# Electrically tunable Berry curvature and strong light-matter coupling in
birefringent perovskite microcavities at room temperature

K. Łempicka-Mirek  M. Król  Institute of Experimental Physics, Faculty of
Physics, University of Warsaw, ul. Pasteura 5, PL-02-093 Warsaw, Poland  H.
Sigurdsson  Science Institute, University of Iceland, Dunhagi 3, IS-107,
Reykjavik, Iceland  Department of Physics and Astronomy, University of
Southampton, Southampton SO17 1BJ, UK  A. Wincukiewicz  Institute of
Experimental Physics, Faculty of Physics, University of Warsaw, ul. Pasteura
5, PL-02-093 Warsaw, Poland  P. Morawiak  Institute of Applied Physics,
Military University of Technology, Warsaw, Poland  R. Mazur  Institute of
Applied Physics, Military University of Technology, Warsaw, Poland  M.
Muszyński  Institute of Experimental Physics, Faculty of Physics, University
of Warsaw, ul. Pasteura 5, PL-02-093 Warsaw, Poland  W. Piecek  Institute of
Applied Physics, Military University of Technology, Warsaw, Poland  P. Kula
Institute of Chemistry, Military University of Technology, Warsaw, Poland  T.
Stefaniuk  Institute of Geophysics, Faculty of Physics, University of Warsaw,
ul. Pasteura 5, PL-02-093 Warsaw, Poland  M. Kamińska  Institute of
Experimental Physics, Faculty of Physics, University of Warsaw, ul. Pasteura
5, PL-02-093 Warsaw, Poland  L. De Marco  CNR NANOTEC, Institute of
Nanotechnology, Via Monteroni, 73100 Lecce, Italy  P. G. Lagoudakis  Hybrid
Photonics Laboratory, Skolkovo Institute of Science and Technology, Territory
of Innovation Center Skolkovo, 6 Bolshoy Boulevard 30, building 1, 121205
Moscow, Russia  Department of Physics and Astronomy, University of
Southampton, Southampton SO17 1BJ, UK  D. Ballarini  CNR NANOTEC, Institute of
Nanotechnology, Via Monteroni, 73100 Lecce, Italy  D. Sanvitto  CNR NANOTEC,
Institute of Nanotechnology, Via Monteroni, 73100 Lecce, Italy  J. Szczytko
B. Piętka<EMAIL_ADDRESS>Institute of Experimental Physics,
Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02-093 Warsaw,
Poland

###### Abstract

The field of spinoptronics is underpinned by good control over photonic spin-
orbit coupling in devices that possess strong optical nonlinearities. Such
devices might hold the key to a new era of optoelectronics where momentum and
polarization degrees-of-freedom of light are interwoven and interfaced with
electronics. However, manipulating photons through electrical means is a
daunting task given their charge neutrality and requires complex electro-optic
modulation of their medium. In this work, we present electrically tunable
microcavity exciton-polariton resonances in a Rashba-Dresselhaus spin-orbit
coupling field at room temperature. We show that a combination of different
spin orbit coupling fields and the reduced cavity symmetry leads to tunable
formation of Berry curvature, the hallmark of quantum geometrical effects. For
this, we have implemented a novel architecture of a hybrid photonic structure
with a two-dimensional perovskite layer incorporated into a microcavity filled
with nematic liquid crystal. Our work interfaces spinoptronic devices with
electronics by combining electrical control over both the strong light-matter
coupling conditions and artificial gauge fields.

There has been surging interest of the condensed matter and solid state
communities in generating artificial gauge fields across various platforms as
means to describe particle properties [1] such as cold atoms [2, 3, 4, 5],
photonic materials [6, 7], acoustics [8], mechanical systems [9], and exciton-
polariton cavities [10, 11, 12]. Gauge fields play an important role in
topological properties of matter [13, 14] and can be intimately linked [15] to
a fundamental band property known as Berry curvature [16] quantifying the
topological invariants of the system. The Berry curvature gives rise to an
anomalous velocity term in a wavepacket’s motion responsible for the Hall
current [17] and also the quantum Hall effect [18], with important
implications in electronic transport [19].

In particular, artificial non-Abelian gauge potentials give rise to effective
spin orbit coupling (SOC) of particles which has seen a lot of investigation
recently in optics [20, 21, 22, 12]. SOC forms an important ingredient in the
field of spintronics [23] and its optical analogue spinoptronics based on
cavity exciton-polaritons [24, 25] seeking to exploit the mixture of internal
(spin or polarization) and external (momentum) degrees of freedom for
information processing. Exciton-polaritons (from here on just "polaritons")
arise in the strong light-matter coupling regime as mixed states of
microcavity photons and excitons [26]. They combine small effective photonic
mass ($\sim 10^{-5}$ of the electron mass) with strong nonlinear effects [27]
and sensitivity to external fields provided by their excitonic matter
component. Moreover, they present an unique opportunity over photonic systems
to study nontrivial band geometry with formation of Berry curvature [11, 28]
and topological effects [29, 30, 31, 32, 33].

The most well known photonic SOC in microcavities is the splitting of
transverse electric and transverse magnetic (i.e., TE-TM splitting) cavity
photon modes [34]. This leads to a double winding effective magnetic field in
the cavity plane which grows quadratically in the photon in-plane momentum
[35]. Recently, photonic analogue of the electronic Dresselhaus [36], and
Rashba-Dresselhaus [37, 38] SOCs have been realized in microcavities. For the
case of cavities that host both Rashba-Dresselhaus (RD) SOC and TE-TM
splitting it has been shown theoretically that reducing the cavity symmetry
could form local concentrations of Berry curvature [39] without the need to
break time reversal symmetry through external magnetic fields acting on the
excitonic component of polaritons [40, 41, 11] or nonzero optical activity for
the photons [42, 28].

Figure 1: Liquid crystal microcavity with 2D perovskite. Schematic dispersion
relation of a bare cavity photon modes in the RD SOC regime and b in the
strong light-matter coupling regime. c Dispersion relation for bottom of lower
polariton branch (the region marked by a dashed square in panel b). d Same as
c with the dispersion subtracted by its mean value $\langle E\rangle$ to more
clearly show the intersection points. Energy of the modes for e positive H-V
splitting and f with broken inversion symmetry. g Schematic representation of
the LC microcavity with the 2D-hybrid organic-inorganic perovskite layer and
LC molecules oriented parallel to the $x$-$y$ plane at zero voltage. h
Rotation of a LC molecule under applied voltage with refractive indices
($n_{\text{o}}$, $n_{\text{e}}$) and angle $\theta$ in the $z$-$x$ plane. We
also show the bare photon i and polariton j dispersions in the linear
horizontal-vertical (H-V) polarization basis in the absence of voltage to
highlight their splitting.

In this work we present a method to electrically tune photonic Berry curvature
in the strong light-matter coupling regime. Our optical system is composed of
a liquid crystal (LC) cavity where both RD SOC and TE-TM splitting effects of
the cavity photons are inherited in the emerging exciton-polariton modes due
to an additional cavity embedded perovskite layer. The strongly bound
perovskite excitons allow observation of strong coupling at room temperature
with high quantum yield and nonlinear effects up to four orders of magnitude
higher than in other photonic systems [12, 43], and thus present a good
material to be interfaced with LCs. Due to the high birefringence and electric
permittivity anisotropy of LCs, making them sensitive to external electric
fields, we achieve unprecedented electric control over an emerging polaritonic
Berry curvature through the specially synthesized photonic SOC and reduced
cavity symmetry.

A cross section of the photonic RD dispersion in the cavity ($k_{x},k_{y}$)
plane is schematically shown in Fig. 1a depicting two opposite circularly
polarized valleys in analogy with spin 1/2 systems. When the photons become
strongly coupled with the perovskite excitons a characteristic anticrossing
behaviour occurs shown in Fig. 1b where the low energy polariton modes (within
the dot-dashed square box) adopt approximately the same dispersion as that of
the photons (Fig. 1a). A surface plot of the polariton dispersion at low
momenta is shown in Fig. 1c. A single degeneracy point at normal incidence
($\mathbf{k}=0$) is highlighted in the RD polariton dispersion (Fig. 1d) where
we have subtracted the dispersion from its mean $\left\langle E\right\rangle$.
When TE-TM splitting and an additional uniform in-plane effective magnetic
field $\mathbf{B}_{x}=\Delta_{HV}\mathbf{\hat{x}}$ are present the RD
degeneracy point morphs into two Dirac cones with degeneracy points known as
diabolical points (Fig. 1e) [33, 11, 42, 44, 45]. The rapidly whirling RD SOC
field around these points leads to a topologically trivial gap opening (Fig.
1f) when an additional perpendicular
$\mathbf{B}_{y}=\Delta_{AD}\mathbf{\hat{y}}$ effective magnetic field is
introduced to break the inversion symmetry of the system with subsequent
formation of a Berry curvature dipole (as shown later in the text).

Figure 2: Strong light-matter coupling regime in liquid crystal microcavity
with 2D perovskite. a Normalized absorption (light green line) and emission
(dark green line) spectra of a polycrystalline 2D-perovskite. b–c Angle-
resolved photoluminescence spectra showing the strong-coupling dispersion in
horizontal (H) and vertical (V) polarization at zero voltage ($\theta=0$).
White dashed curves indicate calculated bare photon dispersion, and green and
magenta curves the calculated polariton dispersion (see Methods). d Real
$n_{\text{o,e}}$ and imaginary $\kappa_{\text{o,e}}$ parts of the ordinary and
extraordinary refractive indices for a thin polycrystaline 2D-perovskite
layer. e–f Berreman simulations corresponding to panels b–c (see Methods).

###  .1 Strong light-matter coupling

The microcavity consists of two distributed Bragg reflectors (DBRs) facing
each other with an embedded 60 nm thick polycrystalline 2D phenylethylammonium
lead iodide (PEAI) perovskite (C6H5C2H4NH3)2PbI4 (hereafter termed: PEPI),
shown schematically in Fig. 1g. The perovskite polycrystalline thin film was
prepared on one of the DBR inner sides using a spin-coating method [46]. The
cavity structure was designed to enhance the photonic field at the position of
the perovskite layer. The cavity was also filled with a highly birefringent
nematic LC which acts as an uniaxial medium with ordinary $n_{\text{o}}$ and
extraordinary $n_{\text{e}}$ refractive indices ($\Delta
n=n_{\text{e}}-n_{\text{o}}=0.4$ [47]). The voltage applied to transparent
electrodes (made of Indium Tin Oxide - ITO) rotates the molecular director,
hence the direction of the optical axis of the LC medium, by an angle $\theta$
in the $x$-$z$ plane (see Fig. 1h). This enables direct control over the
cavity effective refractive indices. Throughout the paper we will refer to the
Stokes parameters of the emitted cavity light (analogous to the polariton
pseudospin) as $S_{1}=(I_{H}-I_{V})/(I_{H}+I_{V})$ and
$S_{2}=(I_{D}-I_{AD})/(I_{D}+I_{AD})$ and
$S_{3}=(I_{\sigma^{+}}-I_{\sigma^{-}})/(I_{\sigma^{+}}+I_{\sigma^{-}})$
corresponding to intensities of horizontal ($I_{H}$), vertical ($I_{V}$),
diagonal ($I_{D}$), antidiagonal ($I_{AD}$), right-hand circular
($I_{\sigma^{+}}$) and left-hand circular ($I_{\sigma^{-}}$) polarized light.

The alignment of the LC molecules inside the cavity at zero voltage is
determined by an ordering polymer layer rubbed along the $x$ (H) direction as
shown schematically in Fig. 1g. Consequently, the bare photon dispersion in
the linear horizontal-vertical (H-V) polarization basis shown in Fig. 1i
displays strongly split bands described by the Hamiltonian:

$\hat{H}_{\phi}(\mathbf{k},\theta)=\frac{\hbar^{2}}{2}\left(\frac{k_{x}^{2}}{m_{x}}+\frac{k_{y}^{2}}{m_{y}}\right)+\hat{S}(\mathbf{k})+\frac{\Delta_{HV}}{2}\hat{\sigma}_{z},$
(1)

where $\mathbf{k}=(k_{x},k_{y})^{\rm T}$ is the in-plane momentum. Here,
$\hat{S}$ describes an effective photonic SOC coming from both cavity TE-TM
splitting [34] and the LC anisotropy [37],

$\hat{S}=2\delta_{xy}k_{x}k_{y}\hat{\sigma}_{x}+\left(\delta_{x}k_{x}^{2}-\delta_{y}k_{y}^{2}\right)\hat{\sigma}_{z}.$
(2)

Physically, this operator introduces direction dependent modification to the
effective masses of linearly polarized modes. When
$\delta_{xy}=\delta_{y}=\delta_{x}$ it realizes the conventional TE-TM
splitting [48]. Finally, $\Delta_{HV}$ describes uniform splitting between H
and V polarized modes (sometimes referred as $X$-$Y$ splitting) which can be
directly controlled in the experiment through the voltage applied to the ITO
electrodes on the cavity, where the electric field drives the molecular
director, hence the optical axis and changes the effective refractive indices
for the H polarized mode. We note that all coefficients in Eqs. (1) and (2)
depend on $\theta$ but not as strongly as $\Delta_{HV}$ [37]. We do not
consider non-Hermitian effects which are weak in our system and only relevant
to the physics of exceptional points [33, 49, 50] which is beyond the scope of
our study.

When an exciton resonance, with energy $E_{\chi}$, from the perovskite layer
is introduced to the cavity (dashed horizontal line in Fig. 1j) the linearly
polarized photonic modes coherently couple to excitons at a rate defined by
the Rabi energies $\Omega_{H,V}$. The strongly coupled system can be described
by a coupled-oscillators model represented by a Hamiltonian:

$\hat{H}_{SC}=\begin{pmatrix}E_{\chi}&0\\\
0&0\end{pmatrix}\otimes\mathds{1}_{2}+\begin{pmatrix}0&0\\\
0&1\end{pmatrix}\otimes\hat{H}_{\phi}+\frac{\hat{\sigma}_{x}}{2}\otimes\begin{pmatrix}\Omega_{\rm
H}&0\\\ 0&\Omega_{\rm V}\end{pmatrix}$ (3)

The four polariton eigenmodes are shown in Figure 1j, exhibiting
characteristic anticrossing behaviour, and labelled as upper (UPH,V) and lower
polaritons (LPH,V).

Figure 3: Gap opening in the Rashba-Dresselhaus regime. a,e Scheme of a
2D-polycrystalline perovskite microcavity with a LC having: two parallel
ordering polymer layers at both sides of the cavity (a) and two ordering
polymer layers at both sides of the cavity with 45∘ tilted rubbing (e) (see
Methods). b,f Dispersion relation of first polariton branch in Rashba-
Dresselhaus SOC regime (b), with the opening of the energy gap (f). c,g
Measured energy resolved PL at $\mathbf{k}=0$ versus applied voltage for
diagonal polarization: without (c) and with (g) gap opening. d,h Measured
angle-resolved PL spectra in R-D regime for $S_{3}$ polarization: without (d)
and with (h) gap opening.

The strong coupling regime in our structure is illustrated in Fig. 2. The
strong emission (dark green line) and absorption (light green line) spectra of
a thin PEPI layer without the cavity are shown in Fig. 2a. The maximum of the
spectra overlap is at 2.38 eV, which corresponds to the most effective
emission and reabsorption processes. The emission spectra from our cavity at
room temperature at zero voltage (i.e., $\theta=0$) are presented in Fig. 2b,c
evidencing strong emission from H-V polariton modes. The characteristic anti-
crossing between the excitonic resonance and the cavity photon modes is
visible in the emission spectra at high emission angles.

Our measurements are compared with the solutions of Eq. (3) plotted with solid
lines in Fig. 2b,c (see Methods). The white dashed lines indicate the fitted
bare photonic branches. Extracted Rabi energies for H and V polarized modes
are: $\Omega_{H}=94.4$ meV and $\Omega_{V}=108.7$ meV. This difference between
the Rabi energies is due to different optical paths for the two linear
polarizations (difference in the LC refractive indices $n_{\rm o}$ and $n_{\rm
e}$).

We also performed numerical simulations on the optical properties of our
cavity using the Berreman matrix method. For this purpose, the real and
imaginary parts of the ordinary and extraordinary refractive indices
$n_{\text{o,e}}$ and $\kappa_{\text{o,e}}$, respectively, were obtained from
ellipsometric measurements and are presented in Fig. 2d. We observe a slight
birefringence of the polycrystalline perovskite in the $z$-axis direction
(perpendicular to the cavity plane). The simulated angle-resolved transmission
spectra are shown in Fig. 2e,f. The theoretical result is fully consistent
with the experiment. The higher energy modes (above 2.39 eV) are not visible
in the experimental spectra due to the strong absorption of the perovskite in
this spectral range.

###  .2 Rashba-Dresselhaus polaritons

By rotating the molecular director of the LC with applied voltage, photonic
modes of different polarization and parities become mixed and form a RD SOC
dispersion relation [37]. In this case, any effective in-plane fields are
minimal and the photonic modes are circularly polarized, forming a dispersion
depicting two shifted valleys of opposite circular polarization (like shown in
Fig. 1a). In this regime, the photonic Hamiltonian can be written:

$\hat{H}_{\phi}^{RD}(\theta,\mathbf{k})=\hat{H}_{\phi}(\theta,\mathbf{k})-2\alpha
k_{y}\hat{\sigma}_{y}$ (4)

where $\alpha$ is the strength of the RD SOC. It is worth noting that in this
low-energy regime polariton and photon lasing was recently demonstrated[51,
52].

The polariton RD SOC regime is illustrated in Fig. 3 for two different cavity
types. In Fig. 3a-d we show the effects of RD SOC photonic modes coupled to
the excitonic resonance. Here, the LC molecules are initially (at 0 V)
oriented by two parallel rubbing polymer layers as shown in Fig. 3a. The
degeneracy point in the RD dispersion (see Fig. 3b at $\mathbf{k}=0$) is
sensitive to H-V splitting that can be directly controlled through the applied
voltage [i.e., $\Delta_{HV}(\theta)$]. We measured the cavity PL as a function
of applied voltage at $\mathbf{k}=0$, shown in Figure 3c, and observed the
crossing of two polariton modes at around 6.7 V. This crossing point
corresponds to the degeneracy point in the RD polariton dispersion, also
evidenced through the measured $S_{3}$ component shown in Fig. 3d.

Figure 4: Nonzero Berry curvature in LC cavity with 2D-perovskite. a Scheme of
orientation of LC molecules inside the cavity in twisted nematic
configuration. Dispersion of polaritonic modes obtained by b experiment and c
Berreman matrix simulations. The pseudocolor scale denotes the $S_{3}$
component. d Measured polariton bands subtracted from their mean energy with
corresponding e Berreman matrix simulations and f solutions of the analytical
model. g–j Experimental polarization and resulting Berry curvature dipole of
the lower energy band compared with k–n results of the Berreman simulations
and o–r analytical two-level model.

###  .3 Berry curvature dipole

Synthesizing polaritonic Berry curvature in optical cavities is a challenging
task since it requires breaking of time-reversal symmetry through external
magnetic fields acting on the excitonic component [40, 41, 11, 28], or nonzero
optical activity for the photons [42]. Such chiral terms open a gap at the
diabolical points in the polariton dispersion, corresponding to a Dirac cone
intersection [53, 12, 54, 44], where the TE-TM SOC field from $\hat{S}$ and
the uniform field $\Delta_{HV}$ cancel each other out. Indeed, with $\alpha=0$
the polariton modes are linearly polarized with a rapidly changing in-plane
SOC field winding around the Dirac cones (i.e., the polariton pseudospin winds
around the $S_{1}$–$S_{2}$ plane when going around the diabolical point [11,
12]). It is worth noting that when non-Hermitian terms are present these
diabolical points become exceptional points which expand to degeneracy lines
(Fermi arcs) when polarization-dependent losses are present [55, 56, 33, 50].

Interestingly, when RD SOC is present it introduces and additional rapidly
varying field around the Dirac cones manifested in the strong $S_{3}$
component in the polariton dispersion. Practically, the diabolical points can
only appear along the ($k_{x},k_{y}=0$) axis where the RD field vanishes (see
Fig. 1e). Therefore, the RD field will only affect the SOC field around these
points but it cannot open the gap. However, because of the combined TE-TM and
RD SOC fields the polariton pseudospin no longer winds just in the
$S_{1}$–$S_{2}$ plane when going around the diabolical points but also around
the $S_{1}$–$S_{3}$ plane. This means that an effective magnetic field in the
$y$-direction corresponding to a finite $S_{2}$ pseudospin component will
break inversion symmetry at the diabolical points and open the gap.
Physically, such an effective magnetic field corresponds to splitting between
the diagonal (D) and antidiagonal (AD) modes written,

$\hat{H}_{\phi}^{RD}(\theta,\mathbf{k})=\hat{H}_{\phi}(\theta,\mathbf{k})-2\alpha
k_{y}\hat{\sigma}_{y}+\frac{\Delta_{AD}}{2}\hat{\sigma}_{x}.$ (5)

Indeed, setting $k_{y}=0$ it can be shown that the presence of the
$\Delta_{AD}$ breaks parity inversion symmetry
$\hat{\sigma}_{z}\hat{H}^{RD}_{\phi}(-k_{x},0)\hat{\sigma}_{z}\neq\hat{H}^{RD}_{\phi}(k_{x},0)$
where $\hat{\sigma}_{z}$ flips the photon circular polarization in the H-V
basis. This leads to opening of the gap with subsequent formation of local
dispersion relation describing massive Dirac particles (see Fig. 1f).

Figure 5: Electrical tuning of polariton Berry curvature. a–d Calculated
dispersion of the cavity modes (energies subtracted by their mean) with
$B_{z}$ marked by a pseudocolor scale from the analytical model for increasing
H-V detuning $\Delta_{HV}$. e Resulting maximum value and center of mass
position of the $B_{z}$ in reciprocal space for varying H-V detuning. f–i
Experimental $B_{z}$ distribution for lower energy mode with increasing
voltage. j Experimentally measured maximum value of $B_{z}$ and $B_{z}$ center
of mass position depending on applied voltage.

For this purpose, we have prepared a second sample with rubbing at the
perovskite layer oriented at an angle of 45∘ with respect to the rubbing at
the DBR surface, Fig. 3e. This results in a twisted nematic LC structure (as
illustrated in Fig. 4a) which introduces a deterministic D-AD splitting. The
splitting is confirmed in Fig. 3g where the PL spectrum at $\mathbf{k}=0$ as a
function of voltage shows a clear anticrossing. Fig. 3h shows the circular
polarization degree of the corresponding angle-resolved dispersion and Fig. 3f
the calculated dispersion.

The concave energy relation indicating the location of the massive Dirac cones
with a clear splitting is illustrated in Fig. 4b-f. The Stokes (pseudospin)
parameters in Fig. 4g-i show that at the minimum of energy-gap the $S_{1}$ and
$S_{3}$ polarizations of the bands whirl around two diagonally polarized
points located along $k_{x}$. This evidences that the splitting between D and
AD polarized modes in the twisted nematic configuration has opened a gap at
these points. More importantly, these points are accompanied with strong
concentration of polaritonic Berry curvature as measured and calculated in
Fig. 4j,n,r. As time reversal symmetry is conserved in the system integral
over the the Berry curvature within a single band is equal to zero, and for
the second band the Berry curvature is expected to be of the opposite sign. We
note that at $\mathbf{k}=0$ in Fig. 4g the mode is predominantly V polarized
due to the applied voltage causing some additional splitting $\Delta_{HV}$.
All of our results are in qualitative agreement with both our analytical model
of Eq. (5) and Berreman simulations. Results of the analytical model in Fig.
4f were fitted to the dispersion relation obtained from Berreman matrix model
(see Methods).

Lastly, the observed polariton Berry curvature can be electrically tuned
through the parameter $\Delta_{HV}$ which is directly controlled by amplitude
of external voltage applied to the cavity. Fig. 5a–e presents calculations
based on Eq. (4) for varying detuning between the H-V modes. The calculated
energies of the two bands are subtracted from their mean value with the Berry
curvature $B_{z}$ marked by a pseudocolor scale. The Berry curvature increases
with H-V detuning as the dispersion relation changes to two gapped Dirac cones
exhibiting a pronounced Berry curvature. Dependence of the maximum value of
Berry curvature and the $B_{z}$ center of mass position in reciprocal space is
summarized in Fig. 5e, which shows that the momentum position of the Berry
curvature maximum can be controlled by $\Delta_{HV}$ detuning and can be
effectively switched off for negative detunings.

Those theoretical predictions can be confirmed experimentally for varying
voltage applied to the cavity which directly controls $\Delta_{HV}$ detuning.
Fig. 5f–i presents Berry curvature extracted from polarization-resolved
transmission measurement at different voltages applied to the ITO electrodes
of the cavity. At 3.88 V (Fig. 5f) corresponding to negative detuning observed
value of the Berry curvatures is low, but rapidly increases for higher
voltages when $\Delta_{HV}$ detuning becomes positive (Fig. 5g). Further
increase of the applied voltage shifts the position of the $B_{z}$ maxima to
higher momenta in reciprocal space up to the limits of the numerical aperture
in the experiment (Fig. 5h,i). Experimental position and value of the
polaritonic Berry curvature summarised in Fig. 5j are in good agreement with
theoretical predictions of Fig. 5e and clearly demonstrate electric tunability
of the band geometry in our system. The polaritonic Berry curvature in our
system is tunable in a continuous manner with the amplitude of external
voltage, rather than magnetic field or temperature [28].

## Conclusion

In conclusion, we have presented a novel architecture for a photonic cavity
heterostructure with a 2D-perovskite incorporated in a highly birefringent
medium. The medium is a nematic liquid crystal that can be arranged in various
ways into the cavity through proper preparation of embedded ordering layers.
We firstly achieve electrically tunable strong light-matter coupling between
the perovskite excitons and cavity photons as a functing of applied external
electric field regime at room temperature with Rabi splitting in the order of
$\sim 100$ meV. We demonstrate how the extreme birefringence coming from the
liquid crystal results in exciton-polariton modes following a Rashba-
Dresselhaus spin-orbit-coupling dispersion relation which opens exciting
perspectives on designing nonlinear valley-optronic devices.

Second, the reduced symmetry of the cavity structure is found to open a gap at
polaritonic diabolical points corresponding to Dirac cones in the cavity
dispersion relation. This allows us to engineer local concentrations of Berry
curvature that can be controlled through applied external voltage which has
only been possible using external magnetic fields or temperature variations
[28]. The importance of the Berry curvature dipole to the quantum nonlinear
Hall effect [57] and rectification in polar semiconductors [58] was recently
stressed in a study demonstrating its electric tunability in a monolayer WTe2
[59]. Our work therefore takes steps towards interfacing polariton
spinoptronic technologies with electronics at room temperature where
nontrivial geometrical properties of the polariton dispersion can be easily
tuned through applied voltage. We note that both electrically pumped polariton
lasers [60] and electro-optic modulated polariton topological lasers [61] are
already possible at cryogenic temperatures. Moreover, with already exciting
work aimed at Rashba-Dresselhaus polariton condensation [51] we expect that
our platform can serve as a testbed for driven-dissipative quantum matter in
unconventional artificial gauge fields.

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Methods

Preparation of liquid crystal microcavity with 2D-perovskite. The studied
microcavity structure consists of two distributed Bragg reflectors (DBRs) made
of 6 SiO2/TiO2 (SiO2 top layer) pairs with maximum reflectance at 520 nm. DBRs
grown on glass substrates with ITO electrodes. The surface of the dielectric
mirror was cleaned with isopropanol and acetone to remove organic residues
from the surface. It was then activated in an oxygen plasma, thanks to which
the 2D-perovskite crystallization precursor solution adhered better to the
substrate. Due to the activation of the DBRs surface, the 2D-perovskite
crystallization precursor solution adheres better to the substrate. The
precursor solution was prepared in a glove box under an argon atmosphere as
follows: phenethylammonium iodide (PEAI) was mixed with lead iodide (PbI2) in
a stoichiometric 2:1 molar ratio and dissolved in N,N-dimethylformamide (DMF)
(mass percent: 10%). The solution was stirred for 4 h at 50 ∘C. All reagents
were purchased from Sigma Aldrich. 40 $\mu$l of solution containing PEAI and
PbI2 was spin-coated in air on a dielectric mirror with rotation speed of
2,000 rpm for 30 sec. Approximately 60 nm thick 2D-layered polycrystalline
lead iodide perovskite [(C6H5(CH2)2NH3)2PbI4] was obtained upon solvent
evaporation on the hotplate for 1 minute at 100 ∘C.

There were two types of microcavities as shown in (Fig. 3). For these samples,
the preparation of the dielectric mirror coated with 2D-polycrystalline
perovskite was the same, as described above. In the case of the first sample
(Fig. 3a), the bottom dielectric mirror with perovskite was spin-coated with a
thin, protective layer (50 nm thick) of PMMA with a 3% methoxybenzene
solution. The upper dielectric mirror was covered with a structured polymer
orienting layer of SE-130. The space between the DBRs is filled with highly
birefringent liquid crystal of $\Delta n=0.4$. Both mirrors were rubbed to
obtain homogeneous LC orientation within the cavity. The total thickness of
this liquid crystal cavity with was approximately 3 $\mu$m. The second sample
type was similar in design to the first one, but the PMMA layer on the
dielectric mirror was arranged at an angle of 45∘ (Fig. 3e,g,h, Fig. 4) or 30∘
(Fig. 5) to the rubbed layer of SE-130 polymer. The thickness of those
cavities was approximately 2.5 $\mu$m. The different thickness of the
presented three perovskite liquid crystal cavities gives a different range of
liquid crystal voltage tuning to the Rashba-Dresselhaus regime (Fig. 3c,g,
Fig. S7, Fig. S8).

Optical measurements. Figure 6 shows the scheme of the experimental setup used
for the optical measurements in both configuration: reflectance and
transmission. The measurements were performed at room temperature using a
continuous wave laser at 405 nm for photoluminescence (PL) excitation and a
white light lamp for the reflectance and transmission spectra. PL measurements
were realized in the reflection configuration.

Figure 6: Scheme of experimental set-up.

Absorbance measurements. Absorption measurements of the PEPI perovskite on the
glass substrate were performed using a CARY 5000 UV-Vis-NIR spectrometer. The
absorption spectrum is presented in Fig. 2a.

Ellipsometry measurements. The refractive index of polycrystalline PEPI
perovskite (presented in Fig. 2d) was estimated based on ellipsometric data
from RC2 ellipsometer (J.A. Woollam Co) in 450-1690 nm spectral range. To
increase the sensitivity of the measurement and retrieve anisotropy more
accurately two different types of samples were prepared. In the first set of
samples, PEPI was spin-coated on the silicon wafer with a 618 nm thick layer
of thermal oxide to utilize the interference enhancement effect. Then all 16
Mueller matrix (MM) elements in reflection were acquired for angles of
incidence from 55∘ to 70∘ by 5∘. In the second case, the PEPI layer was spin-
coated on a 1 mm thick transparent fused silica substrate. For this set of
samples, transmission ellipsometry and transmission intensity data for
illumination angles ranging from 0∘ to 40∘ by 5∘ were acquired additionally.
Such a combination helps to reduce the correlation between the model
parameters, leading to a unique solution. Almost negligible values of off-
diagonal terms and off-diagonal blocks in the MM data collected in the
reflection mode reveal no cross-polarization between $p$\- and $s$\- states
and indicate that the sample under investigation is either isotropic or
uniaxial with a $c$-plane anisotropy. As the sample rotation does not
influence the MM elements, we assume that the c-axis is perpendicular to the
sample surface. All further data analysis was performed using the CompleteEASE
software. The datasets were combined in a multisample model with the same
optical constants for each PEPI layer. We use the general oscillator approach
to retrieve the dielectric permittivity function, constrained with Kramers-
Kronig consistency. Typically the presence of anisotropy alters the shape of
features in the ellipsometric data. In the studied case of PEPI layers,
incorporation of anisotropy into the model leads to 75% improvement of the
Mean Square Error (MSE) of the dielectric permittivity function fit, which
confirms that the spin-coated material exhibits anisotropy.

Coupled oscillators model. Coupled oscillator model presented by solid lines
in Fig. 2b,c was fitted to the experimental data independently for
measurements in horizontal and vertical polarization.

For the horizontal polarization (Fig. 2b) of the polariton modes, the $4\times
4$ Hamiltonian solution is:

$\hat{H}_{\rm
H}=\begin{pmatrix}E_{\chi}&\Omega_{H_{1}}/2&\Omega_{H_{2}}/2&\Omega_{H_{3}}/2\\\
\Omega_{H_{1}}/2&E_{\phi_{H_{1}}}\\!\left(\textbf{k}\right)&0&0\\\
\Omega_{H_{2}}/2&0&E_{\phi_{H_{2}}}\\!\left(\textbf{k}\right)&0\\\
\Omega_{H_{3}}/2&0&0&E_{\phi_{H_{3}}}\\!\left(\textbf{k}\right)\end{pmatrix}.$
(6)

Fitting with exciton energy $E_{\chi}=2.350$ eV resulted in coupling strengths
of: $\Omega_{H_{1}}=67.2$ meV, $\Omega_{H_{2}}=88.4$ meV,
$\Omega_{H_{3}}=94.4$ meV.

For the vertical polarization (Fig. 2c) of the polariton modes, the $3\times
3$ Hamiltonian solution is:

$\hat{H}_{\rm V}=\begin{pmatrix}E_{\chi}&\Omega_{V_{1}}/2&\Omega_{V_{2}}/2\\\
\Omega_{V_{1}}/2&E_{\phi_{V_{1}}}\\!\left(\textbf{k}\right)&0\\\
\Omega_{V_{2}}/2&0&E_{\phi_{V_{2}}}\\!\left(\textbf{k}\right)\end{pmatrix}.$
(7)

Fitting with the same exciton energy leads to coupling strengths of:
$\Omega_{V_{1}}=78.3$ meV, $\Omega_{V_{2}}=108.7$ meV.

Dispersion relations of uncoupled photonic modes $E_{\phi_{V/H_{i}}}$ are
marked in Fig. 2b,c with white dashed lines.

Berreman Method. The spectra presented in Fig. 2e,f show the reflectance
calculated for H and V incident light polarization using Berreman method [62].

Simulated cavity consists of 2 DBR mirrors made of 5 pairs of SiO2 and TiO2
layers calculated for maximal reflectance at 530 nm. Space between the mirrors
consists of LC part and 120 nm thick perovskite layer directly on top SiO2
layer of DBR. To match position of the cavity modes with experiment, the LC
layer is separated into three parts: two interface layers described by an
isotropic ordinary refractive index $n_{\text{o}}$ with thickness of 170 nm
and central anisotropic LC layer with thickness of 845 nm described by
diagonal dielectric tensor with extraordinary refractive index $n_{\text{e}}$
along $x$ direction and $n_{\text{o}}$ for two remaining directions.

$n_{\text{o}}$ and $n_{\text{e}}$ values of the LC are based on Ref. [47] and
refractive indexes of the perovskite layer are presented in Fig. 2d.

In the simulations shown in Fig. 4c,e,k,l,m the total length of the LC cavity
is equal to $L=1362$ nm. Twisted nematic orientation of the LC molecules
around $z$ axis occurs within distances $l=68$ nm at the DBR/LC interface with
total 45 deg reorientation between the top and bottom DBR. Simulations shown
in Fig. 4 were performed with an additional rotation of LC by 42.5 deg around
$y$ axis. Simulations were performed with 5 SiO2/TiO2 DBR pairs centered at
555 nm.

The dispersion of the cavity modes obtained from the Berreman method was
fitted with Eq. (5) with parameters: $E_{\phi}^{0}=2.0997$ eV,
$\Delta_{HV}=13.3$ meV, $\Delta_{AD}=-14.9$ meV, $m_{x}=1.27\times
10^{-5}m_{e}$, $m_{y}=1.38\times 10^{-5}m_{e}$, $\delta_{x}=-0.32$ meV$\mu$m2,
$\delta_{y}=-0.53$ meV$\mu$m2, $\delta_{xy}=0.35$ meV$\mu$m2,
$\alpha=-2.13\times 10^{-3}$ eV$\mu$m.

Calculations presented in Fig. 5a–f were performed for the same parameters but
with $\Delta_{AD}=18.9$ meV observed on the sample with 30 deg rubbing
disorientation (see Supplementary Materials).

To calculate the Berry curvature we follow the procedure described in Ref.
[41]. Based on the Stokes parameters, we define angles:

$\Theta\\!\left(\textbf{k}\right)=\arccos S_{3}\\!\left(\textbf{k}\right),$
(8)
$\Psi\\!\left(\textbf{k}\right)=\arctan\frac{S_{2}\\!\left(\textbf{k}\right)}{S_{1}\\!\left(\textbf{k}\right)},$
(9)

and calculate Berry curvature as

$B_{z}=\frac{1}{2}\sin\Theta\left(\partial_{k_{x}}\Theta\partial_{k_{y}}\Psi-\partial_{k_{y}}\Theta\partial_{k_{x}}\Psi\right).$
(10)

Acknowledgments
This work was supported by the National Science Centre grants
2019/35/B/ST3/04147, 2019/33/B/ST5/02658, 2018/31/N/ST3/03046 and
2017/27/B/ST3/00271. We acknowledge the European Union’s Horizon 2020 program
through a FET Open research and innovation action under the grant agreement
No. 899141 (PoLLoC) and No. 964770 (TopoLight). We also acknowledge the NAWA
Canaletto grant PPN/BIT/2021/1/00124/U/00001. H.S. acknowledges the Icelandic
Research Fund (Rannis), grant No. 217631-051.


# Hyperuniformity in Type-II Superconductors with Point and Planar Defects

Joaquín Puig, Jazmín Aragón Sánchez, Gladys Nieva, Alejandro B. Kolton and
Yanina Fasano∗ Centro Atómico Bariloche and Instituto Balseiro, CNEA, CONICET
and Universidad Nacional de Cuyo, 8400 San Carlos de Bariloche, Argentina

(today)

###### Abstract

We use vortex matter in type-II superconductors as a playground to study how
different types of disorder affect the long wavelength density fluctuations of
the system. We find that irrespective of the vortex-vortex interaction, in the
case of samples with weak and dense point defects the system presents the
hidden order of hyperuniformity characterized by an algebraic suppression of
density fluctuations when increasing the system size. We also reveal that, on
the contrary, for samples with planar defects hyperuniformity is suppressed
since density fluctuations have a tendency to unboundedness on increasing the
system size. Although some of these results were known from previous works,
this paper makes the fundamental discovery that the ability of planar disorder
to suppress hyperuniformity grows on increasing the softness of the structure
for more diluted systems.

††preprint: APS/123-QED

##  I Introduction

Vortex matter in type-II superconducting samples is a system of interacting
elastic objects that can be used as a laboratory playground to study the
growth of hyperuniform materials nucleated in host media with disorder.
Rumi2019 ; Llorens2020b  The quest for hyperuniform material systems is
currently attracting great interest in the condensed matter and materials
science communities due to their unique physical properties. Hyperuniform
materials are endowed with a novel phenomenology that goes against the
conventional wisdom on the effect of structural disorder in systems of
interacting objects. Man2013 ; Chen2018 ; Torquato2018  For instance,
disordered hyperuniform two-dimensional silica structures present a closing of
bandgaps for electrical transport resulting in an enhanced conductivity.
Zheng2020  Also, disordered hyperuniform materials posses complete photonic
bandgaps blocking all directions and polarizations for short wavelengths,
Florescu2009 ; Man2013 ; FroufePerez2016  in contrast to previous assumptions
that periodic or quasiperiodic order was a prerequisite for a material to
present this optical property.

Hyperuniformity is a topological property of a state of matter that is
characterized by strongly-reduced long-wavelength density fluctuations
entailing a decaying structure factor $S(\mathbf{q})$ for small wave-vectors
$\mathbf{q}$. The density of constituents in hyperuniform systems is
homogeneous at large scales, as in a perfect lattice, but it can present
fluctuations at short length scales as in a disordered structure. Torquato2018
; Torquato2003  Hyperuniformity is a structural property defined in an
asymptotic limit and ascertaining this property in real systems is thus
difficult. For this reason most works show that the systems are effectively
hyperuniform. Klatt2019

The structure factor can be directly measured using different X-ray and
neutron diffraction techniques depending on the typical lattice spacing of the
systems. An alternative way of obtaining this magnitude is to compute it from
the real-space positions of the individual constituents considering that
$S(\mathbf{q})=|\hat{\rho}(q_{\rm x},q_{\rm y})|^{2}$, with $\hat{\rho}$ the
Fourier transform of the local density modulation $\rho$. In this work we use
this approach to study the occurrence of hyperuniformity in vortex matter
nucleated in superconducting samples with point and planar crystal defects.
The nature of disorder unavoidably present in the host medium affects whether
the nucleated structure is hyperuniform or non-hyperuniform. In the case of
samples with weak and dense point disorder, some of us revealed that the
vortex structure nucleated in the cuprate superconductor Bi2Sr2CaCu2O8+δ is
effectively hyperuniform at the sample surface. Rumi2019  Later studies report
that disordered hyperuniform vortex structures are nucleated in pnictide
Llorens2020b  and Fe-based Aragon2022  superconductors with point disorder.
Nevertheless, we recently found that the presence of planar correlated defects
suppress hyperuniformity in the vortex structure in an anisotropic fashion.
Puig2021  Thus, attention to the nature of disorder in the host medium has to
be payed in the search for novel hyperuniform materials composed of
interacting objects.

Another relevant parameter that enters into play when trying to nucleate
hyperuniform structures is the magnitude of the interaction between
constituents that when enhanced will tend to decrease the density fluctuations
irrespective of the nature of disorder in the medium. In our case, the vortex
density is controlled by the applied field since the lattice spacing
$a_{0}\propto(B)^{-1/2}$ and, in addition, the interaction between vortices
becomes larger on increasing field. In this work we study how a softening of
the vortex structure affects the density fluctuations at long wavelengths or
short wavevectors for point and correlated disorder in the samples.

##  II Experimental

We image the structure with single vortex resolution at the surface of the
samples by means of the magnetic decoration experiments Fasano2008  performed
at 4.2 K after following a field-cooling process.Fasano2003  As demonstrated
previously, during this cooling process the vortex structure gets frozen at
length-scales of the lattice spacing at a temperature $T_{\mathrm{freez}}\sim
T_{\rm irr}$, the irreversibility temperature at which pinning sets in on
cooling. CejasBolecek2016  At this crossover temperature the bulk pinning
dominates over the vortex-vortex repulsion and the thermal fluctuations.
Pardo1997 ; Fasano1999  Decoration of individual vortices is performed at 4.2
K by evaporating Fe clusters in a controlled-pressure helium atmosphere. These
magnetized clusters are attracted towards the magnetic halo of the vortex core
due to the magnetic force exerted by the local field gradient inherent to
vortices. Once the experiment is performed the sample is warmed up to room
temperature and the Fe clusters that remain attached to the surface due to van
der Waals forces are observed by means of scanning electron microscopy.
Vortices are decorated with the Fe clumps and imaged as white dots in the
images.

The superconducting samples we study are nearly optimally-doped single
crystals of Bi2Sr2CaCu2O8+δ ($T_{\rm c}\sim 90$ K) grown by means of the flux
method. Correa2001 . Some of the crystals present few planar crystallographic
defects consisting in stacking faults in the crystals separating zones with
slightly different orientations of their $c$-axis. Koblischka1995 ;
Herbsommer2001  We also study samples with point disorder only, namely with no
planar defects as revealed by means of magnetic-decoration imaging of the
vortex structure.

Figure 1: Magnetic decoration images of the vortex structure (white dots)
nucleated in Bi2Sr2CaCu2O8+δ samples with point (a),(c) and planar (b),(d)
disorder. Vortex densities correspond to (a),(b) 27 and (c),(d) 4 G. Planar
defects are preferential pinning sites and thus cage vortices in vortex rows,
see highlighted regions in panels (b) and (d). All white scale bars correspond
to 10 $\mu$m.

##  III Results

Figure 1 shows snapshots of vortex structures nucleated in samples with point
(top panels) and planar (bottom panels) defects for two different vortex
densities. Data in the left panels correspond to the vortex structure
nucleated at 27 G and on the right panels to the softer vortex structure
nucleated at 4 G. In the case of samples with point defects, the more diluted
4 G vortex structure presents a larger amount of density fluctuations at short
lengthscales than the one nucleated at 27 G. In samples with planar defects,
they are effective to induce vortex rows aligned along the defects
irrespective of the vortex density, see highlighted regions in the Figs. 1 (b)
and (d). In the region of these rows, the vortex structure present strong
density fluctuations for both magnetic fields.

Figure 2: Angularly-averaged structure factor of the vortex structures
nucleated in Bi2Sr2CaCu2O8+δ with point (full symbols) and planar (open
symbols) disorder for densities of (a) 4 and (b) 27 G. The wavenumber is
normalized by the Bragg wavenumber $q_{0}=2\pi/(a_{0}\cos{\pi/6}$. Full red
lines are fits to the data in samples with point disorder considering $\langle
S(q)\rangle\sim(q/q_{0})^{\alpha}$ when $q/q_{0}\rightarrow 0$ yielding the
$\alpha$ exponents indicated in each case. Data for the 27 G structures are
obtained considering the position of 6189 (2100) vortices in a sample pristine
(with planar defects), whereas data for the 4 G structures come from 500
(2170) vortices for a sample pristine (with planar defects).

In order to quantify how a softening of the structure affects the density
fluctuations at long wavelengths, we calculated the structure factor and then
computed its angular-average at every $q$, $\langle S(\mathbf{q})\rangle$. The
result of this analysis is shown in Fig. 2 for both vortex densities, plotted
as a function of $q/q_{0}$ with $q_{0}=2\pi/(a_{0}\cos{\pi/6}$ the Bragg
wavevector of the hexagonal vortex structure. Data in full symbols correspond
to the case of point disorder whereas open symbols come from structures
nucleated in samples with planar defects. In the first case, $\langle
S(\mathbf{q})\rangle$ decays algebraically when $q\rightarrow 0$ with the
exponents $\alpha=1.1$ for 4 G and 1.2 for 27 G. In both cases these exponents
are yielded by fits of the data in the 0.1-0.4 $q/q_{0}$ range with an error
of 0.3 for 4 G and 0.2 for 27 G. Thus, as already reported on Ref. Rumi2019 ,
density fluctuations in samples with point disorder decay at long wavelengths
in a hyperuniform fashion irrespective of the vortex lattice softness.

In contrast, in the case of samples with planar defects the angularly-averaged
structure factor saturates and even has a tendency to unboundedness when
$q\rightarrow 0$. This indicates that vortex density fluctuations do not tend
to zero at long wavelengths but they rather enhance with distance. Thus, in
samples with correlated disorder the vortex structure does not present the
hyperuniform hidden order and is instead anti-hyperuniform. This phenomenology
takes place even for the harder vortex structure of 27 G with a larger vortex-
vortex interaction, a case where the relative effect of the pinning induced by
planar defects is smaller. Nevertheless, $\langle S(\mathbf{q})\rangle$ grows
faster as $q\rightarrow 0$ when the softness of the structure increases. This
indicates that for softer vortex structures the ability of planar defects to
suppress hyperuniformity is enhanced. Whether this is related to a different
relevance of the finite-size effect that may lead to a recovery of
hyperuniformity in samples with planar defects that are sufficiently thick as
reported in Ref. Puig2022, deserves further theoretical investigation.

##  IV Conclusions

Here we report that planar correlated disorder is effective to enhance long
wavelength density fluctuations as to suppress the hyperuniform hidden order
otherwise present in the vortex structure nucleated in samples with point
disorder. This effect takes place irrespective of the magnitude of vortex-
vortex interaction but it is enhanced in the case of softer structures of
interacting elastic objects such as vortices in type-II superconductors.

## References

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# Conceptual Design of a Low-Energy Ion Beam Storage Ring and
a Recoil Separator to Study Radiative Neutron Capture by Radioactive Ions

Kihong Pak  Department of Nuclear Engineering, Hanyang University,
Seoul 04763, Republic of Korea  Barry Davids<EMAIL_ADDRESS>TRIUMF, 4004
Wesbrook Mall, Vancouver, BC V6T 2A3, Canada  Department of Physics, Simon
Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada  Yong
Kyun Kim<EMAIL_ADDRESS>Department of Nuclear Engineering, Hanyang
University,
Seoul 04763, Republic of Korea

###### Abstract

Recently, the TRIUMF Storage Ring (TRISR), a storage ring for the existing
Isotope Separator and Accelerator-I (ISAC-I) radioactive ion beam facility at
TRIUMF, was proposed. It may be possible to directly measure neutron-induced
radiative capture reactions in inverse kinematics by combining the ring with a
high-flux neutron generator as the neutron target. Herein, we present the
conceptual design of a low-energy ion storage ring as well as a fusion product
extraction system with a Wien filter and recoil separator for detecting
neutron capture products based on ion optical calculations and particle-
tracking simulations.

Suggested keywords

††preprint: APS/123-QED

##  I INTRODUCTION

The origins and abundances of the elements have been quantitatively studied
for more than six decades [1, 2]. It is well known that hydrogen, helium, and
trace quantities of lithium were produced in primordial nucleosynthesis
shortly after the Big Bang [3, 4]. Heavier elements were produced from the
lightest nuclei through nucleosynthetic processes such as fusion reactions in
stars.

Nuclei heavier than iron ($Z=26$) are primarily synthesized by
photodisintegration reactions in the $p$ process and/or by radiative neutron
capture reactions $(n,\gamma)$ occurring in the slow ($s$), rapid ($r$), or
intermediate ($i$) neutron capture processes [5]. Radiative neutron capture
reactions are those in which an atomic nucleus absorbs a neutron to form a
heavier nucleus with the emission of one or more $\gamma$-rays. The classical
$s$ process begins with 56Fe as a seed nucleus and climbs along the valley of
$\beta$ stability up to 209Bi [6]. The $r$ process drives nuclei far to the
neutron-rich side of the stability line [7]. The path of the more recently
introduced $i$ process lies in between, as its neutron density lies between
those of the $s$ and $r$ processes [8, 9].

For several decades, the neutron capture cross-sections of $s$ process
nuclides have been measured using time-of-flight [10] and activation
techniques [11, 12]. Most of the $r$ process involves nuclides far from the
valley of stability, so the products are too short-lived to be directly
measured. Indirect techniques such as the surrogate reaction method have been
applied [13, 14]. Therefore, it is necessary to obtain as much experimental
data as possible away from the stability line to assist in theoretical
extrapolations of nuclear properties. Generally, the shorter the half-life of
the nuclei under investigation, the more difficult it is to prepare a
radioactive sample and perform measurements.

In the case of proton- and $\alpha$-induced capture reactions, inverse
kinematics can be used; however, for neutron-induced reactions, this approach
requires a (radioactive) neutron target. Reifarth _et al._ [15, 16] suggested
combining an ion storage ring with a research reactor as a neutron source to
directly measure neutron-induced reactions in inverse kinematics. With a
thermal neutron density of $2\times 10^{10}$ cm-2 from the reactor and
$10^{13}$ ions s-1 passing through a 500 cm long interaction region ($10^{7}$
stored ions circulating in a ring with a frequency of 1 MHz), the achievable
luminosity of $2\times 10^{23}$ cm-2 s-1 would be large enough to measure
reactions with cross-sections as small as a few mb within a day. Because the
neutron capture reaction product has the same momentum as the primary beam of
mass number $A$ but its speed is only $A/(A+1)$ times as large, it is possible
that Schottky spectroscopy could be used to detect the synthesized ions
through the change in the revolution frequency.

Recently, a storage ring coupled with the existing ISAC facility, combined
with a high-flux neutron generator, was proposed at TRIUMF based on this idea
[17]. This new low-energy storage ring, the TRISR, would be able to utilize
high-intensity radioactive ion beams ($\geq 10^{8}$ s-1) with an energy range
of 0.15 $A$ MeV up to 1.8 $A$ MeV for $A/q\leq 7$. The neutron generator would
need to provide a moderated flux of thermal neutrons with an areal number
density exceeding $10^{8}$ cm-2 in a compact geometry.

A new method was proposed to extract the reaction products from the ring using
a velocity (Wien) filter and a magnetic septum and detect them using a recoil
separator. The maximum electric field strength of such a Wien filter was
assumed to be 35 kV cm-1, on the basis of extensive experience operating Wien
filters at TRIUMF [18]. In this system, ion beams stored in the ring rotate
and pass through the neutron reaction target at frequencies of a few hundred
kHz, sometimes producing neutron capture reaction products. The recoils would
be removed from the ring and transported to the focal plane of the recoil
separator, arriving within $\mu$ s after the reaction occurred, to be
identified and counted. This paper presents conceptual designs of a low-energy
storage ring and recoil separator for the measurement of neutron capture
cross-sections. Ion optical calculations were performed to determine the
locations and specifications of the electromagnetic components of the ring and
recoil separator. Recoil trajectories after the Wien filter were evaluated
using particle-tracking simulations to determine the location and
specifications of a septum magnet to transfer recoils to the recoil separator.

##  II Conceptual Design of
a Storage Ring

Because the TRISR would likely be installed within the existing space in the
ISAC-I experimental hall depicted in Figure 1, the circumference of the ring
cannot exceed 50 m. The size and essential devices required in the lattice are
similar to those of the former Test Storage Ring (TSR) that was operated at
the Max Planck Institute in Heidelberg [19]. However, while the TSR had a
symmetric square design, the TRISR is rectangular in shape because a drift
length is required after the integral Wien filter to increase the separation
of the recoils from the primary beam. The storage ring has an injection system
and a Wien filter, followed by a magnetic septum in a long straight section;
an electron cooler is under consideration. A neutron generator would be placed
in a short section downstream from the injection system, forming the reaction
area. In the other short section, Schottky pickups or other devices may be
placed according to scientific and technical requirements. The designs of
these components are beyond the scope of this work.

Figure 1: Plan view of the ISAC-I experimental hall at TRIUMF. A possible
location of a storage ring that would be coupled to the existing ISAC
radioactive beam facility is shown.

###  II.1 Ion Optical Calculations

To determine the locations and specifications of the electromagnets, such as
dipoles, quadrupoles, and sextupoles, to store the ion beam in the ring, ion
optical calculations were performed using COSY INFINITY 10.0 [20, 21]. The
transfer matrix elements were calculated up to the fifth order and fringe
field effects were included. Fitting was performed by varying the locations,
field strengths, and combinations of magnets to achieve
$(x,x)=(x^{\prime},x^{\prime})=(y,y)=(y^{\prime},y^{\prime})=1$ and
$(x,x^{\prime})=(y,y^{\prime})=(x,\delta_{p})=0$ and minimize the
corresponding higher-order terms at the middle and end of the ring. Here, $x$,
$y$, $x^{\prime}$, $y^{\prime}$, and $\delta_{p}$ are the horizontal and
vertical positions, horizontal and vertical angles, and relative momentum
deviation, respectively. Figure 2 shows the results of the ion optical
calculations for the storage ring. 80Zn${}^{15+}(Z=30,N=50)$ at 0.15
MeV/nucleon with a normalized (4 root mean square) transverse emittance of
$0.2\pi$ $\mu$m [22] was arbitrarily selected as the circulating beam for the
calculations.

Figure 2: The results of ion optical calculations for a single turn in the
ring. The initial beam size and angular divergences are $x,y$: $\pm 1$ mm and
$x^{\prime},y^{\prime}$: $\pm 2.8$ mrad, respectively. The upper and lower
plots show the trajectories in the horizontal and vertical directions,
respectively, with momentum deviations of +2% (blue), 0% (green) and -2%
(red). MQ: quadrupole, MH: sextupole, MS: dipole.

The beam starts in the middle of the first straight section and ends at the
same point after a single rotation with the same angle and no momentum
dispersion. The reaction area was positioned with a 1 m long neutron target in
the second straight section and a Wien filter in the third section. Thus, the
neutron capture products created in the second section can be delivered to the
next section along the same trajectory as that of the primary beam ions, as
they have nearly identical magnetic rigidities $B\rho$. The recoils can then
be separated from the un-reacted beam and deflected by the Wien filter. The
recoils are delivered to the recoil separator through a magnetic septum,
whereas the primary beam ions are stored in the ring and circulated until they
react with neutrons, decay, or are lost via charge-changing collisions. The
ring consists of four 90∘ dipoles, 20 quadrupoles, and eight sextupoles; it
exhibits a mirror-symmetric structure to reduce higher-order aberrations [23].
Its field-free straight sections are 7.2 m and 1.8 m long, and the total
length of the ring is 46.5 m. To verify the properties of the system described
above, we used the Methodical Accelerator Design-X program [24] to calculate
the ion optical parameters, such as $\beta_{x,y}$, $\eta_{x,y}$, and
$\nu_{x,y}$. Figure 3 shows plots of the beta and dispersion functions in the
ring, and Table 1 presents the parameters of the designed ring. The Courant-
Snyder parameters are the same at the beginning and end of the ring.

Figure 3: Beta functions in the horizontal (black) and vertical (red) directions and dispersion function in the horizontal direction (green), which depend on the path length along the optic axis $s$. Table 1: Ion Optical Parameters of the Ion Storage Ring Parameter | Value
---|---
Circumference (m) | 46.5
$\beta_{max,x}$ (m) | 7.3
$\beta_{max,y}$ (m) | 33.3
$\eta_{max,x}$ (m) | 196.4

###  II.2 Evaluation of Recoil Trajectories after the Extraction System

A Wien filter separates particles whose speeds differ from that of the
selected particle, $v_{o}=E/Bc$, where $E$ and $B$ are the strengths of its
orthogonal electric and magnetic fields, respectively. A particle with speed
$v_{o}$ passes through the filter without deflection because the Lorentz force
induced by the $E$ and $B$ fields vanishes; however, a particle with speed
$v\neq v_{o}$ is deflected by the Lorentz force,
$\mathbf{F}=q(\mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{B})$. To separate
the recoils from the beam, a Wien filter with an effective length of 2 m is
placed downstream of a quadrupole triplet at the beginning of the third
straight section. After neutron capture, but prior to the emission of one or
more photons in the radiative capture reaction, the beam and recoil ions have
approximately the same momentum but different masses and speeds, as indicated
in Equation 1.

$p_{0}=p;\ \frac{Av_{o}}{q_{o}}=\frac{(A+1)v}{q_{o}};\
v=\frac{A}{(A+1)}v_{o},$ (1)

where $p_{o}$ and $p$ are the momenta of the beam and recoil, respectively;
$A$ is the mass number of the beam; $v_{o}$ and $v$ are the speeds of the beam
and recoil, respectively; and $q_{o}$ is the charge of the beam.

For a circulating beam of 80Zn at 0.15 MeV/nucleon ($v_{o}\approx 0.018c$),
the 81Zn recoil has a speed of $v\approx 0.9877v_{o}$ after a neutron is
captured. The deflected recoil should be incident on a magnetic septum located
downstream of the Wien filter and delivered to the recoil separator.
Therefore, the trajectories of the beam and recoil after passing through the
Wien filter were evaluated and used to determine the location of the magnetic
septum. The software package G4beamline [25], which is based on the GEANT4
toolkit [26], was used to track the individual beam and recoil ion
trajectories through the storage ring and Wien filter.

The Wien filter was implemented at the beginning of the third straight section
with a length of 2 m and an aperture of $100\times 100$ mm2. Orthogonal
electric and magnetic fields were applied throughout the gap using a three-
dimensional field map. The fringing fields were parameterized using Enge
functions with coefficients ($a_{1},a_{2},...,a_{6}$) derived from the
measured magnetic field of a device recently manufactured by Danfysik. The
effective length of the filter was evaluated as 2000.7 mm. The coefficients
are listed in Table 2, and the electric and magnetic field profiles are shown
in Figure 4.

Table 2: Enge coefficients of the Wien filter fringing fields. Enge coefficient | Value
---|---
$a_{1}$ | $-9.245$
$a_{2}$ | $-0.610$
$a_{3}$ | $0.110$
$a_{4}$ | $-0.003$
$a_{5}$ | $-3.62\times 10^{-5}$
$a_{6}$ | $3.4\times 10^{-6}$
Figure 4: The orthogonal electric and magnetic fields of the Wien filter,
including the fringing fields.

The position distributions of the recoils at the reaction target are the same
as those of the beam; however, the angular divergences are slightly larger,
owing to the emission of one or more photons. We assume that the capture
reaction is followed by the emission of a single photon, thereby maximizing
the associated momentum transfer. The maximum recoil angle $\theta$ and spread
$\Delta p/p$ about the central recoil momentum $p$ can be calculated using
Equation II.2, where $Q$ is the reaction $Q$ value; $E$ is the relative
kinetic energy; and $m_{b}$, $m_{t}$, and $m_{r}$ are the masses of the beam,
target, and recoils, respectively [27].

$\displaystyle\frac{\Delta
p}{p}=\tan\theta_{max}=\frac{m_{r}c^{2}(\sqrt{1+\frac{2(Q+E)}{m_{r}c^{2}}}-1)}{\sqrt{2m_{b}c^{2}(\frac{m_{b}+m_{t}}{m_{t}})E}}$
$\displaystyle\approx\frac{Q+E}{\sqrt{2m_{b}c^{2}(\frac{m_{b}+m_{t}}{m_{t}})E}}$
(2)

The maximum scattering angle for the case of the 81Zn recoils we considered is
2.1 mrad, corresponding to a momentum transfer of 2.77 MeV/c. The trajectories
of the beam and recoil ions exiting the Wien filter were calculated to
determine the location of the magnetic septum. Figure 5 shows the position
distributions of the beam and recoils at different distances from the Wien
filter exit. After a drift length of approximately 4 m, the products are
dispersed by about 40 mm from the beam in the horizontal direction and can be
extracted using a magnetic septum, in which a separation of 2-20 mm between
the homogeneous and field-free regions is typical [28].

Figure 5: Position distributions of the beam (red) and recoil (blue) ions
after passing through the Wien filter at various downstream distances.

The position of the magnetic septum was chosen to be 4.3 m downstream of the
Wien filter, considering the dispersion and potential interference with the
quadrupole. The extraction system consists of a 45∘ magnetic septum, which has
a small aperture and a typical separation between the uniform and vanishing
field regions, followed by a 45∘ dipole magnet to deliver the reaction
products to the recoil separator. Figure 6 shows the trajectories of the beam
ions and recoils simulated using the G4beamline program. The beam is stored in
the ring, continuously circulating while the recoil is separated from it by
the Wien filter and extracted through the subsequent septum. Beam ions with
momenta that differ slightly from the reference momentum are deflected by the
Lorentz force in the Wien filter. However, an electron cooler situated in the
first straight section could compensate for these momentum deviations,
ensuring that the beam continues to circulate stably. The phase space
distributions of the recoils after the extraction system are presented in
Figure 7 and were used as an input for designing the recoil separator. The
properties of the electromagnets in the ring are shown in Table 3.

Table 3: The properties of the electromagnets in the storage ring. $B_{max}$ indicates the magnitude of the peak magnetic field or gradient, $L_{eff}$ is the effective length, and $\rho$ is the radius of curvature. Here $n$ is 0 for the dipole and septum magnets and the Wien filter, 1 for the quadrupoles, and 2 for the sextupoles. Element | $B_{max}$ | Aperture | $L_{eff}$ | Bending | $\rho$ | Quantity
---|---|---|---|---|---|---
| (T/mn) | (mm) | (mm) | Angle (∘) | (mm) |
Dipole | 0.24 | 250$\times$80 | 1963 | 90 | 1250 | 4
Quadrupole | 0.68 | 200 | 350 |  |  | 20
Sextupole | 0.50 | 200 | 250 |  |  | 8
Wien Filter | 0.65 | 100$\times$100 | 2000 |  |  | 1
Septum | 0.59 | 60$\times$100 | 393 | 45 | 500 | 1
Dipole | 0.59 | 60$\times$100 | 393 | 45 | 500 | 1
Figure 6: (a) The ion beam trajectories throughout the storage ring. The beam
starts at the middle of the first (top) straight section with $x,y=\pm 1$ mm,
$x^{\prime},y^{\prime}=\pm 2.8$ mrad, and $\delta_{p}=\pm 2\%$. (b) The recoil
trajectories from the reaction target to the extraction system. The recoils
start at the middle of the second (right) straight section with $x,y=\pm 1$
mm, $x^{\prime},y^{\prime}=\pm 2.8$ mrad, and $\delta_{p}=\pm 2\%$. 10,000
particles were sampled from each distribution. Figure 7: $x-y$ (left),
$x-x^{\prime}$ (middle), and $y-y^{\prime}$ (right) phase space distributions
of the recoils after they are extracted by the magnetic septum and emerge from
the following dipole magnet.

##  III Conceptual design of a recoil separator

In their paper on measuring neutron-induced reactions in inverse kinematics,
Reifarth and Litvinov suggested using Schottky spectroscopy to detect the
appearance of synthesized ions that remain in the storage ring, as well as
particle detectors located after a dipole magnet to detect those that do not
[15]. However, the inclusion of a Wien filter in the ring, followed by a
magnetic septum and recoil separator, would offer an alternative method for
detecting fusion products. The aim would be to continuously extract recoil
ions from the circulating stored beam at a point close to the neutron target.
Thus, reaction products would be removed from the ring within a very short
time after the reaction occurred, on the order of $\mu s$, and transported to
a detection area at the focal plane of the recoil separator, where standard
particle identification techniques would be applied. The advantage of this
approach is that the detection efficiency does not depend on the recoil
lifetime, which can be very short. In addition, negligible energy loss and
scattering would occur for the neutrons in the target, leading to minimal
amounts of scattered or leaky beam that could pass through the recoil
separator.

The design criteria and configuration of the recoil separator were based on
those of the DRAGON and EMMA facilities at TRIUMF [29, 30]. DRAGON is used to
measure proton and $\alpha$ radiative capture reactions by electromagnetically
separating the reaction products from a beam that impinges on a windowless gas
target. The beam and other products with masses and charge states that differ
from those of the selected recoil are removed by slit systems following a
dipole magnet and an electrostatic deflector. EMMA is a vacuum-mode recoil
mass spectrometer that separates the recoils of nuclear reactions from the ion
beam, focuses their energy and angle, and disperses them in a focal plane
according to their mass/charge ($m/q$) ratios. The $m/q$ dispersion of EMMA is
variable and it is typically operated at 10 mm/$\%$.

The TRISR recoil separator presented here was conceptually designed to stop
any ions with charge states that differ from the selected charge state of the
recoils, $q_{0}$, using a slit system located after an electrostatic
deflector, and to disperse recoils with charge state $q_{0}$ according to the
mass at the focal plane downstream of a dipole magnet. The separator consists
of five quadrupoles, two sextupoles, a 40∘ electrostatic deflector, and a 60∘
dipole magnet. The deflection angles of the dipoles were chosen to achieve an
$m/q$ dispersion similar to that of EMMA. The total length of the recoil
separator beam line is 11.0 m. Figure 8 shows the trajectories of recoils with
five different charge states, $q_{0}-2$, $q_{0}-1$, $q_{0}$, $q_{0}+1$, and
$q_{0}+2$. Figure 9 shows the optics of the recoil separator and Table 4 lists
the properties of the separator electromagnets.

Figure 8: The optics of charge state selection in the recoil separator using
an electrostatic deflector in the horizontal direction. The beam ions start
from a spot with a size of $\pm 2.1\ mm$ and angular divergences of $\pm 31.9$
mrad in both transverse directions, a momentum deviation $\Delta p/p$ of $\pm
2\%$, and charge states of $q_{0}-2$, $q_{0}-1$, $q_{0}$, $q_{0}+1$, and
$q_{0}+2$. MQ denotes a quadrupole magnet and ES an electrostatic deflector.
Figure 9: The optics of the recoil separator in the horizontal (top) and
vertical (bottom) directions. The beam starts 1 m before the first quadrupole
magnet with $x=\pm 2.1$ mm, $y=\pm 16.5$ mm, $x^{\prime}=\pm 31.9$ mrad,
$y^{\prime}=\pm 2.7$ mrad, $\Delta p/p=\pm 2\%$, and $\Delta m/m=\pm 1.2\%$.
MQ denotes a quadrupole magnet, MH a sextupole magnet, ES an electrostatic
deflector, and MS a dipole magnet.

The initial beam envelope of $x=\pm 2.1$ mm, $x^{\prime}=\pm 31.9$ mrad,
$y=\pm 16.5$ mm, and $y^{\prime}=\pm 2.7$ mrad was determined from the $\pm
2\sigma$ values of the position and angular distributions of the reaction
products emerging from the dipole magnet downstream of the magnetic septum.
Ions with the selected charge state $q_{0}$ are dispersed according to the
mass by the dipole magnet of the recoil separator and focused in terms of
energy and angle in a focal plane with an $m/q$ dispersion of 11.4 mm/$\%$.
The calculated positions of the recoils and beam ions in the focal plane are
shown in Figure 10, assuming that some scattered beam ions are injected into
the separator.

Figure 10: Calculated position distributions of beam and recoil ions in the focal plane of the recoil separator.  Table 4: The properties of the electromagnets in the recoil separator used in the ion optical calculations. $B_{max}$ and $E_{max}$ indicate the peak strength of the required magnetic field (or gradient) and the maximum electric field, respectively, $L_{eff}$ is the effective length, and $\rho$ is the radius of curvature. Here, $n$ is 0 for the electric and magnetic dipole, 1 for the quadrupoles, and 2 for the sextupoles. Q denotes a quadrupole magnet, ES an electrostatic deflector, and MS a dipole magnet. Element | $B_{max}$ | $E_{max}$ | Aperture | $L_{eff}$ | Deflection | $\rho$ | Quantity
---|---|---|---|---|---|---|---
| (T/mn) | (MV/m) | (mm) | (mm) | angle (∘) | (m) |
Q1 | 1.18 |  | 50 | 250 |  |  | 1
Q2-5 | 0.76 |  | 80 | 350 |  |  | 4
Sextupole | 2.06 |  | 80 | 300 |  |  | 2
ES |  | 1 | 100 | 1396 | 40 | 2 | 1
MS | 0.65 |  | 100 | 2094 | 60 | 2 | 1

##  IV Summary and conclusion

To measure neutron-induced radiative capture reactions using inverse
kinematics, the idea of an ion storage ring with a reactor as the neutron
source was suggested. Based on this idea, a storage ring for TRIUMF, called
the TRISR, was proposed and conceptually designed as a combination of an ion
storage ring, a velocity filter, a compact neutron generator, and a recoil
separator. This system could utilize the existing space and radioactive ion
beams available at the ISAC facility of TRIUMF. If the flux is sufficient, the
reactor could be replaced with a compact neutron generator. Considering the
space limitations of the ISAC-I facility, a storage ring with a Wien filter
was designed and verified through ion optical calculations and particle-
tracking simulations. The trajectories of the beam and fusion products
separated by the velocity filter were calculated and used to determine the
position of the magnetic septum used for recoil extraction. The phase space
distributions of the recoils at the exit of the extraction system were
calculated, and a recoil separator that disperses incident ions according to
$m/q$ was conceptually designed. An electrostatic deflector and a dipole
magnet are used to separate the incident recoils. Figure 11 depicts a
schematic of the storage ring, recoil separator, and other essential devices.
It is possible that the neutron capture cross-sections of radioactive ions
could be measured directly by further developing this conceptual design and
building the facility.

Figure 11: Schematic view of the ion storage ring and the recoil separator.

###### Acknowledgements.

This work was supported by the GRA-NRF Program in Canada through the National
Research Foundation of Korea (NRF), funded by the MSIT (2021K1A3A1A73090699)
and Mitacs Globalink Research Award(IT26791). BD acknowledges the generous
support from the Natural Sciences and Engineering Research Council of Canada.
TRIUMF receives federal funding via a contribution agreement through the
National Research Council of Canada.

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