# Constraints on the maximum densities of neutron stars
from postmerger gravitational waves with third-generation observations

Matteo Breschi1 Sebastiano Bernuzzi1 Daniel Godzieba2 Albino Perego3,4 David
Radice5,2,6 1Theoretisch-Physikalisches Institut, Friedrich-Schiller-
Universität Jena, 07743, Jena, Germany  2Department of Physics, The
Pennsylvania State University, University Park, PA 16802, USA  3Dipartimento
di Fisica, Università di Trento, Via Sommarive 14, 38123 Trento, Italy  4INFN-
TIFPA, Trento Institute for Fundamental Physics and Applications, via
Sommarive 14, I-38123 Trento, Italy  5Institute for Gravitation & the Cosmos,
The Pennsylvania State University, University Park, PA 16802, USA  6Department
of Astronomy & Astrophysics, The Pennsylvania State University, University
Park, PA 16802, USA

###### Abstract

Using data from 289 numerical relativity simulations of binary neutron star
mergers, we identify, for the first time, a robust quasi-universal relation
connecting the postmerger peak gravitational-wave frequency and the value of
the density at the center of the maximum mass nonrotating neutron star. This
relation offers a new possibility for precision equation-of-state constraints
with next-generation ground-based gravitational-wave interferometers. Mock
Einstein Telescope observations of fiducial events indicate that Bayesian
inferences can constrain the maximum density to ${\sim}15\%$ ($90\%$
credibility level) for a single signal at the minimum sensitivity threshold
for a detection. If the postmerger signal is included in a full-spectrum
(inspiral-merger-postmerger) analysis of such signal, the pressure-density
function can be tightly constrained up to the maximum density, and the maximum
neutron star mass can be measured with an accuracy better than $12\%$ ($90\%$
credibility level).

## Introduction.—

Postmerger (PM) gravitational-waves (GWs) emitted from binary neutron star
(BNS) merger remnants are unique probes for the stars’ nuclear equation of
state (EOS) at extreme-densities. During merger, the neutron stars (NSs) fuse
together to form a remnant NS with maximum densities up to two times the
component’s mass reaching ${\sim}3{-}6{\rho_{\rm sat}}$ where ${\rho_{\rm
sat}}\approx 2.7{\times}\,10^{14}\,{\rm g\,cm}^{-3}$ is the nuclear saturation
density. If the remnant does not promptly collapse to black hole, numerical
relativity (NR) simulations predict a loud GW transient emitted on dynamical
timescales of tens of milliseconds with a complex signal morphology and a
characteristic peak frequency $f_{2}\sim 2{-}4~{}{\rm kHz}$, e.g. Shibata and
Taniguchi (2006); Hotokezaka  _et al._ (2013); Bauswein and Stergioulas
(2015); Takami  _et al._ (2014); Bernuzzi  _et al._ (2015); Lehner  _et al._
(2016). These kiloHertz GW transients have not been detected in the two BNS
events GW170817 and GW190425 due to the insufficient sensitivity of the GW
detectors at those frequencies Abbott  _et al._ (2017a, b, 2019, 2018a,
2020a). A first detection is however possible (and expected) with third-
generation Hild  _et al._ (2011); Punturo  _et al._ (2010a) and others
proposed detectors that specifically target the kiloHertz frequencies Ackley
_et al._ (2020). In view of this observational scenario, it is of central
importance to determine what kind of information can be extracted from PM
signals and to which accuracy.

The signal-to-noise (SNR) detection threshold for a PM signal has been
recently studied using different approaches e.g. Clark  _et al._ (2016);
Chatziioannou  _et al._ (2017); Torres-Rivas  _et al._ (2019); Tsang  _et al._
(2019); Breschi  _et al._ (2019); Easter  _et al._ (2019, 2020). These studies
found that SNRs ${\gtrsim}7-8$ at kiloHertz frequencies are typically
necessary to confidently claim the detection of a BNS PM signal. These PM SNRs
correspond to loud inspiral-merger signals with SNR ${\gtrsim}150$ that, in
turn, would provide rather accurate measurements of the binary mass and tidal
polarizability parameters, e.g. Damour  _et al._ (2012); Gamba  _et al._
(2021); Chatziioannou (2021). While the existence of an underlining, unique
EOS for the NS matter allows to estimate the EOS at all densities from GW
observations, inspiral-merger frequencies are mostly informative at the
largest densities of the binary components Abbott  _et al._ (2018b); Landry
and Essick (2019); Agathos  _et al._ (2020). In contrast, PM frequencies are
expected to be mostly informative of the extreme-densities reached by the
remnant, although BNS PM modeling for GW inference is non-trivial. The main
approach is to employ EOS-insensitive relations connecting the peak frequency
$f_{2}$, which is the most robust feature of the PM spectrum as predicted by
simulations, and equilibrium properties of NSs like the radius at fiducial
masses Bauswein  _et al._ (2012, 2016), averaged binary compactnesses Takami
_et al._ (2015), or the binary’s tidal coupling constant ${\kappa^{\rm
T}_{2}}$ Bernuzzi  _et al._ (2015). For example, ${\kappa^{\rm T}_{2}}$ can be
measured from $f_{2}$ to within a factor of two (${\sim}20\%$) at PM SNR
${\sim}8$ (15) Easter  _et al._ (2020), but the inspiral signal of the same
event would deliver a measurement orders of magnitude more accurate Breschi
_et al._ (2019), thus providing more stringent constraints on the pressure-
density EOS function. A PM detection can typically provide only bounds on the
maximum NS mass ${M_{\rm max}^{\rm TOV}}$ Bauswein  _et al._ (2013); Agathos
_et al._ (2020) [solution of the Tolman-Oppenheimer-Volkoff (TOV) equations];
although, multiple PM detections could provide a few percent measurement
Bauswein  _et al._ (2014) complementary to that from an inspiral EOS inference
Abbott  _et al._ (2018b); Landry and Essick (2019); Essick  _et al._ (2020a).
The minimum NS radius ${R_{\rm max}^{\rm TOV}}=R({\rho_{\rm max}^{\rm TOV}})$,
i.e. the radius of a NS at maximum density ${\rho_{\rm max}^{\rm TOV}}$, is a
key information to constrain the NS mass-radius diagram. It can be extracted
from a quasiuniversal relation with the peak PM frequency Bauswein  _et al._
(2014) and measured with an uncertainty of ${\sim}4\,{\rm km}$ at PM SNR 12
Breschi  _et al._ (2019, 2021a). Stronger EOS constraints from BNS mergers
will be possible by combining GW observations with other messengers, e.g.
Margalit and Metzger (2017); Radice  _et al._ (2018a); Bauswein  _et al._
(2017); Coughlin  _et al._ (2018); Raaijmakers  _et al._ (2020); Dietrich  _et
al._ (2020); Essick  _et al._ (2020b); Breschi  _et al._ (2021b).

In this Letter, we propose a new quasiuniversal relation connecting the NS
maximum density to the peak PM frequency. The existence of a EOS-insensitive
${\rho_{\rm max}^{\rm TOV}}(f_{2})$ relation was suggested by previous work
Bauswein  _et al._ (2013, 2014); Lioutas  _et al._ (2021), but the relation
obtained there involves the binary configurations with the largest possible
mass that does not promptly collapse to black hole. Here, by introducing an
effective Keplerian radius associated to $f_{2}$, we propose a more general
approach that can be applied to any binary and delivers a strong constrain
already with a single detection. Employing full Bayesian PM analyses of mock
signals and incorporating the expected information from the inspiral-merger
signal, we demonstrate that a single GW BNS detection with the Einstein
Telescope at the PM detection threshold can already deliver a ${\sim}15$%
measurement of ${\rho_{\rm max}^{\rm TOV}}$ and ${M_{\rm max}^{\rm TOV}}$. The
novel method shows the capabilities of next-generation detectors and introduce
the possibility for direct and accurate constraints on the high-density EOS
properties from observational GW data of BNS mergers.

Figure 1: Empirical relation (black line) for the maximum central density
${\rho_{\rm max}^{\rm TOV}}$ of a non-rotating NS as function of the Keplerian
radius ${R_{f_{2}}}$ (top axis) and the PM peak frequency $f_{2}$ (bottom
axis). The colored markers show the data extracted from 289 NR simulations
with 14 EOSs. Each marker corresponds to a different binary configuration and
the error bars are computed using different numerical resolutions (when
available). The shadowed area reports the 90% credibility region of the fit.

## Effective Keplerian radius.—

The PM peak frequency $f_{2}$ can be associated to an effective remnant radius
using Kepler’s law,

${R_{f_{2}}}^{3}=\frac{GM}{(\pi f_{2})^{2}}\,,$ (1)

where $G$ is the gravitational constant and $M=m_{1}+m_{2}$ is the total mass
of the binary. While this quantity has no direct physical interpretation in
terms of the remnant properties, it correlates to the maximum central density
${\rho_{\rm max}^{\rm TOV}}$ of a non-rotating equilibrium NS with a weak
dependence on the EOS. Figure 1 shows the approximate quasiuniversal relation
${\rho_{\rm max}^{\rm TOV}}({R_{f_{2}}})$ for a NR sample of 289 simulations
of the CoRe collaboration Dietrich  _et al._ (2018); Radice  _et al._ (2018b);
Bernuzzi  _et al._ (2020); Nedora  _et al._ (2021); Prakash  _et al._ (2021).
To explore EOS variation, the simulations were performed with 14 different
EOSs including piecewise polytropes Engvik  _et al._ (1996); Müller and Serot
(1996); Alford  _et al._ (2005); Lackey  _et al._ (2006); Read  _et al._
(2009), 5 finite-temperature nucleonic models Lattimer and Swesty (1991);
Douchin and Haensel (2001); Typel  _et al._ (2010); Steiner  _et al._ (2013);
Logoteta  _et al._ (2020), one finite-temperature hybrid model accounting for
deconfined quark matter Logoteta  _et al._ (2020); Prakash  _et al._ (2021),
and one finite-temperature hadronic model with $\Lambda$ hyperons Banik  _et
al._ (2014); Radice  _et al._ (2017). The majority of the sample
(${\sim}85\%$) is simulated using microphysics, neutrino transport and a
subgrid model for magnetohydrodynamics (MHD) turbulence, that are sufficient
to fully capture the features of the PM signal modeled here. For example, the
highest-to-date resolution simulations of Kiuchi  _et al._ (2018) suggest that
the MHD effective viscosity is small and does not affect the GWs; our subgrid
model is tuned to those simulations. Neutrino induced bulk viscosity might
damp radial oscillations of the remnant (which cause secondary peaks in the
spectrum) and will not impact $f_{2}$. Moreover, recent simulations with
improved neutrino transport suggest that neutrino bulk viscosity is too small
to have a significant impact on dynamics and GW emission Radice  _et al._
(2021).

The sample includes binaries with NS masses in the range $1{-}2\,{\rm
M_{\odot}}$ and dimensionless spin magnitudes $\leq 0.2$. The relation
${\rho_{\rm max}^{\rm TOV}}({R_{f_{2}}})$ fits to the expression:

${\rho_{\rm max}^{\rm
TOV}}=\frac{a_{0}\,c^{6}}{G^{3}M^{2}}\left[1+a_{1}\left(\frac{c^{2}{R_{f_{2}}}}{GM}\right)^{1/4}\right]\,,$
(2)

where $c$ is the speed of light and $(a_{0},a_{1})=(0.135905,-0.59506)$ are
determined by a standard least-squared minimization method at
$\chi^{2}=0.016$. The standard deviations of the calibrated coefficients
$a_{0}$ and $a_{1}$ are respectively equal to $1.35\%$ and $0.22\%$,
corresponding to a relative error of $5.8\%$ on the final prediction.

The r.h.s. of Eq. (2) can be entirely determined from the measurement of the
binary mass $M$ and the PM peak frequency $f_{2}$. Writing the Keplerian
radius ${R_{f_{2}}}$ in terms of $f_{2}$, one can get a direct relation
between the maximum density ${\rho_{\rm max}^{\rm TOV}}$ and the PM frequency,
as

${\rho_{\rm max}^{\rm
TOV}}=\frac{a_{0}c^{6}}{G^{3}M^{2}}\left[1+a_{1}\left(\frac{c^{3}}{\pi
GMf_{2}}\right)^{1/6}\right]\,.$ (3)

Hence, the maximum density ${\rho_{\rm max}^{\rm TOV}}$ can be best inferred
from a full-spectrum BNS GW observation. We next discuss the potential
accuracy of such measurement using a mock Bayesian inference study based on
match filtering techniques and a NR-informed analytical model for the PM
signal.

## Mock inference study.—

We consider the Einstein Telescope (ET) design Hild  _et al._ (2011); Punturo
_et al._ (2010b) and assume the detector is composed by three interferometers
with triangular shape and power spectral density (PSD) configuration D Hild
_et al._ (2011). Mock signals are simulated from two fiducial equal-mass non-
spinning BNS: a binary with mass $M=2.73\,{\rm M_{\odot}}$ and (a “stiff”) EOS
DD2 Typel  _et al._ (2010) and a binary with mass $M=2.6\,{\rm M_{\odot}}$ and
(a “soft”) EOS SLy Douchin and Haensel (2001). Merger and PM waveforms for
these BNSs from NR were already discussed in Refs. Perego  _et al._ (2019);
Breschi  _et al._ (2019) and are employed for the PM analyses. The fiducial
binaries are placed at different distances between $80$ and $200~{}{\rm Mpc}$;
the injected signals have PM SNRs ranging from ${\sim}6$ to $14$,
corresponding to total SNRs from $110$ to ${\sim}260$. For our case study, we
perform full Bayesian analyses of the PM signals at different SNR and employ
the Fisher matrix approach to estimate the uncertainties of the parameters
measured from the inspiral signals.

Bayesian PM analyses are performed using the bajes pipeline Breschi  _et al._
(2021a) and an updated version of the nrpm model for BNS PM signals Breschi
_et al._ (2019). NR data are injected in a segment of $1~{}{\rm s}$ with a
sampling rate of $16~{}{\rm kHz}$. The sampling is performed with dynesty
Speagle (2020) with 3200 live points analyzing the frequency region from
$1~{}{\rm kHz}$ to $8~{}{\rm kHz}$. nrpm is calibrated on a subset of the
numerical data shown in Fig. 1 (see Breschi  _et al._ (2019)) and it employs a
simple analytical prescription for the remnant emission to capture the three
peak frequencies and damping times of typical PM signals. These (complex)
frequencies, here collectively indicated as $Q^{\rm NR}$, are fully determined
by the intrinsic binary properties using NR-informed quasiuniversal relations
in terms of the tidal coupling constant ${\kappa^{\rm T}_{2}}$ and the
symmetric mass ratio $\nu=m_{1}m_{2}/M^{2}$ Bernuzzi  _et al._ (2015); Zappa
_et al._ (2018); Breschi  _et al._ (2019). The prior distribution is assumed
to be uniform in the mass components, spanning the ranges $M\in[1.5,6]~{}{\rm
M_{\odot}}$ and $m_{1}/m_{2}\in[1,2]$, and in the individual NS quadrupolar
tidal polarizability parameters $\Lambda_{i}\in[0,5000]$ for $i=1,2$. Spin
magnitudes are kept fixed to zero. The extrinsic parameters are treated as
discussed in Breschi  _et al._ (2021a), with volumetric prior for the
luminosity distance in the range $[5,500]~{}{\rm Mpc}$. The likelihood
function is analytically marginalized over reference time and phase.

In this work, nrpm is extended with two sets of additional parameters that are
determined by the Bayesian inference. A first set of three parameters
$(\alpha,\beta,{\phi_{\rm PM}})$ is introduced to enhance the flexibility of
the model and to improve fitting factors to the PM signals with complex
morphology. $\alpha$ is a damping time for the PM bursts (see Eq. (9c) of
Breschi  _et al._ (2019)); $\beta$ accounts for a linear contribution in the
frequency evolution as discussed in Easter  _et al._ (2020); and ${\phi_{\rm
PM}}$ is an additional phase-shift corresponding to the first amplitude
minimum Breschi  _et al._ (2019). The prior distribution of these parameters
is chosen uniformly in $\alpha^{-1}\in[1,1000]$, $\beta\in[10^{-4},-10^{-4}]$
and ${\phi_{\rm PM}}\in[0,2\pi]$. A second set of recalibration parameters
$\delta_{Q}$ is introduced to account for the uncertainties of each
quasiuniversal relation $Q^{\rm NR}({\kappa^{\rm T}_{2}},\nu)$. Specifically,
we map $Q^{\rm NR}\mapsto Q^{\rm NR}(1+\delta_{Q})$ and treat $\delta_{Q}$
similarly to standard calibration parameters in GW analyses, e.g. Vitale  _et
al._ (2012). The prior distribution for the $\delta_{Q}$ is assumed to be
normal with zero mean and variance prescribed by the residuals of the
quasiuniversal relations.

This approach aims at combining the strenghts of the nrpm templated analyses,
built on quasiuniversal relations, with those of more agnostic analyses using
minimal assumptions about the signal morphology, e.g. Chatziioannou  _et al._
(2017); Easter  _et al._ (2020). In particular, the improved nrpm model used
here improves the fitting factors by an order of magnitude with respect to
Breschi  _et al._ (2019), and can detect PM GW signals at a SNR comparable to
unmodeled approaches Tsang  _et al._ (2019); Easter  _et al._ (2020). While
relying on a larger number of parameters than unmodeled methods, nrpm can
deliver a measure of the intrisic parameters independently on pre-merger
analyses, and produces narrower $f_{2}$ posteriors for the same PM SNR. The
inference of the tidal coupling constant is also improved: comparing to Easter
_et al._ (2020), ${\kappa^{\rm T}_{2}}$ is inferred to ${\sim}30\%$ for PM SNR
${\sim}12$ instead of ${\sim 15}$. We also note that unmodeled analyses rely
on analogous quasiuniversal relations to infer ${\kappa^{\rm T}_{2}}$, hence
should also consider recalibration parameters.

Fisher matrix analyses of the inspiral-merger signals are performed with
gwbench Borhanian (2021). Considering the large SNR of these signals, the
Fisher matrix provides an approximation of the posterior distributions of the
binary mass and tidal parameters that is sufficiently accurate for our
purposes. For instance, in the lowest SNR case of $110$, the uncertainty on
the binary mass measured from the inspiral signal is ${\sim}10^{-4}\,{\rm
M_{\odot}}$ and the uncertainty on the quadrupolar tidal polarizability
parameters is ${\sim}2\%$. These results imply, as expected, that the
inference on ${\rho_{\rm max}^{\rm TOV}}$ via Eq. (2) has uncertainties
dominated by the measurement of $f_{2}$ from the PM signal and by
recalibration errors.

Figure 2: Bayes’ factors and posterior distributions measured in the mock
inference study as functions of the PM SNR. The DD2 case is reported in green,
the SLy case is reported in red. The top panel shows the recovered Bayes’
factors $\log\mathcal{B}$. Central and bottom panels report the posterior
distributions respectively of $f_{2}$ and ${\rho_{\rm max}^{\rm TOV}}$ and the
dashed-dotted lines denote the injected values. A gray shaded region shows the
detectability threshold.  Figure 3: Posterior distributions for
$\left({\rho_{\rm max}^{\rm TOV}},{R_{\rm max}^{\rm TOV}}\right)$ compared to
the EOS of the injected signals. The contours show the $90\%$ credibility
regions. Green and red lines correspond respectively to the results obtained
with DD2 and SLy EOSs and color intensities refer to the injected SNRs of the
PM data. The dashed lines show the central density as function of the NS
radius and the crosses denote the maximum-density values.

## Maximum density constraint.—

Our analysis indicates that PM signals can be detected with SNRs as low as PM
SNR threshold of ${\sim}7.5$ (total SNR ${\sim}140$). As shown in the top
panel of Fig. 2, the SNR threshold corresponds to a Bayes’ factor of
$\log\mathcal{B}\gtrsim 5$ in favor of the signal (vs. noise) hypothesis. This
detection threshold is comparable to those obtained with minimally modeled PM
waveform, e.g. Easter  _et al._ (2020), and improves over the previous nrpm
results Breschi  _et al._ (2019) as a consequence to the enhanced flexibility
of our fully NR-informed model. Figure 2 (middle panel) also shows the
posterior of $f_{2}$ at different SNRs. The $90\%$ credibility interval of the
$f_{2}$ measurement is of order of ${\sim}12\%$ at the detectability threshold
and decreases to ${\sim}3\%$ at the largest PM SNR ${\sim}12$. The DD2
posteriors show bimodalities for SNR ${\lesssim}9$ due to a loud first peak in
the spectrum (the subdominant frequency peak at frequency $f<f_{2}$), but this
systematic vanishes for increasing SNR.

The posterior distribution for ${\rho_{\rm max}^{\rm TOV}}$ is obtained by
combining the posteriors of $M$ from the inspiral analysis and the posteriors
of $f_{2}$ from the PM analysis. The uncertainties in the fit of Eq. (2) are
taken into account with a resampling similar to the one used for the
recalibration parameters (see also Breschi  _et al._ (2021b)). Analogously to
the $f_{2}$ measurement, the true value is recovered at the detectability
threshold with an error of ${\sim}15\%$ at the $90\%$ credibility level, as
shown in the bottom panel of Fig. 2. Notably, the inference on ${\rho_{\rm
max}^{\rm TOV}}$ hits the theoretical uncertainties on the quasiuniveral
relation in Eq. (2) at SNR ${\sim}10$. A more precise measure is either not
possible, because at that level the relation ${\rho_{\rm max}^{\rm
TOV}}({R_{f_{2}}})$ becomes EOS-dependent, or it requires a more precise
quasiuniversal relation from improved simulations.

In order to further illustrate the accuracy of this approach in constraining
high-density NS properties, we consider the simultaneous measure of
${\rho_{\rm max}^{\rm TOV}}$ and of the minimum NS radius ${R_{\rm max}^{\rm
TOV}}$. Here, the posteriors of $(M,f_{2})$ are mapped into $\left({\rho_{\rm
max}^{\rm TOV}},{R_{\rm max}^{\rm TOV}}\right)$ using Eq. (2) and the
quasiuniversal relation for ${R_{\rm max}^{\rm TOV}}$ in Eq. (23) of Breschi
_et al._ (2019). As above, the uncertainties of the quasiuniversal relation
are taken into account by sampling on appropriate recalibration parameters.
The result is shown in Fig. 3: the injected values, denoted with crosses,
confidently lay within the 90% credibility regions of the recovered posterior
distributions. At the detection threshold, the uncertainty on the radius
${R_{\rm max}^{\rm TOV}}$ are of the order of $30\%$ ($90\%$ credibility
level). Such precision is sufficient to distinguish soft and stiff EOSs within
the 90% credibility regions, as illustrated using the SLy and DD2 EOSs.

Figure 4: Mass-radius diagram constraints from a single full-spectrum Einstein
Telescope (ET) BNS observation with PM SNR 10 (total SNR $180$). The gray area
(prior) corresponds to the two-million EOS sample of Ref. Godzieba  _et al._
(2021). The magenta and cyan areas are the $90\%$ credibility regions given by
inspiral-merger and inspiral-merger-PM inferences respectively. The full-
spectrum (cyan) posterior agrees with the injected EOS (black).

## EOS constraints and mass-radius diagram.—

To illustrate the potential impact of this approach, we show how a single
detection of PM signal at SNR 10 (total SNR 180 and luminosity distance of
$120~{}{\rm Mpc}$) can constrain the mass-radius relation for NSs. The
analysis makes use of the two-million EOS sample of Ref. Godzieba  _et al._
(2021) that are consistent with current constraints on the EOS of dense
matter. In particular, all EOSs predict maximum NS mass above the maximum NS
mass from pulsar observations, ${M_{\rm max}^{\rm TOV}}>1.97~{}{\rm
M_{\odot}}$ Antoniadis  _et al._ (2013), and are consistent with the upper
bound on the tidal coupling constant from GW170817 Abbott  _et al._ (2018b,
2019, 2017a). The only additional assumptions are the validity of general
relativity and that the EOS is causal up to the central density of the maximum
mass NS. We take the SLy binary as fiducial dataset and re-weight the EOS
samples with the posteriors of the masses and the tidal parameters from the
inspiral-merger and with the maximum density and minimum radius posteriors
from the PM measurement.

Figure 4 shows the $90\%$ credibility regions of the posterior distributions
in the mass-radius diagram. The inspiral-merger posteriors give the strongest
constraint. They are mostly informative at the EOS at densities $\rho\simeq
2{\rho_{\rm sat}}$ corresponding to the individual NS components of the binary
Agathos  _et al._ (2015). This inference leads to a measurement of the maximum
NS mass of ${M_{\rm max}^{\rm TOV}}=2.13^{+0.20}_{-0.14}\,{\rm M_{\odot}}$
($90\%$ credibility level), consistent with the injected value ($2.05\,{\rm
M_{\odot}}$). However, the EOS posterior shows a biased behavior for high mass
values, i.e. $M\gtrsim 1.5\,{\rm M_{\odot}}$, excluding the SLy sequence from
the 90% credibility region. This shows that the inspiral-merger signal does
not directly constrain the high-density EOS and the inferred ${M_{\rm
max}^{\rm TOV}}$ represents an extrapolation based on the EOS representation.
The inclusion of PM information strengthens the agreement with the injected
EOS at higher densities allowing a measurements of ${M_{\rm max}^{\rm
TOV}}=2.04^{+0.08}_{-0.06}\,{\rm M_{\odot}}$, which agrees with the injected
value and carries an error lower than $7\%$. The improved constraint reduces
of ${\sim}60\%$ the $p(\rho)$ posterior area of the initial EOS sample
leading, in particular, to tight pressure constraint at fiducial densities:
$\log_{10}\left[p(2{\rho_{\rm sat}})/({\rm dyn}\,{\rm
cm}^{-2})\right]=34.52^{+0.04}_{-0.04}$ and $\log_{10}\left[p(4{\rho_{\rm
sat}})/({\rm dyn}\,{\rm cm}^{-2})\right]=35.39^{+0.04}_{-0.03}$ ($90\%$
credibility level). For a detection at the sensitivity threshold (PM SNR 7.5)
we find that the measured maximum NS mass is ${M_{\rm max}^{\rm
TOV}}=2.07^{+0.15}_{-0.09}\,{\rm M_{\odot}}$ with a relative error of roughly
${\sim}12\%$.

## Outlook.—

This work shows that next-generation GW observatories can deliver strong
constraints on the extreme-density EOS from a single, full-spectrum detection
of a BNS. In particular, the detection of the PM signal from the merger
remnant significantly enhances the measurement of the NS maximum density and
mass. The sensitivity threshold for PM BNS transients is around a luminosity
distance ${\sim}150\,{\rm Mpc}$, consistently with the recent estimates of
Ref. Abbott  _et al._ (2020a). According to our results, the multiple
observation of about five PM BNS transients at sensitivity threshold can lead
to a measurement of the maximum NS mass with an error of ${\sim}5\%$ Torres-
Rivas  _et al._ (2019); Easter  _et al._ (2020). This information provides
narrow observational constraints that would significantly inform nuclear
models in the very-high-density regimes, i.e. $\rho>3{\rho_{\rm sat}}$, which
are unreachable conditions for modern nuclear experiments. Remarkably, the
inclusion of the PM inference contributes in reducing observational errors and
methodological biases.

In real observations, EOS constraints at the accuracy level reported here can
be obtained by incorporating the PM signal in EOS inferences like those
performed in Landry and Essick (2019); Abbott  _et al._ (2019, 2020b); Essick
_et al._ (2020a) for inspiral-merger signals. In this context, the development
of accurate EOS-insensitive relations and the determination of their validity
(or their breaking, e.g. Bauswein  _et al._ (2019); Raithel and Most (2022))
via high-precision NR simulations is a key step to improve the reliability of
the GW measurement. Going beyond the Fisher matrix approach employed here,
future work will consider analyses employing Bayesian methods also for the
inspiral-merger. Full spectrum analyses are currently possible given the
availability of complete inspiral-merger-PM waveform models Breschi  _et al._
(2019). The latter include correlations on the mass and tidal coupling
constant between inspiral-merger and PM, that will further improve the
estimates of this work. However, the most urgent issue for the high-SNR
inspiral-mergers that will be observed by third-generation detectors remains
waveform systematics in the inference of tidal properties Gamba  _et al._
(2021); Samajdar and Dietrich (2018). Nevertheless, the computational
challenges related to full-spectrum Bayesian analyses might be confronted with
parallel methods Smith  _et al._ (2020); Breschi  _et al._ (2021a) and
acceleration techniques for the likelihood evaluation, e.g. Smith  _et al._
(2016); Zackay  _et al._ (2018); Schmidt  _et al._ (2021); Cornish (2021).

###### Acknowledgements.

## Acknowledgments.—

The authors thank Francesco Zappa, Aviral Prakash and Andrea Camilletti for
providing unreleased NR data and Ssohrab Borhanian for providing additional
documentation for gwbench. Moreover, we acknowledge important discussions in
the LIGO-Virgo Extreme matter group and within the Virgo-EGO collaboration, in
particular we thank Nikolaos Stergioulas, Katerina Chatziioannou and Jocelyn
Read. MB and SB acknowledge support by the EU H2020 under ERC Starting Grant,
no. BinGraSp-714626. MB acknowledges support from the Deutsche
Forschungsgemeinschaft (DFG) under Grant No. 406116891 within the Research
Training Group RTG 2522/1. DR acknowledges funding from the U.S. Department of
Energy, Office of Science, Division of Nuclear Physics under Award Number(s)
DE-SC0021177 and from the National Science Foundation under Grants No.
PHY-2011725, PHY-2020275, PHY-2116686, and AST-2108467. Virgo is funded by the
French Centre National de Recherche Scientifique (CNRS), the Italian Istituto
Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef, with
contributions by Polish and Hungarian institutes. Simulations were performed
on ARA, a resource of Friedrich-Schiller-Universtät Jena supported in part by
DFG grants INST 275/334-1 FUGG, INST 275/363-1 FUGG and EU H2020
BinGraSp-714626. The bajes software together with the nrpm model are publicly
available at: https://github.com/matteobreschi/bajes. The NR simulations
employed in this work are publicly available at: http://www.computational-
relativity.org/.

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# Resonances crossing effect and quantum sensor of electric fields

Andrea Sacchetti  Department of Physics, Informatics and Mathematics,
University of Modena and Reggio Emilia, Modena, Italy.
<EMAIL_ADDRESS>
###### Abstract.

While the phenomenon of the exact crossing of energy levels is a rarely
occurring event, in the case of quantum resonances associated with metastable
states this phenomenon is much more frequent and various scenarios can occur.
When there is an exact crossing of the imaginary parts of the resonances in a
two-level quantum system subject to an external DC electric field, then a
damped beating phenomenon occurs, which is absent if there is no such
crossing. This fact, tested numerically on an explicit one-dimensional model,
suggests the possibility of designing quantum sensors to determine in a very
simple way whether the external field strength has an assigned value or not.

Data availability statement. All data generated or analysed during this study
are included in this published article.

Conflict of interest statement. The author has no competing interests to
declare that are relevant to the content of this article.

###### Key words and phrases:

Quantum resonances, time decay of survival amplitude, electrical sensors.

Andrea Sacchetti is member of Gruppo Nazionale per la Fisica Matematica of
Istituto Nazionale di Alta Matematica (GNFM-INdAM). This work is partially
supported by the Next Generation EU - Prin 2022CHELC7 project ”Singular
Interactions and Effective Models in Mathematical Physics” and the UniMoRe-FIM
project “Modelli e metodi della Fisica Matematica”.

Introduction. Quantum sensors are devices that measure physical quantities,
such as electric or magnetic fields, using the laws of Quantum Mechanics.
Among the first devices with these characteristics we can mention the Ramsey
interferometer [1], based on a two-level system. Currently, many other quantum
sensors have been developed, from those based on silicon carbide to those
based on NV centers in nanodiamonds, with many applications, from rechargeable
batteries to biomedical devices (see [2, 3, 4, 5] and references therein).

The typical model of a quantum sensor [6] consists of an Hamiltonian
$H=H_{i}+V_{e}$, where $H_{i}=$ is the internal (or unperturbed) Hamiltonian,
and $V_{e}$ is an external potential that one wants to measure or that
represents an element of control used to set the quantum sensor appropriately;
for example, in some quantum sensors the external field is a DC electric
field, and $V_{e}(x)$ represents a linear Stark potential. Assume that $H_{i}$
is a two-level system with energy levels $E_{1}\leq E_{2}$, and term
$\omega=E_{2}-E_{1}$ is the transition frequency between the two states;
therefore, in the case where $\omega\not=0$, an interference effect generates
a periodic beating effect with period $T=2\pi/\omega$. The basic idea is that
the effect of the perturbation dues to $V_{e}$ is to produce a shift in energy
levels, and a change in the transition frequency $\omega$; so, a periodic
beating motion of the quantum system described by $H$ is still observed, but
with modified period. A relationship is then established between the period of
the beating motion and the external field that can be used to obtain an
estimate of the intensity of the perturbation $V_{e}$.

In this paper, by making use of a similar idea we propose a theoretical model
for a quantum sensor aimed at determining in a simple way whether or not an
external DC electric field strength is equal to a predefined value. In some
ways we have a phenomenological analogy with RLC electrical circuits that
resonate at a specific external frequency; in our model, similarly, only at a
specific value of the DC electric field strength a long time interference
between two metastable states producing a periodic beating effect occurs.

Interference effect at the resonances crossing point. Let the internal one-
dimensional Hamiltonian $H_{i}=-\frac{d^{2}}{dx^{2}}+V_{i}$ be associated with
an asymmetrical double-well potential $V_{i}$, and let the external potential
be a linear Stark potential $V_{e}(x)=-Fx$, where $F$ is the DC electric field
strength. If we consider the ground state of a single well, e.g. the left-
hand-side (l.h.s.) one, the effect of the second well is to slightly perturb
it, and we denote by $E_{1}$ the corresponding energy; similarly, let us
denote by $E_{2}$ the energy of the ground state of the right-hand-side
(r.h.s.) well treating the l.h.s. well as a perturbation. The two-level system
consists by restricting $H_{i}$ to these two states, and the effect of the
linear Stark potential $V_{e}$ is, in the first instance, to change the
splitting $\omega$ as described above. We can observe, however, that a second
effect, associated with crossing energy levels, is generated for an
appropriate choice of parameters. Indeed, if the l.h.s. well of $V_{i}$ is
deeper than the r.h.s. well, then, for $F=0$, the energy level $E_{1}$ is less
than the energy level $E_{2}$.

As $F$ increases, these two levels will come closer together until they almost
cross each other for a critical value of $F$, and we denote by $E$ their
(almost) common value. Since in dimension one energy levels are always non-
degenerate then exact crossing is not possible, and thus we have an avoided-
crossing picture. At this (almost) common value $E$ of the two levels at the
avoided-crossing point the two wells are separated by an inner barrier, where
the ”classical motion” is forbidden since the potential
$V(x)=V_{i}(x)+V_{e}(x)$ exceeds the energy level $E$; also there is an outer
barrier (see fig. 1).

We can measure the length of these barriers by means of the Agmon metric [7]
that in dimension one is simply
$\rho_{i}=\int_{X_{2}}^{X_{3}}\sqrt{V(x)-E}\,dx$, for the inner barrier, and
$\rho_{e}=\int_{X_{4}}^{X_{5}}\sqrt{V(x)-E}\,dx$ that of the outer barrier.

Figure 1. Asymmetrical double-well model with potential $V=V_{i}+V_{e}$; the
inner barrier with Agmon length $\rho_{i}$ is between $X_{2}$ and $X_{3}$, and
the outer barrier with Agmon length $\rho_{e}$ is between $X_{4}$ and $X_{5}$.
There are no stable states for $H$, but metastable ones.

In fact, for any $F>0$ the stationary states become metastable states
associated to quantum resonances $E_{1}(F)$ and $E_{2}(F)$ depending on $F$
and with negative imaginary part, and in the complex plane the crossing
phenomenon between these two levels may be of:

  * -

type I crossing when there is an exact crossing of their imaginary parts $\Im
E_{1}(F)$ and $\Im E_{2}(F)$, and an avoided-crossing of their real parts $\Re
E_{1}(F)$ and $\Re E_{2}(F)$ (see, e.g., fig. 2), which occurs when
$\rho_{i}<2\rho_{e}$;

  * -

type II crossing when there is an exact crossing of their real parts, and an
avoided-crossing of their imaginary parts (see, e.g., fig. 3), which occurs
when $\rho_{i}>2\rho_{e}$.

Crossing phenomenon of resonances was analyzed as early as Avron’s work in
1982 [8], and was later taken up by [9, 10, 11] providing the above criterion
for determining the type of crossing. This theoretical result is valid in the
semiclassical limit, which in the present context means that the parameters
$\alpha_{1}$, $\alpha_{2}$ and $F$ are very large in absolute value so that
the two resonances are very narrow, that is their imaginary parts are very
small in absolute value. However, from a practical point of view, we will see
in a numerical experiment that this criterion is already useful even if the
parameters are not excessively large. We should mention that the question of
resonances crossing is of great interest both in a general context [12, 13],
and even in experimental applications [14, 15]; furthermore, it has also been
analyzed in the case of symmetrical double-well potential [16].

When the imaginary parts of the two resonances $E_{1}(F)$ and $E_{2}(F)$ are
sufficiently different from each other then the lifetime of the two metastable
states are quite different, and the phenomenon of interference will not be
triggered because one of the two metastable states decays much faster than the
other one. In fact, a very slight beating effect can be observed for small
times, as pointed out by [17], but it quickly disappears and only the
exponential decay associated with the narrowest resonance remains as the
dominant term. On the other side, this picture changes markedly at values of
model parameters for which the imaginary parts of the two resonances coincide
$\Im E_{1}(F)=\Im E_{2}(F)$; in this case, which can occur only in the case of
type I crossing, the interference effect is triggered, and thus we observe a
damped beating effect with (pseudo-)period
$T=2\pi/\left(\Re(E_{2})-\Re(E_{1})\right)$ for large time intervals.

In this way, we have a theoretical model for a quantum sensor that makes it
possible to verify whether the intensity $F$ of the external field assumes the
specific value for which type I crossing and beating effect occur; we observe
that this specific value can be appropriately selected by adjusting the value
of the parameters of the internal double-well potential $V_{i}$.

Qualitative analysis in a simple model. To qualitatively verify this idea on
an explicit one-dimensional model, we consider the double-well potential with
two attractive singular interactions due to two Dirac’s $\delta$,

$\displaystyle V_{i}(x)=\alpha_{1}\delta_{x_{1}}+\alpha_{2}\delta_{x_{2}}\,,\
x_{1}<x_{2}\,,\ \alpha_{1}<\alpha_{2}<0\,,$

where $\delta_{y}$ is such that $\int_{{\mathbb{R}}}\delta_{y}f(x)dx=f(y)$.
The idea of using Dirac’s $\delta$ to model wells or barriers goes back to
Enrico Fermi [18], and in this context it has been extensively used for the
study of the Stark effect on both a single well [19, 20, 21, 22, 23, 24], and
on double-well [16, 25, 17, 26] cases.

The spectrum of $H$ is purely continuous and it coincides with the entire real
axis. Therefore, no stable states are possible. It is possible, however, to
have resonances $E$ associated with metastable states $\psi$ such that
$H\psi=E\psi$ where $\psi$ satisfies the outgoing conditions [16]
$\psi(x)=C_{i}^{+}(x)$ for $x>a$, where $C_{i}^{+}(x)=B_{i}(x)+iA_{i}(x)$,
$A_{i}$ and $B_{i}$ being the two Airy functions. Resonances of $H$ can be
equivalently defined as the complex poles of the analytic continuation of the
kernel of the resolvent operator $[H-z]^{-1}$ from the upper half-plane $\Im
z>0$ to the lower half-plane $\Im z<0$ (see [26] and references therein). If
we denote by $K_{0}=K_{0}^{\pm}$, if $\pm\Im z>0$, the kernel of the resolvent
operator $[H_{0}-z]^{-1}$, where $H_{0}=-\frac{d^{2}}{dx^{2}}-Fx$, then it is
known that

$\displaystyle
K_{0}^{\pm}(x,y;z)=\frac{\pi}{F^{1/3}}\left\\{\begin{array}[]{l}Ci^{\pm}\left(-\frac{Fy+{z}}{F^{2/3}}\right)Ai\left(-\frac{Fx+{z}}{F^{2/3}}\right)\,,x\leq
y\\\
Ci^{\pm}\left(-\frac{Fx+{z}}{F^{2/3}}\right)A_{i}\left(-\frac{Fy+{z}}{F^{2/3}}\right)\,,y<x\end{array}\right..$

The kernel $K=K^{\pm}$, if $\pm\Im z>0$, of the resolvent operator
$[H-z]^{-1}$ can be obtained by $K_{0}^{\pm}$ as follows [27]:

$\displaystyle
K^{\pm}(x,y;z)=K_{0}^{\pm}(x,y;z)+\frac{R^{\pm}(x,y;z)}{D^{\pm}(z)}\,,$

where

$\displaystyle
R^{\pm}(x,y;z)=\sum_{n,m=1}^{2}K_{0}^{\pm}(x,x_{n};z)M^{\pm}_{n,m}(z)K_{0}^{\pm}(x_{m},y;z)\,,$

and where, adopting the shortcut notation
$k^{\pm}_{j,\ell}(z)=K_{0}^{\pm}(x_{j},x_{\ell};z)$,

$\displaystyle
D^{\pm}(z)=\frac{\left[{1}+{\alpha_{1}}k^{\pm}_{1,1}(z)\right]\,\left[{1}+{\alpha_{2}}k^{\pm}_{2,2}(z)\right]}{\alpha_{1}\alpha_{2}}-k^{\pm}_{1,2}(z)k^{\pm}_{2,1}(z)\,,$

and

$\displaystyle
M^{\pm}(z):=\left(\begin{array}[]{cc}\frac{1}{\alpha_{2}}+k^{\pm}_{2,2}(z)&-k^{\pm}_{1,2}(z)\\\
-k^{\pm}_{2,1}(z)&\frac{1}{\alpha_{1}}+k^{\pm}_{1,1}(z)\end{array}\right)\,.$

Then resonances $z$ for $H$ are the zeros such that $\Im z<0$ of the analytic
continuation of the function $D^{+}(z)$ from the upper half-plane $\Im z>0$ to
the lower half-plane $\Im z<0$, and it is possible to compute them numerically
once parameter values $F$, $\alpha_{1}$, $\alpha_{2}$, $x_{1}$ and $x_{2}$ are
assigned.

The two resonances have respectively real parts close to the values of the two
single-well ground states: $\Re E_{1}\sim-\alpha_{1}^{2}/4$ and $\Re
E_{2}\sim-\alpha_{2}^{2}/4-Fa$, where $\sim$ means the asymptotic value for
large $|\alpha_{1,2}|$, and where we set, to fix the ideas and without losing
in generality, $x_{1}=0$ and $x_{2}=a>0$. Thus crossing phenomenon occurs when
$F$ is close to the critical value

$\displaystyle F_{C}\sim\frac{\alpha_{1}^{2}-\alpha_{2}^{2}}{4a}\,,$ (3)

and for these values we have that the Agmon lengths of the inner and outer
barriers are given by $\rho_{i}\sim(\alpha_{1}^{3}-\alpha_{2}^{2})/{12F_{C}}$
and $\rho_{e}\sim{\alpha_{2}^{3}}/{12F_{C}}$. Thus a type I crossing occurs
when $\sqrt[3]{3}\gtrsim\alpha_{1}/\alpha_{2}$, and type II when
$\alpha_{1}/\alpha_{2}\gtrsim\sqrt[3]{3}$. Actually, this theoretical result,
which holds true exactly only in the semiclassical limit in which
$|\alpha_{1,2}|$ and $|F|$ go to infinity, retains good validity even when the
parameters are finite. For example, for $a=5$, $\alpha_{1}=-2.8$, and
$\alpha_{2}=-2$, a type I crossing is observed in fig. 2; from the
semiclassical result (3), it turns out that the theoretical value of $F_{C}$
is $0.192$, and this result is in good agreement with the numerical experiment
in which $F_{C}$ takes the value of $0.190(2)$. For $a=5$, $\alpha_{1}=-3.2$,
and $\alpha_{2}=-2$ a type II cross is observed in fig. 3, and again there is
good agreement between (3) and the result of the numerical experiment.

Figure 2. Type I crossing for $a=5$, $\alpha_{1}=-2.8$, and $\alpha_{2}=-2$.
In the l.h.s. panel we plot the real part of $E_{1}(F)$ (full line) and
$E_{2}(F)$ (broken line); in the r.h.s. panel we plot the imaginary part of
$E_{1}(F)$ (full line) and $E_{2}(F)$ (broken line).

Figure 3. Type II crossing for $a=5$, $\alpha_{1}=-3.2$, and $\alpha_{2}=-2$.
In the l.h.s. panel we plot the real part of $E_{1}(F)$ (full line) and
$E_{2}(F)$ (broken line); in the r.h.s. panel we plot the imaginary part of
$E_{1}(F)$ (full line) and $E_{2}(F)$ (broken line).

An important remark concerns the fact that from the type of crossing follows a
different behaviour of observables. Let us consider, for example, the survival
amplitude $A(t)=\langle\psi_{0}|\psi_{t}\rangle$ where $\psi_{0}$ is the wave
function of the initial state, and where $\psi_{t}=e^{-iHt}\psi_{0}$ is the
wave function of the state at instant $t$. In absence of stable states the
survival amplitude decreases in time [26]. This decay is, for a generic
Hamiltonian without stable states, the contribution of two terms: one of
exponential type that is dominant for intermediate time intervals, and one of
power type that instead becomes dominant for longer times; see [28] where it
was numerically conjectured that a transition effect between the two different
types of decay starts around a certain instant $t$, see also [29, 30, 31, 32,
33, 34, 35, 36] for rigorous results, and [37] for the experimental evidence
of the deviation from exponential decay. However, in our model the power law
decay does not play any role because the spectrum of $H$ is not bounded from
below [38], and thus we expect to observe only the exponential decay. In
particular, since there are only two narrow resonances $E_{1}(F)$ and
$E_{2}(F)$ having small imaginary part then from the residue Theorem it
follows that the dominant term of the survival amplitude is given by [21]

$\displaystyle\langle\psi_{0}|e^{-itH_{0}}\psi_{0}\rangle+\sum_{j=1}^{2}c_{j}e^{-itE_{j}(F)}\,,$

where

$\displaystyle
c_{j}=R_{j}\sum_{n,m=1}^{2}M^{+}_{n,m}\bm{(}E_{j}(F)\bm{)}\,q_{n,j}p_{m,j}\,,$

and

$\displaystyle q_{n,j}$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}}K_{0}^{+}\bm{(}x,x_{n};E_{j}(F)\bm{)}\,\bar{\psi}_{0}(x)dx$
$\displaystyle p_{m,j}$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}}K_{0}^{+}\bm{(}x_{m},y;E_{j}(F)\bm{)}\,\psi_{0}(y)dy$

and where $R_{j}$ is the value of the residue of the function $1/D^{+}(z)$ at
$z=E_{j}(F)$. The evolution operator $e^{-itH_{0}}$ is an integral operator
with kernel [39]

$\displaystyle
U_{0}(x,y;t)=\frac{\mbox{exp}\left\\{\frac{i}{2}\left[\frac{(x-y)^{2}}{2t}-\frac{1}{6}F^{2}t^{3}+{tF(x+y)}\right]\right\\}}{\sqrt{4\pi
it}}\,,$

and the term $\langle\psi_{0}|e^{-itH_{0}}\psi_{0}\rangle$ decays more rapidly
than the exponential $e^{-t|\Im E_{j}(F)|}$ in the case of narrow resonances.
Then, the asymptotic behaviour of $A(t)$ is governed only by the contributions
given by the two resonances. If we are in the type II crossing case, where
e.g. $|\Im E_{1}(F)|<|\Im E_{2}(F)|$ for each value of $F$ as in fig. 2, then
the dominant contribution to the exponentially decreasing behaviour of
$|A(t)|$ is given by $|c_{1}|e^{-t|\Im E_{1}(F)|}$. If, on the other hand, we
are in the case of type I crossing, and if we call $F_{C}$ the value of $F$ at
which $\Im E_{1}(F)=\Im E_{2}(F)$ then we still observe an exponentially
decreasing behaviour given by $|c_{j}|e^{-t|\Im E_{j}(F)|}$, where $j=1$ if
$F<F_{C}$, and $j=2$ if $F>F_{C}$ as in fig. 3; eventually, only for $F$ close
to $F_{C}$ there is a damped oscillating behaviour due to the interference
between the two resonances, and in this case the dominant behaviour of
$|A(t)|$ is given by:

$\displaystyle e^{-t|\Im E_{1}(F_{C})|}\left|c_{1}+c_{2}e^{-i\omega
t}\right]\,,$

with pseudo-period $T={2\pi}/{\omega}$ where $\omega=\Re E_{2}(F_{C})-\Re
E_{1}(F_{C})$.

For $a=5$, $\alpha_{1}=-2.8$, and $\alpha_{2}=-2$, we have a type I crossing
when $F$ takes the value $F_{C}=0.190(2)$. We consider a numerical experiment
where $\psi_{0}$ is a normalized Gaussian localized on the l.h.s. well
corresponding to $x=0$:

$\displaystyle\psi_{0}(x)=(2\pi\sigma^{2})^{-1/4}e^{-x^{2}/4\sigma^{2}}\,,\
\mbox{ where }\sigma=1/2\,,$

and we go on to study the behaviour of $A(t)$ for $t\in[10^{2},10^{5}]$ and
for some $F<F_{C}$, $F>F_{C}$, and for $F=F_{C}$. For the values of the
resonances $E_{j}$, and of the coefficients $c_{j}$, $j=1,2$, see table 1.

|  | $F=0.17<F_{C}$ |  | $F=F_{C}=0.190(2)$ |  | $F=0.21>F_{C}$
---|---|---|---|---|---|---
$\Re E_{1}$ |  | $-1.96(3)$ |  | $-1.97(0)$ |  | $-2.06(6)$
$\Im E_{1}$ |  | $-0.35(1)\cdot 10^{-8}$ |  | $-0.36(8)\cdot 10^{-3}$ |  | $-0.13(7)\cdot 10^{-2}$
$\Re E_{2}$ |  | $-1.86(0)$ |  | $-1.95(7)$ |  | $-1.96(3)$
$\Im E_{2}$ |  | $-0.32(9)\cdot 10^{-3}$ |  | $-0.36(8)\cdot 10^{-3}$ |  | $-0.14(6)\cdot 10^{-4}$
$\Re c_{1}$ |  | $0.41(0)\cdot 10^{-2}$ |  | $0.44(8)$ |  | $0.96(5)$
$\Im c_{1}$ |  | $0.85(1)\cdot 10^{-7}$ |  | $-0.027(6)$ |  | $0.13(0)\cdot 10^{-3}$
$\Re c_{2}$ |  | $0.96(6)$ |  | $0.52(1)$ |  | $0.34(9)\cdot 10^{-2}$
$\Im c_{2}$ |  | $-0.11(2)\cdot 10^{-4}$ |  | $0.027(5)$ |  | $-0.14(4)\cdot 10^{-3}$
Table 1. Table of values of the resonances $E_{j}$ and of the coefficients
$c_{j}$, $j=1,2$, for different values of $F$; where $a=5$, $\alpha_{1}=-2.8$,
and $\alpha_{2}=-2$ correspond to a type I crossing. Figure 4. In the case of
type I crossing corresponding to $a=5$, $\alpha_{1}=-2.8$, and
$\alpha_{2}=-2$, we can observe in the interval $t\in[10^{2},10^{4}]$, a
damped beating behaviour (full line) of $|A(t)|$ when $F$ is close to the
critical value $F_{C}=0.190(2)$ for which we have exact crossing of the
imaginary parts of the two resonances; if $F=0.17<F_{C}$ (broken line) or
$F=0.21>F_{C}$ (dot line) no significant oscillation is observed.

As expected, in the l.h.s. picture of fig. 4 a damped oscillating behaviour is
observed for $|A(t)|$ when the intensity $F$ of the external field is close to
$F_{C}$. On the other hand, when $F$ is different from this value a damping
without significant oscillations is observed in the r.h.s. picture of fig. 4
because one of the two metastable states decays much faster than the other
one, and therefore it is not possible to have a long times interference
phenomenon. In fact, oscillations with small amplitude can be seen only for
short times and the damping effect is slower than in the first case, because
one of the two resonances is much narrower than those obtained when $F=F_{C}$.
We remark that the critical value $F_{C}$ does not depend on the initial
state; instead, as the parameters $\alpha_{1}$, $\alpha_{2}$ and $a$ vary, one
can tune the critical value $F_{C}$ as desired.

Conclusions. In conclusion, time evolution of the survival amplitude for a
Schrödinger operator with asymmetrical double-well potential under the effect
of a Stark perturbation is analyzed in detail in this paper. It is verified
that it is a theoretical model useful to design a quantum sensor for which the
response to an external DC electric field has two distinctly different
behaviours depending on whether the intensity of the field is close to or
different from a predetermined value $F_{C}$ of the external field strength.
By means of (3) the value $F_{C}$ can be chosen by tuning the parameters of
the internal potential. The novelty is that it is possible to verify a
specific value of the DC electric field simply by observing whether or not
significant oscillations are present, no matter their period.

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