# Quantum metrology with a squeezed Kerr oscillator

Jiajie Guo  State Key Laboratory for Mesoscopic Physics, School of Physics,
Frontiers Science Center for Nano-optoelectronics, Peking University, Beijing
100871, China  Qiongyi He<EMAIL_ADDRESS>State Key Laboratory for
Mesoscopic Physics, School of Physics, Frontiers Science Center for Nano-
optoelectronics, Peking University, Beijing 100871, China  Collaborative
Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi
030006, China  Peking University Yangtze Delta Institute of Optoelectronics,
Nantong 226010, Jiangsu, China  Hefei National Laboratory, Hefei 230088, China
Matteo Fadel<EMAIL_ADDRESS>Department of Physics, ETH Zürich, 8093
Zürich, Switzerland

###### Abstract

We study the squeezing dynamics in a Kerr-nonlinear oscillator, and quantify
the metrological usefulness of the resulting states. Even if the nonlinearity
limits the attainable squeezing by making the evolution non-Gaussian, the
states obtained still have a high quantum Fisher information for sensing
displacements. However, contrary to the Gaussian case, the amplitude of the
displacement cannot be estimated by simple quadrature measurements. Therefore,
we propose the use of a measurement-after-interaction protocol where a linear
quadrature measurement is preceded by an additional nonlinear evolution, and
show the significant sensitivity enhancement that can be obtained. Our results
are robust when considering realistic imperfections such as energy relaxation,
and can be implemented in state-of-the-art experimental setups.

Figure 1: Illustration of the proposed metrological protocol. Squeezing the
ground state of a nonlinear oscillator results in a non-Gaussian state that
might not have a quadrature with reduced quantum noise. Despite this, the
state can still have high sensitivity to displacements, even if the
information is hidden in high-order moments of quadrature operators. The
method we propose to access the displacement amplitude consists instead in
undergoing a second nonlinear evolution, which allows to keep the measurement
linear.

Continuous variable (CV) quantum systems, such as optical fields or mechanical
oscillators, constitute a platform of primary importance for quantum metrology
applications. Examples include gravitational wave detection [1, 2, 3, 4],
force sensing [5, 6], and the measurement of electric and magnetic fields [7].
Largely explored in this context are squeezed states, namely Gaussian states
that show along some phase-space quadrature an uncertainty that is below the
quantum noise of the vacuum. Besides being relatively easy to be prepared
experimentally, for example through a parametric process [8, 9], such states
are also easy to be measured, as they can be fully characterised by linear
quadrature (i.e. homodyne) measurements.

One of the experimental factors limiting squeezing are the inevitable
nonlinearities present in the system. In fact, highly squeezed states have
large average number of excitations (i.e. energy), as well as wavefunctions
significantly distributed in phase space, which is manifested by the
antisqueezed quadrature. When these go beyond the linear regime of the
considered experimental platform, the state evolution become non-Gaussian, and
squeezing gets degraded by a “wrapping around” of the state [10, 11]. Similar
results are found in the context of ‘crescent states’, namely coherent states
undergoing Kerr evolution [12, 13, 14, 15]. This hinders metrological
applications of the resulting state, despite the fact that non-Gaussian states
can still have high sensitivity to perturbations [16, 17, 18, 19, 20, 21].

One of the main difficulties in doing quantum-enhanced metrology with non-
Gaussian states lies in the fact that the parameter to be retrieved is encoded
in high-order correlators [22]. This requires to access high moments of the
measurement’s probability distribution, or equivalently to perform non-
Gaussian (e.g. number-resolving) measurements, which is of challenging
experimental implementation [23, 24, 25, 26, 27]. In addition, noise
constraints for these observables become also very stringent. Therefore,
nonlinearities are typically seen as limitations, and in experiments one tries
to reduce them as much as possible.

A paradigmatic model where to study the interplay between squeezing and
nonlinear interactions is given by the Hamiltonian of a Kerr oscillator with a
squeezing drive

$\hat{H}=\Delta\hat{a}^{\dagger}\hat{a}+\epsilon(\hat{a}^{\dagger
2}+\hat{a}^{2})-K\hat{a}^{\dagger 2}\hat{a}^{2}\;.$ (1)

Here $\Delta$ is the detuning between oscillator and drive, $\epsilon$ is the
squeezing rate, and $K$ is the Kerr nonlinearity. Besides being interesting
for the study of interesting processes such as chaotic dynamics [28],
tunneling [29], coherent superpositions [30, 31], phase transitions and
blockade effects [32, 33], Hamiltonian (1) also attracted significant
attention in the context of quantum information processing, since its ground
state is a Schrödinger cat state that can be exploited for error-protected
qubit encoding [34, 35, 36]. This observation motivated the recent
experimental implementation of Eq. (1) for electromagnetic modes with
superconductive devices [37, 38], and in mechanical modes with acoustic
resonators [39].

Here we study the use of a squeezed Kerr oscillator for quantum metrology, and
show that when sufficient control over the system’s parameters is available,
the presence of a nonlinearity can significantly improve the metrological
performances even for simple quadrature measurements. The idea relies on
preparing non-Gaussian states that, even when not showing reduced uncertainty
compared to vacuum, can have high sensitivity to displacements. This can be
then accessed by preceding the measurement step by an additional nonlinear
evolution of the state, which we show to result in an effective measurement of
higher-order moments of the quadratures. To conclude, we show that our results
are robust to noise, and of immediate implementation in electrical and
mechanical systems.

##  I Squeezing limits

Let us begin with considering the task of preparing squeezed vacuum states
with Eq. (1), and investigate the limitations posed by the nonlinear term.

We imagine a protocol where a system that is initially in the ground state of
the harmonic oscillator $\ket{0}$ evolves for $t\geq 0$ according to Eq. (1),
due to the application of a parametric drive. During this dynamics, we are
interested in studying the evolution of the state’s minimum uncertainty
quadrature, namely of
$V_{\text{min}}(t)\equiv\min_{\theta}\text{Var}[(\hat{a}e^{-i\theta}+\hat{a}^{\dagger}e^{i\theta})/\sqrt{2}]$
, with
$\text{Var}[\hat{A}]=\langle\hat{A}^{2}\rangle-\langle\hat{A}\rangle^{2}$ the
variance of $A$. Since for coherent states $V_{\text{min}}=1/2$, observing
$V_{\text{min}}<1/2$ implies a reduction of the quantum noise below the
classical limit, and thus implies that the state is squeezed.

For the dynamics we consider, there is in general no known analytic closed-
form expression for $V_{\text{min}}$, which therefore has to be computed
numerically. We show in Fig. 2a the plot of $V_{\text{min}}(t)$ for
$(\Delta,\epsilon)=(0,2)$ and different values of $K$. For $K=0$ it is known
that $V_{\text{min}}(t)=\frac{1}{2}e^{-4\epsilon t}$, meaning that an
arbitrarily small uncertainty is achievable for sufficiently long times. For
$K\neq 0$, however, we observe that $V_{\text{min}}$ attains a minimum value
at a finite time $t_{\text{opt}}$. This is expected, as the nonlinearity
results in a non-Gaussian evolution of the state which limits the achievable
squeezing [10, 11]. We thus define the parameter
$\chi^{-2}_{\text{opt}}=1/V_{\text{min}}(t_{\text{opt}})$, which we will later
show to be related to the state sensitivity, and show in Fig. 2b,c the
dependence of $\chi^{-2}_{\text{opt}}$ and $t_{\text{opt}}$ on $\epsilon/K$,
now also for different values of $\Delta$. Note that, higher squeezing can be
prepared in a shorter time as $\epsilon/K$ increases, since the effect of the
nonlinearity gets relatively less important.

Since squeezed states have a quadrature with reduced uncertainty, they can
provide an advantage in metrological tasks [40, 41, 42]. For this reason, it
may look like as if the best strategy to achieve a larger advantage is to have
$\epsilon/K$ as large as possible and stop the state preparation at
$t_{\text{opt}}$, since longer evolution times degrade $V_{\text{min}}$.
However, as we will now show, this is not necessarily true.

Figure 2: Optimal squeezing of a Kerr oscillator. a) Minimum variance of the
state $e^{-i\hat{H}t}\ket{0}$, fixing $\Delta=0$. For $K>0$ there is an
optimal squeezing point, that we further investigate also as a function of
$\Delta$. b) Squeezing level at the optimal point. c) Optimal squeezing time.

##  II Metrological advantage of non-Gaussian states

Let us remember that, in a typical quantum metrology scheme, the task is to
estimate a parameter $d$ that is encoded in a probe state $\rho$ by a
generator $\hat{G}$ as $\rho_{d}=e^{-id\hat{G}}\rho e^{id\hat{G}}$. A
fundamental limit to the sensitivity is provided by the so-called quantum
Cram$\acute{\text{e}}$r-Rao bound
$\Delta^{2}d\geq\Delta^{2}d_{QCR}\equiv(F_{Q}[\rho,\hat{G}])^{-1}$, where
$F_{Q}[\rho,\hat{G}]$ is the quantum Fisher information (QFI). For a pure
state the QFI is calculated from the variance of the generator as
$F_{Q}[\rho,\hat{G}]=4\text{Var}[\hat{G}]$. Importantly, to achieve the
maximum sensitivity $(\Delta^{2}d_{QCR})^{-1}$ it is necessary to optimize the
measurement that is performed on $\rho_{d}$ in order to estimate $d$. In
general, if one measures $\hat{M}$ then the achieved sensitivity is [22]

$\left(\Delta^{2}d\right)^{-1}=\chi^{-2}[\rho,\hat{G},\hat{M}]\equiv\frac{|\langle[\hat{G},\hat{M}]\rangle|^{2}}{\text{Var}[\hat{M}]}\;,$
(2)

which satisfies $\chi^{-2}[\rho,\hat{G},\hat{M}]\leq F_{Q}[\rho,\hat{G}]$
[43].

To now understand the connection between (2) and squeezing, let us consider
the task of sensing the amplitude of a displacement from the measurement of a
phase-space quadrature. We thus have
$\hat{G}(\phi)=(\hat{a}e^{-i\phi}+\hat{a}^{\dagger}e^{i\phi})/\sqrt{2}$, the
generator of a displacement along direction $\phi+\pi/2$, and
$\hat{M}(\theta)=(\hat{a}e^{-i\theta}+\hat{a}^{\dagger}e^{i\theta})/\sqrt{2}$,
the measurement along direction $\theta$. Looking at the numerator of Eq. (2),
we note that the sensitivity is highest when $\hat{M}$ is perpendicular to
$\hat{G}$, meaning when $\hat{M}$ is along the displacement direction, as we
would expect. In this case we obtain
$\chi^{-2}[\rho,\hat{G}(\theta+\pi/2),\hat{M}(\theta)]=1/\text{Var}[(\hat{a}e^{-i\theta}+\hat{a}^{\dagger}e^{i\theta})/\sqrt{2}]$,
and by further optimizing over the measurement direction $\theta$ we have
$\chi^{-2}\equiv\max_{\phi,\theta}\chi^{-2}[\rho,\hat{G}(\phi),\hat{M}(\theta)]=1/V_{\text{min}}$.

These results shows us two things. First, for sensing displacements from
quadrature measurements then $V_{\text{min}}$ (i.e. the squeezing) is the
correct figure of merit that needs to be optimized, but if the displacement is
estimated from another type of measurement this might not be the case. Second,
depending on the state $\rho$ we are considering then the choice of performing
a quadrature measurement might not be the optimal one saturating the quantum
Cram$\acute{\text{e}}$r-Rao bound, and thus not achieving
$\chi^{-2}[\rho,\hat{G},\hat{M}]=F_{Q}[\rho,\hat{G}]$. This measurement choice
is however proven to be optimal for sensing displacements with Gaussian states
(see Sec. I, II and III of the SM [44]).

To illustrate this last point for the scenario introduced in the previous
section, we compute squeezing and QFI of the states
$\rho=e^{-i\hat{H}t}\ket{0}$ prepared through Eq. (1) at $Kt=0.5$. We plot in
Fig. 3a,b the quantities $\chi^{-2}=1/V_{\text{min}}$ and
$F_{Q}\equiv\max_{\phi}F_{Q}[\rho,\hat{G}(\phi)]=\max_{\phi}4\text{Var}[\hat{G}(\phi)]$,
respectively, for different values of $\Delta/K$ and $\epsilon/K$. Figure 3a
shows that when linear quadrature measurements are performed, then a quantum-
enhanced sensitivity, i.e. $\chi^{-2}>2$, is attained only for a limited set
of states. In general, one can also have $\chi^{-2}<2$, which indicates even
worse sensitivity than the one achieved by coherent states (see also Fig. 2a
for $t>0.25$, when the coloured lines show $V_{\text{min}}>1/2$). When this is
the case, since here we are dealing with pure states, it necessarily implies
that the state is non-Gaussian. On the other hand, Fig. 3b shows that any
state (besides the trivial case $\epsilon/K=0$) has $F_{Q}>2$ and can thus
show a quantum-enhanced sensitivity to displacements if the correct
measurement is performed.

Even if, strictly speaking, linear quadrature measurements are never optimal
for $\epsilon/K>0$, there are regimes in which they are very close to being
optimal. This is indicated by the region below the black line in Fig. 3b,
which corresponds to states for which $(F_{Q}-\chi^{-2})/F_{Q}\leq 0.05$, and
thus to states that are faithfully approximated by being Gaussian. Outside
this region, different measurements are required to approach the maximum
sensitivity, which is what we want to investigate in the next sections by
considering two strategies that are experimentally relevant.

Figure 3: Sensitivity comparison. For the state $e^{-i\hat{H}t}\ket{0}$ with
$Kt=0.5$ we show: a) squeezing $\chi^{-2}$, where the white dotted lines
represents the standard-quantum-limit $\chi_{\text{SQL}}^{-2}=2$. b) QFI
$F_{Q}$, where black dot-dashed line is a boundary
$(F_{Q}-\chi^{-2})/F_{Q}=0.05$. c) Sensitivity for the MAI method
$\chi_{\text{MAI}}^{-2}$. White solid lines are for $\epsilon=|\Delta/2|$, and
indicate the classical phase diagram for a squeezed Kerr oscillator [38].

##  III Nonlinear measurements

The first strategy consists of measuring higher order moments of phase-space
quadratures, and it can be studies systematically in the following way [22].
We define $\textbf{M}^{(k)}$ to be a vector involving up to $k$th-order
moments of the quadrature operators
$\hat{X}=(\hat{a}+\hat{a}^{\dagger})/\sqrt{2}$ and
$\hat{P}=-i(\hat{a}-\hat{a}^{\dagger})/\sqrt{2}$, such that, e.g. linear
quadrature measurements are described by $\textbf{M}^{(1)}=(\hat{X},\hat{P})$,
while 2nd order one by
$\textbf{M}^{(2)}=(\hat{X},\hat{P},\hat{X}^{2},\hat{P}^{2},(\hat{X}\hat{P}+\hat{P}\hat{X})/2)$.
Then, we introduce the nonlinear squeezing parameter as [22]

$\chi_{(k)}^{-2}\equiv\max_{\phi,\vec{m}}\chi^{-2}[\rho,\hat{G}(\phi),\vec{m}\cdot\textbf{M}^{(k)}]\;.$
(3)

Crucially, this optimization task can be casted into an eigenvalue problem of
easy solution (see Section I of the SM [44]). With higher-order moments
involved, these parameters are capable of revealing quantum-enhanced
sensitivites in a wider class of states, even beyond the Gaussian regime.
Moreover, it holds the hierarchy
$\chi_{(1)}^{-2}\leq\chi_{(2)}^{-2}\leq\cdots\leq F_{Q}$, showing that for
sufficiently high $k$ one can attain the maximum sensitivity.

In squeezed Kerr oscillators the nonlinearity can results in highly non-
Gaussian states, whose metrological advantage is unlocked only for
measurements of sufficient high order. Observing $\chi_{(1)}^{-2}=\chi^{-2}<2$
implies that linear quadrature measurements are not sufficient, and thus that
$k>1$ is necessary. Based on this observation, we computed $\chi_{(2)}^{-2}$
but, perhaps surprisingly, did not find any advantage compared to
$\chi_{(1)}^{-2}$. A careful exploration of the terms involved shows that this
is due to the fact that commutators between the generator and second moments
of the quadrature give terms linear in the quadrature, whose expectation value
is zero for the states we consider (see SM [44], Sec. I). This means that, in
our scenario, considering $\textbf{M}^{(2)}$ does not provide any advantage
compared to $\textbf{M}^{(1)}$. In order to see an advantage one would need to
consider at least $\textbf{M}^{(3)}$, which requires a massive increase in the
measurement statistics and low detection noise. This can be of impractical
implementation in several experimental situations, and it is thus viable to
consider also alternative strategies.

##  IV Measurement-After-Interaction (MAI) protocol

The second approach we consider consists of preceding a linear quadrature
measurement by a time-reversed evolution with Eq. (1), i.e. $e^{+i\hat{H}t}$.
A similar idea has been investigated for spin states both theoretically [45,
46, 47, 48, 49, 50] and experimentally [51, 52]. For CV systems, an experiment
with Gaussian states and transformations has been presented in Ref. [53]. In
this framework, reversing the evolution results in an amplification of the
signal to be detected, but also of the quantum noise (see Fig. 1 of Refs. [46,
53]). For this reason, an advantage is obtained only in the presence of
detection noise limiting the measurement resolution, as the ratio between the
signal and the quantum noise level remains constant (see Sec. IV and V of the
SM [44]). In fact, linear quadrature measurements are already optimal for
sensing displacements with Gaussian states, in the sense that they achieve
$\chi^{-2}=F_{Q}$ (see Sec. III of the SM [44]).

In the following we will show that the MAI protocol can provide tremendous
sensitivity enhancements when considering CV non-Gaussian states and
transformations. Necessary conditions for this protocol to be viable are: i)
the capability of implementing a time-reversed evolution and ii) low enough
noise to tolerate a second time evolution of the state. For point i), it is
sufficient to be able to invert the sign of the parameters in the Hamiltonian.
In the specific case of Eq. (1), $\Delta$ and $\epsilon$ are easily tuned by
the parametric drives, while $K$ can be tuned by changing the anharmonicity of
a trapping potential (e.g. in the case of a trapped ion) or the coupling
between the bosonic mode and a two-level system (e.g. in a circuit-QED setup).

Formally, the additional evolution that precedes the measurement can be
absorbed into a redefinition of the measurement operator. In our case we have
$\hat{M}_{\text{MAI}}(\theta)=\hat{U}(ae^{-i\theta}+{a^{\dagger}}e^{i\theta})\hat{U}^{\dagger}/\sqrt{2}$,
where $\hat{U}=e^{-iHt}$, from which we define

$\chi_{\text{MAI}}^{-2}\equiv\max_{\phi,\theta}\chi^{-2}[\rho,\hat{G}(\phi),\hat{M}_{\text{MAI}}(\theta)]\;.$
(4)

We plot this parameter in Fig. (3)c for the states $e^{-iHt}\ket{0}$ prepared
at $Kt=0.5$. From a comparison with the $F_{Q}$ shown in Fig. (3)b we are able
to conclude that the MAI protocol can attain a sensitivity close to optimal.
In particular, this is true also for the non-Gaussian regime, where by looking
at Fig. (3)a we see that by performing linear quadrature without the time-
reversed dynamics one would only get $\chi^{-2}<2$. A more quantitative
comparison will be discussed later in Fig. 4, while an analysis of the scaling
with $N$ can be found in Section VI of the SM [44].

Figure 4: Noise robustness. a) sensitivities
$F_{Q},\chi_{\text{MAI}}^{-2},\chi^{-2}$ obtained for
$\Delta/K=0,\epsilon/K=2$, and different levels of energy relaxation rates
$\gamma/K$. b) Wigner functions at different steps of the MAI protocol for
$Kt=0.4$ and $\gamma/K=0.1$. From left to right: prepared state, displacement,
time-reversed nonlinear evolution. Solid and dashed lines are optimal
directions of the generator and linear quadrature measurement, respectively.

##  V Robustness to losses

To show that the MAI protocol gives an actual advantage in realistic scenarios
we have to consider the effect of experimental imperfections. During both
state preparation and time-reversal dynamics the evolution of the system is
affected by inevitable losses and decoherence, which ultimately limit the
maximum duration of the protocol. If these are too severe compared to the
robustness of the MAI protocol, then no advantage is obtained.

To study this in detail, in our numerical simulations we replace the unitary
time evolutions $\hat{U}$ and $\hat{U}^{\dagger}$ by evolutions according to a
Master equation. We focus on losses (i.e. energy relaxation), which pose a
major limitation in CV systems. These can be described by a jump operator
$\sqrt{\gamma}\hat{a}$, where $\gamma=1/T_{1}$ is the energy relaxation rate.

Since here we are dealing with mixed states and non-unitary evolutions,
calculating the sensitivity becomes more tedious. The QFI for a general state
$\rho$ is computed as
$F_{Q}[\rho,\hat{G}]=\sum_{kl}\frac{(\lambda_{k}-\lambda_{l})^{2}}{(\lambda_{k}+\lambda_{l})}|\bra{k}\hat{G}\ket{l}|^{2}$,
where $\lambda_{k}$ and $\ket{k}$ are the eigenvalues and eigenstates of
$\rho$, respectively. To calculate $\chi^{-2}$ and $\chi^{-2}_{\text{MAI}}$ we
use the fact that
$|\langle[\hat{G},\hat{M}]\rangle|^{2}=\big{|}\frac{\partial}{\partial
d}\text{Tr}[\hat{M}\rho_{d}]\big{|}^{2}_{d=0}$, where
$\rho_{d}=e^{-id\hat{G}}\rho e^{id\hat{G}}$, and then discretize the
derivative numerically by applying a small displacement to the state.

We show a comparison of the sensitivities in Fig. 4a, for $\Delta/K=0$,
$\epsilon/K=2$, and different loss rates $\gamma/K$ (even stronger than the
one observed in experiments [38]). The sensitivity $\chi^{-2}$ obtained from
linear quadrature measurements is almost unaffected by losses, but as we have
seen it approaches $F_{Q}$ only for small times ($Kt\approx 0.1$). On the
other hand, the sensitivity $\chi^{-2}_{\text{MAI}}$ obtained from the MAI
protocol is always significantly larger than $\chi^{-2}$, and it approaches
$F_{Q}$ for a longer time interval ($Kt\approx 0.25$). We thus conclude that
the MAI protocol is robust, in the sense that a large amount of losses is
required before having the maximum of $\chi^{-2}_{\text{MAI}}$ smaller than
the maximum of $\chi^{-2}$ (see also Section VII of the SM [44]). Let us
emphasize that the observed performance of the MAI protocol is remarkable,
considering that doubling the time evolution significantly increases losses.

To have a better understanding of the MAI protocol, we plot in Fig. 4b the
Wigner function of the state at different steps. First, a non-Gaussian state
is prepared by evolving $\ket{0}$ according to Eq. (1) with $\Delta/K=0$,
$\epsilon/K=2$ and $Kt=0.4$. Then, the state is subject to the displacement we
want to sense, followed by an evolution with $-\hat{H}$ for another $Kt=0.4$.
Finally, a linear quadrature measurement is performed on the state to estimate
the displacement amplitude. Solid and dashed lines in Fig. 4b indicate the
optimal generator and measurement directions, respectively.

##  VI Conclusions

We addressed the problem of optimally using a squeezed Kerr oscillator for a
metrological task. In particular, we focus on sensing displacements with the
non-Gaussian states that result from the nonlinear evolution of this system,
and investigate measurement strategies to approach the highest sensitivity. We
show that, while a direct quadrature measurement requires to access high-order
moments, preceding the measurement by a time-reversed nonlinear evolution
allows to achieve high sensitivities even for first moments. Crucially, the
protocol we propose is robust to noise, and can be implemented in current
experiments for quantum-enhanced sensing of e.g. electromagnetic fields [37,
38] or forces [39]. Future works could address the interesting problem of
multiparameter estimation for entangled nonlinear oscillators [54, 55, 56,
57].

Acknowledgments.– We thank S. Liu, F. Sun and M. Tian for helpful discussions.
This work is supported by the National Natural Science Foundation of China
(Grants No. 11975026, No. 12125402, and No. 12147148), and the Innovation
Program for Quantum Science and Technology (Grant No. 2021ZD0301500). MF was
supported by the Swiss National Science Foundation Ambizione Grant No. 208886,
and The Branco Weiss Fellowship – Society in Science, administered by the ETH
Zürich.

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## Supplemental material for “Quantum metrology with a squeezed Kerr
oscillator”

##  VII Metrological sensitivity

In a metrological task, a parameter $d$ is encoded in a probe state $\rho$ by
a generator $\hat{G}$ via $\rho_{d}=e^{-d\hat{G}}\rho e^{id\hat{G}}$. The goal
is then to estimate the parameter $d$ with the smallest uncertainty $\Delta d$
from the measurement of an observable $\hat{M}$ on $\rho_{d}$. The uncertainty
in the estimation can be expressed as

$\displaystyle\Delta^{2}d=\frac{\text{Var}[\hat{M}]}{|\partial_{d}\langle\hat{M}\rangle|^{2}}\Bigg{|}_{d=0}=\frac{\text{Var}[\hat{M}]}{|\langle[\hat{G},\hat{M}]\rangle|^{2}}\;.$
(5)

In our work, we consider a continuous variable scenario where the parameter to
be estimated is the amplitude $d=\sqrt{2}|\alpha|$ of a displacement
$\hat{D}(\alpha)=e^{\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}}$.

It is convenient to expand $\hat{G}$ and $\hat{M}$ as
$\hat{G}=\vec{n}^{T}\cdot\textbf{G}$ and $\hat{M}=\vec{m}^{T}\cdot\textbf{M}$,
where $\textbf{G},\textbf{M}$ are vectors of observables. The sensitivity,
i.e. the inverse of Eq. (5), can now be written as

$\displaystyle\chi^{-2}[\rho,\hat{G},\hat{M}]\equiv\frac{|\langle[\hat{G},\hat{M}]\rangle|^{2}}{\text{Var}[\hat{M}]}=\frac{|\vec{n}^{T}\textbf{C}[\rho,\textbf{G},\textbf{M}]\vec{m}|^{2}}{\vec{m}^{T}\bm{\Gamma}[\rho,\textbf{M}]\vec{m}}\;,$
(6)

where $\textbf{C}[\rho,\textbf{G},\textbf{M}]$ is the commutator matrix with
elements
$\left(\textbf{C}[\rho,\textbf{G},\textbf{M}]\right)_{ij}=-i\langle[\hat{G}_{i},\hat{M}_{j}]\rangle_{\rho}$,
and $\bm{\Gamma}[\rho,\textbf{M}]$ is the covariance matrix with elements
$\left(\bm{\Gamma}[\rho,\textbf{M}]\right)_{ij}=\text{Cov}\left[\hat{M}_{i},\hat{M_{j}}\right]_{\rho}$.
For a given probe state $\rho$, the maximum sensitivity can be obtained by
optimizing Eq. (6) over both vectors $\vec{n}$ and $\vec{m}$, which has been
proved to be equivalent as the maximum eigenvalue of the moment matrix
$\mathcal{M}\equiv\textbf{C}\bm{\Gamma}^{-1}\textbf{C}^{T}$ [22]. In symbols,
the maximum sensitivity is thus

$\displaystyle\chi^{-2}\equiv\max_{\vec{m},\vec{n}}\chi^{-2}[\rho,\hat{G},\hat{M}]=\lambda_{\text{max}}(\mathcal{M})\;,$
(7)

where the optimal direction $\vec{n}_{\text{opt}}$ for the generator is the
eigenvector corresponding to the maximum eigenvalue $\lambda_{\text{max}}$,
and the optimal vector for the measurement operator is
$\vec{m}_{\text{opt}}=\alpha\bm{\Gamma}^{-1}\textbf{C}^{T}\vec{n}_{\text{opt}}$,
with $\alpha$ a normalization constant.

If only linear quadratures measurements are taken into account, the vectors of
operators are $\textbf{G}^{(1)}=\textbf{M}^{(1)}=\\{\hat{X},\hat{P}\\}$. The
generator and the measurement can thus be expressed in terms of two angles,
$\phi$ and $\theta$, namely

$\displaystyle\hat{G}(\phi)=\left(\hat{a}e^{-i\phi}+\hat{a}^{\dagger}e^{i\phi}\right)/\sqrt{2}\;,$
(8)
$\displaystyle\hat{M}(\theta)=\left(\hat{a}e^{-i\theta}+\hat{a}^{\dagger}e^{i\theta}\right)/\sqrt{2}\;.$
(9)

For this choice, the maximum sensitivity is obtained when
$\vec{m}\perp\vec{n}$, so that it can be simplified as
$\chi_{(1)}^{-2}=(\min_{\theta}\text{Var}[\hat{M}(\theta)])^{-1}$.

One way to possibly increase the sensitivity is to include in the vector of
observables M higher-order measurement operators. Let us denote with
$\textbf{M}^{(k)}$ the vector including at most $k$-th order operators. For
example, the second-order $\textbf{M}^{(2)}$ is

$\displaystyle\textbf{M}^{(2)}=\\{\textbf{M}^{(1)},\hat{X}^{2},\hat{P}^{2},(\hat{X}\hat{P}+\hat{P}\hat{X})/2\\}\;,$
(10)

and the third-order $\textbf{M}^{(3)}$ is

$\displaystyle\textbf{M}^{(3)}=\\{\textbf{M}^{(2)},\hat{X}^{3},\hat{P}^{3},(\hat{X}\hat{P}\hat{P}+\hat{P}\hat{X}\hat{P}+\hat{P}\hat{P}\hat{X})/3,(\hat{P}\hat{X}\hat{X}+\hat{X}\hat{P}\hat{X}+\hat{X}\hat{X}\hat{P})/3\\}\;.$
(11)

Note here that, since the generator $\hat{G}$ is associated with a
displacement in phase-space, the vector $\textbf{G}=\textbf{G}^{(1)}$ is
always linear in the quadratures. Combining these object, the sensitivity is

$\displaystyle\chi_{(k)}^{-2}\equiv\max_{\vec{m},\vec{n}}\frac{|\langle[\hat{G},\hat{M}^{(k)}]\rangle|^{2}}{\text{Var}[\hat{M}^{(k)}]}=\lambda_{\text{max}}\left(\mathcal{M}^{(k)}\right)\;,$
(12)

where
$\mathcal{M}^{(k)}=\textbf{C}[\rho,\textbf{G},\textbf{M}^{(k)}]\bm{\Gamma}^{-1}[\rho,\textbf{G},\textbf{M}^{(k)}]\textbf{C}^{T}[\rho,\textbf{G},\textbf{M}^{(k)}]$
is the second-order moment matrix.

For a state $|\psi\rangle=e^{-i\hat{H}t}|0\rangle$, with $H$ the squeezed Kerr
oscillator Hamiltonian considered in the main text, the expectation value of a
linear quadrature measurement along any directions is zero, e.g.
$\langle\hat{X}\rangle=\langle\hat{P}\rangle=0$. Because of this, if we now
calculate the second-order sensitivity $\chi_{(2)}^{-2}$, we will find that
the commutator between the generator $\hat{G}(\phi)$ and any second-order
measurement operators is zero. The resulting commutator matrix is thus

$\displaystyle\textbf{C}[\rho,\textbf{G},\textbf{M}^{(2)}]=-i\left(\begin{matrix}0&1&0&0&0\\\
-1&0&0&0&0\end{matrix}\right)\;,$ (13)

meaning that the moment matrix will be $\mathcal{M}^{(2)}=\mathcal{M}^{(1)}$.
This result leads to the conclusion that, for the case we considered, second-
order measurements are insufficient to provide an advantage over linear
measurements, i.e. $\chi_{(2)}^{-2}=\chi_{(1)}^{-2}$. Hence, to see an
advantage, at least third-order measurements are required, which brings
experimental challenges for the massive measurement data and low detection
noise requirements.

##  VIII Quantum Fisher information

The quantum Fisher information (QFI) $F_{Q}[\rho,\hat{G}]$ quantifies the
sensitivity of a probe state $\rho$ with respect to a perturbation generated
by $\hat{G}$. It can be expressed as [43]

$\displaystyle F_{Q}[\rho,\hat{G}]=\sum_{\begin{subarray}{c}k,l\\\ \text{s.t.
}\lambda_{k}+\lambda_{l}>0\end{subarray}}\frac{(\lambda_{k}-\lambda_{l})^{2}}{(\lambda_{k}+\lambda_{l})}|\bra{k}\hat{G}\ket{l}|^{2}\;,$
(14)

where $\lambda_{k}$ are the eigenvalues of $\rho$, and $\ket{k}$ their
associated eigenvectors. For pure states $\rho=|\psi\rangle\langle\psi|$, Eq.
(14) is simply
$F_{Q}[|\psi\rangle,\hat{G}]=4\text{Var}[\hat{G}]_{|\psi\rangle}$, meaning
that the QFI is proportional to the variance of the generator. In our work,
the perturbations we consider are displacements of the state, meaning that the
associated generator is linear in the quadratures
$\hat{G}=\vec{n}\cdot\textbf{G}^{(1)}$. The maximum QFI is attained for
displacements along the state’s most sensitive direction, and is
$F_{Q}=\max_{\vec{n}}F_{Q}[\rho,\hat{G}]$. This maximization over $\vec{n}$
can be efficiently performed by computing the maximum eigenvalue of the
$2\times 2$ matrix with elements

$\displaystyle[\mathbf{F}_{Q}]_{ij}=2\sum_{\begin{subarray}{c}k,l\\\
\text{s.t.
}\lambda_{k}+\lambda_{l}>0\end{subarray}}\frac{(\lambda_{k}-\lambda_{l})^{2}}{(\lambda_{k}+\lambda_{l})}\langle
k|\hat{G}_{i}|l\rangle\langle l|\hat{G}_{j}|k\rangle\;,$ (15)

where $\hat{G}_{i}\in\\{\hat{X},\hat{P}\\}$, so that we have
$F_{Q}=\lambda_{\text{max}}(\mathbf{F}_{Q})$. We used this result to
efficiently compute the maximum QFI for mixed states, like for Fig. 4(a) in
the main text.

Eq. (14) can be computed analytically for Gaussian states, giving as result
[58]

$\displaystyle
F_{Q}[\rho_{d}]=\frac{1}{2(1+\mathcal{P}(\rho_{d})^{2})}\text{Tr}\left[(\bm{\Gamma}^{-1}[\rho_{d}]\frac{\partial}{\partial
d}\bm{\Gamma}[\rho_{d}])^{2})\right]+\left(\frac{\partial\langle\mathbf{\hat{r}}\rangle_{\rho_{d}}}{\partial
d}\right)^{T}\mathbf{\Gamma}^{-1}[\rho_{d}]\left(\frac{\partial\langle\mathbf{\hat{r}}\rangle_{\rho_{d}}}{\partial
d}\right)\;,$ (16)

where
$\mathcal{P}(\rho)=\text{Tr}[\rho^{2}]=(2\sqrt{\det\mathbf{\Gamma}[\rho]})^{-1}$
is the state purity,
$\langle\mathbf{\hat{r}}\rangle_{\rho_{d}}=(\langle\hat{X}\rangle_{\rho_{d}},\langle\hat{P}\rangle_{\rho_{d}})^{T}$
is the vector of first moments, and $\bm{\Gamma}^{-1}[\rho_{d}]$ is the
covariance matrix. In the most general case, a Gaussian state can be expressed
as a displaced squeezed thermal state, that is

$\rho(\alpha,\xi,\overline{n}_{T})=\hat{D}(\alpha)\hat{S}(\xi)\rho_{T}\hat{S}^{\dagger}(\xi)\hat{D}^{\dagger}(\alpha)\;,$
(17)

where $\rho_{T}$ is a thermal state with average number of thermal excitations
$\overline{n}_{T}$, $\hat{D}(\alpha)$ is the displacement operator, and
$\hat{S}(\xi)\equiv e^{(\xi^{*}\hat{a}^{2}-\xi\hat{a}^{\dagger 2})/2}$ is the
squeezing operator with $\xi=re^{i\zeta}$ . Gaussianity implies that these
states can be fully described by first and second moments of quadrature
operators. For this reason, a Gaussian state is fully determined by its
displacement vector

$\langle\mathbf{\hat{r}}\rangle_{\rho(\alpha,\xi,\overline{n}_{T})}=\begin{pmatrix}\sqrt{2}\Re[\alpha]\\\
\sqrt{2}\Im[\alpha]\end{pmatrix}\;,$ (18)

and the covariance matrix

$\Gamma[\rho(\alpha,\xi,\overline{n}_{T})]=\dfrac{(1+2\overline{n}_{T})}{2}\begin{pmatrix}\cosh(2r)-\sinh(2r)\cos\zeta&-\sinh(2r)\sin\zeta\\\
-\sinh(2r)\sin\zeta&\cosh(2r)+\sinh(2r)\cos\zeta\end{pmatrix}\;.$ (19)

For the case of phase-space displacements,
$\partial_{d}\bm{\Gamma}[\rho_{d}]=0$ because the covariance matrix is
invariant under translations. Eq. (16)

$\displaystyle F_{Q}[\rho,\hat{G}(\phi)]$
$\displaystyle=\left(\frac{\partial\langle\mathbf{\hat{r}}\rangle_{\rho_{d}}}{\partial
d}\right)^{T}\mathbf{\Gamma}^{-1}[\rho_{d}]\left(\frac{\partial\langle\mathbf{\hat{r}}\rangle_{\rho_{d}}}{\partial
d}\right)$
$\displaystyle=\frac{\vec{n}^{T}\mathbf{\Gamma}[\rho_{d}]\vec{n}}{\det\mathbf{\Gamma}[\rho_{d}]}\;,$
(20)

where $\vec{n}=(\cos\phi,\sin\phi)^{T}$, and
$\det\mathbf{\Gamma}[\rho]=\text{Var}[\hat{X}]\text{Var}[\hat{P}]-\text{Cov}[\hat{X},\hat{P}]^{2}$.
In the second line, we used
$\bm{A}^{-1}={\frac{1}{\det\bm{A}}}\Omega^{T}\bm{A}^{T}\Omega$ for a $2\times
2$ matrix $\mathbf{A}$ and $(\partial/\partial
d)\langle\hat{\bm{r}}\rangle_{\rho_{d}}=(-\sin\phi,\cos\phi)^{T}=\Omega^{T}\vec{n}$,
where $\Omega=\bigl{(}\begin{smallmatrix}0&1\\\ -1&0\end{smallmatrix}\bigr{)}$
is the symplectic form. Plugging in Eq. (VIII) the covariance matrix Eq. (19)
we finally obtain that the QFI of a Gaussian state for displacements is

$F_{Q}[\rho,\hat{G}(\phi)]=\dfrac{2\left(\cosh(2r)-\sinh(2r)\cos(\zeta-2\phi)\right)}{(1+2n_{T})}\;.$
(21)

The maximum of this expression is obtained for $\phi=\zeta/2-\pi/2$, and is

$F_{Q}=\dfrac{2e^{2r}}{(1+2\overline{n}_{T})}\;.$ (22)

If we consider a squeezed vacuum state (i.e. $\alpha=0$ and
$\overline{n}_{T}=0$), we can use the relation
$N\equiv\langle\hat{a}^{\dagger}\hat{a}\rangle=\sinh^{2}r$ to derive how the
QFI scales with the average number of particles. This is

$F_{Q}=2\left(1+2N+2\sqrt{N(N+1)}\right)\simeq(4+8N)\quad\text{for}\;N\rightarrow\infty\;.$
(23)

##  IX Optimality of linear quadrature measurements

We might ask whether linear measurements are optimal for detecting
displacements using Gaussian states, in the sense that they allow to saturate
the Cramér-Rao bound $\chi^{-2}=F_{Q}$.

In Eq. (21) we have computed the QFI for Gaussian states under displacements.
We now proceed with calculating $\chi^{-2}$ as defined in Eq. (7).

First, considering a generator $\hat{G}=\vec{n}^{T}\cdot\\{\hat{X},\hat{P}\\}$
with $\vec{n}=(\cos\phi,\sin\phi)^{T}$ and a measurement
$\hat{M}=\vec{m}^{T}\cdot\\{\hat{X},\hat{P}\\}$ with
$\vec{m}=(\cos\theta,\sin\theta)^{T}$ , we have
$|\langle[\hat{G},\hat{M}]\rangle|^{2}=\sin^{2}(\theta-\phi)$. Then, the
variance in the denominator of Eq. (6) is simply
$\text{Var}[\hat{M}]=\vec{m}^{T}\bm{\Gamma}[\rho,\textbf{M}]\vec{m}$, with the
covariance matrix given by Eq. (19). Combining these results imply that the
linear sensitivity to displacements of a Gaussian state is

$\displaystyle\chi^{-2}[\rho,\hat{G}(\phi),\hat{M}(\epsilon)]=\dfrac{2\sin^{2}(\theta-\phi)}{(1+2\overline{n}_{T})\left(\cosh(2r)-\cos(\zeta-2\theta)\sinh(2r)\right)}\;.$
(24)

It is now easy to check that by choosing $\theta=\zeta/2$, such that the
denominator is minimized, and $\phi=\zeta/2-\pi/2$, such that the numerator is
maximized, we obtain for Eq. (7)

$\displaystyle\chi^{-2}=\dfrac{2e^{2r}}{(1+2\overline{n}_{T})}\;.$ (25)

This result coincides with Eq. (22), which allows us to conclude that linear
measurements are optimal for sensing displacements with Gaussian states.

##  X MAI method for Gaussian states

In the scenario of displacement sensing using Gaussian states, Eq. (21) shows
us that the sensitivity be increased by the action of the squeezing operator

$\displaystyle\hat{S}=e^{(\xi^{*}\hat{a}^{2}-\xi\hat{a}^{\dagger 2})/2}\;,$
(26)

where $\xi=re^{i\zeta}$ and $r$ determines squeezing extent and $\zeta$
defines the squeezing direction. The action of this operator is

$\displaystyle\hat{S}^{\dagger}\hat{a}\hat{S}$
$\displaystyle=\cosh(r)\hat{a}-e^{i\zeta}\sinh(r)\hat{a}^{\dagger}\;,$ (27)
$\displaystyle\hat{S}^{\dagger}\hat{a}^{\dagger}\hat{S}$
$\displaystyle=\cosh(r)\hat{a}^{\dagger}-e^{-i\zeta}\sinh(r)\hat{a}\;.$ (28)

For Gaussian states, the anti-squeezing and squeezing directions corresponds
to the optimal generator $\hat{G}_{\text{opt}}$ and the optimal measurement
$\hat{M}_{\text{opt}}$, respectively. As these two directions are orthogonal,
we have $|[\hat{G}_{\text{opt}},\hat{M}_{\text{opt}}]|=1$, which will be
useful later. Note that, upon action of the squeezing operator, these two
directions transform as

$\displaystyle\hat{S}^{\dagger}\hat{G}_{\text{opt}}\hat{S}$
$\displaystyle=e^{r}\hat{G}_{\text{opt}}\;,$ (29)
$\displaystyle\hat{S}^{\dagger}\hat{M}_{\text{opt}}\hat{S}$
$\displaystyle=e^{-r}\hat{M}_{\text{opt}}\;.$ (30)

Let us consider now a pure Gaussian state
$|\psi_{S}\rangle=\hat{D}\hat{S}|0\rangle$, but keeping in mind that the
following discussion can easily be generalized to mixed Gaussian states. Since
the action of the displacement operators $\hat{D}(\alpha)$ gives no
contribution to the sensitivity, we can consider only the case of squeezed
vacuum states $\hat{S}|0\rangle$, whose optimal sensitivity to displacements
is

$\displaystyle\chi^{-2}$
$\displaystyle=\frac{|\langle[\hat{G}_{\text{opt}},\hat{M}_{\text{opt}}]\rangle|^{2}}{\text{Var}[\hat{M}_{\text{opt}}]}$
$\displaystyle=\frac{1}{\langle
0|\hat{S}^{\dagger}\hat{M}_{\text{opt}}^{2}\hat{S}|0\rangle-\langle
0|\hat{S}^{\dagger}\hat{M}_{\text{opt}}\hat{S}|0\rangle}$
$\displaystyle=\frac{1}{\langle
0|\left(\hat{S}^{\dagger}\hat{M}_{\text{opt}}\hat{S}\right)\left(\hat{S}^{\dagger}\hat{M}_{\text{opt}}\hat{S}\right)|0\rangle-\langle
0|\hat{S}^{\dagger}\hat{M}_{\text{opt}}\hat{S}|0\rangle^{2}}$
$\displaystyle=\frac{1}{e^{-2r}\left(\langle
0|\hat{M}_{\text{opt}}^{2}|0\rangle-\langle
0|\hat{M}_{\text{opt}}|0\rangle^{2}\right)}$ $\displaystyle=2e^{2r}\;.$ (31)

We are now interested in comparing the sensitivity Eq. (31) obtained by
performing linear quadrature measurements, with the one achieved by a MAI
protocol. In a MAI protocol, the optimal generator is still the same
$\hat{G}_{\text{opt}}$ as before, but the optimal measurement operator
$\hat{M}_{\text{MAI}}$ takes the form

$\displaystyle\hat{M}_{\text{MAI}}$
$\displaystyle=\hat{S}\hat{M}_{\text{opt}}\hat{S}^{\dagger}\;.$ (32)

The sensitivity of the pure Gaussian state $|\psi_{S}\rangle$ becomes

$\displaystyle\chi^{-2}_{\text{MAI}}$
$\displaystyle=\frac{|\langle[\hat{G}_{\text{opt}},\hat{M}_{\text{MAI}}]\rangle|^{2}}{\text{Var}[\hat{M}_{\text{MAI}}]}$
$\displaystyle=\frac{|\langle
0|\hat{S}^{\dagger}[\hat{G}_{\text{opt}},\hat{S}\hat{M}_{\text{opt}}\hat{S}^{\dagger}]\hat{S}|0\rangle|^{2}}{\langle
0|\hat{S}^{\dagger}\left(\hat{S}\hat{M}_{\text{opt}}\hat{S}^{\dagger}\right)\left(\hat{S}\hat{M}_{\text{opt}}\hat{S}^{\dagger}\right)\hat{S}|0\rangle-\langle
0|\hat{S}^{\dagger}\left(\hat{S}\hat{M}_{\text{opt}}\hat{S}^{\dagger}\right)\hat{S}|0\rangle^{2}}$
$\displaystyle=\frac{|\langle
0|\hat{S}^{\dagger}\hat{G}_{\text{opt}}\hat{S}\hat{M}_{\text{opt}}|0\rangle-\langle
0|\hat{M}_{\text{opt}}\hat{S}^{\dagger}\hat{G}_{\text{opt}}\hat{S}|0\rangle|^{2}}{\langle
0|\hat{M}_{\text{opt}}^{2}|0\rangle-\langle
0|\hat{M}_{\text{opt}}|0\rangle^{2}}$ $\displaystyle=\frac{|e^{r}\langle
0|[\hat{G}_{\text{opt}},\hat{M}_{\text{opt}}]|0\rangle|^{2}}{1/2}$
$\displaystyle=2e^{2r}\;.$ (33)

Perhaps surprisingly, we have obtained the same result as Eq. (31), meaning
$\chi^{-2}=\chi^{-2}_{\text{MAI}}$. This shows that for pure Gaussian states,
the MAI protocol does not actually provide any increase in the sensitivity.
The same conclusion $\chi^{-2}=\chi^{-2}_{\text{MAI}}$ can be derived for
mixed Gaussian states Eq. (17), since a simple calculation gives

$\displaystyle\chi^{-2}=\frac{2e^{2r}}{(1+2\overline{n}_{T})}\;,$ (34)
$\displaystyle\chi_{\text{MAI}}^{-2}=\frac{2e^{2r}}{(1+2\overline{n}_{T})}\;,$
(35)

where we used the thermal state variance
$\text{Var}[\rho_{T},\hat{M}]=(1+2\overline{n}_{T})/2$.

##  XI MAI advantage in the presence of detection noise

In the presence of noise in the detection, the measurement of $M$ can be
written as $\tilde{M}=\hat{M}+\Delta M$, where $\Delta M$ is a random variable
describing the noise. This variable follows the Gaussian distribution with
mean $\langle\Delta M\rangle=0$ and variance $\langle(\Delta
M)^{2}\rangle=\sigma^{2}$.

In the typical scenario without MAI, the sensitivity to displacements for
linear quadrature measurements with detection noise is

$\displaystyle\chi^{-2}$
$\displaystyle=\frac{|\langle[\hat{G}_{\text{opt}},\tilde{M}_{\text{opt}}]\rangle|^{2}}{\text{Var}[\rho,\tilde{M}_{\text{opt}}]}$
$\displaystyle=\frac{|\langle[\hat{G}_{\text{opt}},\hat{M}_{\text{opt}}]\rangle+\langle[\hat{G}_{\text{opt}},\Delta
M]\rangle|^{2}}{\langle\left(\hat{M}_{\text{opt}}+\Delta
M\right)^{2}\rangle-\langle\hat{M}_{\text{opt}}+\Delta M\rangle^{2}}$
$\displaystyle=\frac{|\langle[\hat{G}_{\text{opt}},\hat{M}_{\text{opt}}]\rangle|^{2}}{\left(\langle\hat{M}_{\text{opt}}^{2}\rangle+2\langle\hat{M}_{\text{opt}}\rangle\langle\Delta
M\rangle+\langle\Delta
M^{2}\rangle\right)-\left(\langle\hat{M}_{\text{opt}}\rangle+\langle\Delta
M\rangle\right)^{2}}$
$\displaystyle=\frac{1}{\text{Var}[\rho,\hat{M}_{\text{opt}}]+\sigma^{2}}\;.$
(36)

In the scenario with MAI protocol, the measurement with detection noise can be
expressed as $\tilde{M}_{\text{MAI}}=\hat{U}(\hat{M}+\Delta
M)\hat{U}^{\dagger}=\hat{M}_{\text{MAI}}+\Delta M$ with
$\hat{U}=e^{-i\hat{H}t}$, giving the sensitivity

$\displaystyle\chi_{\text{MAI}}^{-2}$
$\displaystyle=\frac{|\langle[\hat{G}_{\text{opt}},\tilde{M}_{\text{MAI}}]\rangle|^{2}}{\text{Var}[\rho,\tilde{M}_{\text{MAI}}]}$
$\displaystyle=\frac{|\langle[\hat{G}_{\text{opt}},\hat{M}_{\text{MAI}}]\rangle+\langle[\hat{G}_{\text{opt}},\Delta
M]\rangle|^{2}}{\langle
0|\hat{U}^{\dagger}\hat{U}\tilde{M}_{\text{opt}}\hat{U}^{\dagger}\hat{U}\tilde{M}_{\text{opt}}\hat{U}^{\dagger}\hat{U}|0\rangle-\langle
0|\hat{U}^{\dagger}\hat{U}\tilde{M}_{\text{opt}}\hat{U}^{\dagger}\hat{U}|0\rangle^{2}}$
$\displaystyle=\frac{|\langle[\hat{G}_{\text{opt}},\hat{M}_{\text{MAI}}]\rangle|^{2}}{\langle
0|\tilde{M}_{\text{opt}}^{2}|0\rangle-\langle
0|\tilde{M}_{\text{opt}}|0\rangle^{2}}$
$\displaystyle=\frac{|\langle[\hat{G}_{\text{opt}},\hat{M}_{\text{MAI}}]\rangle_{\rho}|^{2}}{\text{Var}[\rho_{0},\hat{M}_{\text{opt}}]+\sigma^{2}}\;.$
(37)

Following the results presented in Section IV, Eqs. (36,37) can be computed
analytically in the Gaussian case, i.e. when $K=\Delta=0$ in $\hat{H}$, such
that $r=2\epsilon t$. For a pure Gaussian state, since the minimum variance is
$\text{Var}[\rho,\hat{M}]=e^{-2r}/2$, the linear sensitivity is

$\chi^{-2}=\frac{1}{\frac{1}{2}e^{-2r}+\sigma^{2}}\;.$ (38)

To calculate the MAI sensitivity, using Eq. (X) we have

$\chi_{\text{MAI}}^{-2}=\frac{e^{2r}}{\frac{1}{2}+\sigma^{2}}\;.$ (39)

We thus have that, in the presence of detection noise, the improvement in
sensitivity due to the MAI protocol is

$\dfrac{\chi_{\text{MAI}}^{-2}}{\chi^{-2}}=\dfrac{1+2e^{2r}\sigma^{2}}{1+2\sigma^{2}}\;.$
(40)

This ratio is grater than one for $\sigma^{2}>0$ and $r>0$, and in the case of
$\sigma^{2}\gg 1$ is $e^{2r}$. Again, as pointed out in Section IV, in the
absence of detection noise $\sigma=0$ and there is no advantage in using a MAI
protocol.

For mixed Gaussian states, the sensitivities in the presence of detection
noises are

$\displaystyle\chi^{-2}$
$\displaystyle=\frac{1}{\frac{(1+2\overline{n}_{T})}{2}e^{-2r}+\sigma^{2}}\;,$
(41) $\displaystyle\chi_{\text{MAI}}^{-2}$
$\displaystyle=\frac{e^{2r}}{\frac{(1+2\overline{n}_{T})}{2}+\sigma^{2}}\;.$
(42)

The same conclusion as for the pure state case can be can be obtained by
looking at the ratio

$\displaystyle\dfrac{\chi_{\text{MAI}}^{-2}}{\chi^{-2}}=\dfrac{1+2\overline{n}_{T}+2e^{2r}\sigma^{2}}{1+2\overline{n}_{T}+2\sigma^{2}}\geq
1\;.$ (43)

##  XII QFI scaling

Figure 5: Sensitivity scaling with $N$. Panel a): Sensitivities $F_{Q}$,
$\chi_{\text{MAI}}^{-2}$, $\chi^{-2}$ as functions of the average number of
excitations $N$ for the same parameters as in Fig. 4 in the main text, namely
$\epsilon/K=2$ and $\Delta/K=0$. Panel b): $F_{Q}$ as a function of $N$ for
different $\epsilon/K$ and $\Delta/K=0$. Here, data points are truncated when
the first maximum of $F_{Q}$ is reached. Solid lines are a linear fit of the
last 20% of points with the model $F_{Q}=aN+4$. Panel c): dependence of the
scaling coefficient $a$ on $\epsilon/K$. In the limit
$\epsilon/K\rightarrow\infty$ we expect $a=8$.

Here, we are interested in investigating how $F_{Q}$, $\chi_{\text{MAI}}^{-2}$
and $\chi^{-2}$ scale with the average number of excitations
$N=\langle\hat{a}^{\dagger}\hat{a}\rangle$ in the system. Contrary to a simple
squeezing transformation, where the one-to-one correspondence between
interaction time and $N$ allows us to derive the scaling Eq. (23), the
presence of a Kerr nonlinearity in our system significantly complicates the
scaling analysis. When the vacuum state $\ket{0}$ evolves according to the
squeezed-Kerr Hamiltonian Eq. (1) in the main text, given values of $\Delta/K$
and $\epsilon/K$ result in a maximum average number of excitations. Moreover,
the evolution of $N$ is in general not monotonic in time.

Taking as an example the scenario presented in Fig. 4 in the main text, we
plot in Fig. 5a the sensitivities as a function of $N$. For this, note that we
stop plotting points after the maximum $N$ (i.e. $N\approx 2.4$) is reached.
We can see that the QFI scales similarly to the ideal squeezing case (black
line, i.e. Eq. (23)), and that the scaling of MAI is also comparable. On the
other hand, the sensitivity obtained from a linear estimate does not show a
favorable scaling with $N$.

The fact that for the scenario we are considering $N$ cannot be arbitrary
large significantly complicates the study of these scalings. Nevertheless, in
order to gain some insight, we proceed in the following way. We first set a
value for $\epsilon/K$, and fix $\Delta/K=0$ for simplicity. Then, we compute
QFI and $N$ for the states $e^{-iHt}\ket{0}$ at many different times $t$. We
repeat this step for several different $\epsilon/K$, and plot the results in
Fig. 5b. The last 20% of points before the maximum QFI value are fitted with
the model $aN+4$, with free parameter $a$. This is motivated by the fact that
the QFI scaling for an ideal squeezed state is $8N+4$, Eq. (23). The values of
$a$ extracted for different values of $\epsilon/K$ are plotted in Fig. 5c,
which shows how the scaling depends on the squeezing rate and Kerr
nonlinearity. We expect the trend to saturate at $a=8$ in the limit
$\epsilon/K\rightarrow\infty$.

##  XIII Robustness to losses

Figure 6: Robustness to losses. Maximum of the sensitivities $F_{Q}$,
$\chi_{\text{MAI}}^{-2}$ and $\chi^{-2}$ as a function of energy relaxation
rate $\gamma/K$, for the same parameters as in Fig. 4 in the main text. The
dashed black line represents the standard quantum limit.

In the analysis of our metrological protocol, we were interested in making
sure that MAI method was giving an advantage also in the presence of losses,
which are inevitable experimentally. Our simulations showed that high levels
of losses are required before the MAI sensitivity would fall below the shot
noise limit, or below the linear measurements sensitivity, see Fig. 4 of the
main text. In this sense, we say that the MAI method is robust to losses.

To further investigate how $F_{Q}$, $\chi_{\text{MAI}}^{-2}$ and $\chi^{-2}$
depend on the level of losses, we look at how the maxima of these quantities
change as a function of $\gamma/K$. This is plotted in Fig. 6. Even if the
linear measurements sensitivity, $\chi^{-2}$, is almost unaffected by the
amount of losses considered, note that this is still surpassed by
$\chi_{\text{MAI}}^{-2}$ for a wide range of experimentally relevant
$\gamma/K$. We thus conclude that the MAI method can indeed provide an
advantage over the linear measurement in concrete experimental scenarios.


# The Be/neutron star system Swift J004929.5-733107 in the Small Magellanic
Cloud - X-ray characteristics and optical counterpart candidates

M. J. Coe1, J. A. Kennea2, P. A. Evans3, L. J. Townsend4,5, A. Udalski6, I. M.
Monageng4,7 and D. A. H. Buckley4,5
1Physics & Astronomy, The University of Southampton, SO17 1BJ, UK
2Department of Astronomy and Astrophysics, The Pennsylvania State University,
525 Davey Lab, University Park, PA 16802, USA
3University of Leicester, Astronomy Research Group, School of Physics &
Astronomy, University Road, Leicester LE1 7RH, UK
4South African Astronomical Observatory, PO Box 9, Observatory, 7935, Cape
Town, South Africa
5Southern African Large Telescope, PO Box 9, Observatory, 7935, Cape Town,
South Africa
6Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478
Warszawa, Poland
7Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch
7701, South Africa
E-mail<EMAIL_ADDRESS>
###### Abstract

Swift J004929.5-733107 is an X-ray source in the Small Magellanic Cloud (SMC)
that has been reported several times, but the optical counterpart has been
unclear due to source confusion in a crowded region of the SMC. Previous works
proposed [MA93] 302 as the counterpart, however we show here, using data
obtained from the S-CUBED project, that the X-ray position is inconsistent
with that object. Instead we propose a previously unclassified object which
has all the indications of being a newly identified Be star exhibiting strong
H$\alpha$ emission. Evidence for the presence of significant I-band
variability strongly suggest that this is, in fact, a Be type star with a
large circumstellar disk. Over 18 years worth of optical monitoring by the
OGLE project reveal a periodic modulation at a period of 413d, probably the
binary period of the system. A SALT optical spectrum shows strong Balmer
emission and supports a proposed spectral classification of B1-3 III-IVe. The
X-ray data obtained from the S-CUBED project reveal a time-averaged spectrum
well fitted by a photon index $\Gamma=0.93\pm 0.16$. Assuming the known
distance to the SMC the flux corresponds to a luminosity $\sim 10^{35}$
erg-s-1. All of these observational facts suggest that this is confirmed as a
Be star-neutron star X-ray binary (BeXRB) in the SMC, albeit one with an
unusually long binary period at the limits of the Corbet Diagram.

###### keywords:

stars: emission line, Be X-rays: binaries

††pubyear: 2021††pagerange: The Be/neutron star system Swift J004929.5-733107
in the Small Magellanic Cloud - X-ray characteristics and optical counterpart
candidates–References

##  1 Introduction

BeXRB are a large sub-group of the well-established category of High Mass
X-ray Binaries (HMXB) characterised by being a binary system consisting of a
massive mass donor star, normally an OBe type, and an accreting compact
object, a neutron star. They are particularly prevalent in the Small
Magellanic Cloud (SMC) which contains the largest known collection of BeXRBs .
Whilst catalogues have been produced listing such systems in the SMC (for
example Coe & Kirk (2015); Haberl & Sturm (2016)), it clear that there are
many more systems yet to be identified. This lack of completeness primarily
arises from the transient nature of their X-ray emission, but also sometimes
from an inaccurate X-ray position making the identity of the optical
counterpart difficult in crowded stellar fields like the SMC. Since the
identity and numbers of HMXBs are important tools in our understanding of star
formation in a low metalicity environments like the SMC (Shtykovskiy &
Gilfanov, 2005), it is important to continue the search for such systems.

In this paper we report on both X-ray and optical recent photometry of the
X-ray system Swift J004929.5-733107. This X-ray source has been previously
reported several times, see for example Sasaki et al. (2000); Haberl et al.
(2008); Haberl & Sturm (2016); Kennea et al. (2018). The possible association
with the nearby proposed Be star [MA93] 302 (Meyssonnier & Azzopardi, 1993)
was first suggested by Haberl & Sasaki (2000) and reiterated in subsequent
publications.

In this work new observations are reported of this X-ray object as part of the
S-CUBED regular monitoring of the SMC (Kennea et al., 2018) using the Neil
Gehrels Swift Observatory (Gehrels et al., 2004). By combining all the many
detections it has been possible to produce an accurate source location and
X-ray lightcurve. The new position excludes the previous proposed optical
counterpart [MA93] 302 and instead points to another nearby star. This star
has not been previously identified nor classified, partially because of some
source confusion with a third nearby star. To resolve this confusion optical
spectra of all three stars were obtained under excellent seeing conditions
with the Southern African Large Telescope (SALT). These spectra combined with
the new accurate X-ray position resolve the previous ambiguity over which star
is the correct counterpart to Swift J004929.5-733107.

In addition historical optical photometric data are reported here from the
OGLE project (Udalski et al., 2015) showing evidence for a probable binary
period of 413d. Furthermore, significant (V–I) colour changes are reported
which are suggested to be related to the changing circumstances in a
circumstellar disk. Therefore it is proposed here that this is the correct
optical counterpart to Swift J004929.5-733107 and it is a newly identified Be
star in the SMC.

##  2 Observations

###  2.1 S-CUBED and Swift Archival Observations

Swift J004929.5-733107 was detected by the S-CUBED survey (Kennea et al.,
2018), a shallow weekly X-ray survey of the optical extent of the SMC by the
Swift X-ray Telescope (XRT; Burrows et al. 2005). Individual exposures in the
S-CUBED survey are typically 60s long, and occur weekly, although
interruptions can occur due to scheduling constraints. Starting with the
S-CUBED observation taken on MJD 57722 (29 Nov 2016), S-CUBED detected it on
many subsequent occasions and internally numbered it SC404.

Through its lifetime, _Swift_ has observed the location of this source on many
occasions in addition to the S-CUBED survey, giving a total of 96 ks of Photon
Counting (PC) mode exposure at the source location. Using the online XRT
analysis tools111https://www.swift.ac.uk/user_objects (Evans et al., 2009) a
spectrum and a position were obtained by stacking all of these data. The
resulting stacked X-ray image is shown in Fig 1. The best position was the
‘astrometric position’, which was obtained by aligning XRT field sources with
2MASS objects (see Evans et al. 2014 for details), and is:

RA(2000)=00h 49m 30.49s, Dec(2000)=-73∘ 31’ 09.3", error radius 1.3" (90%
confidence limit)

We note that this position was covered by a _Chandra_ observation on 2014
April 6 (PI: Predehl). Examination of these data show that SC404 was detected,
at a position that is consistent with this reported S-CUBED position to within
0.5".

The summed PC mode X-ray spectrum is well described by an absorbed power-law
model. With $N_{H}$ fixed at a standard value of $5.2\times 10^{21}\
\mathrm{cm}^{-2}$ Willingale et al. (2013), a photon index of $\Gamma=0.93\pm
0.16$ with a reduced $\chi^{2}=0.89$ (14 degrees of freedom) is found. It is
noted that this hard X-ray spectrum is consistent with other BeXRBs in the
SMC, and given that the companion is a Be-star (see below), this is highly
suggestive that Swift J004929.5-733107 is, in fact, a BeXRB system.

Assuming a standard SMC distance of 62 kpc (Scowcroft et al., 2016) and
correcting for absorption fixed at the value derived from Willingale et al.
(2013), this corresponds to a 0.5-10 keV luminosity of $(9.8\pm 1.7)\times
10^{34}$ erg-$s^{-1}$.

S-CUBED observations taken with the UV/Optical Telescope (UVOT; Roming et al.
2005) were also analysed, utilizing the standard uvotmaghist tool. In addition
to S-CUBED observations we also analyzed UVOT data taken in archival
observations of the field in the same way. Note that S-CUBED UVOT data is
taken entirely utilizing the uvw1 filter, and in order to compare the the
brightness of the source in archival data with those in S-CUBED, we only
present observations taken utilizing that filter.

Figure 1: A deep S-CUBED stacked X-ray image of the field of Swift
J004929.5-733107 which is centred at a position of RA(2000)=00h 49m 30.49s,
Dec(2000)=-73∘ 31’ 09.3". The central object is the subject of this paper and
is consistent with a single point source. The size of the field shown is 12 x
12 arcmin. The source near the western edge of the field is the symbiotic star
system RX J0048.4-7332 (Orio et al., 2007). Figure 2: OGLE I band image of a
66.5 x 66.5 arcsecond field with the three nearby optical objects to Swift
J004929.5-733107 identified. The 90% uncertainty of the X-ray position of
Swift J004929.5-733107 is shown by the circle.

###  2.2 OGLE

Figure 3: OGLE III and IV data for the three candidates for Swift
J004929.5-733107 discussed in this work. The gap in the data around TJD 5000
indicates the transition from OGLE III to OGLE IV. The vertical scale in each
case is the I-band magnitude. Candidate 2 is here proposed to be the correct
optical counterpart.

The OGLE project (Udalski et al., 2015) provides long term I-band photometry
with a cadence of 1-3 days.

Fig 2 is an OGLE I band image and it can be seen that there are 3 objects
close to the position of the X-ray source. The northernmost one, labelled
Candidate 1 in the figure, is [MA93] 302 (Meyssonnier & Azzopardi, 1993) which
has previously been proposed as the counterpart to this X-ray source (Kennea
et al., 2018; Haberl & Sturm, 2016). However, the new much-improved X-ray
position from this work conclusively rules this out now. The southernmost
object, Candidate 3, is probably also ruled out as it lies outside the 90%
uncertainty radius of the X-ray position. Long term data from all three
candidates are presented in Fig 3.

It is therefore concluded that, based upon position, that Candidate 2 is the
most likely optical counterpart to Swift J004929.5-733107.

####  2.2.1 Candidate 1

For Candidate 1, [MA93] 302 (Meyssonnier & Azzopardi, 1993), the OGLE
identifications are SMC 103.5.17819 and SMC 720.20.4879D for OGLE III and IV
respectively. The OGLE III & IV data were investigated for any periodic
behaviour which might indicate the binary period of the system. Lomb-Scargle
techniques were applied to the de-trended data and periods investigated over
the range : 1.1 – 1000d. No significant periodic features were revealed. In
fact over the 18 year period covered by the OGLE observations no fluctuations
greater than 0.05 magnitudes in the I-band were observed. A behaviour pattern
very different to that seen in almost all other Be stars in BeXRB systems.

####  2.2.2 Candidate 2

For Candidate 2 the OGLE identifications are SMC 103.5.17775 and SMC
720.20.4843D for OGLE III and IV respectively. It is difficult to obtain the
brightness of this object at different wavebands from the literature because
of the danger of source confusion with Candidate 3. However, OGLE has observed
this object also in V band during OGLE III and provides average values of V=
14.65 and I=14.45. It is clear that the OGLE data for this source show a long-
term modulation often characteristic of Be stars. This slow modulation is
related to the growth and decay of the circumstellar disk which makes a
significant contribution to the I-band light.

The OGLE III & IV data were de-trended wit a polynomial function and
investigated for any periodic behaviour which might indicate the binary period
of the system. Lomb-Scargle techniques were applied and periods investigated
over the range : 1.1 – 1000d. The former, OGLE III showed no significant peaks
in the power spectrum, but the latter, OGLE IV, revealed a strong peak at
413$\pm$7d. To check that this feature was not associated in any way with a
window function such as the the annual sampling pattern, the simultaneously
acquired OGLE data for Candidate 3 were also analysed in the same manner and
then the two power spectra were subtracted from each other. The result is
shown in Fig 4. The strong peak at 413d is unaffected by this process
indicating that it was unrelated to the data structure. It can also be seen
from this figure that the 413d peak is well separated from any structure
associated with 1 year that may have survived the subtraction process.

Figure 4: Lomb-Scargle timing analysis of the OGLE data for Candidate 2. The
peak is at a period of $413\pm 7$d; the dashed line shows the frequency
corresponding to a period of 1 year.

To investigate the nature of the 413d modulation the OGLE data were then
folded with an ephemeris

$T_{0}$ = 2086.90 + n(413.22) TJD.

The result is shown in Fig 5 and reveals a sinusoidal type of modulation with
an amplitude of $\sim$0.06 mags. This is further discussed below.

Figure 5: OGLE data for Candidate 2 folded at a period of 413d. Top panel
shows the binned data repeated over 2 cycles. Lower panel shows the scatter of
the data points over just one cycle. In both cases the data have been de-
trended with a polynomial fit and the fit then subtracted from the data.

####  2.2.3 Candidate 3

For Candidate 3 the OGLE identifications are SMC 103.5.17780 and SMC
720.20.4842D for OGLE III and IV respectively. Lomb-Scargle analysis of these
data reveals strong peaks at 1.4512d and 3.188d  \- see Fig 6. The latter
period is interpreted as the beat period between the former period and the
sidereal daily period. More careful inspection of the OGLE data reveal that
the true period is, in fact, 2.902d with the folded data revealing an almost
double sinusoidal profile - see Figs 7 and 8.

This system is very likely an ellipsoidal O-type star in a binary system. The
secondary is perhaps a compact object, but more likely a non-interacting
companion (at least the OGLE data show no evidence for interacting behaviour),
maybe a much fainter lower mass star. Several such systems are seen in the SMC
(Pawlak et al., 2016).

Figure 6: Power spectrum of the OGLE IV data for Candidate 3 over the period
range 1–100d.  Figure 7: OGLE III and IV data for Candidate 3 folded at the
period of 2.902d.  Figure 8: OGLE III and IV data for Candidate 3 folded at
the period of 2.902d without the error bars and only showing one cycle.

###  2.3 SALT

The three stars discussed above were observed with the Southern African Large
Telescope (SALT) to help confirm Candidate 2 as the true optical counterpart
to Swift J004929.5-733107. A single broadband spectrum was taken on 22
December 2020 as part of the SALT Transients large science programme. The
observation was made with the Robert Stobie Spectrograph (RSS) in standard
long-slit mode using the PG0900 grating and a 2" slit. The exposure time was
300 s. The primary data reduction steps (which include overscan correlation,
bias subtraction, gain and amplifier cross-talk corrections) were performed
using the SALT science pipeline (Crawford et al. 2010), while the secondary
steps (wavelength calibration, background subtraction and extraction of the
one-dimensional spectrum) were performed using standard IRAF routines. A
second 300 s exposure was made on the same night with the same setup, but
unfortunately the seeing worsened causing the three target traces to become
blurred, making this observation unusable. Fig. 9 shows part of the 2-D image
obtained with SALT during the first observation. The spectral traces of
Candidates 1, 2 and 3 can been seen in the centre of the image, running from
top to bottom respectively. The bright emission is H-alpha, which can be
clearly associated with Candidate 2. This is further strong evidence that
Candidate 2 is the true counterpart - an emission line star.

Figure 9: The region around H-alpha on the 2-D image taken with SALT. The
spectral traces of candidates 1, 2 and 3 shown in Fig. 2 can been seen in the
centre of the image, running from top to bottom respectively. The H-alpha
emission reported in section 2.3 can be clearly associated with candidate 2.

Figure 10: Top: SALT spectrum of candidate 2, with the Balmer lines labelled.
Data taken on 20201222. The chip gaps around 5000 Å and 6000 Å have been
interpolated over (highlighted in grey). Bottom: Normalised blue end section
of the spectrum.

The close proximity of the three stars makes spectral extraction difficult. A
simple projection across the 3 traces in Fig. 9 shows significant overlap
between Candidates 2 and 3, and moderate overlap between Candidates 1 and 2.
The spectra of all three stars were extracted as well as possible with these
restrictions in mind. Due to its faintness, the spectrum of Candidate 1 showed
significant contamination from Candidate 2. Strong H$\alpha$ emission is
present in Candidate 1, which we believe to be contamination from Candidate 2.
Aside from the contamination, Candidate 1 is very red. So much so that the
trace is completely lost blue-ward of around 5000 Å, making characterisation
impossible. We are, however, highly confident that this star (previously
identified in the literature as the optical counterpart to Swift
J004929.5-733107) cannot be associated with the X-ray source.

The spectrum of Candidate 3 is much cleaner than Candidate 1, despite being
closer to Candidate 2. This is likely due to its brightness being more
comparable to Candidate 2. There is moderate H$\alpha$ emission present, which
can, again, be explained as contamination from Candidate 2. The overall
spectrum of Candidate 3 is very blue and shows strong Balmer and He I lines in
absorption. We can also identify several weak lines of ionised oxygen and
helium, suggesting that Candidate 3 is a late O-type star. Combined with the
presence of a 2.9 d period, this makes Candidate 3 an intriguing object.
However, we suggest that it is unlikely to be the true counterpart of the
X-ray source as its position lies significantly outside the Swift 90%
uncertainty.

Fig. 10 shows the full SALT spectrum of Candidate 2 (top) and a zoom-in to the
blue end (bottom). Immediately obvious is the presence of extremely strong
H$\alpha$ emission (as can be seen in the 2-D image - see Fig. 9 ), as well as
weaker emission in H$\beta$ and H$\gamma$, and a rather featureless continuum.
This is indicative of an early-type star with a circumstellar disc, providing
strong evidence that this system is definitely a BeXRB in the SMC. The Balmer
line emission is discussed further in Section 3.2, alongside the spectral
classification and limitations of our analysis based on the likely
contamination of the spectrum from Candidate 3.

Figure 11: Candidate 2 observational data. Top panel : OGLE I-band data;
second panel : OGLE V-band data; third panel : Swift X-ray 2-10 keV; lowest
panel : Swift UVOT data in the uvw1 filter. XRT and UVOT observations before
TJD 7500 are from historical Swift observations prior to the start of the
S-CUBED project.

##  3 Discussion

Since it is extremely likely that Candidate 2 is the correct optical
counterpart to Swift J004929.5-733107 the discussion is restricted to just
that object.

###  3.1 Long term variability

Fig 11 shows the combined data set for this source during the period JD
2453000-2459000: two OGLE bands (I & V), Swift X-ray 2-10 keV, and Swift UV
(filter uvw1 centred on 2600Å). The behaviour seen in the I band is
characteristic of most Be stars found in BeXRB systems, showing long term
fluctuations arising from the alterations in the extent of the circumstellar
disks around the OB star. The V band generally follows the I band behaviour,
but at a lower amplitude ($\sim$0.07 mags compared to $\sim$0.15 mags in the I
band). This is understandable as at least 50% of the V band flux is arising
directly from the OB star and is much more stable. This is further supported
by the negligible fluctuations in the lowest panel - that of the UV band. The
X-ray flux shows little variation and is probably always close to the
detection limit of S-CUBE. Such modest X-ray fluxes are characteristic of
BeXRB with long, low-eccentricity orbital periods.

(a)  (b)

Figure 12: $(V-I)-I$ (left) and $(V-I)-V$ (right) colour-magnitude plots from
OGLE data.

Fig 12 shows the colour variation seen in the disk over the time of the OGLE
data JD 2453326 - 2458745 when simultaneous V and I band observations were
performed. During this period the photometric flux was showing a steady
decline accompanied by a dramatic dip during the period 3600 < TJD < 3800,
which is more pronounced in the V-band. This is possibly an indicator of loss
of the inner regions of the disc, from where the bluer emission originates. At
$\sim$TJD 5500, the system underwent a significant drop in I-band flux while
the V-band magnitude remained constant, suggesting disc-loss in the outer
regions during this time.

Fig 12 (a) reveals general trend of the colour, being redder when the system
is brighter. This possibly indicates a low/intermediate inclination angle of
the disc with a flaring circumstellar disk structure partially obscuring the
bluer light directly from the star. The prominent dip in the V-band emission
during 3600 < TJD < 3800 results in colour variations happening in two
discontinuous ranges, 0.14$\leq$(V-I)$\leq$0.23 and 0.24$\leq$(V-I)$\leq$0.28.
In the second colour range (0.24$\leq$(V-I)$\leq$0.28), the general trend of
the colour being redder when brighter persists in the (V-I)-I plot (Fig 12
(a)), while it is less apparent in the (V-I)-V plot (Fig 12 (b)). The (V-I)-V
distribution also shows more scatter, suggesting more significant changes
occurring in the inner disc regions

###  3.2 Optical spectrum

The equivalent widths of the H$\alpha$ and H$\beta$ lines measured from the
spectrum of Candidate 2 are -40.6 $\pm$ 0.4 Å and -3.6 $\pm$ 0.2 Å,
respectively. The H$\alpha$ equivalent width is significantly larger than the
average seen in the BeXRB population. Using the proposed relationship between
H$\alpha$ equivalent width and orbital period (Reig et al., 1997; Coe & Kirk,
2015), the value measured suggests an orbital period of the order 200-400
days. This is consistent with the 413d period seen in the OGLE data.

Besides the strong Balmer emission, one can see several lines of neutral
helium in absorption in the spectrum (Fig. 10). There is no evidence for any
ionised helium, meaning the spectral type must be later than B 0.5. It is
difficult to determine which metal lines are present, and the strength of
these lines, due to the low signal-to-noise in the blue part of the spectrum.
Mg II seems to be absent, or at least very weak, indicating the spectral type
may be earlier than B3. There is good evidence for OII lines, which seem
stronger than either of the Si IV lines that may be present around 4100 Å.
This would indicate a spectral type of B 1.5 or earlier. However, as mentioned
earlier, there is potential contamination from Candidate 3 to consider. The
strong He I and ionised oxygen lines seen in Candidate 3 may be contributing
signal to our target spectrum, making a precise classification difficult.
Because of this, we are not confident enough to be able to distinguish between
a B2 and B3 late limit for the spectral type. Thus, we suggest a spectral type
between B 1–3, though we stress that a higher quality spectrum taken in better
seeing conditions is needed to confirm this, and to provide a finer
classification.

Given that the metal lines in the spectrum are often not obvious, we are
limited to using the HeI 4121/HeI 4143 ratio as one approach to determine the
luminosity class. This line ratio increases towards more luminous stars,
suggesting that our object has a luminosity class of III-V. This result may be
compared to that obtained by using the absolute magnitude of the star in the
V-band with the distance modulus for the SMC (Graczyk et al., 2014). Using the
tables of Wegner (2006), a V-band magnitude of 14.6 favours a luminosity class
of III-IV for a spectral type of B1-3. A star of luminosity class V would need
to have an earlier spectral type of $\sim$O9 for this magnitude.

Based on the information provided by the optical spectrum and magnitude of the
star, and the limitations of the classification discussed above, our best
estimate of the spectral type of this system is B 1–3 III-IVe.

Finally we note that not only is Candidate 1 ([MA93] 302) positionally
inconsistent as the optical counterpart to Swift J004929.5-733107 but it is
not even an early type star. It is quite possible that light from nearby
Candidate 2 led to its incorrect cataloguing as an early type emission line
star by Meyssonnier & Azzopardi (1993).

###  3.3 Optical periodic modulation

The optical modulation at a period of 413d shown in Fig 5 is interpreted as
the binary period of the system and is very sinusoidal in shape. This is very
different from the sharp outburst profiles seen in other long period BeXRB
systems such as SXP 756 with its similar orbital period of 394d (Coe & Edge,
2004). The most obvious interpretation of these differences must lie in the
eccentricity of the neutron star orbit and the manner in which it interacts
with the circumstellar disk of the Be star. These disks are assumed to be
optically thick, so any variation seen in the OGLE I-band data must be
interpreted as a change in the surface area of the disk. Regular changes being
brought about by the approach of the neutron star at periastron passage. Hence
the more gentle long term changes observed here in Swift J004929.5-733107 must
surely indicate a very low eccentricity orbit for the neutron star. This is
further supported by the small depth of the observed modulation: $\sim$0.06
mags in Swift J004929.5-733107 compared to $\sim$0.5 mags in SXP 756.

If this 413d period is interpreted as the binary period, then an estimate of
the likely spin period of the neutron star may be inferred from the Corbet
Diagram (Corbet, 1984). This is a very long binary period for BeXRB systems
and places Swift J004929.5-733107 right at the edge of the known distribution,
it also suggests an exceptionally long pulse period in the region of
$\geq$1000s. Such a long period is hard to measure since an extended X-ray
observation of many ks would be required whilst the source was reasonably
bright. An example of a source that lies in this extreme zone on the Corbet
Diagram is SAX J2239.3+6116 (in’t Zand et al., 2001) in the Galaxy with an
orbital period of 262d and a pulse period of 1247s. Within the SMC, there are
three systems with proposed pulse periods in excess of 1ks : SXP 1062, SXP
1323 and SXP 4693 (= CXOU J005446.3-722523). The latter is most exceptional
and has no known binary period (Laycock et al., 2010). Such long pulse period
systems tend not to be bright X-ray sources, making the detection of pulse &
binary periods challenging - see Laycock et al. (2010) for a discussion on
such long pulse period systems. Thus it is valuable to add Swift
J004929.5-733107 to the list of these exceptional BeXRB systems.

##  4 Conclusions

Reported here is the confirmation that Swift J004929.5-733107 is a BeXRB
system in the SMC. As a result of many observations arising from the S-CUBED
project a precise position has been determined for this X-ray source. This has
enabled the correct identification of the optical counterpart, resolving a
long-standing confusion in a crowded stellar field. The long-term OGLE data
has made it possible to detect the probable binary period for this system
(413d) making it one of the longest known for a BeXRB system. Such long period
systems are valuable in testing the limits of the correlation between spin &
orbital periods seen in the Corbet diagram, with possible implications for
accretion modes in these well separated binary systems.

##  5 Data availability

All X-ray data are freely available from the Swift archive. The other data
underlying this article will be shared on any reasonable request to the
corresponding author.

## Acknowledgements

The OGLE project has received funding from the National Science Centre,
Poland, grant MAESTRO 2014/14/A/ST9/00121 to AU. PAE acknowledges UKSA
support. IMM acknowledges support from the National Research Foundation (NRF)
and the University of Cape Town (UCT). This work made use of data supplied by
the UK Swift Science Data Centre at the University of Leicester. LJT
acknowledges support from the National Research Foundation (NRF), South Africa
and the SALT consortium. Some of the observations reported in this paper were
obtained with the Southern African Large Telescope, as part of the Large
Science Programme on transients 2018-2-LSP-001 (PI: Buckley).

We are grateful to the anonymous referee for several helpful suggestion to
improve the paper.

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