# Incompressibility and Symmetry Energy of Neutron Star

Ankit Kumar1,2<EMAIL_ADDRESS>H. C. Das1,2<EMAIL_ADDRESS>S. K.
Patra1,2<EMAIL_ADDRESS>1Institute of Physics, Sachivalaya Marg,
Bhubaneswar 751005, India 2Homi Bhabha National Institute, Training School
Complex, Anushakti Nagar, Mumbai 400094, India

###### Abstract

We trace a systematic and consistent method to precisely numerate the
magnitude range for various structural and isospin compositional properties of
the neutron star. Incompressibility, symmetry energy, slope parameter and
curvature of a neutron star are investigated using the relativistic energy
density functional within the framework of coherent density fluctuation model.
The analytical expression for the energy density functional of the neutron
star matter is motivated from the Br$\ddot{u}$ckner functional and acquired by
the polynomial fitting of the saturation curves for three different
relativistic mean-field parameter sets (NL3, G3 and IU-FSU). The modified
functional is folded with the neutron star’s density-dependent weight function
to calculate the numerical values for incompressibility and symmetry energy
using the coherent density fluctuation model. NL3 parameter set, being the
stiffest equation of state, endue us with a higher magnitude of all the
properties compared to the other two parameter sets.

Neutron stars, being the extremely and suprisingly dense objects in the
universe, are astounding to both astrophysicists and nuclear physicists in
their respective manner. Astronuclear physicists perceive the neutron stars as
the enormously large nucleus bound by the gravity and amazed by its short-
range strong nuclear interactions as well as its enormous gravitational and
electromagnetic interactions. An entire and appropriate understanding of the
properties of neutron stars demand a thorough knowledge of both the
astrophysics and the nuclear physics, which are mostly incoherent. An
inevitable conformability exits between the astral properties (i.e.
gravitational potential, central temperature, angular velocity, magnetic
poles) and the nuclear properties (i.e. baryon density, neutrino emissivity,
isospin asymmetry, superconductivity) of the neutron stars Lattimer and
Prakash (2004). However, there are some properties of the neutron stars like
pressure, incompressibility and symmetry energy, which are tremendously
significant for the experimental and theoretical explanations of its
nontrivial behaviour.

The consequences of “Symmetry” are inalienable to many important aspects of
the modern physics. The theoretical modelling to study the analytical and
structural behaviour of highly asymmetric dense nuclear matter depends
significantly on the symmetry energy Lattimer (2014). Also, the parameters
derived by expanding the symmetry energy around saturation density (slope and
curvature parameter) controls the core-crust transition density, transition
pressure and the cooling rate of the neutron stars Alam  _et al._ (2016);
Schneider  _et al._ (2019). The quantitative information on the slope and
curvature parameter of the neutron stars can be applied to constraint the
equation of state (EoS) obtained using different parameter sets of the vast
relativistic framework. The particle fraction in the core of the neutron
stars, which is a critical quantity to the cooling of neutron stars, is also
controlled by the domain of symmetry energy and slope parameter Gil  _et al._
(2021). Since, symmetry energy can not be measured directly, so, the
elucidation of experimentally available data requires a substantial and
consistent theoretical model for nuclear matter at very high densities Zhang
_et al._ (2008); Hagel  _et al._ (2014); Wada  _et al._ (2012). The precise
range of the symmetry energy for the isospin asymmetric nuclear matter and the
neutron stars has been a questionable issue among the researchers incessantly.
Along with the theoretical calculations, laboratory nuclear experiments (i.e.
giant dipole resonance, heavy-ion collisions, neutron skin) and abridged
results of astrophysical observations (i.e. mass-radius profile, dimensional
tidal deformability) has also put some constraints on the domain of symmetry
energy and its derivatives Kumar  _et al._ (2020); Zhang and Chen (2013);
Trippa  _et al._ (2008); Tsang  _et al._ (2009). Another important quantity
which controls the equation of state (EoS) of any dense matter system is the
incompressibility.

Dependence of incompressibility on the isospin asymmetry parameter of the
nuclear system restricts the stiffness of the EoS which indirectly influences
the maximum mass and radius of the neutron star e.g. maximally compressible
dense matter handles the upper limit on the maximum mass of the neutron star
Olson (2000). The quantitative dimension of the incompressibility coefficient
also helps in the determination of angular velocity and the evolution of
cooling stage of the rapidly rotating neutron stars Hashimoto  _et al._
(1994). The collective motion of the nucleons inside a nucleus produce the
nuclear giant resonances, which is a global feature of the finite nuclei
Bertsch  _et al._ (1983); Bertrand (1981). The iso-scalar giant monopole
resonance (ISGMR) is the most common collective oscillation with both the
protons and neutrons being in same phase. This ISGMR is the breathing mode
oscillation related to the incompressibility $K_{A}$ of the finite nucleus of
mass A. Thus, the nuclear incompressibility $K_{A}$ is a vital quantity to
understand the various modes of oscillations of the finite nucleus Youngblood
_et al._ (2004); Buenerd, M. (1984). Similarly the other modes of collective
oscillations, such as iso-vector giant dipole resonance (IVGDR), iso-
scalar/iso-vector giant quadrupole resonances (ISGQR/IVGQR) are also governed
by the nuclear incompressibility and the other symmetry related parameters of
the finite nucleus. The various collective resonances decide the internal
structures of the finite nucleus, for example, the IVGDR tells the shape of
the nucleus and the ISGMR gives information about the compression or expansion
capacity of the nucleus. The calculations of the monopole and quadrupole
excitation energies and their relations with incompressibility using various
sum rule approaches are demonstrated in Centelles  _et al._ (2005); Patra  _et
al._ (2002a). For a particular model, either a relativistic or a non-
relativistic, to be consistent with the constraints set by the terrestrial or
astrophysical experiments, the nuclear incompressibility, symmetry energy and
other related quantities are very much essential. Neutron star (NS), being a
huge nucleus hypothetically, with mass number $A\sim 10^{57}$, it must posses
all the natural properties of practical finite nuclei, like all type of
collective oscillations. So, the incompressibility $K^{star}$, symmetry energy
$S^{star}$ and its higher derivatives like $L^{star}_{sym}$, $K^{star}_{sym}$
etc. are quite informative and necessary to explore the structure of the
neutron star. In the present paper, our aim is to bring the attention of a new
approach, i.e., Coherent Density Fluctuation Model (CDFM) for the evaluation
of these quantities of neutron stars. This model is applicable for neutron
star, as it has a finite surface like that of a standard nucleus. Here, we
first time provide an approach to calculate both of these parameters (symmetry
energy and incompressibility) for a neutron star matter system in a consistent
and accurate manner.

In the last few decades, the theoretical models destined to explore the
behaviour of dense nuclear medium has been proved inevitable to unravel the
properties of compact astrophysical objects i.e. neutron stars or white
dwarfs. The fascinating structural and compositional resemblance of finite
nuclei and neutron stars allude us that the physics of compact objects can be
explored by extrapolating the data of terrestrial experiments and theoretical
formulization of dense matter systems Bednarek  _et al._ (2020). From being
the extraordinary states of highly dense matter in the inner core to the pasta
phases of nuclei at ordinary densities in the outer crust, neutron star
manifests the distribution of matter thoroughly Lattimer (2012). It is
evidential that the neutron star cores are $10^{3}$ times or more denser than
the density at “neutron drip” line, so, we can utilize a consistent, congruous
and ultra-high dense equation of state of nuclear matter for an effectual
perception of neutron star properties Engvik  _et al._ (1994). The
comprehensive knowledge of the EoS of nuclear matter and pure neutron matter
delineate a prominent bridge between the finite nuclei and dense interstellar
bodies. The EoS of strong interacting dense matter is the key component for
the determination of general properties of neutron star (maximum mass, radius,
tidal deformability) and it also controls the cooling rate and dynamics of
core-collapse supernovae remnants Schneider  _et al._ (2019); Bombaci and
Logoteta (2018). Immobilizing the correct EoS for the compact stellar objects
had been a complex task in the nuclear and astro-particle physics over the
last few decades. Several constraints had been enforced on the EoS at high
density with the help of observational gravitational wave data (GW170817)
Abbott  _et al._ (2017), Einstein Observatory (HEAO-2) Boguta (1981) and
various generations of X-ray radio telescopes Observatory ; Greif  _et al._
(2020). There are many non-relativistic (Skyrme Skyrme and Schonland (1961),
Gogny forces Dechargé and Gogny (1980)) and relativistic (Relativistic Mean-
Field Model Walecka (1974); Gambhir  _et al._ (1990)) theoretical approaches
which endue us with a consistent formalism to construct the EoS and calculate
the empirical properties of strongly-interacting dense matter systems, which
we can adopt as a manifestation of compact stars. The relativistic class of
models are an alternative and more factual approach for low-energy Quantum
Chromodynamics with all the built-in non-perturbative properties (i.e. current
conservation, local or global symmetry breaking etc.) where baryons and nuclei
are stabilized as solitons in a mesonic fluid Adam  _et al._ (2020, 2015).
With the advancement in the cumbersome algebra of quantum field theory, the
effective interaction between the nucleons and mesons can be expressed in the
form of energy density functionals., which can be approached with the help of
self-consistent relativistic mean-field (RMF) model.
In this work, we apply the RMF formalism to obtain the EoS for a dense matter
system, where along with the neutrons and protons, electrons are also present
to maintain the charge neutrality. We will denote this kind of infinite dense
matter (consisting neutron, proton and electron) as neutron star matter (NSM)
for our further discussion. Since, our aim here is to calculate the properties
for a neutron star, so it is a necessity to add electrons and muons in our
system to maintain the compositional and neutrality properties of the star.
Now, the Lagrangian for such a dense matter within RMF formalism can be
written as Kumar  _et al._ (2020)

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\sum_{i=p,n}\bar{\psi}_{i}\Bigg{\\{}\gamma_{\nu}(i\partial^{\nu}-g_{\omega}\omega^{\nu}-\frac{1}{2}g_{\rho}\vec{\tau}_{i}\\!\cdot\\!\vec{\rho}^{\,\nu})-(M-g_{\sigma}\sigma$
(1) $\displaystyle-
g_{\delta}\vec{\tau}_{i}\\!\cdot\\!\vec{\delta})\Bigg{\\}}\psi_{i}+\frac{1}{2}\partial^{\nu}\sigma\,\partial_{\nu}\sigma-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{\zeta_{0}}{4!}g_{\omega}^{2}(\omega^{\nu}\omega_{\nu})^{2}$
$\displaystyle\hbox{}-g_{\sigma}\frac{m_{\sigma}^{2}}{M}\Bigg{(}\frac{\kappa_{3}}{3!}+\frac{\kappa_{4}}{4!}\frac{g_{\sigma}}{M}\sigma\Bigg{)}\sigma^{3}+\frac{1}{2}m_{\omega}^{2}\omega^{\nu}\omega_{\nu}-\frac{1}{4}F^{\nu\alpha}F_{\nu\alpha}$
$\displaystyle\hbox{}+\frac{1}{2}\frac{g_{\sigma}\sigma}{M}\Bigg{(}\eta_{1}+\frac{\eta_{2}}{2}\frac{g_{\sigma}\sigma}{M}\Bigg{)}m_{\omega}^{2}\omega^{\nu}\omega_{\nu}+\frac{1}{2}\eta_{\rho}\frac{m_{\rho}^{2}}{M}g_{\sigma}\sigma(\vec{\rho}^{\,\nu}\\!\cdot\\!\vec{\rho}_{\nu})$
$\displaystyle+\frac{1}{2}m_{\rho}^{2}\rho^{\nu}\\!\cdot\\!\rho_{\nu}-\frac{1}{4}\vec{R}^{\nu\alpha}\\!\cdot\\!\vec{R}_{\nu\alpha}-\Lambda_{\omega}g_{\omega}^{2}g_{\rho}^{2}(\omega^{\nu}\omega_{\nu})(\vec{\rho}^{\,\nu}\\!\cdot\\!\vec{\rho}_{\nu})$
$\displaystyle+\frac{1}{2}\partial^{\nu}\vec{\delta}\,\partial_{\nu}\vec{\delta}-\frac{1}{2}m_{\delta}^{2}\vec{\delta}^{\,2}+\sum_{l=e,\mu}\bar{\phi}_{l}\,(i\gamma_{\nu}\partial^{\nu}-m_{l})\phi_{l},$

where the last term stands for the added electrons and muons in the nuclear
matter and $\psi$, $\bar{\psi}$, $\phi_{l}$ are the wave-functions of the
nucleons (proton and neutron), anti-nucleons and leptons respectively. It
includes $\sigma$, $\omega$, $\rho$ and $\delta$ mesons to represent the
interaction of nucleons, self and cross-coupled interactions. $M$ stands for
the mass of the nucleons; $m_{\sigma}$, $m_{\omega}$, $m_{\rho}$,
$m_{\delta}$, $g_{\sigma}$, $g_{\omega}$, $g_{\rho}$, $g_{\delta}$ are the
masses and the self-coupling constants for $\sigma$, $\omega$, $\rho$ and
$\delta$ mesons respectively; $\kappa_{3}$, $\kappa_{4}$, $\zeta_{0}$,
$\eta_{1}$, $\eta_{2}$, $\eta_{\rho}$ and $\Lambda_{\omega}$ are the coupling
constants; $F^{\nu\alpha}$ and $\vec{R}^{\nu\alpha}$ are field strengths and
$\tau_{3}$ is the isospin operator. A more detailed and term by term
explanation of the Lagrangian is discussed in the references Kumar  _et al._
(2020, 2018a, 2017); Avancini  _et al._ (2003). We obtain the EoS for the
defined neutron star matter by applying the Euler-Lagrange’s equation of
motion and the relativistic mean field approximation on the above Lagrangian (
Eq. 1) Kumar  _et al._ (2018a); Singh  _et al._ (2014). To get the
comprehensive idea of the EoS, we derive the EoS for the complete range of
asymmetry factor. The asymmetry parameter ($\alpha$) is defined as
$\alpha=\frac{n_{n}-n_{p}}{n_{n}+n_{p}}$, with $n_{n}$ and $n_{p}$ being the
neutron and proton density, and $\alpha=1$ stands for purely neutron matter.
The number density of electrons and muons in the neutron star matter is kept
equal to the proton number density to maintain the charge neutrality i.e.,

$\displaystyle n_{p}$ $\displaystyle=$ $\displaystyle n_{e}+n_{\mu}.$ (2)

The binding energy per nucleon ($E/n-M$, $E$ being the energy density and $n$
is the total nucleon density) curve as a function of nucleonic number density
is depicted in Fig. 1.

Figure 1: (Color online) The neutron star matter saturation curves as a
function of baryon number density for different asymmetry
$\alpha=\frac{n_{n}-n_{p}}{n_{n}+n_{p}}$ parameter. The solid curve represents
the RMF numerical data and the dotted black curve stands for the fitted
expression.

In the present work, we used three different RMF parameter sets (NL3Lalazissis
_et al._ (1997), G3Kumar  _et al._ (2017) and IU-FSUCarbone and Schwenk
(2019)), with NL3 being the stiffest and the recently developed G3 being the
soft, compose the whole range of equation of state. The values of nuclear
matter properties (i.e. saturation density, binding energy, incompressibility
etc.) for symmetric nuclear matter predicted by the chosen RMF parameter sets
satisfy all the empirical/experimental constraints. The numerical magnitude of
incompressibility of symmetric nuclear matter ($K$) at saturation density for
G3 and IU-FSU forces are $244$ and $231$ $MeV$, and that of symmetry energy
($S$) are $31$ and $32$ $MeV$ respectively Kumar  _et al._ (2020); Das  _et
al._ (2020a), which are appropriate for the range stated by various
theoretical and experimental models Garg and Colò (2018); Danielewicz and Lee
(2014). The three considered parameter sets are widely used in literature and
our aim is to show the variation of the results with different forces. It is
to be noted here that NL3 is one of the most successful parameter set for
finite nuclei. This set is also used to explain the GW190814 data with an
admixture of dark matter inside the neutron star Das  _et al._ (2021a). The
coupling constants and the nuclear matter properties at saturation density
along with the empirical/experimental values are presented in the table 1.

Table 1: The coupling constants and the nuclear matter properties at saturation for the EoS of NL3 Lalazissis  _et al._ (1997), G3 Kumar  _et al._ (2017) and IU-FSU Carbone and Schwenk (2019) parameter sets. The nucleon mass ($M$) is 939.0 MeV. All of the coupling parameters are dimensionless and the NM parameters are in MeV, except $n_{0}$ which is in fm-3. The NM parameters are given at saturation density for NL3, G3 and IU-FSU parameter sets in the lower panel. The references are $[a]$,$[b]$, $[c]$ $\&$ $[d]$ Zyla  _et al._ (2020), $[e]$&$[f]$ Bethe (1971), $[g]$ Garg and Colò (2018), $[h]$&$[i]$ Danielewicz and Lee (2014), and $[j]$ Zimmerman  _et al._ (2020). Parameter | NL3 | G3 | IU-FSU | Empirical/Expt. Value
---|---|---|---|---
$m_{\sigma}/M$ | 0.541 | 0.559 | 0.523 | 0.426 – 0.745 $[a]$
$m_{\omega}/M$ | 0.833 | 0.832 | 0.833 | 0.833 – 0.834 $[b]$
$m_{\rho}/M$ | 0.812 | 0.820 | 0.812 | 0.825 – 0.826 $[c]$
$m_{\delta}/M$ | 0.0 | 1.043 | 0.0 | 1.022 – 1.064 $[d]$
$g_{\sigma}/4\pi$ | 0.813 | 0.782 | 0.793 |
$g_{\omega}/4\pi$ | 1.024 | 0.923 | 1.037 |
$g_{\rho}/4\pi$ | 0.712 | 0.962 | 1.081 |
$g_{\delta}/4\pi$ | 0.0 | 0.160 | 0.0 |
$k_{3}$ | 1.465 | 2.606 | 1.1593 |
$k_{4}$ | -5.688 | 1.694 | 0.0966 |
$\zeta_{0}$ | 0.0 | 1.010 | 0.03 |
$\eta_{1}$ | 0.0 | 0.424 | 0.0 |
$\eta_{2}$ | 0.0 | 0.114 | 0.0 |
$\eta_{\rho}$ | 0.0 | 0.645 | 0.0 |
$\Lambda_{\omega}$ | 0.0 | 0.038 | 0.046 |
$n_{0}$ | 0.148 | 0.148 | 0.154 | 0.148 – 0.185 $[e]$
$B.E.$ | -16.29 | -16.02 | -16.39 | -15.00 – 17.00 $[f]$
$K$ | 271.38 | 243.96 | 231.31 | 220 – 260 $[g]$
$S$ | 37.43 | 31.84 | 32.71 | 30.20 – 33.70 $[h]$
$L_{sym}$ | 120.65 | 49.31 | 49.26 | 35.00 – 70.00 $[i]$
$K_{sym}$ | 101.34 | -106.07 | 23.28 | -174 – -31 $[j]$
$Q_{sym}$ | 177.90 | 915.47 | 536.46 | ———–

To achieve an expressional form of energy functional for the effective
interactions in the neutron star matter explained by RMF formalism, we have
fitted the numerically obtained data. The assumptive form of the fitted energy
functional is motivated by the work of Br$\ddot{u}$ckner et al. Brueckner  _et
al._ (1968a, b). There were several issues with the Br$\ddot{u}$ckner’s energy
functional, for instance, it can not rectify the Coester-band problem Coester
_et al._ (1972) and it is defined purely for the non-relativistic nuclear
matter formalism. We ameliorate the Br$\ddot{u}$ckner’s energy functional in
the local density approximation, so that, it can satisfy the RMF data Kumar
_et al._ (2021), and also added some series of potential function and an extra
term for the lepton’s kinetic energy inclusion. The modified energy density
functional can be stated as,

$\displaystyle{\cal E}$ $\displaystyle=$ $\displaystyle
C_{k}n^{2/3}+C_{e}n^{4/9}+\sum_{i=3}^{14}(b_{i}+a_{i}\alpha^{2})n^{i/3},$ (3)

where
$C_{k}=0.3(\hbar^{2}/2M)(3\pi^{2})^{2/3}[(1+\alpha)^{5/3}+(1-\alpha)^{5/3}]$
Brueckner  _et al._ (1968b) is the kinetic energy coefficient for nucleons and
$C_{e}=b_{e}(1-\alpha)^{5/9}$, with $b_{e}$ as a variable obtained during
fitting procedure, is the kinetic term coefficient for leptons. The last term
stands for the potential interaction of the nucleons and the coefficients
$b_{i}$ and $a_{i}$ has to be obtained by fitting procedure for different RMF
parameter sets. We observed that the accuracy of the fitting mechanism
decreases if we reduce the number of coefficients ($a_{i}$ and $b_{i}$) in the
expansion of potential term of Eq. 3 Kumar  _et al._ (2021). The mean
deviation ‘$\delta$’, which is defined as,
$\delta=[\sum_{j=1}^{N}(E/A)_{j,\mathrm{Fitted}}-(E/A)_{j,\mathrm{RMF}}]/N$, N
being the total number of points, is $18\%$, $6\%$ and $0.5\%$ for 8 (i.e.
‘$i$’ runs from $3$ to $10$ in Eq. 3), 10 and 12 terms respectively. The
fitted curves of the neutron star matter with the above expression (Eq. 3) are
shown in the Fig. 1 through the black dotted lines. We have used this energy
density functional as an input in the coherent density fluctuation method to
achieve the range of certain properties of neutron star (incompressibility,
symmetry energy and slope parameter) for the first time. Coherent density
fluctuation model (CDFM) was first introduced three decades ago by Antonov et
al. and is now a well established formalism to decipher the properties of
finite nuclei Antonov  _et al._ (1980, 1994); Gaidarov  _et al._ (2020). The
mere backbone of the CDFM model is that a coordinate generator ‘$x$’ can be
used to write the one-body density matrix $n(r,r^{\prime})$ of a nucleus as a
sequential superposition of infinite number of one-body density matrices
$n_{x}(r,r^{\prime})$, which are coined as “fluctons” Gaidarov  _et al._
(2011); Kaur  _et al._ (2020). The density of fluctons has the form,

$n_{x}({\bf r})=n_{0}(x)\,\Theta(x-|{\bf r}|),$ (4)

where $n_{0}(x)$ is defined as $n_{0}(x)=3A/4\pi x^{3}$, $A$, being the total
number of nucleons in the finite matter. Within the CDFM approach, the density
distribution of the spherical finite nuclear matter of radius ‘$r$’ can be
expressed as Antonov  _et al._ (2018, 2016),

$\displaystyle n(r)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dx\,|F(x)|^{2}\,n_{0}(x)\,\Theta(x-|{\bf r}|),$
(5)

$|F(x)|^{2}$ is defined as the weight function and it can be obtained
theoretically for a monotonically decreasing local density in the generator
coordinate ‘$x$’ as Antonov  _et al._ (2018),

$|F(x)|^{2}=-\frac{1}{n_{0}(x)}\frac{dn(r)}{dr}\Bigg{|}_{r=x},$ (6)

Now, the nuclear and structural properties of the neutron star, i.e.
incompressibility ($K^{star}$), symmetry energy ($S^{star}$), slope parameter
($L^{star}_{sym}$) and curvature ($K^{star}_{sym}$), can be expressed in terms
of weight function and the expression of the parameters evaluated from the
energy density functional (eq. 3) of infinite star matter system, as, Gaidarov
_et al._ (2011, 2012); Antonov  _et al._ (2016)

$\displaystyle K^{star}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dx\,|F(x)|^{2}\ K^{NSM}(n(x)),$ (7)
$\displaystyle S^{star}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dx\,|F(x)|^{2}\,S^{NSM}(n(x)),$ (8)
$\displaystyle L_{sym}^{star}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dx\,|F(x)|^{2}\,L_{sym}^{NSM}(n(x)),$ (9)
$\displaystyle K_{sym}^{star}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dx\,|F(x)|^{2}\ K_{sym}^{NSM}(n(x)),$ (10)

The profile of the weight function with density holds the key information
about the dependence of the calculated properties on the structure and
composition of the defined matter system. The magnitude of the weight function
at a value of ‘$x$’ will decide the share of that particular region of density
in the overall magnitude of calculated property. The energy density functional
for the neutron star matter can be converted from momentum space to the
coordinate space ‘$x$’ in a local density approximation technique using the
Br$\ddot{u}$ckner method.The expressions for $K^{NSM}$, $S^{NSM}$,
$L^{NSM}_{sym}$ and $K^{NSM}_{sym}$ can be obtained from Eq. 3, by applying
their common derivative definitions Fetter and Walecka (1971); Chen  _et al._
(2009); Chen and Piekarewicz (2014), i.e., the NM parameters $K^{NM}$,
$S^{NM}$, $L_{sym}^{NM}$ and $K_{sym}^{NM}$ are obtained from the following
standard relations:

$\displaystyle K^{NM}$ $\displaystyle=$ $\displaystyle
9\rho_{0}^{2}\frac{\partial^{2}}{\partial\rho^{2}}\bigg{(}\frac{\cal
E}{\rho}\bigg{)}\Big{|}_{\rho=\rho_{0}},$ (11) $\displaystyle S^{NM}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\frac{\partial^{2}({\cal
E}/\rho)}{\partial\alpha^{2}}\Big{|}_{\alpha=0},$ (12) $\displaystyle
L_{sym}^{NM}$ $\displaystyle=$ $\displaystyle 3\rho_{0}\frac{\partial
S(\rho)}{\partial\rho}\Big{|}_{\rho=\rho_{0}}=\frac{3P}{\rho_{0}},$ (13)
$\displaystyle K_{sym}^{NM}$ $\displaystyle=$ $\displaystyle
9\rho_{0}^{2}\frac{\partial^{2}S(\rho)}{\partial\rho^{2}}\Big{|}_{\rho=\rho_{0}}.$
(14)

which are given as follow using Eq. (3)

$\displaystyle K^{NSM}$ $\displaystyle=$
$\displaystyle-150.12\,n_{0}^{2/3}(x)-2.22\,b_{e}\,n_{0}^{4/9}(x)$ (15)
$\displaystyle+\sum_{i=4}^{14}i\,(i-3)\,b_{i}\,n_{0}^{i/3}(x),$ $\displaystyle
S^{NSM}$ $\displaystyle=$ $\displaystyle
41.7\,n_{0}^{2/3}(x)-0.12\,b_{e}\,n_{0}^{4/9}(x)$ (16)
$\displaystyle+\sum_{i=3}^{14}a_{i}\,n_{0}^{i/3}(x),$ $\displaystyle
L_{sym}^{NSM}$ $\displaystyle=$ $\displaystyle
83.4\,n_{0}^{2/3}(x)-0.16\,b_{e}\,n_{0}^{4/9}(x)$ (17)
$\displaystyle+\sum_{i=3}^{14}i\,a_{i}\,n_{0}^{i/3}(x),$ $\displaystyle
K_{sym}^{NSM}$ $\displaystyle=$
$\displaystyle-83.4\,n_{0}^{2/3}(x)+0.266\,b_{e}\,n_{0}^{4/9}(x)$ (18)
$\displaystyle+\sum_{i=4}^{14}i\,(i-3)\,a_{i}\,n_{0}^{i/3}(x),$

The above expressions are derived for symmetric ($\alpha=0$) case. Now, we can
easily calculate the nuclear properties of the neutron star with the help of
weight function and using the above expressions. The weight function
computation demand the slope of density curve with respect to radius of the
neutron star. We compute the mass-radius profile and the density curve of the
neutron star for the three considered RMF parameter sets. The density curve
and the mass-radius profile of the neutron star can be acquired by imposing
the beta equilibrium conditions Pacini (1965); Kumar  _et al._ (2020) in the
neutron star matter and using the obtained EoS as an input for the Tolman-
Oppenheimer-Volkoff (TOV) equations Oppenheimer and Volkoff (1939); Tolman
(1939). The $\beta$\- equilibrium conditions and the TOV equations for the
static isotropic proto-neutron star can be written as,

$\displaystyle\mu_{n}$ $\displaystyle=$ $\displaystyle\mu_{p}+\mu_{e},$
$\displaystyle\mu_{e}$ $\displaystyle=$ $\displaystyle\mu_{\mu},$ (19)
$\displaystyle\mathrm{and}$ $\displaystyle\frac{dP(r)}{dr}$ $\displaystyle=$
$\displaystyle-\frac{[E(r)+P(r)]}{r^{2}\Big{(}1-\frac{2M(r)}{r}\Big{)}}[M(r)+{4\pi
r^{3}P(r)}],$ $\displaystyle\frac{dM(r)}{dr}$ $\displaystyle=$ $\displaystyle
4\pi r^{2}{E(r)}.$ (20)

Here $\mu_{n}$, $\mu_{p}$, $\mu_{e}$, and $\mu_{\mu}$ describe the chemical
potentials of the neutrons, protons, electrons, and muons respectively; $E$
and $P$ are the energy density and pressure of the neutron star. The self
consistent numerical solution of Eq. (2) and Eq. (15) will set the fraction of
neutron, proton, electron and muon number density for a given baryon density
in a neutron star. $M(r)$ is defined as the mass of the neutron star at radius
r and the boundary conditions to solve these equations are $P(R)=0$, for a
particular choice of central density $n_{c}=n(0)$. Finally, we also added the
crust part in the above computed EoS to get a detailed and complete analysis
of the neutron star properties. We extended the surface part of the NS
mathematically by adding the crust energy and pressure calculated by Baym,
Pethick and Sutherland, i.e. BPS crust EoS, in the tail part of all the three
RMF parameter’s main equation of state Baym  _et al._ (1971). A more detailed
formalism to calculate the mass-radius profile of the neutron star using RMF
equation of state can be found in the references Kumar  _et al._ (2020,
2018a); Das  _et al._ (2020b). The mass-radius profile for all the three
assumed parameter sets is shown in fig. 2.

Figure 2: (color online) Mass-radius profile of a neutron star for NL3 (red),
G3 (green) and IU-FSU (blue) parameter sets. The old NICER data given in two
boxes from the two different analysis Miller  _et al._ (2019); Riley  _et al._
(2019). The horizontal line in violet colour represents the new NICER
constraint on the radius of the canonical star Miller  _et al._ (2021).

The maximum mass of the neutron star calculated with the help of G3 and IU-FSU
parameter sets are $2.004M_{\odot}$ and $1.940M_{\odot}$ respectively Kumar
_et al._ (2020); Das  _et al._ (2021b), which fit well in the range of the
observational pulsar data PSR J1614-2230 ($M=1.908\pm 0.04M_{\odot}$)
Arzoumanian  _et al._ (2018), PSR J0348+0432 ($M=2.01\pm 0.04\ M_{\odot})$
Antoniadis  _et al._ (2013) and PSR J0740+6620
($M=2.15^{+0.10}_{-0.09}M_{\odot}$) Cromartie  _et al._ (2019). Recent
evaluation of the pulsar data PSR J0740+6620 done by Fonseca et. al. enumerate
the mass of the star in the range $2.08\pm 0.07M_{\odot}$ with $68\%$
confidence limit Fonseca  _et al._ (2021). The old NICER Miller  _et al._
(2019); Riley  _et al._ (2019) data are satisfied by both G3 and IU-FSU sets.
Also, recently the new equatorial circumferential radius measurements are
reported by Miller at. al. Miller  _et al._ (2021) on the basis of NICER and
XMM-Newton X-ray observation of PSR J0740+6620 within $68\%$ confidence limit.
However, the mass predicted by the NL3 parameter set is quite larger than the
defined limit. The radius constraint put recently by the Miller et. al. using
NICER simulations for the canonical star ($1.4M_{\odot}$) is also well
satisfied by the G3 and IU-FSU parameter sets Miller  _et al._ (2021). Also,
the constraint set by the observational data of GW170817 event for the tidal
deformability of canonical star is satisfied by the G3 parameter set
($\Lambda=582.26$) Kumar  _et al._ (2018b); Abbott  _et al._ (2017); Das  _et
al._ (2021c). So, we can claim that the assumed RMF parameter sets are
consistent with the astrophysical observational data and well-suited to
calculate the various properties of neutron star.

Figure 3: (color online) The neutron star densities ($\rho$) for NL3 (red), G3
(green) and IU-FSU (blue) parameter sets as a function of radius of the
maximum mass star (R). The mass number ($A$) of the maximum mass neutron star
for NL3, G3 and IU-FSU are $3.35\times 10^{54}$, $2.32\times 10^{54}$ and
$2.23\times 10^{54}$ respectively.

The density-radius curve of the neutron star for NL3, G3 and IU-FSU parameter
sets is depicted in Fig. 3. The density-radius curve is computed for the
maximum mass predicted by the corresponding parameter set i.e.
$2.850M_{\odot}$ for NL3, $2.004M_{\odot}$ for G3 and $1.940M_{\odot}$ for IU-
FSU. We observe that the central density of the neutron star is maximum for
the G3 parameter set, while NL3 being the stiffest EoS, have the lowest
central density.
However, with the help of the density-radius curve, we can calculate the
weight function ($|F(x)|^{2}$) of the neutron star. The total number of
nucleons for the neutron star with the maximum mass predicted by NL3, G3 and
IU-FSU parameter sets with mass number A are $3.35\times 10^{54}$, $2.32\times
10^{54}$ and $2.23\times 10^{54}$ respectively Glendenning (1997). The total
number of nucleons computed for a canonical star are $1.53297\times 10^{54}$,
1.53281$\times 10^{54}$, and 1.53286 $\times 10^{54}$ respectively for NL3, G3
and IU-FSU sets. Although, the number difference appears after the third
decimal for all three forces, actually these numbers are quite different from
each other due to the order of magnitude. It is worth mentioning that the
neutron star is a big nucleus with nucleons, electrons and muons, which has
variation of density with radius as shown in Fig. 3. It possesses all
properties of a finite nucleus with mass number A. The giant monopole
excitation energy controls by the incompressibility of the nucleus Patra  _et
al._ (2002b, 2001, c). Here also the $K^{star}$ gives significant information
about the giant resonances of the neutron star.

Table 2: The numerical values of incompressibility, symmetric energy, slope and curvature parameter for maximum mass and the canonical mass star of the corresponding RMF parameter sets. The maximum mass for NL3, G3 and IU-FSU parameter sets are $2.850M_{\odot}$, $2.004M_{\odot}$ and $1.940M_{\odot}$ respectively. All the values are in MeV unit. Parameter Set | NL3 | G3 | IU-FSU
---|---|---|---
| Maximum mass
---
| Canonical mass
---
| Maximum mass
---
| Canonical mass
---
| Maximum mass
---
| Canonical mass
---
$K^{star}$ | 44.956 | 3.934 | 29.480 | 3.590 | 29.827 | 3.389
$S^{star}$ | 146.002 | 15.911 | 66.813 | 8.904 | 60.758 | 7.228
$L^{star}_{sym}$ | 615.854 | 65.038 | 307.015 | 45.392 | 320.321 | 48.225
$K^{star}_{sym}$ | -688.514 | -70.534 | -360.127 | -44.315 | -228.012 | -26.282

With all the input ingredients acquired, we calculate the numerical values of
incompressibility, symmetry energy and its derivatives for maximum mass
neutron star and canonical star represented by all the three parameter sets
using RMF density functional and CDFM model. Till now, to the best of our
knowledge we did not find any work in the literature regarding the
availability of exact numerical values of nuclear matter properties of neutron
star or any theoretical model which can endue us with such formalism. We,
here, are explicating the numerical values of incompressibility, symmetry
energy, slope and curvature parameter (Table 2) of a neutron star for the
chosen RMF forces, which is unaccustomed. We deciphered some interesting
results from the values of Table 2. We observed that the values of all the
nuclear properties for maximum mass star with NL3 is greater in comparison to
G3 and IU-FSU forces. The magnitude of incompressibility coefficient for
maximum mass star with NL3, G3 and IU-FSU parameter sets are $44.956$,
$29.480$ and $29.827$ $MeV$ respectively. This tendency of NL3 predicting the
higher values of incompressibility and other properties for neutron star
justify its nature of stiff EoS, which has also been anticipated for symmetric
nuclear matter case Lalazissis  _et al._ (1997).

Another important dimension of incompressibility is that the incompressibility
of the matter decreases with increase of asymmetricity and density. For
example, the incompressibility of pure nuclear matter, i.e., with equal number
of protons and neutrons at saturation is 243.96 MeV for G3 set and it is
29.827 MeV for neutron star matter, which has a large asymmetricity and
density. This phenomenon of decrement in the incompressibility coefficient as
we keep increasing the density of the system has also been noticed and
reported in the literature priorly Kumar  _et al._ (2020). The validity of a
equation of state can be solely checked by computing its incompressibility
coefficient. The numerical range for incompressibility coefficient is indeed
the most important quantity to calculate as it restricts the stiffness of the
equation of state of the system by checking the compatibility of the equation
of state with causality, which require the adiabatic sound speed not to exceed
the speed of light Das  _et al._ (2020b). The lowest central density for NL3
parameter set despite being the prediction of highest mass is the result of
causality restriction, as the stiffness of the equation of state which is
related to the incompressibility should be be compatible with causality at
highest density Olson (2000). We realise that the maximally incompressible
equation of state can be soften by reducing the $K$, which in turn will reduce
the maximum neutron star mass significantly. To inspect the shift of
incompressibility coefficient with the mass of the neutron star, we extend the
calculations for different masses using G3 parameter set. We observe that the
values of $K^{star}$ with G3, as a representative parameter set, for
$1.4M_{\odot}$, $1.6M_{\odot}$ , $1.7M_{\odot}$, $1.8M_{\odot}$ and
$2.004M_{\odot}$ are $3.590$, $5.203$, $6.302$, $9.163$ and $29.480$ MeV
respectively. However, on the other hand, the incompressibility coefficient of
the canonical star is almost same for all the considered RMF parameter sets,
given in table 2. This particular observation of the incompressibility
coefficient seems to indicate that the value of $K^{star}$ is proportional to
the mass of the star, which sequentially depends on the number of nucleons. A
more detailed study regarding the observance and conclusions of the
correlations between the incompressibility and mass of the neutron star is in
progress and will be published somewhere else Pattnaik  _et al._ (2021).

The magnitude of the symmetry energy for maximum mass neutron star is also a
bit higher in comparison to the magnitude of symmetric nuclear matter at
saturation density for all the RMF parameter sets. The numerical values of
symmetry energy at saturation density for a symmetric nuclear matter with NL3,
G3 and IU-FSU forces are $37.43$, $31.84$ and $32.71$ MeV respectively, while
those for the case of maximum mass neutron star comes out to be $146.002$,
$66.813$ and $60.758$ MeV. The values of symmetry energy for canonical star is
quite smaller in magnitude as compare to maximum mass star of the
corresponding RMF parameter set. Also, contrary to the case of
incompressibility, in spite of being the same mass of canonical star for all
the three parameter sets, the magnitude of the symmetry energy is not equal
for different parameter set. This kind of behaviour reflects the dependence of
symmetry energy on the structure and composition of the neutron star. The
incompressibility $K$ is obtained from the derivative of energy density with
respect to density, but the symmetry energy is derived from the derivative of
the energy density with respect to asymmetricity. Thus, shows a significant
variation in the symmetry energy as compared to incompressibility. As we can
see from fig. 2, the radius of canonical star differ for the NL3, G3 and IU-
FSU parameters, which cause the change in $S^{star}$ for the same mass of the
star.

Although there is no empirical or experimental data available to support the
magnitude of the neutron star’s symmetry energy, but as we know that neutron
star is a highly asymmetric dense object, so, a major change in the symmetry
energy with mass is expected due to its isospin-dependent characteristics and
the structural behaviour. The unexpectedly larger value of $L_{sym}^{star}$
for NL3 parameter set is also supported by the inclination of stiff EoS of the
dense matter towards higher value of slope parameter Bednarek  _et al._
(2020). A precise knowledge of the range of symmetry energy and $L_{sym}$ is
enough to estimate the radius of a neutron star quite perfectly. A well
defined information about the range of symmetry energy and $L_{sym}$ is
relevant to trace a more strong and constrained correlation between the
surface and volume symmetry energy terms in the mass formula Lattimer (2015).
The static dipole polarizability. quadrupole polarizability and the neutron
skin thickness is closely related to the correlation between symmetry energy
and slope parameter Lattimer (2015); Lattimer and Lim (2013). Similar to the
case of $S^{star}$, the magnitude of slope parameter for canonical star is
also significantly smaller in comparison to the maximum mass star for the
considered RMF parameter sets. $K_{sym}^{star}$, being the second order
derivative of symmetry energy, is most sensitive and ambiguous quantity to
calculate precisely. The negative magnitude of the curvature parameter
corroborated by the 1-$\sigma$ constraint and $90\%$ confident bounds on its
value at saturation density of symmetric nuclear matter, derived by Josef
Zimmerman et al. using the observational data of PSR J0030+0451 and GW170817
event Zimmerman  _et al._ (2020); Riley  _et al._ (2019); Abbott  _et al._
(2017). Although, these bounds are not well suited to discuss the curvature
parameter of a neutron star, but it however, implies the possibility of
negative magnitude and allude the proximity range around the $90\%$ confidence
limit of astrophysical observation data. The separation of the contribution of
isovector incompressibility or curvature parameter ($K_{sym}$) part from the
total incompressibility of a matter can be proved very useful for some
teresterrial experiments related to exotic nuclei and heavy-ion collisions
Chen  _et al._ (2007); Centelles  _et al._ (2009). A more confident
theoretical bound on the value of $K_{sym}$ estimated by the present study
through RMF parameter sets for neutron star will path a better way to tune the
experimental techniques related to isoscalar giant resonances towards the
prediction of properties related to astrophysical objects. A more detailed
study about the importance and correlations of the newly introduced parameters
for the neutron star ($X^{star}$) are quite interesting and will be published
in a future work Pattnaik  _et al._ (2021).
Despite the deprivation of direct experimental measurement or empirically
acquirable data for the nuclear properties i.e. incompressibility, symmetry
energy etc. of the neutron star, the numerical values calculated here with the
help of consolidated RMF and CDFM formalism appear justifiable and
irreproachable. The competency of the present theoretical perspective can be
exuberantly validated using various consistent energy density functionals and
relevant RMF parameter sets. The accomplished accessibility to neutron star
properties through a finite nuclei approach favour more dimensions of strongly
correlated bridge betwixt the two unequal size objects. The theoretical
approach employed in this work present new opportunities for the nuclear and
astrophysicist to unearth the wealth of information on the dense astronautical
objects and exotic finite nuclei.

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# Monitoring the energy of a cavity by observing
the emission of a repeatedly excited qubit

H. Hutin  Ecole Normale Supérieure de Lyon, CNRS, Laboratoire de Physique,
F-69342 Lyon, France  A. Essig  Ecole Normale Supérieure de Lyon, CNRS,
Laboratoire de Physique, F-69342 Lyon, France  R. Assouly  Ecole Normale
Supérieure de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France  P.
Rouchon  Laboratoire de Physique de l’Ecole normale supérieure, Mines-Paris -
PSL, Inria, ENS-PSL, CNRS, Université PSL, Paris, France.  A. Bienfait  Ecole
Normale Supérieure de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon,
France  B. Huard  Ecole Normale Supérieure de Lyon, CNRS, Laboratoire de
Physique, F-69342 Lyon, France

###### Abstract

The number of excitations in a large quantum system (harmonic oscillator or
qudit) can be measured in a quantum non demolition manner using a dispersively
coupled qubit. It typically requires a series of qubit pulses that encode
various binary questions about the photon number. Recently, a method based on
the fluorescence measurement of a qubit driven by a train of identical pulses
was introduced to track the photon number in a cavity, hence simplifying its
monitoring and raising interesting questions about the measurement backaction
of this scheme. A first realization with superconducting circuits demonstrated
how the average number of photons could be measured in this way. Here we
present an experiment that reaches single shot photocounting and number
tracking owing to a cavity decay rate 4 orders of magnitude smaller than both
the dispersive coupling rate and the qubit emission rate. An innovative notch
filter and pogo-pin based galvanic contact makes possible these seemingly
incompatible features. The qubit dynamics under the pulse train is
characterized. We observe quantum jumps by monitoring the photon number via
the qubit fluorescence as photons leave the cavity one at a time. Besides, we
extract the measurement rate and induced dephasing rate and compare them to
theoretical models. Our method could be applied to quantum error correction
protocols on bosonic codes or qudits.

Figure 1: (a) Acoustic analogy of the experiment. In order to monitor the
number of apples in a basket, it is attached to a string. By repeatedly
hitting the string, recording the emitted sound reveals the number $n$ at any
time. (b) The heterodyne detection of the driven qubit (red) emission into the
line leads to the monitoring of the photon number of the dispersively coupled
cavity (green). A notch filter centered on $\omega_{c}$ (box) prevents the
cavity from decaying through the qubit. (c) The cavity is a high purity
aluminium $\lambda/4$ coaxial resonator. Two transmon qubits on sapphire chips
stick into the cavity. The filter on the (red) qubit drive line is composed of
two tantalum spurlines and galvanically connected to a transmission line using
a pogo pin (photograph in inset). Wigner tomography of the cavity is performed
using the auxiliary transmon qubit (purple) and its dedicated readout
resonator (not shown).

Resolving the number of photons in an electromagnetic mode is at the core of
many quantum information protocols Knill  _et al._ (2001); Simon  _et al._
(2007); Aaronson and Arkhipov (2014); Hamilton  _et al._ (2017). Most of them
require quantum nondemolition (QND) measurements for measurement based
feedback or heralding. In the microwave domain, such measurements can be
performed using dispersively coupled Rydberg atoms Gleyzes  _et al._ (2007) or
superconducting qubits Schuster  _et al._ (2007). Predetermined Guerlin  _et
al._ (2007); Johnson  _et al._ (2010) or adaptive Peaudecerf  _et al._ (2013,
2014); Wang  _et al._ (2020a); Dassonneville  _et al._ (2020); Curtis  _et
al._ (2021) measurement sequences can monitor the photon number in time and
even detect quantum jumps. Each measurement step yields at most a single bit
of information about the photon number. By adapting each step, it is possible
to reach this upper bound and photocount in a number of cycles that scales
logarithmically with the maximal photon number Wang  _et al._ (2020a);
Dassonneville  _et al._ (2020); Curtis  _et al._ (2021). Recently, another
qubit-based detector was introduced, which is able to track the photon number
using a train of identical qubit pulses forming a frequency comb Essig  _et
al._ (2021). Consequently the frequency of the qubit fluorescence encodes the
photon number at any time. While a proof-of-principle experiment demonstrated
signals proportional to the photon number Essig  _et al._ (2021), the
measurement rate was insufficient compared to the cavity lifetime for single-
shot extraction. Here, we demonstrate photon number tracking in a 3D cavity
using a frequency comb driving a dispersively coupled qubit. We experimentally
compare the photon number measurement rate of our scheme based on heterodyne
detection of the qubit fluorescence to the rate at which the environment could
extract information. This QND photon number monitor, with a fixed drive and
detection scheme, could simplify feedback schemes and quantum error correction
for bosonic codes Cai  _et al._ (2021) or qudits Joo  _et al._ (2019); Wang
_et al._ (2020b).

The detection principle can easily be grasped with a classical analogy (Fig.
1a). Consider a basket (cavity) filled with apples (photons) that can escape.
The number $n$ of apples can be determined by hitting a string (qubit) from
which the basket hangs. The string oscillates at a frequency depending on $n$
and recording the emitted sound (heterodyne measurement of fluorescence
signal) reveals the apple number. Hitting repeatedly the string leads to the
monitoring of $n$: in the frequency domain, it corresponds to driving the
string with a comb.

Experimentally, the basket is an aluminum $\lambda/4$ coaxial cavity Reagor
_et al._ (2016) at frequency $\omega_{c}/2\pi=4.573~{}\mathrm{GHz}$. The qubit
is a transmon at $\omega_{q}/2\pi=6.247~{}\mathrm{GHz}$, dispersively coupled
to the cavity with a frequency shift $-\chi/2\pi=-5.25$ MHz per photon. Photon
number tracking requires to operate in the photon number resolved regime
$\chi>\Gamma_{2}$, with $\Gamma_{2}/2\pi\approx 3.5~{}\mathrm{MHz}$ the qubit
coherence rate Essig  _et al._ (2021). To optimize the information rate, we
maximize the qubit emission rate $1/T_{q}\leq 2\Gamma_{2}$ into the
measurement line under this constraint.

To protect the cavity from decaying through the qubit, we use a notch filter
at the cavity frequency (Fig. 1b). The filter circuit is composed of two on-
chip spurlines in series (Fig. 1c). The qubit and the notch filter are
patterned out of a tantalum film on the same sapphire chip, which is inserted
into the cavity sup . A galvanic connection is ensured between the measurement
line and the on-chip filter using an SMA microwave connector terminated by a
pogo pin 111POGO-PIN-19.0-1 by Emulation Technology, which is a pin connected
to a spring (Fig. 1c). This filter design leads to a high coupling rate
$1/T_{q}=(23\pm 3~{}\mathrm{ns})^{-1}$ between the qubit and the measurement
line, while preserving a cavity lifetime $T_{c}$ larger than
$200~{}\mu\mathrm{s}$ for a single photon. The heterodyne detection benefits
from the large bandwidth of a traveling wave parametric amplifier (TWPA
Macklin  _et al._ (2015)) that covers many $\chi$. An auxiliary transmon qubit
and its readout resonator are used to perform direct Wigner tomography of the
cavity state Lutterbach and Davidovich (1997); Bertet  _et al._ (2002);
Vlastakis  _et al._ (2013); sup .

The qubit is driven with a frequency comb of amplitude $\Omega$ and peaks at
$\omega_{q}+k\Delta\omega$ where $k$ spans all integers between $-K$ and $K$.
In the lab frame, the qubit drive Hamiltonian thus reads

$\hat{H}_{d}=-\hbar\frac{\Omega}{2}\sum_{k=-K}^{K}\cos\left[(\omega_{q}+k\Delta\omega)t\right]\hat{\sigma}_{y}.$
(1)

In the limit of an infinite Dirac comb ($K\rightarrow\infty$), it becomes a
series of Dirac peaks in the time domain with a period $2\pi/\Delta\omega$. In
the frame rotating at $\omega_{q}-\chi\hat{c}^{\dagger}\hat{c}$, and under the
rotating wave approximation, it gives

$\hat{H}_{d}=\hbar\frac{\pi\Omega}{\Delta\omega}\sum_{l=-\infty}^{\infty}\delta\left(t-\frac{2\pi
l}{\Delta\omega}\right)\hat{\sigma}(t),$ (2)

where $\hat{\sigma}(t)=\sin(\hat{c}^{\dagger}\hat{c}\chi
t)\hat{\sigma}_{x}-\cos(\hat{c}^{\dagger}\hat{c}\chi t)\hat{\sigma}_{y}$ and
$\hat{c}^{\dagger}\hat{c}$ is the photon number in the cavity. The dynamics of
the qubit Bloch vector thus consists in periodic kicks every
$2\pi/\Delta\omega$ by an angle $\theta=2\pi\Omega/\Delta\omega$. The natural
choice is to have one peak per possible qubit frequency $(\Delta\omega=\chi)$.
The rotation axis of the kicks would then be the same for any photon number
since $\varphi=2\pi\hat{c}^{\dagger}\hat{c}\chi/\Delta\omega$ is a multiple of
$2\pi$. However, the period between two kicks would then be $2\pi/\chi\approx
190~{}\mathrm{ns}$, which is much longer than $T_{q}$. To limit idle times in
the qubit fluorescence signal, we choose a twice larger peak spacing
$\Delta\omega=2\chi$, which doubles the information rate (Fig. 2a).
Consequently, $\varphi$ is equal to $0$ mod $2\pi$ for even photon numbers and
$\pi$ mod $2\pi$ for odd photon numbers. Therefore the kick direction flips
with each kick for odd photon numbers.

In the experiment, we choose a finite number $2K+1=21$ peaks in the comb. In
the frequency domain the drive is the product of a square window of width
$21\chi/2\pi$ and the infinite comb. Consequently, in the time domain, the
resulting waveform is the convolution of the infinite comb with a sinc
function. This width sets the timescale of each qubit kick to
$\pi/21\chi\approx 5~{}\mathrm{ns}$, which is much shorter than $T_{q}$.
Additionally, this choice guarantees that the qubit frequency remains well
within the bandwidth of the comb, regardless of the desired photon number
ranging from 0 to $N_{\mathrm{max}}=9$. To further minimize boundary effects,
we position the comb center frequency at $\omega_{q}-4\chi$, which corresponds
to the qubit frequency associated with 4 photons in the cavity.

Figure 2: (a) Black peaks: driving comb amplitude $\Omega$ Eq. (1) in the
frequency domain. Colored Lorentzian shapes: qubit frequency spectra
corresponding to cavity photon numbers from 0 to 6. (b) Simulated average
trajectories of the qubit in the Bloch sphere for $n=0$ (blue) and $n=1$
(green). The kick angle is set to $\theta=\pi\Omega/\chi=3\pi/4$, and as in
the experiment, $\chi T_{q}=0.76$. (c) Corresponding average values of
$\langle\hat{\sigma}_{x}\rangle$ as a function of time. (d) Averaged measured
record $\overline{V_{n}}(t)$ when the cavity is prepared in a Fock state
$|n\rangle$ from $n=0$ to 4 from bottom to top, and for a comb amplitude such
that $\theta=\pi\Omega/\chi\approx\pi/2$. The curves are offset by the much
larger driving comb signal $c(t)$. (e) Corresponding quadrature
$\overline{I_{n}}(t)$ of the emitted fluorescence signal reconstructed (see
text) at each qubit frequency $\omega_{q}-n\chi$.

The average predicted dynamics of the qubit is shown in Fig. 2b,c when the
cavity has 0 (blue) or 1 (orange) photon. At intervals of $\pi/\chi$, the
qubit state undergoes a kick lasting approximately $5~{}\mathrm{ns}$, followed
by relaxation as it fluoresces into the measurement line. In contrast with
even photon numbers, the rotation axis flips at every kick for odd photon
numbers. Heterodyne detection of the fluorescence field measures the two
quadratures $(\hat{a}^{\dagger}_{\mathrm{out}}+\hat{a}_{\mathrm{out}})/2$ and
$i(\hat{a}_{\mathrm{out}}-\hat{a}^{\dagger}_{\mathrm{out}})/2$. The emitted
field amplitude can be expressed as
$\hat{a}_{\mathrm{out}}=\hat{a}_{\mathrm{in}}-\hat{\sigma}_{-}/\sqrt{T_{q}}$,
where $\langle\hat{a}_{\mathrm{in}}\rangle$ is the driving comb and
$\hat{\sigma}_{-}=(\hat{\sigma}_{x}-i\hat{\sigma}_{y})/2$ is the qubit
lowering operator Gardiner and Collett (1985). The average dynamics of the
qubit coherences (Fig. 2c) can thus be directly observed in the heterodyne
signal.

We first prepare a Fock state $|n\rangle$ using a coherent excitation on the
cavity followed by heralding using the qubit emission under a drive at a
single tone $\omega_{q}-n\chi$ Gely and Steele (2020); Essig  _et al._ (2021);
sup . We then apply the comb. The qubit fluorescence is amplified and
downconverted by a local oscillator at $\omega_{q}+\omega_{\mathrm{IF}}$, with
$\omega_{\mathrm{IF}}=2\pi\times 66~{}\mathrm{MHz}$. The amplified
fluorescence signal is recorded as a voltage $V(t)$ using an analog-to-
digital-converter. The average $\overline{V_{n}}(t)$ of these records under
the heralding of $n$ photons is shown in Fig. 2d offset by the contribution of
the reflected driving comb
$c(t)\propto\mathrm{Re}(\langle\hat{a}_{\mathrm{in}}\rangle)$ sup . These
signals can be processed to reveal the evolution of
$\langle\hat{\sigma}_{x}\rangle$ and $\langle\hat{\sigma}_{y}\rangle$ in the
qubit frame when there are $n$ photons. To do so, we extract their analytic
representations sup  and demodulate them at $\omega_{\mathrm{IF}}+n\chi$ to
obtain two average quadratures $\overline{I_{n}}(t)$ and
$\overline{Q_{n}}(t)$. The traces of $\overline{I_{n}}(t)$ are shown in Fig.
2e for $n=0$ to $4$ and match the expected evolution of
$\langle\hat{\sigma}_{x}\rangle$. The kicks and the subsequent decays are
visible. The kick direction alternates for odd numbers of photons as expected.
The remaining oscillations in the reconstructed signal may be due to an
imperfect subtraction of the driving comb $c(t)$, or to a distortion of the
driving comb or output signal by the measurement setup.

Decoding the measurement record $V(t)$ in order to infer the photon number is
a task similar to quantum sensing using continuous measurement Gambetta and
Wiseman (2001); Gammelmark and Mølmer (2013, 2014); Catana  _et al._ (2015);
Kiilerich and Mølmer (2016); Albarelli  _et al._ (2018, 2019); Ilias  _et al._
(2022); Descamps  _et al._ (2022). Here we use the average records
$\overline{V_{n}}$ as demodulation weight functions, and define
$N_{\mathrm{max}}$+1 measurement outcomes represented as a vector $\vec{m}$
whose components are

$m_{n}(t)=\int_{t-\tau}^{t}V(t^{\prime})\overline{V_{n}}(t^{\prime})\mathrm{d}t^{\prime},$
(3)

where the integration time $\tau=2~{}\mu\mathrm{s}$ is chosen much shorter
than the cavity lifetime $T_{c}$ and multiple $(21\times)$ of $\pi/\chi$. As a
demonstration, we excite the cavity with more than 20 photons on average using
a strong resonant pulse, then drive the qubit with the frequency comb and
record $\vec{m}$ as a function of time $t$. Quantum jumps can already be
visualized by a simple data processing. We perform a time independent linear
transform sup  $\vec{r}(t)=\mathbf{G}^{-1}\vec{m}(t)$ so that, on average,
$r_{n}(t)=1$ for n photons in the cavity, while all the other components
$r_{k\neq n}$ vanish. Concretely, $\mathbf{G}$ is the Gram matrix of the
average records $\overline{V_{n}}(t)$ so that
$\mathbf{G}_{nm}=\int_{0}^{\tau}\overline{V_{n}}(t)\overline{V_{m}}(t)\mathrm{d}t$.
The evolution of $\vec{r}$ is shown in Fig. 3a for one realization of the
experiment. A faint red trace emerges from the noise, which reveals the
successive losses of single photons in the cavity that here decays from 9 to 0
photons over 1 ms.

Figure 3: (a) Evolution of the measurement outcomes $m_{n}$ as a function of
time $t$ after a linear transform by $\mathbf{G}^{-1}$. The measurement is
performed on one realization using a driving comb whose amplitude corresponds
to an expected kick rotation angle $\theta=\pi/2$. (b) Evolution of the
inferred probability distribution $\mathbb{P}_{t}(n)$ of the photon number $n$
using the outcomes in (a).

To predict the number $n$ of photons at any time of the evolution, the
probability distribution of $n$ is updated conditionally on the outcome
$\vec{m}(t)$ through Bayesian update at every time step $j\tau$. It first
requires determining the likelihood $\mathbb{P}(\vec{m}|n)$ conditioned on the
cavity being in the Fock state $|n\rangle$. $\mathbb{P}(\vec{m}|n)$ can be
approximated by a Gaussian function of $\vec{m}$ owing to the small
measurement efficiency $\eta=0.17\pm 0.02$. Therefore, we characterize its
distribution by the measured mean $\langle\vec{m}\rangle_{|n\rangle}$ and
covariance matrix of $\vec{m}$ for each $|n\rangle$ only sup . Using this
procedure, along with accounting for photon loss during each time step
$j\tau$, the noisy measurement outcomes $\vec{m}(j\tau)$ of Fig. 3a lead to
the probability distribution $\mathbb{P}_{j\tau}(n)$ shown in Fig. 3b for the
same realization. Note that we assume no prior information
($\mathbb{P}_{0}(n)=1/10$), but this choice has anyway no impact on the
quantum trajectory after a few $\tau$. With many realizations, we extract the
average photon number decay. Interestingly, it is not exponential, which
indicates a subtle interplay between cavity dissipation and qubit dynamics sup
.

Figure 4: (a) Red dots: observed average measurement rate $\Gamma_{m}$ as a
function of drive amplitude, expressed as the qubit expected rotation angle
$\theta$ per kick. Orange shadow: measurement rate obtained using a stochastic
master equation with detection efficiency $\eta$ spanning the range
$[0.17,0.20]$. Blue shadow: theoretical bound for an infinite comb and
heterodyne measurement with efficiencies $\eta\in[0.17,0.20]$. (b) Red dots:
observed cavity dephasing rate $\Gamma_{d}$ as a function of $\theta$. Orange:
simulated measurement induced dephasing rate. Black: theoretical accessible
information rate. Blue: theoretical maximal ($\eta=1$) measurement rate
obtained by heterodyne detection.

We now determine the measurement rate $\Gamma_{m}$ of the photon number.
Formally, it is the time derivative of the mutual information between the
photon number $n$ and the outcome $\vec{m}$ at $t=0$ Clerk  _et al._ (2003,
2010). In the weak measurement regime $\tau\Gamma_{m}\ll 1$, it can be
approximated by

$\begin{split}\tau\Gamma_{m}=&-\int\mathbb{P}(\vec{m})\log\mathbb{P}(\vec{m})\mathrm{d}\vec{m}\\\
&+\sum_{n}\mathbb{P}_{0}(n)\int\mathbb{P}(\vec{m}|n)\log\mathbb{P}(\vec{m}|n)\mathrm{d}\vec{m},\end{split}$
(4)

where $\mathbb{P}(\vec{m})=\sum_{n}\mathbb{P}_{0}(n)\mathbb{P}(\vec{m}|n)$. As
shown in Fig. 3b, at most two photon numbers are likely at any time. We thus
choose the prior $\mathbb{P}_{0}(q)=\mathbb{P}_{0}(q+1)=1/2$, and average over
all $q$ values to compute a measurement rate $\Gamma_{m}$ sup . Plugging in
Eq. (4) the measured distributions $\mathbb{P}(\vec{m}|n)$ at various driving
amplitudes $\Omega$, $\Gamma_{m}$ is obtained as a function of the kick angle
$\theta=\pi\Omega/\chi$ (Fig. 4a). It is maximal when the qubit is kicked to
states corresponding to the largest $\langle\hat{\sigma}_{x}\rangle$. It can
be intuitively understood since heterodyne detection probes the quadratures of
the emitted fluorescence signal, which are on average proportional to the
coherence $\langle\hat{\sigma}_{x}-i\hat{\sigma}_{y}\rangle$. Notably, the
rate $\Gamma_{m}$ is much larger than the cavity decay rate $1/T_{c}$ (more
than 16 times), which is well in the single shot measurement regime. Our
complete model sup  reproduces the observed rate $\Gamma_{m}$ using simulated
measurement records by a stochastic master equation with detection efficiency
$\eta=0.185\pm 0.015$ as a fit parameter, in agreement with the independently
measured $\eta$.

It is interesting to compare this measurement rate to the rate at which
information about the photon number leaks into the environment, _i.e._ the
cavity dephasing rate $\Gamma_{d}$. We compute it as the added decay rate of
$\Tr{\hat{\rho}(t)\hat{a}}$ on top of the natural decoherence rate sup .

We use the auxiliary transmon qubit (purple in Fig. 1c) to perform Wigner
tomography on the cavity state and extract
$\Tr{\hat{\rho}(t)\hat{a}}=\int\alpha W_{t}(\alpha)\mathrm{d}^{2}\alpha$, with
$W_{t}(\alpha)$ the Wigner function of $\hat{\rho}(t)$. The dependence of
$\Gamma_{d}$ on kick angle is shown as red dots in Fig. 4b. Strikingly, its
maximum is reached at $\theta\approx\pi$, where the qubit has the largest
energy to emit, and thus the most information to leak out. We note that the
model that successfully predicts the measurement rate (orange) underestimates
the measurement induced dephasing rate at large drive amplitudes, indicating
that the driving comb leads to stronger decoherence than anticipated.

The dephasing rate $\Gamma_{d}$ is about 20 times larger than $\Gamma_{m}$ at
$\theta=\pi/2$ (its maximum). Indeed our measurement setup does not recover
the full information available because of a limited detection efficiency and
the very use of heterodyne detection. To better understand this information
loss, we use a simplified model where the comb is infinite. It reproduces the
measured $\Gamma_{m}$ with $0.17<\eta<0.20$ (blue in Fig. 4a) for small angles
$\theta$. However, the dephasing rate is larger than the measurement rate of
an ideal ($\eta=1$) heterodyne measurement (blue in Fig. 4b). Accordingly,
with the same model, the upper bound on the measurement rate for any detection
scheme – accessible information rate – (black in Fig. 4b) is close to the
measured $\Gamma_{d}$ for small angles $\theta$ and up to about 3 times larger
than what the best heterodyne detector could do: Even an ideal heterodyne
detector would destroy up to about 2/3 of the accessible information. The
experiment provides here a textbook example of destroyed information by a
measurement apparatus, here the heterodyne detector Han  _et al._ (2018). In
contrast to low detection efficiency, heterodyne measurement with $\eta=1$
would reveal information even for $\theta=\pi$. Indeed, while the average
heterodyne signal is zero, its cumulants reveal the photon number. The signal-
to-noise ratio on the cumulants of order $l$ scales as $\eta^{l}$ so that, in
the experiment, the average is the main source of information, hence the
minimum at $\theta=\pi$.

In conclusion, our superconducting circuit and signal processing demonstrate
the possibility to monitor photon numbers in a cavity with a fixed driving.
This result was obtained with a TWPA Macklin  _et al._ (2015) that led to
close to 20% detection efficiency. Information about the photon number is
extracted up to 16 times faster than the cavity decay rate. A simple model
quantitatively explains the dephasing and measurement rates for small drive
amplitudes. As we look ahead to more integrated amplifiers, it would be
interesting to observe how the driving amplitude that maximizes the
measurement rate evolves with increased efficiency. Additionally, the pogo-pin
and spurline filters offer a convenient architecture for achieving galvanic
coupling with quantum circuits within a long lived microwave cavity.
Interesting open questions remain to be explored such as the origin of the
dependence of Fock state decay rates on the comb amplitude. This seems to be a
dual effect to the readout problem of superconducting qubit Sank  _et al._
(2016); Lescanne  _et al._ (2019); Petrescu  _et al._ (2020); Shillito  _et
al._ (2022); Khezri  _et al._ (2023); Burgelman  _et al._ (2022); Thorbeck
_et al._ (2023); Bengtsson  _et al._ (2023).

###### Acknowledgements.

This research was supported by the QuantERA grant ARTEMIS, by ANR under the
grant ANR-22-QUA1-0004, and PR received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No. [884762]). We acknowledge IARPA and Lincoln
Labs for providing a Josephson Traveling-Wave Parametric Amplifier. We thank
Rémy Dassonneville, Pierre Guilmin, Sébastien Jezouin, Perola Milman, Mazyar
Mirrahimi, Klaus Mølmer, Alain Sarlette, Antoine Tilloy, Benoît Vermersch and
Mattia Walschaers for useful discussions.

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