# Giant microwave-optical Kerr nonlinearity via Rydberg excitons in cuprous
oxide

Jon D. Pritchett  Department of Physics, Durham University, Durham DH1 3LE,
United Kingdom  These authors contributed equally to this work.  Liam A. P.
Gallagher  Department of Physics, Durham University, Durham DH1 3LE, United
Kingdom  These authors contributed equally to this work.  Alistair Brewin
Department of Physics, Durham University, Durham DH1 3LE, United Kingdom
Horatio Q. X. Wong  Department of Physics, Durham University, Durham DH1 3LE,
United Kingdom  Wolfgang Langbein  School of Physics and Astronomy, Cardiff
University, Cardiff CF24 3AA, United Kingdom  Stephen A. Lynch  School of
Physics and Astronomy, Cardiff University, Cardiff CF24 3AA, United Kingdom
C. Stuart Adams  Department of Physics, Durham University, Durham DH1 3LE,
United Kingdom  Matthew P. A. Jones  Department of Physics, Durham University,
Durham DH1 3LE, United Kingdom<EMAIL_ADDRESS>
###### Abstract

Microwave-optical conversion is key to future networks of quantum devices,
such as those based on superconducting technology. Conversion at the single
quantum level requires strong nonlinearity, high bandwidth, and compatibility
with a millikelvin environment. A large nonlinearity is observed in Rydberg
atoms, but combining atomic gases with dilution refrigerators is technically
challenging. Here we demonstrate that a strong microwave-optical nonlinearity
in a cryogenic, solid-state system by exploiting Rydberg states of excitons in
Cu2O. We measure a microwave-optical cross-Kerr coefficient of $B_{0}=0.022\pm
0.008$ m V-2 at 4 K, which is several orders of magnitude larger than other
solid-state systems. Our results highlight the potential of Rydberg excitons
for nonlinear optics, and form the basis for a microwave-optical frequency
converter based on Cu2O.

††journal: opticajournal

Superconducting microwave devices play a key role in quantum computation [1].
To eliminate thermal noise, these devices must be cooled to $T\approx 10$ mK,
a requirement which makes direct quantum networking impractical over distances
larger than $\sim 1$ m [2]. Microwave-optical (MO) conversion is therefore a
critical enabling technology [3, 4, 5], with current approaches including
electro-optics [6], rare-earth ions [7], optomechanical systems [8] and
quantum dot molecules [9]. Very strong nonlinearity can be achieved by
exploiting the large microwave dipole moment associated with highly excited
atomic Rydberg states [10], leading to the largest observed cross-Kerr effect
of any material [11]. However interfacing Rydberg atoms with planar
superconducting quantum devices in a millikelvin environment is an outstanding
challenge, and coupling has so far been achieved only at much higher
temperature [12, 13, 14, 15].

$\omega_{\mathrm{MW}}$VBCB$n$P$n$S/D571 nmE

optical

microwave

(a)(c)detector571 nmground
line(b)$-30$$-20$$-10$$0$$10$$20$$30$detuning/$2\pi$
(GHz)$10^{3}$$10^{2}$$10^{1}$$10^{0}$

intensity (103 s-1)

$+2\omega_{\mathrm{MW}}/2\pi$$\omega_{\mathrm{MW}}=2\times
7$GHz$I_{\mathrm{ON}}$$I_{\mathrm{OFF}}$ehCu2O at 4
K$\omega_{\mathrm{MW}}$stripline antennaFPElive
wire$-2\omega_{\mathrm{MW}}/2\pi$ Figure 1: (a) Excitons are created in state
$n$P via laser excitation at $\lambda\approx 571$ nm. A microwave field at
frequency $\omega_{\mathrm{MW}}$ couples odd- and even-parity exciton states.
The exciton wavefunction spans many lattice sites giving rise to a large
dipole moment. (b) Excitons are created in a thin slab of Cu2O located between
the conductors of a microwave stripline. The transmitted light is filtered
with a Fabry-Pérot etalon (FPE) and sent to a detector. (c) Example spectrum
of the transmitted light for $E=hc/\lambda=2.170127$ eV and
$\mathcal{E}_{\mathrm{MW}}=170$ V/m, showing the appearance of sidebands at
$\pm 2\omega_{\mathrm{MW}}/(2\pi)$ when microwaves are applied
($I_{\mathrm{ON}}$).

In this Letter we combine the advantages of atomic Rydberg states (giant
nonlinearity) and the solid state (milliKelvin compatibility [16, 17]) by
using Rydberg states of excitons in the bulk semiconductor Cu2O [18] (Fig. 1).
Excitons are optically excited bound states of an electron and a hole, with an
internal structure that resembles a hydrogen atom. In Cu2O, the resulting
Rydberg series of excitonic states has been measured up to principal quantum
number $n=30$ at $T\approx 40$ mK [17]. Transitions between neighbouring
states of opposite parity are located in the microwave spectral region. The
associated dipole moment scales as $d\propto n^{2}$, reaching 180 $e$ nm at
$n=11$, which is more than thirty times larger than in quantum dot molecules
[9]. Thus Rydberg excitons in Cu2O provide a unique platform for MO coupling
in the solid-state [19], with recent experiments demonstrating microwave
control of the linear and nonlinear optical response [20]. Here we extend this
work to a measurement of the optical phase shift induced by the presence of a
microwave field, characterised by the microwave-optical Kerr coefficient
$B_{0}$

$B_{0}=\frac{\Delta\phi}{2\pi L\left|\mathcal{E}_{\mathrm{MW}}\right|^{2}},$
(1)

where $\Delta\phi$ is the optical phase shift induced by the microwave field,
$L$ is the length of the material and $\mathcal{E}_{\mathrm{MW}}$ is the
amplitude of the applied microwave field.

A schematic of the experiment is shown in Fig. 1. A thin slab of Cu2O
(55$\pm$10 $\mathrm{\SIUnitSymbolMicro m}$) is mounted between the conductors
of a microwave stripline and cooled to $T=4$ K. Excitons in a quantum state
$n$P (where P denotes orbital angular momentum $l=1$) were excited from the
ground state (valence band) using a single-frequency laser at wavelength
$\lambda\approx 571$ nm. A microwave field at angular frequency
$\omega_{\mathrm{MW}}=2\pi\times 7$ GHz couples $n$P states to nearby states
of opposite parity (e.g. $n^{\prime}$S, $n^{\prime}$D, where S and D indicate
$l=0$ and $l=2$). The spectrum of the transmitted laser light was resolved
using a scanning Fabry-Perot etalon. Further experimental details are provided
in Appendix A. Applying the microwave field leads to a change in transmission
at the laser frequency $\omega_{\mathrm{L}}=2\pi c/\lambda$, and the
appearance of sidebands at frequencies $\omega_{\mathrm{L}}\pm
2\omega_{\mathrm{MW}}$ on the light transmitted through the sample (Fig. 1(c))
that are indicative of a cross-Kerr nonlinearity.

The optical absorption spectrum close to the bandgap is shown in Fig. 2(a).
Peaks corresponding to $n$P excitonic states are visible up to $n=16$, limited
by a combination of temperature and sample quality. As Cu2O is
centrosymmetric, the MO coupling absent from the ground-state symmetry but is
created by the Rydberg excitons, as shown in Fig. 2(b), which plots the
spectrum of the microwave-induced change in absorption $\Delta\alpha$. Each
exciton resonance shows decreased absorption on-resonance, and increased
absorption on each side at the location of the $n$S and $n$D states. In
contrast to Rydberg atoms, a response is observed over a wide range of $n$,
even for a single microwave frequency. This is due to non-radiative broadening
of the exciton lines, which means that the microwave field is near-resonant
with many transitions simultaneously. The same effect results in a broad
microwave frequency dependence (see Appendix A and [20]). $|\Delta\alpha|$
increases with $n$ up to $n=13$ before decreasing towards the band edge. As
shown in the inset of Fig. 2(b) $|\Delta\alpha|$ increases linearly with the
microwave power as expected for a Kerr nonlinearity, before saturating at a
value which depends on $n$. Saturation occurs due to multi-photon processes,
and is accompanied by the appearance of higher-order sidebands. For the
remainder of this paper we consider only the linear regime.

The MO Kerr nonlinearity can be described in terms of the dielectric
polarization at the laser frequency

$\mathcal{P}(\omega_{\mathrm{L}})=\epsilon_{0}\left(\chi^{(1)}(\omega_{\mathrm{L}})\mathcal{E}_{\mathrm{L}}+\chi^{(3)}(\omega_{\mathrm{L}})\mathcal{E}_{\mathrm{L}}\mathcal{E}_{\mathrm{MW}}\mathcal{E}_{\mathrm{MW}}+\ldots\right),$

where $\chi^{(1)}$and $\chi^{(3)}$ are the linear and Kerr nonlinear
susceptibilities respectively. Experimentally we observe that the nonlinearity
depends very weakly on the laser and microwave polarizations, and so a scalar
treatment is used. We obtain an expression for $\chi^{(3)}$ by treating the
exciton states as hydrogen-like [20]. $\Delta\alpha$ is related to the
imaginary part of the susceptibility

$\frac{\Delta\alpha
c}{\omega_{\mathrm{L}}{\left|\mathcal{E}_{\mathrm{MW}}\right|^{2}}}=\mathrm{Im}\left(\chi^{(3)}(\omega_{\mathrm{L}})\right)=\mathrm{Im}\left(\sum_{n,n^{\prime},l^{\prime},n^{\prime\prime},\pm}\chi^{(3)}_{n\mathrm{P}n^{\prime}l^{\prime},n^{\prime\prime}\mathrm{P}}(\omega_{\mathrm{L}})\right).$

Each term in the sum is given by

$\begin{split}&\chi^{(3)}_{n\mathrm{P}n^{\prime}l^{\prime}n^{\prime\prime}\mathrm{P}}(\omega_{\mathrm{L}})=\\\
&\frac{N_{\mathrm{d}}}{8\epsilon_{0}\hbar^{3}}\frac{D^{\mathrm{VB}\rightarrow
n\mathrm{P}}d^{n\mathrm{P}\rightarrow
n^{\prime}l^{\prime}}d^{n^{\prime}l^{\prime}\rightarrow
n^{\prime\prime}\mathrm{P}}D^{n^{\prime\prime}\mathrm{P}\rightarrow\mathrm{VB}}}{(\delta_{n\mathrm{P}}-i\Gamma_{n\mathrm{P}})(\delta_{n^{\prime}l^{\prime}}^{\pm\omega_{\mathrm{MW}}}-i\Gamma_{n^{\prime}l^{\prime}})(\delta_{n^{\prime\prime}\mathrm{P}}-i\Gamma_{n^{\prime\prime}\mathrm{P}})},\end{split}$
(2)

where $N_{\mathrm{d}}$ is the effective density of exciton states, $D$ are
effective matrix elements for transitions from the valence band to the P
states, $d$ are dipole matrix elements for transitions between exciton states,
$\Gamma$ are the exciton linewidths and $\delta$ are detunings, given by
$\delta_{n\mathrm{P}}=\omega_{n\mathrm{P}}-\omega_{\mathrm{L}}$ and
$\delta^{\pm\omega_{\mathrm{MW}}}_{n^{\prime}l^{\prime}}=\omega_{n^{\prime}l^{\prime}}-(\omega_{\mathrm{L}}\pm\omega_{\mathrm{MW}})$.
Crucially all of these parameters are either known theoretically or can be
derived from analysis of data without a microwave field applied (see Appendix
C) enabling the use of equation 2 for quantitative predictions of the
nonlinear response.

The predicted $\Delta\alpha$ is compared to the data in Fig. 2(b). The only
adjustable parameter is the amplitude of the microwave electric field
$\mathcal{E}_{\mathrm{MW}}$. The prediction is in excellent agreement with the
data over the full range of $n$. Similar levels of agreement are observed up
to field amplitudes of 80 V/m, above which saturation occurs. From these fits
we obtain a calibration between electric field amplitude and applied microwave
power of $43\pm 3$ (V/m)/mW1/2, in reasonable agreement with a value of 70
(V/m)/mW1/2 obtained from a finite-element simulation of the antenna that
excludes connection losses [20].

Once the microwave field amplitude is known, the real part of the
susceptibility can be used to make a quantitative prediction of the microwave-
optical Kerr coefficient, since

$B_{0}=\frac{\omega_{\mathrm{L}}}{4\eta\pi
c}\mathrm{Re}\left(\chi^{(3)}(\omega_{\mathrm{L}})\right),$

where $\eta=2.8$ is the optical refractive index. Predicted values of $B_{0}$
are shown in Fig. 3(a).

To independently measure $B_{0}$ we make use of the zero-crossings in Fig. 2.
At these points $\Delta\alpha=0$, the imaginary part of the susceptibility is
zero, and sidebands are generated via phase modulation only. By using the
conventional expansion of a phase-modulated wave in terms of Bessel functions,
it can be shown that the ratio of the intensity of the sidebands to the
carrier is $I_{\mathrm{SB}}/I_{\mathrm{C}}=\left|J_{1}(\Delta\phi)\right|^{2}$
where $J_{1}$ is the first-order Bessel function of the first kind. Thus the
microwave-induced phase shift $\Delta\phi$ can be directly extracted from
spectra like Fig. 1(c).

The variation of $\Delta\phi$ with $|\mathcal{E}_{\mathrm{MW}}|^{2}$ is shown
for two zero-crossings in Fig. 3(b). As expected the phase shift increases
linearly in the Kerr regime, before saturating. The Kerr coefficient is
obtained from the gradient in the linear regime using Eq. 1 and the field
amplitude calibration from the absorption data (Fig. 2).

As shown in Fig. 3(a), the measured Kerr coefficient is in agreement with the
prediction from Eq. 2. We emphasise that the phase shift measurement does not
depend on the model for $\chi^{(3)}$ (Eq. 2) and its input parameters; these
affect the measured value of $B_{0}$ only through the field calibration. The
dominant experimental uncertainties come from the electric field calibration
and the thickness of the sample. The analysis also assumes that
$\chi^{(3)}(\omega_{\mathrm{L}}\pm
2\omega_{\mathrm{MW}})=\chi^{(3)}(\omega_{\mathrm{L}})$, which is not strictly
true due to the strong energy dependence near resonance. An indication that
this assumption does not fully hold is the asymmetry in the red and blue
sideband amplitude in Fig. 3. We therefore include the difference between the
red and blue values as part of the uncertainty in $B_{0}$ and present the
average value. Analytic expressions for $\chi^{(3)}(\omega_{\mathrm{L}}\pm
2\omega_{\mathrm{MW}})$ are provided in Appendix B.

2.1704
---
2.1708
---
2.1712
---
2.1716
---
2.0
---
3.0
---
4.0
---
0.00
---
0.02
---

$\Delta\alpha L$
---

$E$ (eV)
---

$\alpha L$
---

$-0.02$
---
$-0.04$
---
2.1704
---
2.1708
---
2.1712
---
2.1716
---
$E$ (eV)
---
$\tiny{10^{-1}}$
---
$\tiny{10^{-2}}$
---
$\tiny{0.1}$
---
$\tiny{10}$
---
$\tiny{100}$
---
$\tiny{1}$
---
$P_{\mathrm{MW}}$ (mW)
---
9P
---
10P
---
12P
---
9P
---
10P
---
11P
---
12P
---
8P
---
(a)
---
(b)
---

$|\Delta\alpha L|$
---

Figure 2: (a) Rydberg series of exciton absorption lines. (b) Measured (red points) and predicted (solid line) change in absorption, $\Delta\alpha L$ at $\mathcal{E}_{\mathrm{MW}}=38$ V/m. Dashed lines indicate the zero crossings used to extract the Kerr coefficient. Inset shows $\Delta\alpha$ versus microwave power $P_{\mathrm{MW}}$ at the $n=9,10,12$P resonances. Solid lines are fits to the linear region. Material |  $B_{0}$ (m V-2) |  $\alpha$ (m-1) |  $\mathcal{F}$ (m2 V-2)
---|---|---|---
glass |  $10^{-14}$ [21] |  $10^{-1}$ [21] | $10^{-13}$
nitrobenzene |  $10^{-12}$ [21] |  $10^{-1}$ [22] | $10^{-11}$
PMN-PT |  $10^{-7}$ [23] |  $10^{2}$ [23] | $10^{-9}$
GO-LC |  $10^{-6}$ [24] |  $10^{3}$ [25] | $10^{-9}$
Rb vapour |  $10^{-6}$ [11] |  $10^{1}$ [26] | $10^{-7}$
Cu2O | $10^{-2}$ | $10^{4}$ | $10^{-6}$
Table 1: Comparison of Kerr ($B_{0}$) and absorption ($\alpha$) coefficients,
and their ratio $\mathcal{F}=B_{0}/\alpha$ for various materials. For
nitrobenzene, glass and Cu2O $\lambda=570$ nm; for PMN-PT (a transparent
ceramic) and GO-LC (graphene oxide liquid crystals) $\lambda=633$ nm; Rb
vapour $\lambda=$ 780 nm.

Our highest measured value of $B_{0}=0.022\pm 0.008$ V m-2 at $n=12$ is
compared to other low-frequency (DC) Kerr coefficients in Table 1. Our
measured Kerr coefficient is extremely large, due to a combination of the
underlying resonant nature of the nonlinearity and the large dipole moment.
The other values in Table 1 were measured far from resonance - a key feature
of Cu2O is that a much larger but still broadband response can be obtained via
multiple resonances. Cu2O also has the highest absorption coefficient among
the materials in Table 1. An alternative figure of merit is the phase shift
per unit absorption length and microwave intensity, described by the ratio
$\mathcal{F}=B_{0}/\alpha$ where $\alpha$ is the linear absorption
coefficient. Here, Cu2O has the largest $\mathcal{F}$, with only Rydberg atoms
in Rb vapour being comparable, highlighting the importance of Rydberg physics
in the solid-state.

Despite the large values of $B_{0}$ and $\mathcal{F}$, the maximum phase shift
remained limited to around 0.1 radians as shown in Fig. 2(b). This is due to a
combination of the saturation of the phase shift at high electric field and
the high background absorption, which together limit the maximum phase shift
that can be measured. The saturation is a consequence of the extremely strong
nonlinearity, and corresponds to the emergence of higher order terms (e.g.
$\chi^{(5)}$) in the susceptibility. In fact the experiment enters the ultra-
strong driving regime, where the coupling strength
$\Omega_{\mathrm{MW}}=d\mathcal{E}_{\mathrm{MW}}/\hbar\gg\omega_{\mathrm{MW}},\Gamma$.
Microwave-optical conversion still occurs in this regime, but it is no longer
given by a simple Kerr effect. A reduction in absorption would enable the use
of thicker samples, resulting in larger phase shifts. The absorption is
dominated by phonon-assisted processes that do not involve Rydberg states, and
which are unaffected by the microwave field [27]. At $n=11$, 80% of the
absorption coefficient is due to the background. As in atomic Rydberg gases,
the background may be reduced by nonlinear spectroscopy techniques such as
second harmonic generation [28, 29] or electromagnetically induced
transparency [26, 30], or by exploiting Rydberg exciton-polaritons [31].

In conclusion, we have demonstrated that the strong microwave-optical
nonlinearity observed in atomic gases can be realised in a cryogenic, solid-
state setting using Rydberg excitons in Cu2O. The experimentally measured Kerr
coefficient $B_{0}=0.022\pm 0.008$ m V-2 is 4 orders of magnitude larger than
other solid-state platforms, with only Rydberg atoms being comparable. The
observations are in agreement with a model based on the conventional hydrogen-
like theory of Wannier-Mott excitons, enabling the quantitative design of
future devices that exploit the giant Kerr effect, additionally we highlight
the effect is visible at relatively low $n$ meaning that synthetic material
could be utilised [32, 33, 34]. Our work complements efforts to prepare
quantum states of light using Rydberg-mediated interactions in Cu2O [35, 36,
37, 38], and opens a potential route to microwave-optical conversion in the
quantum regime.

2.17042.17082.17122.1716$E$(eV)

$B_{0}$(m V-2)

$10^{-4}$$10^{-2}$$10^{-3}$

$I_{\mathrm{SB}}^{\pm}/I_{\mathrm{C}}$

0.10.01

$\Delta\phi$(rad)

$I_{\mathrm{SB}}^{\pm}/I_{\mathrm{C}}$

$10^{-4}$$10^{-2}$$10^{-3}$0.10.01

$\Delta\phi$(rad)

$-0.06$$0.06$$0.00$(b)(a)(i)(ii)$\mathcal{E}_{\mathrm{MW}}^{2}$(V/m)$10^{3}$$10^{4}$$10^{5}$$\mathcal{E}_{\mathrm{MW}}^{2}$(V/m)$10^{3}$$10^{4}$$10^{5}$
Figure 3: (a) Predicted (solid line) and measured (points) Kerr coefficient vs
excitation energy $E$. Shaded background shows exciton spectrum for reference.
(b) $I_{\mathrm{SB}}/I_{\mathrm{C}}$ vs $\mathcal{E}_{\mathrm{MW}}$ for
positive frequency (blue) and negative frequency (red) sidebands. Right-hand
axis gives the corresponding phase shift.

##  Appendix A Experimental details

The Cu2O sample was prepared from a natural gemstone (Tsumeb mine) using the
procedure detailed in [32], and glued to a CaF2 window. The light had an
incident intensity of 20 $\mathrm{\SIUnitSymbolMicro W}$/mm2, propagated along
the [111] crystallographic axis, and was linearly polarized. Finite element
analysis showed that the microwave electric field was aligned in the plane of
the sample and orthogonal to stripline. $\Delta\alpha$ was observed to depend
very weakly on the angle between microwave and optical electric fields with a
less than $10\%$ difference observed when varying the angle by $\pi/2$ [39].
No microwave-induced change in the polarization of the transmitted light was
observed within the experimental uncertainty. The etalon had a free spectral
range $60.1\pm 0.2$ GHz and a finesse of $44.5\pm 0.7$, and was temperature-
tuned. The carrier and sideband amplitudes were extracted by fitting the
Lorentzian response function of the etalon to spectra like those in Fig. 1(c).

As stated in the text, the nonlinear response is very broadband, easily
covering the range 1-20 GHz. However antenna resonances make comparing
multiple frequencies challenging [20]. We therefore fixed
$\omega_{\mathrm{MW}}=2\pi\times 7$ GHz to give good resolution of the
sidebands while remaining in the relevant range for superconducting quantum
circuits.

##  Appendix B Sideband susceptibility

The nonlinear susceptibility at the frequency of the red and blue sidebands is

$\begin{split}&\chi^{(3)}_{n\mathrm{P}n^{\prime}l^{\prime}n^{\prime\prime}\mathrm{P}}(\omega_{\mathrm{L}}\pm
2\omega_{\mathrm{MW}})=\\\
&\frac{N_{\mathrm{d}}}{2\epsilon_{0}\hbar^{3}}\frac{D^{\mathrm{VB}\rightarrow
n\mathrm{P}}d^{n\mathrm{P}\rightarrow
n^{\prime}l^{\prime}}d^{n^{\prime}l^{\prime}\rightarrow
n^{\prime\prime}\mathrm{P}}D^{n^{\prime\prime}\mathrm{P}\rightarrow\mathrm{VB}}}{(\delta_{n\mathrm{P}}-i\Gamma_{n\mathrm{P}})(\delta_{n^{\prime}l^{\prime}}^{\pm\omega_{\mathrm{MW}}}-i\Gamma_{n^{\prime}l^{\prime}})(\delta^{\pm
2\omega_{\mathrm{MW}}}_{n^{\prime\prime}\mathrm{P}}-i\Gamma_{n^{\prime\prime}\mathrm{P}})}.\end{split}$
(3)

The only difference between this and Eq. 2 is the detuning in the third term
on the denominator, which is given by $\delta^{\pm
2\omega_{\mathrm{MW}}}_{n^{\prime}\mathrm{P}}=\omega_{n^{\prime}\mathrm{P}}-(\omega_{\mathrm{L}}\pm
2\omega_{\mathrm{MW}})$. In principle, it is possible to derive the sideband
amplitude by using these susceptibilities as source terms. However this
calculation is strongly dependent on the unknown relative phase of each term
in the summation [29].

##  Appendix C Parameters for model

Here we give details of the parameters in the susceptibility model (Eq. 2).
States from $n=5$ to $17$ and $l=0$ to 2 are included in the model.

Matrix elements for transitions between exciton states, $d$, were calculated
using a spinless hydrogen-like model [30] for the exciton states, with quantum
defects obtained from dispersion relations for the electron and hole [40].

The widths, energies and matrix elements $D$ for transitions from the valence
band to the $n$P exciton state were extracted from experimental absorption
spectra without microwaves. Values for the energy and width of the S and D
states were extracted from [29]. In the two-photon experiments only states up
to $n=12$ were observed and so extrapolation was used to extend the range of
$n$ beyond that available from these experiments. The energies were
extrapolated using a quantum defect model

$E_{nl}=E_{\mathrm{g}}-\frac{R_{\mathrm{X}}}{(n-\delta_{l})^{2}},$ (4)

where $R_{\mathrm{X}}$ is the excitonic Rydberg energy and $\delta_{l}$ is the
quantum defect.

The widths of the P states were fitted using the equation

$\Gamma_{n\mathrm{P}}=\frac{\Gamma}{n^{3}}+\Gamma_{0},$ (5)

where $\Gamma_{0}$ is constant offset attributed to an inhomogenous broadening
due to charges and defects in the material [41]. For the S and D widths, the
$\Gamma_{0}$ was fixed to the value fitted from the P state trend and the high
$n$ states were extrapolated.

Funding This work was supported by the Engineering and Physical Sciences
Research Council (EPSRC), United Kingdom, through research grants
EP/P011470/1, EP/P012000/1, EP/X038556/1 and EP/X03853X/1.

Acknowledgments We thank I. Chaplin and S. Edwards for preparing the samples,
and V. Walther for calculating the dipole matrix elements.

Disclosures The authors declare no conflicts of interest

Data availability Data supporting this work is available at
doi:10.15128/r27d278t10x

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June, 10, 2020

Proposal for Beam-time

for constraining the

Symmetry energy at high densities from
neutron/proton flow excitation functions

Spokespersons: P. Russotto∗, A. Le Fèvre, J. Łukasik<EMAIL_ADDRESS>INFN-LNS, Catania, Italy Ion Species: 197Au @ 250, 400, 600, 1000 AMeV

Main Experimental/Scientific goals

  * •

Determination of the high density behavior of the symmetry energy through the
simultaneous measurement of elliptic flow excitation functions of neutrons,
protons and light clusters. The elliptic flow developed in relativistic heavy
ion collisions has been proven theoretically and experimentally to have a
unique sensitivity and robustness in probing the symmetry energy up to around
$2\rho_{o}$. The knowledge of the density dependence of the symmetry energy in
a broad range of densities will provide a missing link for astrophysical
predictions of the neutron star mass–radius relation. In particular, the data
will provide tighter constraints on the slope parameter L and entirely new
limits on $K_{sym}$, the currently poorly constrained symmetry energy
curvature parameter.

  * •

Enforcement of tight constraints on nuclear transport theories by providing
new data on the symmetry energy and the inter-related phenomena of clustering
and neutron and proton emissions as well as correlations among them.

The proposed campaign represents a unique set of measurements, presently
possible only at the GSI/FAIR facility, because of the available range of beam
energies and the existing instrumentation.

###### Abstract

The proposed experimental program aims at putting new and more stringent
constraints on the density dependence of the symmetry energy at supra-
saturation densities. Densities toward $2\rho_{o}$, indispensable for
obtaining the constraints relevant for astrophysics, are expected to be
reached in the proposed central heavy ion collisions. Proposed systems: Au+Au
at 250, 400, 600 and 1000 AMeV. Such an energy scan can currently be performed
only at the GSI/FAIR facility. Key observables: excitation function of the
neutron/proton elliptic-flow ratios, directed and elliptic flows for n, p and
isotopically resolved light clusters, yields and ratios, energy and angular
distributions, correlation functions. The experimental setup will be based on
the NeuLAND detector for measuring neutrons, protons and light charged
clusters emitted from mid-rapidity. The main novelty of this proposal consists
in using NeuLAND to obtain well resolved proton spectra allowing to probe
effectively about 30% higher densities than with only elemental resolution.
The setup will include also the KRAB plastic barrel surrounding the target for
providing the multiplicity trigger and for estimating the centrality and the
reaction plane orientation by covering polar angles beyond $30^{\circ}$, four
double-rings of CHIMERA, and the R3B New Time-of-Flight Wall TOFD for
extracting event-by-event the centrality and the orientation of the reaction
plane at polar angles up to $30^{\circ}$. The START detector placed upstream
of the target will provide the reference signal for the time of flight
measurement and trigger. The KRATTA triple telescopes and the FARCOS array
will be placed near the target for measuring light charged particles at mid-
rapidity and at backward angles, respectively. Expected value added in nuclear
and astrophysics: providing new constraints on the symmetry energy up to
$2\rho_{o}$ from simultaneous measurement of n, p and isotopically resolved
light charged particles. Specifically, the new measurement is expected to
provide tighter constraints on the slope parameter L and entirely new ones on
the, up to now very weakly constrained, curvature parameter $K_{sym}$. Probing
the densities toward $2\rho_{o}$ is indispensable for meaningful comparisons
with modern nuclear theories. The new data on flow patterns and correlations,
and on competition between cluster and neutron and proton emissions will
provide valuable constraints for the transport models aiming at describing and
explaining these phenomena. Proposed measurement will provide results
complementary to those obtained with the ground and satellite based X-ray
telescopes and with the gravitational wave interferometers. The results are
also expected to be competitive in terms of precision. The knowledge of the
Symmetry Energy in a broad range of densities will provide a missing link for
realistic simulations of astrophysical objects and processes. Expected
instrumental value added: commissioning of the KRAB and FARCOS detectors.

Symmetry energy at high densities from
neutron/proton flow excitation functions

Spokespersons: P. Russotto${}^{\mbox{\footnotesize{1}}}$, A. Le
Fèvre${}^{\mbox{\footnotesize{2}}}$, J. Łukasik${}^{\mbox{\footnotesize{3}}}$

Principal Investigators: K. Boretzky${}^{\mbox{\footnotesize{2}}}$, M.D.
Cozma${}^{\mbox{\footnotesize{4}}}$, E. De
Filippo${}^{\mbox{\footnotesize{5}}}$, I.
Gašparić${}^{\mbox{\footnotesize{6}}}$, A. Le
Fèvre${}^{\mbox{\footnotesize{2}}}$, Y. Leifels${}^{\mbox{\footnotesize{2}}}$,
I. Lihtar${}^{\mbox{\footnotesize{7}}}$, J.
Łukasik${}^{\mbox{\footnotesize{3}}}$, S.
Pirrone${}^{\mbox{\footnotesize{5}}}$, G.
Politi${}^{\mbox{\footnotesize{5,8}}}$, P.
Russotto${}^{\mbox{\footnotesize{1}}}$, W.
Trautmann${}^{\mbox{\footnotesize{2}}}$

Institutions: ${}^{\mbox{\footnotesize{1}}}$INFN-LNS, Catania, Italy;
${}^{\mbox{\footnotesize{2}}}$GSI, Darmstadt, Germany;
${}^{\mbox{\footnotesize{3}}}$IFJ PAN, Kraków, Poland;
${}^{\mbox{\footnotesize{4}}}$IFIN-HH, Bucharest, Romania;
${}^{\mbox{\footnotesize{5}}}$INFN-Sezione di Catania, Italy;
${}^{\mbox{\footnotesize{6}}}$RBI, Zagreb, Croatia;
${}^{\mbox{\footnotesize{7}}}$Univ. of Zagreb, Croatia;
${}^{\mbox{\footnotesize{8}}}$Università di Catania, Italy;

Scientific Context and Motivation. The nuclear matter Equation of State (EoS)
is one of the central topics in contemporary nuclear physics. In general, it
describes the relation between density, pressure, energy, temperature and the
isospin asymmetry $\delta=(\rho_{n}-\rho_{p})/\rho$, where $\rho_{n}$,
$\rho_{p}$, and $\rho$ are the neutron, proton and nuclear matter densities,
respectively. For cold nuclear matter it is conventionally split into a
symmetric matter part independent of $\delta$ and an isospin term, expressed
as a product of the symmetry energy111In the whole document we use the term
”symmetry energy”, instead of the more appropriate term ”asymmetry energy”,
for consistency with what is commonly used by scientific community and in
literature., $E_{sym}(\rho)$, and $\delta^{2}$ [1]:
$E(\rho,\delta)=E(\rho,0)+E_{sym}(\rho)\delta^{2}+o(\delta^{4})$.

Different density dependences of $E_{sym}(\rho)$ can be described
quantitatively by expanding $E_{sym}$ around the normal nuclear matter
density, $\rho_{o}$, leading to the following expression:

$E_{sym}(\rho)=E_{sym,0}+\frac{L}{3}\left(\frac{\rho-\rho_{o}}{\rho_{o}}\right)+\frac{K_{sym}}{18}\left(\frac{\rho-\rho_{o}}{\rho_{o}}\right)^{2}+...$
(1)

where the value of the symmetry energy at normal density $E_{sym,0}\equiv
E_{sym}(\rho=\rho_{o})$, the slope parameter $L\equiv
3\rho_{o}\;\frac{\partial
E_{sym}(\rho)}{\partial\rho}\Big{|}_{\rho=\rho_{o}}$, and the curvature
parameter (symmetry compressibility) $K_{sym}\equiv
9\rho_{o}^{2}\;\frac{\partial^{2}E_{sym}(\rho)}{\partial\rho^{2}}\Big{|}_{\rho=\rho_{o}}$.

A theoretical determination of the nuclear EoS from first principles by
microscopic calculations is challenging and a subject of current scientific
research since several decades [2]. In fact, microscopic calculations of the
density functional of nuclear matter employing different approaches to the
nucleon-nucleon interaction predict rather different forms of the EoS. In
particular, the dependence of $E_{sym}$ on density $\rho$ shows very different
behaviors. Most calculations coincide at or slightly below normal nuclear
matter density, which demonstrates that constraints from finite nuclei are
active for an average density smaller than $\rho_{o}$ and surface effects play
a role. In contrast to that, extrapolations to supra-normal densities diverge
dramatically, calling for more tight experimental constraints in this region.
However, significant progress is currently being made: recently, calculations
based on chiral effective field theory ($\chi EFT$), combined with advanced
statistical methods, have been able to predict the $E_{sym}$ at 2$\rho_{o}$
with about 5$\%$ precision [3]. Nevertheless, heavy-ion laboratory experiments
and astrophysical measurements, see below, are needed to validate theoretical
findings.

The density dependence of the $E_{sym}$ is an important constituent for the
determination of the drip lines, masses, densities, and collective excitations
of neutron-rich nuclei [4, 5], for flows and multi-fragmentation in heavy-ion
collisions at intermediate energies [1, 6], but also for astrophysical
phenomena like supernovae, neutrino emission, and neutron stars [7], where
knowledge on the high-density dependence of the $E_{sym}$ is most important.

In fact, one of the key question of modern physics is the determination of the
mass vs radius relationship of neutron stars. While the (maximum) mass of
neutron stars is mainly governed by the isoscalar nuclear matter equation of
state, $E(\rho,0)$, the radius of a neutron star is governed by the symmetry
energy behavior at high density, around $2\rho_{0}$ [8]. In fact, the pressure
of neutron matter at $2\rho_{0}$ is what is basically needed to determine the
radius of a canonical neutron star; in [9], using a multitude of EoS obtained
by polytrope expansion, a very tight correlation between pressure at
$2\rho_{0}$ and radius of a 1.4 $M_{\odot}$ neutron star was obtained. A
similar result was obtained in ref. [10] using about 100 EoSes of different
kind. In [11] masses and radii of neutron stars were calculated from equations
of state based on recent high-quality chiral nucleon-nucleon potentials. For a
1.4 $M_{\odot}$ neutron star predictions fall between 10.8 and 12.8 km.
Moreover it was shown that the radius of a 1.4 $M_{\odot}$ neutron star is
nearly insensitive to extrapolation beyond $2\rho_{0}$.

The main aim of this proposal is to determine the Symmetry Energy in the
density region toward $2\rho_{0}$ which is relevant for realistic simulations
of astrophysical objects and processes.

The discovery of gravitational waves has permitted a significant step-forward
in this field. In binary neutron star merger events, like the one observed in
the first evidence of gravitational wave GW17082017 [12], one of the key
observable is the so called tidal polarizability $\Lambda$, strictly
correlated to the neutron stars radii. It follows that observation of GW
opened-up new opportunities for determining radii of neutron stars, possible
in the past only in few cases and with larger errors. Thus, the opening of
multi-messenger astronomy including gravitational waves makes the study of the
symmetry energy at high density even more intriguing than in the past,
allowing now a direct and stringent comparison of data from terrestrial
laboratories with astrophysical observations. A second event of binary neutron
star merger GW190425 has been recently reported in [13]; there a pressure of
19-80 $MeV/fm^{3}$ was estimated for neutron star matter at $2\rho_{0}$.

A large amount of studies have been recently published, where the constraint
on symmetry energy from GW observation are compared with the ones early
obtained in terrestrial laboratory; thus, relevant step-forward arises from
comparison of results coming from nuclear and astro-physicists communities. As
an example, in [14] Zhang and Li produced a restricted EoS parameter space
using observational constraints on the radius, maximum mass, tidal
polarizability and causality condition of neutron stars, resulting in an
estimation of the Symmetry Energy at $2\rho_{0}$ of $46.9\pm 10.1~{}MeV$. It
is interesting to note that in a subsequent paper [15], the authors show that
the observation of the 2.17 $M_{\odot}$ neutron star reduces the error to $\pm
9MeV$ and mention that it is unlikely that even heavier neutron stars will be
observed because the value 2.17 $M_{\odot}$ is already close to the
theoretical maximum according to several studies. In [16] a Bayesian analysis
of GW170817 and quiescent low-mass X-ray binaries radii suggested the symmetry
energy at $2\rho_{0}$ to be in the interval $31-51MeV$. In [17, 18] the
authors stated that while the tidal polarizability $\Lambda$ depends strongly
on the details of the symmetry energy, different trends of $E_{sym}(\rho)$
lead to very similar values of $\Lambda$. Thus, measuring $\Lambda$ alone may
not determine completely the density dependence of the symmetry energy; both
nuclear laboratory experiments and astrophysical observations are therefore
necessary to break this degeneracy and determine precisely the details of the
symmetry energy. A similar conclusion comes from [19] where it was shown how
observations of gravitational waves from binary neutron star mergers can be
combined with insights from nuclear physics to obtain useful constraints on
EoS of dense matter between one and two times the nuclear saturation density.
Moreover, first results for the radius of a 1.4 $M_{\odot}$ neutron star from
X-ray pulse-profile modeling have been reported by the NICER (Neutron star
Interior Composition Explorer) collaboration very recently [20, 21]. It will
be interesting to see the impact of the comparably large values of 12.7 km or
13.0 km with errors of $\pm$ 1.2 km in comparison with the data from other
sources.
In the last two decades, meaningful constraints for the nuclear EoS have been
obtained by laboratory experiments. Many results of nuclear structure and
nuclear reaction measurements as well as astrophysical observations were
collected and compared in [22] and [23]. Rather precise values of $E_{sym}$
have been evaluated for $\rho/\rho_{o}\sim 0.6-0.7$ in [24] and [25] by
fitting the properties of doubly closed-shell nuclei. Together with the
results of an analysis of isobaric analogue states [26] and from heavy ion
reaction data [6], one obtains a quite consistent behavior of $E_{sym}$ at low
densities [27].

As mentioned above, the $E_{sym}$ above $\rho_{0}$ can be accessed either by
the determination of the masses and radii of neutron stars [23] or by
employing observables in heavy ion collisions which are related to the early,
high density phase of the reactions. A multitude of observables have been
proposed to be sensitive to the $E_{sym}$ at supra-saturation densities (for a
review see [1]): ratio of multiplicities or spectra of isospin partners (e.g.
$\pi^{-}/\pi^{+}$, n/p or t/3He) and the comparison of their flows. The ratio
of positively and negatively charged pions, as measured in the 1 AGeV regime
for various collision systems by the FOPI collaboration [28], were well
reproduced with the IBUU4 transport code [29] but only with a super-soft
density dependence of $E_{sym}$. The incorporation of in-medium effects like
mass-shifts of the pions, pion potentials, s-wave production of pions, and the
properties of intermediate $\Delta$ resonances has been shown to lead to
different and even opposite conclusions [30, 31], while describing the
experimental data equally well, indicating a strong model dependence in the
interpretation of the pion ratio. As a solution, it has been proposed to study
not the integrated pion yield ratios but the ratios of pion spectra at high
kinetic energies in the center of mass (CM) reference frame [32]. Experiments
with this aim have been carried out at RIKEN with the SAMURAI magnet and the
SPIRIT TPC in 2016 [33], and $\pi^{-}/\pi^{+}$ ratios in neutron rich and
neutron poor Sn+Sn collision have been measured at 270 AMeV. The sensitivity
to the symmetry energy is high around the pion production threshold but the
range of densities effectively probed will be centered below saturation
density [34]. In parallel to this experimental activity, a strong theoretical
effort is being made, comparing pion predictions from several models within
the code comparison project aiming to better understand and eventually reduce
systematic differences between model predictions [35, 36, 37]. The aim is to
support the interpretation of this new set of pion data.

Other observables which are known to be sensitive to $E_{sym}$ at supra-normal
densities are collective flows. At energies below 1.5A GeV the reaction
dynamics is largely determined by the nuclear mean field. The resulting
pressure produces a collective motion of the compressed material whose
strength will be influenced by $E_{sym}$ in isospin-asymmetric systems. The
strengths of collective flows in heavy ion collisions are usually expressed
with a Fourier expansion of the azimuthal distributions of particles around
the reaction plane: $d\sigma(y)/d\phi\propto
1+2(v_{1}(y)\cos{\phi}+v_{2}(y)\cos{2\phi}.....)$. The side flow of particles
is characterized by the coefficient $v_{1}$ and the elliptic flow by $v_{2}$.
The value of $v_{2}$ around mid-rapidity is negative at incident beam energies
between 0.2 and 5 AGeV which signifies that matter is squeezed out
perpendicular to the reaction plane. At intermediate energies (below 1.5A
GeV), the elliptic flow of protons and composite charged particles emitted at
midrapidity in heavy-ion collisions shows the strongest sensitivity to the
nuclear equation of state (EoS) [38]. Thus, $v_{2}$ is a strong tool for
constraining the strength of the mean field, of both isoscalar and isovector
contributions. In fact, as an example, the EoS of symmetric matter was
strongly constrained in [39] by the elliptic flow $v_{2}$ of protons,
deuterons, tritons, ${}^{3}He$ and ${}^{4}He$ emitted around mid-rapidity in
Au+Au collisions [40]. This study concluded in favor of a soft momentum
dependent EoS when comparing with the IQMD model predictions [41]. This result
has been more recently confirmed in a similar comparison, using UrQMD
transport model calculations [42]. For the case of asymmetric matter, the
ratio $v_{2}^{n}/v_{2}^{p(ch)}$ of the elliptic flow strengths of neutrons
with respect to that of protons or light charged particles was recommended in
[43] as a robust observable sensitive to the stiffness of the $E_{sym}$.

Figure 1:  Left panel: constraints deduced for the density dependence of the
symmetry energy from the ASY-EOS (orange band) and FOPI-LAND (yellow band)
experiments, compared also to some low-density results. Right panel: Elliptic
flow ratio of neutrons over charged particles measured in the same acceptance
range for central ($b<7.5$ fm) Au+Au collisions at 400 AMeV as a function of
transverse momentum, $p_{t}/A$. The black squares represent the ASY-EOS
experimental data. The green triangles and purple circles represent the UrQMD
results employing a stiff and soft density dependence of the $E_{sym}$. The
solid line is the result of a linear interpolation between the predictions
leading to the indicated $\gamma=0.75\pm 0.10$. From ref. [48].

Previous Experiments and Background. The first attempt to constrain the
$E_{sym}$ at high densities has been done in [43] by re-analyzing the existing
FOPI-LAND data [44] on neutron and Hydrogen emissions obtained with the LAND
[45] detector. The re-analysis using the $v_{2}^{n}/v_{2}^{H}$ ratio and
comparison to the results of the UrQMD transport model [46] yielded a
moderately soft $E_{sym}$ dependence on density, with the slope parameter
$L=83\pm 26$ MeV. Despite a large uncertainty, the result made it possible to
rule out the extremely soft or stiff density dependencies of the symmetry
energy [47]. It was, in particular, also possible to demonstrate that the
effects of isoscalar-type parameters affecting the individual flows in the
calculations largely cancel in the predictions for the flow ratios.

Motivated by this finding, an attempt was made to improve the accuracy with a
new experiment that was conducted at the GSI laboratory in 2011 (ASY-EOS
experiment S394 [48]). The experimental setup followed the scheme developed
for FOPI-LAND by using LAND as the main instrument for neutron and charged
particle detection. For the event characterization and for measuring the
orientation of the reaction plane, three detection systems had been installed,
the ALADIN Time-of-Flight (AToF)Wall [49], four double rings of the CHIMERA
multidetector [50] carrying together 352 CsI(Tl) scintillators in forward
direction and four rings with 50 thin CsI(Tl) elements of the Washington
University Microball array [51] surrounding the target. A detailed description
of the set-up of the ASY-EOS experiment and of the data analysis procedure is
available in [48]. Constraints on the symmetry energy were obtained by
comparing the experimental $v_{2}^{n}/v_{2}^{ch}$ ratios, neutron over
charged-particles, with those from the UrQMD simulations. Being not able for
technical reasons to well identify protons has been one of the main drawback
of that measurement, reducing also the maximum density effectively probed. A
soft iso-scalar EoS was assumed for the simulations and the $E_{sym}$ was
parametrized with a Fermi-gas-like formula:

$E_{sym}(\rho)=E_{sym}^{pot}(\rho)+E_{sym}^{kin}(\rho)=22(\rho/\rho_{o})^{\gamma}+12(\rho/\rho_{o})^{2/3}\;\mbox{MeV}$
(2)

with $\gamma$ = 0.5 and $\gamma$ = 1.5 corresponding to a soft and a stiff
density dependence, respectively.

The main results of the experiment are shown in Fig. 1. From the fit of the
measured flow ratios with a linear interpolation between the soft and stiff
model predictions the exponent $\gamma=0.75\pm 0.10$ was obtained (right panel
of Fig. 1). After taking into account all corrections and systematic
uncertainties, the final value was found to be $\gamma=0.72\pm 0.19$,
corresponding to a slope parameter $L=72\pm 13$ MeV [48]. The corresponding
density dependence of the Symmetry Energy is shown in left panel of Fig. 1. It
confirms the FOPI-LAND result and represents an improvement of the accuracy by
a factor of two. The sensitivity of the measurement to the density that is
probed with the flow ratio in the studied reaction is shown in Fig. 2. It is
centered at approximately saturation density and extends beyond twice that
value. The sensitivity distribution indicates that the flow ratio
$v_{2}^{n}/v_{2}^{ch}$ at this bombarding energy is well suited to measure the
density dependence of the symmetry energy slightly above the saturation
density $\rho_{0}$.

The ASY-EOS constraints suggest the 51-60 MeV interval for the $E_{sym}$ at
$2\rho_{0}$ in partial overlap with above cited values of the Refs. [14, 16].
The obtained slope parameter $L=72\pm 13$ MeV, corresponding to a symmetry
pressure $p_{0}=3.8\pm 0.7$ $MeVfm^{-3}$, was used to estimate the pressure in
neutron star matter at density $\rho_{0}$ [48]. The obtained value $3.4\pm
0.6$ $MeVfm^{-3}$ is located inside the pressure interval obtained with $95\%$
confidence limit by Steiner et al. from the observation of eight neutron-stars
[52] and inside the $90\%$ confidence interval of the pressure-density
relation presented by the LIGO and Virgo collaborations [53]. The known tight
correlation between the symmetry pressure $p_{0}$ and the radius $R_{1.4}$ of
a canonical neutron star of 1.4 $M_{\odot}$ [9] has, furthermore, been used
and a value $R_{1.4}=12.6\pm 0.7$ km was obtained [54]. It is in amazingly
good agreement with the recently published radii $12.7\pm 1.2$ km [20] and
$13.0\pm 1.2$ km [21] of the NICER collaboration, and the errors are very
competitive. The ASY-EOS $R_{1.4}$ estimation agrees also with the values of
Ref. [11] above reported. Since both GW and X-rays direct observation are in a
starting phase, and ASY-EOS estimations in the $2\rho_{0}$ region should be
taken as an extrapolation, these agreements should be considered as
preliminary.

The comparisons illustrate the value of terrestrial measurements of the EoS of
asymmetric matter to complement and possibly support astrophysical
observations whose uncertainties are related to the complexity of the applied
methods [55] or to serve as starting points for extrapolations to high
densities exceeding $2\rho_{0}$ [56]. It has to be emphasized, however, that
the UrQMD analysis of the ASY-EOS flow ratios relies on two assumptions. The
expression for $E_{sym}$ (Eq. 2) assumes $E_{sym}(\rho_{0})=34$ MeV, leading
to the sharp cross over of the error bands visible in Fig. 1. It does not
reflect the present uncertainty of approximately 3 MeV of our knowledge of the
symmetry energy at saturation [22, 57]. $E_{sym}(\rho_{0})=31$ MeV in the
analysis reduces the resulting slope parameter to $L=63\pm 11$ MeV as reported
in the ASY-EOS publication [48]. The second assumption is that of the
functional form of a power law (Eq. 2) that, with the present results, is
equivalent to assuming -70 MeV to -40 MeV for $K_{sym}$, an interval that does
not at all correspond to our limited knowledge of the curvature term. However,
as shown by Cozma [58], these difficulties can be overcome with measurements
performed at different energies and by exploiting the dependence of the
sensitivity to density on the type of charged particles whose flow is selected
for the comparison with the neutron flow (Fig. 2). The proposed measurements
are intended to serve that purpose. Thus, a new experiment probing with high
effectiveness the region toward $2\rho_{0}$ will allow to get better and more
reliable constraints from terrestrial laboratories to be compared with new and
more systematic data coming form astrophysical observations.

In [58], Cozma reported a more complete analysis of FOPI flow data, FOPI-LAND
and ASY-EOS by using a QMD type transport model supplemented by a phase-space
coalescence model fitted to FOPI experimental multiplicities of free nucleons
and light clusters. Considering that calculation has proven that neutron-to-
proton and neutron-to-charged particles elliptic flow ratios probe on average
different densities, see below, and anchoring symmetry energy parametrization
at the precise value available for $\rho=0.1fm^{-3}$, Cozma extracted both the
slope L and curvature $K_{sym}$ parameters of the Symmetry Energy, as $L=85\pm
22(exp)\pm 20(th)\pm 12(sys)$ MeV and $K_{sym}=96\pm 315(exp)\pm 170(th)\pm
166(sys)$ MeV. Measuring at the same time yields and flow of LCP, as we plan
to do, is a way to reduce the systematic error arising from light-cluster
multiplicities not well reproduced by the models. Also a value of L, free of
systematical theoretical uncertainties, was extracted from the neutron-to-
proton elliptic flow ratio alone, $L=84\pm 30(exp)\pm 19(th)$ MeV.

For the Au+Au case at 400A MeV, the specific density region tested by the flow
ratio observable has been explored in [48] within the TüQMD model [59]. Two
calculations have been performed there using a moderately soft and a stiff
$E_{sym}$ dependencies up to a given density threshold and using a common
intermediate dependence on density above that threshold. The Difference of
Elliptic Flow Ratios, $DEFR$, with these two parametrizations has been then
defined as an “observable” measuring the sensitivity of the density region
below the threshold on the stiffness of the $E_{sym}$. The left top panel of
Fig. 2 shows the $DEFR$ function for the $v_{2}$ ratios of neutrons to charged
particles. It can be seen that the sensitivity achieved with the elliptic flow
ratios increases with the density threshold and then saturates going toward
the $2\rho_{0}$ region. To evaluate this more carefully, bottom panel shows
the derivative of the $DEFR$ (solid line), to be used as a quantitative
estimator of the sensitivity of the v2-ratio observable to a given density
region. Dashed and dash-dotted line report the same quantity but for v2-ratio
of neutrons with respect to Hydrogen’s or protons, respectively. For the Au+Au
case at 400 AMeV, v2-ratio of neutrons with respect to charged-particles or
Hydrogen’s is mainly sensitive to a region centered slightly above saturation
density, while the n/p ratio sensitivity is centered around 1.4$\rho_{0}$.

Thus, the direct measurement of n/p ratio, instead of n/charged particles
ratio, will enable more effective probing of much higher density region than
was possible so far. This will be a very big step forward.

Figure 2: Left panel: (top) TüQMD predictions of the $DEFR$ function for the
ratio of elliptic flows of neutrons and charged particles for the Au+Au system
at 400A MeV; (bottom) corresponding sensitivity density (solid line) together
with the ones obtained from elliptic-flow ratios of neutrons over all hydrogen
isotopes (dashed line) and neutrons over protons (dash-dotted line). From Ref.
[48]. Right panel: IQMD predictions of the incident energy dependence of the
average reduced density $<\rho/\rho_{0}>$ of protons in semi-central (impact
parameter $b=3fm$) collisions of 197Au+197Au (with the soft momentum-dependent
EoS "SM" which best reproduced the experimental data), for various space-time
selections: (triangles) maximum value reached in the central volume of the
collision, (circles) maximum value for protons ending-up in the phase space
selected for constraining the EoS in [39] (reduced transverse velocity ut0 >
0.4 and reduced rapidity |y0| < 0.8), (error bars) spread distribution (one
sigma) of the time averaged value weighted by the force of the mean field felt
by protons falling in the same phase space selection. See [39] for more
details.

This observation gives rise to the expectation that with an isotopic
resolution sufficient for unambiguous proton identification one should be able
not only to constrain the slope of the $E_{sym}$, $L$, but also its curvature,
$K_{sym}$, see eq. (1). The latter is the least constrained experimentally and
theoretically observable so far. The experimental constraint of [60] yields
$K_{sym}=-50\pm$200 MeV. Another compilation [61] locates the theoretical
value of $K_{sym}$ between $-400$ and +466 MeV and its experimental value in a
range from $-566\pm$1350 MeV to +34$\pm$159 MeV. The results of Cozma [58] are
given above. In Fig. 13 of this work it was also shown that for Au+Au semi-
central collisions neutron-proton elliptic flow ratio sensitivities to L and
$K_{sym}$ attain maxima at 600 and 250 AMeV, respectively, thus at quite
different energies. Note that the potential terms that are proportional to L
and $K_{sym}$ have different dependencies on density and, consequently, the
forces generated by these two terms attain their maximum effectiveness at
different regions of density; simplifying, the L term is proportional to the
isospin asymmetry $\delta$, while the $K_{sym}$ term to $\delta\rho$. This
explains why the maximum sensitivity for L and $K_{sym}$ do not occur for the
same incident energies. This is another strong reason for measuring excitation
function of neutron-proton elliptic flow observable.

Microscopic transport calculations predict that for a short time period
($\sim$20 fm/c) densities of up to $\sim$3 times the saturation density can be
reached in the central zone of a heavy-ion collision even at moderate incident
energies $\sim$1A GeV, as demonstrated in Fig. 3. As an example, the right
panel of Fig. 2 provides also convincing arguments that the maximum densities
in the innermost center of the collision may reach up to 3.5$\rho_{0}$ at 1.5
A GeV, while the densities probed by the mid-rapidity protons during the heavy
ion collision may extend up to twice the saturation density, irrespectively of
the transverse momentum cut. This finding comes from [39] and was obtained by
analyzing the elliptic flow data of charged particles and employing the
Quantum Molecular Dynamics code IQMD [41] for Au+Au collisions up to 1.5 AGeV.
Analogously to the above mentioned DERF function used with TüQMD calculations,
this study has given IQMD predictions of ranges of densities that are probed
by the effect of the mean field in the elliptic flow, that is the "force-
weighted" average density shown as error bars in the right panel of Fig. 2. It
shows that for Au+Au collisions at 1A GeV, the typical densities directly
influencing the flow by the way of the mean field span between $\rho_{0}$ and
2.2$\rho_{0}$. Nevertheless, as demonstrated in [38], higher densities reached
in the compressed central region of the colliding system (dubbed as
"fireball") play also an indirect role in the strength of the elliptic flow,
because they determine how fast the later expansion of the fireball will occur
and modify the elliptic flow by the interplay between expanding fireball and
flying away spectators. And the speed of this expansion is ruled by the
strength of the mean field. Therefore, at 1A GeV, the influence of the EoS at
3$\rho_{0}$ on $v_{2}$ cannot be ruled out. This conclusion applies also to
the isovector part of the EoS. In the most compressed phase of the collision,
it is expected that due to the symmetry energy, the neutron part in excess of
the fireball will expand faster than protons, then will interact differently
with the spectators, which will result in a different elliptic flow.

Figure 3: Evolution of the central baryon density in 132Sn+124Sn collisions at
beam energies from 200 to 2000 AMeV for b = 1 fm, as predicted by the hadronic
transport model of [62].

Therefore, according to the models, it is possible to access the $E_{sym}$ at
higher densities by raising the beam energy. However, as the incident energy
is increased the fraction of nucleons excited into baryonic resonances, mainly
$\Delta(1232)$ at energies of 1 AGeV and below, in the highly compressed phase
of the collision reaches values in the neighborhood of $20\%$ [63] with
potential impact on the time evolution of the reaction that may leave a
comparable imprint on the $E_{sym}$ at 2-3 $\rho_{0}$ depending on the chosen
observables. Thus, one should be aware that the highest density reached during
a heavy-ion collision is not necessarily equivalent to the density that can
safely be probed, i.e. without the occurrence of unmanageable systematic
theoretical uncertainties, for the purpose of constraining the high density
dependence of the $E_{sym}$, as discussed in [48], [39] and [38]. The ASY-EOS
II measurements will provide powerful data to explore these delicate aspects.

New Approach and Relevance to the Field. The main novelty of the proposal
consists in using the NeuLAND detector for measuring neutrons, isotopically
resolved H and He isotopes in a broad energy range within the same acceptance.
This will be the first measurement of this kind and quality, allowing the high
density $E_{sym}$ to be pinned down with an unprecedented precision. Using the
12 double planes of NeuLAND to obtain isotopically well resolved proton
spectra will allow us to effectively probe about 30% higher densities than was
possible so far. New detectors, KRAB and FARCOS, will improve the precision
and quality of the data as detailed below. Measuring excitation functions of
flow observables will provide additional constraints on $E_{sym}$ by scanning
through a broad range of densities.

New tight constraints on the symmetry energy for densities reaching
$2\rho_{o}$ will complement the results obtained with the X-ray telescopes and
with the GW interferometers and are also expected to be competitive in terms
of precision. The knowledge of the symmetry energy in a broad range of
densities will provide the missing information for astrophysical predictions
of the neutron star mass–radius relation and for realistic simulations of
neutron stars, supernova explosions and nucleosynthesis.

Precise data on the symmetry energy and the inter-related phenomena of
clustering and neutron and proton emissions as well as correlations between
them will present strong constraints to nuclear transport theories. The data
should allow theorists to address problems such as delta and pion production
and dynamics, importance of three-body forces at high densities, cluster
formation, and effective neutron and proton mass splitting related to the
momentum dependence of the nuclear mean field.

Objectives, Expected Results and Theory Background. The ASY-EOS experimental
results proved the effectiveness of the $v_{2}^{n}/v_{2}^{ch}$ ratio in
constraining the high-density behavior of the $E_{sym}$. The method appears to
be very robust and precise; we can notice that statistical errors smaller than
$10\%$ can be obtained. As seen in Fig. 2 (right panel),  although the
accurate ASY-EOS determination of the $E_{sym}$ is estimated to have reached
the supra-saturation density region, which was already a unique achievement,
there remains a strong need for constraining it further, with an even better
precision given by the more effective $v_{2}^{n}/v_{2}^{p}$ ratio, toward
$\sim$2 $\rho_{0}$. SIS@GSI provides a unique tool to probe such densities
with heavy-ion collisions. Simulations of semi-central Au+Au collisions at
250, 400, 600, 800, 1000 and 1500 AMeV and, for comparison, neutron rich
132Sn+124Sn, 124Sn+124Sn and Pb+Pb systems at 400, 600 and 800 AMeV have been
carried out by using the same version of the UrQMD transport model already
used in [48]. The neutron-to-proton elliptic flow ratio,
$v_{2}^{n}/v_{2}^{p}$, at mid-rapidity ($0.4<y_{lab}/y_{proj}<0.6$), with a
stiff ($\gamma$=1.5) and a soft ($\gamma$=0.5) parametrization of the
potential part of the $E_{sym}$ for semi-central $(b_{red}<0.54)$ collisions
is shown, as a function of the incident beam energy, in the left panel of Fig.
4. The difference of such $v_{2}^{n}/v_{2}^{p}$ ratios between the stiff and
soft choices can be taken as the sensitivity of the proposed observable, and
is shown in the right panel of the same figure.

Figure 4: Left panel: Excitation functions of neutron-to-proton elliptic flow
ratios, $v2_{n}/v2_{p}$, at mid-rapidity for semi-central Au+Au, 132Sn+124Sn,
124Sn+24Sn and Pb+Pb collisions, as predicted by the UrQMD model for stiff and
soft $E_{sym}(\rho)$. Right panel: differences between the stiff and soft
results.

The obtained sensitivity decreases with increasing the beam energy, because
the mean-field contribution decreases at higher energies where the two-body
collisions start to dominate. Nevertheless, up to 1 AGeV the sensitivity of
the proposed observable is $\sim$15%, while a measurement can easily reach a
$\sim$5% accuracy, allowing clear discrimination between stiff and soft
choices.

A similar conclusion can be drawn from a recent paper [64] where UrQMD
simulation with 11 selected Skyrme forces was performed. In Fig. 3 of that
paper, the ratio between the elliptic flow parameter of free neutrons and
protons was plotted as a function of the slope parameter L for Au+Au collision
from 400 to 1000 AMeV. The highest sensitivity was there obtained at 400 AMeV,
while sensitivity at 1000 AMeV was reduced by a factor 2.

Sensitivity of the Au+Au systems is similar to the one of the other neutron
rich systems, even in the case of 132Sn radioactive ion beams. Heavier
systems, Pb and Au, present smaller statistical errors, for the same numbers
of simulated events. It is also important to stress the differences in trends
(slopes) observed in left panel of Fig. 4. For the soft EoS the ratios
increase with the energy while for the stiff EoS the trend is opposite. This
proves the needs for measuring the excitation functions of these observables
and the importance of using neutron rich beams where the effect is stronger.
We have verified that by filtering the simulations for the acceptance of the
common neutron and proton detector (NeuLAND, see below) does not change the
results shown for the sensitivity of the proposed observable.

In order to convince ourselves that this level of sensitivity to the symmetry
energy is not just peculiar to the UrQMD approach, we have performed
simulations with other transport models, among them with the IQMD [41]. We
show predictions in left panel of Fig. 5 for the same systems as in Fig. 4.
Comparing with UrQMD, we observe qualitatively similar trends, except for a
higher predicted sensitivity for the highest incident energy. The right panel
of Fig. 5 shows the predictions of the TüQMD transport model [59] which have
been performed within the acceptance cut of the FOPI-LAND experiment, but
using two extreme cases for the $E_{sym}$ parametrizations, i.e. the x=-2
super-stiff and x=2 super-soft cases. Also in this case we find a decreasing
sensitivity with the increasing beam energy, with a strong sensitivity of the
flow ratios even at the highest beam energies between the two extreme super-
stiff and super-soft choices. Note here that when using hydrogen isotopes
instead of protons, this sensitivity is reduced. This emphasizes again the
necessity to experimentally separate the hydrogen isotopes.

Figure 5: Left panel: same as right panel of Fig. 4 but with the predictions
of the IQMD model [41] at b=6 fm. Right panel: Similar to right panel of Fig.
4 with the predictions of the TüQMD model [59] for mid-central collisions
($b<7.5fm$) of the Au+Au system. The magenta curve provides in addition the
flow ratio obtained when taking all hydrogen isotopes instead of protons.

Measuring the excitation functions will provide an additional constraint on
$E_{sym}(\rho)$ through the trends. Availability of the high resolution
neutron detector with a capability to resolve LCPs and relativistic stable
heavy ions makes the upcoming FAIR-Phase-0 facility a unique place to perform
proposed measurements. In the future, once the new R3B-cave will be available,
we plan to pursue this kind of studies by profiting from the radioactive heavy
ions and detectors there installed.

Experimental Design, Methods, Set-Up, Technical Requirements The set-up that
we propose to use in the experiment is sketched in Fig. 6. Eight rings of
CsI(Tl) of CHIMERA [50], covering the polar angles between 7∘ and 20∘ will
permit the measurement of charge and energy of forward emitted LCPs, up to Z
$\sim$ 4\. These rings fulfill the requirements of both high granularity, 352
independent modules, and the cylindrical symmetry around the beam axis. This
configuration was already used in the ASY-EOS experiment and proved to be
adequate in order to determine the impact parameter and the reaction plane
orientation of the events [48].

Figure 6: Scale drawing of the proposed set-up for the experiment in cave C.

Charged particles emitted between $\sim$1∘ and 7∘ will be detected by the R3B
TOFD. The TOFD consists of two frames. Each frame has two planes of 44
scintillator paddles with the dimensions $1000\times 27\times
5~{}\mbox{mm}^{3}$. The paddles of successive planes are shifted by half a
paddle width. TOFD will be able to measure charge and velocity of particles
emitted at small angles, mainly projectile spectator-residue and its decay
products. This will be useful to measure global variables for characterizing
the collision centrality, such as $Z_{max}$, charge of the biggest fragment,
and $Z_{Bound}$, amount of charge in clusterized fragments; the TOFD will
serve the same purpose as the Aladin ToFWall in the ASY-EOS experiment.

The NeuLAND detector [65] will give a unique opportunity to measure the
neutron and LCP in the same angular regions. The outstanding calorimetric
properties of NeuLAND will allow protons and other hydrogen isotopes to be
relatively well separated, and will give access to the neutron vs proton
observables. The NeuLAND demonstrator was a part of the SPiRIT experiment [33]
carried out at RIKEN in 2016 and the capability of resolving both protons and
neutrons was clearly demonstrated there. The identification plot of hydrogen
isotopes in the demonstrator (4 double planes, 40 cm total thickness) is
presented in the left panel of Fig. 7. The p, d, t lines are clearly resolved
up to the punch through energy (about 260 MeV for protons) above which the
characteristic back-bendings occur.

The foreseen 12 double planes of NeuLAND, resulting in a total depth of 120
cm, will assure stopping of protons up to about 500 MeV. A simulated
identification plot for the Au+Au collisions at 400 AMeV is presented in the
right panel of Fig. 7. Indeed, no punch-through segments are observed at this
energy and the p, d, t lines clearly stick out of the secondary reaction and
multi-hit background. The simulations include tracking, the secondary reaction
losses, multiple Coulomb scattering, light propagation in plastic
scintillators and quenching effects. The estimated efficiency for proton
identification amounts to about 64% at 200 MeV and 36% at 400 MeV. Taking into
account the thickness of the NeuLAND calorimeter and the secondary reaction
and scattering probability, the estimated efficiencies are still impressive.
The identification capability within the punch-through segments at higher
energies can be improved by applying statistical methods including regularized
decomposition as shown in [66].

With NeuLAND in its start-up version, consisting of 120 cm detector depth (12
double planes), the one neutron interaction probability is about 70% at 400
MeV [67]. Taking into account also the reconstruction efficiency a five-
neutron event is recognized with correct neutron multiplicity with a
probability of about 20 to 30% (200 to 1000 MeV).

NeuLAND will be placed 5.8 m away from the target covering effectively the
mid-rapidity regions, i.e., the polar angles between 33∘ and 57∘. In order to
better discriminate neutrons from protons/LCP a second TOFD will be placed at
a distance of 2 m from the target, in geometrical correspondence with the
NeuLAND, playing a role of the veto wall.

Figure 7: Left panel: Time-of-flight (in ns) vs Total light deposit in the
NeuLAND demonstrator as obtained from Sn+Sn @ 270 AMeV measurement at RIKEN;
Right panel: Time-of-flight vs Total light collected in 12 double-planes of
NeuLAND from the simulation of Au+Au reaction at 400 AMeV (by I. Gasparic).

The KRAB (KRAkow Barrel) detector [68] has been designed to provide a fast
trigger signal based on the multiplicity threshold as well as precise
azimuthal distributions for charged particles beyond the angular acceptance of
the CHIMERA+TOFD setup. The main features of the KRAB detector are: 5 rings of
4$\times$4 mm2 fast scintillating fibers (BCF-10, SciFi) read out by the
MicroFJ-30035-TSV SiPMs, covering polar angles from $30^{\circ}$ to
$165^{\circ}$ with $\sim$87% geometrical efficiency and with $\sim$5% multi-
hit probability. It will be sufficiently large for radioactive beams and
sufficiently small and lightweight in order not to disturb neutrons, having
the min and max internal radii of 6.9 and 11.5 cm and a length of $\sim$50 cm.
It will consist of 4$\times$160 segments in forward rings and 96 segments in
the backward ring with a total of 736 channels. The mechanical structure
holding the SciFi segments and the front end electronics has been 3D-printed
with the ABS filament. The SiPMs will be read out and controlled using the 32
channel CITIROC ASICs. In the ASY-EOS experiment the backward region was
covered by the MicroBall detector consisting of 50 CsI(Tl) crystals arranged
in 4 rings. It was used to roughly define the reaction-plane orientation in
the backward region. In addition the correlation between the impact vectors
deduced from the backward and forward azimuthal distributions measured by the
MicroBall and CHIMERA detectors was found to be of fundamental importance to
efficiently reject the off-target upstream reactions in air. Thanks to its
high segmentation, the KRAB detector will allow to measure precisely the
azimuthal distributions which are indispensable for high resolution estimates
of the reaction plane. It will also produce a fast trigger based on total
multiplicity in an angular region of $\theta>30^{\circ}$ where a strong
correlation between the multiplicity and the magnitude of the impact parameter
is expected from the model predictions. This will allow for more precise
centrality estimates than with the CHIMERA detector alone. Moreover, the
design of KRAB assumes construction of a “helium sleeve”, with the target
holder inside. Simulations indicate that this should reduce the number of
unwanted hits caused by delta-electrons by a factor of about 30. Thus, it is
expected that KRAB should greatly improve the quality of the data with respect
to the first ASY-EOS experiment. The design and the actual view of the KRAB
detector are presented in Fig. 8.

Figure 8: Left panel: The design of the KRAB detector including the CITIROC
boards; Right panel: The actual view of the KRAB as of the end of March 2020.

In addition, yields and flows of LCPs at mid-rapidity and at backward angles
will be measured by using the KRATTA triple telescope array [69] and the
Femtoscope Array for Correlations and Spectroscopy (FARCOS) [70],
respectively. Measuring precisely yields and isotopic compositions of clusters
emitted at mid-rapidity and from the target-spectator is of fundamental
importance for advanced tuning of clusterization algorithms used in the
transport models to get more realistic predictions from them. In fact
clustering influences also the predictions of neutron/proton yields and flows,
with clusters acting as absorbers of the otherwise free nucleons. Moreover,
transport models predict a sensitivity of the ratio of yields and flows of
light isobar nuclei to the high density behavior of the $E_{sym}$ [71], and
measuring isotopic composition of clusters will allow the use of thermodynamic
methods (thermometry) to study properties of the emitting sources, such as
their temperature. The high angular resolution of FARCOS, $\sim$0.25∘ will
perfectly suit the measurements of particle-particle correlation functions
[72] and, through interferometry, will allow for characterization of space-
time properties of emitting sources.

Justification of Beamtime Request. The system/energies we want to measure in
the proposed campaign are:

197Au + 197Au | at | 250, 400, 600, 1000 AMeV
---|---|---

Motivation of the 4 energies is the following:

  * •

the 250 AMeV energy, according to calculations shown in Fig. 13 of [58], is
the energy showing the highest sensitivity on $K_{sym}$; this energy
corresponds to the dynamic range of KRATTA assuring clean p, d, t
identification without punch-through segments;

  * •

the 400 AMeV energy is the one measured in the past ASY-EOS experiment and is
necessary as a reference point, capable of unveiling systematic difference
between the new and old measurements; this is the energy of the maximum
squeeze-out, it also assures no punch-through segments in NeuLAND
identification maps;

  * •

the 600 AMeV energy, according to calculations shown in Fig. 13 of [58], is
the energy showing the highest sensitivity on L;

  * •

the 1000 AMeV energy is the energy allowing to explore the highest densities
where the neutron/proton elliptic flow observable keeps a significant
sensitivity ($\sim 15\%$) to symmetry energy parametrization, according to the
UrQMD calculations of Fig. 4;

Let us stress again the importance of measuring all these energies to provide
additional constraint on the Symmetry Enegy through the observed trends, which
are expected to be opposite for the soft and stiff assumptions (see Fig. 4).

Acknowledgement:
Work supported by Polish National Science Centre, contract No.
UMO-2017/25/B/ST2/02550.

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