# Generating Hermite polynomial excited squeezed states by means of
conditional measurements on a beam splitter

Xue-xiang Xu1,†, Hong-chun Yuan2 and Hong-yi Fan3 1College of Physics and
Communication Electronics, Jiangxi Normal University, Nanchang 330022, China
2College of Optoelectronic Engineering, Changzhou Institute of Technology,
Changzhou 213002, China
3Department of Material Science and Engineer, University of Science and
Technology of China, Hefei 230026, China
†Corresponding author<EMAIL_ADDRESS>
###### Abstract

A scheme for conditional generating a Hermite polynomial excited squeezed
vacuum states (HESVS) is proposed. Injecting a two-mode squeezed vacuum state
(TMSVS) into a beam splitter (BS) and counting the photons in one of the
output channels, the conditional state in the other output channel is just a
HESVS. To exhibit a number of nonclassical effects and non-Guassianity, we
mainly investigate the photon number distribution, sub-Poissonian
distribution, quadrature component distribution, and quasi-probability
distribution of the HPESVS. We find that its nonclassicality closely relates
to the control parameter of the BS, the squeezed parameter of the TMSVS, and
the photon number of conditional measurement. These further demonstrate that
performing the conditional measurement on a BS is an effective approach to
generate non-Guassian state.

ocis:  (270.5570) Quantum detectors; (270.4180) Multiphoton processes;
(270.5290) Photon statistics

Keywords:Conditional measurement; beam splitter; Wigner function;
Nonclassicality

##  I Introduction

Quantum state engineering has been a subject of increasing interest to
construct various novel nonclassical states in quantum optics and quantum
information processinga1 ; a2 . From a theoretical point of view, the simplest
way of generating nonclassical field states is to apply the photon creation
operation to classical states such as the thermal and coherent statesa3 ; a4 ;
a5 . These nonclassical states, such as the single-photon added coherent
statea6  and single-photon-added thermal statea7 , have been realized
experimentally. Subsequently, it has been demonstrated that subtracting
photons from traditional quantum states exhibit an abundance of nonclassical
propertiesa8 ; a9 ; a10 ; a11 . Photon subtraction or addition can improve
entanglement between Guassian statesa12 , loophole-free tests of Bell’s
inequalitya13 , and quantum computinga14 .

To meet the requirement of the development of quantum optics and quantum
information tasks, some nonclassical states are explored by performing the
different combination of photon subtraction and photon additiona15 ; a16 ; a17
; a18 ; a19 ; a20 , which have different properties. Kim et ala21  discussed
single photon adding then subtracting (or single photon subtracting then
adding) coherent state (or thermal state) to probe quantum commutation rules
$\left[a,a^{{\dagger}}\right]=1$. Lee et ala17  investigated the
nonclassicality of field states when photon subtraction-then-addition
operation or the photon addition-then-subtraction operation is applied to the
coherent state (or thermal state), respectively. Yang and Lia18  analyzed
multiphoton addition followed by multiphoton subtraction
($a^{l}a^{{\dagger}k}$) and its inverse ($a^{{\dagger}l}a^{k}$) on an
arbitrary state. Recently, Lee and Nhaa23  proposed a coherent superposition
of photon addition and subtraction, $ta+ra^{{\dagger}}$
($\left|t\right|^{2}+\left|r\right|^{2}=1$) acting on a coherent state and a
thermal state. More recently, we investigated the nonclassical properties of
optical fields generated by Hermite-excited coherent statea24  and Hermite-
excited squeezed thermal statesa25 .

On the other hand, another promising method for generating highly nonclassical
states of optical fields is known to be conditional measurementa25a ; a26 ;
a27 ; a28 ; a28a . Namely, when a system is prepared in an entangled state of
two subsystems and a measurement is performed on one subsystem, then the
quantum state of the other subsystem can be reduced to a new state. In
particular, it turned out that conditional measurement on a beam splitter may
be advantageously used for generating new classes of quantum statesa28 ; a28a
. Dakna’s group used conditional measurement on the BS to generate cat-like
statea29 . Podoshvedov et ala27  proposed optical scheme for generating both a
displaced photon and a displaced qubit via conditional measurement. In Ref.a30
, they proposed to create arbitrary Fock states via conditional measurement on
the BS. In addition, conditional output measurement on the BS may be used to
produce photon-added states for a large class of signal-mode quantum states,
such as thermal state, coherent state, and squeezed statesa31 . Similarly,
photon-subtracted states can be produced by means of conditional measurement
on the BSa32 . Therefore, based on conditional measurement on the BS, it is
possible to generate and manipulate various nonclassical optical fields in a
real laboratory.

In this paper, we study the Hermite polynomial excited squeezed vacuum state
(HESVS), a kind non-Gaussian quantum state, generated by conditional output
measurement on a BS. The calculations show that when a two-mode squeezed
vacuum state (TMSVS) is injected in the input channels and the photon number
of the mode in one of the output channels is measured, then the mode in the
other output channel is prepared in a conditional state that has the typical
features of a Hermite polynomial excited squeezed state. To exhibit the
nonclassical properties of this conditional state, we mainly analyze the
states in terms of the photon number distribution, sub-Poissonian
distribution, quadrature component distribution, and Quasi-probability
distribution including the Wigner function(WF) and Husimi function(HF). The
paper is organized as follows. Section 2 presents the basic scheme for
generation of the HESVS and its normalization related to Legendre polynomial.
The nonclassical properties of the HESVS are analytically and numerically
studied in Section 3-4. The results indicate that the conditional HESVS is
strongly noncassical and non-Gaussian due to the presence of the partial
negative WF. Finally, a summary and concluding remarks are given in Section 5.

##  II Generation of Hermite polynomial excited squeezed state

It is well known that the input-output relations at a lossless beam splitter
can be characterized by the SU(2) Lie algebra. In the Schrödinger picture, the
role played by the beam splitter (BS) upon the input state $\rho_{in}$ results
in the output state

$\rho_{out}=\hat{B}\rho_{in}\hat{B}^{{\dagger}},$ (1)

where
$\hat{B}=\exp\left[\theta\left(a^{{\dagger}}b-ab^{{\dagger}}\right)\right]$
corresponds to the unitary operator in terms of the creation (annihilation)
operator $a^{{\dagger}}$($a$) and $b^{{\dagger}}$($b$) for mode $a$ and $b$,
whose transformations satisfya30

$\displaystyle\hat{B}a\hat{B}^{{\dagger}}$
$\displaystyle=a\cos\theta-b\sin\theta,$
$\displaystyle\hat{B}b\hat{B}^{{\dagger}}$
$\displaystyle=a\sin\theta+b\cos\theta.$ (2)

Moreover, $\cos\theta$ and $\sin\theta$ are the transmittance and reflectance
of the beam splitter, respectively. Note that the globe phase factor of BS may
be omitted without loss of generality. For the sake of simplicity, we also
assume that $\theta$ is tunable in the range of $\left[0,\pi/2\right]$. Under
special circumstances, when $\theta=0$ or $\theta=\pi/2$, the BS corresponds
to the cases of total transmission and total reflection, respectively. For
$\theta=\pi/4$, the BS is just the symmetrical, i.e. 50/50 BS.

###  II.1 Hermite polynomial excited squeezed state

A two-mode squeezed vacuum state (TMSVS) is the correlated state of two field
modes $a$ and $b$ (signal and idle) that can be generated by a nonlinear
medium. Theoretically, the TMSVS is obtained by applying the unitary operator
$S_{2}\left(r\right)$ on the two-mode vacuum,

$\left|\Psi\right\rangle_{ab}=S_{2}\left(r\right)\left|0,0\right\rangle=\cosh^{-1}re^{a^{{\dagger}}b^{{\dagger}}\tanh
r}\left|0,0\right\rangle,$ (3)

where
$S_{2}\left(r\right)=\exp\left[r\left(a^{{\dagger}}b^{{\dagger}}-ab\right)\right]$
is the two-mode squeezed operator and the values of $r$ determines the degree
of squeezing. The larger $r$, the more the state is squeezed. Especially, when
$r=0$, $\left|\Psi\right\rangle_{ab}$ reduces to two-mode vacuum state
$\left|0,0\right\rangle$.

Figure 1: Preparation scheme of HPESVS. When a TMSVS is mixed by a beam
splitter and the number of photons $\left|m\right\rangle$is measured in one of
the output channels, then the conditional quantum state in the other output
channel is generated.

The conceptual scheme of the experimental setup is depicted in Fig.1. The two
input modes prepared in the two-mode squeezed state
($\rho_{in}=\left|\Psi\right\rangle_{ab}\left\langle\Psi\right|$) is mixed at
BS, so the output-state density operator can be given by
$\rho_{out}=\hat{B}\left|\Psi\right\rangle_{ab}\left\langle\Psi\right|\hat{B}^{{\dagger}}.$
In fact the output modes in $\rho_{out}$ are generally highly correlated. When
the photon number of the mode in the second output channel is measured and $m$
photons are detected, then the mode in the first output channel is prepared in
a quantum state, whose density operator $\rho_{out}^{a}$ reads as

$\rho_{out}^{a}=N_{m}^{-1}\left.{}_{b}\left\langle
m\right|\hat{B}\left|\Psi\right\rangle_{ab}\left\langle\Psi\right|\hat{B}^{{\dagger}}\left|m\right\rangle_{b}\right.=\left|\Psi_{m}\right\rangle\left\langle\Psi_{m}\right|,$
(4)

where $\left|\Psi_{m}\right\rangle$ is the normalized output conditional state
(a pure state) and $N_{m}$ is the normalization factor determined by
$\mathrm{Tr}\left(\rho_{out}^{a}\right)=1$.

Next, using the integration of the TMSVSa33 ,

$\left|\Psi\right\rangle_{ab}=\frac{1}{\sinh
r}\int\frac{d^{2}\alpha}{\pi}e^{-\left|\alpha\right|^{2}/\tanh r+\alpha
a^{{\dagger}}+\alpha^{\ast}b^{{\dagger}}}\left|0_{a},0_{b}\right\rangle,$ (5)

where
$\left|0_{a},0_{b}\right\rangle=\left|0\right\rangle_{a}\otimes\left|0\right\rangle_{b}$
is two-mode vacuum state, and the transformation relation in Eq.(2), after
some algebra we derive that

${}_{b}\left\langle m\right|\hat{B}\left|\Psi\right\rangle_{ab}$
$\displaystyle=\frac{1}{\sinh
r}\int\frac{d^{2}\alpha}{\pi}e^{-\left|\alpha\right|^{2}/\tanh
r}e^{\left(\alpha\cos\theta+\alpha^{\ast}\sin\theta\right)a^{{\dagger}}}\left|0\right\rangle_{a}$
$\displaystyle\times\frac{1}{\sqrt{m!}}\frac{\partial^{m}}{\partial\tau^{m}}e^{\tau\left(\alpha^{\ast}\cos\theta-\alpha\sin\theta\right)}|_{\tau=0}$
$\displaystyle=\frac{1}{\cosh
r\sqrt{m!}}\frac{\partial^{m}}{\partial\tau^{m}}e^{\frac{\mu}{2}a^{{\dagger}2}-\frac{\mu}{2}\tau^{2}+\tau\nu
a^{{\dagger}}}\left|0\right\rangle_{a}|_{\tau=0},$ (6)

where $\mu=\sin 2\theta\tanh r$ and $\nu=\cos 2\theta\tanh r$. Hence the
output conditional state $\left|\Psi_{m}\right\rangle$ is explicitly expressed
as

$\left|\Psi_{m}\right\rangle=\Omega_{m}^{1/2}H_{m}\left(\frac{\nu
a^{{\dagger}}}{\sqrt{2\mu}}\right)S_{1}\left(\lambda\right)\left|0\right\rangle$
(7)

with $\Omega_{m}=\mu^{m}\cosh\lambda/(2^{m}m!N_{m}\cosh^{2}r)$, where we have
used the generating function of the single-variable $m$-order Hermite
polynomial
$H_{m}\left(x\right)=\partial_{\tau}^{m}e^{2x\tau-\tau^{2}}|_{\tau=0}\ $and
the expression of single-mode squeezed vacuum
$S_{1}\left(\lambda\right)\left|0\right\rangle=\cosh^{-1/2}\lambda
e^{\left(\tanh\lambda/2\right)a^{{\dagger}2}}\left|0\right\rangle$ with the
single-mode squeezed operator
$S_{1}\left(\lambda\right)=\exp\left[\lambda\left(a^{{\dagger}2}-a^{2}\right)/2\right]$.
Eq.(7) indicates that the conditional state $\left|\Psi_{m}\right\rangle$ is
actually a single-mode $m$-order Hermite polynomial excited squeezed vacuum
state. It is worth noticing that the degree of squeezing $\lambda$ of the
conditional state is not the same squeezing parameter $r$ of the TMSVS but
related to the parameters of the TMSVS and the BS satisfying
$\tanh\lambda=\sin 2\theta\tanh r$. For the symmetrical case, i.e.,
$\theta=\pi/4$,
$\hat{B}\left|\Psi\right\rangle_{ab}|_{\theta=\pi/4}=S_{1a}\left(r\right)\left|0\right\rangle\otimes
S_{1b}\left(-r\right)\left|0\right\rangle$ is just the product state of two
separate single-mode squeezed vacuum state. In this case, when we detect $m$
photons in the second output channel, the conditional state is always
$S_{1}\left(r\right)\left|0\right\rangle$ with the same squeezed parameter
$r$. This is that the effect of the symmetrical BS splits the entangled TMSVS
into two independent single-mode squeezed vacuum state. Note that when no
photons are detected, $m=0$, then $\left|\Psi_{m}\right\rangle$ also reduces
to $S_{1}\left(\lambda\right)\left|0\right\rangle$.

###  II.2 Normalization via probability of such event

In addition, the normalization factor $N_{m}$ is determined by the probability
$p\left(m\right)$ of such an event given by

$\displaystyle N_{m}$ $\displaystyle=$ $\displaystyle p\left(m\right)$ (8)
$\displaystyle=$ $\displaystyle\mathrm{Tr}\left(\left.{}_{b}\left\langle
m\right|\hat{B}\left|\Psi\right\rangle_{ab}\left\langle\Psi\right|\hat{B}^{{\dagger}}\left|m\right\rangle_{b}\right.\right)$
$\displaystyle=$
$\displaystyle\frac{1}{m!\sqrt{A}\cosh^{2}r}\allowbreak\frac{\partial^{2m}}{\partial
s^{m}\partial\tau^{m}}e^{-B_{1}s^{2}/2-B_{1}\tau^{2}/2+\allowbreak
B_{2}s\tau}\allowbreak|_{s=\tau=0}$ $\displaystyle=$
$\displaystyle\frac{\left(-\sqrt{B_{3}}\right)^{m}}{\cosh^{2}r\sqrt{A}}P_{m}\left(\sqrt{B_{4}}\right)$

where we have set $A=1-\mu^{2}$, $B_{1}=\mu/\left(A\cosh^{2}r\right)$,
$B_{2}=\nu^{2}/A$, $B_{3}=\left(\tanh^{4}r-\mu^{2}\right)/A$,
$B_{4}=\nu^{4}/\left(A^{2}B_{3}\right)$, and in the last step we have used the
formula of $m$-order Legendre polynomial P${}_{m}\left(x\right)$, i.e.,

$\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}e^{-t^{2}-\tau^{2}+\frac{2x}{\sqrt{x^{2}-1}}\tau
t}|_{t,\tau=0}=\frac{2^{m}m!}{\left(x^{2}-1\right)^{m/2}}\text{P}_{m}\left(x\right).$
(9)

Especially, when $r=0$,
$\left|\Psi\right\rangle\rightarrow\left|0,0\right\rangle$, there is no
photons in the output channels. In this case, there is no necessary to making
conditional measurement. So the event is happen only for $r\neq 0$. When
$\theta=0$ or $\theta=\pi/2$, leading to $\mu=0$ and $\nu=\tanh r$ or
$\nu=-\tanh r$ then $A=1$, $B_{1}=0$, $B_{2}=\tanh^{2}r$, $B_{3}=\tanh^{4}r$,
and $B_{4}=1$, $N_{m}|_{\theta=0or\theta=\pi/2}=\tanh^{2m}r/\cosh^{2}r$ and
$\left|\Psi_{m}\right\rangle$ is just the Fock $\left|m\right\rangle$, which
is rational because of the inherent properties of the TMSVS. If
$\theta=\pi/4$, the BS is just the symmetrical, i.e., 50/50 BS, leading to
$\mu=\tanh r,$ $\nu=0$ then $A=1-\tanh^{2}r$, $B_{1}=-\frac{\tanh r}{2}$,
$B_{2}=0$, then the output states is the the product of two independent
single-mode SVS.

According to Eq.(8), we discuss the probability of observing such a
conditional $m$-order HPESVS. In Fig.2 the probability $p\left(m\right)$ is
plotted for two parameter values of the BS. For a given transmittance of BS,
$p\left(m\right)$ as a function of the input squeezing parameter $r$ can
attain a maximum and the maximum is shifted towards larger values of $r$ when
$m$ is increased (see Fig.2a). In this figure, for each $r$, we should use the
transmittance of the BS in a way to optimize the success probability. The
ideal procedure would use the appropriate transmittance for each value of $r$
and $m$. By tuning the parameters of the interaction, namely, the control
parameter of the BS, the squeezed parameter $r$ of the TMSVS, and the photon
number of conditional measurement $m$, the HPESVS may be modulated, generating
a wide range of nonclassical phenomena, as described below.

Figure 2: The probability of producing $\left|\Psi_{m}\right\rangle$ is shown
as a function of the parameter $r$ of the TMSVS for two parameter values of
the BS [(a) $\theta=\pi/7$; (b) $\theta=2\pi/7$] and various values of $m$,
where $m=1,2,3,4$correspond to the solid, dashed, dotted and dotdashed lines,
respectively.

##  III Observable nonclassical effects of the conditional HPESVS

To study the nonclassical properties of the conditional states in more detail,
we shall calculate the photon number distribution, sub-Poissonian
distribution, and quadrature component distribution.

###  III.1 Photon number distribution

The photon number distribution (PND), the probability of finding $n$ photons,
is a key characteristic of every quantum state. Recalling Eq.(7), the PND of
the conditional HPESVS reads as

$P\left(n|m\right)=\left|\left\langle
n\right.\left|\Psi_{m}\right\rangle\right|^{2}.$ (10)

Using the unnormalized coherent
$\left|z\right\rangle=\exp[za^{{\dagger}}]\left|0\right\rangle$, leading to
$\left|n\right\rangle=\frac{1}{\sqrt{n!}}\frac{\partial^{n}}{\partial
z^{n}}\left|z\right\rangle|_{z=0}$, and combining with Eq.(7), we finally
obtain

$\displaystyle P\left(n|m\right)$
$\displaystyle=\frac{1}{m!n!N_{m}\cosh^{2}r}\allowbreak\left|\frac{\partial^{2m}}{\partial\tau^{m}\partial
s^{n}}e^{\frac{\mu}{2}s^{2}-\frac{\mu}{2}\tau^{2}+\nu
s\tau}|_{s=\tau=0}\right|^{2}$
$\displaystyle=\frac{m!n!}{N_{m}\cosh^{2}r}\allowbreak\left|\sum_{g=0}^{\min[m,n]}\frac{\left(-1\right)^{\frac{m-g}{2}}\left(\allowbreak\frac{\mu}{2}\right)^{\frac{m+n-2g}{2}}\nu^{g}}{\left(\frac{n-g}{2}\right)!\left(\frac{m-g}{2}\right)!g!}\right|^{2},$
(11)

where we have used $\frac{\partial^{m}}{\partial
x^{m}}x^{n}|_{x=0}=m!\delta_{mn}$ ($\delta_{mn}$ is Krocher function) and
$N_{m}$ is given in Eq.(8). In the summation of Eq.(11), the value of $g$ must
make $\frac{n-g}{2}$ and $\frac{m-g}{2}$ be integer. To see clearly the
variation of the PND, in Fig.3 we plot the bar graph of the PND for the
conditional HPESVS with different values of parameters $m$, $\theta$, and $r$.
From Fig.3 we easily see that when the number $m$ of the conditional
measurement is odd (even), then the photon-number distribution is nonzero only
for odd (even) photon numbers. The probability $P\left(n|m\right)$ with
different parity between $m$ and $n$ is zero. For given $m$ and $\theta$, the
bigger the squeezing parameter $r$, the wider the distribution (see Figs.3(a)
and 3(b)).

Figure 3: Photon-number distribution of the conditional HPESVS for (a)
$m=1,\theta=2\pi/7,r=0.5$; (b) $m=1,\theta=2\pi/7,r=1.0$; (c)
$m=4,\theta=2\pi/7,r=1.0$; (d) $m=4,\theta=3\pi/7,r=1.0$, respectively.

###  III.2 Sub-Poissonian distribution

In order to study the photon-number statistics of this conditional state, we
first calculate $\left\langle
a^{k}a^{{\dagger}l}\right\rangle=\left\langle\Psi_{m}\right|a^{k}a^{{\dagger}l}\left|\Psi_{m}\right\rangle$.
Using the completeness of coherent state
$\int\frac{d^{2}\alpha}{\pi}\left|\alpha\right\rangle\left\langle\alpha\right|=1$
as well as $y^{k}=\frac{\partial^{k}}{\partial t^{k}}e^{ty}|_{t=0}$ yields

$\displaystyle\left\langle a^{k}a^{{\dagger}l}\right\rangle$
$\displaystyle=\frac{1}{N_{m}m!\cosh^{2}r\sqrt{A}}\frac{\partial^{2m}}{\partial
s^{m}\partial\tau^{m}}e^{-B_{1}s^{2}/2-B_{1}\tau^{2}/2+B_{2}\tau s}$
$\displaystyle\times\frac{\partial^{k+l}}{\partial x^{k}\partial
y^{l}}e^{\left[\mu\nu\left(xs+y\tau\right)+\nu\left(ys+x\tau\right)+\mu\left(x^{2}+y^{2}\right)/2+xy\right]/A}|_{s=\tau=x=y=0}.$
(12)

Thus, the mean photon number

$\left\langle n\right\rangle=\left\langle
a^{{\dagger}}a\right\rangle=\left\langle aa^{{\dagger}}\right\rangle-1,$ (13)

can be determined by Eq.(12) with $k=l=1$. Examples are shown in Figs.4(a) and
4(c). We see that when $\theta=\pi/5$, the number of photons that can be found
in $\left|\Psi_{m}\right\rangle$ increases with $m$ for the given larger $r$,
and $\left\langle n\right\rangle$ as a function of $\theta$ is symmetric
distribution for $\theta=\pi/4$. This is simply a consequence of the BS
transformation. In particular, when no photons are detected, $m=0$, then
$\left\langle n\right\rangle=\sinh^{2}r$ reduces to the mean photon number of
single-mode squeezed vacuum state.

A measure of the deviation of the photon number distribution from a Poissonian
is the Mandel $Q$ factor defined bya34

$\displaystyle Q$ $\displaystyle=\frac{\left\langle
n^{2}\right\rangle-\left\langle n\right\rangle^{2}}{\left\langle
n\right\rangle}-1$ $\displaystyle=\frac{\left\langle a^{2}a^{\dagger
2}\right\rangle-\left\langle aa^{\dagger}\right\rangle^{2}-2\left\langle
aa^{\dagger}\right\rangle+1}{\left\langle aa^{\dagger}\right\rangle-1},$ (14)

It holds that $Q\geqslant 0$ and the equality is achieved for the Fock state.
The light is sub-Poissonian when the photon-number variance $\left\langle
n^{2}\right\rangle-\left\langle n\right\rangle^{2}$ is less than $\left\langle
n\right\rangle$. This is indicated by a negative value of $Q$. The statistics
are Poissonian when $Q=0$, and super- (sub-) Poissonian if $Q>0$ ($Q<0$).

Figure 4: (a) Mean photon number $\left\langle n\right\rangle$ and (b) Mandel
${\small Q}$ factor versus $r$ for different $m=1$(solid line), $m=2$(dashed
line), $m=3$(dotted line), and $m=4$(dotdashed line) with the same
$\theta=\pi/5$. (c) Mean photon number $\left\langle n\right\rangle$ and (d)
Mandel Q factor versus $\theta$ for different $m=1$(solid line), $m=2$(dashed
line), $m=3$(dotted line), and $m=4$(dotdashed line) with the same $r=0.5$.
Figure 5: The sub-Poissonian properties of the conditional state with
$Q=0$(solid line), $Q=-0.2$(dashed line), $Q=-0.5$(dotted line), and
$Q=-0.8$(dotdashed line) in the plane space of two parameters ($\theta$and
$r$) with different number of conditional measurement: (a) $m=1$; (b) $m=2$;
(c) $m=3$; (d) $m=4$, respectively.

According to Eqs.(12) and (14), we plot the variation of $Q$ for HPESVS versus
$r$ or $\theta$ for different $m=1,2,3,4$ in Fig.4(b) and 4(d). It is clearly
seen that $Q$ as a function of $\theta$ is also symmetric distribution for
$\theta=\pi/4$ and the HPESVS has sub-Poissonian statistics behavior due to
the emergence of the negativity of $Q$. With the increasing values of $m$, the
increasing intrend of $Q$ is accelerated for larger $r.$ To further exhibit
the high nonclassicality, Fig.5 shows the dependence of this conditional state
on $\theta$ and $r$ for four different $Q$ factors. Especially, we consider
first the boundary case of the Poissonian distribution, $Q=0$ (see the solid
line in Fig.5).

###  III.3 Quadrature component distribution

Next we pay attention to the conditional quadrature component distribution
(QCD)a28a

$P\left(x,\varphi|m\right)=\left|\left\langle
x,\varphi\right.\left|\Psi_{m}\right\rangle\right|^{2},$ (15)

which can be measured in balanced homodyne detection. Here
$\left|x,\varphi\right\rangle$ is the eigenstate of the quadrature component
$X\left(\varphi\right)=\left(ae^{-i\varphi}+a^{{\dagger}}e^{i\varphi}\right)$,
expressed as in the Fock basis

$\left|x,\varphi\right\rangle=\pi^{-1/4}e^{-\frac{x^{2}}{2}+\sqrt{2}xa^{{\dagger}}e^{i\varphi}-\frac{a^{{\dagger}2}e^{2i\varphi}}{2}}\left|0\right\rangle.$
(16)

Using Eqs.(7) and (16) and inserting the completeness of coherent state
$\int\frac{d^{2}\alpha}{\pi}\left|\alpha\right\rangle\left\langle\alpha\right|=1$,
after integration, the wave function $\left\langle
x,\varphi\right.\left|\Psi_{m}\right\rangle$ reads

$\left\langle
x,\varphi\right.\left|\Psi_{m}\right\rangle=\frac{\pi^{-1/4}\left(\sqrt{\Gamma/2}\right)^{m}e^{-\frac{\Pi}{2}x^{2}}}{\sqrt{N_{m}m!\left(1+\mu
e^{-2i\varphi}\right)}\cosh
r}H_{m}\left(\frac{\Delta}{\sqrt{\Gamma}}x\right),$ (17)

where we have set $\Theta=1+\mu e^{-2i\varphi}$, $\Pi=\left(1-\mu
e^{-2i\varphi}\right)/\Theta$,
$\Gamma=\left(\mu+e^{-2i\varphi}\tanh^{2}r\right)/\Theta$, and
$\Delta=e^{-i\varphi}\nu/\Theta$.

As a result of Eq.(17), we easily obtain the conditional QCD defined by
Eq.(15) and plot the variation of $P\left(x,\varphi|m\right)$ for the HPESVS
as a function of $x$ or $\varphi$ for different $m=1,2,3,4$ in Fig.6. Ones see
that for $\varphi$ near $\pi/2$ the QCD $P\left(x,\varphi|m\right)$ with $m>0$
exhibits two separated peaks, wheresas for $\varphi$ close to $0$ and $\pi$ an
interference pattern is observed.

Figure 6: Quadrature-component distribution $P(x,\varphi|m)$of the conditional
state $\left|\Psi_{m}\right\rangle$for $\theta=\pi/7,$$r=0.5$and various
numbers ${\small m}$of measured photons with (a) $m=1$, (b) $m=2$, (c) $m=3$,
and (d) $m=4$, respectively.

##  IV Quasi-probability distribution of the conditional HPESVS

Quasi-probability distribution function in the phase space is a very useful
tool for a comprehensive description of the nonclassical state. Thus, in this
section, we shall analytically discuss several quasi-probability
distributions, including Wigner function and Husimi function to characterize
the nonclassicality of the conditional HPESVS.

###  IV.1 Wigner function

The WF was first introduced by Wigner in 1932 to calculate quantum correction
to a classical distribution function of a quantum-mechanical system. The
presence of negativity of the WF is a signature of its nonclassicalitya36 ;
a37 . For a single-mode density operator $\rho$, the WF in the coherent state
representation $\left|z\right\rangle$ can be expressed as

$W(\alpha)=\frac{2e^{2\left|\alpha\right|^{2}}}{\pi}\int\frac{d^{2}z}{\pi}\left\langle-z\right|\rho\left|z\right\rangle
e^{-2\left(z\alpha^{\ast}-z^{\ast}\alpha\right)},$ (18)

where $\alpha=\left(x+ip\right)/\sqrt{2}$. The Wigner function of the
conditional state
$\rho_{out}^{a}=\left|\Psi_{m}\right\rangle\left\langle\Psi_{m}\right|$, can
be calculated in a straightforward way.

$\displaystyle W(x,p|m)$ $\displaystyle=\frac{2}{\pi
N_{m}m!\cosh^{2}r\sqrt{A}}e^{-2\Xi\left|\alpha\right|^{2}+\frac{2\mu}{A}\alpha^{2}+\frac{2\mu}{A}\alpha^{\ast
2}}$ $\displaystyle\times\frac{\partial^{2m}}{\partial\tau^{m}\partial
s^{m}}e^{Rs+R^{\ast}\tau-B_{2}\tau
s-\frac{B_{1}}{2}s^{2}-\frac{B_{1}}{2}\tau^{2}}|_{s=\tau=0}$
$\displaystyle=\frac{2m!}{\pi
N_{m}\cosh^{2}r\sqrt{A}}e^{-2\Xi\left|\alpha\right|^{2}+\frac{2\mu}{A}\alpha^{2}+\frac{2\mu}{A}\alpha^{\ast
2}}$
$\displaystyle\times\sum_{l=0}^{m}\frac{\left(-B_{2}\right)^{l}\left(-B_{1}/2\right)^{m-l}}{l!\left[\left(m-l\right)!\right]^{2}}\left|H_{m-l}\left(-\frac{R}{\sqrt{2B_{1}}}\right)\right|^{2},$
(19)

where we have set $\Xi=\left(1+\mu^{2}\right)/\left(1-\mu^{2}\right)$ and
$R=2\nu\left(\alpha-\mu\alpha^{\ast}\right)/A$. Especially, when no photons
are detected, $m=0$, then $W(x,p|0)$
$\rightarrow\exp\left(-p^{2}e^{-2\lambda}-x^{2}e^{2\lambda}\right)/\pi$ is a
Gaussian form in phase space, which is just WF of single-mode SVS, as
expected.

Figure 7: Wigner functions $W(x,p|m)$of the conditional state
$\left|\Psi_{m}\right\rangle$for $\theta=\pi/7,$$r=0.5$and various numbers m
of measured photons with (a) $m=1$, (b) $m=2$, (c) $m=3$, (d) $m=4$,
respectively.

The WFs of the conditional HPESVS $\left|\Psi_{m}\right\rangle$ in Fig.7 are
plotted for the same parameters as in Fig.6. The figures indicate that the
conditional HPESVS is a noncassical non-Gaussian state, since the partial
negative regions in phase space are observed in Fig.7. This further
demonstrates that performing the conditional output measurement on a BS is an
effective approach to generate non-Guassian state. In addition, it is seen
from Fig.7 that for odd $m$ there exists a negative valley in the center
region, whereas for even $m$ there exists a main peak. In fact, for the center
region $W(0,0|m)=\frac{2}{\pi}\left(-1\right)^{m}$, as expected.

###  IV.2 Husimi function

The Husimi function $Q(x,p|m)$ of the state $\left|\Psi_{m}\right\rangle$ is
defined bya28

$Q\left(x,p|m\right)=\frac{1}{\pi}\left|\left\langle\beta\right.\left|\Psi_{m}\right\rangle\right|^{2},$
(20)

where $\left|\beta\right\rangle$ is a coherent state and
$\beta=\left(x+ip\right)/\sqrt{2}$. Using Eq.(7), the scalar product
$\left\langle\beta\right.\left|\Psi_{m}\right\rangle$ can be easily calculated
as follow

$\left\langle\beta\right.\left|\Psi_{m}\right\rangle=\frac{\allowbreak
e^{-\frac{\left|\beta\right|^{2}}{2}+\allowbreak\frac{\mu}{2}\beta^{\ast
2}}}{\sqrt{m!N_{m}}\cosh
r}\frac{\partial^{m}}{\partial\tau^{m}}e^{-\tau^{2}\frac{\mu}{2}+\tau\nu\beta^{\ast}}|_{\tau=0},$
(21)

So we find that $Q\left(x,p|m\right)$ can be written as

$Q\left(x,p|m\right)=\frac{\left(\mu/2\right)^{m}e^{-\left|\beta\right|^{2}+\allowbreak\frac{\mu}{2}\left(\beta^{2}+\beta^{\ast
2}\right)}}{\pi N_{m}m!\cosh^{2}r}\left|\allowbreak
H_{m}\left(\frac{\nu}{\sqrt{2\mu}}\beta^{\ast}\right)\right|^{2}.$ (22)

Figure 8: Husimi functions ${\small Q(x,p|m)}$ of the conditional state
$\left|\Psi_{m}\right\rangle$ for $\theta{\small=}\pi{\small/7}$, ${\small
r=0.5}$and various numbers ${\small m}$ of measured photons with (a)${\small
m=1}$, (b) ${\small m=2}$, (c) ${\small m=3}$, (d) ${\small m=4}$,
respectively.

As expected, for $m=0$ the Husimi function is Gaussian, while for odd $m$ a
two-peak structure and for even m a single peak are observed in Fig.8, whose
parameters are the same as WFs in Fig.7. Note that the Husimi function is a
phase-space function that can be measured in multiport balanced homodyning.
Since the Husimi function can be regarded as a smoothed Wigner function, it is
always non-negative and the oscillating behavior, typical of WF (see Fig.7),
cannot be observed.

##  V Conclusions and Discussions

In summary, we have shown that Hermite polynomial excited squeezed states can
be generated by conditional measurements using a simple beam splitter scheme.
When a two-mode squeezed vacuum state is mixed by a beam splitter and the
number of photons is measured in one of the output channels, then the
conditional quantum state in the other output channel reveals all properties
of a Hermite polynomial excited squeezed state. Then, we also have numerically
analyzed the conditional HPESVS in terms of the photon-number statistics,
quadrature-component distribution and quasi-probability distribution such as
the Wigner and Husimi functions. The results show that by tuning the
parameters of the interaction, namely, the control parameter of the BS, the
squeezed parameter $r$ of the TMSVS, and the photon number of conditional
measurement $m$, the HPESVS may be modulated, generating a wide range of
nonclassical phenomena. This further demonstrates that performing the
conditional measurement on a BS is an effective approach to generate non-
Guassian state.

###### Acknowledgements.

This project was supported by the Naitional Nature Science Foundation of China
(Nos.11264018 and 11447002).

## References

  * (1) M.S. Kim, “Recent developments in photon-level operations on travelling light fields,” J. Phys. B: At. Mol. Opt. Phys. 41, 133001 (2008).
  * (2) F. Dell’Anno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53-168 (2006).
  * (3) G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492-497 (1991).
  * (4) X. X. Xu, L. Y. Hu, and H. Y. Fan, “Photon-added squeezed thermal states: statistical properties and its decoherence in a photon-loss channel,” Opt. Commun. 283, 1801-1809 (2010).
  * (5) J. Lee, J. Kim, and H. Nha, “Demonstrating higher-order nonclassical effects by photon-added classical states: realistic schemes,” J. Opt. Soc. Am. B 26, 1363-1369 (2009).
  * (6) A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660-662 (2004).
  * (7) A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
  * (8) A. Biswas and G. S. Agarwal, “Nonclassicality and decoherence of photon-subtracted squeezed states,” Phys. Rev. A 75, 032104 (2007).
  * (9) L. Y. Hu and H. Y. Fan, “Statistical properties of photonsubtracted squeezed vacuum in thermal environment,” J. Opt. Soc. Am. B 25, 1955–1964 (2008).
  * (10) L. Y. Hu, X. X. Xu, Z. S. Wang, and X. F. Xu, “Photon-subtracted squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842-043853 (2010).
  * (11) J. Fiurášek, “Engineering quantum operations on traveling light beams by multiple photon addition and subtraction,” Phys. Rev. A 80, 053822 (2009).
  * (12) A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
  * (13) H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401 (2004).
  * (14) S. D. Bartlett and B. C. Sanders, “Universal continuousvariable quantum computation: requirement of optical nonlinearity for photon counting,” Phys. Rev. A 65, 042304 (2002).
  * (15) Y. Yang and F. L. Li, “Entanglement properties of non-Gaussian resources generated via photon subtraction and addition and continuous-variable quantum-teleportation improvement,” Phys. Rev. A 80, 022315 (2009).
  * (16) S. Y. Lee, S. W. Ji, H. J. Kim, and H. Nha, “Enhancing quantum entanglement for continuous variables by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 84, 012302 (2011).
  * (17) S.Y. Lee, J. Park, S.W. Ji, C.H.R. Ooi, and H.W. Lee, “Nonclassicality generated by photon annihilation-then-creation and creation-then-annihilation operations,” J. Opt. Soc. Am. B 26, 1532-1537 (2009).
  * (18) S.Y. Lee and H. Nha, “Second-order superposition operations via Hong-Ou-Mandel interference,” Phys. Rev. A 85, 043816 (2012).
  * (19) S. Wang, X.X. Xu, H.C. Yuan, L.Y. Hu, and H.Y. Fan, “Coherent operation of photon subtraction and addition for squeezed thermal states: analysis of nonclassicality and decoherence,” J. Opt. Soc. Am. B 28, 2149-2158 (2011).
  * (20) L.Y. Hu and Z.M. Zhang, “Statistical properties of coherent photon-added two-mode squeezed vacuum and its inseparability,” J. Opt. Soc. Am. B 30, 518-519 (2011).
  * (21) M.S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401-260404 (2008).
  * (22) Y. Yang and F.L. Li, “Nonclassicality of photon-subtracted and photon-added-then-subtracted Gaussian states,” J. Opt. Soc. Am. B 26, 830-835 (2009).
  * (23) S.Y. Lee and H. Nha, “Engineering quantum operations on traveling light beams by multiple photon addition and subtraction,” Phys. Rev. A 82, 053812 (2010).
  * (24) G. Ren, J.M. Du, H.J. Yu, and Y.J. Xu, “Nonclassical properties of Hermite polynomial’s coherent state,” J. Opt. Soc. Am. B 29, 3412-3418 (2012).
  * (25) Z. Wang, H.M. Li, and H.C Yuan, “Quasi-probability distributions and decoherence of Hermite-excited squeezed thermal states,” J. Opt. Soc. Am. B 31, 2163-2174 (2014).
  * (26) M. Ban, ”Quasicontinuous measurements of photon number,” Phys. Rev. A 49, 5078 (1994).
  * (27) J. Fiurášek, “Conditional generation of N-photon entangled states of light,” Phys. Rev. A 65, 053818 (2002).
  * (28) S.A. Podoshvedov, J. Kim, and J. Lee, “Generation of a displaced qubit and entangled displaced photon state via conditional measurement and their properties,” Opt. Commun. 218, 3748-3754 (2008).
  * (29) M. Dakna, L. Knöll, and D.G. Welsch, “Quantum state engineering using conditional measurement on a beam splitter,” Eur. Phys. J. D. 3, 295-308 (1998).
  * (30) J. Clausen, M. Dakna, L. Knöll, and D.G. Welsch, “Conditional quantum-state transformation at a beam splitter,” J. Opt. B: Quantum Semiclass. Opt. 1, 332-328 (1999).
  * (31) M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D. G. Welsch “Generating Schrodinger cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184-3194 (1997).
  * (32) B.M. Escher, A.T. Avelar, and B. Baseia, “Synthesis of arbitrary Fock states via conditional measurement on beam splitters,” Phys. Rev. A 72, 045803 (2005).
  * (33) M. Dakna, L. Knöll, and D.G. Welsch, “Photon-added state preparation via conditional measurement on a beam splitter,” Opt. Commun. 145, 309-321 (1998).
  * (34) S. Olivares and M.G.A. Paris, “Squeezed Fock state by inconclusive photon subtraction,” J. Opt. B: Quantum Semiclassical Opt. 7, S616-S621 (2005).
  * (35) M.O. Scully and M.S. Zubairy, Quantum Optics (Cambidge U. Press, 1997).
  * (36) L. Mandel, “Sub-Poissonian photon statistics in resonance fluorescence,” Opt. Lett. 4, 205-207 (1979).
  * (37) H.-Y. Fan, H.-L. Lu, and Y. Fan, “Newton–Leibniz integration for ket–bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480-494 (2006).
  * (38) A. Kenfack and K. Zyczkowski, “Negativity of the Wigner function as an indicator of non-classicality,” J. Opt. B: Quamtum Semiclass. Opt. 6, 396-404 (2004).
  * (39) J. Li, G. Li, J. M. Wang, S. Y. Zhu, and T. C. Zhang, “A comparison of two nonclassical measures, entanglement potential and the negativity of the Wigner function,” J. Phys. B: At. Mol. Opt. Phys. 43, 085504 (2010).


These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

1]Warsaw University of Technology, Warsaw, Poland

2]Jan Kochanowski University, Kielce, Poland

3]Goethe University, Frankfurt am Main, Germany

4]Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine

5]Frankfurt Institute for Advanced Studies, Giersch Science Center, Frankfurt
am Main, Germany

6]Department of Physics, School of Applied Sciences, Kalinga Institute of
Industrial Technology, Bhubaneswar, Odisha, India

# Large isospin symmetry breaking in kaon production at high energies

Wojciech Brylinski<EMAIL_ADDRESS>Marek Gazdzicki
<EMAIL_ADDRESS>Francesco Giacosa<EMAIL_ADDRESS>Mark
Gorenstein<EMAIL_ADDRESS>Roman Poberezhnyuk
<EMAIL_ADDRESS>Subhasis Samanta<EMAIL_ADDRESS>Herbert
Stroebele<EMAIL_ADDRESS>[  [  [  [  [  [

###### Abstract

It is well known that isospin symmetry is fulfilled to a good approximation in
strong interactions, as confirmed in low-energy scattering experiments and in
mass spectra of both light and heavy hadrons. In collisions of nuclei with an
equal number of protons and neutrons, isospin symmetry imposes that the number
of produced charged kaons should equal the number of neutral ones. The
NA61/SHINE experiment at CERN recently reported an excess of charged over
neutral kaon production in high-energy nucleus-nucleus collisions. Here, we
argue that the measured charge-to-neutral kaon ratio of about 1.2 indicates an
unexpectedly large violation of isospin symmetry. Using well-established
models for hadron production, we demonstrate that known symmetry-breaking
effects and the initial nuclei containing more neutrons than protons lead only
to a small (few per cent) deviation from unity at high energies. Thus, they
cannot explain the measurements. The significance of the isospin symmetry
violation beyond the known effects is 5.5$\cdot\sigma$ when errors quoted by
the experiments are used and 8.1$\cdot\sigma$ for the PDG-like scaled errors.
New systematic, high-precision measurements and theoretical efforts are needed
to establish the origin of the observed large isospin-symmetry breaking.

###### keywords:

isospin symmetry breaking, QCD, kaon production

In 1932, Heisenberg suggested that the neutron and the proton could be seen as
different manifestations of the same particle, the nucleon [1], explaining
their similar masses. Wigner referred to this feature as ‘isotopic spin’, or
‘isospin’, due to the formal similarity with the more familiar spin [2].
Today, isospin is an important element of nuclear physics and nuclei
classification [3]. The concept of isospin was promptly extended to the other
strongly interacting particles, the hadrons. In particular, Kemmer [4] applied
it to the pion triplet postulated by Yukawa in 1935 [5]. For a historical
outline, see Refs. [6, 7].

Nowadays, hadrons are viewed as emergent within Quantum Chromodynamics (QCD)
[8], the theory of strong interactions between the elementary carriers of
colour charge, the quarks and the gluons. Because of confinement, QCD is based
on a local colour gauge symmetry that is hidden in colourless hadrons. Hadrons
are classified into mesons with integer spin, typically quark-anti-quark
states such as pions and kaons, and baryons with semi-integer spin, typically
three-quark states such as protons and neutrons.

Within QCD, the isospin symmetry could be understood as a part of a more
general symmetry, denoted as flavour symmetry, which arises because the gluons
interact ‘democratically’ with the light quark flavours $u$, $d$, and $s$.
Moreover, in the limit where the quarks can be considered massless (the so-
called chiral limit), these interactions do not depend on their chirality,
thus realizing a larger symmetry called chiral symmetry. The latter is
spontaneously broken in the QCD vacuum generating (quasi-) Goldstone bosons,
pions, kaons, and the $\eta$-meson.

In more specific terms, the isospin operator triplet
$\hat{\mathbb{I}}=(\hat{I}_{x},\hat{I}_{y},\hat{I}_{z})$ refers to an $SU(2)$
rotation of the ‘isodoublet’ $(u,d)$, where $u$ has eigenvalue $I_{z}=1/2$ and
$d$ $I_{z}=-1/2$, thus an isospin transformation changes e.g. a proton into a
neutron or generates an $O(3)$ rotation of the pion triplet. Kaons, which are
grouped into the isodoublet $(I=1/2)$ states $(K^{+}\equiv
u\bar{s},K^{0}\equiv d\bar{s})$ and $(\bar{K^{0}}\equiv s\bar{d},K^{-}\equiv
s\bar{u})$ correspond naturally to the underlying isospin doublets $(u,d)$ and
$(-\bar{d},\bar{u})$ and shall play a crucial role in the following. Isospin
symmetry implies that only transitions preserving total isospin $I$ and its
$z$-component $I_{z}$ are allowed. This property is referred to as the isospin
conservation.

QCD would strictly obey the isospin symmetry, having equal light quark masses,
$m_{u}=m_{d}$. Since the current quark masses of $u$ and $d$ quarks differ
only by a few MeV [9], only small deviations from this symmetry are expected.
The relative mass differences of charge states of pions and kaons are small,
$\left(m_{\pi^{+}}-m_{\pi^{0}}\right)/\left(m_{\pi^{+}}+m_{\pi^{0}}\right)\simeq
0.017$ and
$\left(m_{K^{+}}-m_{K^{0}}\right)/\left(m_{K^{+}}+m_{K^{0}}\right)\simeq-0.004$.
The elastic cross sections for pion-pion, pion-nucleon, and nucleon-nucleon
scattering closely follow the predictions of isospin symmetry [10, 11].

Thus, one also expects that isospin symmetry is obeyed in good approximation
in hadron production processes at high energies. However, recently reported
NA61/SHINE experimental results on producing charged and neutral kaons,
supported by the compilation of the world data reported in Ref. [12], seem to
contradict those expectations [13]. Here, we address this question in detail.

Isospin symmetry and kaon production in nucleus-nucleus collisions.
Let us consider, for definiteness, an initial state with two identical nuclei
having equal proton and neutron numbers ($Z=N=A/2$) in their ground
states111Considering two distinct nuclei, each of them having an equal number
of protons and neutrons, would lead to the same results.. This initial state
is denoted as $\left|A+A\right\rangle$. In conserved quantities, the condition
$Z=N$ is given by $Q/B=1/2,$ where $Q=Z$ and $B=A$ are the nuclei’s electric
charge and baryon number, respectively. Then, each nucleus’ third component of
isospin is zero222For almost all nuclei of this type, the total isospin is
also zero [14], hence the total isospin of the $\left|A+A\right\rangle$ state
vanishes..

The isospin transformation is parameterized by the unitary operator
$\hat{O}=e^{i\theta_{j}\hat{I}_{j}}$ and, in general, (approximate) isospin
symmetry of QCD means that
$\hat{O}^{\dagger}\hat{H}_{QCD}\hat{O}\simeq\hat{H}_{QCD}$, where
$\hat{H}_{QCD}$ refers to the Hamiltonian of QCD. A particularly useful
unitary operator is the special isospin transformation, called charge symmetry
transformation, $\hat{C}=e^{i\pi\hat{I}_{y}}$. It corresponds to an inversion
of the third component of isospin:
$\hat{C}\left|u\right\rangle=-\left|d\right\rangle$ and
$\hat{C}\left|d\right\rangle=\left|u\right\rangle$, schematically
$u\leftrightarrow d$. At the hadronic level, the $\hat{C}$ transformation
implies $p\leftrightarrow n$, $\pi^{+}\leftrightarrow\pi^{-}$,
$K^{+}\leftrightarrow K^{0}$, $\bar{K}^{0}\leftrightarrow K^{-}$, etc.

Without referring to a detailed mathematical formalism, charge symmetry means
that strong interactions are invariant under the inversion of the third
component of the isospin of every nucleus and hadron of the initial and final
states. Let us consider an ensemble of initial states being invariant under
the charge transformation $\hat{C}$ \- probabilities of having initial states
related by this transformation are equal. This is indeed the case of nucleus-
nucleus collisions where each nucleus has an equal number of protons and
neutrons (thus, $I_{z}=0$). Then, the invariance under
$\hat{C}$-transformation holds also for the final state ensemble, implying
that the mean multiplicities of $\hat{C}$-related quantities, such $K^{+}$ and
$K^{0}$ and $\bar{K}^{0}$ and $K^{-}$, coincide:

$\langle K^{+}\rangle=\langle K^{0}\rangle\text{ , }$ (1)

and

$\langle K^{-}\rangle=\langle\bar{K}^{0}\rangle\text{ . }$ (2)

Note, since models predict only properties of ensembles of events, we need to
consider mean multiplicities for testing charge invariance.

The relations (1) and (2) are in agreement with the result first formulated in
1955 by Shmushkevich [15, 16], see also Refs. [17, 18, 19]. This result states
that an initial ‘uniform’ ensemble of hadronic state (that is, one with an
equal mean number of each member of any isospin multiplet) evolves into a
uniform final-state ensemble.

The neutral kaons are eigenstates of strong interaction, oscillating
($K^{0}\leftrightarrow\bar{K}^{0}$) due to weak interaction. The weak
eigenstates are the long-lived ‘K-long’ $K_{L}^{0}$ and short-lived ‘K-short’
$K_{S}^{0}$, which decay into three and two pions with a mean lifetimes of 15
m/$c$ and2.8 cm/$c$, respectively. By neglecting the small CP violation, the
weak and strong kaonic eigenstates are related by:

$\left(\begin{array}[c]{c}\left|K_{S}^{0}\right\rangle\\\
\left|K_{L}^{0}\right\rangle\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cc}1&-1\\\
1&1\end{array}\right)\left(\begin{array}[c]{c}\left|K^{0}\right\rangle\\\
\left|\bar{K}^{0}\right\rangle\end{array}\right)\text{ ,}$ (3)

hence, for the mean number of $K_{S}^{0}$ and $K_{L}^{0}$ one gets:

$\langle K_{S}^{0}\rangle=\frac{1}{2}\langle
K^{0}\rangle+\frac{1}{2}\langle\bar{K}^{0}\rangle=\langle
K_{L}^{0}\rangle\text{ .}$ (4)

Thus, we have an equal number of long-lived and short-lived kaons, even if
$\langle K^{0}\rangle$ and $\langle\bar{K}^{0}\rangle$ are different due to
the non-zero baryon number of the initial state.

By combining Eqs. (1) and (2) and using Eq. (4) one finds:

$\langle K^{+}\rangle+\langle K^{-}\rangle=2\langle K^{0}_{S}\rangle~{},$ (5)

where also the r.h.s. in accessible to experiments. Consequently, the exact
isospin symmetry prediction for the charged-to-neutral kaon ratio in nucleus-
nucleus collisions with $Q/B=1/2$ reads:

$R_{K}\equiv\frac{\langle K^{+}\rangle+\langle K^{-}\rangle}{\langle
K^{0}\rangle+\langle\bar{K}^{0}\rangle}=\frac{\langle K^{+}\rangle+\langle
K^{-}\rangle}{2\langle K_{S}^{0}\rangle}=1~{}.$ (6)

This prediction serves as the initial reference for testing the isospin
symmetry with the results provided by the NA61/SHINE experiment [12] that
measured $R_{K}=1.262\pm 0.017(stat)\pm 0.062(sys)$ in collisions of argon and
scandium nuclei at the energy of 11.9 GeV per nucleon pair. This result is
consistent with the compilation of the world data, even if each previous
measurement, taken individually, was also consistent with $R_{K}=1$ within its
large uncertainty. Yet, as discussed below, all data taken together clearly
show a significant deviation from unity.

Neither is the isospin symmetry exact nor does experimental data refer to
$A+A$ collisions with $Q/B=1/2$. In the following, we estimate possible
deviations of $R_{K}$ from unity due to the known effects: isospin symmetry-
breaking processes and $Q/B\neq 1/2$, using popular and well-established
statistical and dynamical models. Then, we quantify the difference between the
experimental results and the model predictions and discuss possible
implications.

Known isospin symmetry-breaking effects. The isospin symmetry is explicitly
broken in weak and electromagnetic processes and is an approximate symmetry of
strong interactions. Here, we consider the known isospin symmetry-breaking
effects one by one.

  1. (A)

Weak processes. The weak interaction does not obey the isospin symmetry. The
mean lifetimes [9] of charged and neutral kaons are
$\tau(K^{+})=\tau(K^{-})=(1.238\pm 0.001)\cdot 10^{-8}$ s and
$\tau(K^{0}_{S})=(8.954\pm 0.004)\cdot 10^{-11}$ s, $\tau(K^{0}_{L})=(5.116\pm
0.021)\cdot 10^{-8}$ s. The charged kaons are typically measured by
reconstructing their trajectories in a detector. Due to the large mean
lifetime, the corrections for their losses caused by weak decays are small. In
contrast, the neutral $K^{0}_{S}$ kaons are measured by reconstructing their
decays into two charged pions. Typically, the corrections for the losses
caused by weak decays are large. This is because the decay should be far
enough from the interaction point to separate the decay point from the
background in high multiplicity $A+A$ collisions. Assuming that the
$K^{0}_{S}$ mesons are measurable when their proper lifetime is larger than
the mean lifetime (typical for NA61/SHINE [12]), one estimates the maximum
relative bias of the mean multiplicity as

$\frac{\Delta(\langle K_{S}^{0}\rangle)}{\langle
K_{S}^{0}\rangle}=\frac{3\cdot\sigma(\tau(K_{S}^{0})}{\tau(K_{S}^{0})}\approx
0.0013.$ (7)

Thus, the maximum deviation of $R_{K}$ from unity due to the uncertainty of
the mean $K^{0}_{S}$ lifetime is 0.13 %.

  2. (B)

Electromagnetic process. Electromagnetic interactions do not conserve isospin
because the electromagnetic coupling constant differs for the quark flavours
$u$ and $d$. Yet, electromagnetic decays of hadrons are typically suppressed
by a factor $\alpha\simeq 1/137$ compared to strong ones. Consequently,
processes that involve the production of virtual photons and their subsequent
decay into charged kaons are suppressed by a factor $\alpha^{2}$ and thus
negligible.

  3. (C)

Strong processes within QCD. The isospin symmetry is not exact because $u$ and
$d$ quark masses are different, $\approx 2.2$ and $\approx 4.7$ MeV,
respectively. The different quark masses lead to different masses of hadrons
within the isospin multiplets, particularly different masses of charged and
neutral kaons. The ratio $R_{K}$ can be expanded as a function of the mass
difference of the $u$ and $d$ quarks as:

$R_{K}=R_{K}^{(0)}+\frac{m_{d}-m_{u}}{\Lambda_{QCD}}R_{K}^{(1)}+...\text{ ,}$
(8)

where $R_{K}^{(0)}$ is the value of the ratio for the isospin symmetry
($m_{d}=m_{u}$) being equal to one for $Q/B=1/2$, and $\Lambda_{QCD}\simeq
250$ MeV is the QCD energy scale [20, 21]. The question is whether
$R_{K}^{(1)}+...$ terms can reproduce the experimental data. We list below the
considered effects and quantify their impact on $R_{K}$ using the statistical
hadron resonance gas model (HRG) [22]. The results are cross-checked with the
microscopic, UrQMD transport model [23, 24, 25]. For details, see Methods.

    1. (i)

Smaller masses of charged kaons than neutral ones, $m_{K^{+}}=m_{K^{-}}=493.7$
MeV and $m_{K^{0}}=m_{\bar{K}^{0}}=497.6$ MeV, lead to an increase of $R_{K}$
resulting from direct kaon production by about 0.02. This was estimated by
removing resonances from the particle list of HRG.

    2. (ii)

Different kaon masses impact the branching ratios of hadronic resonances
decaying into kaons. The most striking example is $\phi(1024)$ meson, which
decays twice as frequently to charged kaons as into neutral ones. This large
difference is because the $\phi(1024)$ mass is just above the kaon-kaon
thresholds. Including the kaon production from resonance decays increases
$R_{K}$ by about 0.01.

    3. (iii)

Mass differences of hadrons belonging to other isospin multiplets also break
the isospin symmetry and affect $R_{K}$. The largest effect comes from the
mass difference between proton and neutron, which reduces $R_{K}$ at the
lowest collision energies. At energies larger than 10 GeV, this effect is
negligible.

Interestingly, HRG for $Q/B=1/2$ and in the exact isospin-symmetry limit
reproduces Eq. (1) exactly.

Figure 1:  The charged-to-neutral kaon ratio $R_{K}$ as a function of
collision energy. The symbols show the experimental data. The black line shows
the HRG predictions for $Q/B=0.4$. The blue dots indicate the HRG predictions
for $Q/B$ values corresponding to the ones in experiments. For different
nuclei, $Q/B$ corresponds to the electric charge over baryon number of the
whole system. The grey rectangles show the UrQMD model results.

Collisions with $Q/B<1/2$. For nuclei with $Q/B<1/2$ the third component of
isospin equals $|I_{z}|=|B/2-Q|$ and therefore the total isospin is limited as
$|I_{z}|\leq I\leq B/2$.

The compiled experimental results in nucleus-nucleus collisions correspond to
the $Q/B$ ranging from about 0.4 (Pb+Pb and Au+Au collisions) to 0.5 (Si+Si
and S+S collisions); see Fig. 1. The limits correspond to the normalized per
baryon third component and total isospin $|I_{z}|/B=0.1$, $0.1<I/B<1/2$ and
$|I_{z}|/B=I/B=0$, respectively. The non-zero $I_{z}$ and $I$ for heavier
nuclei can impact the charged-to-neutral kaon ratio in two ways:

  1. (a)

The larger fraction of neutrons than protons for heavy nuclei having
$|I_{z}|/B\approx 0.1$ enhances neutral kaon production compared to charged
ones and thus reduces $R_{K}$. This increases the disagreement between the
experimental results and the model predictions calculated for $|I_{z}|=I=0$.
The reduction is significant at low collision energies and is small compared
to other effects at energies larger than 10 GeV; see Fig. 1 and Methods.

  2. (b)

For $|I_{z}|/B\approx 0.1$, the total isospin is limited by $0.1\leq I/B\leq
1/2$. Generally, nuclei in the ground state have the lowest possible value of
the total isospin [14]. This rule extends to a state of two identical nuclei
in the ground state, which, for the considered case, implies
$I/B\approx|I_{z}|/B\approx 0.1$. Thus, in the considered cases, a possible
dependence of $R_{K}$ on $I$ and $I_{z}$ reduces to the dependence on
$I=|I_{z}|$. The latter is discussed in (a) above.

The experimental data also allow a rough estimate of the influence of the
normalized isospin small but non-zero value on the charged-to-neutral kaon
ratio. Results for of heavy nuclei, 208Pb+207Pb and 197Au+197Au ($R_{K}\approx
1.15$ for $I/B=|I_{z}|/B\approx 0.1$), are similar to those for intermediate
nuclei, 40Ar+45Sc ($I/B=|I_{z}|/B\approx 0.08$), and 28Si+28Si
($I/B=|I_{z}|/B=0$), see Fig. 1. This suggests that the sensitivity of the
ratio to the $I/B=|I_{z}|/B$ close to zero ($I/B=|I_{z}|/B\leq 0.1$) is low.
Note, however, that there are large uncertainties in the experimental results.

In the case of central collisions, the charge-to-baryon ratio of the
interacting nucleon system (participant nucleons) can be higher than the total
proton-to-nucleon ratio in colliding nuclei. This is because protons tend to
be distributed closer to the centre of a nucleus. However, this effect is
expected to be rather small and relevant only for the lowest energies
($\lesssim 4$ GeV), where $R_{K}$ is sensitive to the small changes of $Q/B$.

In summary, at energies larger than 10 GeV, the mass difference between
charged and neutral kaons, implying isospin symmetry-breaking processes,
increases the $R_{K}$ ratio by about 0.03. Other isospin-breaking effects can
be neglected. The ratio $R_{K}$ is reduced for collisions with $Q/B<1/2$, but
this is insignificant for energies larger than 10 GeV. At energies lower than
10 GeV, the $R_{K}$ ratio is significantly more sensitive to the known
isospin-breaking effects and the $Q/B$ ratio; see Methods for more details.

The above findings are illustrated in Fig. 1 showing predictions of the HRG
and UrQMD models as functions of collision energy. Importantly, the plot
includes the experimental results concerning the $R_{K}$ ratio from NA61/SHINE
[12] and from other experiments, namely CERES [26], STAR BES [27, 28], STAR
[29, 30, 31, 32], ALICE [33, 34], NA35 [35, 36, 37], NA49 [38], HADES [39, 40,
41, 42] and FOPI [43, 44]. The compilation is taken from the NA61/SHINE
publication [12]. The HRG calculations were performed with $Q/B=0.4$ (solid
line) and $Q/B$ values corresponding to collisions studied in the experiments
(black dots). The UrQMD results were obtained for central Au+Au collisions.
See Methods for details.

The two main observations are that the model predictions agree well with each
other but are systematically lower than the experimental data.

Figure 2:  The experimental data for the charged to neutral kaon ratio divided
by the HRG baseline $R_{K}/R_{K}^{\rm HRG}$ as a function of collision energy.
The solid black line shows the weighted average of the experimental data, and
the shaded area shows the error of the weighted average.

Quantification of the isospin-symmetry breaking beyond known effects. To
quantify the measured isospin-symmetry breaking (in short, isospin-breaking)
beyond the known effects, the ratio of the measured $R_{K}$ to the
corresponding HRG one is shown in Fig. 2. We do not consider the three lowest
energy data points (HADES and FOPI) because the known isospin-breaking and
$Q/B<1/2$ effects are significant at the low collision energies. Thus, the
predictions may be model-dependent. We also do not consider two NA35 points
because, unlike other measurements, charged kaons were identified by
reconstructing their decays, leading to large statistical uncertainties and
possible biases. Thus, the number of selected measurements at different
collision energies (from AGS to LHC) is $15$. Out of them, only one is below
unity. No significant dependence of the double ratio on collision energy and
nuclear mass number of colliding nuclei is visible.

The weighted average of all results is $1.143\pm 0.026$, where the second
number stands for the error calculated following the Particle Data Group
procedure [9]. The weighted average deviates from the expectation for the
isospin symmetry broken by the known effects by $5.55$ errors. The $\chi_{\rm
min}^{2}/\text{dof}=0.47$ may indicate an overestimation of the uncertainties.
The estimated scaling factor for the uncertainty is 2.15, and the
corresponding scaled error is 0.018. The weighted average deviates from the
expectation for the isospin symmetry, broken by the known effects by 8.14
scaled errors.

Closing remarks. Several comments are in order here.

  1. (I)

The known effects of the isospin-symmetry breaking fail to explain the
experimental results on the charged-to-neutral kaon ratio reported recently by
NA61/SHINE [12]. A dedicated experimental and theoretical effort is needed to
solve the puzzle of this unexpected large isospin symmetry breaking.

  2. (II)

Concerning measurements, reviewing the validity of the past results and
confirming them with new high-precision data is important. Systematic results
on the collision energy and nuclear mass dependence of the isospin-breaking
effect should help us understand its nature. Measurements of the charged-to-
neutral kaon ratio in collisions of an equal number of protons and neutrons
would eliminate uncertainty in its interpretation. In particular, measurements
of the ratio in deuterium-deuterium interactions, the lightest nuclei with
$I=0$, are important.

  3. (III)

An interesting experimental test is also possible by considering $\pi^{-}+C$
and $\pi^{+}+C$ interactions. While the ensemble of only one of them is not
invariant under charge transformation, the ensemble having an equal number of
$\pi^{-}+C$ and $\pi^{+}+C$ interactions is invariant. Thus, for the joint
ensemble, the exact charge symmtery predicts $R_{K}=1$.

  4. (IV)

Kaons play a special role due to the simple isospin structure and easy-to-
measure measure. This explains why the first results on a large isospin-
symmetry breaking are reported for kaons. Yet, it is important to perform a
similar study for other isospin multiplets in the future. For example, using
the same methods, we have checked that the (anti-)proton to (anti)-neutron
ratio is even less sensitive to the known isospin symmetry-breaking effects
and, thus, is predicted to be almost exactly one in nucleus-nucleus collisions
with $Q/B=0.5$ in a broad range of collision energies including the low
energies. Considering collision-by-collision fluctuations of charged and
neutral kaon numbers would also be interesting. A puzzling result was recently
reported by ALICE at LHC [45] and discussed in Ref. [46].

  5. (V)

The relationship between the production rates of charged and neutral kaons in
hadronic collisions was used to estimate the flux of secondary neutral kaons
produced in $p$+Be collisions at beam momenta of several 100 GeV/$c$ in
neutral kaon and neutrino beams [47]. Recently, the charged-to-neutral kaon
ratio was studied in p+p interactions at SPS energies, and no excess in
comparison to a simple quark counting model emerged [48]. On the other hand,
$\pi^{-}$+C interactions at SPS energies yield the ratio of charged-to-neutral
kaons, which is significantly larger than unity and predictions of popular
models [49].

  6. (VI)

Concerning models, one may argue that there are known isospin-breaking effects
which are improperly implemented in the popular models. For example, an
isospin-breaking in production and decay of heavy resonances could sum up a
significant result. While this seems hard to hold since we considered known
effects of this type, careful and independent verification is required.
Independently, one can extend the models by introducing new isospin-breaking
processes and fitting their parameters to the data. This can be done either
for the quark-gluon processes or the hadron-resonance processes. For example,
within the statistical hadronization models, one can introduce the quark
fugacity factors for $u$ and $d$ quarks separately [50, 51, 52]. This could
allow us to make predictions for other hadron ratios.

  7. (VII)

Concerning unknown effects violating isospin symmetry, the possibility of
having a phase of strongly interacting matter with a significant isospin
violation was suggested by Pisarski and Wilczek within a linear $\sigma$ model
of QCD [53]. They expect masses of $\pi^{0}$ and $\eta$ mesons to decrease if
the $U_{A}(1)$ symmetry is effectively restored at lower temperatures than the
chiral phase transition. Can the chiral anomaly [54, 55] or other non-
perturbative phenomena within QCD significantly affect the charge-to-neutral
kaon ratio?

Summary. In this work, we have investigated the ratio $R_{K}$ of charged-to-
neutral kaon yields in nucleus-nucleus collisions at high energies. In the
limit of exact isospin symmetry, this ratio is unity if the initial state of
nuclei having an equal number of protons and neutrons ($Q/B=1/2$) is
considered. The known isospin-symmetry breaking effects included in popular
models predict $R_{K}\approx 1.03$ at high collision energy. Many experimental
results correspond to collisions of nuclei with more neutrons than protons
($Q/B<1/2$, with $Q/B\simeq 0.4$). This reduces $R_{K}$ significantly only at
low collision energies, where the ratio is lower than one. We conclude that
the known effects violating isospin symmetry cannot explain experimental
results on nucleus-nucleus collisions, for which $R_{K}$ is mostly within the
range 1.1-1.2. The significance of the isospin symmetry violation beyond the
known effects is 5.55$\sigma$ when errors quoted by the experiments are used
and 8.14$\sigma$ when the PDG-like scaled errors are used.

The evidence for unexpected large isospin symmetry breaking for kaons that
cannot be explained by otherwise successful models seems well upheld. Assuming
it passes future experimental tests, the quest is to understand its origin.

Acknowledgments

The authors thank Wojciech Broniowski, Katarzyna Grebieszkow, Stanislaw
Mrowczynski, Owe Philippsen, Rob Pisarski, Krishna Rajagopal, Jan Steinheimer,
Leonardo Tinti, Ludwik Turko and Volodymyr Vovchenko for fruitful discussions
and comments. This work is partially supported by the Polish National Science
Centre (NCN) grants 2018/30/A/ST2/00226 and 2019/33/B/ST2/00613, the German
Research Foundation grants GA1480/8-1 and the Alexander von Humboldt
Foundation.

Methods

Hadron Resonance Gas Model

We use the Hadron Resonance Gas model implementation from Ref. [22] to
quantify the isospin-breaking effects and their interplay. HRG includes all
hadrons and resonances with confirmed status in the Particle Data Tables [56].
The PDG-listed masses, charges, lifetimes, and decay modes are used. Thus, HRG
includes the isospin-symmetry violation due to masses and branching ratios of
hadrons and resonances.

In HRG calculations, the exact net strangeness conservation is enforced, i.e.,
the calculations are done within the strangeness canonical ensemble (SCE) [57,
58]. The model parameters are baryo-chemical potential, $\mu_{B}$,
temperature, $T$, volume of the system $V$, and the strangeness under-
saturation parameter $\gamma_{S}$ (see Ref. [59]). We adopt a simple
parametrization of $\mu_{B}$ and $T$ as a function of collision energy
introduced in Ref. [60]. We have checked that $R_{K}$ is weakly sensitive to
the strangeness suppression effect introduced by having the parameter
$\gamma_{S}<1$. Therefore, for simplicity, calculations are done for
$\gamma_{S}=1$. The energy-dependent Breit-Wigner spectra [61] model the
resonance widths.

Figure 1 (black line) shows the HRG predictions for $R_{K}$ as a function of
collision energy $\sqrt{s_{NN}}$ for $Q/B=0.4$. At low collision energies,
$R_{K}<1$ due to the enhancement of neutral kaon production caused by a larger
number of neutrons than protons, $Q/B<1/2$. On the other hand, at high
collision energies, HRG predicts $R_{K}\approx 1.018$ due to the mass
difference between charged and neutral kaons produced directly. Finally,
$R_{K}\approx 1.032$ when kaon production via resonance decays is included.
This is mostly due to $\phi$ decays, which strongly prefer the decay into
charged kaons over the one into neutral kaons.

We have checked that the system’s electric to baryon charge ratio $Q/B$
significantly affects $R_{K}$ up to $\sqrt{s_{NN}}\approx 10$ GeV. At higher
energies, pions dominate, and total electric and baryon charges are
significantly larger than the corresponding net charges. Thus, $R_{K}$ becomes
increasingly less sensitive to $Q/B$ with an increase in $\sqrt{s_{NN}}$.

We have checked that the strangeness grand-canonical ensemble and other
popular parametrizations [62, 63] of the model parameters as a function of
collision energy lead to quantitatively similar results for $R_{K}$, for
$\sqrt{s_{NN}}\gtrsim 4$ GeV. The uncertainties of $R_{K}$ estimated by
changing the parametrization of model parameters [60] are less than $1$% for
$\sqrt{s_{NN}}>3$ GeV. The effect of including light nuclei in the particle
list is negligible.

UrQMD model

The UrQMD transport model [23, 24, 25] describes A+A collisions by explicitly
propagating hadrons in phase space. During the propagation, rescattering among
hadrons takes place. The particle production in this model happens either via
resonance decay or via a string excitation and fragmentation following the
LUND model [64]. The grey rectangles in Fig. 1 indicate the UrQMD model
predictions. Here, we have considered central Au+Au collisions ($A=197,Z=79$,
$Q/B\approx 0.4$). The predictions are shown within the $\sqrt{s_{NN}}$ range
of 2.4 to 20 GeV. At each energy, $10^{4}$ events are used for the analysis.

One sees that for $\sqrt{s_{NN}}\lesssim 7$ GeV, the predictions of UrQMD and
HRG are similar. At higher energies, the $R_{K}$ ratio in HRG is
systematically higher than the one predicted by UrQMD. This is likely caused
by UrQMD assuming $\phi$-meson decays to be exactly isospin symmetric instead
of taking the branching ratios for PDG. This is the reason for showing the
UrQMD predictions only up to 20 GeV.

## References

  *   * Heisenberg [1932] Heisenberg, W.: On the structure of atomic nuclei. Z. Phys. 77, 1–11 (1932) https://doi.org/10.1007/BF01342433
  * Wigner [1937] Wigner, E.: On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei. Phys. Rev. 51, 106–119 (1937) https://doi.org/10.1103/PhysRev.51.106
  * Warner et al. [2006] Warner, D.D., Bentley, M.A., Isacker, P.V.: The role of isospin symmetry in collective nuclear structure. Nature Physics 2(5), 311–318 (2006) https://doi.org/10.1038/nphys291
  * Kemmer [1939] Kemmer, N.: The particle aspect of meson theory. Proc. Roy. Soc. Lond. A 173, 91–116 (1939) https://doi.org/10.1098/rspa.1939.0131
  * Yukawa [1935] Yukawa, H.: On the Interaction of Elementary Particles I. Proc. Phys. Math. Soc. Jap. 17, 48–57 (1935) https://doi.org/10.1143/PTPS.1.1
  * Kemmer [1982] Kemmer, N.: Isospin. J. Phys. Colloq. 43(C8), 359–393 (1982) https://doi.org/10.1051/jphyscol:1982824
  * Webb [2008] Webb, R..: Spin’s nuclear sibling. Nature Phys 4 (Supll. 1), 8–9 (2008) https://doi.org/10.1038/nphys861
  * Fritzsch et al. [1973] Fritzsch, H., Gell-Mann, M., Leutwyler, H.: Advantages of the Color Octet Gluon Picture. Phys. Lett. B 47, 365–368 (1973) https://doi.org/10.1016/0370-2693(73)90625-4
  * Workman et al. [2022] Workman, R.L., et al.: Review of Particle Physics. PTEP 2022, 083–01 (2022) https://doi.org/10.1093/ptep/ptac097
  * Pennington [2005] Pennington, M.R.: Swimming with quarks. J. Phys. Conf. Ser. 18, 1–73 (2005) https://doi.org/10.1088/1742-6596/18/1/001 arXiv:hep-ph/0504262
  * Alarcon et al. [2012] Alarcon, J.M., Martin Camalich, J., Oller, J.A.: The chiral representation of the $\pi N$ scattering amplitude and the pion-nucleon sigma term. Phys. Rev. D 85, 051503 (2012) https://doi.org/10.1103/PhysRevD.85.051503 arXiv:1110.3797 [hep-ph]
  * Adhikary et al. [2023] Adhikary, H., Adrich, P., Allison, K.K., Amin, N., Andronov, E.V., Arsene, I.C.: Excess of charged over neutral $K$ meson production in high-energy collisions of atomic nuclei (2023) arXiv:2312.06572 [nucl-ex]
  * Gazdzicki and Hansen [1991] Gazdzicki, M., Hansen, O.: Hadron production in nucleon-nucleon collisions at 200-GeV/c: A Compilation. Nucl. Phys. A 528, 754–770 (1991) https://doi.org/10.1016/0375-9474(91)90261-4
  * Lenzi and Bentley [2009] Lenzi, S.M., Bentley, M.A.: In: Al-Khalili, J.S., Roeckl, E. (eds.) Test of Isospin Symmetry Along the N=Z Line, pp. 57–98. Springer, Berlin, Heidelberg (2009). https://doi.org/10.1007/978-3-540-85839-3_3
  * Shmushkevich [1955] Shmushkevich, I.: . Dokl. Akad. Nauk SSSR 103, 235 (1955)
  * Dushin and Shmushkevich [1956] Dushin, N., Shmushkevich, I.: . Dokl. Akad. Nauk SSSR 106, 801 (1956)
  * MacFarlane et al. [1965] MacFarlane, A.J., Pinski, G., Sudarshan, G.: Shmushkevich’s method for a charge independent theory. Phys. Rev. 140, 1045 (1965) https://doi.org/10.1103/PhysRev.140.B1045
  * Wohl [1982] Wohl, C.G.: Isospin relations by counting. American Journal of Physics 50(8), 748–753 (1982) https://doi.org/10.1119/1.12743
  * Pal [2014] Pal, P.: An Introductory Course of Particle Physics -CRC Press, (2014)
  * Kneur and Neveu [2012] Kneur, J.-L., Neveu, A.: $Lambda^{\rm QCD}_{\rm MS}$ from Renormalization Group Optimized Perturbation. Phys. Rev. D 85, 014005 (2012) https://doi.org/10.1103/PhysRevD.85.014005 arXiv:1108.3501 [hep-ph]
  * Deur et al. [2016] Deur, A., Brodsky, S.J., Teramond, G.F.: The QCD Running Coupling. Nucl. Phys. 90, 1 (2016) https://doi.org/10.1016/j.ppnp.2016.04.003 arXiv:1604.08082 [hep-ph]
  * Vovchenko and Stoecker [2019] Vovchenko, V., Stoecker, H.: Thermal-FIST: A package for heavy-ion collisions and hadronic equation of state. Comput. Phys. Commun. 244, 295–310 (2019) https://doi.org/10.1016/j.cpc.2019.06.024 arXiv:1901.05249 [nucl-th]
  * Bass et al. [1998] Bass, S.A., et al.: Microscopic models for ultrarelativistic heavy ion collisions. Prog. Part. Nucl. Phys. 41, 255–369 (1998) https://doi.org/10.1016/S0146-6410(98)00058-1 arXiv:nucl-th/9803035
  * Bleicher et al. [1999] Bleicher, M., et al.: Relativistic hadron hadron collisions in the ultrarelativistic quantum molecular dynamics model. J. Phys. G 25, 1859–1896 (1999) https://doi.org/10.1088/0954-3899/25/9/308 arXiv:hep-ph/9909407
  * Bleicher and Bratkovskaya [2022] Bleicher, M., Bratkovskaya, E.: Modelling relativistic heavy-ion collisions with dynamical transport approaches. Prog. Part. Nucl. Phys. 122, 103920 (2022) https://doi.org/10.1016/j.ppnp.2021.103920
  * Kalisky [2008] Kalisky, M.: Reconstruction of charged kaons in the three pion decay channel in Pb+Au 158 AGeV collisions by the CERES experiment. PhD thesis, Darmstadt, Tech. U. (2008)
  * Adam et al. [2020] Adam, J., et al.: Strange hadron production in Au+Au collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}$ = 7.7, 11.5, 19.6, 27, and 39 GeV. Phys. Rev. C 102(3), 034909 (2020) https://doi.org/10.1103/PhysRevC.102.034909 arXiv:1906.03732 [nucl-ex]
  * Adamczyk et al. [2017] Adamczyk, L., et al.: Bulk Properties of the Medium Produced in Relativistic Heavy-Ion Collisions from the Beam Energy Scan Program. Phys. Rev. C 96(4), 044904 (2017) https://doi.org/10.1103/PhysRevC.96.044904 arXiv:1701.07065 [nucl-ex]
  * Adler et al. [2004] Adler, C., et al.: Kaon production and kaon to pion ratio in Au+Au collisions at s(NN)**1/2 = 130-GeV. Phys. Lett. B 595, 143–150 (2004) https://doi.org/10.1016/j.physletb.2004.06.044 arXiv:nucl-ex/0206008
  * Aggarwal et al. [2011] Aggarwal, M.M., et al.: Strange and Multi-strange Particle Production in Au+Au Collisions at $\sqrt{s_{NN}}$ = 62.4 GeV. Phys. Rev. C 83, 024901 (2011) https://doi.org/10.1103/PhysRevC.83.024901 arXiv:1010.0142 [nucl-ex]
  * Agakishiev et al. [2012] Agakishiev, G., et al.: Strangeness Enhancement in Cu+Cu and Au+Au Collisions at $\sqrt{s_{NN}}=200$ GeV. Phys. Rev. Lett. 108, 072301 (2012) https://doi.org/10.1103/PhysRevLett.108.072301 arXiv:1107.2955 [nucl-ex]
  * Abelev et al. [2009] Abelev, B.I., et al.: Systematic Measurements of Identified Particle Spectra in $pp,d^{+}$ Au and Au+Au Collisions from STAR. Phys. Rev. C 79, 034909 (2009) https://doi.org/10.1103/PhysRevC.79.034909 arXiv:0808.2041 [nucl-ex]
  * Abelev et al. [2013a] Abelev, B.B., et al.: $K^{0}_{S}$ and $\Lambda$ production in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV. Phys. Rev. Lett. 111, 222301 (2013) https://doi.org/10.1103/PhysRevLett.111.222301 arXiv:1307.5530 [nucl-ex]
  * Abelev et al. [2013b] Abelev, B., et al.: Centrality dependence of $\pi$, K, p production in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV. Phys. Rev. C 88, 044910 (2013) https://doi.org/10.1103/PhysRevC.88.044910 arXiv:1303.0737 [hep-ex]
  * Gazdzicki and Rohrich [1996] Gazdzicki, M., Rohrich, D.: Strangeness in nuclear collisions. Z. Phys. C 71, 55–64 (1996) https://doi.org/10.1007/s002880050147 arXiv:hep-ex/9607004
  * Alber et al. [1994] Alber, T., et al.: Strange particle production in nuclear collisions at 200-GeV per nucleon. Z. Phys. C 64, 195–207 (1994) https://doi.org/10.1007/BF01557391
  * Baechler et al. [1993] Baechler, J., et al.: Production of charged kaons in proton - nucleus and nucleus-nucleus collisions at 200-GeV/Nucleon. Z. Phys. C 58, 367–374 (1993) https://doi.org/10.1007/BF01557694
  * [38] Strabel, C.: Energieabhangigkeit der K0S-Produktion in zentralen Pb+Pb Reaktionen. PhD thesis, Johann Wolfgang Goethe-Universitat, 2006. https://edms.cern.ch/document/2958534/1
  * Adamczewski-Musch et al. [2019] Adamczewski-Musch, J., et al.: Sub-threshold production of K${}^{0}_{s}$ mesons and ${\Lambda}$ hyperons in Au+Au collisions at $\sqrt{s_{NN}}$ = 2.4 GeV. Phys. Lett. B 793, 457–463 (2019) https://doi.org/10.1016/j.physletb.2019.03.065 arXiv:1812.07304 [nucl-ex]
  * Adamczewski-Musch et al. [2018] Adamczewski-Musch, J., et al.: Deep sub-threshold $\phi$ production in Au+Au collisions. Phys. Lett. B 778, 403–407 (2018) https://doi.org/10.1016/j.physletb.2018.01.048 arXiv:1703.08418 [nucl-ex]
  * Agakishiev et al. [2009] Agakishiev, G., et al.: Phi decay: A Relevant source for K- production at SIS energies? Phys. Rev. C 80, 025209 (2009) https://doi.org/10.1103/PhysRevC.80.025209 arXiv:0902.3487 [nucl-ex]
  * Agakishiev et al. [2010] Agakishiev, G., et al.: In-Medium Effects on K0 Mesons in Relativistic Heavy-Ion Collisions. Phys. Rev. C 82, 044907 (2010) https://doi.org/10.1103/PhysRevC.82.044907 arXiv:1004.3881 [nucl-ex]
  * Gasik et al. [2016] Gasik, P., et al.: Strange meson production in Al+Al collisions at 1.9 A GeV. Eur. Phys. J. A 52(6), 177 (2016) https://doi.org/10.1140/epja/i2016-16177-y arXiv:1512.06988 [nucl-ex]
  * Lopez et al. [2010] Lopez, X., et al.: Measurement of $K^{*}(892)^{0}$ and $K^{0}$ mesons in Al+Al collisions at 1.9$A$ GeV. Phys. Rev. C 81, 061902 (2010) https://doi.org/10.1103/PhysRevC.81.061902 arXiv:1006.1905 [nucl-ex]
  * Acharya et al. [2022] Acharya, S., et al.: Neutral to charged kaon yield fluctuations in Pb – Pb collisions at sNN=2.76 TeV. Phys. Lett. B 832, 137242 (2022) https://doi.org/10.1016/j.physletb.2022.137242 arXiv:2112.09482 [nucl-ex]
  * Kapusta et al. [2023] Kapusta, J.I., Pratt, S., Singh, M.: Confronting anomalous kaon correlations measured in Pb-Pb collisions at sNN=2.76 TeV. Phys. Rev. C 107(1), 014913 (2023) https://doi.org/10.1103/PhysRevC.107.014913 arXiv:2210.03257 [hep-ph]
  * Bonesini et al. [2001] Bonesini, M., Marchionni, A., Pietropaolo, F., Fatis, T.: On Particle production for high-energy neutrino beams. Eur. Phys. J. C 20, 13–27 (2001) https://doi.org/10.1007/s100520100656 arXiv:hep-ph/0101163
  * Stepaniak and Pszczel [2023] Stepaniak, J., Pszczel, D.: On the relation between $K^{0}_{s}$ and charged kaon yields in proton-proton collisions (2023) arXiv:2305.03872 [hep-ph]
  * Adhikary et al. [2023] Adhikary, H., et al.: Measurement of hadron production in $\pi$–C interactions at 158 and 350 GeV/c with NA61/SHINE at the CERN SPS. Phys. Rev. D 107(6), 062004 (2023) https://doi.org/10.1103/PhysRevD.107.062004 arXiv:2209.10561 [nucl-ex]
  * Petrán et al. [2013] Petrán, M., Letessier, J., Petráček, V., Rafelski, J.: Hadron production and quark-gluon plasma hadronization in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV. Phys. Rev. C 88(3), 034907 (2013) https://doi.org/10.1103/PhysRevC.88.034907 arXiv:1303.2098 [hep-ph]
  * Rafelski and Petran [2014] Rafelski, J., Petran, M.: Universal QGP Hadronization Conditions at RHIC and LHC. EPJ Web Conf. 78, 06004 (2014) https://doi.org/10.1051/epjconf/20147806004 arXiv:1406.1871 [nucl-th]
  * Rafelski and Petran [2015] Rafelski, J., Petran, M.: QCD phase transition studied by means of hadron production. Phys. Part. Nucl. 46(5), 748–755 (2015) https://doi.org/10.1134/S1063779615050238 arXiv:2212.13302 [hep-ph]
  * Pisarski and Wilczek [1984] Pisarski, R.D., Wilczek, F.: Remarks on the Chiral Phase Transition in Chromodynamics. Phys. Rev. D 29, 338–341 (1984) https://doi.org/10.1103/PhysRevD.29.338
  * Giacosa et al. [2018] Giacosa, F., Koenigstein, A., Pisarski, R.D.: How the axial anomaly controls flavor mixing among mesons. Phys. Rev. D 97(9), 091901 (2018) https://doi.org/10.1103/PhysRevD.97.091901 arXiv:1709.07454 [hep-ph]
  * Giacosa et al. [2023] Giacosa, F., Jafarzade, S., Pisarski, R.: Anomalous interactions for heterochiral mesons with $J^{PC}=1^{+-}$ and $2^{-+}$ (2023) arXiv:2309.00086 [hep-ph]
  * Patrignani et al. [2016] Patrignani, C., et al.: Review of Particle Physics. Chin. Phys. C 40(10), 100001 (2016) https://doi.org/10.1088/1674-1137/40/10/100001
  * Braun-Munzinger et al. [2002] Braun-Munzinger, P., Cleymans, J., Oeschler, H., Redlich, K.: Maximum relative strangeness content in heavy ion collisions around 30-GeV/A. Nucl. Phys. A 697, 902–912 (2002) https://doi.org/10.1016/S0375-9474(01)01257-X arXiv:hep-ph/0106066
  * Cleymans et al. [2016] Cleymans, J., Hippolyte, B., Oeschler, H., Redlich, K., Sharma, N.: Thermal Model Description of Collisions of Small Nuclei (2016) arXiv:1603.09553 [hep-ph]
  * Rafelski [2015] Rafelski, J.: Melting Hadrons, Boiling Quarks. Eur. Phys. J. A 51(9), 114 (2015) https://doi.org/10.1140/epja/i2015-15114-0 arXiv:1508.03260 [nucl-th]
  * Poberezhnyuk et al. [2019] Poberezhnyuk, R., Vovchenko, V., Motornenko, A., Gorenstein, M.I., Stoecker, H.: Chemical freeze-out conditions and fluctuations of conserved charges in heavy-ion collisions within quantum van der Waals model. Phys. Rev. C 100(5), 054904 (2019) https://doi.org/10.1103/PhysRevC.100.054904 arXiv:1906.01954 [hep-ph]
  * Vovchenko et al. [2018] Vovchenko, V., Gorenstein, M.I., Stoecker, H.: Finite resonance widths influence the thermal-model description of hadron yields. Phys. Rev. C 98(3), 034906 (2018) https://doi.org/10.1103/PhysRevC.98.034906 arXiv:1807.02079 [nucl-th]
  * Cleymans et al. [2006] Cleymans, J., Oeschler, H., Redlich, K., Wheaton, S.: Comparison of chemical freeze-out criteria in heavy-ion collisions. Phys. Rev. C 73, 034905 (2006) https://doi.org/10.1103/PhysRevC.73.034905 arXiv:hep-ph/0511094
  * Becattini et al. [2006] Becattini, F., Manninen, J., Gazdzicki, M.: Energy and system size dependence of chemical freeze-out in relativistic nuclear collisions. Phys. Rev. C 73, 044905 (2006) https://doi.org/10.1103/PhysRevC.73.044905 arXiv:hep-ph/0511092
  * Andersson et al. [1983] Andersson, B., Gustafson, G., Soderberg, B.: A General Model for Jet Fragmentation. Z. Phys. C 20, 317 (1983) https://doi.org/10.1007/BF01407824