# Preparing and Analyzing Solitons in the sine-Gordon Model with Quantum Gas
Microscopes

Elisabeth Wybo  Technical University of Munich, TUM School of Natural
Sciences, Physics Department, 85748 Garching, Germany  Munich Center for
Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München,
Germany  Alvise Bastianello  Technical University of Munich, TUM School of
Natural Sciences, Physics Department, 85748 Garching, Germany  Munich Center
for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München,
Germany  Monika Aidelsburger  Munich Center for Quantum Science and Technology
(MCQST), Schellingstr. 4, 80799 München, Germany  Fakultät für Physik, Ludwig-
Maximilians-Universität, 80799 Munich, Germany  Immanuel Bloch  Munich Center
for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München,
Germany  Fakultät für Physik, Ludwig-Maximilians-Universität, 80799 Munich,
Germany  Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
Michael Knap  Technical University of Munich, TUM School of Natural Sciences,
Physics Department, 85748 Garching, Germany  Munich Center for Quantum Science
and Technology (MCQST), Schellingstr. 4, 80799 München, Germany

###### Abstract

The sine-Gordon model emerges as a low-energy theory in a plethora of quantum
many-body systems. Here, we theoretically investigate tunnel-coupled Bose-
Hubbard chains with strong repulsive interactions as a realization of the
sine-Gordon model deep in the quantum regime. We propose protocols for quantum
gas microscopes of ultracold atoms to prepare and analyze solitons, that are
the fundamental topological excitations of the emergent sine-Gordon theory.
With numerical simulations based on matrix product states we characterize the
preparation and detection protocols and discuss the experimental requirements.

##  I Introduction

Universality forms one of the pillars for the classification of quantum phases
of matter. Upon coarse-graining, microscopic details become irrelevant and
only symmetry and topology determine the essential properties. Seemingly
different looking microscopic systems can then be described by the same set of
collective degrees of freedom that are captured by the same emergent effective
field theory. A prominent example is the relativistic sine-Gordon field theory
[1, 2] which emerges as the low-energy description of various physical
systems, including among others the massive Thirring model [1, 3], the Coulomb
gas [4], spin chains [5, 6, 7], bosonic and fermionic Hubbard models [8, 9],
and circuit quantum electrodynamics [10, 11]; and thus is of central interest
for a multifaceted community.

In the strongly interacting regime, the sine-Gordon model possesses a complex
quasiparticle spectrum, consisting of solitons, that are massive topological
excitations, and breathers, that are bound states of these solitons.
Furthermore, the sine-Gordon model is a renowned example of an integrable
field theory [12, 13], which implies infinitely long-lived quasiparticles,
unconventional relaxation dynamics [14], and unconventional transport [15,
16]. Elegant analytical methods can be used to obtain exact results which have
improved our understanding of the model [12, 13, 17, 18, 19, 20, 21, 22, 23].
Despite these advances from integrability, analytical predictions for
correlation functions and the full-counting statistics of many observables,
both in equilibrium and out of equilibrium, are very difficult to access.
Moreover, it is pertinent to understand in which dynamical regimes the
effective sine-Gordon field theory is realized in one of the microsocopic
models described above [24]. This can lead to the development of highly-
tunable sine-Gordon quantum simulators, that are for instance based on
ultracold atoms in optical lattices, Rydberg atoms, or trapped ions.

Pioneering theoretical work [25, 26] motivated the experimental realization of
the sine-Gordon model with tunnel-coupled, one-dimensional quasi-condensates
[27, 28, 29]. In these atom chip implementations, the sine-Gordon mass scale
can be tuned and correlation functions can be efficiently characterized by
matter-wave interferometry [30, 31, 32, 27, 33, 34], but the interactions in
the one-dimensional gases have been restricted to be rather weak. As a
consequence, the emergent sine-Gordon field theory is approximately
semiclasscial [27, 35, 36]. From this perspective, tunnel-coupled optical
lattice systems can be very convenient: they allow the kinetic energy to be
quenched and, hence, the effective interactions to be enhanced. Although in a
previous theoretical work [24] we have shown that the sine-Gordon model
describes the low-energy physics of two coupled spin chains very well,
identifying an experimental realization of such a setup and developing the
measurement protocols to extract the information of the emergent field theory
remain unaddressed challenges that motivate our present work.

Figure 1: Creation of solitons in coupled Bose-Hubbard chains. (a) A soliton
can be created by applying a chemical-potential gradient between the two
chains in one half of the system
$\sum_{j}\Delta_{j}(\tau)(n_{j\uparrow}-n_{j\downarrow})$, as indicated by the
dashed region. Ramping up the tunnel coupling $t_{\perp}$ simultaneously,
which decreases the barrier on the rung, enables us to further stabilize the
phase imprinting process. (b) The spatial profile of the applied chemical
potential gradient $\Delta_{j}$ for the preparation of one (1S) and two (2S)
solitons. (c) The time dependence of the chemical potential difference
$\Delta_{j}$ and tunnel coupling $t_{\perp}$. (d,e) The resulting profiles of
the mean local current $\langle J_{j}\rangle$ suggest the creation of one (d)
and two (e) solitons, respectively. The time-evolution is performed with the
TEBD algorithm for bond dimension $\chi=1000$ and $L=128$ rungs.

Here, we show how solitons, which are the fundamental excitations of the sine-
Gordon model, can be created in a setting of ultracold bosons trapped in an
optical lattice [see Fig. 1(a)]. The solitons are most directly observed when
measuring the current between the chains by first applying beam splitter
operations and then taking snapshots with a quantum gas microscope. Using
similar beam-splitter operations, we furthermore show how the full counting
statistics of the topological charge, that counts the number of excited
solitons, can be obtained. We use numerical simulations based on matrix
product states, to assess the experimental requirements of the proposed
protocols.

The paper is organized as follows: in Section II we briefly introduce the
sine-Gordon model and its realization using coupled one-dimensional Bose-
Hubbard chains. The creation of a soliton is discussed in Section III. In
Section IV we present a detection method to capture the topological charge of
solitons. The requirements for the experimental preparation of a low-energy
state of the sine-Gordon field theory is discussed in Section V. Conclusions
are presented in Section VI and the appendixes contain a couple of technical
details.

##  II Tunnel-coupled Bose-Hubbard chains

The sine-Gordon model emerges as the low-energy description of two tunnel-
coupled, interacting, one-dimensional systems. Here, we consider two identical
Bose-Hubbard chains with Hamiltonians $H^{\text{BH}}_{\uparrow}$ and
$H^{\text{BH}}_{\downarrow}$, coupled with a tunneling term $H_{\perp}$. The
dynamics is therefore set by the total Hamiltonian
$H=H^{\text{BH}}_{\uparrow}+H^{\text{BH}}_{\downarrow}+H_{\perp}$. Here,

$H_{\alpha}^{\text{BH}}=-t\sum_{j=0}^{L-2}(b^{{\dagger}}_{j\alpha}b_{j+1\alpha}+\mathrm{h.c.})+\frac{U}{2}\sum_{j=0}^{L-1}n_{j\alpha}(n_{j\alpha}-1)\,,$
(1)

where $b_{j\alpha}$ ($b_{j\alpha}^{\dagger}$) annihilates (creates) a boson on
rung $j$ in chain $\alpha=\\{\uparrow,\downarrow\\}$ and
$n_{j\alpha}=b^{\dagger}_{j\alpha}b_{j\alpha}$ is the density. The tunnel
coupling Hamiltonian reads

$H_{\perp}=-t_{\perp}\sum_{j=0}^{L-1}(b^{{\dagger}}_{j\uparrow}b_{j\downarrow}+\mathrm{h.c.}).$
(2)

We use open boundary conditions throughout our work. The derivation of the
low-energy description of the coupled wires closely follows the original
proposal [25]. We consider each of the chains to be at non-integer filling to
realize a superfluid state for arbitrary values of the hopping $t$ and the
interaction strength $U$.

In the absence of inter-chain coupling $t_{\perp}=0$, the low-energy behavior
is obtained by bosonization. One first introduces the phase field
$\phi_{\alpha}(x)$, and its conjugate field $\Pi_{\alpha}(x)$. These two
fields are related to the microscopic operators as
$b^{\dagger}_{j,\alpha}\simeq\gamma e^{i\phi_{\alpha}(x)}$ and as
$n_{j,\alpha}\simeq n+\Pi_{\alpha}(x)$, respectively, where $\gamma$ is a non-
universal prefactor and $n$ is the average density. Within bosonization, the
two one-dimensional chains are governed by the Luttinger-Liquid Hamiltonian

$H^{\mathrm{LL}}_{\alpha}=\int\differential x\frac{\hbar
v_{s}}{2}\left(\frac{\pi}{K}\Pi_{\alpha}^{2}(x)+\frac{K}{\pi}(\partial_{x}\phi_{\alpha}(x))^{2}\right)\,,$
(3)

where, $K$ is the Luttinger parameter and $v_{s}$ is the sound velocity. The
Luttinger parameter is larger than one, $K\geq 1$, for repulsive interactions
$U>0$ [9], and approaches $K=1$ in the limit of infinite repulsive
interactions $U/t\to\infty$. For weak tunneling $t_{\perp}\ll t$, the
interchain coupling can be reintroduced perturbatively [25]. Upon bosonizing
the transverse Hamiltonian reads as
$H_{\perp}=-2|\gamma|^{2}t_{\perp}\int\differential
x\cos(\phi_{\uparrow}-\phi_{\downarrow})$. Performing a rotation to symmetric
$\psi=\phi_{\uparrow}+\phi_{\downarrow}$ and antisymmetric
$\phi=\phi_{\uparrow}-\phi_{\downarrow}$ degrees of freedom, results in the
explicit decoupling of the two sectors. The symmetric combination is governed
by a gapless Luttinger Liquid, and the sine-Gordon field theory (4) emerges
for the relative degree of freedom $\phi$:

$H^{\text{SG}}=\int\differential x\,\frac{\hbar
v_{s}}{2}\left(\frac{2\pi}{K}\Pi^{2}+\frac{K}{2\pi}(\partial_{x}\phi)^{2}\right)-2t_{\perp}|\gamma|^{2}\cos\phi\,.$
(4)

The integrability of the sine-Gordon model allows for an exact determination
of its particle content, together with an analytical expression for the two-
body scattering matrix [13]. The fundamental excitations are topological
solitons and anti-solitons, for which the phase field $\phi$ interpolates
between the degenerate minima of the $\cos\phi-$interaction. Therefore, the
phase field $\phi$ winds up by $2\pi$ when traversing a soliton. These
quasiparticles have a relativistic dispersion law
$E_{s}(k)=\sqrt{v_{s}^{4}M^{2}+v_{s}^{2}k^{2}}$. The soliton mass $M$ has a
complicated dependence on the bare parameters of the Hamiltonian [37], but
scales with the transverse hopping $t_{\perp}$ in a rather simple manner
$M\propto t_{\perp}^{\frac{2K}{4K-1}}$.

Depending on the interaction $K$, solitons and anti-solitons can form bound
states of infinite lifetime, which are called breathers. The interaction $K$
sets the number of breathers in the excitation spectrum, $N=\lfloor
4K-1\rfloor$, and their mass as
$M_{B_{n}}=2M\sin\left(\frac{\pi}{2}\frac{n}{4K-1}\right)$. Crucially, when
$K$ is increased the relative mass difference between two consecutive
breathers is reduced: in the limit of large $K$, or weak interactions, the
breather’s mass spectrum collapses to a continuum and the quantum model is
well-approximated by the classical sine-Gordon theory [35, 38, 39]. In
contrast, the deep quantum regime of the field theory is realized when only
few breathers are present in the spectrum. In tunnel-coupled Bose-Hubbard
chains strong interactions can be achieved by quenching the kinetic energy of
the atoms. Thereby, Luttinger parameters $K$ of order one, are reachable when
considering non-integer fillings, which avoids the Mott insulating regimes. To
this end, we will focus in our numerical simulations presented below on the
regime of infinitely strong interactions, corresponding to $K=1$. Deviations
from this regime, will lead to changes in the microscopic parameters of the
field theory. However, the qualitative behavior will remain. In the infinitely
repulsive regime, the Bose-Hubbard ladder maps to coupled Heisenberg chains.
In such spin chains the regime of validity of the sine-Gordon description has
been thoroughly analyzed in Ref. [24].

##  III Controlled soliton imprinting

Creating, manipulating, and detecting the fundamental excitations of an
emergent sine-Gordon field theory is a challenge. One possibility is acting on
the microscopic model with an inhomogeneous perturbation such that the ground
state of the emergent field theory hosts initially localized solitons.
Proposals for achieving this are based on Raman-coupled quasi-condensates
[40], and motivated further theoretical analysis of the propagation of
solitons [41, 42]. By contrast, for the experimental realization we are
proposing, the goal is to prepare the ground state of the field theory, and
subsequently to dynamically imprint the soliton.

In this section, we propose the protocol for imprinting a soliton on the exact
ground state of the two tunnel-coupled Bose-Hubbard chains. Protocols for
approximately realizing the ground state are discussed in Section V. Our
strategy aims at carefully tuning the spatial profile of the relative phase
$\phi(x)$ such that it undergoes a $2\pi-$phase slip when traversing the
system. In order to avoid a massive creation of excitations during the
preparation, smooth and slow parameter changes are paramount.

We begin by introducing a space-time dependent chemical-potential gradient
between the two chains $H\rightarrow
H+\sum_{j}\Delta_{j}(\tau)(n_{j\uparrow}-n_{j\downarrow})$ that induces a
relative phase drift between the two halves of the system [29]. This is best
seen from bosonizing the density gradient
$(n_{j\uparrow}-n_{j\downarrow})\to\Pi(x)$. The sine-Gordon Hamiltonian thus
gets an additional term $H^{\text{SG}}\to H^{\text{SG}}+\int\differential
x\,\Delta(\tau,x)\Pi(x)$ leading to a deformed equation of motion
$\partial_{t}\phi=\frac{v_{s}\pi}{K}\Pi+\Delta(\tau,x)$, where
$\Delta(\tau,x)$ denotes the potential imbalance in the continuum limit. The
gradient $\Delta(\tau,x)$ hence acts as a source for the accumulation of the
relative phase $\phi$. When winding up the phase in half of the system, while
leaving the other untouched, a relative phase difference of $2\pi$ can be
achieved. This configuration is then stabilized by the cosine potential of the
relative phase in the sine-Gordon model (4). At this point, the potential
imbalance needs to be switched off in the preparation protocol. To prevent the
phase to slip into the next minimum or to strongly oscillate, we
simultaneously deepen the potential barrier by increasing the inter chain
hopping $t_{\perp}$, which helps pinning down the phase to the desired value.
The result of our protocol is the creation of a localized wavepacket of
excitations, primarily containing the desired number of solitons. The
wavepacket is approximately contained within the regions where the chemical
potential gradient $\partial_{x}\Delta(\tau,x)$ is not zero. A graphical
summary of the soliton creation protocol can be found in Fig. 1 (b-c).

Within bosonization the rung current is proportional to the sine of the
relative phase: $J_{j}\equiv-
ib^{\dagger}_{j\uparrow}b_{j\downarrow}+ib^{\dagger}_{j\downarrow}b_{j\uparrow}\simeq
2|\gamma|^{2}\sin\phi$. We can thus optimize the parameters of the protocol by
measuring the current between the two rungs and requiring a smooth sine-shaped
profile without additional oscillations [see Fig. 1 (d,e)]. The data is
obtained numerically using the time-evolving block decimation (TEBD) algorithm
[43] for two coupled Bose-Hubbard chains of length $L=128$ at filling $n=1/8$
and Luttinger parameter $K=1$. In Appendix A, we show the current profiles for
different filling fractions. Depending on the strength of the chemical-
potential gradient, we can also control the number of zero-crossings in the
current profile. Therefore, our data suggests that a quantized number of
solitons can be created in the quantum limit of the sine-Gordon theory in a
controlled way using this protocol.

We furthermore numerically compute the energy of the state before and after
the soliton imprinting. For the data shown in Fig. 1 (d-e), we find that the
additional energy is $\Delta_{1S}=0.50t$ and $\Delta_{2S}=0.92t$. This should
be compared to the cost of creating a soliton measured by the energy gap. For
$K=1$, one has that the dispersion relations of the soliton and the first
breather are equal, $E_{B_{1}}=E_{S}$. Hence, we can extract the energy of the
soliton by measuring the energy of the lowest breather in the spectral
function [24]. From that we estimate $E_{S}(k=0)=(0.24\pm 0.04)t$. Therefore,
slightly more energy is pumped into the system by the preparation protocol,
which is reasonable as there will also be excitations of the gapless symmetric
modes and furthermore slightly-dispersing wave packets of the solitons are
created. Nonetheless, it would be desirable, to have access to other more
direct observables to characterize the solitons.

One approach could be to track the phase from individual snapshots of the
current and locate phase jumps therein. This strategy has been successfully
applied deep in the semiclassical limit, characterized by a large Luttinger
parameter $K$ for coupled quasi-condensates [27], where solitons have a large
spatial extent and can therefore be directly imaged. In the setting considered
here, however, the local current can only take three values $+,0,-$ depending
on the configuration of the hardcore bosons on a single rung. Even in the case
of finite Hubbard interactions, the local current is still quantized to
integer values.

Therefore, coarse-graining over spatially extended regions has to be
performed. It is crucial that averaging on scales larger than the soliton
itself has to be avoided, because then the phase slip cannot be resolved
either. We estimate the size of the soliton using a classical soliton profile
that at rest obeys $\ell\partial_{x}\phi=\sqrt{1-\cos\phi}$, with $\ell=2\hbar
K/(\pi v_{s}M)$. For $K=1$ and $n=1/8$, we obtain $Mv_{s}^{2}\simeq 0.25t$,
giving $\ell\simeq 2$ lattice sites. From these estimates, it can be deduced
that the phase profile of the soliton cannot be directly extracted, as coarse-
graining is required to resolve the phase, but the soliton itself is very
localized in space. This is a direct consequence of being deep in the quantum
regime of the sine-Gordon model.

To confirm this picture we perform numerically projective measurements of the
current: upon coarse-graining all phase slips are averaged out rapidly, see
Appendix B. Due to the small size of the classical soliton one may question
the validity of the field theory description. However, it should be emphasized
that the classical estimate is only a rough qualitative indicator. Indeed, a
careful quantitative study of coupled spin-chains [24] shows that the sine-
Gordon model is indeed a faithful description of strongly-interacting ladder
systems. As an alternative to characterize and count the solitons we propose
instead to investigate the topological charge.

##  IV Detecting solitons from the topological charge

Among the infinite set of conservation laws of the sine-Gordon model, that
arise from integrability [44, 12], is the topological charge, which is an
integral over a total derivative,

$Q=\int\frac{\differential x}{2\pi}\,\partial_{x}\phi(x)\,.$ (5)

Using the canonical commutation relations $[\phi(x),\Pi(y)]=i\delta(x-y)$ one
can readily obtain $[Q,H^{\text{SG}}]=0$. The topological charge is quantized
in integer units: each soliton contributes $+1$, anti-soliton $-1$, and
breathers, which are bound states of a soliton and an anti-soliton, do not
contribute at all. For our analysis it will be useful to introduce the
accumulated topological charge $Q(x)=\int^{x}\frac{\differential
y}{2\pi}\partial_{y}\phi$, which is only conserved (and quantized) as
$x\to\infty$. The accumulated topological charge has the advantage that it is
automatically coarse-grained over a large portion of the system.

Our goal is now to find a microscopic realization of the accumulated
topological charge. To this end, it is useful to introduce the local current
$J_{j}=(-ib^{{\dagger}}_{j\uparrow}b_{j\downarrow}+\mathrm{h.c.})$ and energy
$E_{j}=(b^{{\dagger}}_{j\uparrow}b_{j\downarrow}+\mathrm{h.c.})$ operators
acting on a single rung. Using bosonization, the topological charge density
can be identified as the most relevant contribution to the following combined
observable

$E_{j}J_{j+1}-J_{j-1}E_{j+2}\simeq 8|\tilde{\gamma}|^{4}\partial_{x}\phi\,.$
(6)

The energy operators are measured on even rungs and currents on odd ones and
$\tilde{\gamma}\simeq\gamma$; a possible difference between the two may arise
due to UV field theory renormalization. The actual value of $\tilde{\gamma}$
is however unimportant for us. With this identification, we can readily
introduce a lattice version of the accumulated topological charge as

$Q_{j}\equiv\sum_{i\text{ even}}^{j}(E_{i}J_{i+1}-J_{i-1}E_{i+2})\simeq
16\pi|\tilde{\gamma}|^{4}Q(x)=\\\
=16\pi|\tilde{\gamma}|^{4}[\phi(x)-\phi(0)]\,.$ (7)

We numerically evaluate this lattice regularization of the topological charge
for the state with one and two solitons, respectively, discussed in the
previous section; see Fig. 2. The accumulated topological charge captures the
phase difference between a given site in our system and the left boundary. The
chemical potential gradient $\Delta_{j}$ has been chosen to act on the right
half of the system. Therefore, the left boundary does not evolve and $\phi(0)$
is a mere constant. As a consequence, $Q_{j}$ probes the dynamics of the field
$\phi(x)$. We find that during the soliton creation process ($\tau<10\hbar/t$)
the value of the topological charge is still drifting. However, as the soliton
creation process has finished, the topological charge remains remarkably
stable in time, as demonstrated by the plateau in Fig. 2. By comparing the
one-soliton and two-soliton imprinting, respectively shown in Fig. 2(a) and
Fig. 2(b), we observe the total phase slip of the two-soliton state is about
twice the one of the one-soliton state, as expected. The stability of the
soliton is a clear signature of the emergent sine-Gordon field theory, because
the accumulated charge is just a sum of local energy and current operators
without any evident topological property. Thus the presence of the underlying
soliton is quite remarkable.

The topological charge density as defined in Eq. (6) can be directly measured
with Quantum Gas Microscopes [45]. To this end, a proper local basis rotation
has to be performed with beam-splitting operations on individual rungs of the
Bose-Hubbard ladder. The simpler operator is the current, which can be
obtained as follows [46]. After the hopping between the rungs is frozen by
strongly increasing the depth of the optical lattice, each of the individual
rungs are evolved in time for a duration of $\pi/4$. This realizes a unitary
$U_{j}^{J}=\exp(i\frac{\pi}{4}E_{j})$ which transforms the relative density to
the rung current

Figure 2: Accumulated topological charge. The accumulated topological charge
is shown for one-soliton (a) and two solitons (b) at various times (legend).
The soliton creation protocol and system size are the same as in Fig. 1 and
takes a total time of $10\hbar/t$. Thus for the earliest times shown, the
imprinting has not yet completed. Once the solitons have been created, the
total topological charge remains remarkably constant. The lattice
regularization of the topological charge is not quantized to unity, due to the
non-universal prefactor $\tilde{\gamma}$ in Eq. (7). However, the topological
charge for the two-solitons (b) is approximately twice the one of the single
soliton (a), indicating a quantization of the excitation.

$(U^{J}_{j})^{{\dagger}}(n_{j\downarrow}-n_{j\uparrow})U^{J}_{j}=J_{j}\,.$ (8)

Here, crucially, we assume the absence of double occupancies. In the case of
infinite repulsion, which we consider, this is trivially the case. However,
also in the experimentally accessible limit of strong but finite interactions
real double occupancies can be neglected, because we are focusing on the
dilute limit and strong interactions. As an alternative, arbitrary occupancies
of the double well will be allowed if one switches off the interactions during
the rotation for example by Feshbach resonances.

In a similar way, the rung energy $E_{j}$ can be transformed to the relative
density, by first applying a potential imbalance, which couples to the density
difference, for period $\pi/4$ and then applying the beam splitter operation,
leading to
$U_{j}^{E}=U_{j}^{J}\exp[i\frac{\pi}{4}(n_{j\uparrow}-n_{j\downarrow})]$. This
transformation maps the density difference to the local energy density

$(U^{E}_{j})^{{\dagger}}(n_{j\uparrow}-n_{j\downarrow})U^{E}_{j}=E_{j}\,.$ (9)

By first applying a potential gradient on every even rung and then applying
the beam-splitter operation $U^{J}$ globally, as depicted in Fig. 3(a),
projective measurements in the density basis yield the staggered string
operators ‘$J_{0}E_{1}J_{2}E_{3}J_{4}E_{5}...$’. From those the accumulated
charge $Q_{j}$ (7) is then directly obtained. By numerically sampling
snapshots after performing the beam-splitter operations [47, 48], we generate
the full counting statistics of the topological charge; see Appendix C for
details. The average over the accumulated topological charge over 8000
snapshots are shown in Fig. 3(b), where we find good agreement with the direct
measurement of the state. Moreover, we show a difference of the full
distribution function of the topological charge $Q_{L-4}$ for a state in which
solitons are prepared compared to the ground state which does not carry any
solitonic excitations; Fig. 3(c). We note that the $x$-axis of this histogram
is quantized to integer values, as $Q_{j}$ is an integer when it is taken from
a single snapshot. Deep in the quantum regime, the soliton manifests itself as
a skewness in the broad distribution of $Q_{j}$. Deep in the classical regime,
by contrast, the full distribution function of the topological charge would
exhibit a sharp peak.

Figure 3: Snapshots of the topological charge. (a) After freezing the hopping
within the chain, a density gradient is applied for a period of $\pi/4$ to
even sites, pink, followed by a global beam-splitter tunneling operation for a
time $\pi/4$. This is effectively described by the unitaries $U^{J}$ and
$U^{E}$. (b) The accumulated topological charge extracted from the snapshots
for the one- and two-soliton state prepared as in Fig. 1. The average is taken
over $\approx 8000$ snapshots. The errorbars represent the standard error of
the mean, and the grey line is the expectation value of the state. (c) The
differential histograms of the total topological charge between the soliton
states and the ground state. Deep in the quantum regime, the soliton manifests
itself in the skewness of this distribution.

##  V Ground state preparation

So far we have assumed that the ground state of the coupled chain has already
been prepared. Here, we suggest an adiabatic ground state preparation protocol
that can be implemented using ultracold atoms in optical lattices with tunable
local potentials. The finite mass gap of the sine-Gordon field theory, and the
decoupling between the symmetric and anti-symmetric sector at low energies,
enables us to prepare the sine-Gordon ground state in reasonable time. Of
course, this only remains true as long as the sine-Gordon model is a good
description of the coupled chains, which may not be always the case during the
state-preparation process.

We propose the following adiabatic ground state protocol for creating an
approximate ground state of the two tunnel-coupled Bose-Hubbard chains at
filling $n=1/2$:

  1. (i)

Initialize a single chain with unit filling while the intra-chain hopping is
switched off.

  2. (ii)

Split the potential on every site to prepare the ground state of a single
particle in a double well.

  3. (iii)

Linearly ramp up the intra-chain hopping $t$ to spread correlations through
the system.

We simulate this ground state preparation protocol numerically using matrix
product states; Fig. 4(a). To this end, we start out from the ground state of
each rung and then increase the intra-chain hopping $t$ over a duration of
$50\hbar/t$, such that the ratio of the inter- and intra-chain hopping is
$t_{\perp}/t=0.2$ at the end of the protocol. We have not attempted to
optimize these numbers too carefully. Rather this should be seen as a guide
for the approximate requirements of the adiabatic state preparation protocol.

The whole system is gapless due to the symmetric degrees of freedom. Hence a
large amount of energy is injected in the system even for slow protocols.
However, the symmetric modes are expected to quickly decouple from the sine-
Gordon Hamiltonian, which in turn has a finite mass gap. Therefore, the
antisymmetric sector remains overall close to its ground state. The total
energy it is thus not a good indicator of the quality of the sine-Gordon’s
state preparation, due to the large contribution from the symmetric modes.

In order to assess the quality of the adiabatiacally prepared state, it can
for example be probed spectroscopically, see Appendix D. We numerically find
that the low-energy excitations as measured by the spectral function are in
good agreement with the spectral function evaluated from the exact ground
state. This demonstrates that the asymmetric sector is well prepared using
this procedure.

Figure 4: Adiabatic ground state preparation. (a) The ground state preparation
protocol starts out by a one-dimensional Mott insulator with negligible
hopping, $t\approx 0$. Then the tube is split adiabatically into two,
realizing the ground state on each of the single rungs. Now correlations build
up across the system by adiabatically ramping up the intra-chain hopping $t$
over a time $50\hbar/t$, such that after completing the process
$t_{\perp}/t=0.2$. (b,c) The subsequent soliton imprinting protocol is closely
related to the one discussed previously. Before applying the chemical
potential gradient profile (b), however, we first lower $t_{\perp}/t$ from
$0.2$ to $0.02$ (c). The grey step-like function then shows the time interval
during which the relative phase is accumulated. (d) Current profiles around
the soliton imprinting stage. (e) Accumulated topological charge shown at
time-slices indicated by the grey dashed lines in (d). The lightest shade
represents the earliest time and the darkest one the latest.

In addition, we can even imprint a soliton on top of the adiabatically
prepared ground state, following our suggestion discussed earlier in Section
III; for details see Fig. 4 (b,c). The resulting current profiles are again
indicative of having a soliton imprinted (d). This is furthermore supported by
a reasonably stable accumulated topological charge (e). This demonstrates that
the adiabatically prepared ground state has the essential features of the
ground state of the emergent sine-Gordon field theory.

##  VI Conclusion & Outlook

The sine-Gordon field theory emerges as the effective description in a wealth
of physical systems. Here, we propose access to the deep quantum regime of the
sine-Gordon model by realizing tunnel coupled Bose-Hubbard chains. We discuss
adiabatic ground state preparation protocols, as well as protocols for
creating the fundamental excitations – the solitons – of the sine-Gordon
model. The soliton can be most directly seen in the inter-chain current and
the topological charge, which both can be measured with quantum gas
microscopes using beam-splitter operations.

For future work, it would be interesting to explore scattering of solitonic
wave packets. Fundamental information of the excitations and their mutual
interactions can be extracted from scattering events. Information about the
integrable field theory can thereby be obtained, in which multi-particle
scattering processes are rather peculiar [13, 12, 24]. Moreover, it would be
exciting to find ways for creating initial states with a finite density of
solitons. In such a true many-body regime thermalization dynamics [14] and the
emergent generalized hydrodynamics [49, 50, 16] of the strongly interacting
sine-Gordon field theory could be explored.

## Acknowledgements

We thank F. Essler for useful discussions. Matrix product state simulations
were performed using the TeNPy package [51]. We acknowledge support from the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy–EXC–2111–390814868 and DFG grants No.
KN1254/1-2, KN1254/2-1, the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (Grant Agreement No.
851161), as well as the Munich Quantum Valley, which is supported by the
Bavarian state government with funds from the Hightech Agenda Bayern Plus.

Data availability: Raw data and data analysis are available on Zenodo [52].

##  Appendix A Soliton profiles at different densities

We compare the soliton profiles and the accumulated topological charge
obtained for states at different densities; Fig. 5. Apart from tuning the
strength of the chemical potential, we also reduce the imprinting time with
increasing density. Concretely, the time scales for imprinting are chosen
inversely proportional to the density to compensate for the higher sound
velocity in systems with higher density. We also numerically find that for a
higher density the soliton is more dispersive; Fig. 5.

We also evaluate the accumulated topological charge, which depends
approximately linearly on the density; Fig. 5(d–f). There are small drifts of
the topological charge in time, signaling a weak breaking of the conservation
law of the total topological charge. As previously pointed out, the
conservation of the topological charge is not directly implemented in the
coupled Hubbard chain, but emerges from the effective field theory. This
furthermore confirms that the parameter regime of Fig. 2 realizes a high-
quality sine-Gordon soliton, while more important corrections beyond sine-
Gordon are present at higher densities, as shown in Fig. 5. One reason for the
worse stability is the reduced preparation time which is required due to the
increased sound velocity, while keeping the system size fixed. It would be
interesting to optimize the soliton imprinting to further reduce these
undesired effects in future work.

Figure 5: Solitons at different densities. (a-c) The current profiles when
applying the soliton creation protocol for different densities in the initial
ground state of $L=64$ rungs. (a,d) $n=1/8$, (b,e) $n=1/4$ and (c,f) $n=1/2$.
The soliton spreading is significantly faster for higher densities due to the
larger sound velocity. (d-f) The corresponding accumulated topological charge
for some time slices.

##  Appendix B Attempting to extract the phase of the soliton from current
snapshots

In this section directly analyze the snapshots of the current
$J_{j}=(-ib^{{\dagger}}_{j\uparrow}b_{j\downarrow}+\mathrm{h.c.})$ and show
that deep in the quantum regime one cannot extract the phase directly upon
coarse-graining. As we discuss in the main text, the reason is that in this
regime the soliton itself is of very small extent and hence is averaged out
upon coarse-graining. Let us describe the formal procedure here. Within
bosonization, the current operator can be written in terms of the phase field
as

$J\simeq 2|\gamma|^{2}\sin(\phi),$ (10)

where $\gamma$ is a non-universal prefactor. When taking snapshots of the
current $JJJJ\dots$, it is conceivable in principle that one can extract the
phase profile in space. The challenges are the following: first a reliable
estimate of the non-universal prefactor $\gamma$ is needed. Secondly (and more
importantly) coarse graining has to be performed to obtain the phase. A
snapshot of the current on the ladder takes the values

$N_{j\downarrow}-N_{j\uparrow}\in\\{-1,0,1\\}\,,$ (11)

where $N_{j\alpha}=0,1$ is the measurement outcome of $n_{j\alpha}$. We do not
consider higher occupancy as we are in the limit $K\rightarrow 1$ and at low
average filling. Then we coarse-grain the current over a certain number of
sites, and invert the relation (10) while flattening the grouped current with
the value of $2|\gamma|^{2}$. We extracted a value of $2|\gamma|^{2}\approx
0.7$ from the spectral function, however, the exact value does not
considerably affect the outcome of the procedure. There is always an ambiguity
in the inversion of (10), we choose to make a ‘jump’ in the phase when the
absolute value of the current hits it maximum (here taken to be 0.7) once.
Another maximum with the same sign corresponds to a jump back. In Fig. 6(a,c),
we show 20 snapshots of the coarse-grained current

$\overline{J_{j}}=\frac{1}{N}\sum_{i=0}^{N-1}(N_{i+j\downarrow}-N_{i+j\uparrow})$
(12)

over respectively $N=4$ and $N=6$ sites. In Fig. 6(b,d), we then show show the
extracted phase

$\overline{\phi_{j}}=\arcsin\left(\frac{\overline{J_{j}}}{2|\gamma|^{2}}\right),$
(13)

where $\overline{J_{j}}$ is thus flattened to $2|\gamma|^{2}$ in case
$\overline{J_{j}}>2|\gamma|^{2}$. From this it is already clear that the rare
phase jumps, completely disappear upon coarse graining already over a couple
of sites. In the random subset of 20 snapshots that are shown, there is not
even a single one that winds up a phase of $2\pi$ according to our procedure
when grouping $N=6$ sites.

Figure 6: Snapshots of the coarse-grained current. (a,c) Illustration of 20
snapshots of the coarse-grained current for respectively $N=4$ and $N=6$
grouped sites of the 1 soliton state. There are huge fluctuations in this data
because we are considering the deep quantum regime of the emergent sine-Gordon
theory. (b,d) The relative phase extracted from the snapshots of the current.
The rare jumps that are present for $N=4$ disappear when taking $N=6$. The
horizontal grey lines are drawn at $\pm\pi/2$ and $\pm 3\pi/2$. The snapshots
have been obtained from the one soliton state, presented in the main text in
Fig. 1 at time $\tau=12.5\hbar/t$.

In order to have a larger sample size of snapshots, we create a histogram of
the phase accumulation obtained in around $40000$ snapshots; Fig. 7(a,b).
These histograms are symmetric, and do not show special features. To
strengthen this point, we plot the differential histograms between the ground
state and the state containing one soliton; Fig. 7(c,d). Except for
statistical fluctuations we find that there is no difference between these
states. This illustrates that that no special features can be obtained from
the coarse grained data. Taking the estimated size of the soliton of two
lattice sites, which we give in the main text, this finding is not surprising.
It just reflects that the regime we are considering is dominated by quantum
fluctuations.

Figure 7: Histograms. (a,b) Histograms of the phase accumulation across the
system obtained from $\approx 40\ 000$ snapshots of the one soliton state for
respectively $N=4$ and $N=6$ grouped sites. (c,d) The differential histograms
comparing the soliton data to the ground state.

##  Appendix C Extracting the topological charge from snapshots

In this section, we show more in detail how to extract the soliton profile
from snapshots of the topological charge density. After performing the
rotations corresponding to $U^{J}$ and $U^{E}$, see respectively Eqs. (8) and
(9) in the main text, and taking the snapshots in that basis $JEJE\dots$, we
can reconstruct the expectation value of $\expectationvalue{J_{j}E_{j+3}}$
(for even $j$) and $\expectationvalue{E_{j}J_{j+1}}$ (for odd $j$). This is
shown in Fig. 8(a). These expectation values deviate from zero around the
position of the soliton. Taking then the difference corresponds to the
topological charge density which exhibits a bump at the position of the
soliton; Fig. 8(b). Taking the cumulative sum then reveals the accumulated
topological charge, which is quantized when traversing the full system; Fig.
8(c).

Figure 8: Snapshots for determining the topological charge. (a) The average
data extracted from the snapshots. For an odd rung index $j$, we show
$E_{i}J_{i+1}$, and for even rung index $j$, we show $J_{j}E_{i+3}$. These
expectation values deviate from zero around the position of the soliton. The
averages are taken over $N\approx 8000$ snapshots. (b) From the difference
between the two expectation values, we compute the topological charge density.
(c) The cumulative profile of it then gives the topological charge $Q_{j}$.

##  Appendix D Breather spectroscopy of the adiabatically prepared state

Here, we analyze the breather spectrum of the adiabatically prepared ground
state following the protocol of Sec. V, and compare it to the spectrum of the
exact ground state on the coupled chains. To this end, we compute the spectral
function at momentum $k=0$

$S(\omega,k=0)\sim\int\differential te^{i\omega
t}\sum_{n=0}^{L-1}\expectationvalue{O_{n}(t)O_{L/2}}{\tilde{\psi}_{0}},$ (14)

where $\ket{\tilde{\psi}_{0}}$ is the (adiabatically prepared) ground state.
The operators $O_{n}$ are defined on the rungs, to couple to the asymmetric
sector in which the sine-Gordon theory is emerging. We observe a nice
correspondence between the spectral functions and the expected breather
energies; Fig. 9. Data is shown for three different operators: (i)
$O_{i}=b^{{\dagger}}_{i\uparrow}b_{i\downarrow}$ coupling to all breathers,
(ii) $O_{i}=b^{{\dagger}}_{i\uparrow}b_{i\downarrow}+\mathrm{h.c.}$ coupling
to the even breathers only, and (iii)
$O_{i}=ib^{{\dagger}}_{i\uparrow}b_{i\downarrow}+\mathrm{h.c.}$ coupling to
the odd breathers.

Figure 9: Breather spectroscopy. The spectral function at zero momentum,
$k=0$, as defined in Eq. (14) is computed for the ground state and the
adiabatically prepared ground state at $t_{\perp}/t=0.2$. This state is
prepared as described in the main text for a system of $L=32$ rungs within a
time of $50\hbar/t$. The spectrum is computed separately for the different
parity sectors of the breather. The grey dashed lines show the location of the
exact energies of the three breathers present for $K=1$. The spectra are in
very good agreement, demonstrating that the ground state of the anti-symmetric
sector is well prepared by our procedure.

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2022

[1,2]C. Fougères [1]F. de Oliveira Santos

1]Grand Accélérateur National d’Ions Lourds (GANIL), CEA/DRF-CNRS/IN2P3, Caen,
France 2]Physics Division, Argonne National Laboratory, Lemont (IL), USA
3]Departament de Física, EEBE, Universitat Politècnica de Catalunya,
Barcelona, Spain 4]Institut d’Estudis Espacials de Catalunya, Barcelona, Spain
5]Institut Laue-Langevin, Grenoble, France 6]Center for Exotic Nuclear
Studies, Institute for Basic Science, Daejeon, Republic of Korea 7]Instituto
de Física, Universidade de São Paulo, CEP 05508-090, São Paulo, Brazil 8]CERN,
Geneva, Switzerland 9]Helmholtz-Zentrum Dresden-Rossendorf, Institute of Ion
Beam Physics and Materials Research, Dresden, Germany 10]Istituto Nazionale di
Fisica Nucleare, Sezione di Milano, Milano, Italy 11]Oliver Lodge Laboratory,
University of Liverpool, Liverpool, United Kingdom 12]Vinca Institute of
Nuclear Sciences, University of Belgrade, Belgrade, Serbia 13]Department of
Physics, KTH Royal Institute of Technology, Stockholm, Sweden 14]Institute of
Nuclear Physics, Kraków, Poland 15]Laboratoire de Physique des 2 Infinis Irène
Joliot-Curie, CNRS/IN2P3, Université Paris-Saclay, Orsay, France 16]Instituto
de Física Corpuscular, CSIC-Universidad de Valencia, Valencia, Spain
17]Institut de Physique des 2 Infinis de Lyon, CNRS/IN2P3, Université Claude
Bernard de Lyon 1, Villeurbanne, France 18]Institut für Kernphysik,
Universität zu Köln, Köln, Germany 19]Institute for Nuclear Research (ATOMKI),
Debrecen, Hungary 20]Departamento de Ingeniería Electrónica, Universitat de
Valencia, Valencia, Spain 21]Instituto de Estructura de la Materia, CSIC,
Madrid, Spain 22]Department of Physics, Faculty of Science, Ankara University,
Ankara, Turkey 23]Istituto Nazionale di Fisica Nucleare, Sezione di Padova,
Padova, Italy 24]Dipartimento di Fisica e Astronomia, Universitá degli Studi
di Padova, Padova, Italy 25]Dipartimento di Fisica, Università di Milano,
Milano, Italy 26]Nuclear Physics Institute of the Czech Academy of Sciences,
Řež, Czech Republic 27]Laboratori Nazionali di Legnaro INFN, Legnaro (PD),
Italy 28]Department of Physics and Astronomy, Uppsala University, Uppsala,
Sweden 29]Department of Physics, University of Surrey, Guildford, United
Kingdom 30]Laboratorio de Radiaciones Ionizantes, Departamento de Física
Fundamental, Universidad de Salamanca, Salamanca, Spain 31]Institut
pluridisciplinaire Hubert Curien, Université de Strasbourg, CNRS, Strasbourg,
France 32]Irfu, CEA, Université Paris-Saclay, Gif-sur-Yvette, France
33]Department of Integrated Sciences, Centro de Estudios Avanzados en Fisica,
Matematicas y Computacion (CEAFMC), University of Huelva, Huelva, Spain
34]Université de Bordeaux, CNRS/IN2P3, LP2IB, Gradignan, France 35]Horia
Hulubei National Institute for R&D in Physics and Nuclear Engineering, IFIN-HH
Bucharest, Măgurele, Romania

# Search for 22Na in novae supported by a novel method for measuring
femtosecond nuclear lifetimes

<EMAIL_ADDRESS><EMAIL_ADDRESS>J. José  C. Michelagnoli  E. Clément  Y.H.
Kim  A. Lemasson  V. Guimar$\tilde{\text{a}}$es  D. Barrientos  D. Bemmerer
G. Benzoni  A.J. Boston${}^{\text{11}}$ R. Böttger  F. Boulay  A. Bracco  I.
Celikovic  B. Cederwall  M. Ciemala  C. Delafosse  C. Domingo-Pardo  J.
Dudouet  J. Eberth  Z. Fülöp  V. González  J. Goupil  H. Hess  A. Jungclaus
A. Kaşkaş${}^{\text{22}}$ A. Korichi  S.M. Lenzi  S. Leoni  H. Li  J.
Ljungvall  A. Lopez-Martens  R. Menegazzo  D. Mengoni  B. Million  J. Mrázek
D.R. Napoli  A. Navin  J. Nyberg  Zs. Podolyak  A. Pullia  B. Quintana  D.
Ralet  N. Redon  P. Reiter  K. Rezynkina  F. Saillant  M.D. Salsac  A.M.
Sánchez-Benítez${}^{\text{33}}$ E. Sanchis  M.
Şenyi$\tilde{\text{g}}$it${}^{\text{22}}$ M. Siciliano  N.A. Smirnova  D.
Sohler  M. Stanoiu  Ch. Theisen  J.J. Valiente-Dobón  P. Ujić  M. Zielińska  [
[  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [
[  [  [  [  [  [  [  [

Classical novae are thermonuclear explosions in stellar binary systems, and
important sources of 26Al and 22Na1, 2, 3. While $\gamma$ rays from the decay
of the former radioisotope have been observed throughout the Galaxy, 22Na
remains untraceable. The half-life of 22Na (2.6 yr4) would allow the
observation of its 1.275 MeV $\gamma$-ray line from a cosmic source. However,
the prediction of such an observation requires good knowledge of the nuclear
reactions involved in the production and destruction of this nucleus5. The
22Na($p,\gamma)^{23}$Mg5 reaction remains the only source of large uncertainty
about the amount of 22Na ejected. Its rate is dominated by a single resonance
on the short-lived state at 7785.0(7) keV in 23Mg6. In the present work, a
combined analysis of particle-particle correlations and velocity-difference
profiles is proposed to measure femtosecond nuclear lifetimes. The application
of this novel method to the study of the 23Mg states, combining magnetic and
highly-segmented tracking $\gamma$-ray spectrometers, places strong limits on
the amount of 22Na produced in novae, explains its non-observation to date in
$\gamma$ rays (flux $<$ 2.5$\times$10-4 ph.cm-2s-1 7), and constrains its
detectability with future space-borne observatories.

Nuclear reactions between charged particles in astrophysical environments
proceed by quantum tunneling. The measurement of these reactions in the
laboratory is very difficult because of the very small cross sections
($\sigma~{}\lesssim$ 1 nb).  To date, about ten reactions involving
radioactive nuclei and charged particles have been directly measured at low
energies. Among them, the 22Na($p,\gamma)^{23}$Mg reaction has a direct impact
on the amount of radioactive 22Na produced in novae 8, 9, 10, 11, 6. The
astrophysical relevance of 22Na for diagnosis of nova outbursts was first
mentioned by Clayton and Hoyle3, in the context of its decay into a short-
lived excited state of 22Ne, and the subsequent emission of a $\gamma$-ray
photon of 1.275 MeV released during its de-excitation. A precise determination
of the 22Na($p,\gamma)^{23}$Mg thermonuclear rate is necessary to improve the
predicted abundances of nuclei in the mass region $A\geq 20$ during nova
outbursts, including the 22Na abundance in the ejecta that impacts in turn the
predicted 20Ne/22Ne ratios in presolar grains12 of a putative nova origin, as
well as, the corresponding $\gamma$-ray emission flux. The rate of this
reaction is mainly dictated by a single resonance in 23Mg at 7785.0(7) keV 6.
A direct measurement of its strength ($\omega\gamma$) has been performed in
three different studies with conflicting results:
$\omega\gamma=5.7_{-0.9}^{+1.6}$ meV 6, 1.4(3) meV 13 and $<0.36$ meV14.
Indirect experimental methods, such as lifetime measurement, have also been
employed to determine the strength of this resonance. It has been shown that
the ejected mass of 22Na in novae is inversely proportional to the lifetime of
the key state 6. The lifetime of this state ($\tau$) was previously measured
to be $\tau=10(3)$ fs11, consistent with the recently determined upper limit
of $12$ fs15. Yet, this is at odds with the predictions of the nuclear Shell
Model ($SM$), $\tau_{SM}\approx$ 1 fs. Here we propose an experimental method
for measuring such short lifetimes. This method has been applied to the key
state in 23Mg in order to obtain an independent measurement and, so, to reduce
systematic errors.

The experiment was performed at GANIL, France. A 24Mg beam was accelerated to
110.8(4) MeV and impinged on a 3He target of $\approx 2\times 10^{17}$
atoms/cm2 uniformly implanted up to a depth of 0.1 $\mu$m below the surface of
a 5.0(5) $\mu$m gold foil, producing the 23Mg nucleus by the 3He(24Mg,4He)23Mg
reaction. Both 24Mg and 23Mg nuclei were then stopped in a 20.0(5) $\mu$m gold
foil. About twenty states were populated in 23Mg with excitation energies
between 0 and 8 MeV. The light particles produced by the reaction were
identified and measured with the VAMOS++ magnetic spectrometer16 placed at 0∘
with respect to the beam direction, using 2 drift chambers, a plastic
scintillator and 2 small drift chambers placed at the entrance of the
spectrometer. This led to an unambiguous identification of the 4He particles.
The 4He particles were detected up to an angle in the laboratory of 10.0(5)∘
relative to the beam axis. Excitation energy ($E_{x}$), velocity at the time
of the reaction ($\beta_{reac}=v_{reac}/c$), and angle of the 23Mg nuclei
($\theta_{recoil}$), have been determined by measuring the momentum of the 4He
ejectiles, with a resolution (FWHM) of $\approx$ 500 keV, $\approx 0.0005$ and
$\approx 0.05^{\circ}$ respectively.

The excited states in 23Mg have been also clearly identified via their
$\gamma$-ray transitions measured with the AGATA $\gamma$-ray spectrometer.
The present experimental method has taken advantage of the highest angle
sensitivity of AGATA17, 18. This $\gamma$-ray spectrometer is based on
$\gamma$-ray tracking techniques in highly-segmented HPGe detectors. It
consisted of 31 crystals covering an angular range from 120∘ to 170∘, with a
total geometrical efficiency $\approx 0.6~{}\pi$. With AGATA, it is possible
to measure accurately the energy and the emission angle of the $\gamma$ rays
with a resolution of 4.4(1) keV at 7 MeV and 0.7(1)∘, respectively. The
$\gamma$ rays were observed Doppler shifted since they were emitted from a
23Mg nucleus moving at a certain velocity $\beta_{ems}=v_{ems}/c$. The
measured energy $E_{\gamma}$ is a function of the rest energy $E_{\gamma,0}$
and of the angle $\theta$ between the $\gamma$ ray and the 23Mg emitting
nucleus:
$E_{\gamma}=E_{\gamma,0}\sqrt{1-\beta^{2}_{ems}}/(1-\beta_{ems}\text{cos}(\theta))$.
With AGATA, this relativistic Doppler effect can be observed continuously as a
function of the angle, as shown in Fig. 1 for the case of the
$E_{\gamma,0}=4840.0^{+0.2}_{-0.4}$ keV $\gamma$-ray transition emitted from
the $E_{x}=5292.0(6)$ keV excited state in 23Mg moving with
$\beta_{ems}\approx 0.075$.

Figure 1: Identification of $\gamma$-ray transitions in 23Mg. Left: The energy
of the measured $\gamma$ rays is plotted as a function of the angle between
the $\gamma$ ray and the 23Mg emitter. This matrix is conditioned with the
detection of an $\alpha$ particle at $5.2<E_{x}<5.4$ MeV in the VAMOS++
magnetic spectrometer. The $\gamma$-ray transition
($E_{\gamma,0}=4840.0^{+0.2}_{-0.4}$ keV) from the $E_{x}=5292.0(6)$ keV
excited state in 23Mg is clearly observed. Its energy is Doppler shifted. The
background observed here is mostly due to random coincidences between $\gamma$
rays from the Compton background and $\alpha$ particles produced in fusion-
evaporation reactions between the beam and 12C and 16O impurities deposited on
the target. Right: Picture of the AGATA $\gamma$-ray spectrometer used to
detect the $\gamma$ rays emitted during the reaction.

Conversely, if the rest energy of the $\gamma$ ray is known, it is possible to
determine the nucleus velocity at the time of emission $\beta_{ems}$ from the
measured $E_{\gamma}$ and $\theta$ (see Methods, section _Determination of
velocities_). Their precise measurement, enabled by the combination of both
AGATA and VAMOS++, has allowed for the accurate determination of
$\beta_{ems}$, on a event-by-event basis.

In the present work, both velocities $\beta_{ems}$ and $\beta_{reac}$ were
measured simultaneously, on a event-by-event basis. The two velocities are not
identical since $\mathit{(i)}$ the 23Mg nucleus slows down in the target
before being stopped ($\approx$500 fs) and $\mathit{(ii)}$ there is a time
difference between the reaction and the $\gamma$-ray emission due to the
finite lifetime of the state. Thus, the profile of the velocity differences,
$\Delta\beta=\beta_{reac}-\beta_{ems}$, is a function of the lifetime of the
state - a longer lifetime gives a larger value of $\Delta\beta$. The technique
proposed here, for measuring femtosecond lifetimes, is based on the analysis
of velocity-difference profiles.

The results obtained in this work are shown in Fig. 2 for three different
excited states of 23Mg. On the left side, $\beta_{reac}$ is shown as a
function of $\beta_{ems}$, and on the right side, the yields of the
corresponding $\gamma$ rays as a function of the velocity difference
$\Delta\beta$.

Figure 2: Angle-integrated velocity-difference profiles.  Left: the 23Mg
velocity at the time of reaction ($\beta_{reac}$) is shown against the
velocity at the time of the $\gamma$-ray emission ($\beta_{ems}$), for three
excited states. The line corresponds to the prompt $\gamma$-ray emission when
$\beta_{ems}$=$\beta_{reac}$. The points observed on the left of the line
($\beta_{ems}<\beta_{reac}$) correspond to delayed $\gamma$-ray emissions.
Right: the corresponding angle-integrated velocity-difference profiles for the
three states compared with simulations (continuous lines). It shows
unambiguously that the astrophysical state (in red) has a lifetime $4<\tau<40$
fs. The red shaded area corresponds to 1$\sigma$ uncertainty on the lifetime,
the horizontal error bars correspond to the width of the bins, which are
larger than the real experimental uncertainty, and the vertical error bars to
the statistical uncertainty.

The overall shape of the $\Delta\beta$ measured profiles can be explained in a
simple way. The $\gamma$ rays are emitted following the exponential decay law,
i.e. $N(t)=N_{0}\times e^{-\Delta t/\tau}$, with $\tau$ being the lifetime of
the state and $\Delta t$ the time elapsed after the reaction. For short
lifetimes ($\tau\lesssim$100 fs), in the non-relativistic approximation, the
deceleration force acting on the 23Mg ions is almost constant, and
consequently $\Delta\beta\propto\Delta t$. Therefore, the right side tail of
the curves in Fig. 2 (right) would follow the exponential decay function
$N(t)\propto e^{-\Delta\beta/\tau}$. This explains the asymmetric shape of the
profile shown in blue, which corresponds to $\tau=40$ fs. Moreover, the
experiment has a resolution in $\Delta\beta$, i.e. 0.0032(1), which is
measured by the width of the almost Gaussian profile observed for very short
lifetimes, see the green profile for $\tau=4$ fs in Fig. 2 (right). The
overall shape of the measured $\Delta\beta$ distributions is the convolution
of the exponential decay with a Gaussian function. The sensitivity of the
method, governed by the statistics, can be better than the resolution of
$\Delta\beta$. This makes the method sensitive to very short lifetimes, down
to 0.8 fs according to simulations. From comparison of the data with Monte
Carlo simulations, shown in Fig. 2 (right) by the continuous lines (see also
the Supplementary section), the best agreement is obtained with lifetimes of
$4^{+1}_{-3}$ fs and $40^{+6}_{-7}$ fs for the states $E_{x}=5292.0(6)$ keV
($E_{\gamma,0}=4840.1^{+0.2}_{-0.4}$ keV) and $E_{x}=3796.0(1.2)$ keV
($E_{\gamma,0}=3344.8^{+0.8}_{-1.0}$ keV), respectively, which is in excellent
agreement with their known values of 5(2) fs4 and 41(6) fs4. This shows that
the present method is a powerful tool to determine lifetimes in the
femtosecond range. With the same method, it is also possible to accurately
measure the rest energy of the $\gamma$-ray transitions. For the transition
from the key state, we measured $E_{\gamma,0}=7333.0^{+0.5}_{-0.2}$ keV, in
good agreement with the referenced value of $E_{\gamma,0}=7333.2(11)$ keV 4.

The present method has many advantages. Since the excitation energy of the
state is precisely selected with VAMOS++, we can ignore any possible top-
feeding contribution to the state, and, therefore, the measure is not affected
by the lifetime of higher-lying states. Moreover, this method is independent
of the reaction mechanism populating the states, i.e. direct transfer or
compound-nucleus formation (24Mg+3He
$\rightarrow^{27}$Si${}^{*}\rightarrow^{23}$Mg+4He). Furthermore, the result
does not depend on the angular distribution of the emitted $\gamma$ ray and of
the charged particles, nor on the angle-dependent detection efficiency. A
single spectrum concentrates all the statistics of the experiment, which
maximizes the signal-to-noise ratio. The main sources of uncertainties and
their associated contributions for the 7785.0(7) keV state are presented in
Table 1.

Table 1: Systematic errors (in $\%$) on the lifetime of the $E_{x}=7785.0(7)$ keV state.  Source of error |  Error on $\tau$
---|---
| in %
$\gamma$-ray energy resolution111Measured during the experiment. | 15
$\gamma$-ray absolute angle (1 deg)222Including the uncertainty in the position of the target. | 12
$\gamma$-ray angle resolution111Measured during the experiment. | 12
Stopping powers333From SRIM19-EVASIONS simulations, considering 20 $\%$ uncertainty in stopping powers tables. | 10
Beam energy dispersion111Measured during the experiment. | 6.5
Implementation profile of 3He ions333From SRIM19-EVASIONS simulations, considering 20 $\%$ uncertainty in stopping powers tables. | 6.0
$\gamma$-ray energy shift during runs444From the energy calibration stability measured throughout the experiment. | 3.5
Transverse spatial dispersion of the beam111Measured during the experiment. | 0.3
Total | 26
|

The result obtained for the lifetime of the astrophysical state is
$\tau=11^{+7}_{-5}$ fs, including statistical and systematic uncertainties,
see Fig. 2, confirming the measured value of Jenkins et al11 ($\tau=10(3)$
fs). The reduced magnetic dipole transition probability $B(M1)$ was deduced
from the measured lifetime assuming a negligible $E2$ electric quadrupole
contribution: $B(M1)=0.017^{+0.011}_{-0.008}~{}\mu_{N}^{2}$. This value is
among the lowest values measured for $M1$ transitions. To get an insight from
theory, we have performed shell-model calculations in the $sd$ shell with the
well-established phenomenological USDA and USDB20, 21 interactions using the
NushellX@MSU code22. These calculations confirm that the key state is
characterized by a low $B(M1)$ value of 0.14 $\mu_{N}^{2}$, assuming $7/2^{+}$
spin and parity assignment (see the discussion below) and optimum values of
the effective proton and neutron g-factors. Although this value is almost a
factor of 10 larger than the experimentally determined one, the difference is
not far from the typical rms error on $M1$ transition probabilities
established for the $sd$ shell with these interactions23, 24. Therefore, it
can be said that the measured values of the lifetime are compatible with each
other and with shell-model calculations.

The measured lifetime was used to determine the strength of the resonance:
$\omega\gamma=\frac{2J_{{}^{23}\text{Mg}}+1}{(2J_{{}^{22}\text{Na}}+1)(2J_{p}+1)}\times\frac{\Gamma_{\gamma}\Gamma_{p}}{\Gamma_{\gamma}+\Gamma_{p}}=\omega
BR_{p}(1-BR_{p})\frac{\hbar}{\tau}$, with $J$ and $\Gamma$ being the spin and
the partial widths of the key state. Actually, the spin of this state has also
been a subject of debate 11, 25, where two values have been proposed:
$J_{{}^{23}\text{Mg}}^{\pi}=7/2^{+}$11 or $5/2^{+}$25. We assume it to be
$7/2^{+}$. This state is only 18(1.5) keV away from the known $5/2^{+}$
isobaric analog state of the 23Al ground state. In the case of $5/2^{+}$
assignment, the states would interfere, as predicted by the nuclear shell-
model calculations24, and such a mixing has not been observed 26, 27, 13, 6.
Moreover, concerning the $\gamma$-ray decay pattern, shell-model calculations
agree with a $7/2^{+}$ spin24. The proton branching ratio, $BR_{p}$, was also
measured in this work by detecting protons emitted from the 23Mg unbound
states with an annular silicon detector placed downstream of the target. The
obtained value, $BR_{p}=0.68(6)$%, is in excellent agreement with the latest
published value, $BR_{p}=0.65(8)$%27.

Combining all measured values into the new recommended values:
$BR_{p}=0.67(5)$% and $\tau=10.2(26)$ fs, results in a consolidated resonance
strength of $\omega\gamma=0.24^{+0.11}_{-0.04}$ meV, which is compatible with
the direct measurement 14 ($\omega\gamma<0.36$ meV) but not with the other two
direct measurements. It should be pointed out that these direct measurements
using radioactive targets are not in agreement with each other, and also that
the obtained value is very low, close to the sensitivity limit of these direct
experiments. The new value of the resonance strength and a Monte-Carlo
approach 28 were used to deduce a new rate for the 22Na($p,\gamma)^{23}$Mg
reaction (see Fig. 3 and Table 1 in Supplementary). The reliably estimated
experimental uncertainties of the present method allowed a more accurate
determination of this rate. This new rate was found to be very reliable at
maximum nova temperatures, with uncertainties reduced to 40 % (10 %) at T=0.1
GK (0.5 GK). It is worth to mention that if we consider the two possible
assignments $J_{{}^{23}\text{Mg}}^{\pi}=(5/2^{+},7/2^{+})$, only a limited
increase in the uncertainties is obtained, at most 11$\%$ below 0.25 GK.

To quantitatively assess the impact of the new 22Na($p,\gamma)^{23}$Mg
reaction rate obtained from this work on nova nucleosynthesis, a series of
hydrodynamic simulations have been performed. Four physical magnitudes
determine the strength of a nova outburst, and, in turn, the synthesis of
22Na: the white dwarf mass M${}_{\text{WD}}$ (or radius R${}_{\text{WD}}$),
its initial luminosity $L_{\rm WD,ini}$, the mass-accretion rate $\dot{M}$,
and the metallicity of the accreted material. The influence of the white dwarf
mass on the synthesis of 22Na has been analyzed. Three different values for
the white dwarf mass have been considered: 1.15 M⊙, 1.25 M⊙, and 1.35 M⊙. In
these simulations, the star hosting the nova outbursts is assumed to be an ONe
white dwarf, with initial luminosity, $L_{\rm WD,ini}=10^{-2}$ L⊙, and
accreting solar composition material 29 from the secondary star at a constant
mass-accretion rate of $\dot{M}$ = $2\times 10^{-10}$ M⊙/yr. The accreted
matter is assumed to mix with material from the outermost white dwarf
layers30. For consistency and completeness, the nova outbursts on 1.15 M⊙
white dwarfs have been computed with two different, one-dimensional stellar
evolution codes: MESA31, 32, 33, 34, 35, 36 and SHIVA37, 1.

At the early stages of the outburst, when the temperature at the base of the
envelope reaches $T_{\rm base}\sim 5\times 10^{7}$ K, the chain of reactions
20Ne($p,\gamma)^{21}$Na($\beta^{+}$)21Ne($p,\gamma)^{22}$Na powers a rapid
rise in the 22Na abundance. When the temperature reaches $T_{\rm base}~{}\sim
8~{}\times~{}10^{7}$ K, 22Na($p,\gamma)^{23}$Mg becomes the main destruction
channel. At $T_{\rm base}\sim 10^{8}$ K, 22Na($p,\gamma)^{23}$Mg becomes the
most important reaction involved in the synthesis and destruction of 22Na, and
therefore, its abundance begins to decrease. When $T_{\rm base}\sim~{}2\times
10^{8}$ K, 21Na($p,\gamma)^{22}$Mg becomes faster than 21Na($\beta^{+}$)21Ne,
such that 20Ne($p,\gamma)^{21}$Na($p,\gamma)^{22}$Mg($\beta^{+}$)22Na takes
over as the dominant path, favoring the synthesis of 22Na, which achieves a
second maximum after the temperature peak T${}_{\text{peak}}$. This pattern
continues up to 22Na($p,\gamma)^{23}$Mg becomes again the most relevant
reaction when the temperature drops. During the subsequent and final expansion
and ejection stages, as the temperature drops dramatically, the evolution of
22Na is fully governed by 22Na($\beta^{+}$)22Ne. The most relevant results
obtained in these simulations are summarized in Table 2.

Table 2: Abundances of (Ne, Na) isotopes in the ejecta obtained for different nova models calculated with the MESA and SHIVA codes. See text for the parameters. Model | 115a | 115b | 125 | 135
---|---|---|---|---
HD Code | MESA | SHIVA | SHIVA | SHIVA
M${}_{\text{WD}}$ (M⊙) | 1.15 | 1.15 | 1.25 | 1.35
R${}_{\text{WD}}$ (km) | 4428 | 4334 | 3797 | 2258
T${}_{\text{peak}}$ (108 K) | 2.12 | 2.27 | 2.48 | 3.13
M${}_{\text{ejec}}$ (10-5 M⊙) | 4.63 | 2.46 | 1.90 | 0.46
X(20Ne) |  1.4$\times 10^{-1}$ |  1.8$\times 10^{-1}$ |  1.8$\times 10^{-1}$ |  1.5$\times 10^{-1}$
X(21Ne) |  2.3$\times 10^{-5}$ |  3.6$\times 10^{-5}$ |  3.9$\times 10^{-5}$ |  4.1$\times 10^{-5}$
X(22Ne) |  1.9$\times 10^{-3}$ |  1.3$\times 10^{-3}$ |  6.4$\times 10^{-4}$ |  3.3$\times 10^{-5}$
X(22Na) | 3.1$\mathbf{\times 10^{-4}}$ | 3.2$\mathbf{\times 10^{-4}}$ | 3.7$\mathbf{\times 10^{-4}}$ | 9.1$\mathbf{\times 10^{-4}}$
X(23Na) |  8.2$\times 10^{-4}$ |  9.9$\times 10^{-4}$ |  1.1$\times 10^{-3}$ |  3.4$\times 10^{-3}$
|  |  |  |

The larger peak temperatures achieved during nova outbursts on more massive
white dwarfs yield larger mean, mass-averaged abundances of 22Na in the
ejecta. However, it is worth noting that the total mass of 22Na ejected in a
nova outburst decreases with the white dwarf mass ($8\times 10^{-9}$ M⊙, for
Model 115b, $7\times 10^{-9}$ M⊙, for Model 125, and $4.2\times 10^{-9}$ M⊙,
for Model 135), since more massive white dwarfs accrete and eject smaller
amounts of mass, see M${}_{\text{ejec}}$ in Table 2.

The new and reliable reaction rate obtained in this work also opens the door
to further advanced sensitivity studies on astrophysical parameters, performed
here for the first time. The flux has been obtained in a series of nova
simulations for a large range of mass-accretion rates $\dot{M}$ and white
dwarf luminosities $L_{\rm WD,ini}$. For example, Fig. 3 shows the expected
flux of 22Na for a nova event located 1 kpc from the Earth, with
M${}_{\text{WD}}$=1.2 M⊙ and accreting solar composition material. This figure
shows that the amount of 22Na does depend on the two parameters, varying
smoothly by up to a factor of 3. The change is abrupt only on the lower right
side of the figure. In this region, the 22Na/22Mg and 22Na/21Ne ratios around
the peak temperatures have the same values as in the other regions, which
means that the production pathways are unchanged. On the contrary, the
23Mg/22Na ratio is found 4 times larger, indicating that the destruction by
the 22Na($p,\gamma)^{23}$Mg reaction is more intense in this region. This
graph provides a link between the unknown astrophysical parameters of the
novae and the measured 22Na flux. In the future, precise measurements of the
22Na $\gamma$-ray flux will constrain these astrophysical parameters.

Figure 3: Prediction of the 1.275 MeV $\gamma$-ray flux emitted from a nova.
The 22Na $\gamma$-ray emission flux is shown as a function of the white dwarf
initial luminosity and the mass-accretion rate. This is calculated from the
22Na mass-averaged abundance within the ejected shells. Computations were done
for a 1.2 M⊙ ONe white dwarf located 1 kpc from the Earth, using the MESA
code31, 32, 33, 34, 35, 36.

New instruments for $\gamma$-ray astronomy are under study or under
construction: ESA’s enhanced e-ASTROGAM38 and NASA’s COmpton Spectrometer and
Imager, COSI39. Both are, in principle, capable of detecting the 1.275 MeV
$\gamma$ rays released in the decay of 22Na produced in nova outbursts, since
they are being designed with a higher efficiency than the past missions,
INTEGRAL-SPI40 and COMPTEL-CGRO41. The very high precision in the
determination of the 22Na($p,\gamma)^{23}$Mg rate reported in this work allows
to derive, for the first time, realistic estimates of the maximum
detectability distance of novae in $\gamma$ rays, through the 1.275 MeV line.
With the expected sensitivities of e-ASTROGRAM ($3\times 10^{-6}$ photons cm-2
s-1) and COSI ($1.7\times 10^{-6}$ photons cm-2 s-1) at 1 MeV, and assuming
that $\sim 7\times 10^{-9}$ M⊙ of 22Na are ejected in a typical nova outburst
(model 125, Table 2), new maximum detectability distances of 2.7 kpc and 4.0
kpc, respectively, have been derived. These new estimates suggest a large
chance for the possible detection of the 22Na $\gamma$ rays produced in an ONe
nova by the next generation of space-borne $\gamma$-ray observatories.

In addition to the results obtained for the astrophysical state, new
experimental results were obtained for about twenty 23Mg states observed in
this study and will be the subject of a future publication. The present method
is more universal than the conventional Doppler Shift Attenuation Method
(DSAM)42, 43, 44. Unlike DSAM where $\beta_{reac}$ is not experimentally
accessed and its change in the target a source of uncertainty, in the present
method a thick target can be used since $\beta_{reac}$ and $\beta_{ems}$ are
determined for each event. This feature explains the low sensitivity of the
obtained results to the 3He initial implementation profile and its evolution
with time, as shown in Table 1. Indeed, measuring $\beta_{reac}$ during the
experiment, event by event, is equivalent to measuring the reaction point, and
therefore, to an on-line measurement of the 3He profile in the target and its
evolution. Moreover, if the state decays mainly by proton emission instead of
$\gamma$ ray, it is possible to determine its lifetime with the same method
using $\alpha-p$ correlations. The protons are kinetically boosted at forward
angles as the $\gamma$ rays are Doppler shifted. From this kinetic-boost
effect, it is possible to determine the velocity $\beta_{ems}$ of the nucleus
at the proton-emission time and to deduce the lifetime of the state with
femtosecond accuracy. In addition, it is possible to constrain the spin of the
state using the particle-particle correlation analysis method45, by measuring
the angular distribution of the second emitted particle, proton or $\gamma$,
over a wide range of angles. It might be interesting to also measure the 23Mg
recoil nuclei. On the one hand the magnetic rigidity of 23Mg is very different
from that of the primary beam, so it could be easy to measure these nuclei
with a magnetic spectrometer. On the other hand, the background noise
generated by fusion-evaporation reactions could be much less with the
measurement and identification of the recoil nuclei. All these advantages,
obtained or achievable with remarkable detectors like AGATA and VAMOS++, make
this method a promising way to measure in the future other excited states with
femtosecond lifetimes.

## Methods

Determination of velocities The velocity $\beta_{reac}$ of the 23Mg nuclei at
the reaction time was derived from the momentum of the $\alpha$ particles
measured with the VAMOS++ magnetic spectrometer and from the kinematics laws
of energy and momentum conservation in the case of the 3He(24Mg,$\alpha$)23Mg
two-body reaction

$\beta_{reac}=\sqrt{1-\gamma_{reac}^{-2}}\\\ $

with

$\gamma_{reac}=\sqrt{1+\frac{\frac{m_{\alpha}^{2}\beta_{\alpha}^{2}}{1-\beta_{\alpha}^{2}}+\frac{m_{beam}^{2}\beta_{beam}^{2}}{1-\beta_{beam}^{2}}-2\cos(\theta_{\alpha})m_{beam}m_{\alpha}\sqrt{\gamma_{\alpha}^{2}-1}\sqrt{\gamma_{beam}^{2}-1}}{m_{recoil}^{2}}}$
(1)

where $m_{\alpha}$, $m_{beam}$ and $m_{recoil}$ are the rest-mass energies of
the $\alpha$, 24Mg and 23Mg nuclei, $\theta_{\alpha}$ the angle between the
$\alpha$ particle and the beam axis. The parameter $\gamma_{\alpha}$ was
measured with VAMOS++ and corrected for the energy losses in the target using
the SRIM code 19. The parameter $\gamma_{beam}$, measured prior to the
experiment, was then determined from the measured $\gamma_{\alpha}$ and a
$\gamma$-ray transition detected in coincidence.

The velocity $\beta_{ems}$ of the 23Mg nuclei at the $\gamma$-ray emission
time was derived from the measured $\gamma$ rays. Since the 23Mg nuclei are
moving at the time of the $\gamma$-ray emission, the $\gamma$-ray energy is
Doppler shifted, with the measured energies $E_{\gamma}$ shifted from the rest
energy $E_{\gamma,0}$ according to

$E_{\gamma}=E_{\gamma,0}\frac{\sqrt{1-\beta^{2}_{ems}}}{1-\beta_{ems}\text{cos}(\theta)}$
(2)

It follows that

$\beta_{ems}=\frac{R^{2}\text{cos}(\theta)+\sqrt{1+R^{2}\text{cos}^{2}(\theta)-R^{2}}}{R^{2}\text{cos}^{2}(\theta)+1}$
(3)

with $R=E_{\gamma}/E_{\gamma,0}$. Here, $R<1$ since the AGATA detector was
located upstream of the target. The angle $\theta$ between the $\gamma$ ray
and the 23Mg recoil nucleus was derived from the measured ($\theta$, $\phi$)
of the $\gamma$ ray and the $\alpha$ particle using the formulas

$\displaystyle\cos(\theta)=$
$\displaystyle\sin(\theta_{\gamma})\sin(\theta_{recoil})[\cos(\phi_{\gamma})\cos(\phi_{recoil})+\sin(\phi_{\gamma})\sin(\phi_{recoil})]$
$\displaystyle+\cos(\theta_{\gamma})\cos(\theta_{recoil})$

where

$\displaystyle\theta_{recoil}=$
$\displaystyle\text{acos}(\frac{m_{beam}\sqrt{\gamma_{beam}^{2}-1}-m_{\alpha}\cos(\theta_{\alpha})\sqrt{\gamma_{\alpha}^{2}-1}}{m_{recoil}\sqrt{\gamma_{recoil}^{2}-1}})$
$\displaystyle\phi_{recoil}=$ $\displaystyle\pi+\phi_{\alpha}$ (4)

Fit of velocity-difference profiles

Velocity-difference profiles were numerically simulated with a Monte-Carlo
approach developed in the EVASIONS C++/ROOT code. The principle of these
numerical simulations will be the subject of a future publication. Simulated
velocity-difference profiles were normalized to the measured ones via the
profile integrals. The goodness of fit between experimental and simulated
profiles was quantified with the Pearson $\chi^{2}$ tests where the lifetime
and the $\gamma$-ray rest energy are taken as free parameters.

Branching ratios The $E_{x}=7785.0(7)$ keV astrophysical state can decay via
proton or $\gamma$-ray emission. Therefore, after applying a selection on
$E_{x}$ in 23Mg by using the measured $\alpha$ particles, the number of
detected protons and $\gamma$ rays allowed us to determine the proton and
$\gamma$-ray branching ratios. These values were corrected for the detection
efficiencies. On the one hand, the geometrical efficiency of the silicon
detector was estimated by numerical simulations. The angular distribution was
considered isotropic for the emitted $\ell$=0 protons. On the other hand, the
AGATA efficiency was measured at low energies with a radioactive 152Eu source,
and simulated with the AGATA Geant4 code library 46, 47 to determine the
efficiency at high energies after scaling the simulations to the measured
efficiencies at low energies.

Determination of the 22Na flux in novae The amount of 22Na ejected in a nova
outburst was obtained with the simulation codes MESA31, 32, 33, 34, 35 and
SHIVA37, 1 from its abundance in the ejected layers. MESA and SHIVA use
several criteria for the ejection of a specific layer, based either on
achieving escape (velocity) or a luminosity above the Eddington limit where
the radiative pressure exceeds the gravitational force.

Data availability The experimental data of this work related to particles
measurements are available upon reasonable request by contacting the primary
author. Due to the large size of the data, they cannot be hosted publicly. The
ownership of data generated by the AGATA $\gamma$-ray spectrometer resides
with the AGATA collaboration as detailed in the AGATA Data Policy48.

Codes availability The EVASIONS code used in this study is available from the
primary author upon reasonable request. Other codes employed here, i.e.
SRIM19, NushellX${@}$MSU22, MESA31, 32, 33, 34, 35 and RatesMC 28, are freely
available.

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Acknowledgments The authors thank the GANIL accelerator staff for their beam
delivery and support as well as the AGATA collaboration. This work was
supported by the Normandie Region, the European project ChETEC-INFRA
(101008324) and SPIRAL2-CZ. Y.H.K. would like to acknowledge the support of
the Institute for Basic Science (IBS-R031-D1). M.S. would like to acknowledge
that this work has been partially supported by the OASIS project no.
ANR-17-CE31-0026 and by the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics, under contract number DE-AC02-06CH11357.

Authors’ contributions F.O.S. and C.M. were PIs of the experiment during which
F.O.S., C.M., E.C., Y.H.K., A.L., V.G., D.B., F.B., I.C., M.C., C.D., J.D.,
Z.F., S.L., J.L., A.L.M., J.M., D.R., N.R., A.M.S.B, M.S. and P.U.
participated in person. Targets making was performed by R.B.. C.F., C.M.,
E.C., A.L., D.B., G.B., A.J.B., A.B., B.C., C.D.P., J.D., J.E., V.G., J.G.,
H.H., A.J., A.K., A.K., S.M.L., S.L., H.L., J.L., A.L.M., R.M., D.M., B.M.,
D.R.N., J.N., Zs.P., A.P., B.Q., D.R., P.R., K.R., F.S., M.D.S., E.S., M.S.,
M.S., D.S., Ch.T., J.J.V.D. and M.Z. have been involved to the AGATA
collaboration. Experimental analysis and simulations were undertook by C.F.
and F.O.S.. Experimental results were in particular discussed by C.F., F.O.S.,
E.C., A.L. and A.N.. Theoretical calculations on nuclear structure were done
by C.F., N.S. and F.O.S.. Astrophysical calculations and analysis were
performed by C.F., J.J. and F.O.S.. F.O.S., C.F. and J.J. were involved in the
primary manuscript. All co-authors discussed the final results and contributed
to the final manuscript. Ethics declarations: competing interests The authors
declare no competing interests. Supplementary information Additional
information is available in the supplementary section of the paper, in
particular fit results, numerical simulations and reaction rate used in the
present study.