††thanks: +These authors contributed equally to this work.

# Microwave Signatures of Topological Superconductivity in Planar Josephson
Junctions

Barış Pekerten+ 0000-0002-5794-9706<EMAIL_ADDRESS>Department of
Physics, University at Buffalo, State University of New York, Buffalo, New
York 14260, USA  David Brandão+ 0000-0002-5772-1832<EMAIL_ADDRESS>Department of Physics, University at Buffalo, State University of New York,
Buffalo, New York 14260, USA  Bassel Heiba Elfeky  0000-0003-3200-0040 Center
for Quantum Information Physics, Department of Physics, New York University,
New York 10003, USA  Tong Zhou  0000-0003-4588-5263 Department of Physics,
University at Buffalo, State University of New York, Buffalo, New York 14260,
USA  Eastern Institute for Advanced Study, Eastern Institute of Technology,
Ningbo, Zhejiang 315200, China  Jong E. Han  0000-0002-5518-2986 Department of
Physics, University at Buffalo, State University of New York, Buffalo, New
York 14260, USA  Javad Shabani  0000-0002-0812-2809 Center for Quantum
Information Physics, Department of Physics, New York University, New York
10003, USA  Igor Žutić  0000-0003-2485-226X<EMAIL_ADDRESS>Department of
Physics, University at Buffalo, State University of New York, Buffalo, New
York 14260, USA

###### Abstract

Planar Josephson junctions provide a platform to host topological
superconductivity which, through manipulating Majorana bound states (MBS),
could enable fault-tolerant quantum computing. However, what constitutes
experimental signatures of topological superconductivity and how MBS can be
detected remains strongly debated. In addition to spurious effects that mimic
MBS, there is a challenge to discern the inherent topological signals in
realistic systems with many topologically-trivial Andreev bound states,
determining the transport properties of Josephson junctions. Guided by the
advances in microwave spectroscopy, we theoretically study Al/InAs-based
planar Josephson junction embedded into a radio-frequency superconducting
quantum interference device to identify microwave signatures of topological
superconductivity. Remarkably, by exploring the closing and reopening of a
topological gap, we show that even in a wide planar Josephson junction with
many Andreev bound states, such a topological signature is distinguishable in
the resonance frequency shift of a microwave drive and the “half-slope”
feature of the microwave absorption spectrum. Our findings provide an
important step towards experimental detection of non-Abelian statistics and
implementing scalable topological quantum computing.

The interest in topological superconductors reflects their fascinating
fundamental properties and potential applications using manipulation of
Majorana bound states (MBS) Kitaev (2003); Nayak  _et al._ (2008); Alicea  _et
al._ (2011); Sarma  _et al._ (2015); Matos-Abiague  _et al._ (2017); Güngördü
and Kovalev (2022); Laubscher and Klinovaja (2021). However, detecting such
topological superconductivity and MBS remains a long-standing challenge
Amundsen  _et al._ (2024); Yu  _et al._ (2021); Chen  _et al._ (2019); Lee
_et al._ (2012). Revisiting Sr2RuO4, expected for decades to be $p$-wave spin-
triplet which could support MBS Mackenzie and Maeno (2003); Rice (2004);
Nelson  _et al._ (2004); Das Sarma  _et al._ (2006), or quantized zero-bias
conductance peak (ZBCP) Sengupta  _et al._ (2001) as an experimental MBS
signature Mourik  _et al._ (2012) both provide cautionary guidance for various
difficulties in detection of superconducting properties Žutić and Mazin
(2005); Chen  _et al._ (2019); Das Sarma and Pan (2021).

We focus on planar Josephson junctions (JJs) Shabani  _et al._ (2016); Hell
_et al._ (2017); Pientka  _et al._ (2017); Fornieri  _et al._ (2019); Ren  _et
al._ (2019); Setiawan  _et al._ (2019); Dartiailh  _et al._ (2021a); Banerjee
_et al._ (2023), where topological superconductivity is realized through
proximity effects in the normal region (N) of two-dimensional electron gas
(2DEG), which separates superconducting (S) regions, shown in Fig. 1(a). Such
JJs, with a phase or gate control and multi-terminal geometries suitable for
probing non-Abelian statistics Fu and Kane (2008); Zhou  _et al._ (2020,
2022), could overcome limitations for topological superconductivity in 1D
nanowires requiring fine-tuned parameters Laubscher and Klinovaja (2021);
Lutchyn  _et al._ (2010); Oreg  _et al._ (2010) and MBS detection through
ZBCP.

Our approach builds on advances in circuit quantum electrodynamics and highly-
sensitive microwave spectroscopy of Andreev bound states (ABS) Chtchelkatchev
and Nazarov (2003); Despósito and Levy Yeyati (2001); Bretheau  _et al._
(2013a); Reynoso  _et al._ (2012); Tosi  _et al._ (2019). Due to electron-hole
symmetry, ABS come in pairs and with energies $\pm E_{\mathrm{ABS}}(\phi)$,
that depend on the relative superconducting phase, $\phi$, between the two S
regions. In contrast, MBS are topologically-protected ABS with $E\approx 0$
localized to the ends of the junction that cannot be doubly occupied. To
investigate these states, we theoretically consider an experimentally accurate
model system where the planar JJ is embedded into a radio frequency
superconducting quantum interference device (rf-SQUID), driven at a microwave
frequency, $f_{d}$. The absorbed photons with $E=hf_{d}$ cause transitions
between the ABS and change the JJ’s critical current, $I_{c}$. By coupling the
rf-SQUID to a microwave resonator, one can calculate this effect from the
resulting change in the resonant frequency, $f_{r}$, of the resonator via a
capacitively coupled transmission line Park  _et al._ (2020); Pozar (2012);
Elfeky  _et al._ (2023a), as shown in Fig. 1. This approach, using numerical
discretization, allows us to calculate the ABS spectrum and topological phase
diagram of the planar JJ, as well as the microwave spectra and the shift in
the critical current.

Figure 1: (a) A planar JJ formed by two Al superconducting (S) regions
covering InAs-based 2DEG normal region (N). Coordinate axes and JJ dimensions
are indicated. Alternating current, $I$, flows along $\pm\hat{x}$; the applied
magnetic field is ${\bm{B}}$. (b)-(d) Optical images of an example fabricated
SQUID resonator, which we consider in our theoretical results. Gold: Al, dark
brown: the etched mesa, and light brown: gate and a flux line. (c) Device
diagram with resonator capacitively coupled to a transmission line and
inductively to a superconducting loop with a flux $\Phi$ and a single JJ. (d)
Schematic description from (c) of the JJ, SQUID, microwave drive line inducing
fluctuations in $\Phi\rightarrow\Phi+\delta\Phi(t)$, resonator, and
transmission line with input (output) power $P_{\text{in}}$
($P_{\text{out}}$).

Realistic planar JJs, in contrast with many microwave studies focusing on a
few ABS which is a characteristic for nanowires, atomic contacts, or
constrictions, can have hundreds of topologically-trivial ABS, which
complicate detecting inherent properties of topological superconductivity and
MBS Deacon  _et al._ (2010); Zhang  _et al._ (2022); Janvier  _et al._ (2015);
Wallraff  _et al._ (2004); Tosi  _et al._ (2019); van Woerkom  _et al._
(2017); Hays  _et al._ (2021); Pekker  _et al._ (2013); Matute-Cañadas  _et
al._ (2022). Based on the experimental parameters of our fabricated JJs, we
theoretically identify microwave signatures for closing and reopening of a
topological gap. Our calculations show that such signatures are
distinguishable in the transitions between the ABS due to a microwave drive of
frequency, $f_{d}$, which change the current-phase relation (CPR),
$I(\phi)=\Sigma_{n}I_{n}^{\mathrm{ABS}}$ that depends on the occupation of the
ABS levels. This change in the CPR results in a shift in $f_{r}$, measured in
the microwave circuit Park  _et al._ (2020); Pozar (2012); Elfeky  _et al._
(2023a).

In microwave detection, a quality factor $\gtrsim 10,000$ and a linewidth of
$0.1-1\;$MHz yield fast ($\sim\mu$s) and neV energy resolution of ABS Tosi
_et al._ (2019); Hays  _et al._ (2018). In contrast to ZBCP measurements, this
high sensitivity and fermion-parity conservation, which enables the probing of
quasiparticle poisoning and dynamical properties of superconducting devices
Kurilovich  _et al._ (2021); Kos  _et al._ (2013); Hays  _et al._ (2018);
Elfeky  _et al._ (2023a), make our findings important beyond signatures of
topological superconductivity and MBS. These findings could also advance the
microwave studies in many other systems with a large number of ABS, for
example, in superconducting spintronics Linder and Robinson (2015); Eschrig
(2015); Cai  _et al._ (2023); Amundsen  _et al._ (2024), in nonreciprocal
phenomena such as diode effects Amundsen  _et al._ (2024); Baumgartner  _et
al._ (2022); Dartiailh  _et al._ (2021a); Pekerten  _et al._ (2022);
Lotfizadeh  _et al._ (2024), or in emerging superconducting qubits Krantz  _et
al._ (2019); Tafuri (2019) (ed.); Pita-Vidal  _et al._ (2020).

We consider a model Al/InAs-based JJ with $L=4\;\mu$m and $W_{N}=80\;$nm [see
Fig. 1], embedded in an rf-SQUID, that is inductively coupled to a hanger
$\lambda/4$ coplanar waveguide resonator for our calculations. The model
resonator is capacitively coupled to a transmission feed line, allowing for
the complex transmission and the resonant response to be measured. We use
$f_{r}\sim 7\;$GHz, estimated from experimentally relevant circuit parameters
Park  _et al._ (2020); Pozar (2012). We therefore present our numerical
results with $f_{d}$ up to a few $f_{r}$ ($\sim 40$ GHz), unless showing the
full range of $h\,f_{d}\lesssim 2\Delta_{0}\sim 120$ GHz is instructive.

Our model for the microwave-driven JJ is described by the Bogoliubov-de Gennes
(BdG) Hamiltonian Pientka  _et al._ (2017); Zhou  _et al._ (2022)

$\displaystyle H_{0}$
$\displaystyle=\left[\frac{\mathbf{p}^{2}}{2m^{\ast}}-\mu\left(x,y\right)+\frac{\alpha}{\hbar}\left(p_{x}\sigma_{y}-p_{y}\sigma_{x}\right)\right]\tau_{z}$
$\displaystyle\quad-\frac{g^{\ast}\mu_{B}}{2}\mathbf{B}\cdot\bm{\sigma}+\Delta\left(x,y\right)\tau_{+}+\Delta^{\ast}\left(x,y\right)\tau_{-}\;,$
(1)

for which numerically solve the corresponding discretized eigenvalue problem
SM_  to obtain ABS. Here, ${\bf p}$ is the momentum, $\mu\left(x,y\right)$ is
the chemical potential, $\alpha$ is the Rashba spin-orbit coupling (SOC)
strength, $\mathbf{B}=B\,\hat{y}$ is the in-plane magnetic field and $\mu_{B}$
is the Bohr magneton, while $m^{\ast}$ and $g^{\ast}$ are the effective mass
and $g$-factor, respectively. The proximity-induced superconductivity in the
InAs 2DEG is described by the pair potential
$\Delta(x,y)=\Delta_{0}\,\Theta(|x-W_{N}/2|)\,\exp{(i\,\phi(x,y)/2})$, where
$\Theta$ is the step function and in $\phi(x,y)=\phi_{0}\,\mathrm{sgn}{(x)}$,
$\phi_{0}$ is uniform in each S region. Similarly, we take
$\Delta_{0}\rightarrow\Delta_{0}(B,T)$ to be real and uniform in the S
regions, having the usual BCS dependence on $B$ and temperature $T$ Tinkham
(1996); Zhou  _et al._ (2022). $\sigma_{i}$ ($\tau_{i}$) are Pauli (Nambu)
matrices in the spin (particle-hole) space, and $\tau_{\pm}=(\tau_{x}\pm
i\tau_{y})/2.$

Figure 2: (a), (c) ABS spectra as a function of $\phi$ for JJ from Fig. 1(a).
$W_{S}=600\;$nm, $W_{N}=80\;$nm, $L=4\;\mu$m, and $B=0.2\;$T. (b), (d) The
corresponding topological charge, $Q_{D}$, as a function of $B$ and $\phi$.
(a), (b) For $\mu=0.1\;$meV, the gapped ABS are trivial for all $\phi$. (c)
For $\mu=0.1\;$meV, there is a gap closing and reopening. The inset: MBS at
$E\approx 0$ (red solid line). (d) A topological transition as $\phi$ is
varied at $B=0.2\;$T, corresponding to the gap closing points in (c). Figure
3: (a), (b) Transition lines $\widetilde{\mathcal{T}}(f,\phi)$, calculated
using Eq. (4), corresponding to $hf_{d}=\Delta E=|E_{n}-E_{m}|$ between ABS
from Figs. 2(a) and (c), with a Lorentzian broadening parameter
$\varepsilon=30\;$mK. In the schematic of pair (c) and single-particle (d)
transitions where the ground state is denoted by a blue line, the sum of the
arrow lengths in (c) gives $\Delta E$. Resolving (e) pair and (f) single-
particle transitions from (a) up to $40\;$GHz shows no gap closing and
opening. In contrast a similarly resolved (g) pair and (h) single-particle
transitions from (b) reveal signatures of topological superconductivity. The
inset: an enlarged part of the Fig. 2(c) spectra. The transitions in the
circles “A” and “B” of the inset lead to the “V”-shaped features “A” and “B”
in (g), signaling a gap closing and reopening. Transitions in the circle “C”
lead to the “V”-shaped feature “C” in (h).

We choose material parameters consistent with our fabricated epitaxial Al/InAs
JJs which support robust proximity-induced superconductivity and topological
transitions Dartiailh  _et al._ (2021a): $\Delta_{0}=0.23\;$meV,
$m^{\ast}=0.027m_{0}$ where $m_{0}$ is the electron mass, $g^{\ast}=10$ for
InAs, $\alpha=10$ meVnm, critical magnetic field $B_{c}(T=0)=1.6\;$T and
temperature $T=30\;$mK. The corresponding ABS spectra are shown in Fig. 2. In
the top row, for $\mu=0.1\;$meV and $B=0.2\;$T, the JJ is in the topologically
trivial phase for all $\phi$. In Fig. 2(a) there is no gap closing at
$B=0.2\;$T, while in Fig. 2(b) the calculated topological charge Nayak  _et
al._ (2008), $Q_{D}=1$, confirms this trivial phase.The mirror symmetry in the
system with respect to the middle of the junction along the $x$-direction
allows for an effective time reversal symmetry, leading to a system that can
support MBS in a wider range of $\phi$ values around $\phi=\pi$, as seen in
Figure 2(b) and (d) Pientka  _et al._ (2017). For a finite $W_{S}$, there are
normal reflections and an imperfect transparency at the N/S interface, causing
the topological phase diagram to deviate from the typical “diamond shape”
Pientka  _et al._ (2017); Pekerten  _et al._ (2022, 2024) with a trivial phase
even at $\phi=\pi$ and small $B$. In contrast, for a larger $\mu$, the JJ in
our model undergoes topological transitions even for $B\sim 0$ at $\phi=\pi$,
leading to a wider range of $\phi$ that can support MBS for higher $B$ [Fig.
2(d)]. At $B=0.2\;$T, MBS near $E\sim 0$ is seen in the spectrum in Fig. 2(c)
and the inset for $0.8\pi<\phi<1.2\pi$. 2 2 In Supplemental Material (SM) SM_
, we complement these results with CPR, topological gap, the MBS localization,
as well as topological phase transitions for larger $\mu$ as $B$ is varied.

We next discuss the signatures of the topological gap closing and reopening in
Fig. 2(c) using the microwave spectroscopy with the setup from Figs. 1(b)-(d).
The microwave drive induces a flux fluctuation,
$\Phi(t)=\Phi(0)+\delta\Phi(t)$, in the rf-SQUID around its unperturbed value
$\Phi(0)$, which leads to the JJ phase fluctuation $\delta\phi(t)$, described
by the perturbed Hamiltonian Despósito and Levy Yeyati (2001); Zazunov  _et
al._ (2003); Janvier  _et al._ (2015); Hays  _et al._ (2021); Olivares  _et
al._ (2014); Väyrynen  _et al._ (2015)

$\displaystyle H(\phi+\delta\phi(t))$
$\displaystyle=H_{0}(\phi)+\delta\phi(t)\,H_{\mathrm{MW}},$ (2)

where the perturbation $H_{\mathrm{MW}}=\partial H_{0}(\phi)\,/\,\partial\phi$
is given by

$\displaystyle
H_{\mathrm{MW}}=-i\,(\Delta_{0}/2)\,\left(e^{-i\phi(\mathbf{r})/2}\,\tau_{+}-e^{i\phi(\mathbf{r})/2}\,\tau_{-}\right).$
(3)

For our model experimental setup based on fabricated JJs embedded in microwave
resonator circuits, we can examine the validity of this approach by a simple
estimate, $\delta\phi/2\pi\sim
M\,I_{\mathrm{MW}}/\Phi_{0}<M\,I_{c}/\Phi_{0}\approx 0.05$, where we use an
experimentaly relevant value of $M\sim 120\;$pH for the mutual inductance and
a critical current of $I_{c}\lesssim 1\;\mu$A as an upper bound for the
microwave driving current $I_{\mathrm{MW}}$ Dartiailh  _et al._ (2021b); Phan
_et al._ (2023), thus justifying the treatment of the effects of the microwave
drive as a perturbation Väyrynen  _et al._ (2015); Wesdorp  _et al._ (2024);
Chidambaram  _et al._ (2022). The transitions caused by this perturbation
conserve the fermion parity Bretheau  _et al._ (2017); Janvier  _et al._
(2015); Kos  _et al._ (2013); Väyrynen  _et al._ (2015); SM , which reduces
the number of the allowed transition lines shown in Figs. 3(a) and (b). We can
then distinguish the pair transitions, between a Cooper pair and two ABS, or
the single-particle transitions between two ABS, as denoted in Figs. 3(c) and
(d). In the BdG Hamiltonian, pair (single-particle) transitions correspond to
transitions between the states with opposite (same) energy sign.

With high energy- and time-resolution, microwave spectroscopy provides
information about the allowed transitions lines,
$h\,f_{d}^{nm}=|E_{m}-E_{n}|$, with the transition probability
$|\mathcal{M}_{nm}|^{2}$ governed by the matrix element
$\mathcal{M}_{nm}=\braket{\psi_{m}}{H_{\mathrm{MW}}}{\psi_{n}}$, where
$\psi_{n}$ is the eigenstate with eigenenergy $E_{n}$ of $H_{0}$ [Eq. (1)].
Our calculated results for planar JJs in Fig. 3 show the allowed transition
lines

$\displaystyle\mathcal{T}(f_{d},\phi)=\sum_{mn}|\mathcal{M}_{nm}|^{2}\mathcal{F}(E_{m},E_{n})\,\mathcal{L}(E_{mn}-hf_{d}),$
(4)

where $\mathcal{F}(E_{m},E_{n})$ is the thermal occupation factor and
$\mathcal{L}(E)=(\epsilon/\pi)/\left(E^{2}+\epsilon^{2}\right)$ the Lorentzian
broadening, parameterized by $\epsilon$. Figures 3(a) and (b), with the
transition lines normalized to their maximum value,
$\widetilde{\mathcal{T}}=\mathcal{T}/\mathcal{T}_{\mathrm{max}}$, correspond
to the system described by Figs. 2(a) and (c), respectively. While Fig. 3(a)
which clearly shows the bulk bandgap, as we see a very limited transition
probability around $E\approx 0$ and $\phi\approx\pi$, we can conclude that
Fig. 3(b) displays the gap closing signature of topological superconductivity.

Unlike the majority of the microwave studies with a much smaller number of ABS
Janvier  _et al._ (2015); Wallraff  _et al._ (2004); Tosi  _et al._ (2019);
van Woerkom  _et al._ (2017); Hays  _et al._ (2021) with theoretical results
that we can recover by reducing the system size of our model JJs Bretheau  _et
al._ (2017); Dartiailh  _et al._ (2021b); Elfeky  _et al._ (2023b), the
several hundred ABS considered in Fig. 3 complicate the identification of
topological superconductivity. To address this, we resolve the calculated
transition lines into pair and single-particle transitions, now replotted up
to $f_{d}=40\;$GHz, compatible with relevant experimental setups for microwave
detection, instead of the full ABS energy range of $2\Delta_{0}$ or $\sim
120\;$GHz.

By comparing these two types of transitions for the trivial [Figs. 3(e) and
(f)] and topological phase [Figs. 3(g) and (h)], we see some striking
differences. This is further illustrated in the inset of Fig. 3(g), which is
the enlarged version of Fig. 2(c) focusing on low-energy ABS spectra near the
topological gap closing and reopening, and the matching features in Fig. 3(g)
and (h). The features marked by “A” and “B” in the inset denote the locations
of topological gap closing and reopening and the MBS at $E\approx 0$ between
these points. The matching “V”-shaped features seen in Fig. 3(g), also marked
by “A” and “B,” resolve the pair transition lines around the gap closing and
reopening, which therefore mark the microwave signatures of the topological
superconductivity. On the other hand, the “V”-shaped single-particle
transition lines seen in Fig. 3(h), marked by “C”, correspond to the
transitions between the ABS levels in a region far from $E\approx 0$ state
(marked by “C” in the inset) and therefore are not related to topological
superconductivity. We also note that the features near zero frequency at
$\phi\sim\pi/2,3\pi/2$ in Fig. 3(a), (b), (f) and (h) are related to single-
particle transitions between ABS near $E\sim\Delta_{0}$, and are not topology
related.

While, to the best of our knowledge, these here identified microwave
signatures of topological superconductivity were not previously discussed, it
is reassuring that for the trivial phase, we recover the behavior known from
the previous JJ studies. For example, by reducing the number of ABS that we
consider, we retain all the main features of the previous work on the weak-
link JJs Tosi  _et al._ (2019). This also includes the observed band gap
around $\phi=\pi$ in Figs. 3(e) and (f), as expected for the trivial phase.

Figure 4: (a)-(c) The transition lines and (d)-(f) $\delta I(f,\phi)$ for the
ABS from Fig. 2(c) with suppressed higher-energy single-particle transitions.
The lines $\widetilde{\mathcal{T}}(f,\phi)$ in (a) are resolved into (b) pair
and (c) single-particle transitions. The remaining transition lines in (c) are
due to the energy-split overlapping MBS. (d) $\delta I(f,\phi)$ displays the
gap closing and reopening. $\delta I(f,\phi)$ shown from (e) pair and (f)
single-particle transitions, reveals the gap closing and opening as well as
the transitions from $E\approx 0$ split MBS.

To provide further guidance to experimental work and previous $T=0$ results
for a small number of Andreev bound states Blais  _et al._ (2021); Bretheau
_et al._ (2014); Olivares  _et al._ (2014), we consider the effects of the
microwave drive on the JJ’s CPR, which affects the $f_{r}$ as discussed above.
Without the microwave drive, the free energy of the JJ is
$F=-k_{B}T\sum_{E_{n}}\mathrm{ln}\left[2\,\mathrm{cosh}\left(E_{n}/2k_{B}T\right)\right]$
Tinkham (1996), where $k_{B}$ is the Boltzmann constant and $\\{E_{n}\\}$ is
the spectrum for the JJ. The CPR, $I(\phi)=(2e/\hbar)\,\partial
F\,/\,\partial\phi$, is then

$\displaystyle I(\phi)$
$\displaystyle=-\frac{e}{\hbar}\,\sum_{E_{n}}\,\frac{\partial
E_{n}}{\partial\phi}\,\mathrm{tanh}\left(\frac{E_{n}}{2k_{B}T}\right).$ (5)

Equation (5) shows the contribution of each energy level to the supercurrent,
$(e/\hbar)\,\partial E_{n}/\partial\phi$, weighted by a thermal occupation
$\mathrm{tanh}(E_{n}/2k_{B}T)$. We consider a change to $I(\phi)$ in Eq. (5)
as a function of $f_{d}$, due to the microwave drive. At short timescales, we
phenomenologically express the change in $I(\phi)$ as

$\displaystyle\delta I(f,\phi)$
$\displaystyle=\Gamma\,\sum_{m,n}\,\left(\frac{\partial
E_{m}}{\partial\phi}-\frac{\partial
E_{n}}{\partial\phi}\right)\,|\mathcal{M}_{mn}|^{2}$
$\displaystyle\quad\times\,\mathcal{F}(E_{m},E_{n})\,\mathcal{L}(E_{mn}-hf_{d}),$
(6)

We choose a special form of the thermal occupation
$\mathcal{F}(E_{m},E_{n})=f_{\mathrm{FD}}(E_{n})\,\left(1-f_{\mathrm{FD}}(E_{m})\right),$
where $f_{\mathrm{FD}}$ is the Fermi-Dirac function, to achieve the effect of
suppressing single-particle transitions between the levels away from $E\approx
0$, due to the doubling of the fermionic degrees of freedom in the BdG
Hamiltonian. This form of $\mathcal{F}(E_{m},E_{n})$ allows us to simulate the
suppression of quasiparticle transitions between energy levels $E_{(}m,n)$
away from 0. The identification of topological signatures is simplified in the
resulting transition lines $\widetilde{\mathcal{T}}$ and the shift in the
current $\delta I(f,\phi)$. A counterpart plot with a conventional thermal
occupation factor is discussed in SM SM_ . In the limit of $T=0\;$K and
$\epsilon=0$, our Eq. (6) recovers the same form as in the published
literature Blais  _et al._ (2021); Bretheau  _et al._ (2014); Olivares  _et
al._ (2014). While Eq. (6) does not include the $\phi$-dependent change of the
slope and curvature of the ABS, our key results and trends are preserved by
this simplification Park  _et al._ (2020); SM .

For the JJ from Fig. 2(c), with single-particle transitions where $|E|\gtrsim
k_{B}T$, we use this approach to plot the transition lines in Figs. 4(a)-(c)
and $\delta I(f,\phi)$ of Eq. (6) in Figs. 4(d)-(f). Both the transition lines
and $\delta I(f,\phi)$ reveal the signatures of the gap closing and opening,
even for a system with hundreds of ABS. While these low-frequency transitions
near the gap closing points appear in Figs. 4(b) and (e) for the pair
transition, an even more striking feature becomes visible in Figs. 4(c) and
(f) for the single-particle transitions. The suppression of the higher-energy
single-particle transitions, including the feature “C” in Fig. 3(h), removes
the background that competes with the signal from transitions between the
energy-split MBS and higher energies, now observed prominently in Fig. 4(c) as
a transition line between $\phi\sim 0.8\pi$ and $\phi\sim 1.2\pi$ with half of
the slope of the “V” shapes marked by “A” and “C” in Fig. 3(g) Väyrynen  _et
al._ (2015); Murthy  _et al._ (2020). This “half-slope” feature is also
observed in Figs. 3–6 in the panels that depict topological transitions. The
time-resolved capabilities of microwave spectroscopy could then also provide a
detection of the non-Abelian statistics in planar JJs Zhou  _et al._ (2020,
2022), which is supported by experimentally measured parameters, but is very
challenging using conventional transport measurements.

Our overall features in Fig. 4 are consistent with other work on ABS in
quantum dots Deacon  _et al._ (2010); Zhang  _et al._ (2022), constrictions
Janvier  _et al._ (2015); Wallraff  _et al._ (2004) and nanowires Tosi  _et
al._ (2019); van Woerkom  _et al._ (2017); Hays  _et al._ (2021). This is
important for our effort to expand the microwave spectroscopy as a sensitive
probe for topological superconductivity in systems with a large number of ABS,
where additional signatures become more pronounced by resolving pair and
single-particle transitions. For example, one could apply our proposed
detection to topological JJs based on 2D materials Xie  _et al._ (2023), or
systems with a dominant cubic SOC where the topological superconductivity is
expected to be $f$-wave Alidoust  _et al._ (2021); Luethi  _et al._ (2023). It
would be interesting to test if the enhanced signatures, obtained by removing
the single-particle transitions (violating fermion parity conservation), could
be implemented experimentally by identification and dynamical selection of
parity in JJs, demonstrated in atomic contacts van Woerkom  _et al._ (2017);
Bretheau  _et al._ (2013b) and weak links Ackermann  _et al._ (2023); Wesdorp
_et al._ (2024).

Figure 5: (a) ABS spectrum as a function of $B$ showing the lowest 240 levels
with $\mu=1$ meV and $\phi=0.75\pi$. The gap closing and reopening, and the
appearance of the zero-energy MBS at $B=0.24\;$T match the topological phase
transition (see SM SM_ ). (b) Transition lines $\widetilde{\mathcal{T}}$ as a
function of $B$. “V” feature indicating that the gap is closing and reopening
is visible at $B=0.24\;$T near $f_{d}\sim 0$.

The “V” feature of topological gap closing and reopening is also seen in Fig.
5 from our numerical results for an experimental system where the JJ is phase
biased and the Zeeman field is varied. We demonstrate the case where the phase
bias is $\phi=0.75\pi$ (at $\phi=\pi$ the transition would be very close to
$B=0\;$T). The ABS spectrum in Fig. 5(a) and the topological phase diagram in
SM SM_  indicate a transition at $B=0.24\;$T. A corresponding “V” feature in
the transition lines is observed in Fig. 5(b). The resolved pair transitions,
and the $\delta\,I(f,B)$ plots in SM SM_  also demonstrate the “V” feature and
the gap reopening, indicating that topological transitions may be observed in
phase-biased microwave experiments.

Figure 6: (a) ABS spectrum as a function of $\phi$ showing the lowest 240
levels with $\mu=5$ meV and $B=0.5\;$T, for $L=4\;\mu$m. (b) Transition lines
$\widetilde{\mathcal{T}}$ as a function of $B$. The double “V” feature,
indicating the gap closing and reopening, is less sharp but visible around
$\phi=0.75\pi$ and $\phi=1.25\pi$ near $f_{d}\sim 0$.

We next examine the influence of a higher chemical potential in a JJ with many
ABS levels. The corresponding calculated results are shown in Fig. 6. By
comparing the results for the spectra in Figs. 2(c) and 6(a), we see that at a
higher $\mu$ will be less defined at a constant $B$ as a function of $\phi$,
since a wider range of ABS levels will cross $E=0$. This will result in a
reduced topological gap and less protected MBS, and will lead to a less sharp
“V”-shaped signature in the transition lines [Fig. 6(b)]. A larger $B$ will
shift the gap closing points further apart by shifting the ABS lines in the
spectra, making the gap reopening signature more distinct. Similarly, a larger
finite $L$ will open the topological gap further, having the same effect of
making the signatures more distinct (see SM SM_ ). In our simulations with
$\mu=5\;$meV and $L=4\;\mu$m, $\alpha=10\;$meV nm at $B=0.5\;$T, the gap
reopening signature remains prominently visible but less sharp in the
transition lines.

Our findings have various implications beyond topological superconductivity.
Microwave spectroscopy of ABS could provide a sensitive probe of nonreciprocal
phenomena in JJs Amundsen  _et al._ (2024), while its temporal resolution
could elucidate novel driving mechanism with time-dependent gate-control of
SOC Monroe  _et al._ (2022). As a result, even in the absence of any bias
current in the JJ, it is possible to consider enhanced opportunities for
superconducting electronics and neuromorphic computing using voltage pulses
corresponding to the single flux quantum Tafuri (2019) (ed.); Holmes  _et al._
(2013); Monroe  _et al._ (2022); Likharev and Semenov (1991); Crotty  _et al._
(2010). With decades of transport studies in junctions with only a single
superconductor Kashiwaya and Tanaka (2000), microwave detection also offers a
complementary approach to revisit various manifestations of ABS and
unconventional superconductivity.

###### Acknowledgements.

We thank Prof. Leonid Glazman for valuable discussions. This work is supported
by US ONR under awards MURI N000142212764, N000141712793. Computational
resources were provided by the UB Center for Computational Research.

## References

  * Kitaev (2003) A. Kitaev, “Fault-tolerant quantum computation by anyons,” Ann. Phys. 303, 2 (2003).
  * Nayak  _et al._ (2008) C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083 (2008).
  * Alicea  _et al._ (2011) J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, “Non-Abelian statistics and topological quantum information processing in 1D wire networks,” Nat. Phys. 7, 412 (2011).
  * Sarma  _et al._ (2015) S. Das Sarma, M. Freedman, and C. Nayak, “Majorana zero modes and topological quantum computation,” npj Quant. Inf. 1, 150001 (2015).
  * Matos-Abiague  _et al._ (2017) A. Matos-Abiague, J. Shabani, A. D. Kent, G. L. Fatin, B. Scharf, and I. Žutić, “Tunable magnetic textures: From Majorana bound states to braiding,” Solid State Commun. 262, 1 (2017).
  * Güngördü and Kovalev (2022) U. Güngördü and A. A. Kovalev, “Majorana bound states with chiral magnetic textures,” J. Appl. Phys. 132, 041101 (2022).
  * Laubscher and Klinovaja (2021) K. Laubscher and J. Klinovaja, “Majorana bound states in semiconducting nanostructures,” J. Appl. Phys. 130, 081101 (2021).
  * Amundsen  _et al._ (2024) M. Amundsen, J. Linder, J. W. A. Robinson, and I. Žutić, “Colloquium: Spin-orbit effects in superconducting hybrid structures,” Rev. Mod. Phys. 96, 021003 (2024).
  * Yu  _et al._ (2021) P. Yu, J. Chen, M. Gomanko, G. Badawy, E. P. A. M. Bakkers, K. Zuo, V. Mourik, and S. M. Frolov, “Non-Majorana states yield nearly quantized conductance in proximatized nanowires,” Nat. Phys. 17, 482 (2021).
  * Chen  _et al._ (2019) J. Chen, B. D. Woods, P. Yu, M. Hocevar, D. Car, S. R. Plissard, E. P. A. M. Bakkers, T. D. Stanescu, and S. M. Frolov, “Ubiquitous Non-Majorana Zero-Bias Conductance Peaks in Nanowire Devices,” Phys. Rev. Lett. 123, 107703 (2019).
  * Lee  _et al._ (2012) E. J. H. Lee, X. Jiang, R. Aguado, G. Katsaros, C. M. Lieber, and S. De Franceschi, “Zero-Bias Anomaly in a Nanowire Quantum Dot Coupled to Superconductors,” Phys. Rev. Lett. 109, 186802 (2012).
  * Mackenzie and Maeno (2003) A. P. Mackenzie and Y. Maeno, “The Superconductivity of Sr2RuO4 and the Physics of Spin-Triplet Pairing,” Rev. Mod. Phys. 75, 657 (2003).
  * Rice (2004) T. M. Rice, “Superfluid Helium-3 Has a Metallic Partner,” Science 306, 1142 (2004).
  * Nelson  _et al._ (2004) K. D. Nelson, Z. Q. Mao, Y. Maeno, and Y. Liu, “Odd-Parity Superconductivity in Sr2RuO4,” Science 306, 1151 (2004).
  * Das Sarma  _et al._ (2006) S. Das Sarma, C. Nayak, and S. Tewari, “Proposal to stabilize and detect half-quantum vortices in strontium ruthenate thin films: Non-Abelian braiding statistics of vortices in a ${p}_{x}+i{p}_{y}$ superconductor,” Phys. Rev. B 73, 220502(R) (2006).
  * Sengupta  _et al._ (2001) K. Sengupta, I. Žutić, H.-J. Kwon, V. M. Yakovenko, and S. Das Sarma, “Midgap edge states and pairing symmetry of quasi-one-dimensional organic superconductors,” Phys. Rev. B 63, 144531 (2001).
  * Mourik  _et al._ (2012) V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, “Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices,” Science 336, 1003 (2012).
  * Žutić and Mazin (2005) I. Žutić and I. Mazin, “Phase-Sensitive Tests of the Pairing State Symmetry in Sr2RuO4,” Phys. Rev. Lett. 95, 217004 (2005).
  * Das Sarma and Pan (2021) S. Das Sarma and H. Pan, “Disorder-induced zero-bias peaks in Majorana nanowires,” Phys. Rev. B 103, 195158 (2021).
  * Shabani  _et al._ (2016) J. Shabani, M. Kjaergaard, H. J. Suominen, Younghyun Kim, F. Nichele, K. Pakrouski, T. Stankevic, R. M. Lutchyn, P. Krogstrup, R. Feidenhans’l, S. Kraemer, C. Nayak, M. Troyer, C. M. Marcus, and C. J. Palmstrøm, “Two-dimensional epitaxial superconductor-semiconductor heterostructures: A platform for topological superconducting networks,” Phys. Rev. B 93, 155402 (2016).
  * Hell  _et al._ (2017) M. Hell, M. Leijnse, and K. Flensberg, “Two-Dimensional Platform for Networks of Majorana Bound States,” Phys. Rev. Lett. 118, 107701 (2017).
  * Pientka  _et al._ (2017) F. Pientka, A. Keselman, E. Berg, A. Yacoby, A. Stern, and B. I. Halperin, “Topological Superconductivity in a Planar Josephson Junction,” Phys. Rev. X 7, 021032 (2017).
  * Fornieri  _et al._ (2019) A. Fornieri, A. M. Whiticar, F. Setiawan., E. Portolés, A. C. C. Drachmann, A. Keselman, S. Gronin, C. Thomas, T. Wang, R. Kallaher, G. C. Gardner, E. Berg, M. J. Manfra, A. Stern, C. M. Marcus, and F. Nichele, “Evidence of topological superconductivity in planar Josephson junctions,” Nature 569, 89 (2019).
  * Ren  _et al._ (2019) H. Ren, F. Pientka, S. Hart, A. Pierce, M. Kosowsky, L. Lunczer, R. Schlereth, B. Scharf, E. M. Hankiewicz, L. W. Molenkamp, B. I. Halperin, and A. Yacoby, “Topological superconductivity in a phase-controlled Josephson junction,” Nature 569, 93 (2019).
  * Setiawan  _et al._ (2019) F. Setiawan, C.-T. Wu, and K. Levin, “Full proximity treatment of topological superconductors in Josephson-junction architectures,” Phys. Rev. B 99, 174511 (2019).
  * Dartiailh  _et al._ (2021a) M. C. Dartiailh, W. Mayer, J. Yuan, K. S. Wickramasinghe, A. Matos-Abiague, I. Žutić, and J. Shabani, “Phase Signature of Topological Transition in Josephson Junctions,” Phys. Rev. Lett. 126, 036802 (2021a).
  * Banerjee  _et al._ (2023) A. Banerjee, O. Lesser, M. A. Rahman, H.-R. Wang, M.-R. Li, A. Kringhøj, A. M. Whiticar, A. C. C. Drachmann, C. Thomas, T. Wang, M. J. Manfra, E. Berg, Y. Oreg, Ady Stern, and C. M. Marcus, “Signatures of a topological phase transition in a planar Josephson junction,” Phys. Rev. B 107, 245304 (2023).
  * Fu and Kane (2008) L. Fu and C. L. Kane, “Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator,” Phys. Rev. Lett. 100, 096407 (2008).
  * Zhou  _et al._ (2020) T. Zhou, M. C. Dartiailh, W. Mayer, J. E. Han, A. Matos-Abiague, J. Shabani, and I. Žutić, “Phase Control of Majorana Bound States in a Topological X Junction,” Phys. Rev. Lett. 124, 137001 (2020).
  * Zhou  _et al._ (2022) T. Zhou, M. C. Dartiailh, K. Sardashti, J. E. Han, A. Matos-Abiague, J. Shabani, and I. Žutić, “Fusion of Majorana bound states with mini-gate control in two-dimensional systems,” Nat. Commun. 13, 1738 (2022).
  * Lutchyn  _et al._ (2010) R. M. Lutchyn, Jay D. Sau, and S. Das Sarma, “Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures,” Phys. Rev. Lett. 105, 077001 (2010).
  * Oreg  _et al._ (2010) Y. Oreg, G. Refael, and F. von Oppen, “Helical Liquids and Majorana Bound States in Quantum Wires,” Phys. Rev. Lett. 105, 177002 (2010).
  * Chtchelkatchev and Nazarov (2003) N. M. Chtchelkatchev and Y. V. Nazarov, “Andreev Quantum Dots for Spin Manipulation,” Phys. Rev. Lett. 90, 226806 (2003).
  * Despósito and Levy Yeyati (2001) M. A. Despósito and A. Levy Yeyati, “Controlled dephasing of Andreev states in superconducting quantum point contacts,” Phys. Rev. B 64, 140511(R) (2001).
  * Bretheau  _et al._ (2013a) L. Bretheau, C. O. Girit, H. Pothier, D. Esteve, and C. Urbina, “Exciting Andreev pairs in a superconducting atomic contact,” Nature 499, 312 (2013a).
  * Reynoso  _et al._ (2012) A. A. Reynoso, G. Usaj, C. A. Balseiro, D. Feinberg, and M. Avignon, “Spin-orbit-induced chirality of Andreev states in Josephson junctions,” Phys. Rev. B 86, 214519 (2012).
  * Tosi  _et al._ (2019) L. Tosi, C. Metzger, M. F. Goffman, C. Urbina, H. Pothier, S. Park, A. Levy Yeyati, J. Nygård, and P. Krogstrup, “Spin-Orbit Splitting of Andreev States Revealed by Microwave Spectroscopy,” Phys. Rev. X 9, 011010 (2019).
  * Park  _et al._ (2020) S. Park, C. Metzger, L. Tosi, M. F. Goffman, C. Urbina, H. Pothier, and A. Levy Yeyati, “From adiabatic to dispersive readout of quantum circuits,” Phys. Rev. Lett. 125, 077701 (2020).
  * Pozar (2012) D. M. Pozar,  _Microwave engineering_ , 4th ed. (Wiley, Hoboken, NJ, 2012).
  * Elfeky  _et al._ (2023a) B. H. Elfeky, W. M. Strickland, J. Lee, J. T. Farmer, S. Shanto, A. Zarassi, D. Langone, M. G. Vavilov, E. M. Levenson-Falk, and J. Shabani, “Quasiparticle dynamics in epitaxial Al-InAs planar Josephson junctions,” PRX Quantum 4, 030339 (2023a).
  * Deacon  _et al._ (2010) R. S. Deacon, Y. Tanaka, A. Oiwa, R. Sakano, K. Yoshida, K. Shibata, K. Hirakawa, and S. Tarucha, “Tunneling Spectroscopy of Andreev Energy Levels in a Quantum Dot Coupled to a Superconductor,” Phys. Rev. Lett. 104, 076805 (2010).
  * Zhang  _et al._ (2022) P. Zhang, H. Wu, J. Chen, S. A. Khan, P. Krogstrup, D. Pekker, and S. M. Frolov, “Signatures of Andreev Blockade in a Double Quantum Dot Coupled to a Superconductor,” Phys. Rev. Lett. 128, 046801 (2022).
  * Janvier  _et al._ (2015) C. Janvier, L. Tosi, L. Bretheau, C. O. Girit, M. Stern, P. Bertet, P. Joyez, D. Vion, D. Esteve, M. F. Goffman, H. Pothier, and C. Urbina, “Coherent Manipulation of Andreev States in Superconducting Atomic Contacts,” Science 349, 1199 (2015).
  * Wallraff  _et al._ (2004) A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162 (2004).
  * van Woerkom  _et al._ (2017) D. J. van Woerkom, A. Proutski, B. van Heck, D. Bouman, J. I. Väyrynen, L. I. Glazman, P. Krogstrup, J. Nygård, L. P. Kouwenhoven, and A. Geresdi, “Microwave spectroscopy of spinful Andreev bound states in ballistic semiconductor Josephson junctions,” Nat. Phys. 13, 876 (2017).
  * Hays  _et al._ (2021) M. Hays, V. Fatemi, D. Bouman, J. Cerrillo, S. Diamond, K. Serniak, T. Connolly, P. Krogstrup, J. Nygård, A. Levy Yeyati, A. Geresdi, and M. H. Devoret, “Coherent Manipulation of an Andreev Spin Qubit,” Science 373, 430 (2021).
  * Pekker  _et al._ (2013) D. Pekker, C.-Y. Hou, V. E. Manucharyan, and E. Demler, “Proposal for Coherent Coupling of Majorana Zero Modes and Superconducting Qubits Using the 4$\pi$ Josephson Effect,” Phys. Rev. Lett. 111, 107007 (2013).
  * Matute-Cañadas  _et al._ (2022) F. J. Matute-Cañadas, C. Metzger, S. Park, L. Tosi, P. Krogstrup, J. Nygård, M. F. Goffman, C. Urbina, H. Pothier, and A. Levy Yeyati, “Signatures of Interactions in the Andreev Spectrum of Nanowire Josephson Junctions,” Phys. Rev. Lett. 128, 197702 (2022).
  * Hays  _et al._ (2018) M. Hays, G. de Lange, K. Serniak, D. J. van Woerkom, D. Bouman, P. Krogstrup, J. Nygård, A. Geresdi, and M. H. Devoret, “Direct Microwave Measurement of Andreev-Bound-State Dynamics in a Semiconductor-Nanowire Josephson Junction,” Phys. Rev. Lett. 121, 047001 (2018).
  * Kurilovich  _et al._ (2021) V. D. Kurilovich, C. Murthy, P. D. Kurilovich, B. van Heck, L. I. Glazman, and C. Nayak, “Quantum critical dynamics of a Josephson junction at the topological transition,” Phys. Rev. B 104, 014509 (2021).
  * Kos  _et al._ (2013) F. Kos, S. E. Nigg, and L. I. Glazman, “Frequency-dependent admittance of a short superconducting weak link,” Phys. Rev. B 87, 174521 (2013).
  * Linder and Robinson (2015) J. Linder and J. W. A. Robinson, “Superconducting spintronics,” Nat. Phys. 11, 307 (2015).
  * Eschrig (2015) M. Eschrig, “Spin-polarized supercurrents for spintronics: A review of current progress,” Rep. Prog. Phys. 78, 104501 (2015).
  * Cai  _et al._ (2023) R. Cai, I. Žutić, and W. Han, “Superconductor/Ferromagnet Heterostructures: A Platform for Superconducting Spintronics and Quantum Computation,” Adv. Quantum Technol. 6, 2200080 (2023).
  * Baumgartner  _et al._ (2022) C. Baumgartner, L. Fuchs, A. Costa, S. Reinhardt, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, P. E. Faria Junior, D. Kochan, J. Fabian, N. Paradiso, and C. Strunk, “Supercurrent rectification and magnetochiral effects in symmetric Josephson junctions,” Nat. Nanotechnol. 17, 39 (2022).
  * Pekerten  _et al._ (2022) B. Pekerten, J. D. Pakizer, B. Hawn, and A. Matos-Abiague, “Anisotropic topological superconductivity in Josephson junctions,” Phys. Rev. B 105, 054504 (2022).
  * Lotfizadeh  _et al._ (2024) N. Lotfizadeh, W.F. Schiela, B. Pekerten, P. Yu, W. Strickland, A. Matos-Abiague, and J. Shabani, “Superconducting diode effect sign change in epitaxial Al-InAs Josepshon junctions,” Commun. Phys. 7, 120 (2024).
  * Krantz  _et al._ (2019) P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, “A quantum engineer’s guide to superconducting qubits,” Appl. Phys. Rev. 6, 021318 (2019).
  * Tafuri (2019) (ed.) F. Tafuri (ed.),  _Fundamentals and Frontiers of the Josephson Effect_ (Springer Nature, Cham, 2019).
  * Pita-Vidal  _et al._ (2020) M. Pita-Vidal, A. Bargerbos, C.-K. Yang, D. J. van Woerkom, W. Pfaff, N. Haider, P. Krogstrup, L. P. Kouwenhoven, G. de Lange, and A. Kou, “Gate-Tunable Field-Compatible Fluxonium,” Phys. Rev. Applied 14, 064038 (2020).
  * (61) “Supplementary material,” See Supplemental Material at [URL] and references therein. Groth  _et al._ (2014); Tewari and Sau (2012); Nayak  _et al._ (2008); Pientka  _et al._ (2017); Pekerten  _et al._ (2022); Dartiailh  _et al._ (2021a); Elfeky  _et al._ (2023a).
  * Tinkham (1996) M. Tinkham,  _Introduction to Superconductivity_ (McGraw-Hill, New York, 1996).
  * Pekerten  _et al._ (2024) B. Pekerten, D. Brandão, B. Bussiere, D. Monroe, T. Zhou, J. E. Han, J. Shabani, A. Matos-Abiague, and I. Žutić, “Beyond the standard model of topological Josephson junctions: From crystalline anisotropy to finite-size and diode effects,” unpublished (2024).
  * Zazunov  _et al._ (2003) A. Zazunov, V. S. Shumeiko, E. N. Bratus’, J. Lantz, and G. Wendin, “Andreev Level Qubit,” Phys. Rev. Lett. 90, 087003 (2003).
  * Olivares  _et al._ (2014) D. G. Olivares, A. Levy Yeyati, L. Bretheau, C. O. Girit, H. Pothier, and C. Urbina, “Dynamics of quasiparticle trapping in Andreev levels,” Phys. Rev. B 89, 104504 (2014).
  * Väyrynen  _et al._ (2015) J. I. Väyrynen, G. Rastelli, W. Belzig, and L. I. Glazman, “Microwave signatures of Majorana states in a topological Josephson junction,” Phys. Rev. B 92, 134508 (2015).
  * Dartiailh  _et al._ (2021b) M. C. Dartiailh, J. J. Cuozzo, B. H. Elfeky, W. Mayer, J. Yuan, K. S. Wickramasinghe, E. Rossi, and J. Shabani, “Missing Shapiro steps in topologically trivial Josephson junction on InAs quantum well,” Nat. Commun. 12, 78 (2021b).
  * Phan  _et al._ (2023) D. Phan, P. Falthansl-Scheinecker, U. Mishra, W.M. Strickland, D. Langone, J. Shabani, and A.P. Higginbotham, “Gate-Tunable Superconductor-Semiconductor Parametric Amplifier,” Phys. Rev. Appl. 19, 064032 (2023).
  * Wesdorp  _et al._ (2024) J. J. Wesdorp, F. J. Matute-Cañadas, A. Vaartjes, L. Grünhaupt, T. Laeven, S. Roelofs, L. J. Splitthoff, M. Pita-Vidal, A. Bargerbos, D. J. van Woerkom, P. Krogstrup, L. P. Kouwenhoven, C. K. Andersen, A. Levy Yeyati, B. van Heck, and G. de Lange, “Microwave spectroscopy of interacting Andreev spins,” Phys. Rev. B 109, 045302 (2024).
  * Chidambaram  _et al._ (2022) V. Chidambaram, A. Kringhoj, L. Casparis, F. Kuemmeth, T. Wang, C. Thomas, S. Gronin, G. C. Gardner, Z. Cui, C. Liu, K. Moors, M. J. Manfra, K. D. Petersson, and M. R. Connolly, “Microwave sensing of Andreev bound states in a gate-defined superconducting quantum point contact,” Phys. Rev. Res. 4, 023170 (2022).
  * Bretheau  _et al._ (2017) L. Bretheau, J. I-J. Wang, R. Pisoni, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, “Tunnelling spectroscopy of Andreev states in graphene,” Nat. Phys. 13, 756 (2017).
  * (72) “Supplementary material,” See Supplemental Material at [URL].
  * Elfeky  _et al._ (2023b) B. H. Elfeky, J. J. Cuozzo, N. Lotfizadeh, W. F. Schiela, S. M. Farzaneh, W. M. Strickland, D. Langone, E. Rossi, and J. Shabani, “Evolution of 4$\pi$-Periodic Supercurrent in the Presence of an In-Plane Magnetic Field,” ACS Nano 17, 4650 (2023b).
  * Blais  _et al._ (2021) A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, “Circuit quantum electrodynamics,” Rev. Mod. Phys. 93, 025005 (2021).
  * Bretheau  _et al._ (2014) L. Bretheau, C. O. Girit, M. Houzet, H. Pothier, D. Esteve, and C. Urbina, “Theory of microwave spectroscopy of Andreev bound states with a Josephson junction,” Phys. Rev. B 90, 134506 (2014).
  * Murthy  _et al._ (2020) C. Murthy, V. D. Kurilovich, P. D. Kurilovich, B. van Heck, L. I. Glazman, and C. Nayak, “Energy spectrum and current-phase relation of a nanowire Josephson junction close to the topological transition,” Phys. Rev. B 101, 224501 (2020).
  * Xie  _et al._ (2023) Y.-M. Xie, É. Lantagne-Hurtubise, A. F. Young, S. Nadj-Perge, and J. Alicea, “Gate-Defined Topological Josephson Junctions in Bernal Bilayer Graphene,” Phys. Rev. Lett. 131, 146601 (2023).
  * Alidoust  _et al._ (2021) M. Alidoust, C. Shen, and I. Žutić, “Cubic spin-orbit coupling and anomalous Josephson effect in planar junctions,” Phys. Rev. B 103, L060503 (2021).
  * Luethi  _et al._ (2023) M. Luethi, K. Laubscher, S. Bosco, D. Loss, and J. Klinovaja, “Planar Josephson junctions in germanium: Effect of cubic spin-orbit interaction,” Phys. Rev. B 107, 035435 (2023).
  * Bretheau  _et al._ (2013b) L. Bretheau, C. O. Girit, C. Urbina, D. Esteve, and H. Pothier, “Supercurrent Spectroscopy of Andreev States,” Phys. Rev. X 3, 041034 (2013b).
  * Ackermann  _et al._ (2023) N. Ackermann, A. Zazunov, S. Park, R. Egger, and A. Levy Yeyati, “Dynamical parity selection in superconducting weak links,” Phys. Rev. B 107, 214515 (2023).
  * Monroe  _et al._ (2022) D. Monroe, M. Alidoust, and I. Žutić, “Tunable Planar Josephson Junctions Driven by Time-Dependent Spin-Orbit Coupling,” Phys. Rev. Applied 18, L031001 (2022).
  * Holmes  _et al._ (2013) D. S. Holmes, A. L. Ripple, and M. A. Manheimer, “Energy-Efficient Superconducting Computing-Power Budgets and Requirements,” IEEE Trans. Appl. Supercond. 23, 1701610 (2013).
  * Likharev and Semenov (1991) K. K. Likharev and V. K. Semenov, “RSFQ Logic/Memory Family: A New Josephson-Junction Technology for Sub-Terahertz-Clock-Frequency Digital Systems,” IEEE Trans. Appl. Supercond. 1, 3 (1991).
  * Crotty  _et al._ (2010) P. Crotty, D. Schult, and K. Segall, “Josephson junction simulation of neurons,” Phys. Rev. E 82, 011914 (2010).
  * Kashiwaya and Tanaka (2000) S. Kashiwaya and Y. Tanaka, “Tunnelling effects on surface bound states in unconventional superconductors,” Rep. Prog. Phys. 63, 1641 (2000).
  * Groth  _et al._ (2014) C. W. Groth, A. R. Akhmerov, and X. Waintal, “Kwant: a software package for quantum transport,” New J. Phys. 16, 063065 (2014).
  * Tewari and Sau (2012) S. Tewari and J. D. Sau, “Topological Invariants for Spin-Orbit Coupled Superconductor Nanowires,” Phys. Rev. Lett. 109, 150408 (2012).


with $\epsilon\in\\{\pm\\}$, $0\leq s\leq p$, $1\leq s^{\prime}\leq p-1$ and
$\xi^{-1}=\frac{1-i}{2\sqrt{p}}\frac{\hat{q}^{p-1}}{[p-1]!}(-1)^{p}q^{-(p-3)/2}$.

###### Proof.

The formulas for $v_{A}^{-1}$ are easily deduced from Proposition 4.13, (6.4)
and (6.5). Computing the action of $v_{B}^{-1}$ is more difficult. We will use
the commutation relations of $v_{B}^{-1}$ with the $A,B$-matrices (which
follow from Theorem 5.2), namely

$\displaystyle
v_{B}^{-1}\overset{I}{A}=\overset{I}{v}\overset{I}{B}{{}^{-1}}\overset{I}{A}v_{B}^{-1},\qquad
v_{B}^{-1}\overset{I}{B}=\overset{I}{B}v_{B}^{-1}$ (6.6)

to compute the action of $v_{B}^{-1}$ by induction. The multiplication rules
of the GTA basis (see [21, Section 5]) will be used several times. Let us
denote

$v_{B}^{-1}\triangleright\chi^{\epsilon}_{s}=\sum_{\sigma\in\\{\pm\\}}\sum_{\ell=1}^{p}\lambda^{\sigma}_{\ell}(\epsilon,s)\chi^{\sigma}_{\ell}+\sum_{j=1}^{p-1}\delta_{j}(\epsilon,s)G_{j}.$

Relation (6.6) implies
$v_{B}^{-1}\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!\\!_{A}=\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!_{vB{{}^{-1}}A}v_{B}^{-1}$
(recall (6.2)). On the one hand, we obtain by (6.3)

$\displaystyle
v_{B}^{-1}\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!\\!_{A}\triangleright\chi^{\epsilon}_{s}$
$\displaystyle=v_{B}^{-1}\triangleright\chi^{\epsilon}_{s}\big{(}{-}\hat{q}^{2}C?\big{)}$
$\displaystyle=\sum_{\begin{subarray}{c}\ell=1\\\
\sigma\in\\{\pm\\}\end{subarray}}^{p}-\epsilon\big{(}q^{s}+q^{-s}\big{)}\lambda^{\sigma}_{\ell}(\epsilon,s)\chi^{\sigma}_{\ell}+\sum_{j=1}^{p-1}-\epsilon\big{(}q^{s}+q^{-s}\big{)}\delta_{j}(\epsilon,s)G_{j}.$

On the other hand, we use Lemma 6.4 and the multiplication rules

$\displaystyle\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!_{vB{{}^{-1}}A}v_{B}^{-1}\triangleright\chi^{\epsilon}_{s}=\sum_{\sigma\in\\{\pm\\}}\sum_{\ell=1}^{p}\lambda^{\sigma}_{\ell}(\epsilon,s)\chi^{+}_{2}\chi^{\sigma}_{\ell}+\sum_{j=1}^{p-1}\delta_{j}(\epsilon,s)\chi^{+}_{2}G_{j}$
$\displaystyle\hphantom{\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!_{vB{{}^{-1}}A}v_{B}^{-1}\triangleright\chi^{\epsilon}_{s}}{}=\sum_{\sigma\in\\{\pm\\}}\big{(}\lambda^{\sigma}_{2}(\epsilon,s)+2\lambda^{-\sigma}_{p}(\epsilon,s)\big{)}\chi^{\sigma}_{1}+\sum_{\ell=2}^{p-2}\big{(}\lambda^{\sigma}_{s-1}(\epsilon,s)+\lambda^{\sigma}_{s+1}(\epsilon,s)\big{)}\chi^{\sigma}_{\ell}$
$\displaystyle\hphantom{\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!_{vB{{}^{-1}}A}v_{B}^{-1}\triangleright\chi^{\epsilon}_{s}=}{}+\big{(}\lambda^{\sigma}_{p-2}(\epsilon,s)+2\lambda^{\sigma}_{p}(\epsilon,s)\big{)}\chi^{\sigma}_{p-1}+\lambda^{\sigma}_{p-1}(\epsilon,s)\chi^{\sigma}_{p}+\frac{\delta_{2}(\epsilon,s)}{[2]}G_{1}$
$\displaystyle\hphantom{\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!_{vB{{}^{-1}}A}v_{B}^{-1}\triangleright\chi^{\epsilon}_{s}=}{}+\sum_{j=2}^{p-2}[j]\left(\frac{\delta_{j-1}(\epsilon,s)}{[j-1]}+\frac{\delta_{j+1}(\epsilon,s)}{[j+1]}\right)\\!G_{j}+\frac{\delta_{p-2}(\epsilon,s)}{[2]}G_{p-1}.$

This gives recurrence equations between the coefficients which are easily
solved

$\displaystyle
v_{B}^{-1}\triangleright\chi^{\epsilon}_{s}=\lambda(\epsilon,s)\left(\sum_{\ell=1}^{p-1}(-1)^{s}(-\epsilon)^{p-\ell}\big{(}q^{\ell
s}+q^{-\ell
s}\big{)}\big{(}\chi^{+}_{\ell}+\chi^{-}_{p-\ell}\big{)}+\chi^{+}_{p}+(-\epsilon)^{p}(-1)^{s}\chi^{-}_{p}\right)$
$\displaystyle\hphantom{v_{B}^{-1}\triangleright\chi^{\epsilon}_{s}=}{}+\delta(\epsilon,s)\sum_{j=1}^{p-1}(-\epsilon)^{j+1}\frac{[j][js]}{[s]}G_{j}.$

The coefficients $\lambda(\epsilon,s)=\lambda^{+}_{p}(\epsilon,s)$ and
$\delta(\epsilon,s)=\delta_{1}(\epsilon,s)$ are still unknown. In order to
compute them by induction, we use the relation
$v_{B}^{-1}\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!\\!_{B}=\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!\\!_{B}v_{B}^{-1}$,
which is another consequence of (6.6). Before, note that

$\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!\\!_{B}\triangleright\chi^{\epsilon}_{s}=\big{(}\chi^{+}_{2}(\chi^{\epsilon}_{s})^{v}\big{)}^{v^{-1}}=\frac{v_{\mathcal{X}^{\epsilon}(s)}}{v_{\mathcal{X}^{\epsilon}(s-1)}}\chi^{\epsilon}_{s-1}+\frac{v_{\mathcal{X}^{\epsilon}(s)}}{v_{\mathcal{X}^{\epsilon}(s+1)}}\chi^{\epsilon}_{s+1}=-\epsilon
q^{-s+\frac{1}{2}}\chi^{\epsilon}_{s-1}-\epsilon
q^{s+\frac{1}{2}}\chi^{\epsilon}_{s+1},$

with $1\leq s\leq p-1$ and the convention that $\chi^{\pm}_{0}=0$. It follows
that

$\displaystyle v_{B}^{-1}\triangleright\chi^{\epsilon}_{s+1}=-\epsilon
q^{-s-\frac{1}{2}}\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!\\!_{B}\triangleright\big{(}v_{B}^{-1}\triangleright\chi^{\epsilon}_{s}\big{)}-q^{-2s}v_{B}^{-1}\triangleright\chi^{\epsilon}_{s-1}.$
(6.7)

Due to (6.4), (6.5) and the multiplication rules, we have

$\displaystyle\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!\\!_{B}\triangleright\big{(}v_{B}^{-1}\triangleright\chi^{\epsilon}_{s}\big{)}$
$\displaystyle=\big{(}\chi^{+}_{2}\big{(}v_{B}^{-1}\triangleright\chi^{\epsilon}_{s}\big{)}^{v}\big{)}^{v^{-1}}$
$\displaystyle=\frac{v_{\mathcal{X}^{+}(p-1)}}{v_{\mathcal{X}^{+}(p)}}\big{(}\lambda^{+}_{p-1}(\epsilon,s)+\hat{q}\delta_{p-1}(\epsilon,s)\big{)}\chi^{+}_{p}+\frac{v_{\mathcal{X}^{+}(2)}}{[2]}\delta_{2}(\epsilon,s)G_{1}+\cdots,$

where the dots ($\cdots$) mean the remaining of the linear combination in the
GTA basis. After replacement by the values found previously and insertion in
relation (6.7), we obtain

$\displaystyle\lambda(\epsilon,s+1)\chi^{+}_{p}+\delta(\epsilon,s+1)G_{1}+\cdots$
$\displaystyle\qquad{}=\big{(}q^{-(s+1)}\big{(}q^{s}+q^{-s}\big{)}\lambda(\epsilon,s)-q^{-2s}\lambda(\epsilon,s-1)+(-\epsilon)^{p-1}(-1)^{s-1}\hat{q}q^{-(s+1)}\delta(\epsilon,s)\big{)}\chi^{+}_{p}$
$\displaystyle\qquad\quad{}+\big{(}{-}q^{-(s+2)}\big{(}q^{s}+q^{-s}\big{)}\delta(\epsilon,s)-q^{-2s}\delta(\epsilon,s-1)\big{)}G_{1}+\cdots.$

These are recurrence equations. It just remains to determine the first values
$\lambda(\epsilon,1)$, $\delta(\epsilon,1)$. Observe that, since
$\overline{U}_{q}$ is unibalanced:

$\displaystyle
v_{B}^{-1}\triangleright\chi^{+}_{1}=\big{(}\varphi_{v^{-1}}\big{(}\chi^{+}_{1}\big{)}^{v}\big{)}^{v^{-1}}=\mu^{l}(v)^{-1}\mu^{l}\big{(}K^{p-1}v?\big{)}^{v^{-1}}=\mu^{l}(v)^{-1}\mu^{r}\big{(}K^{p+1}?\big{)}.$
(6.8)

In [21, Section 4.3] the decomposition of $\mu^{r}(K^{p+1}?)$ in the GTA basis
has been found (when $\mu^{r}$ is suitably normalized). Thanks to this, we
obtain

$v_{B}^{-1}\triangleright\chi^{+}_{1}=\lambda(+,1)\chi^{+}_{p}+\delta(+,1)G_{1}+\dots=\xi(-1)^{p-1}\chi^{+}_{p}-\xi
G_{1}+\cdots$

and

$\displaystyle v_{B}^{-1}\triangleright\chi^{-}_{1}$
$\displaystyle=v_{\mathcal{X}^{-}(1)}v_{B}^{-1}\overset{\mathcal{X}^{-}(1)}{W}\\!\\!\\!\\!\\!_{B}\triangleright\chi^{+}_{1}=v_{\mathcal{X}^{-}(1)}\overset{\mathcal{X}^{-}(1)}{W}\\!\\!\\!\\!\\!_{B}\triangleright
v_{B}^{-1}\triangleright\chi^{+}_{1}=v_{\mathcal{X}^{-}(1)}\big{(}\chi^{-}_{1}\big{(}v_{B}^{-1}\triangleright\chi^{+}_{1}\big{)}^{v}\big{)}^{v^{-1}}$
$\displaystyle=-\xi\chi^{+}_{p}+\xi
G_{1}+\dots=\lambda(-,1)\chi^{+}_{p}+\delta(-,1)G_{1}+\cdots.$

The scalar $\xi$ does not depend on the choice of $\mu^{r}$ thanks to the
factor $\mu^{l}(v)^{-1}=\mu^{l}\circ S(v)^{-1}=\mu^{r}(v)^{-1}$ in (6.8).
Thanks to the formulas [24] for $\mu^{r}$ and $v$ in the PBW basis we compute
$\mu^{r}(v)$ and this gives the value of $\xi$. We are now in position to
solve the recurrence equations. It is easy to check that the solutions are

$\delta(\epsilon,s)=\xi\epsilon(-1)^{s}q^{-(s^{2}-1)}[s],\qquad\lambda(\epsilon,s)=\xi\epsilon(-\epsilon)^{p-1}sq^{-(s^{2}-1)}.$

We now proceed with the proof of the formula for $G_{s^{\prime}}$. Relation
(6.6) implies $v_{B}^{-1}H_{B}^{1}=H_{B}^{1}v_{B}^{-1}$ (recall (6.2)). By
(6.4), (6.5) and the multiplication rules, we have on the one hand

$v_{B}^{-1}H_{B}^{1}\triangleright\chi^{+}_{s}=v_{B}^{-1}\triangleright\big{(}G_{1}\big{(}\chi^{+}_{s}\big{)}^{v}\big{)}^{v^{-1}}=[s]v_{B}^{-1}\triangleright
G_{s}-\hat{q}(p-s)v_{B}^{-1}\triangleright\chi^{+}_{s}+\hat{q}sv_{B}^{-1}\triangleright\chi^{-}_{p-s},$

whereas on the other hand

$H_{B}^{1}v_{B}^{-1}\triangleright\chi^{+}_{s}=\big{(}G_{1}\big{(}v_{B}^{-1}\triangleright\chi^{+}_{s}\big{)}^{v}\big{)}^{v^{-1}}=\hat{q}p\sum_{j=1}^{p-1}\delta_{j}(+,s)\left(G_{j}-\hat{q}\frac{p-j}{[j]}\chi^{+}_{j}+\hat{q}\frac{j}{[j]}\chi^{-}_{p-j}\right).$

Equalizing both sides and inserting the previously found values, we obtain the
desired formula. ∎

######  Remark 6.6.

The guiding principle of the previous computations was that the mutiplication
of two symmetric linear forms in the GTA basis is easy when one of them is
$\chi^{+}_{2}$, $\chi^{-}_{1}$ or $G_{1}$ (see [21, Section 5]), and that all
the formulas can be derived from $v_{B}^{-1}\triangleright\chi^{+}_{1}$ using
only such products.

Recall that the standard representation $\mathbb{C}^{2}$ of
$\mathrm{SL}_{2}(\mathbb{Z})=\mathrm{MCG}(\Sigma_{1,0})$ is defined by

$\tau_{a}\mapsto\begin{pmatrix}1&0\\\
-1&1\end{pmatrix},\qquad\tau_{b}\mapsto\begin{pmatrix}1&1\\\
0&1\end{pmatrix}.$

######  Lemma 6.7.

Let $V$ be a (projective) representation of $\mathrm{SL}_{2}(\mathbb{Z})$
which admits a basis $\left(x_{s},y_{s}\right)$ such that

$\displaystyle\tau_{a}x_{s}=\sum_{\ell}a_{\ell}(s)x_{\ell},\qquad$
$\displaystyle\tau_{b}x_{s}=\sum_{\ell}b_{\ell}(s)(x_{\ell}+y_{\ell}),$
$\displaystyle\tau_{a}y_{s}=\sum_{\ell}a_{\ell}(s)(y_{\ell}-x_{\ell}),\qquad$
$\displaystyle\tau_{b}y_{s}=\sum_{\ell}b_{\ell}(s)y_{\ell}.$

Then there exists a $($projective$)$ representation $W$ of
$\mathrm{SL}_{2}(\mathbb{Z})$ such that $V\cong\mathbb{C}^{2}\otimes W$. More
precisely, $W$ admits a basis $(w_{s})$ such that

$\tau_{a}w_{s}=\sum_{\ell}a_{\ell}(s)w_{\ell},\qquad\tau_{b}w_{s}=\sum_{\ell}b_{\ell}(s)w_{\ell}.$

###### Proof.

It is easy to check that the formulas for $\tau_{a}w_{s}$ and $\tau_{b}w_{s}$
indeed define a $\mathrm{SL}_{2}(\mathbb{Z})$-representation on $W$. Let
$(e_{1},e_{2})$ be the canonical basis of $\mathbb{C}^{2}$. Then

$e_{1}\otimes w_{s}\mapsto y_{s},\qquad e_{2}\otimes w_{s}\mapsto x_{s}$

is an isomorphism which intertwines the $\mathrm{SL}_{2}(\mathbb{Z})$-action.
∎

######  Theorem 6.8.

The $(p+1)$-dimensional subspace
$\mathcal{V}=\operatorname{vect}\big{(}\chi^{+}_{s}+\chi^{-}_{p-s},\chi^{\pm}_{p}\big{)}_{1\leq
s\leq p-1}$ is stable under the $\mathrm{SL}_{2}(\mathbb{Z})$-action of
Theorem 6.5. Moreover, there exists a $(p-1)$-dimensional projective
representation $\mathcal{W}$ of $\mathrm{SL}_{2}(\mathbb{Z})$ such that

$\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}=\mathcal{V}\oplus\big{(}\mathbb{C}^{2}\otimes\mathcal{W}\big{)}.$

###### Proof.

By [21, Corollary 5.1], $\mathcal{V}$ is an ideal of
$\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}$. It is easy to see that
$\mathcal{V}$ is moreover stable under the action (6.5) of
$\mathcal{Z}\big{(}\overline{U}_{q}\big{)}$. Thus we deduce without any
computation that $\mathcal{V}$ is $\mathrm{SL}_{2}(\mathbb{Z})$-stable. Next,
in view of the formulas in Theorem 6.5, it is natural to define

$x_{s}=\hat{q}\frac{p-s}{[s]}\chi^{+}_{s}-\hat{q}\frac{s}{[s]}\chi^{-}_{p-s},\qquad
y_{s}=G_{s}-x_{s}.$

Then

$\displaystyle v_{A}^{-1}\triangleright
x_{s}=v_{\mathcal{X}^{+}(s)}^{-1}x_{s},\qquad$ $\displaystyle
v_{B}^{-1}\triangleright
x_{s}=\xi(-1)^{s}q^{-(s^{2}-1)}\frac{\hat{q}p}{[s]}\sum_{j=1}^{p-1}(-1)^{j+1}[j][js]\\!\left(x_{j}+y_{j}\right),$
$\displaystyle v_{A}^{-1}\triangleright
y_{s}=v_{\mathcal{X}^{+}(s)}^{-1}(y_{s}-x_{s}),\qquad$ $\displaystyle
v_{B}^{-1}\triangleright
y_{s}=\xi(-1)^{s}q^{-(s^{2}-1)}\frac{\hat{q}p}{[s]}\sum_{j=1}^{p-1}(-1)^{j+1}[j][js]y_{j}.$

The result follows from Lemma 6.7. ∎

The structure of the Lyubashenko–Majid representation on
$\mathcal{Z}\big{(}\overline{U}_{q}\big{)}$ has been described in [24]. By
Theorem 5.12, this projective representation is equivalent to the one
constructed here and Theorem 6.8 is in perfect agreement with the
decomposition given in [24].

Note that the subspace $\mathcal{V}$ is generated by the characters of the
finite-dimensional projective $\overline{U}_{q}$-modules:
$\mathcal{V}=\mathrm{vect}\big{(}\chi^{P}\big{)}_{P\in\mathrm{Proj}\big{(}\overline{U}_{q}\big{)}}$.
We precise that, explicitly, the projective representation $\mathcal{W}$ has a
basis $(w_{s})_{1\leq s\leq p-1}$ such that the action of $\tau_{a}$,
$\tau_{b}$ is

$\tau_{a}w_{s}=v_{\mathcal{X}^{+}(s)}^{-1}w_{s},\qquad\tau_{b}w_{s}=\xi(-1)^{s}q^{-(s^{2}-1)}\frac{\hat{q}p}{[s]}\sum_{j=1}^{p-1}(-1)^{j+1}[j][js]w_{j}.$

###  6.4 A conjecture about the representation of
$\boldsymbol{\mathcal{L}_{1,0}^{\mathrm{inv}}\big{(}\overline{U}_{q}\big{)}}$
on $\boldsymbol{\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}}$

Another natural (but harder) question is to determine the structure of
$\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}$ under the action of
$\mathcal{L}_{1,0}^{\mathrm{inv}}\big{(}\overline{U}_{q}\big{)}$. As mentioned
in the proof of Theorem 6.8, the subspace

$\mathcal{V}=\operatorname{vect}\big{(}\chi^{+}_{s}+\chi^{-}_{p-s},\chi^{\pm}_{p}\big{)}_{1\leq
s\leq p-1}$

is quite “stable”. We propose the following conjecture.

######  Conjecture 6.9.

$\mathcal{V}$ is a
$\mathcal{L}_{1,0}^{\mathrm{inv}}\big{(}\overline{U}_{q}\big{)}$-submodule of
$\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}$.

In order to prove this conjecture one needs to find a basis or a generating
set of $\mathcal{L}_{1,0}^{\mathrm{inv}}\big{(}\overline{U}_{q}\big{)}$, and
then to show that $\mathcal{V}$ is stable under the action of the basis
elements (or of the generating elements). Both tasks are difficult.

Let us mention that since $\mathcal{V}$ is an ideal of
$\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}$, it is stable under the
action of $z_{A}$, $z_{B}$ and $z_{B^{-1}A}$ for all
$z\in\mathcal{Z}\big{(}\overline{U}_{q}\big{)}=\mathcal{L}^{\mathrm{inv}}_{0,1}\big{(}\overline{U}_{q}\big{)}$
(see Proposition 4.13 and Lemma 6.4). Also recall the wide family of
invariants given in (4.3); we can try to test the conjecture with them. A long
computation (which is not specific to $\overline{U}_{q}$) shows that

$\mathrm{tr}_{12}\Big{(}\overset{I\otimes
J}{g}\\!\\!\\!_{12}\Phi_{12}\overset{I}{A}_{1}\overset{IJ}{(R^{\prime})}_{12}\overset{J}{B}_{2}\overset{IJ}{R}_{12}\Big{)}\triangleright\chi^{K}=v_{J}\mathrm{tr}_{13}\Big{(}\overset{I\otimes
K}{T}\\!\\!\\!\\!_{13}\overset{I\otimes
K}{v}{{}^{-1}}\\!\\!\\!\\!\\!\\!\\!\\!\\!_{13}\>\>\>s_{IJ,K}(\Phi)_{13}\Big{)},$

where $\chi^{K}$ is the character of $K$, $\overset{J}{v}=v_{J}\mathrm{id}$
(note that we may assume that $I$, $J$, $K$ are simple modules) and

$s_{IJ,K}(\Phi)=\mathrm{tr}_{2}\Big{(}\overset{J}{g}_{2}\overset{JK}{R}_{23}\Phi_{12}\overset{JK}{(R^{\prime})}_{23}\Big{)}.$

Proving that $\mathcal{V}$ is stable under the action of these invariants
amounts to show symmetry properties between $s_{IJ,\mathcal{X}^{+}(s)}$ and
$s_{IJ,\mathcal{X}^{-}(p-s)}$ for all simple $\overline{U}_{q}$-modules $I$,
$J$. We have checked that it is true if $\Phi=\mathrm{id}_{I\otimes J}$ (in
this case $s_{IJ,K}(\mathrm{id}_{I\otimes J})=s_{J,K}\mathrm{id}_{I\otimes
K}$, where $s_{J,K}$ is the usual $S$-matrix) for all simple modules $I$, $J$,
and also that it holds for $I=J=\mathcal{X}^{+}(2)$ with every $\Phi$.

######  Proposition 6.10.

  1. $1)$

$\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}$ is indecomposable as a
$\mathcal{L}_{1,0}^{\mathrm{inv}}\big{(}\overline{U}_{q}\big{)}$-module.

  2. $2)$

Assume that the Conjecture holds. Then the
$\mathcal{L}_{1,0}^{\mathrm{inv}}\big{(}\overline{U}_{q}\big{)}$-modules
$\mathcal{V}$ and
$\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}/\mathcal{V}$ are simple. It
follows that $\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}$ has length $2$
as a $\mathcal{L}_{1,0}^{\mathrm{inv}}\big{(}\overline{U}_{q}\big{)}$-module.

###### Proof.

These are basically consequences of (6.5) and of the multiplication rules in
the GTA basis [21, Section 5]. To avoid particular cases, let
$\chi^{\epsilon}_{0}=0$, $\chi^{\epsilon}_{p+1}=\chi^{-\epsilon}_{1}$,
$\chi^{\epsilon}_{-1}=\chi^{-\epsilon}_{p-1}$ and $e_{-1}=e_{p+1}=0$.

1) Observe that $\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}$ is
generated by $\chi^{+}_{1}=\varepsilon$ as a
$\mathcal{L}_{1,0}^{\mathrm{inv}}\big{(}\overline{U}_{q}\big{)}$-module: this
is a general fact which follows immediately from Lemma 6.4. Explicitly (see
(5.2))

$\overset{\mathcal{X}^{\epsilon}(s)}{W}\\!\\!\\!\\!_{vB^{-1}A}\triangleright\chi^{+}_{1}=\chi^{\epsilon}_{s}\chi^{+}_{1}=\chi^{\epsilon}_{s},\qquad
H^{s}_{vB^{-1}A}\triangleright\chi^{+}_{1}=G_{s}\chi^{+}_{1}=G_{s}.$

Write $\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}=U_{1}\oplus U_{2}$. At
least one of the two subspaces $U_{1},U_{2}$ necessarily contains an element
of the form $\varphi=G_{1}+\sum_{i\neq
1}\lambda_{i}G_{i}+\sum_{j,\epsilon}\eta^{\epsilon}_{j}\chi^{\epsilon}_{j}$;
assume that it is $U_{1}$. Then
$\big{(}w_{1}^{+}\big{)}_{A}\triangleright\varphi=\varphi^{w_{1}^{+}}=\chi^{+}_{1}\in
U_{1}$ thanks to (6.5). It follows that
$U_{1}=\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}$ and $U_{2}=\\{0\\}$,
as desired.

2) Let $0\neq U\subset\mathcal{V}$ be a submodule, and let
$\psi=\sum_{j=0}^{p}\lambda_{j}\big{(}\chi^{+}_{j}+\chi^{-}_{p-j}\big{)}\in U$
with $\lambda_{s}\neq 0$ for some $s$. Then using Proposition 4.13 and (6.5),
we get
$(e_{s})_{A}\triangleright\psi=\psi^{e_{s}}=\lambda_{s}\big{(}\chi^{+}_{s}+\chi^{-}_{p-s}\big{)}$,
and thus $\chi^{+}_{s}+\chi^{-}_{p-s}\in U$. Apply
$\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!_{vB^{-1}A}$ (we use Lemma 6.4 and
Proposition 6.3)

$\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!_{vB^{-1}A}\triangleright\big{(}\chi^{+}_{s}+\chi^{-}_{p-s}\big{)}=\chi^{+}_{2}\big{(}\chi^{+}_{s}+\chi^{-}_{p-s}\big{)}=\big{(}\chi^{+}_{s-1}+\chi^{-}_{p-s+1}\big{)}+\big{(}\chi^{+}_{s+1}+\chi^{-}_{p-s-1}\big{)}.$

Hence

$\displaystyle(e_{s-1})_{A}\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!_{vB^{-1}A}\triangleright\big{(}\chi^{+}_{s}+\chi^{-}_{p-s}\big{)}=\chi^{+}_{s-1}+\chi^{-}_{p-s+1},$
$\displaystyle(e_{s+1})_{A}\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!_{vB^{-1}A}\triangleright\big{(}\chi^{+}_{s}+\chi^{-}_{p-s}\big{)}=\chi^{+}_{s+1}+\chi^{-}_{p-s-1}.$

It follows that
$\chi^{+}_{s-1}+\chi^{-}_{p-s+1},\chi^{+}_{s+1}+\chi^{-}_{p-s-1}\in U$.
Continuing like this, one gets step by step that all the basis vectors belong
to $U$, hence $U=\mathcal{V}$.

Next, let $\overline{G}_{s}$ and $\overline{\chi}^{+}_{s}$ be the classes of
$G_{s}$ and $\chi^{+}_{s}$ modulo $\mathcal{V}$ (with
$\overline{\chi}^{+}_{0}=\overline{\chi}^{+}_{p}=0$). Let $0\neq
U\subset\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}/\mathcal{V}$ be a
submodule and
$\omega=\sum_{j=1}^{p-1}\nu_{j}\overline{G}_{j}+\sigma_{j}\overline{\chi}^{+}_{j}\in
U$ be non-zero. If all the $\nu_{j}$ are $0$, then there exists
$\sigma_{s}\neq 0$ and
$(e_{s})_{A}\triangleright\omega=\sigma_{s}\overline{\chi}^{+}_{s}\in U$. If
one of the $\nu_{j}$, say $\nu_{s}$, is non-zero, then
$(w^{+}_{s})_{A}\triangleright\omega=\nu_{s}\overline{\chi}^{+}_{s}\in U$. In
both cases we get $\overline{\chi}^{+}_{s}\in U$. Now we proceed as previously

$(e_{s-1})_{A}\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!_{vB^{-1}A}\triangleright\overline{\chi}^{+}_{s}=\overline{\chi}^{+}_{s-1},\qquad(e_{s+1})_{A}\overset{\mathcal{X}^{+}(2)}{W}\\!\\!\\!\\!_{vB^{-1}A}\triangleright\overline{\chi}^{+}_{s}=\overline{\chi}^{+}_{s+1}.$

Thus we get step by step that $\overline{\chi}^{+}_{j}\in U$ for all $j$.
Apply $H^{1}_{vB^{-1}A}$

$H_{vB^{-1}A}^{1}\triangleright\overline{\chi}^{+}_{j}=G_{1}\chi^{+}_{j}+\mathcal{V}=[j]G_{j}+\mathcal{P}.$

It follows that $\overline{G}_{j}\in U$ for all $j$, and thus
$U=\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}/\mathcal{V}$ as desired. ∎

In order to determine the structure of
$\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}$ if the Conjecture is true,
it will remain to determine whether $\mathcal{V}$ is a direct summand of
$\operatorname{SLF}\big{(}\overline{U}_{q}\big{)}$ or not.

### Acknowledgements

I am grateful to my advisors, Stéphane Baseilhac and Philippe Roche, for their
regular support and their useful remarks. I thank the referees for carefully
reading the manuscript and for many valuable comments which improved the
paper.

## References

  * [1] Aghaei N., Gainutdinov A.M., Pawelkiewicz M., Schomerus V., Combinatorial quantisation of ${\rm GL}(1|1)$ Chern–Simons theory I: The torus, arXiv:1811.09123.
  * [2] Alekseev A.Yu., Integrability in the Hamiltonian Chern–Simons theory, St. Petersburg Math. J. 6 (1995), 241–253, arXiv:hep-th/9311074.
  * [3] Alekseev A.Yu., Grosse H., Schomerus V., Combinatorial quantization of the Hamiltonian Chern–Simons theory. I, Comm. Math. Phys. 172 (1995), 317–358, arXiv:hep-th/9403066.
  * [4] Alekseev A.Yu., Grosse H., Schomerus V., Combinatorial quantization of the Hamiltonian Chern–Simons theory. II, Comm. Math. Phys. 174 (1996), 561–604, arXiv:hep-th/9408097.
  * [5] Alekseev A.Yu., Schomerus V., Representation theory of Chern–Simons observables, Duke Math. J. 85 (1996), 447–510, arXiv:q-alg/9503016.
  * [6] Alekseev A.Yu., Schomerus V., Quantum moduli spaces of flat connections, arXiv:q-alg/9612037.
  * [7] Arike Y., A construction of symmetric linear functions on the restricted quantum group $\overline{U}_{q}(\mathfrak{sl}_{2})$, Osaka J. Math. 47 (2010), 535–557, arXiv:0807.0052.
  * [8] Atiyah M.F., Bott R., The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523–615.
  * [9] Baseilhac S., Roche Ph., Unrestricted moduli algebras: the case of punctured spheres, in preparation.
  * [10] Ben-Zvi D., Brochier A., Jordan D., Integrating quantum groups over surfaces, J. Topology 11 (2018), 874–917, arXiv:1501.04652.
  * [11] Blanchet C., Beliakova A., Gainutdinov A.M., Modified trace is a symmetrised integral, arXiv:1801.00321.
  * [12] Brochier A., Jordan D., Fourier transform for quantum $D$-modules via the punctured torus mapping class group, Quantum Topol. 8 (2017), 361–379, arXiv:1403.1841.
  * [13] Buffenoir E., Noui K., Roche Ph., Hamiltonian quantization of Chern–Simons theory with ${\rm SL}(2,{\mathbb{C}})$ group, Classical Quantum Gravity 19 (2002), 4953–5015, arXiv:hep-th/0202121.
  * [14] Buffenoir E., Roche Ph., Two-dimensional lattice gauge theory based on a quantum group, Comm. Math. Phys. 170 (1995), 669–698, arXiv:hep-th/9405126.
  * [15] Buffenoir E., Roche Ph., Link invariants and combinatorial quantization of Hamiltonian Chern–Simons theory, Comm. Math. Phys. 181 (1996), 331–365, arXiv:q-alg/9507001.
  * [16] Bullock D., Frohman C., Kania-Bartoszyńska J., Topological interpretations of lattice gauge field theory, Comm. Math. Phys. 198 (1998), 47–81, arXiv:q-alg/9710003.
  * [17] Chekhov L., Mazzocco M., Rubtsov V., Algebras of quantum monodromy data and character varieties, in Geometry and Physics, Vol. I, A Festschrift in honour of Nigel Hitchin, Oxford University Press, Oxford, 2018, 39–68, arXiv:1705.01447.
  * [18] Curtis C.W., Reiner I., Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. 11, Interscience Publishers, New York – London, 1962.
  * [19] Drinfeld V.G., Almost cocommutative Hopf algebras, Leningrad Math. J. 1 (1990), 321–342.
  * [20] Etingof P., Gelaki S., Nikshych D., Ostrik V., Tensor categories, Mathematical Surveys and Monographs, Vol. 205, Amer. Math. Soc., Providence, RI, 2015.
  * [21] Faitg M., A note on symmetric linear forms and traces on the restricted quantum group $\overline{U}_{q}(\mathfrak{sl}(2))$, Osaka J. Math., to appear, arXiv:1801.07524.
  * [22] Faitg M., Projective representations of mapping class groups in combinatorial quantization, Comm. Math. Phys., to appear, arXiv:1812.00446.
  * [23] Farb B., Margalit D., A primer on mapping class groups, Princeton Mathematical Series, Vol. 49, Princeton University Press, Princeton, NJ, 2012\.
  * [24] Feigin B.L., Gainutdinov A.M., Semikhatov A.M., Tipunin I.Yu., Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center, Comm. Math. Phys. 265 (2006), 47–93, arXiv:hep-th/0504093.
  * [25] Fock V.V., Rosly A.A., Flat connections and polyubles, Theoret. and Math. Phys. 95 (1993), 526–534.
  * [26] Fock V.V., Rosly A.A., Poisson structure on moduli of flat connections on Riemann surfaces and the $r$-matrix, in Moscow Seminar in Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 191, Amer. Math. Soc., Providence, RI, 1999, 67–86, arXiv:math.QA/9802054.
  * [27] Gainutdinov A.M., Tipunin I.Yu., Radford, Drinfeld and Cardy boundary states in the $(1,p)$ logarithmic conformal field models, J. Phys A: Math. Theor. 42 (2009), 315207, 30 pages, arXiv:0711.3430.
  * [28] Goldman W.M., Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85 (1986), 263–302.
  * [29] Ibanez E., Evaluable Jones–Wenzl idempotents at root of unity and modular representation on the center of $\overline{U}_{q}{\mathfrak{sl}}(2)$, Ph.D. Thesis, Université de Montpellier, France, 2016, arXiv:1604.03681.
  * [30] Kassel C., Quantum groups, Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York, 1995.
  * [31] Kondo H., Saito Y., Indecomposable decomposition of tensor products of modules over the restricted quantum universal enveloping algebra associated to ${\mathfrak{sl}}_{2}$, J. Algebra 330 (2011), 103–129, arXiv:0901.4221.
  * [32] Lyubashenko V., Invariants of $3$-manifolds and projective representations of mapping class groups via quantum groups at roots of unity, Comm. Math. Phys. 172 (1995), 467–516, arXiv:hep-th/9405167.
  * [33] Lyubashenko V., Majid S., Braided groups and quantum Fourier transform, J. Algebra 166 (1994), 506–528.
  * [34] Meusburger C., Wise D.K., Hopf algebra gauge theory on a ribbon graph, arXiv:1512.03966.
  * [35] Montgomery S., Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, Vol. 82, Amer. Math. Soc., Providence, RI, 1993\.
  * [36] Radford D.E., Hopf algebras, Series on Knots and Everything, Vol. 49, World Sci. Publ. Co. Pte. Ltd., Hackensack, NJ, 2012.
  * [37] Reshetikhin N.Yu., Semenov-Tian-Shansky M.A., Quantum $R$-matrices and factorization problems, J. Geom. Phys. 5 (1988), 533–550.
  * [38] Schomerus V., Deformed gauge symmetry in local quantum physics, Habilitation Thesis, Hamburg, Germany, 1998.
  * [39] Suter R., Modules over $\overline{U}_{q}(\mathfrak{sl}_{2})$, Comm. Math. Phys. 163 (1994), 359–393.