like scalar, or dark scalar), $\theta$ is the mixing angle, and
$\alpha$ is the quartic coupling constant. The scalar with mass in GeV range
may be a mediator between the SM and dark matter particles, or serve as a
light inflaton Bezrukov:2009yw . The phenomenology of the scalar portal at
accelerator experiments has been intensively studied in Boiarska:2019jym ;
Bird:2004ts ; Batell:2009jf ; Bezrukov:2009yw ; Clarke:2013aya ; Schmidt-
Hoberg:2013hba ; Evans:2017lvd ; Bezrukov:2018yvd ; Monin:2018lee ;
Winkler:2018qyg ; Frugiuele:2018coc ; Helmboldt:2016zns  as well as in
Voloshin:1985tc ; Raby:1988qf ; Truong:1989my ; Donoghue:1990xh ;
Willey:1982ti ; Willey:1986mj ; Grzadkowski:1983yp ; Leutwyler:1989xj ;
Haber:1987ua ; Chivukula:1988gp  in the context of the light Higgs boson.

At the LHC, Higgs bosons are copiously produced. In particular, during the
high luminosity phase, around $10^{8}$ bosons will be generated. In the case
$\alpha\neq 0$, they may decay into a pair of scalars through the process
$h\to SS$.

Current constraints on $\alpha$ are not very restrictive for the model of
scalars. Indeed, the strongest bound on $\alpha$ comes from searches for
invisible decays $h\to\text{inv}$ at ATLAS and CMS, constraining
$\text{Br}(h\to\text{inv})<0.15$ Sirunyan:2018owy ; Aaboud:2018sfi . During
the high luminosity phase of the LHC, it would be possible to probe the
branching ratio down to the values $\text{Br}(h\to\text{inv})=0.05$
Bechtle:2014ewa .

The number of Higgs bosons that would be produced during the High Luminosity
phase of the LHC is $N_{h}\simeq 2\cdot 10^{8}$. Therefore, given the current
constraints on $\text{Br}(h\to\text{inv})$, the production channel $h\to SS$
allows to significantly extend the reach of the LHC and LHC-based experiments,
making it possible to search for the scalars with masses up to $m_{S}\simeq
m_{h}/2$.

The parameter space of dark scalars excluded by past experiments and probed by
proposed LHC experiments (we choose FASER2 and MATHUSLA as a representative
example), assuming $\text{Br}(h\to SS)=0.05$, is shown in the left panel of
Fig. 1. In the paper Boiarska:2019vid , it has been demonstrated that FASER2
has a limited potential to probe this model, mainly due to the suppressed
small angular coverage and short length of the decay volume.

###  2.2 Heavy Neutral Leptons

The Lagrangian of the fermion portal is

$\mathcal{L}_{f}=F_{\alpha I}\bar{L}_{\alpha}\tilde{H}N_{I}+\text{h.c.},$ (2)

where $N_{I},i=1,2,\dots$ is an HNL, $\tilde{H}=i\sigma_{2}H^{*}$ is the Higgs
doublet in the conjugated representation, $L_{\alpha}$ is the SM lepton
doublet ($\alpha=e,\mu,\tau$), and $F_{\alpha I}$ are complex couplings. HNLs
interact with SM particles similarly to a neutrino, but suppressed by the
mixing angle $U_{\alpha I}=F_{\alpha I}/\sqrt{2}v$, where $v$ is the Higgs
VEV.

The parameter space of HNLs is shown in the right panel of Fig. 1. Among the
proposed experiments, we show SHiP, FASER2 and MATHUSLA. At the LHC, HNLs
heavier than kaons may be produced in decays of D, B mesons, and W bosons. The
last channel allows to extend the maximal HNL mass reach from
$m_{N}=m_{B}\simeq 5\text{ GeV}$ to $m_{N}\simeq m_{W}$ as compared the
dedicated beam experiments such as SHiP. However, this is not the case for
neither MATHUSLA nor FASER2, since they are located too far from the HNL
production point, and the HNLs produced from $W$ bosons in the accessible
parameter space are too short-lived to reach the decay volume
Bondarenko:2019yob .

##  3 FACET experiment

FACET (Forward Aperture CMS ExTension) Cerci:2021nlb  is a recent proposal of
a subsystem of CMS to be added to search for long-lived particles during the
High Luminosity (HL) phase of the LHC.

The schematic layout of FACET is shown in Fig. 2. Given $z$ to be the distance
from the CMS experiment along the beam axis, FACET will be located between the
35 T$\cdot$m superconducting beam separation dipole D1 at $z=80$ m and the
TAXN absorber at $z=128$ m. The decay volume is an enlarged proton beam pipe
with radius $r=0.5$ m located from $z=101\text{ m}$ to $z=119\text{ m}$.

The detector part is located right after the decay volume. It has shape of the
annulus with the inner radius $r_{\text{in}}=18\text{ cm}$ and outer radius
$r_{\text{out}}=50\text{ cm}$. The detector covers polar angles
$1.5<\theta<4\text{ mrad}$ and consists of $\simeq 3$ m of silicon tracker
with the transversal resolution of $\sigma_{x,y}=30\ \mu\text{m}$, the timing
layer with Low-Gain Avalanche Detectors (LGAD) having resolution
$\sigma_{t}\sim 30\text{ ps}$, and a high granularity EM and hadronic
calorimeter.

The background is greatly reduced because of 200-300 $\lambda_{\text{int}}$ of
magnetized iron in the LHC quadrupole magnets Q1–Q3. Detailed FLUKA
simulations predict $\simeq 30$ charged particles with momentum $p>1\text{
GeV}$ and $\simeq 1$ light neutral hadron ($K^{0}_{S},K^{0}_{L},\Lambda$) per
bunch crossing Cerci:2021nlb . Charged particles and decays of neutral
hardrons may mimic decays of new physics particles. The combination of the
precision hodoscope (with inefficiency of $10^{-5}$) and precision tracking
reduces the background for most of the new physics decay channels down to a
negligible level for the mass of a new physics particle $m_{X}\gtrsim
0.8\text{ GeV}$. Nevertheless, decays of neutral hadrons create a background
in the region $m_{X}\lesssim 0.8\text{ GeV}$, making it complicated to search
for new particles in this mass range.

Figure 2: The schematic layout of the FACET experiment (see text for details).
The figure is taken from Cerci:2021nlb .

##  4 FACET vs FASER2: qualitative comparison of the sensitivity

In this section, we discuss the comparison of sensitivities of the FACET and
FASER2 experiments. For this purpose, we will use semi-analytic estimates
similar to the ones presented in Bondarenko:2019yob .

We estimate the number of events with decays of a particle $Y=S,N$ at the
FACET and FASER2 experiments (see Table 1) in the following way:

$N_{\text{events}}=\sum_{X}N_{X}\times\chi_{S/N}^{(X)}\cdot\text{Br}(X\to
Y)\times\\\
\times\int\frac{dP_{\text{decay}}(\gamma_{Y},z)}{dz}f_{\theta_{Y},E_{Y}}\cdot\epsilon_{\text{decay}}(\gamma_{Y},\theta_{Y},z)d\theta_{Y}dE_{Y}dz$
(3)

Here, $X$ corresponds to the decaying particle producing scalars, with $N_{X}$
being the total number of SM particles produced during the high luminosity
phase of the LHC. We assume $N_{h}=1.7\cdot 10^{8}$ from Cepeda:2019klc , take
$N_{B}=2.4\cdot 10^{15}$, $N_{D}=5\cdot 10^{16}$, from FONLL Cacciari:1998it ;
Cacciari:2001td ; Cacciari:2012ny ; Cacciari:2015fta  at the upper bound of
uncertainties (see a discussion in Bondarenko:2019yob ), and $N_{W}=3.3\cdot
10^{11}$ from ATLAS:2016nlr .

$\chi_{Y}^{(X)}=1$ or $2$ is the number of particles $Y$ produced per decay of
$X$.

The integration in (3) is performed over $Y$ energies $E_{Y}$, polar angles
$\theta_{Y}$, and the distance from the collision point along the beam axis
$z$ within $l_{\text{min}}<z<l_{\text{min}}+l_{\text{fid}}$, where
$l_{\text{min}}$ is the distance to the beginning of the decay volume, and
$l_{\text{fid}}$ is the decay volume length.
$dP_{\text{decay}}/dz=\frac{e^{-z/c\tau_{Y}\gamma_{Y}}}{c\tau_{Y}\gamma_{Y}}$
is the differential decay probability.

$f^{(X)}_{\theta_{Y},E_{Y}}$ is the angle-energy distribution of $Y$ produced
by decays of $X$. To derive it, we have followed the semi-analytic approach
summarized in Boiarska:2019vid ; namely, we have integrated the differential
distribution $d\text{Br}(X\to Y)$ multiplied with the distribution of the
mother particle $X$ over $X$ energy and angles. The mother particle
distributions have been obtained from FONLL Cacciari:1998it ; Cacciari:2001td
; Cacciari:2012ny ; Cacciari:2015fta  ($B,D$), by the method described in
Boiarska:2019vid  (for $h$), and from Kling:2021fwx  (for $W$).

$\epsilon_{\text{decay}}(\gamma_{S},\theta_{S},z)$ is the decay acceptance –
the fraction of decay products from $Y$ intersecting the front plane of the
detector. We estimate it using a simple Monte Carlo simulation by
approximating the decay of scalars into two massless particles, and of HNLs
into three massless particles via the charged current.111The approximation
works within 25% accuracy for the whole scalar/HNL mass range.

Further, we will assume that both FACET and FASER2 are background-free
experiments. Taking into account considerable background from neutral hadron
decays on FACET for $m_{Y}\lesssim 0.8\text{ GeV}$, discussed in Section 3,
the obtained results for FACET are only valid above this mass. Parameters of
the experiments are summarized in Table 1.

Experiment |  $l_{\text{min}},$ m |  $l_{\text{fid}},$ m |  $\theta_{\text{min}},\theta_{\text{max}}$, mrad |  $\Omega$, sr
---|---|---|---|---
FASER2 | 480 | 5 | $0,2.1$ | $1.3\cdot 10^{-5}$
FACET | 101 | 18 | $1.6,4.1$ | $4.5\cdot 10^{-5}$
Table 1: Parameters of FASER2 and FACET configurations: the distance to the
beginning of the decay volume $l_{\text{min}}$, the length of the decay volume
$l_{\text{fid}}$, the polar angle coverage of detectors
$\theta_{\text{min}},\theta_{\text{max}}$, and the solid angle $\Omega$
covered by the detectors.

Let us compare the sensitivity of FACET and its modification at the lower
bound (the regime $c\tau_{Y}\gamma_{Y}\gg l_{\text{max}}$) and the upper bound
(regime $c\tau_{Y}\gamma_{Y}\lesssim l_{\text{min}}$) with the sensitivity of
FASER2. The lifetime scales as $\tau_{Y}\propto g_{Y}^{-2}$, where $g_{Y}$ is
the mixing angle. The production branching ratio scales as Br

At the upper bound, the number of events behaves as $N_{\text{events}}\propto
g^{2}_{Y}\times\exp[-l_{\text{min}}/c\tau_{Y}\gamma_{Y}]$. Therefore,
neglecting the power $g^{2}_{Y}$, for ratio of the maximal probed mixing
angles we get

$\frac{g^{2}_{\text{upper,$\text{FACET}$}}}{g^{2}_{\text{upper,$\text{FASER2}$}}}\simeq\frac{l_{\text{min}}^{\text{$\text{FASER2}$}}}{l_{\text{min}}^{\text{$\text{FACET}$}}}\cdot\frac{\langle\gamma_{Y}\rangle_{\text{$\text{FACET}$}}}{\langle\gamma_{Y}\rangle_{\text{$\text{FASER2}$}}}\approx
5,$ (4)

given $\approx 5$ times smaller $l_{\text{min}}$ at FACET and
$\langle\gamma_{Y}\rangle_{\text{$\text{FACET}$}}\approx\langle\gamma_{Y}\rangle_{\text{$\text{FASER2}$}}$.

At the lower bound, $e^{-z/c\tau_{Y}\gamma_{Y}}\approx 1$, and the ratio of
the probed mixing angles is

$\frac{g^{2}_{\text{lower,$\text{FACET}$}}}{g^{2}_{\text{lower,$\text{FASER2}$}}}\approx\left(\frac{\epsilon^{\text{$\text{FASER2}$}}_{\text{geom}}}{\epsilon^{\text{$\text{FACET}$}}_{\text{geom}}}\times\frac{l_{\text{det}}^{\text{$\text{FASER2}$}}}{l_{\text{det}}^{\text{$\text{FACET}$}}}\right)^{\kappa},$
(5)

Here, $\kappa=1/2$ in the case when both $Y$ production and decay are
controlled by $\theta$, and $\kappa=1$ in the case when the production is
controlled by different couplings. $\epsilon_{\text{geom}}$ is the averaged
geometric acceptance at the lower bound:

$\epsilon_{\text{geom}}\approx\frac{1}{l_{\text{det}}}\int
f_{\theta_{Y},E_{Y}}\cdot\epsilon_{\text{decay}}(\gamma_{Y},\theta_{Y},z)d\theta_{Y}dE_{Y}dz$
(6)

###  4.1 Scalar portal

The production processes of scalars at the LHC are $h\to S+S$ for the Higgs
bosons, and $B^{+/0}\to S+S+X_{s}$, $B_{s}\to S+S$, $B^{+/0}\to S+X_{s}$, for
the $B$ mesons Boiarska:2019jym , see also Fig. 3. The first three processes
are mediated by the quartic coupling $\alpha$, while the process $B^{+/0}\to
S+X_{s}$ by the mixing angle $\theta$.

(a)

(b)

(c)

Figure 3: Diagrams of the production of the scalar $S$ in the model (1): meson
decay $X\to X^{\prime}+S$ (a) mediated by the mixing $\theta$; and the Higgs
boson decay $h\to S+S$ (b), the mesons decays $X\to S+S$, $X\to
X^{\prime}+S+S$ (c) mediated by the quartic coupling $\alpha$.

####  4.1.1 Geometric acceptance

Let us discuss the geometric acceptance. For the moment, we will drop the
decay acceptance.

The solid angle distribution $df/d\cos(\theta)\sim df/d\Omega$ of Higgs
bosons, B mesons, and light scalars produced in their decays is shown in Fig.
4.

Figure 4: The angular distribution of B mesons, Higgs bosons (solid lines) and
light scalars with $m_{S}=50\text{ MeV}$ produced by their decays (dashed
lines). The arrows indicate polar angle coverage of FASER2 and FACET
experiments.

The distribution of $B,h$ remains constant in the angular coverage of FASER2
and gradually drops by a factor 1.5-2 for the angular coverage of the FACET
experiment. While the distribution of scalars with mass close to the kinematic
threshold is the same as for their mother particles, the distribution of light
scalars $m_{S}\ll m_{B},m_{h}/2$ gets broadened due to acquiring transverse
momentum, of order of $p_{T}\simeq m_{B/h}/2$. Given that the typical $B,h$
energy in the far-forward direction is $\mathcal{O}(1\text{ TeV})$, the
smearing is $\Delta\theta\sim p_{T}/1\text{ TeV}$ – smaller than the angular
coverage of FACET and FASER2 for scalars from $B$ mesons, and much larger for
scalars from $h$. As a result, the angular distribution of light scalars from
$B$ remains very similar to the distribution of $B$, while in the case of
scalars from $h$ it is isotropic up to the angles $30\text{ mrad}$. This in
particular suggests that FACET already has an optimal placement and size to
search for particles from $B$ mesons.

Therefore, if not including the decay acceptance in Eq. (6), for the geometric
acceptance of scalars from $X=h,B$ one has

$\frac{\epsilon_{\text{geom},S}^{\text{FACET}}}{\epsilon_{\text{geom},S}^{\text{FASER2}}}\approx\begin{cases}\Omega_{\text{FACET}}/\Omega_{\text{FASER2}},\quad
m_{S}\ll m_{B},m_{h}/2\\\
\epsilon_{\text{geom},X}^{\text{FACET}}/\epsilon_{\text{geom},X}^{\text{FASER2}},\quad
m_{S}\to m_{B}\text{ or }m_{h}/2\end{cases}$ (7)

Experiment | $\epsilon_{\text{geom},S},\ h\to SS$ | $\epsilon_{\text{geom},S},\ B\to KS$
---|---|---
$m_{S}=50\text{ MeV}$ | $m_{S}=62\text{ GeV}$ | $m_{S}=50\text{ MeV}$ | $m_{S}=5.1\text{ GeV}$
FASER2 | $5\cdot 10^{-5}$ | $1\cdot 10^{-3}$ | $2.8\cdot 10^{-3}$ | $7.5\cdot 10^{-3}$
FACET | $1.7\cdot 10^{-4}$ | $6.1\cdot 10^{-4}$ | $6.9\cdot 10^{-3}$ | $1\cdot 10^{-2}$
Table 2: Geometric acceptances (Eq. (6) for scalars produced by decays of
Higgs bosons and $B$ mesons, for various choices of the scalar mass.

Let us now discuss the effect of the decay acceptance. It becomes important if
the characteristic angle between the decay products in a 2-body decay,
$\langle\alpha\rangle\simeq 1\arcsin(2/\gamma_{S})$, exceeds the angle covered
by the detector as seen from the beginning of the decay volume, which is
$0.4\text{ rad}$ for FASER2 and $0.1\text{ rad}$ for FACET.

Figure 5: The behavior of the geometric acceptance (6) for scalars produced in
decays $h\to SS$ and $B\to X_{s}+S$. The solid lines are obtained with taking
the decay acceptance, $\epsilon_{\text{decay}}$, into account, whereas the
dashed lines correspond to $\epsilon_{\text{decay}}=1$.

Given the typical scalar energies $E_{S}=\mathcal{O}(1\text{ TeV})$, for light
scalars with $m_{S}\ll m_{h/2},m_{B}$ the decay acceptance is 1. However, with
the increase of the scalar mass, more and more decay products produced in the
beginning of the decay volume fly in directions outside the detector coverage.
This feature effectively shrinks $l_{\text{det}}$. For FACET, this effect is
much more important than for FASER2. As a result, in dependence on the scalar
mass, the geometric acceptance at FACET drops even below the geometric
acceptance at FASER2 for $h$ and becomes very close to the geometric
acceptance at FASER2 for $B$, see Fig. 5 and Table 2.

Given the ratio
$l_{\text{det}}^{\text{$\text{FASER2}$}}/l_{\text{det}}^{\text{$\text{FACET}$}}\approx
4$ and the behavior of the geometric acceptances (see Fig. 5), we conclude
that the overall increase of the number of events in the regime of the lower
bound at FACET as compared to FASER2 reaches a factor $2-15$ and $5-15$ for
the case of the production from $h$ and $B$ respectively, in dependence on the
scalar mass.

####  4.1.2 Maximal number of events

In the case of the production from Higgs bosons, it is also useful to compare
the maximal possible number of events at FACET and FASER2. A factor of two
estimate is given by

$N_{\text{events,max}}\approx N_{\text{h}}\cdot(2\cdot\text{Br}(h\to
SS))\cdot\epsilon_{\text{geom}}\cdot P_{\text{decay,max}},$ (8)

where $\epsilon_{\text{geom}}$ is given by Eq. (6), while
$P_{\text{decay,max}}$ is the maximal value of the decay probability as a
function of $l_{\text{decay,S}}=c\tau_{S}\gamma_{S}$, which depends only on
$l_{\text{min}},l_{\text{max}}$:222In reality, the true maximal events number
is even somewhat smaller, since for each given $m_{S},\theta^{2}$ there is the
distribution in $l_{\text{decay}}$ due to the energy distribution of scalars.

Figure 6: The maximal possible number of events (eq. (10)) at FASER2 and FACET
as a function of the scalar mass. The dashed gray line denotes 3 events
defining the sensitivity domain of the experiments.

$P_{\text{decay,max}}=\left[\left(\frac{l_{\text{min}}+l_{\text{fid}}}{l_{\text{min}}}\right)^{-\frac{l_{\text{min}}}{l_{\text{fid}}}}-\left(\frac{l_{\text{min}}+l_{\text{fid}}}{l_{\text{min}}}\right)^{-\frac{l_{\text{min}}+l_{\text{fid}}}{l_{\text{fid}}}}\right]\approx\begin{cases}3.8\cdot
10^{-3},\quad\text{$\text{FASER2}$},\\\
0.06,\quad\text{$\text{FACET}$}\end{cases}$ (9)

Plugging in all numbers, we get

$N_{\text{events,max}}\approx\begin{cases}3.8\cdot\left(\frac{\epsilon_{\text{geom}}}{4.8\cdot
10^{-5}}\right),\quad\text{$\text{FASER2}$},\\\
200\cdot\left(\frac{\epsilon_{\text{geom}}}{1.8\cdot
10^{-4}}\right),\quad\text{$\text{FACET}$},\end{cases}$ (10)

see also Fig. 6.

From Eq. (10), we see that $N_{\text{events,max}}$ at FASER2 is very close to
the number of events required at 95% C.L. to observe one event in background
free regime. More accurate estimates Boiarska:2019vid  that included the
energy distribution of scalars (which decreases the value of
$P_{\text{decay,max}}$) showed that it is even lower, dropping below 3. This
explains why FASER2 has no sensitivity to scalars from Higgs bosons in the
domain $m_{S}\lesssim 45\text{ GeV}$.

###  4.2 Comparison for HNLs

Consider now the case of HNLs. The interaction vertices of HNLs with SM
particles are similar to the vertices of active neutrinos $\nu_{\alpha}$, but
are suppressed by the mixing angle $U_{\alpha}\ll 1$ Asaka:2005an ;
Asaka:2005pn .

At the LHC, the HNLs may be copiously produced in decays of $D,B$ mesons and
$W$ bosons Bondarenko:2018ptm . In this Section, we consider HNLs that mix
predominantly with $\nu_{e}$, keeping in mind that the results for the other
mixings are similar.

For the qualitative comparison, we will consider the following production
channels: $D_{s}\to N+e$, $B_{c}\to N+e$, $W\to N+e$, which respectively
dominate the production of HNLs from $D$ mesons above $m_{N}\simeq 0.5\text{
GeV}$, from $B$ mesons above $m_{N}\simeq 3\text{ GeV}$, and from $W$ bosons.
The angular distributions for these particles, as well as for light HNLs with
mass $m_{N}=50\text{ MeV}$ produced by their decays, are shown in Fig. 7.

Figure 7: The angular distribution of $D_{s},B_{c}$ mesons, W bosons bosons
(solid lines) and light HNLs with $m_{N}=50\text{ MeV}$ produced by their
decays (dashed lines). The arrows indicate polar angle coverage of FASER2 and
FACET experiments.

The values of the geometric acceptances are given in Table 3.

Experiment | $\epsilon_{\text{geom}},D_{s}\to e+N$ | $\epsilon_{\text{geom}},B_{c}\to e+N$ | $\epsilon_{\text{geom}},W\to e+N$
---|---|---|---
FASER2 | $8.5\cdot 10^{-3}$ | $4.9\cdot 10^{-3}$ | $4.2\cdot 10^{-4}$
FACET | $1.4\cdot 10^{-2}$ | $1.3\cdot 10^{-2}$ | $1.3\cdot 10^{-3}$
Table 3: Geometric acceptances (6) for HNLs produced in decays of
$D_{s},B_{c}$, and $W$ bosons (the decay acceptance is included). The HNL
masses are $m_{N}=1.5\text{ GeV}$ for the production from $D_{s}$, $3\text{
GeV}$ for the production from $B_{c}$, and $5\text{ GeV}$ for the production
from $W$.

The production from $W$ bosons is not important for the lower bound of the
sensitivity of FACET and FASER2. To demonstrate this, let us consider two mass
ranges $m_{N}\lesssim m_{B_{c}}$ and $m_{N}>m_{B_{c}}$. In the mass range
$m_{N}\lesssim m_{B_{c}}$, the production from $W$ competes with the
production from $D$ and $B$. Let us compare the total number of HNLs produced
by $B$ mesons and by $W$ bosons for FACET (for FASER2, the situation is
similar):

$\frac{N_{\text{prod}}^{\text{from W}}}{N_{\text{prod}}^{\text{from
B}}}=\frac{N_{\text{W}}\cdot\epsilon_{\text{geom}}^{\text{from
W}}}{N_{\text{B}}\cdot\epsilon_{\text{geom}}^{\text{from
B}}}\times\frac{\text{Br}(W\to N+e)}{\sum_{i}f_{b\to B_{i}}\text{Br}(B_{i}\to
N+X)}\simeq\\\ \simeq 10^{-5}\times\frac{\text{Br}(W\to N+e)}{\sum_{i}f_{b\to
B_{i}}\text{Br}(B_{i}\to N+X)}\ll 1$ (11)

where $f_{b\to B_{i}}$ is the fragmentation fraction of $b$ quark into a meson
$B_{i}$, and we have taken into account that the second multiplier remains
$\ll 10^{5}$ for practically all HNL masses below $m_{B_{c}}$
Bondarenko:2019yob .

For the mass range $m_{N}>m_{B_{c}}$, HNLs from $W$ are too short-lived and
cannot reach the detector. Indeed, let us estimate the number of events with
HNLs with mass $m_{B_{c}}$ produced by $W$ decays at
$l_{\text{decay}}=c\tau_{N}\gamma_{N}\simeq l_{\text{min}}$. Number of events
increases with decreasing $l_{\text{decay}}$ for $l_{\text{decay}}\gtrsim
l_{\text{min}}$, so this should give an upper bound of events from $W$ for the
lower bound of sensitivity. The corresponding mixing angle is $U^{2}\simeq
l_{\text{min}}/c\tau_{N,U^{2}=1}\gamma_{N}\approx 6\cdot 10^{-8}$, where we
used the results of Bondarenko:2018ptm  for $\tau_{N}$ and $E_{N}=1\text{
TeV}$. The number of events is thus

$N_{N}^{(W)}\bigg{|}_{m_{N}=m_{B_{c}}}^{c\tau_{N}\gamma_{N}=l_{\text{min}}}=N_{W}\times\epsilon_{\text{geom},N}^{(W)}\times\text{Br}(W\to
N)\times P_{\text{decay}}(c\tau_{N}\gamma_{N}=l_{\text{min}})<1,$ (12)

where we have used $\epsilon_{\text{geom},N}^{(W)}$ from Table 3.

##  5 Results and discussion

Using Eq. (3) and requiring $N_{\text{events}}>3$, corresponding to 95% C.L.
in the background free-regime of observing 1 event, we obtain the sensitivity
of FACET and FASER2 to HNLs and Higgs-like scalars shown in Fig. 1.

The results are in agreement with the estimates from Sec. 4. Namely, as
compared to FASER2, detectors of FACET covers $\simeq 3$ larger solid angle,
while the decay volume of FACET is $\simeq 4$ times longer and located $\simeq
5$ times closer. Because of this, for dark scalars, FACET may probe the whole
mass range $m_{S}<m_{h}/2$, while at FASER2 it is impossible to search for
scalars in the mass range $m_{B}-m_{\pi}<m_{S}\lesssim 45\text{ GeV}$ due to
the suppression of the geometric acceptance (see the discussion in Sec.
4.1.2). For HNLs, FACET may probe masses up to $m_{N}\simeq 6\text{ GeV}$,
while FASER2 only up to $\simeq 4\text{ GeV}$, which is again due both to
better sensitivity of FACET at the lower and upper bounds.

As a cross check of our results, we compare the sensitivity to dark scalars
obtained in this work with Cerci:2021nlb , which used FORESEE package
Kling:2021fwx . Namely, we compared the sensitivities of FASER2 to scalars
with zero quartic coupling, and the sensitivities of FACET assuming
$\text{Br}(h\to SS)=0.05$, see Fig. 8. The sensitivities agree well for low
masses $m_{S}\lesssim 10\text{ GeV}$, but disagree by a factor of 2-3 at
higher masses. The differences may be due to smaller decay width in
Cerci:2021nlb  (which explains the discrepancy at the upper bound) and absence
of the decay acceptance in their estimates.

Figure 8: Comparison of the sensitivity of FACET to dark scalars obtained in
our work and in Cerci:2021nlb , assuming $\text{Br}(h\to SS)=0.025$ (for
FACET) and $\text{Br}(h\to SS)=0$ (for FASER2).

An important feature shown in Fig. 1 is that the sensitivity of FACET at the
upper bound is better than the sensitivity of other experiments. The reason is
the following: the upper bound is controlled by the ratio $\langle
p\rangle/l_{\text{min}}$, where $\langle p\rangle$ is the mean momentum of
decaying particles, and $l_{\text{min}}$ is the distance from the production
point to the decay volume. While FACET has $l_{\text{min}}$ comparable to
experiments such as SHiP and MATHUSLA, $\langle p\rangle$ is much higher –
$\simeq 10$ times higher than at SHiP, and $\simeq 100$ times higher than at
MATHUSLA.

##  6 Conclusions

In this paper, we have estimated the potential of FACET, an experiment located
in the far-forward direction at the LHC, to probe new physics, considering the
models of scalar and fermion portals as an example. Using semi-analytic
estimates, we have compared it with another proposed far-forward experiment,
FASER2, see Sec. 4. FACET has larger decay volume, allowing to probe the
parameter space of long-lived particles, and is located closer to the
interaction point (Table 1), which allows searching for short-lived particles.
The combination of these features improves the sensitivity compared to FASER2,
significantly extending the probed mass range for both models, see Fig. 1. In
particular, for dark scalars that are produced by decays of Higgs bosons,
FACET may probe the whole kinematically allowed mass range $m_{S}<m_{h}/2$,
whereas FASER2 has no sensitivity at scalar masses $m_{S}\lesssim 45\text{
GeV}$. For HNLs, FACET may probe masses up to the kinematic threshold for the
production from $B_{c}$ mesons, while FACET has the sensitivity limited by
$m_{N}\simeq 4\text{ GeV}$.

In addition, FACET is complementary to other LHC-based experiments (such as
MATHUSLA) and SHiP. Indeed, the latter may search for new physics particles
with much smaller mixing angles due to larger geometrical acceptance and decay
volume. FACET, on the other hand, is better suited for probing particles with
large couplings due to its on-axis placement: particles produced in the far-
forward direction at the LHC have large $\gamma$ factors, which significantly
increases their lifetime and makes possible to reach the decay volume before
decaying.

##  Appendix A Mixing and quartic coupling at the lower bound of the
sensitivity

Below the $B$ meson mass, scalars may be produced by decays $h\to S+S,B\to
X_{s}+S+S$, mediated by the quartic coupling, and $B\to X_{s}+S$ mediated by
the mixing. Let us establish which of the production channels determine the
lower bound of the sensitivity. Namely, let us compare the numbers of scalars
produced by decays of $h$ and $B$ in the direction of the FACET experiment:

$\frac{N_{\text{prod}}(B\to S)}{N_{\text{prod}}(h\to
SS)}\sim\frac{N_{B}\cdot\chi^{(B)}_{S}\text{Br}(B\to
S)\cdot\epsilon_{\text{geom},S}^{(B)}}{N_{h}\cdot\chi^{(h)}_{S}\text{Br}(h\to
SS)\cdot\epsilon_{\text{geom},S}^{(h)}}\simeq\\\
\simeq\begin{cases}\mathcal{O}(1)\cdot\frac{0.05}{\text{Br}(h\to
SS)}\frac{\text{Br}(B\to
X_{s}S)}{3\theta^{2}}\frac{\theta^{2}}{10^{-10}},\quad B\to X_{s}S\\\
\mathcal{O}(10)\frac{\text{Br}(B\to X_{s}SS)}{5\cdot 10^{-10}}\quad B\to
X_{s}SS,\end{cases}$ (13)

where we have normalized the branching ratios $\text{Br}(B\to
X_{s}S),\text{Br}(B\to X_{s}SS)$ by their characteristic values, see
Boiarska:2019jym . We conclude that below the kinematic threshold for $B\to
X_{s}SS$, the lower bound is determined by the quartic production from $B$,
whereas above there are two competing contributions from the mixing production
from $B$ and quartic from $h$.

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