# Generating Large Cats with Nine Lives: Long-Lived Macroscopically Distinct
Superposition States in Atomic Ensembles

Wei Qin  Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering
Research, Wako-shi, Saitama 351-0198, Japan  Adam Miranowicz  Theoretical
Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi,
Saitama 351-0198, Japan  Institute of Spintronics and Quantum Information,
Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland  Hui
Jing  Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering
Research, Wako-shi, Saitama 351-0198, Japan  Key Laboratory of Low-Dimensional
Quantum Structures and Quantum Control of Ministry of Education, Department of
Physics and Synergetic Innovation Center for Quantum Effects and Applications,
Hunan Normal University, Changsha 410081, China  Franco Nori  Theoretical
Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi,
Saitama 351-0198, Japan  Department of Physics, The University of Michigan,
Ann Arbor, Michigan 48109, USA

###### Abstract

We propose to create and stabilize long-lived macroscopic quantum
superposition states in atomic ensembles. We show that using a fully quantum
parametric amplifier can cause the simultaneous decay of two atoms and, in
turn, create stabilized atomic Schrödinger cat states. Remarkably, even with
modest parameters these intracavity atomic cat states can have an extremely
long lifetime, up to _4 orders of magnitude_ longer than that of intracavity
photonic cat states under the same parameter conditions, reaching _tens of
milliseconds_. This lifetime of atomic cat states is ultimately limited to
_several seconds_ by extremely weak spin relaxation and thermal noise. Our
work opens up a new way toward the long-standing goal of generating large-size
and long-lived cat states, with immediate interests both in fundamental
studies and noise-immune quantum technologies.

_Introduction.—_ Schrödinger cat states, which are macroscopically distinct
superposition states, express the essence of quantum mechanics. Such states
are appealing not only for fundamental studies of quantum mechanics Zurek
(2003); Haroche (2013), but also for wide applications ranging from quantum
metrology Kira  _et al._ (2011); Pezzè  _et al._ (2018) to quantum computation
Ralph  _et al._ (2003); Gilchrist  _et al._ (2004); J. (2014); Ofek  _et al._
(2016); Cai  _et al._ (2021). So far, a large number of approaches Agarwal
_et al._ (1997); Sackett  _et al._ (2000); Leibfried  _et al._ (2005);
Ourjoumtsev  _et al._ (2006, 2007); Vlastakis  _et al._ (2013); Monz  _et al._
(2011); Lau  _et al._ (2014); Etesse  _et al._ (2015); Bradley  _et al._
(2019); Lu  _et al._ (2019); Figgatt  _et al._ (2019); Omran  _et al._ (2019);
Song  _et al._ (2019); Wei  _et al._ (2020); Chen  _et al._ (2021) have been
proposed to generate cat states. However, these cat states (especially of
large size) are extremely fragile in a noisy environment, and their fast
decoherence makes them impractical for applications. Thus, the ability to
stabilize cat states, as an essential prerequisite for their various
applications, is highly desirable. To address this problem, a two-photon loss
has been engineered Gerry and Hach III (1993); Gilles  _et al._ (1994);
Karasik  _et al._ (2008); Mirrahimi  _et al._ (2014) and recently
experimentally demonstrated Everitt  _et al._ (2014); Leghtas  _et al._
(2015); Touzard  _et al._ (2018); Lescanne  _et al._ (2020); Grimm  _et al._
(2020). Such a nonlinear loss can protect cat states against photon dephasing
Cohen (2017); Mirrahimi  _et al._ (2014) but, unfortunately, not against the
unavoidable single-photon loss. This implies a significantly limited cat-state
lifetime. Single-photon loss has been considered to be the major source of
noise in fault-tolerant quantum computation based on cat states Ralph  _et
al._ (2003); Gilchrist  _et al._ (2004); J. (2014); Ofek  _et al._ (2016); Cai
_et al._ (2021). Thus, the stabilization of large-size cat states for an
extended time remains challenging.

Ensembles of atoms or spins have negligible spin relaxation; consequently,
their major source of noise is spin dephasing, i.e., collective dephasing,
local dephasing, and inhomogeneous broadening. This motivates us to engineer
the _simultaneous_ decay of two atoms of an ensemble (here denoted as “two-
atom decay”) and then use it to stabilize atomic cat states. Such cat states
could have a very long lifetime if the two-atom decay is possible. This is
because such a decay may protect these atomic cat states against spin
dephasing, which is a close analogy to the mechanism of using two-photon loss
to suppress photon dephasing.

However, it seems to us that the two-atom decay, which is fundamentally
different from two-photon loss, is still lacking. To implement it, here we
propose to exploit fully quantum degenerate parametric amplification. More
importantly, the lifetime of the resulting atomic cat states can be made up to
_4 orders of magnitude_ longer than that of common intracavity photonic cat
states (see Table I in sup ), i.e., equal superpositions of two opposite-phase
coherent states. To ensure a fair comparison, these photonic cat states need
to have the same size as our atomic cat states and also suffer from single-
photon loss of the same rate as given for the signal mode. With a modest
cavity decay time ($\sim 16$ $\mu$s), our cat-state lifetime can reach $\sim
20$ ms. This is comparable to 17 ms Deleglise  _et al._ (2008), which is the
longest lifetime of intracavity photonic cat states to date but which was
achieved with an extreme cavity decay time ($\sim 0.13$ sec). As the cavity
decay time increases, our cat-state lifetime can further increase but
ultimately is limited to a maximum value determined by spin relaxation and
thermal noise. For a typical spin relaxation time $\sim 40$ sec Amsüss  _et
al._ (2011); Grezes  _et al._ (2014), we can predict a maximum cat-state
lifetime of $\sim 3$ sec.

Figure 1: (a) Schematic setup of our proposal. The pump and signal cavities
are coupled via a parametric coupling $J$, and the atomic ensemble is coupled
to the signal cavity with a single-atom coupling $g$. The pump cavity is
subject to a coherent drive with amplitude $\Omega$ and frequency
$\omega_{d}$. Here, $\omega_{p}$, $\omega_{s}$ are the resonance frequencies
of the pump and signal cavities, $\kappa_{p}$, $\kappa_{s}$ are their
respective single-photon loss rates, and $\omega_{q}$ is the atomic transition
frequency. (b), (c) Quantum Monte Carlo trajectory pictured through the
probabilities of the system being in the states $|m_{p}0\rangle$$|n\rangle$.
Initially, only two atoms in the ensemble are excited. Here,
$\kappa_{p}=0.2\chi$ and $\kappa_{s}=\Omega=0$. (d) Time evolution of the
preparation error $\eta$ for a cat size $\left|\alpha\right|^{2}=1$. Here,
$\kappa_{p}=5\chi$, $\kappa_{s}=0.3\kappa_{p}$, and the ensemble is
initialized in the ground state $|0\rangle$, the single-excitation state
$|1\rangle$, and a spin coherent state $|\theta_{0},0\rangle$ with
$\sqrt{N}\tan\left(\theta_{0}/2\right)=1$ for the states
$|\mathcal{C}_{+}\rangle$, $|\mathcal{C}_{-}\rangle$, and $\rho_{\rm ens}^{\rm
ss}$, respectively. In (b)-(d), we assume that $N=100$, $J=3g_{\rm col}$, and
both cavities are initialized in the vacuum.

_Physical model.—_ The central idea is illustrated in Fig. 1(a). To consider
degenerate parametric amplification in the fully quantum regime, our system,
inspired by recent experimental advances Leghtas  _et al._ (2015); Touzard
_et al._ (2018); Chang  _et al._ (2020); Vrajitoarea  _et al._ (2020);
Lescanne  _et al._ (2020), contains two parametrically coupled cavities: one
as a pump cavity with frequency $\omega_{p}$ and the other as a signal cavity
with frequency $\omega_{s}$. We assume that the pump cavity is subject to a
coherent drive with amplitude $\Omega$ and frequency $\omega_{d}$. The
intercavity parametric coupling $J$ stimulates the conversion between pump
single photons and pairs of signal photons. Furthermore, an ensemble of $N$
identical two-level atoms is placed in the signal cavity, and the atomic
transition, of frequency $\omega_{q}$, is driven by a coupling $g$ to the
signal photon. When $2\omega_{q}\approx\omega_{p}\ll 2\omega_{s}$, a pair of
excited atoms can jointly emit a pump photon. The subsequent loss of the pump
photon gives rise to the two-atom decay, which in turn stabilizes large-size,
extremely long-lived cat states in the ensemble.

The system Hamiltonian in a frame rotating at $\omega_{d}$ is

$\displaystyle H=$
$\displaystyle\sum_{i=p,s}\delta_{i}a^{{\dagger}}_{i}a_{i}+\delta_{q}S_{z}+J\left(a_{p}a^{{\dagger}2}_{s}+a^{{\dagger}}_{p}a^{2}_{s}\right)$
$\displaystyle+g\left(a_{s}S_{+}+a^{{\dagger}}_{s}S_{-}\right)+\Omega\left(a_{p}+a^{{\dagger}}_{p}\right),$
(1)

where $a_{p}$, $a_{s}$ are the annihilation operators for the pump and signal
modes, $S_{\pm}=S_{x}\pm iS_{y}$, $\delta_{p}=\omega_{p}-\omega_{d}$,
$\delta_{s}=\omega_{s}-\omega_{d}/2$, and
$\delta_{q}=\omega_{q}-\omega_{d}/2$. The collective spin operators are
$S_{\alpha}=\frac{1}{2}\sum_{j=1}^{N}\sigma_{j}^{\alpha}$, with
$\sigma_{j}^{\alpha}$ ($\alpha=x,y,z$) the Pauli matrices for the $j$th atom.
The Lindblad dissipator, $\mathcal{L}\left(o\right)\rho=o\rho
o^{{\dagger}}-\frac{1}{2}o^{{\dagger}}o\rho-\frac{1}{2}\rho o^{{\dagger}}o$,
describes the dissipative dynamics determined by

$\dot{\rho}=-i\left[H,\rho\right]+\sum_{i=p,s}\kappa_{i}\mathcal{L}\left(a_{i}\right)\rho,$
(2)

where $\kappa_{p}$ and $\kappa_{s}$ are the photon loss rates of the pump and
signal modes. Spin dephasing, spin relaxation, and thermal noise are discussed
below.

Figure 2: (a) Time evolution of the preparation error $\eta$ of the states
$|\mathcal{C}_{+}\rangle$, $|\mathcal{C}_{-}\rangle$, and $\rho_{\rm ens}^{\rm
ss}$ under the bosonic approximation for different cat sizes
$\left|\alpha\right|^{2}=2$, $4$, and $6$. The initial states are chosen as in
Fig. 1(d). (b) Wigner function at times $t_{1},\ldots,t_{5}$ shown on top of
panel (a) for the $\left|\alpha\right|^{2}=4$ cat size. The first, second, and
third rows correspond to the states $|\mathcal{C}_{+}\rangle$,
$|\mathcal{C}_{-}\rangle$, and $\rho_{\rm ens}^{\rm ss}$, respectively. For
all plots, we set $J=3g_{\rm col}$, $\delta_{p}=J^{2}/\left(20g_{\rm
col}\right)$, $\kappa_{p}=5\chi$, and $\kappa_{s}=0.3\kappa_{p}$.

We assume that $2\omega_{q}\approx\omega_{p}\approx\omega_{d}$, and the
detuning $\Delta=\omega_{s}-\omega_{q}\gg\left\\{g_{\rm col},J\right\\}$.
Here, $g_{\rm col}=\sqrt{N}g$ represents the collective coupling of the
ensemble to the signal mode. Then, we can predict a parametric coupling,
$\chi=g^{2}_{\rm col}J/\Delta^{2}$, between atom pairs and pump single
photons. Accordingly, the Hamiltonian $H$, after time averaging Gamel and
James (2010); Shao  _et al._ (2017), becomes

$H_{\rm
avg}=\frac{\chi}{N}\left(a_{p}S_{+}^{2}+a^{{\dagger}}_{p}S_{-}^{2}\right)+\Omega\left(a_{p}+a^{{\dagger}}_{p}\right),$
(3)

which describes a third-order process. The stronger second-order process has
been eliminated with an appropriate detuning between $\omega_{p}$ and
$2\omega_{q}$ (see sup ). To derive $H_{\rm avg}$, we have considered the low-
excitation regime, where the average number of excited atoms is much smaller
than the total number of atoms.

We now adiabatically eliminate the pump mode $a_{p}$, yielding an effective
master equation

$\displaystyle\dot{\rho}_{\rm ens}=$ $\displaystyle\;-i\left[H_{\rm
ens},\rho_{\rm ens}\right]$ $\displaystyle+\frac{\kappa_{\rm
1at}}{N}\mathcal{L}\left(S_{-}\right)\rho_{\rm ens}+\frac{\kappa_{\rm
2at}}{N^{2}}\mathcal{L}\left(S_{-}^{2}\right)\rho_{\rm ens},$ (4)

where $H_{\rm ens}=i\chi_{\rm 2at}\left(S_{-}^{2}-S^{2}_{+}\right)/N$, and
$\rho_{\rm ens}$ represents the reduced density matrix of the ensemble. Here,
$\kappa_{\rm 2at}=4\chi^{2}/\kappa_{p}$ and $\chi_{\rm
2at}=2\Omega\chi/\kappa_{p}$ are the rates of the simultaneous decay and
excitation of two atoms, respectively. Moreover, $\kappa_{\rm
1at}=\left(g_{\rm col}/\Delta\right)^{2}\kappa_{s}$ is the rate of the Purcell
single-atom decay (see sup ), and we can tune it to be $\ll\kappa_{\rm 2at}$,
as long as $\kappa_{s}\ll(g_{\rm col}J/\Delta)^{2}/\kappa_{p}$.

We note that the methods of Refs. Garziano  _et al._ (2016); Macrì  _et al._
(2020); Garziano  _et al._ (2020) can lead to a Hamiltonian formally similar
to $H_{\rm avg}$. However, contrary to our method, the two-atom decay _cannot_
be realized in those methods assuming the strong cavity-photon loss. This is
because those methods require a virtual cavity photon to mediate a third-order
process, which indicates that the cavity-photon loss cannot be allowed to be
strong; moreover, they also depend on a longitudinal coupling, which cannot be
collectively enhanced in atomic ensembles.

To gain more insights into the engineered two-atom decay, we use the quantum
Monte Carlo method Scully and Zubairy (1997). In Figs. 1(b), (c) we plot a
single quantum trajectory with the Hamiltonian $H$ and an initial state
$|00\rangle$$|2\rangle$ (see sup  for more cases). Here, the first ket
$|m_{p}m_{s}\rangle$ ($m_{p},m_{s}=0,1,2,\ldots$) in the pair refers to the
cavity state with $m_{p}$ pump photons and $m_{s}$ signal photons, and the
second $|n\rangle$ ($n=0,1,2,\ldots$) refers to the collective spin state
$|S=N/2,m_{z}=-N/2+n\rangle$, corresponding to $n$ excited atoms in the
ensemble. The non-Hermitian Hamiltonian $H_{\rm
NH}=H-\frac{i}{2}\kappa_{p}a_{p}^{{\dagger}}a_{p}$ drives Rabi oscillations
between $|00\rangle$$|2\rangle$ and $|10\rangle$$|0\rangle$, as shown in Fig.
1(b). The Rabi oscillations are then interrupted by a quantum jump $a_{p}$. We
find from Fig. 1(c) that the jump leaves the system in its ground state
$|00\rangle$$|0\rangle$, implying that single-photon loss of the pump mode
causes the two-atom decay.

_Stabilized manifold of atomic cat states.—_ When $\kappa_{\rm 1at}=0$, the
dynamics of the effective master equation, Eq. (Generating Large Cats with
Nine Lives: Long-Lived Macroscopically Distinct Superposition States in Atomic
Ensembles), describes a pairwise exchange of atomic excitations between the
ensemble and its environment, thus conserving the excitation-number parity. As
demonstrated in sup , the ensemble is driven to an even cat state
$\mbox{$|\mathcal{C}_{+}\rangle$}=\mathcal{A}_{+}\left(\mbox{$|\theta,\phi\rangle$}+\mbox{$|\theta,\phi+\pi\rangle$}\right)$
if initialized in an even parity state, or to an odd cat state
$\mbox{$|\mathcal{C}_{-}\rangle$}=\mathcal{A}_{-}\left(\mbox{$|\theta,\phi\rangle$}-\mbox{$|\theta,\phi+\pi\rangle$}\right)$
if initialized in an odd parity state. Here, $|\theta,\phi\rangle$, where
$\phi=\pi/2$ and $\theta=2\arctan(\left|\alpha\right|/\sqrt{N})$, refers to a
spin coherent state, and
$\mathcal{A}_{\pm}=1/\\{2[1\pm\exp(-2\left|\alpha\right|^{2})]\\}^{1/2}$.
Moreover, $\alpha=i\sqrt{\Omega/\chi}$ is the coherent amplitude. The average
number of excited atoms, $\left|\alpha\right|^{2}$, of the states
$|\mathcal{C}_{\pm}\rangle$ characterizes the cat size Deleglise  _et al._
(2008). When assuming the initial state to be a spin coherent state
$|\theta_{0},\phi_{0}\rangle$, the steady state of the ensemble is confined
into a quantum manifold spanned by the states
$\left\\{\mbox{$|\mathcal{C}_{+}\rangle$},\mbox{$|\mathcal{C}_{-}\rangle$}\right\\}$,
and is expressed as $\rho_{\rm ens}^{\rm
ss}=c_{++}\mbox{$|\mathcal{C}_{+}\rangle$}\mbox{$\langle\mathcal{C}_{+}|$}+c_{--}\mbox{$|\mathcal{C}_{-}\rangle$}\mbox{$\langle\mathcal{C}_{-}|$}\
+c_{+-}\mbox{$|\mathcal{C}_{+}\rangle$}\mbox{$\langle\mathcal{C}_{-}|$}+c_{+-}^{*}\mbox{$|\mathcal{C}_{-}\rangle$}\mbox{$\langle\mathcal{C}_{+}|$}$,
where $c_{++}=\frac{1}{2}[1+\exp(-2\left|\alpha_{0}\right|^{2})]$ with
$\alpha_{0}=\sqrt{N}\exp\left(i\phi_{0}\right)\tan\left(\theta_{0}/2\right)$,
$c_{--}=1-c_{++}$, and $c_{+-}$ is given in sup . To confirm these
predictions, we numerically integrate Johansson  _et al._ (2012, 2013), the
master equation in Eq. (2), to simulate the time evolution of the preparation
error $\eta=1-F$ in Fig. 1(d). Here, $F$ is the fidelity between the actual
and ideal states. It is seen that, as expected, the ensemble states are
steered into a stabilized 2D cat-state manifold with a high fidelity.

In the low-excitation regime considered above, the collective spin in fact
behaves as a quantum harmonic oscillator. This allows us to map $S_{-}$ to a
bosonic operator $b$, i.e., $S_{-}\approx\sqrt{N}b$, and thus to investigate
cat states of large size ($\left|\alpha\right|\geqslant 2$) in large
ensembles. The spin coherent state $|\theta,\phi\rangle$ accordingly becomes a
bosonic coherent state $|\alpha\rangle$, such that the states
$|\mathcal{C}_{\pm}\rangle$ become
$\mbox{$|\mathcal{C}_{\pm}\rangle$}=\mathcal{A}_{\pm}\left(\mbox{$|\alpha\rangle$}+\mbox{$|-\alpha\rangle$}\right)$.
With the master equation in Eq. (2) and under the bosonic approximation, we
plot the time evolution of the preparation error $\eta$ in Fig. 2(a), and the
Wigner function $W\left(\beta\right)$ for different times in Fig. 2(b). We
find that a cat state of size $\left|\alpha\right|^{2}=4$ is obtained after
time $t\sim 250/g_{\rm col}$, or more specifically, $t\sim 4$ $\mu$s, for a
typical collective coupling strength $g_{\rm col}/2\pi=10$ MHz Kubo  _et al._
(2010); Amsüss  _et al._ (2011); Kubo  _et al._ (2012); Putz  _et al._ (2014);
Astner  _et al._ (2017).

_Suppressed spin dephasing.—_ So far, we have assumed a model where there is
no spin dephasing; however, there will always be some spin dephasing. Before
discussing spin dephasing, let us first consider the rate $\gamma$ of
convergence, i.e., how rapidly the steady cat states can be reached. To
determine $\gamma$, we introduce the Liouvillian spectral gap,
$\lambda=\left|{\rm Re}\left[\lambda_{1}\right]\right|$, of the effective
master equation in Eq. (Generating Large Cats with Nine Lives: Long-Lived
Macroscopically Distinct Superposition States in Atomic Ensembles) for
$\kappa_{\rm 1at}=0$. Here, $\lambda_{1}$ is the Liouvillian eigenvalue with
the smallest modulus of the real part. Since the gap $\lambda$ determines the
slowest relaxation of the Liouvillian Minganti  _et al._ (2018), we thus
conclude that $\gamma>\lambda$. In the inset of Fig. 3(a), we numerically
calculate the gap $\lambda$ and find
$\lambda\approx\left|\alpha\right|^{2}\kappa_{\rm 2at}$ for
$\left|\alpha\right|^{2}\geq 2$.

Below we consider collective spin dephasing $\gamma_{\rm
col}\mathcal{L}(S_{z})\rho_{\rm ens}$, local spin dephasing $\gamma_{\rm
loc}\sum_{j=1}^{N}\mathcal{L}(\sigma_{j}^{z})\rho_{\rm ens}$, and
inhomogeneous broadening $\frac{1}{2}\sum_{j=1}^{N}\delta_{j}\sigma_{j}^{z}$.
Here, $\gamma_{\rm col}$ and $\gamma_{\rm loc}$ are the collective and local
dephasing rates, respectively. Moreover, $\delta_{j}=\omega_{j}-\omega_{q}$,
where $\omega_{j}$ 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 sup . We assume that the distribution of $\delta_{j}$ has a
linewidth $\Delta_{\rm inh}$. The three sources of dephasing noise conserve
the excitation-number parity of the superradiant subspace, where the cat
states are created and stabilized. Thus, all these dephasing processes can be
strongly suppressed by the two-atom decay as long as $\gamma\gg\\{\gamma_{\rm
col},\gamma_{\rm loc},\Delta_{\rm inh}\\}$ (i.e., $|\alpha|^{2}\kappa_{\rm
2at}\gg\\{\gamma_{\rm col},\gamma_{\rm loc},\Delta_{\rm inh}\\}$) (see sup
for more details). Figure 3(a) shows the dependence of such a dissipative
suppression on the ratio $\kappa_{\rm 2at}/\gamma_{\rm deph}$, assuming
$\gamma_{\rm col}=\gamma_{\rm loc}=\Delta_{\rm inh}\equiv\gamma_{\rm deph}$.
It is seen that for $\kappa_{\rm 2at}=10\gamma_{\rm deph}$, corresponding to
an ensemble coherence time of $\gamma_{\rm deph}^{-1}\sim 27$ $\mu$s, a steady
cat state is generated, implying a significant suppression of spin dephasing.
We note that in Fig. 3(a) the error $\eta$ is limited by a small $N$,
especially for $\kappa_{\rm 2at}=10\gamma_{\rm deph}$, and a larger $N$ could
lead to a smaller $\eta$ until the bosonic approximation is well satisfied.

Figure 3: (a) Effects of collective dephasing, local dephasing, and
inhomogeneous broadening on the preparation error $\eta$ of the state
$|\mathcal{C}_{+}\rangle$ of size $|\alpha|^{2}=2$. We integrated the
effective master equation, Eq. (Generating Large Cats with Nine Lives: Long-
Lived Macroscopically Distinct Superposition States in Atomic Ensembles), with
an additional spin dephasing $\gamma_{\rm col}\mathcal{L}(S_{z})\rho_{\rm
ens}$, local spin dephasing $\gamma_{\rm
loc}\sum_{j=1}^{N}\mathcal{L}(\sigma_{j}^{z})\rho_{\rm ens}$, and
inhomogeneous broadening $\frac{1}{2}\sum_{j=1}^{N}\delta_{j}\sigma_{j}^{z}$.
The frequency shifts $\delta_{j}$ are randomly given according to a Lorentzian
distribution of linewidth $\Delta_{\rm inh}$. For simplicity, we here set
$N=10$, $\gamma_{\rm col}=\gamma_{\rm loc}=\Delta_{\rm inh}\equiv\gamma_{\rm
deph}$ and $\kappa_{\rm 1at}=0$ so that only the effects of these dephasing
processes are shown. Inset: the Liouvillian spectral gap $\lambda$ of the
master equation (Generating Large Cats with Nine Lives: Long-Lived
Macroscopically Distinct Superposition States in Atomic Ensembles) versus the
cat size $\left|\alpha\right|^{2}$ for $\kappa_{\rm 1at}=0$ under the bosonic
approximation. (b) Ratio $\kappa_{\rm 2at}/\gamma_{\rm deph}$ versus the
parameter $\Delta/g_{\rm col}$ for $\gamma_{\rm deph}^{-1}=10$ $\mu$s, $100$
$\mu$s, and $1$ ms for $\kappa_{p}=5\chi$ and $J/2\pi=30$ MHz. The yellow
shaded area represents the $\kappa_{\rm 2at}\geq 10\gamma_{\rm deph}$ regime,
where spin dephasing is strongly suppressed by the two-atom decay. The solid
green line shows $\tau_{\rm at}/\tau_{\rm ph}$ versus $\Delta/g_{\rm col}$.
Other parameters in (a) and (b) are chosen as in Fig. 2.

_Cat-state lifetime.—_ Let us now consider the cat-state lifetime $\tau_{\rm
at}$. According to the above discussions, the effects of spin dephasing on
$\tau_{\rm at}$ can be excluded. This lifetime is thus determined by the
Purcell decay rate $\Gamma_{\rm 1at}=2\left|\alpha\right|^{2}\kappa_{\rm
1at}$, such that

$\tau_{\rm at}=\Gamma_{\rm 1at}^{-1}=\left(\frac{\Delta}{g_{\rm
col}}\right)^{2}\frac{1}{2\left|\alpha\right|^{2}\kappa_{s}}.$ (5)

Note that intracavity photonic cat states, i.e., equal superpositions of two
opposite-phase coherent states, rapidly decohere into statistical mixtures due
to single-photon loss. The lifetime of such photonic cat states is thus given
by $\tau_{\rm ph}=1/2\left|\alpha\right|^{2}\kappa_{s}$ Haroche and Raimond
(2006). Here, for a fair comparison, we have assumed the same cat size
$\left|\alpha\right|^{2}$ as our atomic cat states, and the same single-photon
loss rate $\kappa_{s}$ as given for the signal cavity. It is seen that
$\tau_{\rm at}$ is longer by a factor of $\left(\Delta/g_{\rm col}\right)^{2}$
compared to $\tau_{\rm ph}$. To make $\tau_{\rm at}/\tau_{\rm ph}$ larger, it
is essential to increase $\Delta/g_{\rm col}$. However, the rate $\kappa_{\rm
2at}$, which needs to be $\gg\gamma_{\rm deph}$ as mentioned already,
decreases as $\Delta/g_{\rm col}$ increases. Thus, the ratio $\Delta/g_{\rm
col}$ has an upper bound for a given $\gamma_{\rm deph}$. Experimentally, the
coherence time $\gamma_{\rm deph}^{-1}$ of NV-spin ensembles has reached $\sim
1$ ms with spin-echo pulse sequences Hahn (1950); Stanwix  _et al._ (2010),
and if dynamical-decoupling techniques are employed, it can be even close to
$1$ sec Bar-Gill  _et al._ (2013). In Fig. 3(b), the ratio $\kappa_{\rm
2at}/\gamma_{\rm deph}$ for different $\gamma_{\rm deph}$, as well as the
ratio $\tau_{\rm at}/\tau_{\rm ph}$, is plotted versus $\Delta/g_{\rm col}$.
Assuming a realistic parameter of $\gamma_{\rm deph}^{-1}=1$ ms, we find from
Fig. 3(b) that in stark contrast to previous work on intracavity photonic cat
states (see Table I in sup ), our approach can lead to an increase in the cat-
state lifetime of up to 4 orders of magnitude for $\kappa_{\rm 2at}\approx
15\gamma_{\rm deph}$ and a very large cat size of
$\left|\alpha\right|^{2}\geqslant 4$. Correspondingly, for a typical single-
photon loss rate of $\kappa_{s}/2\pi=10$ kHz (i.e., a cavity decay time $\sim
16$ $\mu$s) Leghtas  _et al._ (2015), the lifetime of the
$\left|\alpha\right|^{2}=4$ cat states resulting from our approach is $\sim
20$ ms.

As the cavity loss rate $\kappa_{s}$ decreases, the lifetime $\tau_{\rm at}$
further increases and ultimately reaches its maximum value, limited by spin
relaxation and thermal noise (see sup  for more details). This maximum
lifetime is given by $\tau_{\rm at}^{\rm max}=\Gamma_{\rm relax}^{-1}$. Here,
$\Gamma_{\rm relax}=[2|\alpha|^{2}(1+2n_{\rm th})+2n_{\rm th}]\gamma_{\rm
relax}$ Kim and Bužek (1992) is the cat-state decay rate arising from spin
relaxation with a rate $\gamma_{\rm relax}$ and thermal noise with a thermal
average boson number $n_{\rm th}$. For realistic parameters of $\gamma_{\rm
relax}=2\pi\times 4$ mHz Amsüss  _et al._ (2011); Grezes  _et al._ (2014) and
$T=100$ mK, we can predict a maximum lifetime of $\tau_{\rm at}^{\rm max}\sim
3$ sec, which is more than 2 orders of magnitude longer than the longest
lifetime, $17$ ms, of the intracavity photonic cat states reported in Ref.
Deleglise  _et al._ (2008).

_Conclusions.—_ We have introduced a method to create and stabilize large-
size, long-lived Schrödinger cat states in atomic ensembles. This method is
based on the use of fully quantized degenerate parametric amplification to
facilitate the simultaneous decay of two atoms, i.e., the two-atom decay. The
resulting atomic cat states can last _an extremely long time_ , because of
strongly suppressed spin dephasing and extremely weak spin relaxation and
thermal noise. These long-lived cat states are promising for both fundamental
tests and practical applications of quantum mechanics. Our work can further
stimulate more efforts to create and protect macroscopic cat states or other
fragile quantum states and to use them to improve the performance of various
modern quantum technologies.

###### Acknowledgements.

We thank Carlos Sánchez Muñoz and Fabrizio Minganti for their valuable
discussions. H.J. is supported by the National Natural Science Foundation of
China (Grants No. 11935006 and No. 11774086) and the Science and Technology
Innovation Program of Hunan Province (Grant No. 2020RC4047). A.M. is 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) program, the Moonshot
R&D Grant No. JPMJMS2061, and the Centers of Research Excellence in Science
and Technology (CREST) Grant No. JPMJCR1676], the Japan Society for the
Promotion of Science (JSPS) [via the Grants-in-Aid for Scientific Research
(KAKENHI) Grant No. JP20H00134 and the JSPS–RFBR Grant No. JPJSBP120194828],
the Army Research Office (ARO) (Grant No. W911NF-18-1-0358), 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.

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##  I Supplemental Material

## Introduction

> Here, we first compare the lifetimes of our atomic cat states and common
> intracavity photonic cat states. We next present how to eliminate the
> second-order effect and as a result to make the desired third-order effect
> dominant. Then, the Purcell single-atom decay induced by single-photon loss
> of the signal cavity is derived, and quantum Monte-Carlo trajectories of the
> ensemble-cavity system are shown. Furthermore, we give the detailed
> derivation of atomic cat states stabilized by the two-atom decay, and show
> the strong suppression of spin dephasing by this nonlinear two-atom decay.
> We then discuss the source of spin dephasing due to inhomogeneous broadening
> in nitrogen-vacancy center ensembles. Finally, we discuss the effects of
> spin relaxation and thermal noise on the cat state lifetime, and also show
> the maximum cat state lifetime limited by them.

###  I.1 S1. Comparison of the lifetimes of our atomic cat states and
intracavity photonic cat states

Table 1: Some relevant parameters of experimentally implemented intracavity photonic cat states $\mbox{$|\mathcal{C}_{\pm}\rangle$}_{\rm ph}$. Here, $\left|\alpha\right|^{2}$ characterizes the cat size, $T_{c}$ is the cavity photon lifetime, $\kappa_{s}=1/T_{c}$ is the cavity photon loss rate, $\tau_{\rm exp}$ is the cat state lifetime measured in experiments, and $\tau_{\rm theor}=1/(2\left|\alpha\right|^{2}\kappa_{s})$ is the theoretical prediction of the cat state lifetime. For comparison, we also list at the end of the table the corresponding theoretical predictions for our atomic cat states $|\mathcal{C}_{\pm}\rangle$. Ref. | approach type | $\left|\alpha\right|^{2}$ |  $T_{c}$ ($\mu$s) |  $\kappa_{s}/{2\pi}$ (kHz) |  $\tau_{\rm exp}$ ($\mu$s) |  $\tau_{\rm theor}$ ($\mu$s)
---|---|---|---|---|---|---
Deleglise  _et al._ (2008) | unitary evolution | 3.0 | $1.3\times 10^{5}$ | $1.2\times 10^{-3}$ | $1.7\times 10^{4}$ | $2.2\times 10^{4}$
Lescanne  _et al._ (2020) | reservoir engineering | 5.8 | 3.0 | 53.0 | 0.2 | 0.26
Vlastakis  _et al._ (2013) | unitary evolution | 28 | 22.1 | 7.2 | — | 0.4
Leghtas  _et al._ (2015) | reservoir engineering | 2.4 | 20 | 8.0 | — | 4.1
Touzard  _et al._ (2018) | reservoir engineering | 5 | 92 | 1.7 | 8 | 9.2
Brune  _et al._ (1996) | unitary evolution | 3.3 | 160 | 1.0 | 38.4 | 35
Wang  _et al._ (2019) | unitary evolution | 1.4 | 0.14 | $1.1\times 10^{3}$ | — | $5.3\times 10^{-2}$
Assemat  _et al._ (2019) | unitary evolution | 11.3 | $8.1\times 10^{3}$ | $2.0\times 10^{-2}$ | 200 | 360
Xu  _et al._ (2020) | unitary evolution | 2 | 692 | 0.2 | — | 173
our results | reservoir engineering | 4 | 16 | 10 | — | $2\times 10^{4}$
$5.3\times 10^{3}$ | $3.0\times 10^{-2}$ | — | $2\times 10^{6}$

The cat state lifetime can be defined as the inverse cat state decoherence
rate. Sec. S8 shows how to derive the cat state decoherence rate and then
obtain the cat state lifetime. In this section, let us first compare the
lifetime of intracavity atomic cat states resulting from our approach with
that of common intracavity photonic cat states, under some realistic
parameters. Our atomic cat states refer to superpositions of two spin coherent
states, i.e.,

$\mbox{$|\mathcal{C}_{\pm}\rangle$}=\;\mathcal{A}_{\pm}\left(\mbox{$|\theta,\phi\rangle$}\pm\mbox{$|\theta,\phi+\pi\rangle$}\right),$
(S1)

Here,
$\mathcal{A}_{\pm}=1/\\{2[1\pm\exp(-2\left|\alpha\right|^{2})]\\}^{1/2}$, and
the state $|\theta,\phi\rangle$, where $\phi=\pi/2$ and
$\theta=2\arctan(\left|\alpha\right|/\sqrt{N})$, is the spin coherent state
that is obtained by rotating the ground state of the ensemble by an angle
$\theta$ about the axis $\left(\sin\phi,-\cos\phi,0\right)$ of the collective
Bloch sphere. For a large ensemble, we can apply the bosonic approximation,
which maps the collective spin of the ensemble to a quantum harmonic
oscillator. Under this approximation, the spin coherent states
$|\theta,\phi\rangle$ and $|\theta,\phi+\pi\rangle$ become bosonic coherent
states $|\alpha\rangle$ and $|-\alpha\rangle$, respectively, with coherent
amplitudes $\pm\alpha$. The atomic cat states in Eq. (S1) likewise become

$\mbox{$|\mathcal{C}_{\pm}\rangle$}=\mathcal{A}_{\pm}\left(\mbox{$|\alpha\rangle$}\pm\mbox{$|-\alpha\rangle$}\right).$
(S2)

Furthermore, the intracavity photonic cat states refer to

$\mbox{$|\mathcal{C}_{\pm}\rangle$}_{\rm
ph}=\mathcal{A}_{\pm}\left(\mbox{$|\alpha\rangle$}_{\rm
ph}\pm\mbox{$|-\alpha\rangle$}_{\rm ph}\right),$ (S3)

where $\mbox{$|\pm\alpha\rangle$}_{\rm ph}$ are the photonic coherent states
with coherent amplitudes $\pm\alpha$. It is seen, from Eqs. (S2) and (S3),
that $\left|\alpha\right|^{2}$ is the average number of excited atoms or
photons and, thus, can characterize the cat size.

In Table 1, we list some parameters of intracavity photonic cat states
$\mbox{$|\mathcal{C}_{\pm}\rangle$}_{\rm ph}$ implemented in experiments. For
comparison, we also show the corresponding results of our atomic cat states
$|\mathcal{C}_{\pm}\rangle$ at the end of the table. With modest parameters
the lifetime of our atomic cat states is predicted to be longer, by up to
_four orders of magnitude_ , compared to those photonic cat states under the
same parameter conditions. For a modest single-photon loss rate of
$\kappa_{s}/2\pi=10$ kHz (i.e., a cavity decay time of $T_{c}\sim 16$ $\mu$s),
the lifetime of our atomic cat states can reach $\sim 20$ ms for a cat size of
$\left|\alpha\right|^{2}=4$. This lifetime is comparable in length to that
($\sim 17$ ms) reported in Ref. Deleglise  _et al._ (2008) in Table 1, which,
to our best knowledge, is the longest lifetime of intracavity photonic cat
states to date. We stress that in such a comparison our cat state lifetime is
achieved with a modest cavity decay time of $T_{c}\sim 16$ $\mu$s. This is in
stark contrast to the cat state lifetime reported in Ref. Deleglise  _et al._
(2008), which was achieved with an extreme cavity decay time of $T_{c}=0.13$
sec. This means that our approach can stabilize (_for an extremely long time_)
large-size cat states, even with common setups.

When decreasing the single-photon loss rate $\kappa_{s}$, i.e., increasing the
cavity decay time $T_{c}$, our atomic cat state lifetime can further increase.
For example, a single-photon loss rate $\kappa_{s}/2\pi=3.0\times 10^{-2}$
kHz, corresponding to a cavity decay time $T_{c}\sim 5.3$ ms, results in a cat
state lifetime of $\sim 2$ sec, _more than two orders of magnitude longer than
the lifetime_ , i.e., $17$ ms, reported in Ref. Deleglise  _et al._ (2008) in
Table 1. Ultimately, the maximum value of our cat state lifetime is determined
by extremely weak spin relaxation and thermal noise, reaching $\sim 3$ sec.

The essential reason for such an improvement in the cat state lifetime is
because, as shown in Fig. S1, single excitation loss of ensembles (i.e., spin
relaxation) is extremely weak compared to that of cavities (i.e., single-
photon loss). At the same time, spin dephasing, though stronger than photon
dephasing, is greatly suppressed by the engineered two-atom decay. This is in
close analogy to the mechanism of using two-photon loss to suppress photon
dephasing.

Figure S1: Comparison of the effects of noise on intracavity photonic cat
stats and our atomic cat states. Solid arrows represent the strong effects,
and dashed arrows represent the extremely weak or strongly suppressible
effects. While the lifetime of photonic cat states is $\sim 10$ $\mu$s, our
atomic cat states can have a $\sim 0.1$ sec lifetime.

###  I.2 S2. Elimination of the second-order effect

The time-averaged Hamiltonian $H_{\rm avg}$ in Eq. (3) in the main article
describes a third-order process, and there exists a stronger second-order
process, which is described by the Hamiltonian

$\displaystyle
H^{\left(2\right)}=-\frac{g^{2}}{\Delta}\left(2a_{s}^{{\dagger}}a_{s}S_{z}+S_{+}S_{-}\right)-\frac{J^{2}}{\Delta^{\prime}}\left(2a_{p}^{{\dagger}}a_{p}-a_{s}^{{\dagger}}a_{s}^{{\dagger}}a_{s}a_{s}+4a^{{\dagger}}_{p}a_{p}a_{s}^{{\dagger}}a_{s}\right),$
(S4)

where $\Delta^{\prime}=2\omega_{s}-\omega_{p}$. In order to make the third-
order $H_{\rm avg}$ dominant, we need to eliminate the second-order
$H^{\left(2\right)}$. Since the signal cavity is initialized in the vacuum
state, the Hamiltonian $H^{\left(2\right)}$ is thus reduced to

$H^{\left(2\right)}=-\frac{g^{2}}{\Delta}S_{+}S_{-}-\frac{2J^{2}}{\Delta^{\prime}}a_{p}^{{\dagger}}a_{p}.$
(S5)

We further focus our attention on the low-excitation regime, where the average
number of excited atoms is much smaller than the total number of atoms. In
this regime, the operator $S_{z}$ can be expressed as
$S_{z}=-N/2+\delta\\!S_{z}$, where $\delta\\!S_{z}$ is a small fluctuation. As
a result, we find

$S_{+}S_{-}\approx N\delta\\!S_{z},$ (S6)

according to the identity $N\left(N/2+1\right)/2=S_{z}^{2}-S_{z}+S_{+}S_{-}$,
and then obtain

$H^{\left(2\right)}=-\frac{g^{2}_{\rm
col}}{\Delta}\delta\\!S_{z}-\frac{2J^{2}}{\Delta^{\prime}}a_{p}^{{\dagger}}a_{p}.$
(S7)

It is seen that the second-order process causes a Lamb shift (i.e., the first
term), and a dispersive resonance shift for the pump cavity (i.e., the second
term). These additional shifts can be compensated by properly detuning the
pump cavity resonance $\omega_{p}$ from twice the atomic resonance
$\omega_{q}$. Hence, the second-order process can be strongly suppressed, such
that the third-order process becomes dominant.

###  I.3 S3. Purcell single-atom decay induced by single-photon loss of the
signal cavity

Since the signal cavity is largely detuned from both the ensemble and the pump
cavity, the average number of photons inside the signal cavity is thus very
low. In this case, we can only consider the vacuum state $|0\rangle$ and the
single-photon state $|1\rangle$ of the signal cavity. We work within the limit
where $\delta_{s}\approx\Delta\gg\left\\{\delta_{p},\delta_{q},g_{\rm
col},J\right\\}$, and the Hamiltonian in Eq. (1) in the main article can thus
be rewritten as $H=H_{e}+H_{g}+V+V^{\dagger}$. Here,

$\displaystyle H_{e}=$
$\displaystyle\;\delta_{s}\mbox{$|1\rangle$}\mbox{$\langle 1|$},$ (S8)
$\displaystyle H_{g}=$
$\displaystyle\;\delta_{p}a^{{\dagger}}_{p}a_{p}+\delta_{q}S_{z}+\Omega\left(a_{p}+a^{{\dagger}}_{p}\right),$
(S9)

represents the interactions inside the excited- and ground-state subspaces,
and

$V=gS_{-}\mbox{$|1\rangle$}\mbox{$\langle 0|$}$ (S10)

describes the perturbative interaction between the excited- and ground-state
subspaces. Then, according to the formalism of Ref. Reiter and Sørensen
(2012), we can define a non-Hermitian Hamiltonian $H_{\rm
NH}^{e}=H_{e}-i\kappa_{s}\mbox{$|1\rangle$}\mbox{$\langle 1|$}/2$, and obtain
an effective Lindblad dissipator for the ensemble

$\kappa_{s}\mathcal{L}\left[\mbox{$|0\rangle$}_{s}\mbox{$\langle
1|$}\left(H_{\rm NH}^{e}\right)^{-1}V\right]\rho_{\rm ens}=\frac{\kappa_{\rm
1at}}{N}\mathcal{L}\left(S_{-}\right)\rho_{\rm ens},$ (S11)

where

$\kappa_{\rm 1at}=\frac{\kappa_{s}g^{2}_{\rm
col}}{\delta_{s}^{2}+\kappa_{s}^{2}/4}\approx\left(\frac{g_{\rm
col}}{\Delta}\right)^{2}\kappa_{s}.$ (S12)

This means that the single-photon loss process of the signal cavity gives rise
to the single-atom decay of the ensemble. Importantly, the resulting decay
rate $\kappa_{\rm 1at}$ is smaller than the cavity decay rate $\kappa_{s}$ by
a factor of $\left(g_{\rm col}/\Delta\right)^{2}$. Thus, our atomic cat states
have an extremely long lifetime.

###  I.4 S4. Quantum Monte-Carlo trajectory for the initial states
$|00\rangle$$|3\rangle$ and $|00\rangle$$|4\rangle$

Figure S2: Quantum Monte-Carlo trajectory pictured through the probabilities
of the system being in the states $|m_{p}0\rangle$$|n\rangle$ for the initial
states (a, b) $|00\rangle$$|3\rangle$ and (c, d) $|00\rangle$$|4\rangle$. A
single quantum jump $a_{p}$ gives rise to the two-atom decay in the ensemble.
In all plots, we used the full Hamiltonian $H$ in Eq. (1) in the main article,
and set $N=100$, $J=3g_{\rm col}$, $\delta_{p}=J^{2}/20g_{\rm col}$, and
$\kappa_{p}=0.2\chi$. In order to show more clearly the quantum jump
responsible for the two-atom decay, we further set $\kappa_{s}=\Omega=0$.

The dynamics described by the time-averaged $H_{\rm avg}$ in Eq. (3) of the
main article implies that pairs of atoms can jointly convert their excitations
into pump single photons, and then the subsequent single-photon loss process
of the pump cavity results in the simultaneous decay of two atoms, i.e., the
two-atom decay.

In Fig. S2, we plot single quantum trajectories, utilizing the quantum Monte
Carlo method, for the initial states $|00\rangle$$|3\rangle$ and
$|00\rangle$$|4\rangle$. Here, the first ket $|m_{p}m_{s}\rangle$
($m_{p},m_{s}=0,1,2,\ldots$) in the pair refers to the cavity state with
$m_{p}$ pump photons and $m_{s}$ signal photons, and the second $|n\rangle$
($n=0,1,2,\ldots$) refers to the collective spin state
$|S=N/2,m_{z}=-N/2+n\rangle$, corresponding to $n$ excited atoms in the
ensemble.

For the former case, where initially the ensemble has three excited atoms, we
find from Figs. S2(a, b) that two excited atoms, as a pair, decay via a
single-photon loss process of the DPA pump (corresponding to a quantum jump),
and one excited atom is kept in the ensemble because alone it cannot emit a
single photon. If there are initially four excited atoms as shown in Figs.
S2(c, d), all excited atoms, as two pairs, can decay sequentially via two
single-photon loss processes of the DPA pump (corresponding to two quantum
jumps).

###  I.5 S5. Stabilized atomic cat states by the two-atom decay

In this section we show a detailed derivation of atomic cat states stabilized
by the engineered two-atom decay. We begin with the effective master equation
given in Eq. (4) of the main text

$\dot{\rho}_{\rm ens}=\;i\left[\rho_{\rm ens},H_{\rm
ens}\right]+\frac{\kappa_{\rm 1at}}{N}\mathcal{L}\left(S_{-}\right)\rho_{\rm
ens}+\frac{\kappa_{\rm 2at}}{N^{2}}\mathcal{L}\left(S_{-}^{2}\right)\rho_{\rm
ens},$ (S13)

Here,

$\displaystyle H_{\rm ens}=\;$ $\displaystyle\frac{i}{N}\chi_{\rm
2at}\left(S_{-}^{2}-S^{2}_{+}\right),$ (S14) $\displaystyle\chi_{\rm 2at}=\;$
$\displaystyle\frac{2\Omega\chi}{\kappa_{p}},$ (S15) $\displaystyle\kappa_{\rm
1at}=\;$ $\displaystyle\left(\frac{g_{\rm col}}{\Delta}\right)^{2}\kappa_{s},$
(S16) $\displaystyle\kappa_{\rm 2at}=\;$
$\displaystyle\frac{4\chi^{2}}{\kappa_{p}}.$ (S17)

To proceed, we assume that $\kappa_{\rm 1at}=0$, such that the single-atom
decay induced by the signal cavity is subtracted. Then, we obtain in the
steady state

$\left(S_{-}^{2}-N\alpha^{2}\right)\mbox{$|D\rangle$}\mbox{$\langle
D|$}S_{+}^{2}-S_{+}^{2}\left(S_{-}^{2}-N\alpha^{2}\right)\mbox{$|D\rangle$}\mbox{$\langle
D|$}+{\rm H.c.}=0,$ (S18)

where $|D\rangle$ is the dark state of the ensemble, and

$\alpha=i\sqrt{2\chi_{\rm 2at}/\kappa_{\rm 2at}}=i\sqrt{\Omega/\chi}.$ (S19)

This indicates that the dark state $|D\rangle$ satisfies

$\left(S_{-}^{2}-N\alpha^{2}\right)\mbox{$|D\rangle$}=0.$ (S20)

We now express $|D\rangle$, in terms of the eigenstates
$|S=N/2,m_{z}=-N/2+n\rangle$ of the collective spin operator $S_{z}$, as

$\mbox{$|D\rangle$}=\sum_{n}c_{n}\mbox{$|n\rangle$},$ (S21)

where, for simplicity, we have defined
$\mbox{$|n\rangle$}\equiv\mbox{$|S=N/2,m_{z}=-N/2+n\rangle$}$. Here, $n$
refers to the number of excited atoms in the ensemble. The condition in Eq.
(S20) gives two recursion relations as follows

$\displaystyle c_{2n+k}=\;\frac{\varepsilon^{n}}{\sqrt{(2n+k)!}}c_{k},$ (S22)

where $k=0,1$. Here, we have worked within the low-excitation regime, in which
$\mbox{$\langle S_{z}\rangle$}\approx-N/2$, such that the main contributions
to the dark state $|D\rangle$ are from these components with $n\ll N$.

The recursion relation in Eq. (S22) reveals that, when the ensemble is
initially in a collective spin state $|n\rangle$ with an even $n$, e.g., in
the ground state $|0\rangle$ (i.e., a spin coherent state with all atoms in
the ground state), the dark state $|D\rangle$ can be expressed as,

$\displaystyle\mbox{$|D\rangle$}_{\rm
even}=\frac{1}{\sqrt{\cosh\left|\alpha\right|^{2}}}\sum_{n}\frac{\alpha^{2n}}{\sqrt{\left(2n\right)!}}\mbox{$|2n\rangle$}.$
(S23)

Similarly, when the ensemble is initially in a collective spin state
$|n\rangle$ with an odd $n$, e.g., in the single-excitation state $|1\rangle$
(i.e., a state with only one atom is excited), the dark state $|D\rangle$
becomes

$\displaystyle\mbox{$|D\rangle$}_{\rm
odd}=\frac{1}{\sqrt{\sinh\left|\alpha\right|^{2}}}\sum_{n}\frac{\alpha^{2n+1}}{\sqrt{\left(2n+1\right)!}}\mbox{$|2n+1\rangle$}.$
(S24)

On the other hand, the spin coherent state $|\theta,\phi\rangle$ is defined as

$\mbox{$|\theta,\phi\rangle$}=R\left(\theta,\phi\right)\mbox{$|0\rangle$}.$
(S25)

Here,

$R\left(\theta,\phi\right)=\exp\left(\tau
S_{+}\right)\exp\left[\ln\left(1+|\tau|^{2}\right)S_{z}\right]\exp\left(-\tau^{*}S_{-}\right),$
(S26)

is a rotation operator with
$\tau=\exp\left(i\phi\right)\tan\left(\theta/2\right)$. In the low-excitation
limit, $S_{+}^{n}\mbox{$|0\rangle$}\approx\sqrt{n!N^{n}}\mbox{$|n\rangle$}$,
and then

$\mbox{$|\theta,\phi\rangle$}\approx\exp\left(-N\left|\tau\right|^{2}/2\right)\sum_{n}\frac{\left(\sqrt{N}\tau\right)^{n}}{\sqrt{n!}}\mbox{$|n\rangle$}.$
(S27)

By setting $\sqrt{N}\tau=\alpha$, we further have

$\displaystyle\mbox{$|D\rangle$}_{\rm
even,odd}=\mathcal{A}_{\pm}\left(\mbox{$|\theta,\phi\rangle$}\pm\mbox{$|\theta,\phi+\pi\rangle$}\right)=\mbox{$|\mathcal{C}_{\pm}\rangle$},$
(S28)

where
$\mathcal{A}_{\pm}=1/\\{2[1\pm\exp(-2\left|\alpha\right|^{2})]\\}^{1/2}$. This
is what we have already given in Eq. (S1).

We now consider the case when the atomic ensemble is initialized in a spin
coherent state $|\theta_{0},\phi_{0}\rangle$. In this case, the atomic
ensemble evolves into a subspace spanned by the cat states
$\left\\{\mbox{$|\mathcal{C}_{+}\rangle$},\mbox{$|\mathcal{C}_{-}\rangle$}\right\\}$
and, thus, its steady state is

$\rho_{\rm ens}^{\rm
ss}=c_{++}\mbox{$|\mathcal{C}_{+}\rangle$}\mbox{$\langle\mathcal{C}_{+}|$}+c_{--}\mbox{$|\mathcal{C}_{-}\rangle$}\mbox{$\langle\mathcal{C}_{-}|$}+c_{+-}\mbox{$|\mathcal{C}_{+}\rangle$}\mbox{$\langle\mathcal{C}_{-}|$}+c_{+-}^{*}\mbox{$|\mathcal{C}_{-}\rangle$}\mbox{$\langle\mathcal{C}_{+}|$}.$
(S29)

To obtain the amplitudes $c_{++}$, $c_{--}$, and $c_{+-}$, we follow the
method in Refs. Albert and Jiang (2014); Mirrahimi  _et al._ (2014), and after
straightforward calculations, find that

$\displaystyle c_{++}=$
$\displaystyle\frac{1}{2}\left[1+\exp\left(-2\left|\alpha_{0}\right|^{2}\right)\right],$
(S30) $\displaystyle c_{--}=$ $\displaystyle
1-c_{++}=\frac{1}{2}\left[1-\exp\left(-2\left|\alpha_{0}\right|^{2}\right)\right],$
(S31) $\displaystyle c_{+-}=$
$\displaystyle-\frac{\alpha_{0}^{*}\left|\alpha\right|\exp\left(-\left|\alpha_{0}\right|^{2}\right)}{\sqrt{2\sinh\left(2\left|\alpha\right|^{2}\right)}}\int_{0}^{\pi}d\varphi
I_{0}\left(\left|\alpha^{2}-\alpha_{0}^{2}\exp\left(i2\varphi\right)\right|\right)\exp\left(-i\varphi\right),$
(S32)

where
$\alpha_{0}=\sqrt{N}\exp\left(i\phi_{0}\right)\tan\left(\theta_{0}/2\right)$,
and
$I_{0}\left(\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\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.

The above results show that the ensemble states are steered into a 2D quantum
manifold spanned by the cat states $|\mathcal{C}_{+}\rangle$ and
$|\mathcal{C}_{-}\rangle$. In typical atomic ensembles, spin relaxation is
extremely weak, such that the dominant noise source is spin dephasing.
However, the engineered two-atom decay can protect the cat states of the
quantum manifold against spin dephasing. As a result, these cat states have a
very long lifetime even with modest parameters, and thus, can be used for
fundamental studies of quantum physics. Moreover, this atomic-cat-state
manifold stabilized by the two-atom decay could also be used to encode logical
qubits (i.e., cat qubits) for fault-tolerant quantum computation, as an
alternative to the photonic-cat-state manifold stabilized by two-photon loss
Mirrahimi  _et al._ (2014).

###  I.6 S6. Strongly suppressed spin dephasing

Figure S3: Dicke space for an ensemble consisting of $N$ identical two-level
atoms or spins. The full space can be separated into the superradiant subspace
of total spin $S=N/2$ and the subradiant subspace of total spin $S<N/2$. The
blue solid double-headed arrow represents the two-atom excitation ($\chi_{\rm
2at}$), and the dashed arrows represent the dissipative processes, including
single-atom decay ($\kappa_{\rm 1at}$), two-atom decay ($\kappa_{\rm 2at}$),
collective dephasing ($\gamma_{\rm col}$), local dephasing ($\gamma_{\rm
loc}$), and inhomogeneous broadening ($\Delta_{\rm inh}$). The two-atom decay
and excitation only act inside the superradiant subspace, and thus the
resulting cat states are stored inside this subspace. While collective
dephasing does not couple the superradiant subspace to the subradiant
subspace, local dephasing and inhomogeneous broadening couple these subspaces.

In this section, we discuss the strong suppression of spin dephasing of atomic
ensembles by the engineered two-atom decay. In general, the ensemble dephasing
noise can be classified into three different types, i.e., collective spin
dephasing, local spin dephasing, and inhomogeneous broadening. Below we show
that as long as the rate $\gamma$ of convergence of cat states
($\gamma>|\alpha|^{2}\kappa_{\rm 2at}$) is much stronger than the collective
dephasing rate $\gamma_{\rm col}$, the local dephasing rate $\gamma_{\rm
loc}$, and the inhomogeneous linewidth $\Delta_{\rm inh}$, the engineered two-
atom decay is capable of suppressing all of these dephasing processes. Here,
the rate $\gamma$ describes how rapidly the steady cat states can be reached.
As a result, steady cat states can be achieved with high fidelity.

To proceed, we note that our atomic cat states are stored in the superradiant
subspace, rather than in the subradiant subspace. Here, the superradiant
(subradiant) subspace refers to the manifold of total spin $S=N/2$ ($S<N/2$),
as shown in Fig. S3.

###  I.7 A. Collective spin dephasing

We first consider collective spin dephasing, which can be described with the
Lindblad dissipator,

$\gamma_{\rm col}\mathcal{L}(S_{z})\rho_{\rm ens}=\gamma_{\rm
col}\left(S_{z}\rho_{\rm ens}S_{z}-\frac{1}{2}S_{z}S_{z}\rho_{\rm
ens}-\frac{1}{2}\rho_{\rm ens}S_{z}S_{z}\right).$ (S33)

It arises when the atoms or spins of the ensemble are simultaneously coupled
to a common bath. For example, the coupling to the collective phonon modes of
the diamond can lead to collective dephasing for NV spin ensembles Prasanna
Venkatesh  _et al._ (2018). Such a dephasing process does not couple the
superradiant to subradiant subspace as shown in Fig. S3, and as a result, the
excitation-number parity of the superradiant subspace is conserved. Thus,
collective spin dephasing can be suppressed by the two-atom decay, as long as
the condition

$|\alpha|^{2}\kappa_{\rm 2at}\gg\gamma_{\rm col}$ (S34)

is satisfied. This dissipative suppression can be better understood from the
quantum-jump approach. The jump operator $S_{z}$, when acting, e.g., on the
state $|\mathcal{C}_{+}\rangle$, excites a state

$\mbox{$|\psi\rangle$}=\mathcal{A}_{+}\left[R\left(\theta,\phi\right)-R\left(\theta,\phi+\pi\right)\right]\mbox{$|1\rangle$},$
(S35)

according to

$S_{z}\mbox{$|\mathcal{C}_{+}\rangle$}=\left(-\frac{N}{2}+\left|\alpha\right|^{2}\right)\mbox{$|\mathcal{C}_{+}\rangle$}+\alpha\mbox{$|\psi\rangle$},$
(S36)

where $\mathcal{A}_{+}=1/\\{2[1+\exp(-2\left|\alpha\right|^{2})]\\}^{1/2}$,
and $R\left(\theta,\phi\right)$ is defined in Eq. (S26). It is seen that the
state $|\psi\rangle$ still has even parity, and thus can be autonomously
driven back to the target state $|\mathcal{C}_{+}\rangle$ by the two-atom
decay. As shown in Fig. S4, a steady cat state is achieved in the presence of
collective spin dephasing.

Figure S4: Effects of collective spin dephasing on the preparation error
$\eta$ of the state $|\mathcal{C}_{+}\rangle$ of size $|\alpha|^{2}=2$. We
integrated the effective master equation (4) in the main article, with an
additional collective spin dephasing $\gamma_{\rm
col}\sum_{j=1}^{N}\mathcal{L}(\sigma_{j}^{z})\rho_{\rm ens}$. For simplicity,
we set $\kappa_{\rm 1at}=0$, so that only the effects of collective spin
dephasing are shown. Other parameters are: $N=10$, $J=3g_{\rm col}$,
$\delta_{p}=J^{2}/(20g_{\rm col})$, $\kappa_{p}=5\chi$, and
$\kappa_{s}=0.3\kappa_{p}$.

###  I.8 B. Local spin dephasing

We now consider local spin dephasing, described by the Lindblad dissipator

$\gamma_{\rm loc}\sum_{j=1}^{N}\mathcal{L}\left(\sigma_{j}^{z}\right)\rho_{\rm
ens}=\gamma_{\rm loc}\sum_{j=1}^{N}(\sigma_{j}^{z}\rho_{\rm
ens}\sigma_{j}^{z}-\rho_{\rm ens}).$ (S37)

The quantum jump, $\sigma_{j}^{z}$, when acting on the superradiant state
$\mbox{$|n\rangle$}\equiv\mbox{$|S=N/2,m_{z}=-N/2+n\rangle$}$, results in a
superposition of the state $|n\rangle$ with a subradiant state
$\mbox{$|n\rangle$}^{\perp}_{j}$. This indicates a dissipative coupling of the
superradiant to subradiant subspace, as shown in Fig. S3, yielding

$\sigma_{j}^{z}\mbox{$|n\rangle$}=\left(1-\frac{2n}{N}\right)\mbox{$|n\rangle$}-2\sqrt{\frac{n}{N}}\mbox{$|n\rangle$}^{\perp}_{j}.$
(S38)

Here, the subradiant state $\mbox{$|n\rangle$}^{\perp}_{j}$ is orthogonal to
the superradiant state $|n\rangle$ and has the same magnetic quantum number
$m_{z}$ as $|n\rangle$.

Figure S5: Effects of local spin dephasing on the preparation error $\eta$ of
the state $|\mathcal{C}_{+}\rangle$ of size $|\alpha|^{2}=2$. We integrated
the effective master equation (4) in the main article, with an additional
local spin dephasing $\gamma_{\rm
loc}\sum_{j=1}^{N}\mathcal{L}(\sigma_{j}^{z})\rho_{\rm ens}$. For simplicity,
we set $\kappa_{\rm 1at}=0$, so that only the effects of local spin dephasing
are shown. Other parameters are set the same as in Fig. S4.

As an example, we consider the action of the quantum jump $\sigma_{j}^{z}$ on
the even cat state $|\mathcal{C}_{+}\rangle$. Note that similar results hold
for the odd cat state $|\mathcal{C}_{-}\rangle$. According to Eq. (S38), we
obtain

$\sigma_{j}^{z}\mbox{$|\mathcal{C}_{+}\rangle$}=\sum_{n}c_{2n}\sigma_{j}^{z}\mbox{$|2n\rangle$}=\sum_{n}c_{2n}\left(1-\frac{4n}{N}\right)\mbox{$|2n\rangle$}-2\sum_{n}c_{2n}\sqrt{\frac{2n}{N}}\mbox{$|2n\rangle$}^{\perp}_{j}.$
(S39)

It is seen that the quantum jump $\sigma_{j}^{z}$ distorts the cat state
$|\mathcal{C}_{+}\rangle$, but conserves the excitation-number parity of the
superradiant subspace, although it carries some information about the cat
state $|\mathcal{C}_{+}\rangle$ away from the superradiant to subradiant
subspace. Thus as long as

$|\alpha|^{2}\kappa_{\rm 2at}\gg\gamma_{\rm loc},$ (S40)

the two-atom decay and excitation, which act only inside the superradiant
subspace (see Fig. S3), can autonomously steer the dephasing-distorted cat
state [i.e., the superradiant component
$\sum_{n}c_{2n}\left(1-4n/N\right)\mbox{$|2n\rangle$}$] back to the target
state $|\mathcal{C}_{+}\rangle$. This indicates that, as confirmed in Fig. S5,
local spin dephasing can be strongly suppressed and, consequently, that a
steady cat state can be achieved in the superradiant subspace.

###  I.9 C. Inhomogeneous broadening

Let us now consider inhomogeneous broadening of the ensemble. Its detrimental
effects can, in principle, be completely canceled by spin-echo pulses Hahn
(1950). For ensembles of ultracold atoms Hattermann  _et al._ (2017), these
detrimental effects can also be minimized through spin self-rephasing
collisions, even without the need for spin-echo pulse sequences Deutsch  _et
al._ (2010); Kleine Büning  _et al._ (2011).

Figure S6: Effects of inhomogeneous broadening on the preparation error $\eta$
of the state $|\mathcal{C}_{+}\rangle$ of size $|\alpha|^{2}=2$. We integrated
the effective master equation (4) in the main article, with an additional
inhomogeneous broadening $H_{\rm
inh}=\frac{1}{2}\sum_{j=1}^{N}\delta_{j}\sigma_{j}^{z}$. For simplicity, we
set $\kappa_{\rm 1at}=0$, so that only the effects of inhomogeneous broadening
are shown. The frequency shifts $\delta_{j}$ are randomly given according to a
Lorentzian distribution of linewidth $\Delta_{\rm inh}$. Other parameters are
set the same as in Fig. S4.

The Hamiltonian modeling inhomogeneous broadening is given by

$H_{\rm inh}=\frac{1}{2}\sum_{j=1}^{N}\delta_{j}\sigma_{j}^{z},$ (S41)

where $\delta_{j}=\omega_{j}-\omega_{q}$. Here, $\omega_{j}$ is the transition
frequency of the $j$th qubit spin, and $\omega_{q}$ can be viewed as the
average of transition frequencies of all the qubit spins. Under the time
evolution, each constituent of the symmetric superradiant state $|n\rangle$
acquires a random phase originating from inhomogeneous broadening. As a
result, the superradiant state $|n\rangle$ is coupled to a subradiant state as
shown in Fig. S3, thus destroying the cat states.

Nevertheless, according to the action of the operator $\sigma_{j}^{z}$ on the
cat state $|\mathcal{C}_{+}\rangle$, as given in Eq. (S39), inhomogeneous
broadening conserves the excitation-number parity of the superradiant
subspace. Thus, the two-atom decay can strongly suppress inhomogeneous
broadening when

$|\alpha|^{2}\kappa_{\rm 2at}\gg\Delta_{\rm inh}.$ (S42)

To confirm the suppression of inhomogeneous broadening, we perform numerical
simulations, as shown in Fig. S6. Inhomogeneous broadening is assumed to be,
as an example, the Lorentzian distribution with a width $\Delta_{\rm inh}$,
and similar results hold for other spectra. It is seen from Fig. S6 that a cat
state is stabilized in the presence of inhomogeneous broadening, as expected.

###  I.10 D. Total effects of collective dephasing, local dephasing, and
inhomogeneous broadening

In Fig. S7, we show the total effects of collective dephasing, local
dephasing, and inhomogeneous broadening on the superposition, $\rho_{\rm
ens}^{\rm ss}$, of the even and odd cat states, as a supplement to Fig. 3(a)
in the main article which shows the case of the state
$|\mathcal{C}_{+}\rangle$. As expected, the steady 2D cat-state manifold can
be obtained, even when these three sources of dephasing noise are present
simultaneously.

Note that in Fig. S7, the preparation error $\eta$, especially for the
$\kappa_{\rm 2at}=10\gamma_{\rm deph}$ case, is limited by the small number
$N$ which is chosen for the convenience of numerical simulations. Here, we
have assumed that $\gamma_{\rm deph}\equiv\gamma_{\rm col}=\gamma_{\rm
loc}=\Delta_{\rm inh}$. A larger $N$ leads to a smaller $\eta$, until the
bosonic approximation is valid well, i.e., until the collective behavior of
the ensemble can be well approximated by a harmonic oscillator. A similar
increase in $\eta$ can also be observed in Figs. S4, S5, and S6.

Figure S7: Total effects of collective dephasing, local dephasing, and
inhomogeneous broadening on the preparation error $\eta$ of the state
$\rho_{\rm ens}^{\rm ss}$ of size $|\alpha|^{2}=2$. We integrated the
effective master equation (4) in the main article, with an additional spin
dephasing $\gamma_{\rm col}\mathcal{L}(S_{z})\rho_{\rm ens}$, local spin
dephasing $\gamma_{\rm loc}\sum_{j=1}^{N}\mathcal{L}(\sigma_{j}^{z})\rho_{\rm
ens}$, and inhomogeneous broadening
$\frac{1}{2}\sum_{j=1}^{N}\delta_{j}\sigma_{j}^{z}$. The frequency shifts
$\delta_{j}$ are randomly given according to a Lorentzian distribution of
linewidth $\Delta_{\rm inh}$. For simplicity, we set $\gamma_{\rm
col}=\gamma_{\rm loc}=\Delta_{\rm inh}\equiv\gamma_{\rm deph}$, and
$\kappa_{\rm 1at}=0$, so that only the effects of these dephasing processes
are shown. Other parameters are set the same as in Fig. S4.

###  I.11 S7. Inhomogeneous broadening in nitrogen-vacancy center ensembles

In Sec. S6, we have discussed three types of dephasing noise for our model.
Different types of atomic or spin ensembles have different dephasing
mechanisms. Below, we take ensembles of nitrogen-vacancy (NV) center electron
spins in diamond, as an example, to discuss the source of dephasing. In these
systems, the source of dephasing is inhomogeneous broadening of the NV
transition.

The electronic ground state of NV centers is a spin triplet, which has
$m_{s}=0$ and $\pm 1$ sublevels. We use $|0\rangle$ and $|\pm 1\rangle$ to
label these three sublevels. The zero-field splitting between the states
$|0\rangle$ and $|\pm 1\rangle$ is $\sim 2.87$ GHz. In the presence of a
static field, the degenerate states $|\pm 1\rangle$ are split with a Zeeman
splitting $\Delta_{\rm zm}$. In order to encode a two-level atom or qubit
here, we assume that the state $|0\rangle$ is used as the ground state and the
state $|+1\rangle$ as the excited state. Inhomogeneous broadening of the spin
transition can be described by the Hamiltonian in Eq. (S41), which for
convenience is recalled here

$H_{\rm inh}=\frac{1}{2}\sum_{j=1}^{N}\delta_{j}\sigma_{j}^{z},$ (S43)

where
$\sigma_{j}^{z}=\mbox{$|+1\rangle$}_{j}\mbox{$\langle+1|$}-\mbox{$|0\rangle$}_{j}\mbox{$\langle
0|$}$, and $\delta_{j}=\omega_{j}-\omega_{q}$. Here, $\omega_{j}$ is the
transition frequency of the $j$th qubit spin, and $\omega_{q}$ can be viewed
as the average of transition frequencies of all the qubit spins.

In general, inhomogeneous broadening of NV ensembles originates from the
interactions of the NV centers with (A) the local strain field, (B) the 13C
and 14N nuclear spins, and (C) the P1 centers. Thus, the frequency shift
$\delta_{j}$ can be separated into three parts, i.e.,

$\delta_{j}=\delta_{j}^{\rm str}+\delta_{j}^{\rm nuc}+\delta_{j}^{\rm P1},$
(S44)

which includes contributions from the strain field ($\delta_{j}^{\rm str}$),
the 13C and 14N nuclear spins ($\delta_{j}^{\rm nuc}$), and the P1 centers
($\delta_{j}^{\rm P1}$).

###  I.12 A. Local strain

The local strain field breaks the $C_{3v}$ symmetry of the NV center, and as a
result shifts the frequency of the states $|\pm 1\rangle$. The NV electronic
spin is coupled to the local strain field via the Hamiltonian Doherty  _et
al._ (2013)

$\displaystyle H_{\rm strain}=\;$ $\displaystyle d_{\parallel}\mathcal{E}_{\rm
str}^{z}\mathcal{S}_{z}^{2}+d_{\perp}\mathcal{E}_{\rm
str}^{x}\left(\mathcal{S}_{y}^{2}-\mathcal{S}_{x}^{2}\right)+d_{\perp}\mathcal{E}_{\rm
str}^{y}\left(\mathcal{S}_{x}\mathcal{S}_{y}+\mathcal{S}_{y}\mathcal{S}_{x}\right)$
$\displaystyle=\;$
$\displaystyle\Pi_{z}(\mbox{$|+1\rangle$}\mbox{$\langle+1|$}+\mbox{$|-1\rangle$}\mbox{$\langle-1|$})+\left(\Pi_{\perp}\mbox{$|+1\rangle$}\mbox{$\langle-1|$}+{\rm
H.c.}\right),$ (S45)

where $\Pi_{z}=d_{\parallel}\mathcal{E}_{\rm str}^{z}$,
$\Pi_{\perp}=-d_{\perp}\left(\mathcal{E}_{\rm str}^{x}+i\mathcal{E}_{\rm
str}^{y}\right)$, and
$\vec{\mathcal{S}}=(\mathcal{S}_{x},\mathcal{S}_{y},\mathcal{S}_{z})$ is the
NV spin operator. Here, $\vec{\mathcal{E}}_{\rm str}=(\mathcal{E}_{\rm
str}^{x},\mathcal{E}_{\rm str}^{y},\mathcal{E}_{\rm str}^{z})$ represents the
strain field, and $d_{\parallel}\sim 2\pi\times 0.35$ Hz cm/V, $d_{\perp}\sim
2\pi\times 17$ Hz cm/V are the axial and non-axial components of the ground-
state electric dipole moment. The first term in Eq. (I.12) corresponds to the
frequency shifts of the states $|\pm 1\rangle$, and the second term describes
their coupling. Due to the Zeeman splitting $\Delta_{\rm zm}$, the coupling
between the states $|\pm 1\rangle$ becomes largely detuned. As a result, the
transition frequency of the qubit spin (i.e., the transition
$\mbox{$|0\rangle$}\rightarrow\mbox{$|+1\rangle$}$) is shifted by

$\delta^{\rm str}=\Pi_{z}+\frac{\left|\Pi_{\perp}\right|^{2}}{\Delta_{\rm
zm}}.$ (S46)

For a realistic parameter $|\Pi_{\perp}|=2\pi\times 5$ MHz Kubo  _et al._
(2010), and a common Zeeman splitting $\Delta_{\rm zm}=2\pi\times 100$ MHz, an
estimate of $\delta^{\rm str}$ is therefore given by $\delta^{\rm str}\sim
2\pi\times 0.3$ MHz.

###  I.13 B. Nuclear spins

Natural diamond samples consist of $\sim 98.9\%$ spinless 12C atoms and $\sim
1.1\%$ 13C isotopes of nuclear spin $\mathcal{I}_{\rm C}=1/2$. These 13C atoms
are randomly distributed in the diamond lattice. Moreover, the 14N atoms
constituting the NV centers each have a nuclear spin $\mathcal{I}_{\rm N}=1$.
The NV centers are coupled to these 14N and 13C nuclear spins through
hyperfine interactions, given by

$H_{\rm nuc}=\vec{\mathcal{S}}\cdot\mathbb{A}_{\rm
N}\cdot\vec{\mathcal{I}}_{N}+\vec{\mathcal{S}}\cdot\sum_{j}\mathbb{A}_{\rm
C}\cdot\vec{\mathcal{I}}_{\rm C}^{j},$ (S47)

where $\vec{\mathcal{I}}_{N}$ and $\vec{\mathcal{I}}_{\rm C}^{j}$ are the spin
operators for the 14N atom and the $j$th 13C atom, respectively, while
$\mathbb{A}_{\rm N}$ and $\mathbb{A}_{\rm C}^{j}$ are the corresponding
hyperfine interaction tensors. Working under the secular approximation, i.e.,
neglecting the $\mathcal{S}_{x}$ and $\mathcal{S}_{y}$ terms, the Hamiltonian
$H_{\rm nuc}$ becomes approximated by $H_{\rm nuc}\approx\delta^{\rm
nuc}\mathcal{S}_{z}$ Dréau  _et al._ (2012); Zou  _et al._ (2014), with

$\delta^{\rm nuc}=\mathbb{A}_{\rm N}m_{N}+\sum_{j}\mathbb{A}_{\rm C}^{j}m_{\rm
C}^{j},$ (S48)

where $m_{\rm N}=0$, $\pm 1$ and $m_{\rm C}^{j}=\pm 1/2$ are magnetic quantum
numbers.

The coupling to the 14N nuclear spin splits the state $|+1\rangle$ (or
$|-1\rangle$) into three hyperfine sublevels, equally spaced by
$\mathbb{A}_{\rm N}\sim 2\pi\times 2.16$ MHz. This results in a linewidth
broadening $\sim 2\pi\times 4.3$ MHz.

The 13C hyperfine splitting depends on the positions of the 13C nuclear spins
relative to the NV center. According to the studies in Ref. Zhao  _et al._
(2012), the coherence time induced by the 13C hyperfine coupling is $\sim 2$
$\mu$s, implying a linewidth broadening of $\sim 2\pi\times 80$ kHz. If
12C-enriched methane is used as a carbon source to prepare the diamond samples
Grezes  _et al._ (2015), then the concentration of 13C nuclear spins (and as a
result the corresponding linewidth broadening) can be significantly reduced.

###  I.14 C. P1 centers

In diamond samples, single substitutional nitrogen atoms (so-called P1
centers), which were not converted into the NV centers, are the main
paramagnetic impurities and each of them has an unpaired electron. These
inevitable impurities form an electron spin bath, and the NV center is coupled
to it through the dipole-dipole interaction, which is described by the
Hamiltonian Taylor  _et al._ (2008):

$\displaystyle H_{\rm
P1}=\sum_{j}\frac{\mu_{0}g_{s}^{2}\mu_{B}^{2}}{4\pi\left|\vec{r}_{j}\right|^{3}}\mathcal{S}_{z}\left[\vec{n}_{z}-3\left(\vec{n}_{z}\cdot\vec{n}_{j}\right)\vec{n}_{j}\right]\cdot\vec{\mathcal{S}}_{j},$
(S49)

where $\vec{\mathcal{S}}_{j}$ is the bath spin located at position
$\vec{r}_{j}$, and $\vec{n}_{j}=\vec{r}_{j}/|\vec{r}_{j}|$. In most
experiments implementing the strong coupling of a high-density NV ensemble to
a superconducting resonator Kubo  _et al._ (2010); Amsüss  _et al._ (2011);
Kubo  _et al._ (2011, 2012); Grezes  _et al._ (2014); Putz  _et al._ (2014);
Astner  _et al._ (2017), the residual P1 centers are the main source of
decoherence of the NV ensemble, and a typical linewidth broadening is
$\delta^{\rm P1}\sim 2\pi\times 7$ MHz Kubo  _et al._ (2010, 2011); Putz  _et
al._ (2014).

A solution to reduce the inhomogeneous linewidth induced by the P1 centers is
to improve the efficient conversion of the P1 centers to the NV centers. The
inhomogeneous linewidth would therefore be dominated by the hyperfine
interaction with the 14N nuclear spin [i.e., the first term on the right-hand
side of Eq. (S48)]. That is, the inhomogeneous linewidth would be limited to
$\sim 2\pi\times 4.3$ MHz, as experimentally reported in Refs. Grezes  _et
al._ (2015); Angerer  _et al._ (2018).

###  I.15 D. Short summary

The detrimental effects of inhomogeneous broadening mentioned above are
reversible and can in principle be completely eliminated by spin-echo
techniques or dynamical decoupling pulse sequences. The residual inhomogeneous
broadening can be further suppressed by the engineered two-atom decay in our
proposal (see Sec. S6). Note that although we discuss the ensembles of NV
spins, our model is generic and can be implemented with other types of
ensembles, e.g., ensembles of trapped ultracold atoms. Inhomogeneous
broadening of ultracold-atom ensembles, which arises mainly due to the
trapping potential and the atomic interactions Hattermann  _et al._ (2017),
can be strongly reduced through spin self-rephasing collisions without the use
of spin-echo or dynamical decoupling pulses Deutsch  _et al._ (2010); Kleine
Büning  _et al._ (2011).

###  I.16 S8. Spin relaxation, thermal noise, and the maximum cat state
lifetime

In the main article, we discussed the effects of spin dephasing, and also
showed that it can be strongly suppressed by the engineered two-atom decay. In
this section, let us consider the effects of spin relaxation and thermal
noise, and also the maximum cat state lifetime limited by them. Here, we
proceed with the bosonic approximation. Such an approximation maps the spin
coherent states $|\theta,\phi\rangle$ and $|\theta,\phi+\pi\rangle$ to the
bosonic coherent states $|\pm\alpha\rangle$, respectively. Correspondingly,
the cat states
$\mbox{$|\mathcal{C}_{\pm}\rangle$}=\mathcal{A}_{\pm}\left(\mbox{$|\theta,\phi\rangle$}\pm\mbox{$|\theta,\phi+\pi\rangle$}\right)$
become
$\mbox{$|\mathcal{C}_{\pm}\rangle$}=\mathcal{A}_{\pm}\left(\mbox{$|\alpha\rangle$}\pm\mbox{$|-\alpha\rangle$}\right)$,
as given in Eq. (S2).

Spin relaxation and thermal noise can be described by the Lindblad
dissipators, $\gamma_{\rm relax}(n_{\rm th}+1)\mathcal{L}\left(b\right)\rho$
and $\gamma_{\rm relax}n_{\rm th}\mathcal{L}\left(b^{\dagger}\right)\rho$.
Here, $\gamma_{\rm relax}$ is the spin relaxation rate, and $n_{\rm
th}=\left[\exp\left(\hbar\omega_{q}/k_{B}T\right)-1\right]^{-1}$ is the
thermal average boson number at temperature $T$. The Purcell decay rate of the
cat state coherence, which is induced by single-photon loss of the signal
cavity, is given by

$\Gamma_{\rm 1at}=2\left|\alpha\right|^{2}\kappa_{\rm 1at},$ (S50)

with $\kappa_{\rm 1at}=\left(g_{\rm col}/\Delta\right)^{2}\kappa_{s}$ as given
in Eq. (S12). At the same time, for a thermal background at $T\neq 0$, an
additional decay rate of the cat state coherence, which is induced by spin
relaxation and thermal noise, is given by Kim and Bužek (1992)

$\Gamma_{\rm relax}=\left[2\left|\alpha\right|^{2}\left(1+2n_{\rm
th}\right)+2n_{\rm th}\right]\gamma_{\rm relax}.$ (S51)

By assuming realistic parameters $\omega_{q}=2\pi\times 3$ GHz, $T=100$ mK,
$\left|\alpha\right|^{2}=4$, and $\gamma_{\rm relax}=2\pi\times 4$ mHz Amsüss
_et al._ (2011); Grezes  _et al._ (2014), we have $\Gamma_{\rm relax}\approx
2\pi\times 54$ mHz, much smaller the decay rate, $\Gamma_{\rm 1at}\approx
2\pi\times 8.0$ Hz, which is obtained with $\kappa_{s}=2\pi\times 10$ kHz and
$\Delta/g_{\rm col}=100$. This means that the effects of both spin relaxation
and thermal noise on the cat states $|\mathcal{C}_{\pm}\rangle$ can be safely
neglected. In this case, the lifetime of these cat states is determined only
by the Purcell single-atom decay rate $\kappa_{\rm 1at}$, and is given by

$\tau_{\rm at}=\Gamma_{\rm 1at}^{-1}=\left(\frac{\Delta}{g_{\rm
col}}\right)^{2}\frac{1}{2\left|\alpha\right|^{2}\kappa_{s}}.$ (S52)

On the other hand, the intracavity photonic cat states
$\mbox{$|\mathcal{C}_{\pm}\rangle$}_{\rm ph}$ in Eq. (S3) mainly suffer from
single-photon loss, e.g, with a rate $\kappa_{s}$, and thus their lifetime is
given by Haroche and Raimond (2006),

$\tau_{\rm ph}=\frac{1}{2\left|\alpha\right|^{2}\kappa_{s}}.$ (S53)

It is found from Eqs. (S52) and (S53) that $\tau_{\rm at}$ is longer than
$\tau_{\rm ph}$ by a factor of $\left(\Delta/g_{\rm col}\right)^{2}$, i.e.,

$\frac{\tau_{\rm at}}{\tau_{\rm ph}}=\left(\frac{\Delta}{g_{\rm
col}}\right)^{2}.$ (S54)

According to the analysis in the main article, the factor $\left(\Delta/g_{\rm
col}\right)^{2}$ can be tuned to be $\sim 10^{4}$ under modest parameters.
This indicates that _the lifetime of our atomic cat states is longer than that
of intracavity photonic cat states by up to four orders of magnitude for cat
sizes of $\left|\alpha\right|^{2}\geq 4$._

In fact, the decoherence rate $\Gamma_{\rm 1at}$ can be further decreased with
the smaller single-photon loss rate $\kappa_{s}$ (i.e., the longer $T_{c}$).
This results in a longer cat state lifetime. When $\Gamma_{\rm 1at}$ is
comparable to or even smaller than $\Gamma_{\rm relax}$, the lifetime
$\tau_{\rm at}$ is given by

$\tau_{\rm at}=\left(\Gamma_{\rm 1at}+\Gamma_{\rm relax}\right)^{-1}.$ (S55)

For a single-photon loss rate of $\kappa_{s}/2\pi=30$ Hz, we have $\Gamma_{\rm
1at}=2\pi\times 24$ mH, which is smaller than $\Gamma_{\rm relax}\sim
2\pi\times 54$ mHz. In this case, Eq. (S55) gives a cat state lifetime of
$\tau_{\rm at}\sim 2$ sec. Ultimately, when decreasing the rate $\kappa_{s}$,
the lifetime $\tau_{\rm at}$ increases to its maximum value,

$\tau_{\rm at}^{\rm max}=\Gamma_{\rm relax}^{-1}.$ (S56)

Using the parameters given above, we can predict a maximum lifetime of
$\tau_{\rm at}^{\rm max}\sim 3$ sec.

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