# Universal relation for supernova gravitational waves

Hajime Sotani<EMAIL_ADDRESS>Astrophysical Big Bang Laboratory,
RIKEN, Saitama 351-0198, Japan  Interdisciplinary Theoretical & Mathematical
Science Program (iTHEMS), RIKEN, Saitama 351-0198, Japan  Tomoya Takiwaki
Division of Science, National Astronomical Observatory of Japan, 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan  Center for Computational Astrophysics, National
Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Hajime Togashi  Department of Physics, Tohoku University, Sendai 980-8578,
Japan

###### Abstract

Using the numerical simulation data for two-dimensional core-collapse
supernova, we examine the protoneutron star (PNS) asteroseismology with the
relativistic Cowling approximation. As shown in the previous study, the
gravitational wave signals appearing in the numerical simulation can be well
identified with the gravity (fundamental) oscillation in the early (later)
phase before (after) the avoided crossing between the gravity and fundamental
oscillations. On the other hand, the time evolution of supernova gravitational
waves strongly depends on the PNS models, such as the progenitor mass and the
equation of state for dense matter. Nevertheless, we find that the fundamental
and gravity mode frequencies according to the gravitational wave signals
appearing in the numerical simulations can be expressed as a function of the
protoneutron star average density independently of the PNS models. Using the
average density, we derive the empirical formula for supernova gravitational
wave frequency. In addition, we confirm that the dependence of the PNS surface
density on the PNS average density is almost independent of the PNS models and
also discuss how the different treatment of the non-uniform matter in the
equation of state affects the observables.

###### pacs:

04.40.Dg, 97.10.Sj, 04.30.-w

##  I Introduction

Direct detection of gravitational waves starts the era of gravitational wave
astronomy and adds as a new messenger to electromagnetic waves and neutrinos
with the observation of GW170817, which is the gravitational wave event from
the binary neutron star merger GW6 ; EM . Now, we become able to use not only
electromagnetic waves and neutrinos but also gravitational waves as a tool for
probing astronomical phenomena. The observation of electromagnetic waves is
complementary important in gravitational wave astronomy, especially for
determining the source position, because the angular resolution with
gravitational wave detectors is poor, spatially if only a limited set of
detectors are used. For the case of the GW170817, searches within the sky
location area provided by the gravitational wave detectors actually were
crucial to finding the faint kilonova by electromagnetic observations, while
it was possible to improve the sky location of the gravitational wave event,
assuming that it was at the location of the kilonova. The third observing run
(O3) by LIGO and Virgo has already finished, where KAGRA finally joined this
collaboration network at the end of O3, and these gravitational wave detectors
are now preparing for O4, which may start after August 2022. During the O3,
the gravitational waves from two merger events composed of a neutron star with
a black hole, which are the last piece in the compact binary merger event,
have also been detected BHNS . Owing to the development of detectors’
sensitivity, the observable distance is growing and the number of events is
expected to significantly increase.

The next frontier in gravitational wave astronomy after the compact binary
mergers must be the (galactic) core-collapse supernova, which accompanies the
death of massive star. Since, compared to the compact binary mergers, the
supernova is almost spherically symmetric, the radiation energy of
gravitational waves may not be so strong, which leads to a difficulty of
detection from the supernova explosion quite far from us. Considering the
event rate of supernova, e.g, $\sim 1$ per century in our galaxy, the
detection of the gravitational wave from such a system seems to be quite rare.
Nevertheless, once one would detect it, one could simultaneously detect not
only the gravitational waves but also the electromagnetic waves and neutrinos.
This must be a good opportunity for understanding the supernova explosion
mechanism, and we have to wait for such an opportunity with thoroughgoing
conditions.

The gravitational waves from supernova explosions have been mainly studies via
numerical simulations (e.g., Murphy09 ; MJM2013 ; Ott13 ; CDAF2013 ; Yakunin15
; KKT2016 ; Andresen16 ; Richers2017 ; Takiwaki2017 ; OC2018 ; RMBVN19 ;
VBR2019 ; PM2020 ; Andersen21 ; Pajkos21 ; Shibagaki21 ). These simulations
tell us the existence of the gravitational wave signals, whose frequency
increases from a few hundred hertz up to a few kilohertz in $\sim 1$ second
after corebounce. This ramp up signal is originally considered as a result of
the Brunt-Väisälä frequency at the surface of the protoneutron star (PNS), the
so-called surface gravity ($g$-) mode MJM2013 ; CDAF2013 . Meanwhile, since
the Brunt-Väisälä frequency determined with local stellar properties strongly
depends on the radial position inside the star, the gravitational wave signals
appearing in the numerical simulations may come from specific global
oscillations of the dense region. In fact, the signals can be identified with
the frequency of the PNS fundamental ($f$-) mode MRBV2018 ; SKTK2019 ; ST2020b
, where the signals in very early phase just after corebounce is identified
with the $g_{1}$-mode oscillations ST2020b , or with the $g$-mode like
oscillations of the region inside the shock radius TCPF2018 ; TCPOF2019a ;
TCPOF2019b . Anyway, the identification of the gravitational wave signals with
a specific oscillation mode must be important for understanding the physics
behind the gravitational wave radiations.

In order to study the gravitational waves from the core-collapse supernovae,
the perturbative analysis is considered as another approach. In general, the
oscillation frequencies strongly depend on the interior properties of the
source object, where the excited frequencies respectively correspond to the
input physics KS1999 . So, as an inverse problem, one could extract the
interior properties, when one would detect the frequency. This technique is
known as asteroseismology, which is similar to seismology on the Earth and
helioseismology on the Sun. For compact objects, asteroseismology on cold
neutron stars is extensively studied up to now. For instance, by identifying
the frequencies of quasi-periodic oscillations observed in the afterglow
following the giant flares with the crustal torsional oscillations, the crust
properties in neutron stars are constrained SA09 ; GNHL2011 ; SNIO2012 ;
Deibel14 ; Gabler2018 . In a similar way, once one would detect the
oscillation frequencies in gravitational waves from neutron stars, one might
constrain the properties of the neutron stars, such as the mass, radius,
equation of state (EOS), and so on (e.g., AK1996 ; AK1998 ; STM2001 ; SH2003 ;
SYMT2011 ; PA2012 ; DGKK2013 ; Sotani2020 ).

On the other hand, the studies about asteroseismology on PNSs are gradually
increasing (e.g., MRBV2018 ; SKTK2019 ; ST2020b ; TCPF2018 ; TCPOF2019a ;
TCPOF2019b ; FMP2003 ; Burgio2011 ; FKAO2015 ; ST2016 ; Camelio17 ; SKTK2017 ;
SS2019 ; WS2019 ; WS2020 ; Mezzacappa20 ; ST2020a ; ST2020c ). This may be
partially thanks to progress of the numerical simulation of core-collapse
supernovae, which can provide the background PNS models for making a linear
analysis. In these studies, using the numerical data obtained by the
simulation of core-collapse supernovae, first the spherically symmetric
background models are prepared by averaging in the angular direction, if the
multi-dimensional simulation has been done. With using such a background
model, mainly two different approaches have been done in the PNS
asteroseismology. The first approach is that the PNS surface is defined by a
specific surface density, e.g., $\rho_{s}=10^{11}$ g/cm3, and one considers
the characteristic oscillations inside the PNS model, where the boundary
condition imposed at the PNS surface is assumed to be the vacuum boundary
condition, i.e., the Lagrangian perturbation of pressure should be zero
MRBV2018 ; SKTK2019 ; ST2020b ; ST2016 ; FMP2003 ; Burgio2011 ; FKAO2015 ;
Camelio17 ; SKTK2017 ; SS2019 , even though the matter still exists outside
the PNS surface selected here. With this approach, one can classify the
oscillation modes with the number of radial nodes in the eigenfunctions
according to the standard asteroseismology, while there is uncertainty in the
selection of the surface density. But, it has been shown that at least the
$f$\- and $g_{1}$-mode frequencies in the later phase (e.g., since $\sim$ 300
msec after core bounce) are almost independent of the selection of surface
density MRBV2018 ; ST2020b . In the second approach, one adopts the background
data up to the shock radius and imposes the boundary condition that the
position of the shock radius is fixed during the oscillations TCPF2018 ;
TCPOF2019a ; TCPOF2019b . With the second approach, by definition one can
uniquely fix the position of the outer boundary, but one has to reclassify the
oscillation modes because the eigenvalue problem to solve is mathematically
different from the standard asteroseismology due to the different boundary
condition at the outer boundary. In this study, we simply adopt the first
approach, because we mainly focus on the $f$\- and $g_{1}$-mode frequencies,
which correspond to the gravitational wave signals appearing in the numerical
simulation ST2020b .

Nonetheless, since the $f$\- and $g_{1}$-mode frequencies generally depend on
the PNS models (e.g., Andersen21 ), it may be difficult to extract a kind of
physical properties of PNSs via direct observation of the supernova
gravitational waves. To solve this difficulty, the universal relation between
the $f$\- and $g_{1}$-mode frequencies and PNS properties, which is
independent of the PNS models, must be quite important, if it exists. In fact,
with the same motivation the universal relation of the oscillation frequency
as a function of the PNS surface gravity is proposed in Refs. TCPOF2019b ;
Bizouard2021 . On the other hand, we alternatively proposed another fitting
formula for the supernova gravitational waves as a function of the PNS average
density in Ref. ST2020b , which fits well the gravitational wave frequencies
just after the corebounce. However, we did not discuss how universal our
fitting formula can predict the supernova gravitational wave frequency. Thus,
in this study we mainly focus on the discussion for the universality of our
fitting formula. In addition, we will confirm another universal relation
between the PNS surface gravity and PNS average density, which is originally
found in Ref. ST2020a . Furthermore, we also discuss how the observables, such
as the PNS mass, radius, and gravitational waves, depend on the different
treatment of the non-uniform matter in PNS.

This paper is organized as follows. In Sec. II, we describe the PNS models
considered in this study. In Sec. III, we show the eigenfrequencies of
gravitational waves from the PNS with the relativistic Cowling approximation
and show the universal relation of gravitational wave frequencies appearing in
the numerical simulations as a function of the PNS average density. Then, we
make a conclusion in Sec. IV.

##  II PNS Models

In order to identify the gravitational wave signals appearing in the core-
collapse supernova simulations with a specific PNS oscillation frequency, we
examine asteroseismology on the PNSs. For this purpose, first we have to
prepare the PNS model as a background model, on which the linear analysis is
done. As in the previous our studies ST2020a ; ST2020b ; ST2020c , such
background PNS models are provided by the two-dimensional (2D) neutrino
radiation hydrodynamic simulations with 3DnSNe code, which has been already
applied for several studies takiwaki2016 ; oconnor2018 ; kotake2018 ;
matsumoto2020 . In this numerical code, the neutrino transport is solved by
adopting isotropic diffusion source approximation liebendoerfer2009 ;
takiwaki2014 , where we adopt the same procedure for the set of neutrino
reactions as in Ref. oconnor2018 . The numerical simulations have been done
with the resolution in the spherical polar grid of $512\times 128$, where the
radial grid covers up to $5000$ km from the center, adopting the $20M_{\odot}$
progenitor model given in Ref. WH2007 .

With respect to the EOS for high density region, we adopt four different EOSs
in this study, i.e., DD2, SFHo, TGLD, and TGTF. For uniform matter, DD2 is
constructed based on the relativistic mean field approach with the density-
dependent parametrization DD2 , while SFHo is constructed with another
parametrization that is fitted to the neutron star mass and radius constrained
from the x-ray observations SFHo . Non-uniform matter in both EOSs are
described by the nuclear statistical equilibrium (NSE) model, where nuclei are
regarded as classical particles. TGTF Togashi17  and TGLD Furusawa17
basically have the same EOS properties for uniform matter, which are
constructed via variational method with AV18 and UIX potential. The difference
is how to construct a non-uniform matter density region, i.e., TGTF is adopted
single nucleus approximation (SNA) with Thomas-Fermi model, while TGLD is
constructed by the NSE treatment with liquid drop model. The EOS parameters
adopted in this study are listed in Table 1, where $K_{0}$ and $L$ are
respectively incompressibility of the symmetric nuclear matter and the density
gradient of the symmetry energy at the saturation point, while $\eta$ given by
the combination of $K_{0}$ and $L$ as $\eta\equiv(K_{0}L^{2})^{1/3}$ is a
suitable parameter for expressing the low-mass (cold) neutron stars SIOO14 .
Considering the constraint from GW170817 on the radius of a $1.4M_{\odot}$
neutron star Annala18 , i.e., $R\leq 13.6$ km, $\eta$ may be also constrained
as
$\eta\,\,\raisebox{-3.44444pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$
}125$ MeV. In the same table, the maximum mass of a corresponding cold neutron
star is also listed.

In addition to the above PNS models, for reference we also consider the
results in Ref. ST2020b , which has been studied with completely different PNS
model. That is, the $2.9M_{\odot}$ helium star (He2.9) given in Ref. He29  is
adopted as a progenitor model and LS220 LS220  is adopted for the EOS, with
which the 2D numerical simulation has been done. Anyway, in order to prepare
the spherically symmetric background model, the numerical data obtained by 2D
simulations is averaged in the angular ($\theta$) direction. In this study, we
especially set the PNS surface density to be $\rho_{s}=10^{11}$ g/cm3.

Table 1: EOS parameters, $K_{0}$, $L$, and $\eta$, adopted in this study. In the rightmost column, the maximum mass, $M_{\rm max}/M_{\odot}$, for the cold neutron star constructed with each EOS is listed. For reference, the properties for LS220 are also shown.  EOS |  $K_{0}$ (MeV) |  $L$ (MeV) |  $\eta$ (MeV) | $M_{\rm max}/M_{\odot}$
---|---|---|---|---
DD2 | 243 | 55.0 | 90.2 | 2.42
SFHo | 245 | 47.1 | 81.6 | 2.06
TGLD/TGTF | 245 | 38.7 | 71.6 | 2.21
LS220 | 220 | 73.8 | 106.2 | 2.05
Figure 1: Time evolution of the PNS mass (top) and radius (bottom) for various
models. The surface density is chosen to be $10^{11}$ g/cm3.  Figure 2: The
PNS surface gravity (top) and compactness (bottom) are shown as a function of
the square root of the normalized PNS average density for various models. The
thick-solid line in the top panel is the fitting formula given by Eq. (1).

In Fig. 1 we show the PNS mass (top) and radius (bottom) evolution for various
PNS models. One can clearly observe that the difference of the progenitor mass
significantly changes the PNS mass evolution, but it affects the PNS radius
evolution not so much. In addition, we find a deviation between the results
with TGTF and TGLD, i.e., the PNS mass and radius evolution depends on the
treatment of non-uniform matter (SNA or NSE). This dependence will discuss
later together with that of the gravitational wave frequencies.

Additionally, in Fig. 2, we show the surface gravity, $M_{\rm PNS}/R_{\rm
PNS}^{2}$, in the top panel and PNS compactness, $M_{\rm PNS}/R_{\rm PNS}$, in
the bottom panel as a function of the square root of the normalized PNS
average density, $(M_{\rm PNS}/1.4M_{\odot})^{1/2}(R_{\rm PNS}/10\ {\rm
km})^{-3/2}$, for various PNS models. In the both panels, the PNSs evolve from
the bottom-left to the top-right. From this figure, in particular, we can
confirm that the surface gravity is quite weak dependence on the square root
of the normalized PNS average density, which is originally pointed out in Ref.
ST2020a . With the results, we derive the fitting formula as

$M_{\rm PNS}/R_{\rm PNS}^{2}\ ({\rm
km^{-1}})=\left[-1.985-0.465\ln(x)+19.247x\right]\times 10^{-3},$ (1)

where $x$ denotes the square root of the normalized PNS average density, i.e.,

$x\equiv\left(\frac{M_{\rm PNS}}{1.4M_{\odot}}\right)^{1/2}\left(\frac{R_{\rm
PNS}}{10\ {\rm km}}\right)^{-3/2},$ (2)

which is shown in the top panel of Fig. 2 with the thick-solid line. To be
honest, currently we can not understand why this relationship appears in the
PNS models, because this type of the relation can not be seen in cold neutron
stars, but this feature may be a key for understanding the physics of PNSs.

##  III gravitational wave asteroseismology

On the PNS models discussed in the previous section, we make a linear
analysis, where we simply adopt the relativistic Cowling approximation in this
study, i.e., the metric perturbations are neglected during the fluid
oscillations. The perturbation equations are derived by linearizing the energy
conservation law, which finally become the ordinary differential equations for
the Lagrangian displacement of fluid element. By imposing appropriate boundary
conditions, the problem to solve becomes an eigenvalue problem with respect to
the eigenvalue of $\omega$, which directly corresponds to the eigenfrequency,
$f$, via $f=\omega/(2\pi)$. The imposed boundary conditions are the regularity
condition at the center and the condition that the Lagrangian perturbation of
pressure should be zero at the PNS surface. The concrete perturbation
equations and boundary conditions are the same as shown in Ref. SKTK2019 . In
this study, we focus on only the $\ell=2$ oscillation modes, because they are
considered to become energetically dominant in the gravitational wave
emission. We note that one can qualitatively discuss the behavior of
gravitational wave frequency even with the relativistic Cowling approximation,
but the frequencies with the approximation deviate $\sim 20\%$ from those
without the approximation ST2020c .

In Fig. 3 we show the PNS oscillation frequencies determined by solving the
eigenvalue problem with open marks on the contour, which denotes the
gravitational wave signals appearing in the numerical simulation, for the PNS
model with TGLD (see in appendix A for the other PNS models), focusing on only
the $f$-, $g_{i}$-, and $p_{i}$-mode frequencies with $i=1,2$. Here, the
gravitational wave signals are calculated with the same procedure as in Ref.
Marek06 , using the numerical data obtained by simulations. From this figure,
as in Ref. ST2020b , one can obviously see that the gravitational wave signals
in numerical simulation is identified by the $g_{1}$-mode ($f$-mode)
oscillation from the PNS before (after) the avoided crossing between the $f$\-
and $g_{1}$-mode. In Fig. 4, we also plot the time evolution of the $f$\- and
$g_{1}$-mode frequencies for various PNS models. As in Ref. Andersen21 , one
can observe that the time evolution of the gravitational waves strongly
depends on the PNSs models (see also Fig. 9).

Figure 3: Comparing the gravitational wave signals appearing in the numerical
simulation (background contour) to the PNS frequencies (open-marks) determined
by solving the eigenvalue problem for the PNS model with TGLD.  Figure 4: The
$f$\- and $g_{1}$-mode frequencies for various PNS models are shown as a
function of the postbounce time.

On the other hand, in the left panel of Fig. 5, we show the $f$\- and
$g_{1}$-mode frequencies as a function of the square root of the normalized
PNS average density, $x$. From this figure, the $f$\- and $g_{1}$-mode
frequencies according to the gravitational wave signals appearing in the
numerical simulation are well fitted, such as

$f\ {\rm(kHz)}=-1.410-0.443\ln(x)+9.337x-6.714x^{2},$ (3)

independently of the PNS models. The predicted values from Eq. (3) are also
plotted in the left panel of Fig. 5 with the thick-solid line. That is, once
one would detect the supernova gravitational waves, which could be the same as
gravitational wave signals appearing in the numerical simulations, one can
extract the evolution of the PNS average density by using Eq. (3). In the same
figure, we also show the empirical relation for the $f$-mode frequency derived
in Ref. SS2019 , which is

$f_{f}{\rm(kHz)}=0.9733-2.7171x+13.7809x^{2},$ (4)

with the thick-dashed line. We remark that this relation is obtained for the
case of the failed supernova with general relativistic simulation, i.e., the
PNS considered in Ref. SS2019  would eventually collapse to a black hole,
focusing on the region of $(M_{\rm PNS}/1.4M_{\odot})^{1/2}(R_{\rm PNS}/10\
{\rm
km})^{-3/2}\,\raisebox{-3.44444pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$
}0.1$. By comparing this empirical relation to the gravitational wave
frequencies obtained in this study and the fitting formula given by Eq. (3),
one can observe a significant deviation for the later phase. Unfortunately, we
can not identify why this deviation comes from, but it may be because the
dependence of the gravitational wave frequencies on the PNS average density in
a black hole formation is simply different from that for the case of
successful supernova, or it may comes from the treatment of the general
relativistic effect in the simulation. On the other hand, it is also suggested
that the gravitational wave frequency is expressed as a function of the PNS
surface gravity in Refs. TCPOF2019b ; Bizouard2021 . In a similar way, we also
show the gravitational wave frequencies for various PNS models as a function
of the PNS surface gravity in the right panel of Fig. 5, where the thick-solid
line denotes the fitting formula given by

$f{\rm(kHz)}=-0.0752-0.2600\ln(\bar{x}_{3})+0.7446\bar{x}_{3}-0.0600\bar{x}_{3}^{2},$
(5)

where $\bar{x}_{3}$ denotes $\bar{x}/0.001$ and $\bar{x}$ is the PNS surface
gravity defined by $\bar{x}\equiv M_{\rm PNS}/R_{\rm PNS}^{2}$ in the unit of
$M_{\odot}$ km-2. For reference, we also show the universal relation derived
in Ref. TCPOF2019b  with the thick-dotted line, where the standard deviation
of the data is 76 Hz. We note that the universal relation in Ref. TCPOF2019b
had a missing factor and the amended relation is plotted in this figure
private , instead of the original relation. Since we have already shown that
the relation between the PNS surface gravity and the average density weakly
depends on the PNS model, as shown in the top panel of Fig. 2, we expected
that the gravitational wave frequencies corresponding to the signals in
numerical simulation could be also expressed as a function of the PNS surface
gravity almost independently of the PNS models. However, the gravitational
wave frequencies seem to depend a little on the PNS models, if they are
considered as a function of the PNS surface gravity. That is, to characterize
the gravitational wave frequency, the PNS average density may be better than
the surface gravity, based on the eigenfrequencies computed using the
linearised theory with the Cowling approximation. Anyway, as shown in Figs. 3
and 9, the resultant eigenfrequencies systematically deviate from the
gravitational wave signals appearing in the numerical simulation, which is the
same order of magnitude as the differences seen between modes when using
surface gravity.

Figure 5: In the left panel, the $f$\- and $g_{1}$-mode frequencies for
various PNS models are shown as a function of the square root of the
normalized PNS average density. The thick-solid line is the fitting formula
for the $g_{1}$-mode ($f$-mode) frequency before (after) the avoided crossing
between the $f$\- and $g_{1}$-modes, which is given by Eq. (3), while thick-
dashed line denotes the empirical formula derived in Ref. SS2019  (Eq. (4)).
On the other hand, in the right panel the same frequencies shown in the left
panel are shown as a function of the surface gravity, where the thick-solid
line denotes the fitting formula given by Eq. (5). The thick-dotted line is
the universal relation derived in Ref. TCPOF2019b , but it is amended private
.

Furthermore, we discuss the impact on the observables due to the different
treatment of non-uniform matter in EOS. That is, as mentioned before, TGTF and
TGLD are constructed with the same nuclear properties for uniform matter, but
the treatment in non-uniform matter is different from each other. This
difference hardly affects the cold neutron star properties Togashi17 ;
Furusawa20 , while we find that it strongly affects the evolution of the PNS
properties and gravitational wave frequencies in this study. The treatment of
non-uniform matter affects the nuclear composition at the PNS surface, as in
Fig. 6, where the left and right panels correspond to the models with TGTF and
TGLD at $T_{\rm pb}=0.07$ sec. In the figure, the mass fractions of the
neutron, proton, alpha particle, and representative single nucleus for the
models with TGTF are shown as $X_{n}$, $X_{p}$, $X_{\alpha}$, and $X_{\rm A}$,
while those of the neutron, proton, deuteron, triton, helion, alpha particle,
other light nuclei with $Z\leq 5$, and nuclei with $Z\geq 6$ for the models
with TGLD are shown as $X_{n}$, $X_{p}$, $X_{d}$, $X_{t}$, $X_{h}$,
$X_{\alpha}$, $X_{\rm Z\leq 5}$, and $X_{\rm Z\geq 6}$, respectively. From
this figure, one can observe that for the models with TGLD not only the
neutron, proton, and alpha particle, but also various nuclei, deuteron,
triton, and helion can appear even inside the PNS region, which corresponds to
the density region for $\rho\geq 10^{11}$ g/cm3.

The difference between the results with TGTF and TGLD may be understood as
follows. The nuclear composition changes the strength of the neutrino cooling.
At the PNS surface, the absorption and scattering by nucleon are the dominant
opacity source. Since we especially treat light elements as alpha particles in
our hydrodynamic simulations, the appearance of the deuteron and other light
elements in TGLD reduces the fraction of nucleon and hence their opacity at
the PNS surface. The top panel of Fig. 7 shows the total neutrino luminosity
at the PNS surface given by $L_{\rm
tot}=L_{\nu_{e}}+L_{\bar{\nu}_{e}}+4L_{\nu_{X}}$, where $\nu_{e}$,
$\bar{\nu}_{e}$ and $\nu_{X}$ represent electron type, anti-electron type, and
heavy lepton type neutrinos, respectively. The higher neutrino luminosity at
TGLD is consistent with their fast contraction than TGTF (see Fig. 1). We also
evaluate the Kelvin-Helmholtz timescale, which is roughly the timescale of the
contraction of the stars Kippenhahn12 , given by

$\tau_{\rm KH}=|E_{\rm g}|/L_{\rm tot}.$ (6)

In this equation, $E_{\rm g}$ is the the gravitational binding energy of the
PNS defined by $E_{\rm g}=\frac{1}{2}\int\rho\Phi{\rm d}V$, where $\Phi$ is
the gravitational potential and the volume integration is performed in PNS.
The bottom panel of Fig. 7 shows this timescale. Reflecting the larger
luminosity of TGLD, the timescale is smaller in TGLD in early phase. The range
of the timescale, which is from 0.3 sec to several seconds, is roughly
consistent with the typical value shown in Ref. Scheck06  (see also Refs.
Nakazato19 ; Nakazato20 ; Nakazato21  for later epoch). Note that if the
emission and absorption process of deuteron and other light elements are
considered as in Refs. Nagakura19 ; Fischer20 , which increases the opacity of
PNS, the difference of the timescale between TGLD and TGTF could be smaller
than that shown in this study.

To quantify how the evolution of observables depends on the different
treatment of non-uniform matter in EOS, we calculate the relative deviation of
quantity with TGLD to that with TGTF via

$\Delta\xi=(\xi_{\rm TF}-\xi_{\rm LD})/\xi_{\rm TF},$ (7)

where $\xi$ denotes the PNS mass, radius, and gravitational wave frequencies,
while $\xi_{\rm TF}$ and $\xi_{\rm LD}$ denote the corresponding properties
calculated with TGTF and TGLD. The obtained result is shown in Fig. 8, where
the relative deviation in the PNS mass and radius is shown in the top panel,
while that in the $f$\- and $g_{1}$-mode frequencies are shown in the bottom
panel. From this figure, one can observe that the PNS mass weakly depends on
the different treatment in EOS, but the PNS radius and gravitational wave
frequencies significantly depend on it.

Figure 6: Mass fraction for the models with TGTF (left panel) and TGLD (right
panel) at $T_{\rm pb}=0.07$ sec.  Figure 7: Total neutrino luminosity at the
PNS surface, $L_{\rm tot}$, and the Kelvin-Helmholtz timescale, $\tau_{\rm
KH}$, given by Eq. (6) are shown as a function of $T_{\rm pb}$ in the top and
bottom panels, respectively. The right panels are just extended view of the
region enclosed by dashed line in the left panels.  Figure 8: Relative
deviation of a quantity with TGLD to that with TGTF, which is calculated by
$\Delta\xi=(\xi_{\rm TF}-\xi_{\rm LD})/\xi_{\rm TF}$ for some property ($\xi$)
calculated with TGTF ($\xi_{\rm TF}$) and TGLD ($\xi_{\rm LD}$). The top panel
corresponds to $\Delta M$ and $\Delta R$, while the bottom panel is $\Delta
f_{f}$ and $\Delta f_{g_{1}}$.

##  IV Conclusion

The supernova gravitational waves must be the next frontier in the
gravitational wave astronomy. To prepare for such a crucial opportunity with
watertight conditions, one has to study the supernova gravitational waves and
understand the physics behind them. In this study, we primarily focus on the
dependence of gravitational wave frequency on the PNS models and derive the
universal relation for the gravitational wave frequency as a function of the
PNS average density, with which one may extract the evolution of the PNS
average density independently of the PNS models by observing the supernova
gravitational waves. In addition, we confirm that the PNS surface gravity can
be expressed as a function of the square root of the PNS average density
almost independently of the PNS models. Furthermore, we also discuss how the
different treatment of the non-uniform matter in EOS affects the observables.
In practice, we show that the PNS mass hardly depends on the treatment of the
non-uniform matter, while the PNS radius, the $f$-, and $g_{1}$-mode
frequencies strongly depend on it.

Similar universal relations between the frequency of gravitational waves and
the observables of the PNS have been proposed in several studies TCPOF2019b ;
Bizouard2021 ; SS2019 ; ST2020b . The universal relation in Refs. TCPOF2019b ;
Bizouard2021  relates the frequencies with PNS surface gravity. This relation
has been derived using more PNS models than ours. On the other hand, as
discussed in this paper, we prefer to adopt the average density and have
updated the relation proposed in Ref. ST2020b , increasing PNS models. Our new
universal relation can predict the gravitational wave frequencies not only for
the later phase but also for the early phase. A caveat is that the new
universal relation would not be applicable for a black hole formation SS2019 ,
since our PNS models are calculated with phenomenological general relativity.
Numbers of the general relativistic PNS models are necessary to obtain more
robust predictions.

The time evolution of mass and radius of PNS is strongly affected by entropy
and electron fraction profiles ST2016 ; Preau21 , that are evolved with
neutrino cooling processes. The evolution of them depends on the progenitor
mass Warren20 ; Schneider20  and EOS’s properties such as incompressibility
and effective mass Yasin20 ; Schneider20 ; Andersen21 . In this paper, we have
discussed the impact of non-uniform matter treatment, which changes neutrino’s
opacity via the composition of nuclei. To make our prediction of the peak
frequency more accurate, we need to construct a general formula that is valid
in these vast parameter spaces.

###### Acknowledgements.

This work is supported in part by Japan Society for the Promotion of Science
(JSPS) KAKENHI Grant Numbers JP17H06357, JP17H06364, JP18H01212, JP19KK0354,
JP20H04753, and JP21H01088 and by Pioneering Program of RIKEN for Evolution of
Matter in the Universe (r-EMU). This research has been also supported by MEXT
as “Program for Promoting Researches on the Supercomputer Fugaku” (Toward a
unified view of the universe: from large scale structures to planets,
JPMXP1020200109) and JICFuS, the National Institutes of Natural Sciences
(NINS) program for cross-disciplinary study (Grant Numbers 01321802 and
01311904) on Turbulence, Transport, and Heating Dynamics in Laboratory and
Solar/Astrophysical Plasmas: “SoLaBo-X”. Numerical computations were in part
carried out on Cray XC50, PC cluster and analysis server at Center for
Computational Astrophysics, National Astronomical Observatory of Japan.

##  Appendix A Comparison between the gravitational wave signals and PNS
eigenfrequencies

In the main text, we show the comparison between the gravitational wave
signals appearing in the numerical simulation and the gravitational wave
frequencies from the PNS only for the case with TGLD in Fig. 3. In Fig. 9, we
show the similar comparison for the other PNS models considering in this
study. From this figure, for any PNS models, one can observe that the
gravitational wave signals appearing in the numerical simulation are
identified with the $g_{1}$-mode ($f$-mode) oscillations before (after) the
avoided crossing between the $f$\- and $g_{1}$-mode frequencies. In addition,
one may be see a signature of the gravitational wave signal corresponding to
the $p_{1}$-mode frequency in some PNS models, e.g., with SFHo, even though it
is relatively weak. In this figure, one can observe the systematical
differences ($\sim 100$ Hz) between the $f$-mode frequency and the trace in
the spectrograms at late times, but unfortunately we could not identify the
origin of this difference.

Figure 9: Comparison between the gravitational wave signals appearing in the
numerical simulations (contour) and the PNS specific oscillation frequencies
(open marks) for various PNS models. The circles, squares, and diamonds
correspond to the $f$-, $g_{i}$-, and $p_{i}$-modes with $i=1,2$,
respectively. The result for the PNS model with LS220 and the $2.9M_{\odot}$
progenitor is taken from Ref. ST2020b .

## References

  * (1) B. P. Abbott et al. (LIGO Scientific and Virgo Collaboration), Phys. Rev. Lett. 119, 161101 (2017).
  * (2) B. P. Abbott et al. (LIGO Scientific and Virgo Collaboration), Astrophys. J. 848, L12 (2017).
  * (3) R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaboration), Astrophys. J. 915, L5 (2021).
  * (4) J. W. Murphy, C. D. Ott, and A. Burrows, Astrophys. J. 707, 1173 (2009).
  * (5) B. Müller, H. -T. Janka, and A. Marek, Astrophys. J. 766, 43 (2013).
  * (6) C. D. Ott, E. Abdikamalov, P. Mösta, R. Haas, S. Drasco, E. P. O’Connor, C. Reisswig, C. A. Meakin, and E. Schnetter, Astrophys. J. 768, 115 (2013).
  * (7) P. Cerdá-Durán, N. DeBrye, M. A. Aloy, J. A. Font, and M. Obergaulinger, Astrophys. J. Lett. 779, L18 (2013).
  * (8) K. N. Yakunin, A. Mezzacappa, P. Marronetti, S. Yoshida, S. W. Bruenn, W. R. Hix, E. J. Lentz, O. E. Bronson Messer, J. A. Harris, E. Endeve, J. M. Blondin, and E. J. Lingerfelt, Phys. Rev. D 92, 084040 (2015).
  * (9) T. Kuroda, K. Kotake, and T. Takiwaki, Astrophys. J. Lett. 829, L14 (2016).
  * (10) H. Andresen, B. Müller, E. Müller, and H.-T. Janka, Mon. Not. R. Astron. Soc. 468, 2032, (2017).
  * (11) S. Richers, C. D. Ott, E. Abdikamalov, E. O’Connor, and C. Sullivan, Phys. Rev. D 95, 063019 (2017).
  * (12) T. Takiwaki and K. Kotake, Mon. Not. R. Astron. Soc. 475, L91 (2018).
  * (13) E. P. O’Connor and S. M. Couch, Astrophys. J. 865, 81 (2018).
  * (14) D. Radice, V. Morozova, A. Burrows, D. Vartanyan, and H. Nagakura, Astrophys. J. Lett. 876, L9 (2019).
  * (15) D. Vartanyan, A. Burrows, and D. Radice, Mon. Not. R. Astron. Soc. 489, 2227 (2019).
  * (16) J. Powell and B. Müller, Mon. Not. R. Astron. Soc. 494, 4665 (2020).
  * (17) O. E. Andersen, S. Zha, A. S. Schneider, A. Betranhandy, S. M. Couch, E. P. O’Connor, arXiv:2106.09734.
  * (18) M. A. Pajkos, M. L. Warren, S. M. Couch, E. P. O’Connor, and K. -C. Pan, Astrophys. J. 914, 80 (2021).
  * (19) S. Shibagaki, T. Kuroda, K. Kotake, and T. Takiwaki, Mon. Not. R. Astron. Soc. 502, 3066 (2021).
  * (20) V. Morozova, D. Radice, A. Burrows, and D. Vartanyan, Astrophys. J. 861, 10 (2018).
  * (21) H. Sotani, T. Kuroda, T. Takiwaki, and K. Kotake, Phys. Rev. D 99, 123024 (2019).
  * (22) H. Sotani and T. Takiwaki, Mon. Not. R. Astron. Soc. 498, 3503 (2020).
  * (23) A. Torres-Forné, P. Cerdá-Durán, A. Passamonti, and J. A. Font, Mon. Not. R. Astron. Soc. 474, 5272 (2018).
  * (24) A. Torres-Forné, P. Cerdá-Durán, A. Passamonti, M. Obergaulinger, and J. A. Font, Mon. Not. R. Astron. Soc. 482, 3967 (2019).
  * (25) A. Torres-Forné, P. Cerdá-Durán, M. Obergaulinger, B. Müller and J. A. Font, Phys. Rev. Lett. 123, 051102 (2019).
  * (26) K. D. Kokkotas and B. G. Schmidt, Living Rev. Relativ. 2, 2 1999.
  * (27) L. Samuelsson and N. Andersson, Class. Quant. Grav. 26, 155016 (2009).
  * (28) M. Gearheart, W. G. Newton, J. Hooker, and B. -A. Li, Mon. Not. R. Astron. Soc. 418, 2343 (2011).
  * (29) H. Sotani, K. Nakazato, K. Iida, and K. Oyamatsu, Phys. Rev. Lett. 108, 201101 (2012); Mon. Not. R. Astron. Soc. 428, L21 (2013); Mon. Not. R. Astron. Soc. 434, 2060 (2013).
  * (30) A. T. Deibel, A. W. Steiner, and E. F. Brown, Phys. Rev. C 90, 025802 (2014).
  * (31) M. Gabler, P. Cerdá-Durán, N. Stergioulas, J. A. Font, and E. Müller, Mon. Not. R. Astron. Soc. 476, 4199 (2018).
  * (32) N. Andersson and K. D. Kokkotas, Phys. Rev. Lett. 77, 4134 (1996).
  * (33) N. Andersson and K. D. Kokkotas, Mon. Not. R. Astron. Soc. 299, 1059 (1998).
  * (34) H. Sotani, K. Tominaga, and K. I. Maeda, Phys. Rev. D 65, 024010 (2001).
  * (35) H. Sotani and T. Harada, Phys. Rev. D 68, 024019 (2003); H. Sotani, K. Kohri, and T. Harada, ibid. 69, 084008 (2004).
  * (36) H. Sotani, N. Yasutake, T. Maruyama, and T. Tatsumi, Phys. Rev. D 83 024014 (2011).
  * (37) A. Passamonti and N. Andersson, Mon. Not. R. Astron. Soc. 419, 638 (2012).
  * (38) D. D. Doneva, E. Gaertig, K. D. Kokkotas, and C. Krüger, Phys. Rev. D 88, 044052 (2013).
  * (39) H. Sotani, Phys. Rev. D 102, 063023 (2020); 103021 (2020); 103, 123015 (2021).
  * (40) V. Ferrari, G. Miniutti, and J. A. Pons, Mon. Not. R. Astron. Soc. 342, 629 (2003).
  * (41) G. F. Burgio, V. Ferrari, L. Gualtieri, and H.-J. Schulze, Phys. Rev. D 84, 044017 (2011).
  * (42) J. Fuller, H. Klion, E. Abdikamalov, and C. D. Ott, Mon. Not. R. Astron. Soc. 450, 414 (2015).
  * (43) H. Sotani and T. Takiwaki, Phys. Rev. D 94, 044043 (2016).
  * (44) G. Camelio, A. Lovato, L. Gualtieri, O. Benhar, J. A. Pons, and V. Ferrari, Phys. Rev. D 96, 043015 (2017).
  * (45) H. Sotani, T. Kuroda, T. Takiwaki, and K. Kotake, Phys. Rev. D 96, 063005 (2017).
  * (46) H. Sotani and K. Sumiyoshi, Phys. Rev. D 100, 083008 (2019).
  * (47) J. R. Westernacher-Schneider, E. O’Connor, E. O’Sullivan, I. Tamborra, M.-R. Wu, S. M. Couch, and F. Malmenbeck, Phys. Rev. D 100, 123009 (2019).
  * (48) J. R. Westernacher-Schneider, Phys. Rev. D 101, 083021 (2020).
  * (49) A. Mezzacappa, P. Marronetti, R. E. Landfield, E. J. Lentz, K. N. Yakunin, S. W. Bruenn, W. R. Hix, O. E. B. Messer, E. Endeve, J. M. Blondin, and J. A. Harris, Phys. Rev. D 102, 023027 (2020).
  * (50) H. Sotani and T. Takiwaki, Phys. Rev. D 102, 023028 (2020).
  * (51) H. Sotani and T. Takiwaki, Phys. Rev. D 102, 063025 (2020).
  * (52) M. A. Bizouard, P. Maturana-Russel, A. Torres-Forné, M. Obergaulinger, P. Cerdá-Durán, N. Christensen, J. A. Font, and R. Meyer, Phys. Rev. D 103, 063006 (2021).
  * (53) T. Takiwaki, K. Kotake, and Y. Suwa, Mon. Not. R. Astron. Soc. 461, L112 (2016).
  * (54) E. O’Connor et al., J. Phys. G 45, 104001 (2018).
  * (55) K. Kotake, T. Takiwaki, T. Fischer, K. Nakamura, and G. Martínez-Pinedo, Astrophys. J. 853, 170 (2018).
  * (56) J. Matsumoto, T. Takiwaki, K. Kotake, Y. Asahina and H. R. Takahashi, Mon. Not. Roy. Astron. Soc. 499, 4174-4194 (2020).
  * (57) M. Liebendoerfer, S. Whitehouse, and T. Fischer, Astrophys. J. 698, 1174 (2009).
  * (58) T. Takiwaki, K. Kotake, and Y. Suwa, Astrophys. J. 786, 83 (2014).
  * (59) S. E. Woosley and A. Heger, Phys. Rep. 442, 269 (2007).
  * (60) S. Typel, G. Ropke, T. Klahn, D. Blaschke, and H. H. Wolter, Phys. Rev. C 81, 015803 (2010).
  * (61) A. W. Steiner, M. Hempel, and T. Fischer, Astrophys. J. 774, 17 (2013).
  * (62) H. Togashi, K. Nakazato, Y. Takehara, S. Yamamuro, H. Suzuki, and M. Takano, Nucl. Phys. A961, 78 (2017).
  * (63) S. Furusawa, H. Togashi, H. Nagakura, K. Sumiyoshi, S. Yamada, H. Suzuki, and M. Takano, J. Phys. G: Nucl. Part. Phys. 44, 094001 (2017).
  * (64) H. Sotani, K. Iida, K. Oyamatsu, and A. Ohnishi, Prog. Theor. Exp. Phys. 2014, 051E01 (2014).
  * (65) E. Annala, T. Gorda, A. Kurkela, and A. Vuorinen, Phys. Rev. Lett. 120, 172703 (2018).
  * (66) T. J. Moriya, P. A. Mazzali, N. Tominaga, S. Hachinger, S. I. Blinnikov, T. M. Tauris, K. Takahashi, M. Tanaka, N. Langer, and P. Podsiadlowski, Mon. Not. R. Astron. Soc. 466, 2085 (2017).
  * (67) J. M. Lattimer and F. D. Swesty, Nucl. Phys. A535, 331 (1991).
  * (68) A. Marek, H. Dimmelmeier, H.-T. Janka, E. Müller, and R. Buras, Astron. Astrophys. 445, 273 (2006).
  * (69) Private communication with P. Cerda-Duran.
  * (70) S. Furusawa, H. Togashi, K. Sumiyoshi, K. Saito, S. Yamada, and H. Suzuki, Prog. Theor. Exp. Phys. 2020, 013D05 (2020).
  * (71) R. Kippenhahn, A. Weigert, and A. Weiss, “Stellar Structure and Evolution”, Berlin, Heidelberg: Springer Berlin Heidelberg, 2012.
  * (72) L. Scheck, K. Kifonidis, H. T. Janka, and E. Mueller, Astron. Astrophys. 457, 963 (2006).
  * (73) K. Nakazato and H. Suzuki, Astrophys. J. 878, 25 (2019).
  * (74) K. Nakazato and H. Suzuki, Astrophys. J. 891, 156 (2020).
  * (75) K. Nakazato, F. Nakanishi, M. Harada, Y. Koshio, Y. Suwa, K. Sumiyoshi, A. Harada, M. Mori, and R. A. Wendell, arXiv:2108.03009.
  * (76) H. Nagakura, S. Furusawa, H. Togashi, S. Richers, K. Sumiyoshi, and S. Yamada, Astrophys. J. Suppl. 240, 38 (2019).
  * (77) T. Fischer, S. Typel, G. Röpke, N. U. F. Bastian, and G. Martínez-Pinedo, Phys. Rev. C 102, 055807 (2020).
  * (78) E. Preau, A. Pascal, J. Novak, and M. Oertel, Mon. Not. Roy. Astron. Soc. 505, 939 (2021).
  * (79) M. L. Warren, S. M. Couch, E. P. O’Connor, and V. Morozova, Astrophys. J. 898, 139 (2020).
  * (80) A. da Silva Schneider, E. O’Connor, E. Granqvist, A. Betranhandy, and S. M. Couch, Astrophys. J. 894, 4 (2020).
  * (81) H. Yasin, S. Schäfer, A. Arcones, and A. Schwenk, Phys. Rev. Lett. 124, 092701 (2020).


]http://www.mathnet.ru/eng/person17991

# Non-commutative graphs based on finite-infinite system couplings: quantum
error correction for a qubit coupled to a coherent field

G.G. Amosov<EMAIL_ADDRESS>Steklov Mathematical Institute of Russian
Academy of Sciences, Gubkina str. 8, Moscow 119991, Russia  A.S. Mokeev
<EMAIL_ADDRESS>Steklov Mathematical Institute of Russian Academy of
Sciences, Gubkina str. 8, Moscow 119991, Russia  A.N. Pechen
<EMAIL_ADDRESS>[ Department of Mathematical Methods for Quantum
Technologies, Steklov Mathematical Institute of Russian Academy of Sciences,
Gubkina str. 8, Moscow 119991, Russia  National University of Science and
Technology ”MISIS”, Leninsky prosp. 6, Moscow 119991, Russia

###### Abstract

Quantum error correction plays a key role for quantum information transmission
and quantum computing. In this work, we develop and apply the theory of non-
commutative operator graphs to study error correction in the case of a finite-
dimensional quantum system coupled to an infinite dimensional system. We
consider as an explicit example a qubit coupled via the Jaynes-Cummings
Hamiltonian with a bosonic coherent field. We extend the theory of non-
commutative graphs to this situation and construct, using the Gazeau-Klauder
coherent states, the corresponding non-commutative graph. As the result, we
find the quantum anticlique, which is the projector on the error correcting
subspace, and analyze it as a function of the frequencies of the qubit and the
bosonic field. The general treatment is also applied to the analysis of the
error correcting subspace for certain experimental values of the parameters of
the Jaynes-Cummings Hamiltonian. The proposed scheme can be applied to any
system that possess the same decomposition of spectrum of the Hamiltonian into
a direct sum as in JC model, where eigenenergies in the two direct summands
form strictly increasing sequences.

##  I Introduction

Quantum error correcting codes (or, in other terminology, quantum
anticliques), introduced theoretically in the pioneering papers [1, 2, 3], and
implemented experimentally, e.g., in [4], play an important role in quantum
information theory [5]. Analog error correction for continuous variables, such
as position and momentum, was considered [6], symmetry breaking in open
quantum systems for photonic cat qubits [7], etc. In particular, bosonic codes
which use encoding of information in states of bosonis field are of high
interest [8, 9, 10, 11, 12].

Error correction theory studies the possibility of encoding information in
quantum states in a way to allow zero-error decoding in the presence of a
given fixed acceptable set of errors. Mathematically each error is described
by some completely positive map acting on the set of states of the quantum
system. In the general setting [13], for any given set of errors it is
possible to define a unique non-commutative operator graph [14] such that the
knowledge of this graph allows to define all error correcting codes for this
set of errors. The correspondence between sets of errors and non-commutative
operator graphs is not one-to-one, but in the case of separable Hilbert space
each non-commutative operator graph describes codes for some set of errors
[15, 16, 17]. As was established in the finite-dimensional [15, 16] and
infinite-dimensional [17] cases, the noncommutative graph describing errors in
information transmission is always generated by some positive operator-valued
measure.

Various models of error correction were analyzed using the approach based on
the use of non-commutative graphs. It was applied for quantum error correction
for the models of coupled finite-dimensional systems [18, 19, 20], and for
coupled infinite-dimensional system [21]. In [21], the non-commutative graph
generated by the dynamics of a bipartite bosonic quantum system in an infinite
dimensional Hilbert space was defined. The graph consists of orbits driven by
the unitary group which is the solution of the Schröedinger equation for a two
interacting bosonic oscillators. In this framework possible error correcting
codes are given by coherent states in the bosonic Fock space. In all these
cases, it was possible to find quantum anticliques, which are projectors onto
error correcting subspaces.

In this work, we extend the theory of operator graphs to the case when one
system is finite dimensional while another is infinite dimensional. Currently,
several quantum error-correction [22, 23] and entanglement protection [24]
techniques was inroduced for systems of such structure. Explicitly, we
consider the situation where the information is encoded in a joint state of a
qubit (a two-level quantum system) coupled via the Jaynes-Cummings Hamiltonian
[25] with a bosonic oscillator or bosonic coherent field. The Jaynes-Cummings
Hamiltonian is the key model for the various theoretical and experimental
works in quantum optics and for studying the interactions between light and
matter, see, e.g. [41, 34, 35, 37, 40, 36, 38, 39, 32, 33, 31, 43, 44, 42],
including in strong [32], ultra-strong [34, 35, 36] and deep strong coupling
regimes [34, 35, 37, 36, 38, 39].

We develop for this model the theory of non-commutative operator graphs and
apply it to find the corresponding quantum anticlique. The construction is
based upon Gazeau–Klauder coherent states [26]. With this setting, we
explicitly find the error correcting subspace for any values of the parameters
of the Jaynes-Cummings model.

The structure of the paper is the following. In section 2, we discuss the
problem of finding the existence of an error correcting procedure for a
quantum channel, and also describe quantum channels corresponding to operator
graphs of the class which includes the non-commutative graphs later
constructed in section 5. In section 3, the Jaynes-Cummings model is
discussed. Section 4 describes the construction of Gazeau-Klauder coherent
states. In section 5, using the Gazeau-Klauder coherent states for the Jaynes-
Cummings model, we construct the non-commutative operator graphs that have
quantum error correcting codes, and find quantum anticlique and error
correction subspaces.

##  II Quantum channels and non-commutative graphs

Consider encoding information in states of a quantum system with Hilbert space
$\cal H$. The (convex) set $\mathfrak{S}({\cal H})$ of quantum states is the
set of positive unit trace operators in $\cal H$. Errors which can occur under
information transmission can be described by a quantum channel
$\Phi:\mathfrak{S}({\cal H})\to\mathfrak{S}({\cal H})$ which is a completely
positive trace preserving (CPTP) map. As any CPTP map, it possesses the Kraus
operator-sum representation (Kraus OSR) [27]

$\Phi(\rho)=\sum\limits_{k\in K}V_{k}\rho
V_{k}^{*},\quad\rho\in\mathfrak{S}(\mathcal{H}).$ (1)

The Kraus operators $\left\\{V_{k},\ k\in K\right\\}$ are parametrized by some
set $K$. They should satisfy the property

$\sum\limits_{k\in K}V_{k}^{*}V_{k}=\mathbb{I}.$ (2)

to preserve trace of density matrix. In infinite dimensional spaces $K$ is not
necessary countable and the sum in (1) – (2) can be replaced by an integral
(see e.g. [28]). Nevertheless for any channel $\Phi$ there exists a countable
set $K$ parametrizing Kraus operators such that (1) holds true. Note that the
Kraus OSR of a quantum channel is non-unique. The same quantum channel also
has a Kraus OSR with operators $\tilde{V}_{j}=\sum\limits_{i}U_{ji}V_{i}$,
$j=1,\dots,m$ ($m\geq|K|$), where $U^{+}U=\mathbb{I}$ (unitary matrix).

The linear space, $\mathcal{V}$, plays an important role in the theory of
optimal coding

$\mathcal{V}=\overline{span}\left\\{V_{k}^{*}V_{j},\ k,j\in K\right\\}.$ (3)

The linear space $\mathcal{V}$ does not depend on the choice of the operators
$\\{V_{i}\\}$ used for Kraus OSR of a given quantum channel; despite the non-
uniqueness of the Kraus OSR, it is unique for a given quantum channel. Notice
that in the finite dimensional case there is not need to take the closure in
(3). For the infinite dimensional case see [21, 17].

The linear space $\mathcal{V}$ has the properties of a non-commutative graph.
Such objects were introduced in [29] as operator systems and recently
redefined as non-commutative graphs in quantum information theory [14]. A non-
commutative graph is a linear subspace $\mathcal{V}$ of bounded operators in a
Hilbert space ${\cal H}$ possessing the properties

  * •

${\bf V}\in\mathcal{V}$ implies that ${\bf V}^{*}\in\mathcal{V}$;

  * •

${\bf I}\in{\mathcal{V}}$

The famous Knill–Laflamme condition[3, 13] claims that a zero error
transmission via some channel $\Phi$ is possible iff for some orthogonal
projector $P$ for all $A\in\mathcal{V}$ holds $PAP=\alpha(A)P$, where
$\alpha(A)\in\mathbb{C}$. Here $P$ is the projector on the subspace generated
by error correction code [3]. The optimal code belongs to the subspace ${\cal
H}_{P}=P{\cal H}$. The dimension of the subspace ${\cal H}_{P}$ is the maximal
amount of quantum information that could be transmitted via $\Phi$ with zero
error. An orthogonal projection $P$ such that $\dim\left(P{\cal H}\right)\geq
2$ is a quantum anticlique for a non-commutative graph $\mathcal{V}$ if it
satisfies:

$\dim P{\mathcal{V}}P=1.$ (4)

The most natural quantum channel is given by the projection measurement

$\Phi_{\mathcal{P}}(\rho)=\sum\limits_{k\in K}P_{k}\rho
P_{k},\quad\rho\in\mathfrak{S}({\cal H}),$ (5)

where $\mathcal{P}=(P_{k})$ is the orthogonal resolution of identity

$\sum\limits_{k\in K}P_{k}=\mathbb{I}.$

For the channel (5), the non-commutative graph (3) is

$\mathcal{V}=\overline{span}\left\\{P_{k},k\in K\right\\}.$

In the finite dimensional case, it is enough to consider only discrete sets
$K$, while for the infinite dimensional case $P_{k}=E(B_{k})$ are generated by
some projection valued measure on the real line, where
$B_{k}\subset{\mathbb{R}}$ are some Borel sets possessing the property
$\cup_{k\in K}B_{k}={\mathbb{R}}$. In this case, different choices of
$B_{k}$’s produce different projectional measurements and a reachable set of
admissible errors.

Suppose that some unitary group ${\mathcal{U}}=\\{U_{t}=e^{-itG},\ t\geq 0\\}$
acts in the Hilbert space $H$. Then the possible set of errors can be extended
to all possible projection measurements

$\Phi^{t}(\rho)=\sum\limits_{k\in K}U_{t}P_{k}U_{t}^{*}\rho
U_{t}P_{k}U_{t}^{*},\ t\in{\mathbb{R}},$ (6)

For the goal of constructing a quantum anticlique allowing to correct errors
of the form (6) for any fixed $t$, it is natural to define the non-commutative
graph corresponding to all these errors as follows

${\mathcal{V}}=\overline{span}\\{U_{t}P_{k}U_{t}^{*},\ t\in{\mathbb{R}},\ k\in
K\\}.$

Our interpretation of such a graph is that the quantum system distorts the
transmitted information by a set of time-dependent errors.

Based upon this interpretation, suppose that there is a set of orthogonal
projections $\left\\{P_{\alpha},\ \alpha\in\mathfrak{A}\right\\}$
parameterized by some set $\mathfrak{A}$, and the operator space is generated
by orbits of some unitary group $\mathcal{U}$ as follows

${\mathcal{V}}=\overline{span}\\{U_{t}P_{\alpha}U_{t}^{*},\ t\in{\mathbb{R}},\
\alpha\in\mathfrak{A}\\}.$ (7)

It is known [15, 17] that (7) is a non-commutative operator graph
corresponding to some channel iff $\mathbb{I}\in\mathcal{V}$. We shall
construct the explicit example of graph for the Jaynes-Cummings model.
Moreover, we shall show that there exists an anticlique for this graph.

##  III Jaynes-Cummings model

We consider a two-level quantum system (qubit) coupled to a coherent field.
Hilbert space of the qubit is ${\cal H}_{\rm s}=\mathbb{C}^{2}$. Ground and
excited basis states of the qubit are denoted as $\\{\ket{g},\ket{e}\\}$.
Hilbert space of the coherent field is ${\cal H}_{\rm
f}=L^{2}(\mathbb{R})=\\{f:\mathbb{R}\to\mathbb{C}\,|\,\int_{\mathbb{R}}|f(x)|^{2}dx<\infty\\}$.
Fock states of the field are denoted as $\\{\ket{k},k\in\mathbb{N}_{0}\\}$. We
use the set of natural numbers including zero
$\mathbb{N}_{0}={0}\cup\mathbb{N}$ to enumerate the states. The qubit and the
field are assumed to be coupled via the Jaynes-Cummings Hamiltonian acting in
${\cal H}={\cal H}_{\rm s}\otimes{\cal H}_{\rm f}$

$H=\omega_{f}a^{+}a^{-}+\frac{\omega_{s}}{2}\sigma_{z}+\frac{\kappa}{2}(\sigma^{-}a^{+}+\sigma^{+}a^{-}),$
(8)

Here $\omega_{s},\omega_{f}\in\mathbb{R_{+}}$ are the frequencies of the qubit
and the field, respectively, $\kappa\geq 0$ is the coupling constant,
$\sigma_{z}$ is the Pauli matrix, $\sigma^{+},\sigma^{-}$ are the rising and
lowering operators of the qubit and the $a^{+},a^{-}$ are the creation and
annihilation operators of the field. The detuning parameter is
$\Delta=\omega_{f}-\omega_{s}$. We use the normalization of the physical units
such that $\hbar=1$. Denote the basis in ${\cal H}$ as
$\ket{q}\otimes\ket{p}=\ket{q,p}$, where the first number $q\in\mathbb{N}_{0}$
denotes the coherent state and the second number $p\in\\{e,g\\}$ denotes the
qubit state. The schematic picture of the Jaynes-Cumming model states
interaction is provided in Fig. 1.

Figure 1:  Jaynes-Cummings model of a qubit interacting with bosonic
reservoir.

The Schröedinger equation with the Jaynes-Cummings Hamiltonian has an exact
solution. The Hamiltonian has the following eigenstates

$\displaystyle\ket{0,g},$
$\displaystyle\ket{n,+}=\cos\left(\frac{\theta_{n}}{2}\right)\ket{n-1,e}+\sin\left(\frac{\theta_{n}}{2}\right)\ket{n,g},$
$\displaystyle\ket{n,-}=\sin\left(\frac{\theta_{n}}{2}\right)\ket{n-1,e}-\cos\left(\frac{\theta_{n}}{2}\right)\ket{n,g},$

where $\theta_{n}=\tan^{-1}(\kappa\sqrt{n}/\Delta)$ and $n\in\mathbb{N}$, for
the non-resonant case $\Delta\neq 0$. For the resonant case $\Delta=0$ the
eigenstates are

$\displaystyle\ket{0,g},$ $\displaystyle\ket{n,+}=\ket{n-1,e}+\ket{n,g},$
$\displaystyle\ket{n,-}=\ket{n,g}-\ket{n-1,e}.$

In both cases the corresponding eigenenergies are

$\displaystyle E_{0,g}$ $\displaystyle=$
$\displaystyle\frac{\omega_{f}+\Delta}{2}$ $\displaystyle E_{n,\pm}$
$\displaystyle=$
$\displaystyle\omega_{f}\left(n-\frac{1}{2}\right)\pm\frac{1}{2}\sqrt{\Delta^{2}+\kappa^{2}n},\quad
n\in\mathbb{N}.$

Below we follow closely to the method provided in [30], where a new class of
coherent states was constructed for the Jaynes-Cummings model with strictly
increasing sequences of the eigenenergies $E_{n,+}$ and $E_{n,-}$. Our goal is
to divide the space $\cal H$ into three direct summands, two of which are
generated by eigenstates corresponding to strictly increasing sequences of
eigenenergies and one is finite dimensional. The sequence

$J_{k}=E_{k+1,+},\quad k\in\mathbb{N}_{0}$ (9)

is known to be strictly increasing. On the other hand, the sequence
$S_{0}=E_{0,g},S_{k}=E_{k,-},\ k\in\mathbb{N}$ may have degenerate levels
$S_{k_{1}}=S_{k_{2}},\ k_{1}\neq k_{2}$. We want to keep only a strictly
increasing tail of the sequence $S_{k}$. Let us show that there exists
$M_{0}\in\mathbb{N}$ such that for all $l_{2}>l_{1}\geq M_{0}$ one gets
$S_{l_{2}}>S_{l_{1}}$. It is equivalent to

$S_{n+1}-S_{n}=\omega_{f}-\frac{1}{2}\left(\sqrt{\Delta^{2}+\kappa^{2}(n+1)}-\sqrt{\Delta^{2}+\kappa^{2}n}\right)>0,\quad\forall
n\geq M_{0}.$ (10)

From this one gets that $M_{0}$ is the minimal integer solution of the
inequality

$\left(\sqrt{\Delta^{2}+\kappa^{2}(M_{0}+1)}+\sqrt{\Delta^{2}+\kappa^{2}M_{0}}\right)^{-1}<\frac{2\omega_{f}}{\kappa^{2}}.$
(11)

Thus, the sequence

$S_{k}=E_{k,-},\quad k\geq M_{0},$ (12)

becomes strictly increasing.

Let us fix any number $K_{0}\in\mathbb{N}$, $K_{0}\geq M_{0}$. Then the
sequence $S_{k}=E_{k,-}$, $k\geq K_{0}$ will also be strictly increasing and
we can separate the pieces where we assured to have strictly increasing
eigenenergies. This allows to represent the Hilbert space $\cal H$ as the
direct sum ${\cal H}={\cal H}_{1}\oplus{\cal H}_{2}\oplus{\cal H}_{3}$, where

$\displaystyle{\cal H}_{1}$ $\displaystyle=$ $\displaystyle span\\{\ket{n,+},\
n\in\mathbb{N}\\},$ (13) $\displaystyle{\cal H}_{2}$ $\displaystyle=$
$\displaystyle span\\{\ket{n,-},\ n\geq K_{0}\\},$ (14) $\displaystyle{\cal
H}_{3}$ $\displaystyle=$ $\displaystyle span\\{\ket{g,0}\\}\cup\\{\ket{n,-},\
1\leq n<K_{0}\\},$ (15)

The subspace ${\cal H}_{3}$ later will be shown to be the error correcting
subspace for this system. The error dynamics will be shown to interchange
states in ${\cal H}_{1}$ and ${\cal H}_{2}$, while keeping states in
$\mathcal{H}_{3}$ unchanged.

In Section 5, following the ideas of [30] we will define the Gazeau-Klauder
coherent states in ${\cal H}_{1}$ and ${\cal H}_{2}$.

##  IV Gazeau-Klauder coherent states

Here we introduce the construction of Gazeau-Klauder coherent states [26]. Let
us consider an infinite dimensional Hilbert space ${\cal H}$ with the basis
$\ket{k},k\in\mathbb{N}_{0}$ and a self-ajoint operator $G$ which is diagonal
in this basis. In [26] Gazeau and Klauder defined the generalized coherent
states corresponding to the operator $G$ as a two-parameter system of vectors
$\\{\ket{x,y},x\in\mathbb{R_{+}},\ y\in\mathbb{R}\\}\subset\mathcal{H}$ with
the following properties

  1. 1.

Continuity:$\
(x,y)\rightarrow(x_{0},y_{0})\Rightarrow\ket{x,y}\rightarrow\ket{x_{0},y_{0}}$.

  2. 2.

Resolution of identity: $\int\ket{x,y}\bra{x,y}d\nu(x,y)=\mathbb{I}_{\cal H}$.

  3. 3.

Temporal stability: $e^{-itG}\ket{x,y}=\ket{x,y+\omega t}$.

  4. 4.

Action identity: $\bra{x,y}G\ket{x,y}=\omega x$.

for some real constant $\omega$ and some measure $\nu$.

Consider the set of eigenvalues $h_{k}=\bra{k}G\ket{k}$ for the operator $G$.
In the case $h_{0}=0$ and strictly increasing $h_{k}$, Gazeau and Klauder gave
the explicit construction for the system of coherent states. If $h_{0}>0$ and
the sequence $h_{k}$ is strictly increasing, their construction describes the
set of vectors that satisfy the first two properties and the following version
of the time stability condition:

$e^{-itG}\ket{x,y}\bra{x,y}e^{itG}=\ket{x,y+\omega t}\bra{x,y+\omega t}.$

The main property of generalized coherent states is that they form the
resolution of identity. It was shown that the measure $\nu$ has the form

$d\nu(x,y)=\tau(x)dxdy,$

where $\tau(x)$ is some probability distribution density on the half-axis.

Below we give an explicit description of this construction. Consider a
sequences of weights

$c_{k}>0,\quad k\in\mathbb{N}_{0},$

with the convergence condition

$\limsup_{k\rightarrow\infty}\sqrt[k]{c_{k}}=R>0.$ (16)

Suppose that these weights are the moments of probability distributions with
density $\rho(x)>0$ on the interval $[0,R)$,

$c_{k}=\int\limits_{0}^{R}\rho(x)x^{k}dx<+\infty,\quad k\in\mathbb{N}_{0}.$

We also need the normalization factor and the density defined by the formulae

$\displaystyle N^{2}(x)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{x^{k}}{c_{k}},\quad 0\leq x<R,$ (17)
$\displaystyle\tau(x)$ $\displaystyle=$ $\displaystyle N^{2}(x)\rho(x).$

The radius of convergence in (17) is equal to $R$ by the property (16). Now
the Gazeau–Klauder coherent states are defined as follows

$\ket{x,y}=\frac{1}{N(x)}\sum_{k=0}^{\infty}\frac{x^{k/2}e^{-ih_{k}y}}{\sqrt{c_{k}}}\ket{k}$
(18)

We suppose the constant $\omega$ is equal to one. Following the definition in
[26], for $f:\mathbb{R}\rightarrow B(\mathcal{H})$, where $B(\mathcal{H})$ is
the set of bounded linear operators on $\mathcal{H}$, we introduce its
integration as the weak-limit of averages of weak integrals,

$I(f)=\int\limits_{-\infty}^{+\infty}f(y)d\mu(y)=\lim_{R\rightarrow+\infty}\frac{1}{2R}\int\limits_{-R}^{R}f(y)dy.$

Note that if such integral converges for some $f$ with the image lying in the
weakly-closed subspace $Im(f)\subset\mathcal{W}\subset B(H)$, then the
integral also lies in this subspace, $I(f)\in\mathcal{W}$. The resolution of
identity property for coherent states results in

$\int\limits_{0}^{R}\int\limits_{-\infty}^{+\infty}\ket{x,y}\bra{x,y}\tau(x)dxd\mu(y)=\mathbb{I}_{\cal
H}.$ (19)

##  V Graphs generated by Gezeau-Klauder coherent states

Now we are able to define systems of Gazeau–Klauder coherent states [26] in
the Hilbert spaces ${\cal H}_{1}$ and ${\cal H}_{2}$. Take two sequences of
weights

$c_{k}^{(j)}>0,\quad d\in\mathbb{N}_{0},\ j=1,2,$

with the same convergence condition

$\limsup_{k\rightarrow\infty}\sqrt[k]{c_{k}^{(j)}}=R>0.$

Suppose that the weights have the corresponding probability densities
$\rho_{1}(x),\rho_{2}(x)>0$ on the interval $[0,R]$ such that

$c_{k}^{(j)}=\int\limits_{0}^{R}\rho_{j}(x)x^{k}dx<+\infty,\quad
k\in\mathbb{N}_{0},\ j=1,2.$

Then, the normalization factors and the densities for measures defining
resolutions of identity are given by the formulae

$\displaystyle N_{j}^{2}(x)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{x^{k}}{c_{k}^{(j)}},\quad 0\leq x<R,$
$\displaystyle\tau_{j}(x)$ $\displaystyle=$ $\displaystyle
N_{j}^{2}(x)\rho_{j}(x),\quad j=1,2.$

Consider the Gazeau–Klauder coherent states

$\displaystyle\ket{J,x,y}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{1}(x)}\sum_{k=0}^{\infty}\frac{x^{k/2}e^{-iJ_{k}y}}{\sqrt{c_{k}^{(1)}}}\ket{k+1,+},$
$\displaystyle\ket{S,x,y}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{2}(x)}\sum_{k=0}^{\infty}\frac{x^{k/2}e^{-iS_{k+K_{0}}y}}{\sqrt{c_{k}^{(2)}}}\ket{k+K_{0},-},$

where the strictly increasing sequences of eigenenergies are given by (9) and
(12) respectively.

Since the Gazeau–Klauder coherent states section IV form the resolution of
identity, we get for the projections $P_{{\cal H}_{1}},P_{{\cal H}_{2}}$ on
the subspaces ${\cal H}_{1},{\cal H}_{2}$ that

$\int\limits_{0}^{R}\int\limits_{-\infty}^{+\infty}\ket{J,x,y}\bra{J,x,y}\tau_{1}(x)dxd\mu(y)=P_{{\cal
H}_{1}},$ (20)
$\int\limits_{0}^{R}\int\limits_{-\infty}^{+\infty}\ket{S,x,y}\bra{S,x,y}\tau_{2}(x)dxd\mu(y)=P_{{\cal
H}_{2}}.$ (21)

Consider the unitary group ${\mathcal{U}}=\\{U_{t}=e^{-it{H}},\
t\in{\mathbb{R}}\\}$, where the Hamiltonian $H$ is determined by (8). Systems
$\ket{J,x,y},\ket{S,x,y}$ satisfy the temporal stability property (IV) with
respect to $\mathcal{U}$,

$U_{t}\ket{J,x,y}\bra{J,x,y}U_{t}^{*}=\ket{J,x,y+t}\bra{J,x,y+t}$ (22)
$U_{t}\ket{S,x,y}\bra{S,x,y}U_{t}^{*}=\ket{S,x,y+t}\bra{S,x,y+t}$ (23)

Consider the two families of orthogonal projections

$P_{x}^{1}=\ket{J,x,0}\bra{J,x,0},\quad P_{x}^{2}=\ket{S,x,0}\bra{S,x,0},$

for $x\in[0,R]$. The projections $P_{x}^{1}$, $P_{x}^{2}$ and $P_{x}^{3}\equiv
P_{{\cal H}_{3}}$ are pairwise orthogonal for any fixed value of $x\in[0,R]$.

Theorem 1. The subspace

$\mathcal{V}=\overline{span}\\{U_{t}P_{x}^{j}U_{t}^{*},\ t\in\mathbb{R},\
x\in[0,R],\ j\in\\{1,2,3\\}\\}$

is a non-commutative operator graph with the anticlique $P_{{\cal H}_{3}}.$

Proof. Consider the operator

$Q_{x}=P_{x}^{1}+\frac{\tau_{2}(x)}{\tau_{1}(x)}P_{x}^{2}+\frac{1}{\tau_{1}(x)}P_{{\cal
H}_{3}}$

It follows from (22) and (23) that

$\displaystyle U_{t}Q_{x}U_{t}^{*}$
$\displaystyle=\ket{J,x,t}\bra{J,x,t}+\frac{\tau_{2}(x)}{\tau_{1}(x)}\ket{S,x,t}\bra{S,x,t}+\frac{1}{\tau_{1}(x)}P_{{\cal
H}_{3}}.$

Then, (20) and (21) result in

$\int\limits_{0}^{R}\int\limits_{-\infty}^{+\infty}\tau_{1}(x)\left(\ket{J,x,t}\bra{J,x,t}+\frac{1}{\tau_{1}(x)}\mathbb{I}_{{\cal
H}_{2}}+\frac{\tau_{2}(x)}{\tau_{1}(x)}\ket{S,x,t}\bra{S,x,t}\right)dxd\mu(t)=\mathbb{I}_{\cal
H}\in{\mathcal{V}}.$ (24)

Since $K_{0}$ is given by the rule (25) the dimension of ${\cal H}_{3}$ is at
least 2. From the equalities

$\displaystyle P_{{\cal H}_{3}}\ket{J,x,t}\bra{J,x,t}P_{{\cal H}_{3}}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle P_{{\cal
H}_{3}}\ket{S,x,t}\bra{J,x,t}P_{{\cal H}_{3}}$ $\displaystyle=$ $\displaystyle
0$

we obtain that $P_{{\cal H}_{3}}$ is an anticlique. $\Box$

##  VI The error correcting subspace

As Theorem 1 states, the subspace ${\cal H}_{3}$ is the error correcting
subspace. In some cases the number is $M_{0}=1$, so in this case for
$K_{0}=M_{0}$ the error correcting subspace would be empty (since its
dimension is $K_{0}-1$). However, in our construction one can take any natural
$K_{0}\geq M_{0}$. To satisfy this condition for our coding procedure, that is
the dimension of error correcting subspace is greater or equal to two, we
should take any

$K_{0}\geq K_{0}^{*}=\max\\{3,M_{0}\\}.$ (25)

To analyze the minimal dimension of the error correcting subspace for various
parameters of the Jaynes-Cummings Hamiltonian, consider the coupling rates
$\gamma_{f}=\kappa/\omega_{f}$ and $\gamma_{s}=\kappa/\omega_{s}$. In terms of
these quantities the inequality (11) takes the following form

$\sqrt{(\gamma_{f}^{-1}-\gamma_{s}^{-1})^{2}+M_{0}+1}+\sqrt{(\gamma_{f}^{-1}-\gamma_{s}^{-1})^{2}+M_{0}}>\frac{\gamma_{f}}{2}.$
(26)

Now it is evident that the key quantity defining the dimension for the
resonant case (when $\Delta=\omega_{f}-\omega_{s}=0$ and hence
$\gamma_{f}=\gamma_{s}$) is the coupling rate $\gamma_{f}$. For having $M_{0}$
equal or larger than $4$ or greater, we need the $\gamma_{f}$ to be at least
$2(2+\sqrt{3})$. For non-resonant case for fixed $\gamma_{f}$ decreasing the
value $\gamma_{s}$ will decrease $M_{0}$. Figure 2 shows the behavior of the
minimal possible dimension $D_{\rm min}=K_{0}^{*}-1$ of the error correcting
subspace ${\cal H}_{3}$ vs coupling rates $\gamma_{s}$ and $\gamma_{f}$. The
figure clearly shows that the behavior is non-symmetric with respect to
$\gamma_{s}$ and $\gamma_{f}$, as is also evident from inequality (26). The
resonant case is the extremal case in the inequality (26), what means if
$M_{0}$ satisfies it for some $\gamma_{f}$, then for the same parameter
$\gamma_{f}$ the number $M_{0}$ also solves the inequality in the resonant
case. Figure 3 shows the dependence of $D_{\rm min}$ on the coupling rate
$\gamma_{f}$ for equal frequencies.

Figure 2:  Behavior of the minimal dimension of the error correcting subspace
vs coupling rates of the Jaynes-Cummings Hamiltonian. Figure 3:  Behavior of
the minimal dimension of the error correcting subspace for the resonant case
$\Delta=0$ vs coupling rate of the field.

The Jaynes-Cummings Hamiltonian is used in various theoretical and
experimental analysis in quantum optics, cavity QED, e.g. [41, 34, 35, 37, 40,
36, 38, 39, 32, 33, 31, 43, 44, 42], including in strong [32], ultra-strong
[34, 35, 36] and deep strong coupling regimes [34, 35, 37, 36, 38, 39]. We
take for example of the weak coupling regime the parameters used in
experiments performed in the group of Serge Haroche [31]. In the setup of that
experiments the cavity is designed to have the frequency equal to the
frequency of the atom, i.e. $\Delta=0$. The approximate parameters for the
experiment are $\kappa=2\pi\cdot 47$ KHz and $\omega_{f}=\omega_{s}=2\pi\cdot
51.1$ GHz. For this case the inequality (11) becomes

$\sqrt{M_{0}+1}+\sqrt{M_{0}}>\frac{\kappa}{2\omega_{f}}=\frac{47}{2\cdot
51.1\cdot 10^{6}}\approx 0.46\cdot 10^{-6}$ (27)

The minimal natural solution of this inequality is $M_{0}=1$, so $K_{0}=3$ and
the minimal dimension is $D_{\rm min}=K_{0}-1=2$. This minimal two-dimensional
error correcting subspace is spanned by the two vectors $\ket{g,0}$ and
$\ket{1,-}=\ket{1,g}-\ket{0,e}$. We remark that this is the error correcting
subspace of minimal dimension. One could choose arbitrary large $K_{0}$ and
the corresponding non-commutative operator graph will have the error
correcting subspace $H_{3}$ of dimension $K_{0}-1$.

Jaynes-Cummings model is derived from Rabi model via the rotating wave
approximation (RWA). This approximation is typically valid for
$\gamma_{f}<0.1$ and $\omega_{f}\approx\omega_{s}$. In this case the minimal
dimension is $D_{\rm min}=2$. As it can be seen from Fig. 2 and Fig. 3, to
have $D_{\rm min}>2$ one has to consider values of $\gamma_{f}$ in the range
of deep strong coupling regime [34, 35, 37]. This regime, as well as less
intense ultrastrong regime, is of interest now. In these regimes the Rabi
model is non-integrable and investigating these regimes motivates describing
eigen-energies approximations for this model [37, 40]. One can show [36] that
introducing a special type of frequency modulations applied to the field and
the qubit will give the dynamics governed by Jaynes-Cummings Hamiltonian with
$\gamma_{f}$ in the range of deep strong coupling regime. In circuit-QED
simulations rates of $\gamma_{f}$ for Rabi model up to 2.1 are achieved [38,
39]. Thus value is lower, while not that much, than the minimal
$\gamma_{f}\approx 7.5$ that is necessary to see the effect in which minimal
dimension of the error correcting code in the proposed scheme will be 3 or
greater. Our analysis allows to construct non-trivial quantum error correcting
codes for all possible values $\gamma_{f},\gamma_{s}$.

We remark that our scheme could be applied to any system that possess the same
decomposition into the direct sum, where eigenenergies in the two direct
summands form strictly increasing sequences. Potentially, this property could
be exploited for more complex Hamiltonian beyound the Jaynes-Cummings model,
as for example for Jaynes-Cummings-Hubbard Hamiltonian, describing interaction
of several qubit-cavity systems, or for perhaps directly for Rabi Hamiltonian.

##  VII Conclusion

In this work, the theory of non-commutative operator graphs has been developed
for error correction in the case of a finite-dimensional quantum system
coupled to an infinite-dimensional quantum system. We have constructed the
non-commutative operator graph generated by orbits of the unitary group driven
by Hamiltonian (8) of the Jaynes-Cummings model. We have shown that for a
positive integer $K_{0}$ that satisfies (25), using for encoding the
eigenstates $\ket{g,0}$ with $\ket{n,-}$ for $0\leq n<K_{0}$ (III) allows to
transmit information with zero error via quantum channels with operator graphs
belonging to the constructed graph. Thus the error correcting subspace is
explicitly computed for all values of the parameters of the Jaynes-Cummings
model. Our scheme could be applied to any system that possess the same
decomposition of eigenenergies into the direct sum as for JC Hamiltonian,
where eigenenergies in the two direct summands form strictly increasing
sequences.

###### Acknowledgements.

This work was funded by the Ministry of Science and Higher Education of the
Russian Federation (grant number 075-15-2020-788) and performed at the Steklov
Mathematical Institute of the Russian Academy of Sciences.

## References

  * Shor [1995] P. Shor, Phys. Rev. A 52, R2493 (1995).
  * Ekert and Macchiavello [1996] A. Ekert and C. Macchiavello, Phys. Rev. Lett. 77, 2585 (1996).
  * Knill and Laflamme [1997] E. Knill and R. Laflamme, Phys. Rev. A 55, 900 (1997).
  * Cory  _et al._ [1998] D. G. Cory, M. D. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo, Phys. Rev. Lett. 81, 2152 (1998).
  * Mancini and Winter [2020] S. Mancini and A. Winter,  _A Quantum Leap in Information Theory_ (World Scientific Publishing Co., Singapore 2020).
  * Lloyd and Slotine [1998] S. Lloyd and J. J. Slotine, Phys. Rev. Lett. 80, 4088 (1998).
  * Lieu  _et al._ [2020] S. Lieu, R. Belyansky, J. T. Young, R. Lundgren, V. V. Albert, and A. V. Gorshkov, Phys. Rev. Lett. 125, 240405 (2020).
  * Cochrane  _et al._ [1999] P. T. Cochrane, G. J. Milburn, and W. J. Munro, Phys. Rev. A 59, 2631 (1999).
  * Gottesman  _et al._ [2001] D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A 64, 012310 (2001).
  * [10] V. V. Albert, K. Noh, K. Duivenvoorden, D. J. Young, R. T. Brierley, P. Reinhold, C. Vuillot, L. Li, C. Shen, S. M. Girvin, B. M. Terhal, and L. Jiang, Phys. Rev. A 97, 032346 (2018).
  * Terhal  _et al._ [2020] B. M. Terhal, J. Conrad, and C. Vuillot, Quantum Science and Technology 5, 043001 (2020).
  * Albert  _et al._ [2020] V. V. Albert, J. P. Covey, and J. Preskill, Phys. Rev. X. 10, 031050 (2020).
  * Knill  _et al._ [2000] E. Knill, R. Laflamme, and L. Viola, Phys. Rev. Lett. 84, 2525 (2000).
  * Duan  _et al._ [2013] R. Duan, S. Severini, and A. Winter, IEEE Trans. Inf. Theory 59, 1164–1174 (2013).
  * Duan [2009] R. Duan (2009), arxiv.org/abs/0906.2527.
  * Shirokov and Shulman [2015] M. E. Shirokov and T. Shulman, Commun. Math. Phys. 335, 1159–1179 (2015).
  * Yashin [2020] V. I. Yashin, Quantum Information Processing 19, 195 (2020).
  * Amosov and Mokeev [2018a] G. G. Amosov and A. S. Mokeev, Int. J. Theor. Phys. (2018a), https://doi.org/10.1007/s10773-018-3963-4.
  * Amosov and Mokeev [2018b] G. G. Amosov and A. S. Mokeev, Quantum Information Processing 17, 325 (2018b).
  * Amosov and Mokeev [2013] G. G. Amosov and A. S. Mokeev, Lobachevskii Journal of Mathematics 41, 592–596 (2020).
  * Amosov  _et al._ [2020] G. G. Amosov, A. S. Mokeev, and A. N. Pechen, Quantum Information Processing 19, 95 (2020).
  * Kyaw  _et al._ [2015] T. H. Kyaw, S. Herrera-Martí, E. Solano, G. Romero, and L. C. Kwek, Phys. Rev. B 91, 064503 (2015).
  * Chunfeng Wu  _et al._ [2019] Chunfeng Wu, Yimin Wang, Chu Guo, Yingkai Ouyang, Gangcheng Wang, and Xun-Li Feng, Phys. Rev. A 99, 012335 (2019).
  * Fasihi  _et al._ [2019] M. A. Fasihi and B. Mojaveri, Quantum Information Processing 18, 75 (2019).
  * Jaynes and Cummings [1963] E. T. Jaynes and F. W. Cummings, Proc. of the IEEE 51, 89–109 (1963).
  * Gazeau and Klauder [1999] J. P. Gazeau and J. R. Klauder, Journal of Physics A 32, 123 (1999).
  * Holevo [2012] A. S. Holevo,  _Quantum System, Channels, Information_ (Walter De Gruyter, Berlin, 2012).
  * Solomon  _et al._ [2011] I. J. Solomon, K. Sabapathy, and R. Simon, Phys. Rev. A 84, 022341 (2011).
  * Choi and Effros [1977] M. Choi and E. Effros, J. Funct. Anal. 24, 156–209 (1977).
  * Daoud and Hussin [2002] M. Daoud and V. Hussin, J. Phys. A 35, 7381 (2002).
  * Brune  _et al._ [1996] M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 76, 1800 (1996).
  * Cirac  _et al._ [1993] J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, Phys. Rev. Lett. 70, 762 (1993).
  * Cirac  _et al._ [1994] J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, Phys. Rev. A. 49, 1202 (1994).
  * Forn-Díaz  _et al._ [2019] P. Forn-Díaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Rev. Mod. Phys. 91, 025005 (2019).
  * Kockum  _et al._ [2019] A. F. Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, Nat. Rev. Phys. 1, 19–40 (2019).
  * Huang  _et al._ [2010] Jin-Feng Huang, Jie-Qiao Liao, and Le-Man Kuang, Phys. Rev. A. 101, 043835 (2020).
  * Casanova  _et al._ [2010] J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, Phys. Rev. Lett. 105, 263603 (2010).
  * Yoshihara  _et al._ [2017] F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and K. Semba, Nat. Phys. 13, 44–47 (2017).
  * Langford  _et al._ [2017] N. K. Langford, R. Sagastizabal, M. Kounalakis, C. Dickel, A. Bruno, F. Luthi, D. J. Thoen, A. Endo, and L. DiCarlo, Nat. Commun. 8, 1715 (2017).
  * Irish [2007] E. K. Irish, Phys. Rev. Lett. 99, 173601 (2007).
  * Rempe  _et al._ [1987] G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
  * Saffman  _et al._ [2010] M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod. Phys. 82, 2313 (2010).
  * Blais  _et al._ [2004] A. Blais, Ren-Shou Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A. 69, 062320 (2004).
  * Larson [2007] J. Larson, Phys. Scr. 76, 146 (2007).