# Manipulating anyons in quantum Hall droplets of light using dissipations

Yangqian Yan, Qi Zhou  Department of Physics and Astronomy, Purdue University,
West Lafayette, IN, 47907

###### Abstract

Whereas anyons are the building blocks in topological quantum computation, it
remains challenging to create and control each anyon individually. Here, we
point out that dissipative dynamics in cavities deterministically deliver
droplets of light in desired fractional quantum Hall states. In these quantum
Hall droplets, both the number and locations of anyons are precisely
controllable without requiring extra potentials to imprint and localize such
quasiparticles. Using the density profile of light, the anyonic statistics is
readily accessible. Moreover, entangling a quantum spin valve with these
quantum Hall droplets establishes a direct readout of the braiding statistics.
Our work unfolds a promising route for quantum optics to solve challenging
problems in quantum Hall physics.

It has been a long-lasting goal of physicists to detect anyonic statistics,
i.e., exchanging two quasi-particles leads to a phase that is neither 0 nor
$\pi$ Paredes  _et al._ (2001); Kitaev (2003); Nayak  _et al._ (2008); Dutta
and Mueller (2018). Recent experiments have reported direct observations of
anyonic statistics in two-dimensional electron gases using either Fabry-Perot
interferometers or collisions between anyons at a beamsplitter Bartolomei  _et
al._ (2020); Nakamura  _et al._ (2020). In spite of these exciting
developments, to fully utilize anyons in quantum computation, it is required
to have the capability to manipulate a single anyon in experiments. Such a
task remains challenging in two-dimensional electron gases since it is
difficult to precisely control the number of anyons in the sample. For
instance, neither the number of anyons confined by the gates nor that in the
edge currents is unambiguous in typical solid materials. It is thus desired to
explore other platforms, in which anyons can be deterministically created and
controlled.

Recent progress in quantum optics has provided physicists with an
unprecedented opportunity to explore quantum Hall physics using highly
controllable photon-atom interactions Jia  _et al._ (2018a); Ozawa  _et al._
(2019); Clark  _et al._ (2020). Synthetic Landau levels have been realized in
twisted cavities to access quantum Hall physics in both a flat plane or a cone
Lu  _et al._ (2014); Schine  _et al._ (2016); Jia  _et al._ (2018b); Ozawa
_et al._ (2019). In the latter case, only a fraction of the lowest Landau
levels is occupied, for instance, $z_{j}^{3m}$, where $z_{j}=(x_{j}+iy_{j})$
is the complex coordinate of the $j$th particle and $m$ is an integer. Strong
interactions between photons have been introduced by hybridizing photons with
atoms Lukin (2003); Birnbaum  _et al._ (2005); Fushman  _et al._ (2008);
Peyronel  _et al._ (2012); Chang  _et al._ (2014); Firstenberg  _et al._
(2016); Hartmann (2016). Furthermore, two photons uploaded to a twisted cavity
have been prompted to the quantum Hall regime Clark  _et al._ (2020). In
parallel, it has been found that the internal state of a single atom in a
cavity has been entangled with photons, enabling a new means of quantum spin-
valve to control hybridized photon-atom systems Reiserer  _et al._ (2013).
Based on these currently available techniques, here, we point out a scheme to
deterministically deliver desired quantum Hall droplets of light, in which
both the number and locations of anyons are controllable.

We considered a system identical to that in the Chicago experiment, where $N$
atoms are uploaded to a twisted cavity. In this experiment, these photons are
prepared at an initial state $\sim a^{\dagger N}_{m}|0\rangle$, where
$a^{\dagger}_{m}$ is the creation operator of a photon at the lowest Landau
level (LLL) with an angular momentum $m$. Whereas a dissipative dynamics turns
this trivial initial state into a one in the quantum Hall regime, only one
anyon has been created and the position of the anyon is fixed at the origin.
This scheme applies to both a cone and a flat plane as shown in Fig. 1(a-b).
To levitate constraints to the number and location of anyons in this scheme,
we consider an initial state
$\sim(\sum_{m}c_{m}a^{\dagger}_{m})^{N}|0\rangle$, i.e., each photon is
prepared at a superposition of multiple angular momenta. This can be easily
realized using a pumping laser with multiple angular momentum modes
[Forexample:]Fontaine2019. Such a simple generalization leads to conceptual
differences in the resultant quantum Hall droplets. For instance, two anyons
could emerge in the droplet, one placed at the origin of a flat plane [Fig.
1(c-d)] or the apex of a cone [Fig. 1(e-f)] and the other separated by a
certain distance. A unique advantage of this scheme is that external
potentials are not required to localize anyons. In fact, the position of the
second anyon is automatically determined by $\\{c_{m}\\}$, i.e., the initial
condition of the dissipative dynamics. As such, by changing $\\{c_{m}\\}$, the
position of the second anyon is highly tunable.

Figure 1:  (a-b) An initial state of photons occupying a single LLL dissipates
to a quantum Hall droplet with an anyon fixed at the origin. An appropriate
linear combination of multiple LLLs allows the initial state to dissipate to a
quantum Hall droplet with two anyons in a flat plane (c-d) and a cone (e-f).
The red dots mark the locations of anyons. Density distributions of the
droplets are highlighted by colors. The allowed LLLs are constraint between
$m_{min}$ and $m_{max}$.

We define $\ket{vac}$ as the state with all atoms occupying the $s$ orbital
and no photon in the cavity. The time evolution of the density matrix, $\rho$,
of this open system is determined by a master equation Cohen-Tannoudji  _et
al._ (1992); Clark  _et al._ (2019),
$\dot{\rho}=\frac{i}{\hbar}[\rho,H_{0}+H_{pump}+U]+\mathcal{D}[\rho]$, where

$\displaystyle H_{0}$ $\displaystyle=\sum_{m}(gp_{m}^{\dagger}a_{m}+\Omega
r_{m}^{\dagger}p_{m}+h.c.),$ (1) $\displaystyle H_{pump}$
$\displaystyle=\sum_{m}\Omega_{m}^{pump}\left(a^{\dagger}_{m}+a_{m}\right).$
(2) $\displaystyle\mathcal{D}[\rho]$
$\displaystyle=\sum_{m}\gamma\left(p_{m}\rho
p_{m}^{\dagger}-\frac{1}{2}\\{\rho,p_{m}^{\dagger}p_{m}\\}\right)$ (3)

Replacing $m$ by $3m$, the master equation applies to a cone. The first term
in Eq. (1) describes a resonant coupling of the cavity mode with the atoms. To
simplify notations, we have used $p^{\dagger}_{m}$ ($r^{\dagger}_{m}$) and
$p_{m}$ ($r_{m}$) to denote the atomic transitions between the $s$ orbital and
the short-lived $p$ orbital (long-lived Rydberg orbital). We have also
considered a constant coupling, $g$. Our results can be directly generalized
to $m$-dependent couplings. The second term in Eq. (1) is induced by an
external laser that couples the $p$ state to the Rydberg state with a coupling
strength $\Omega$. The subscript of the operators, $m$, denotes collective
excitations of atoms. For instance,
${a}_{m}^{\dagger}=\int_{z}dz\varphi_{m}(z){a}_{z}^{\dagger}$, where
$z=(x+iy)/\ell_{m}$ is the real space coordinate,
$\varphi_{m}(z)=\frac{1}{\sqrt{\pi m!}\ell_{m}}z^{m}e^{-|z|^{2}/2}$ is the
wavefunction with an angular momentum $m$ in the LLL, and ${a}^{\dagger}_{z}$
is the operator for photons at position $z$. The magnetic length $\ell_{m}$
has been set to 1. ${p}_{m}^{\dagger}$ and ${r}_{m}^{\dagger}$ can be
expressed in terms of operators in the real space in a similar manner.

Because of the diluteness of the atoms inside the cavity, the interaction
between two Rydberg atoms is well modeled by a contact interaction $U$ Popp
_et al._ (2004) and the interactions between two atoms at the $s$ and $p$
orbitals are negligible,

$\displaystyle
U=U_{0}\sum_{\\{m_{i}\\}}U_{m_{1},m_{2}}^{m_{3},m_{4}}r_{m_{1}}^{\dagger}r_{m_{2}}^{\dagger}r_{m_{3}}r_{m_{4}},$
(4) $\displaystyle
U_{m_{1},m_{2}}^{m_{3},m_{4}}=\frac{\left(m_{1}+m_{2}\right)!}{2\pi
2^{m_{1}+m_{2}}\sqrt{m_{1}!m_{2}!m_{3}!m_{4}!}}.$ (5)

In Eq. (3), $\gamma$ describes the decay of atoms in the short-lived $p$
orbital, and $\Omega_{pump}$ denotes the strength of an additional field that
continuously pump photons to the cavity. Again, $\gamma$ has been treated as a
constant for simplicity. We first consider a vanishing pump, $\Omega_{m}=0$.
The initial state,
$|\Psi_{\text{I}}\rangle=(\sum_{m}c_{m}a^{\dagger}_{m})^{N}|vac\rangle/\sqrt{N!}$,
can be rewritten in the coordinate space as

$|\Psi_{\text{I}}\rangle=\mathcal{N}_{\text{I}}^{-\frac{1}{2}}\int\prod_{i}dz_{i}\Psi_{\text{I}}(\\{z_{i}\\})a^{\dagger}_{z_{i}}|vac\rangle,$
(6)

where
$\Psi_{\text{I}}(\\{z_{i}\\})=\prod_{i=1}^{N}(\sum_{m}{c}_{m}\varphi_{m}(z_{i}))$,
$\\{z_{i}\\}$ is a short-hand notation for $\\{z_{1},z_{2},...z_{N}\\}$, and
$\mathcal{N}_{\text{I}}$ is the normalization factor. For $N=1$, it is easy to
see that all bright states vanish at long times, and the initial state decays
to a one-body zero-energy dark polariton state,

$\displaystyle\rho_{\text{I}}=|\Psi_{\text{I}}\rangle\langle\Psi_{\text{I}}|\stackrel{{\scriptstyle
t\gg\gamma^{-1}}}{{\xrightarrow{\hskip
17.07182pt}}}\rho^{o}_{\text{F}}=|\Psi_{\text{F}}^{o}\rangle\langle\Psi_{\text{F}}^{o}|,$
(7) $\displaystyle|\Psi_{\text{F}}^{o}\rangle=\lambda\int
dz\Big{(}\sum_{m}c_{m}\varphi_{m}\Big{)}d^{\dagger}_{z}|vac\rangle,$ (8)

where $\lambda=\frac{\Omega}{\sqrt{\Omega^{2}+g^{2}}}$ and
$d^{\dagger}_{z}=\lambda
a_{z}^{\dagger}-\frac{g\lambda}{\Omega}r_{z}^{\dagger}$ is the creation
operator for a dark state at $z$ (Supplemental Materials). Since the $p$
orbital is not involved, $|\Psi_{\text{F}}^{o}\rangle$ as a superposition of
eigenstates of $H_{0}$ survives the dissipative dynamics.

When $N>1$, a $N$-polariton zero-energy dark states, $|D_{l}\rangle$, is
labeled by $l$, the angular momentum that is conserved in the dissipative
dynamics. $|D_{l}\rangle$ satisfies two criteria. First, it must be
constructed by only one-body dark polariton states. Otherwise, the $p$ orbital
will be present and causes decay. Second, its spatial wavefunction must vanish
when two dark polaritons sit on top of each other. Otherwise, the finite
interaction energy leads to an off-resonant condition such that this state
cannot be excited. As such, in the long time limit, the dissipative dynamics
in the cavity amounts to projecting the initial state to a superposition of
$|D_{l}\rangle$ with zero-energies,

$\displaystyle|\Psi_{\text{F}}\rangle=\sum_{l}\tilde{u}_{l}|D_{l}\rangle,$ (9)
$\displaystyle|D_{l}\rangle=\int\prod_{i=1}^{N}dz_{i}\phi_{l}(\\{z_{i}\\})\prod_{i}d_{z_{i}}^{\dagger}|vac\rangle,$
(10)

where $\tilde{u}_{l}=\langle D_{l}|\Psi_{\text{I}}\rangle$ Juliá-Díaz and Graß
(2012), and $\phi_{l}(\\{z_{i}\\})=0$ when $z_{j}=z_{k}$ for any $j\neq k$
(Supplemental Materials). The above discussions show that, if an appropriate
$|\Psi_{\text{F}}\rangle$ is chosen, the corresponding $u_{l}$ could deliver a
target state of interest.

As an example, we show how to obtain a $N$-polariton state in a flat plane
with two anyons, i.e., two quasi-holes,

$\displaystyle|\Phi\rangle=\int\prod_{i}dz_{i}\Phi(\\{z_{i}\\})\prod_{i}d^{\dagger}_{z_{i}}|vac\rangle,$
(11)
$\displaystyle\Phi(\\{z_{i}\\})=\mathcal{N}^{-\frac{1}{2}}\prod_{i=1}^{N}z_{i}(z_{i}-\eta)\prod_{i<j}(z_{i}-z_{j})^{2}\prod_{i}e^{-\frac{|z_{i}|^{2}}{2}},$

where

the first anyon is located at $z=0$ and the other at $z=\eta$. All results can
be immediately generalized to a cone by replacing $z_{i}$ by $z_{i}^{3}$.

Expanding $\Phi(\\{z_{i}\\})$ in terms of $\eta$, we see that it is a
superposition of multiple zero-energy states,

$\Phi(\\{z_{i}\\})=\mathcal{N}^{-\frac{1}{2}}\Big{(}\sum_{l=N^{2}}^{N^{2}+N}(-\eta)^{N^{2}+N-l}\phi_{l}\Big{)},$
(12)

where

$\displaystyle\phi_{N^{2}}(\\{z_{i}\\})=\prod_{i=1}^{N}z_{i}\prod_{i<j}(z_{i}-z_{j})^{2}\prod_{i}e^{-|z_{i}|^{2}/2},$
(13)
$\displaystyle\phi_{l}(\\{z_{i}\\})=\left(\sum_{i_{1},i_{2},...,i_{N-(N^{2}-l)}}\prod_{k=1}^{N-(N^{2}-l)}z_{i_{k}}\right)\phi_{N^{2}}(\\{z_{i}\\}.)$
(14)

Apparently, each $\phi_{l}(\\{z_{i}\\})$ vanishes whenever two particles have
identical coordinates and thus have a zero interaction energy. It carries a
unique total angular momentum $l$, and $\Phi$ is a superposition of multiple
states with different angular momenta, $l=N^{2},N^{2}+1,...,N^{2}+N$, when
$\eta\neq 0$. Interestingly, each $\phi_{l}(\\{z_{i}\\})$ is precisely a Jack
polynomial, a powerful tool that has been extensively studied in quantum Hall
physics (Supplemental Materials).

To access $|\Phi\rangle$ at long times in the dissipative dynamics, we
required that the allowed single-particle states in the LLL are
$m=1,2,\dots,N$. As such, the initial state is written as

$\Psi_{\text{I}}(\\{z_{i}\\})=\mathcal{N}_{\text{I}}^{-\frac{1}{2}}\prod_{i=1}^{N}\left(z_{i}^{(N+1)}+\sum_{m=1}^{N}c_{m}{z_{i}^{m}}\right)\prod_{i}e^{-|z_{i}|^{2}/2},$
(15)

which depends on $N$ parameters. Apparently, $\Psi_{\text{I}}$ is also a
superposition of multiple states with angular momenta $N$, …, $N^{2}+N$,

$\Psi_{\text{I}}=\sum_{l=N}^{N^{2}+N}\psi_{l}(\\{z_{i}\\}).$ (16)

When $l<N^{2}$, there does not exist a state in LLL with vanishing contact
interaction energy as long as single-particle $m=0$ states are excluded. Thus,
any $\psi_{l<N^{2}}(\\{z_{i}\\})$ can only be turned into bright states and
eventually be dissipated away. When $N^{2}\leq l\leq N^{2}+N$,
$\phi_{l}(\\{z_{i}\\})$ is only zero energy eigenstate for each $l$, provided
that the available single-particle states are constrained between
$m_{\text{min}}=1$ and $m_{\text{max}}=N+1$. The cutoffs, $m_{\text{min}}=1$
and $m_{\text{max}}=N+1$, can be provided by either the finite size of the
mirror or purposely limiting available angular momentum channels Clark  _et
al._ (2019), as shown in Fig. 1. Therefore, $\psi_{l}(\\{z_{i}\\})$, which has
a fixed angular momentum $l$, must decay to $\phi_{l}(\\{z_{i}\\})$ in long
times.

We denote the overlap between $\psi_{l}(\\{z_{i}\\})$ and $\phi_{l}$ as
$\mu_{l}$,

$u_{l}=\frac{\int\prod_{i}dz_{i}\phi^{*}_{l}(\\{z_{i}\\})\psi_{l}(\\{z_{i}\\})}{\int\prod_{i}dz_{i}\phi^{*}_{l}(\\{z_{i}\\})\phi_{l}(\\{z_{i}\\})}$
(17)

To ensure that $\Psi_{\text{I}}(\\{z_{i}\\})$ in Eq. (16) eventually
dissipates to $\Phi(\\{z_{i}\\})$ as shown in Eq. (12), it is required that

$\displaystyle u_{l}(c_{1},c_{2},...,c_{N})$ $\displaystyle=-\eta
u_{l+1}(c_{1},c_{2},...,c_{N}),$ (18) $\displaystyle l$
$\displaystyle=N^{2},...,N^{2}+N-1.$ (19)

Since there are in together $N$ such equations, we could then uniquely
determine the $N$ unknown parameters in Eq. (15). For instance, if $N=2$, we
find that $c_{1}=-4\eta^{2}/25$, $c_{2}=-9\eta/10$. A few other examples of
the explicit expressions of $c_{m}$ are given in the supplemental materials.
In fact, $\psi_{l}(\\{z_{i}\\})$ in the initial state can also be written in
terms of Jack polynomials so as to simplify numerics (Supplemental Materials).
When Eq. (19) is satisfied, we conclude that

$\displaystyle|\Psi_{\text{I}}\rangle\stackrel{{\scriptstyle
t\gg\gamma^{-1}}}{{\xrightarrow{\hskip
17.07182pt}}}\lambda^{N}\mathcal{N}_{\text{I}}^{-\frac{1}{2}}\mathcal{N}^{\frac{1}{2}}|\Phi\rangle,$
(20)

i.e., the dissipative dynamics turns initial state into desired quantum Hall
states. Here the $\lambda^{N}$ factor originates from the overlap between the
single-particle photonic state and the dark-polariton state.

The same scheme could be applied to a quasi-hole state of $N$ polaritons on a
cone by tuning $\delta_{m}$, or the creation of a single quasi-hole state at
location $\eta$ (Supplemental Materials).

Figure 2:  Radial density $\rho(r)$ for one-quasihole (ac) state and two-
quasihole (bd) state. Panel (ab) are for the flat space and (cd) are for a
cone with $\delta m=3$. The radial spacing is $2l_{m}$. The density is
normalized to the number of particles $N=20$ ($N=8$) for panel (ab) [(cd)].
Panel (e) [(f)] show the statistical phase factor as a function of $N$ in a
flat space (cone) while fixing $r=4.4$ ($r=6.6$). Insets show the statistical
phase factor as a function of $r$. Black, orange, red, green, blue, brown,
purple, cyan lines are for $N=4,6,8,10,12,14,16,18$. The red dots mark the
locations of anyons. Due to the curvature singularity at the cone tip, the
density at the origin of panel (c) reaches $5.6l_{m}^{-2}/(4\pi)$, which
exceeds the color scale. Thus, this region is excluded from the density plot.

The dissipative dynamics at long times as discussed above can be verified by
numerically solving the master equation, from which the density matrix at any
given time is obtained. Since the numerics is heavy for $N>2$, we focus on
$N=2$ when solving the master equation. We do observe that starting from a
$2$-body state, $(a_{1}^{\dagger})^{2}|vac\rangle$, the final $2$-body state
that survives the dissipative dynamics is indeed the quasihole state at the
origin (Supplemental Materials). Nevertheless, we would like to point out that
the full solution of the master equation includes a mixture of density
matrices with different total particle numbers. In addition to the $N$-body
sector, other sectors with particle numbers ranging from $0$ to $N$ also
exist. The same analysis can be performed for each sector, which hosts certain
quantum Hall states with less than $N$ particles. Alternatively, a simpler
solution is to turn on $\Omega^{pump}$. The steady state is a superposition of
states with the number of photons centered around $N$
[Itisknownthattheinterplaybetweenthepumpingfieldandthedissipationcouldestablishaquasi-
equilibriuminacavitysuchthatthenumberofphotonsorpolaritionsisconcentratedaroundanarrowpeakat$N$withvariance$\sqrt{N}$.]Fox2006.
Choosing the appropriate single momentum cutoffs, the $N+i$ photon sector with
$i>0$ does not have zero energy state, under strong interaction and large
$p$-wave decay, they will decay thus $N$-photon event must come from the
$N$-body sector.

We now turn to the measurement of anyonic statistics. The simplest way is to
measure the density profile of the quasi-hole state. It is known that moving a
single hole around a loop, the many-body wavefunction accumulates a phase
proportional to the total particle number enclosed by the loop with a radius
$R$ Arovas  _et al._ (1984). Adding an extra hole inside this loop, half of
the change of such geometric phase precisely provides us with the statistic
phase of the anyons,

$\displaystyle\phi_{\text{stat}}=\int_{0}^{R}\pi(\rho_{1,\eta}(r)-\rho_{2,\eta}(r))dr.,$
(21)

where $\rho_{1,\eta}$ is the density of the state with one hole located at
$z=\eta$ and $\rho_{2,\eta}$ is that of the state with an additional hole at
the origin 111 The density is defined as
$\rho_{j,\eta}(r)=\int_{0}^{\infty}|\Psi_{j,\eta}|^{2}\sum_{i}\delta(|z_{i}|-r)\prod_{i=1}^{N}dz_{i}$.
The normalization gives $N$, i.e., $\int_{0}^{\infty}\rho_{i,\eta}=N$. The
geometric phase difference is
$\Delta\phi=\int_{0}^{R}2\pi(\rho_{1,\eta}(r)-\rho_{2,\eta}(r))dr$, Circling
one hole around another by a loop is equivalent to exchange twice, i.e.,
$\phi_{\text{stat}}=\Delta\phi/2$. . Thus, comparing the density profiles of
these two quantum Hall states readily allows us to extract the abelian
statistic phase. In practice, however, a key question is the finite size
effect as Eq. (21) is obtained in the thermodynamic limit where the size of
the hole is negligible. As shown in Fig. 2, on both the flat space and the
cone, when $N=20$, the density at the plateau is readily close to $1/2$,
indicating that such particle number is readily a good approximation for the
results in the thermodynamic limit. The inset of Fig. 3(c)

Figure 3: (a) Tilting the single-particle angular momentum levels results in
the rotation of hole (b). (c-d) illustrates the coupling of a quantum spin
valve with a many-body state. (c) While the spin is down, light can enter the
cavity. (d) While the spin is up, light is reflected off due to off-resonance
condition. As the number of particles increases, the statistical phase factor
approaches $0.5\pi$. Inset shows the statistical phase factor as a function of
$R$ for $N=20$.

shows the statistical phase factor over $\pi$ as a function of cutoff radius
$R$. A plateau where anyonic statistics is robust is already clear. A more
rigorous means is to trace the dependence of the height of the plateau as a
function of $N$, from which the result at $N\rightarrow\infty$ can be
extrapolated [Fig. 3(c)]. Considering the cone leads to smaller required $N$
Wu  _et al._ (2017) [Fig 3(d)]. For example, $\phi_{\text{stat}}$ on a cone
shows a plateau at $N=8$ and 10. For $N=4$ and $N=6$, we can already find a
regime of $r$ where $\phi_{\text{stat}}$ is close to 0.5.

A more direct means of measuring the anyonic statistics is to braid two
anyons, i.e., moving one around the other. The anyonic statistics encoded by
the overlap between two many-body wavefunctions, one with anyons braided and
the other without such a braiding, is then transferred to a quantum spin-
valve. Similar ideas of using a single spin to measure quantum coherence in
many-body systems have been studied for various purposes Quan  _et al._
(2006); Hanson  _et al._ (2008); Wei and Liu (2012); Reiserer  _et al._
(2013); Vasilyev  _et al._ (2020). Here, measuring the spin coherence of the
quantum spin-valve, the anyonic statistics can be extracted. Whereas this
scheme is very generic, here, we consider a quantum spin-valve in a cavity to
concretize the discussions Reiserer  _et al._ (2013). To this end, we consider
the quasi-hole state leaked out from the cavity where it is created. Then only
the photonic part of the state is relevant and the spatial part of the
wavefunction remains unchanged. Such quasi-hole state of photons is
transferred to another cavity, which includes a quantum spin-valve. This
quantum spin-valve is made of a single atom, whose hyperfine spin state
controls whether the quantum Hall state of photons could enter the cavity.
When it is spin-up (down), photons can (cannot) enter the cavity, similar to
the experiment in reference Reiserer  _et al._ (2013). Thus, when the quantum
spin-valve is prepared at a superposition of up and down, two copies of the
quasi-hole state of photons is created,
$|\Phi\rangle=\frac{1}{\sqrt{2}}\left(|\uparrow\rangle|\Phi_{\eta}\rangle+\right|\downarrow\rangle|\Phi_{\eta}\rangle)$,
where the copy combined with spin-up(down) is inside (outside) the cavity.

Inside the cavity, we now tilt LLL such that it is no longer flat and the
single particle energy becomes $\sum_{m}mE_{\delta}a^{\dagger}_{m}a_{m}$,
where $E_{\delta}$ is a constant energy. After tilting LLL for a certain time
$t$, each $\phi_{l}$ in Eq. (II) acquires a dynamical phase,
$e^{-iE_{\delta}lt}$. This phase can then be absorbed by $\eta$ such that
$\eta\rightarrow\eta^{\prime}=\eta e^{-i\omega t}$. Thus, as time goes by, one
hole encircle the other with an angular frequency, $\omega=E_{\delta}$,
without requiring an external force to drag it, as shown in Fig. 1(d).
Meanwhile, the copy outside the cavity remains unchanged. After a certain time
$t$, the wavefunction becomes
$|\Phi^{\prime}\rangle=\frac{1}{\sqrt{2}}\left(|\uparrow\rangle|\Phi_{\eta^{\prime}}\rangle+\right|\downarrow\rangle|\Phi_{\eta}\rangle)$
At this time, the coherence of the quantum spin-valve is given by
$\langle\downarrow|\uparrow\rangle=\frac{1}{2}\mathcal{F}\equiv\frac{1}{2}\langle\Phi_{\eta}|\Phi_{\eta^{\prime}}\rangle$.
In other words, the overlap between two quasi-hole states, $\mathcal{F}$, is
encoded in the quantum spin-valve. In particular, when $\eta$ and
$\eta^{\prime}$ are close to each other, i.e., $t$ is small, the Berry
connection can be obtained from $\mathcal{F}$. In this particular braiding
scheme, only the amplitude of $\eta$ changes, and $\vec{A}=A\hat{\theta}$,
where $\theta=\arg\eta$, $\hat{\theta}$ is the unit vector in the angular
direction, and
$A=i\langle\Phi_{\eta}|\nabla_{\theta}|\Phi_{\eta}\rangle/|\eta|$. It is
straightforward to see that
${A}=i(1-\mathcal{F})/(\arg\eta-\arg\eta^{\prime})$. Once $\vec{A}$ is
obtained for each $\theta$, a loop integral of $\vec{A}$ then delivers the
total geometric phase accumulated by braiding the anyons.

To summarize, we propose to use dissipation to transform a trivial product
state with mixed angular momentums to an entangled few-body Laughlin states
with quasihole excitations. Tuning the initial mixing of the angular momentum
states, up to two quasiholes could be generated. The anyonic statics could
then be verified indirectly by measuring the density distribution or directly
by braiding. We hope the few-body version of the non-abelian anyonic state
could be engineered in the future.

###### Acknowledgements.

This work is supported by the Air Force Office of Scientific Research under
award number FA9550-20-1-0221 and a seed grant from PQSEI.

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Supplemental Material of “Manipulating anyons in quantum Hall droplets of
light using dissipations”

In this supplemental material, we present results of the one-body dark state,
Jack polynomials for both one and two quasi-hole states, mapping initial state
to monimial basis, and solutions to the master equation.

##  I The one-body dark state

The spatial wave function in first quantization form reads
$\Psi_{\text{I}}(\\{z_{i}\\})=\prod_{i=1}^{N}(\sum_{m}{c}_{m}\varphi_{m}(z_{i}))$.
Though Eq. (3) from the main text can be solved numerically, it is useful to
analytically study what state survive the dissipative dynamics at long times.
When $N=1$, we see from Eq. (1) from the main text that the eigenstate of
$H_{0}$ can be written as $|D_{m}\rangle=d^{\dagger}_{m}\ket{vac}$,
$|B_{m,\pm}\rangle=b^{\dagger}_{m,\pm}\ket{vac}$, and

$\displaystyle d^{\dagger}_{m}=\lambda
a_{m}^{\dagger}-\frac{g\lambda}{\Omega}r_{m}^{\dagger},$ $\displaystyle
b^{\dagger}_{m,\pm}=\frac{g\lambda}{\sqrt{2}\Omega}a_{m}^{\dagger}+\frac{\lambda}{\sqrt{2}}r_{m}^{\dagger}\pm\frac{1}{\sqrt{2}}p_{m}^{\dagger}.$
(S1)

where $\lambda=\frac{\Omega}{\sqrt{\Omega^{2}+g^{2}}}$. Apparently, the dark
state, $|D_{m}\rangle$, of a polariton is immune to the dissipation as it does
not contain the $p$ orbital. The two bright states of polaritons,
$|B_{m,\pm}\rangle$, eventually decay and will not contribute to the final
density matrix when $t\gg 1/\gamma$. If we use $|\Psi^{1}_{\text{I}}\rangle$
to denote the initial state of a single particle, all bright states vanish at
long times, and

$|\Psi^{o}_{\text{I}}\rangle\stackrel{{\scriptstyle
t\gg\gamma^{-1}}}{{\xrightarrow{\hskip 17.07182pt}}}\lambda\int
dz\Big{(}\sum_{m}c_{m}\varphi_{m}(z)\Big{)}d^{\dagger}_{z}|vac\rangle,$ (S2)

where $d^{\dagger}_{m}=\int dz\varphi_{m}(z)d^{\dagger}_{z}$, and
$d^{\dagger}_{z}=\sum_{m}\varphi_{m}^{*}(z)d^{\dagger}_{m}$.

##  II Jack polynomials and quantum Hall states

The wave function of the quasihole state carrying a single quasihole is
written as

$\displaystyle\Phi_{1,\eta}(\\{z_{i}\\})=\mathcal{N}^{-\frac{1}{2}}_{1}\prod_{i=1}^{N}(z_{i}-\eta)\prod_{i<j}(z_{i}-z_{j})^{2}\prod_{i}e^{-\frac{|z_{i}|^{2}}{2}},$
(S3)

where

$\eta$ denotes the location of quasihole. When $\eta$ is finite, this state is
a superposition of multiple angular momentum states. Expanding Eq. (S3) in
power series of $\eta$, we obtain

$\Phi_{1,\eta}(\\{z_{i}\\})=\mathcal{N}_{1}^{-\frac{1}{2}}\Big{(}\sum_{l=N^{2}-N}^{N^{2}}(-\eta)^{N^{2}-l}\phi_{l}(\\{z_{i}\\})\Big{)}\prod_{i}e^{-|z_{i}|^{2}/2},$
(S4)

where

$\displaystyle\phi_{l}(\\{z_{i}\\})=\left(\sum_{i_{1},i_{2},...,i_{N-(N^{2}-l)}}z_{i_{1}}z_{i_{2}}...z_{i_{N-(N^{2}-l)}}\right)\phi_{N^{2}-N}(\\{z_{i}\\}.$

Each state $\phi_{l}(\\{z_{i}\\})$ has a fixed angular momentum, $l$. For
instance, when $N=3$,
$\Phi_{1,\eta}(z_{1},z_{2},z_{3})=\mathcal{N}_{1}^{-\frac{1}{2}}\Big{(}\sum_{l=6}^{9}(-\eta)^{N^{2}-l}\phi_{l}(z_{1},z_{2},z_{3})\Big{)}e^{-(|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2})/2}$,
and

$\displaystyle\phi_{6}=(z_{1}-z_{2})^{2}(z_{2}-z_{3})^{2}(z_{3}-z_{1})^{2},$
$\displaystyle\phi_{7}=(z_{1}+z_{2}+z_{3})(z_{1}-z_{2})^{2}(z_{2}-z_{3})^{2}(z_{3}-z_{1})^{2},$
$\displaystyle\phi_{8}=(z_{1}z_{2}+z_{2}z_{3}+z_{3}z_{1})(z_{1}-z_{2})^{2}(z_{2}-z_{3})^{2}(z_{3}-z_{1})^{2},$
$\displaystyle\phi_{9}=z_{1}z_{2}z_{3}(z_{1}-z_{2})^{2}(z_{2}-z_{3})^{2}(z_{3}-z_{1})^{2}.$
(S5)

It turns out that $\phi_{l}(\\{z_{i}\\})$ corresponds to a Jack polynomial
$J^{\alpha}_{\mathbf{\lambda}}(\\{z_{i}\\})$ Bernevig and Haldane (2008). A
Jack polynomial Lapointe  _et al._ (2000), which has $N$ arguments,
$\\{z_{i}\\}\equiv\\{z_{1},z_{2},...,z_{N}\\}$, depends on a number of
indices. One of them is denoted by $\alpha$. For relevant states in our case,
$\alpha=-2$. For the Moore-Read and other quantum Hall states, $\alpha$ takes
different values. $\lambda$, a short-hand notation for a set of indices,
$\\{\lambda_{1},\lambda_{2},...,\lambda_{N}\\}$, where
$\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq...\geq\lambda_{N}$, depends on
the angular momentum, $l$, and the total particle number, $N$. If we denote
$l$ as $l=N(N-1)+l_{r}$ with $l_{r}>0$, the explicit expression of
$\lambda_{i}$ can be written as

$\lambda_{i}=\left\\{\begin{array}[]{ll}2N-2i+1,&i<l_{r},\\\ 2N-2i,&i\geq
l_{r}.\\\ \end{array}\right.$

$\lambda=\\{\lambda_{1},\lambda_{2},...,\lambda_{N}\\}$ specifies one of the
monomials in the Jack polynomial,

$m_{\lambda}=\frac{1}{\prod_{k=0}^{\infty}(n_{k}!)}\text{Permanant}(z_{i}^{\lambda_{i}}),$
(S6)

where $n_{k}$ is the number of times for the integer $k$ to appear in
$\\{\lambda_{1},\lambda_{2},...,\lambda_{N}\\}$. For instance,
$m_{420}=(z_{1}^{4}z_{2}^{2}z_{3}^{0}+z_{2}^{4}z_{3}^{2}z_{1}^{0}+z_{3}^{4}z_{1}^{2}z_{2}^{0})$.
Each of these three integers, $4$, $2$, $0$, shows up in $\lambda=\\{4,2,0\\}$
once and any other integer does not exist. As such,
$\prod_{k=0}^{\infty}(n_{k}!)=1$ in this example. All other monomials in a
Jack polynomial can be derived from $m_{\lambda}$ by squeezings.

The explicit expression of a Jack Polynomial can be written in terms of a
determinant,

$J_{\lambda}^{\alpha}=\frac{1}{\mathcal{N}_{J_{\lambda}^{\alpha}}}\left|\begin{array}[]{cccccc}m_{\lambda^{(1)}}&m_{\lambda^{(2)}}&\cdots&\cdots&m_{\lambda^{(n-1)}}&m_{\lambda^{(n)}}\\\
d_{\lambda^{(1)}}-d_{\lambda^{(n)}}&C_{\lambda^{(2)}\lambda^{(1)}}&\cdots&\cdots&C_{\lambda^{(n-1)}\lambda^{(1)}}&C_{\lambda^{(n)}\lambda^{(1)}}\\\
0&d_{\lambda^{(2)}}-d_{\lambda^{(n)}}&&&\cdots&C_{\lambda^{(n)}\lambda^{(2)}}\\\
\vdots&0&\ddots&&&\vdots\\\ \vdots&\vdots&\ddots&\ddots&&\vdots\\\
0&0&\cdots&0&d_{\lambda^{(n-1)}}-d_{\lambda^{(n)}}&C_{\lambda^{(n)}\lambda^{(n-1)}}\end{array}\right|,$
(S7)

where $\lambda^{(n)}\equiv\lambda$. A squeezing operation, which allows two
particles to exchange a certain amount of angular momentum, changes
$m_{\lambda^{(n)}}$ to another monomial, such as $m_{\lambda^{(n-1)}}$. For
instance, $m_{420}$ becomes $m_{411}$ after applying a squeezing operation in
which the angular momentum of one particle decreases from 2 to 1 and
correspondingly the angular momentum of the other one increases from 0 to 1,
i.e., $\lambda^{(n)}=\\{4,2,0\\}$ has become $\lambda^{(n-1)}=\\{4,1,1\\}$.
Repeating the squeezing operations to $\lambda^{(n)}$, or other states
obtained from previous squeezings, all states in the first row of the above
equation are obtained. Fig. S1

Figure S1:  Repeatedly applying the squeeze operator to the Jack polynomial of
$m_{420}$ generates all other polynomials for total angular momentum $l=6$.

shows an example of how to obtain all states from squeezing $m_{420}$. These
$m_{\lambda^{(i)}}$ are ordered based on
$\lambda^{(1)}<\lambda^{(2)}<...<\lambda^{(n)}$. $\lambda^{(j)}<\lambda^{(k)}$
means that there exist a $i^{*}$ such that
$\lambda^{(j)}_{i}\leq\lambda^{(k)}_{i}$ for any $i\leq i^{*}$. For instance,
the states in Fig. S1 are ordered as
$m_{222},m_{321},m_{330},m_{411},m_{420}$.

In Eq.(S7), the upper triangle excluding the first row is defined as

$C_{\lambda^{(j)}\lambda^{(k)}}=\left\\{\begin{array}[]{ll}\left(\lambda^{(j)}_{i_{1}}-\lambda^{(j)}_{i_{2}}\right)\left(\begin{array}[]{c}n(\lambda^{(k)}_{i_{1}})\\\
2\end{array}\right)&\text{ if }\lambda^{(k)}_{i_{1}}=\lambda^{(k)}_{i_{2}}\\\
\left(\lambda^{(j)}_{i_{1}}-\lambda^{(j)}_{i_{2}}\right)n(\lambda^{(k)}_{i_{1}})n(\lambda^{(k)}_{i_{2}})&\text{
if }\lambda^{(k)}_{i_{1}}\neq\lambda^{(k)}_{i_{2}}\end{array}\right.$ (S8)

where
$\lambda^{(k)}_{i_{1}}=\lambda^{(j)}_{i_{1}}-k,\lambda^{(k)}_{i_{2}}=\lambda^{(j)}_{i_{2}}+k$,
$0\leq\lambda^{(j)}_{i_{2}}<\lambda^{(j)}_{i_{1}}$ and
$0<k\leq(\lambda^{(j)}_{i_{1}}-\lambda^{(j)}_{i_{2}})/2$. As shown in Fig. S1,
$\lambda^{(j)}_{i_{1}}$ and $\lambda^{(j)}_{i_{2}}$ are the ones in
$\lambda^{(j)}=\\{\lambda^{(j)}_{1},\lambda^{(j)}_{2},...,\lambda^{(j)}_{N}\\}$
that change to $\lambda^{(k)}_{i_{1}}$ and $\lambda^{(k)}_{i_{2}}$ in a
squeezing from $\lambda^{(j)}$ to $\lambda^{(k)}$. $n(\lambda^{(k)}_{i_{1}})$
is the number of times that $\lambda^{(k)}_{i_{1}}$ shows up in
$\lambda^{(k)}=\\{\lambda^{(k)}_{1},\lambda^{(k)}_{2},...,\lambda^{(k)}_{N}\\}$.

The final next-to-diagonal line is defined as

$d_{\lambda^{(j)}}-d_{\lambda^{(k)}}=\sum_{i=1}^{N}\left(\frac{\alpha}{2}\left((\lambda^{(j)}_{i})^{2}-(\lambda^{(k)}_{i})^{2}\right)-i\left(\lambda^{(j)}_{i}-\lambda^{(k)}_{i}\right)\right).$
(S9)

Finally, the normalization factor, $1/\mathcal{N}_{J^{\alpha}_{\lambda}}$ is
chosen such that
$J^{\alpha}_{\lambda}=m_{\lambda^{(n)}}+\sum_{j=1}^{n-1}c_{j}m_{\lambda^{(j)}}$,
i.e., the coefficient in front of $m_{\lambda^{(n)}}$ is one.

As an example, the explicit expression of $J^{2}_{420}$ is written as,

$J^{2}_{420}=\frac{1}{\mathcal{N}_{J^{2}_{420}}}\left|\begin{array}[]{ccccc}m_{222}&m_{321}&m_{330}&m_{411}&m_{420}\\\
-4&6&0&0&12\\\ 0&-4&3&3&4\\\ 0&0&-1&0&2\\\ 0&0&0&-1&2\\\ \end{array}\right|$
(S10)

Compare the above equation to $\phi_{6}$ that we previously obtained, we find
that $\phi_{6}=J^{2}_{420}$.

Similarly, we could conclude that $\phi_{7}=J_{520}$, $\phi_{8}=J_{530}$,
$\phi_{9}=J_{531}$.

##  III Mapping initial state to monimial basis

we start from an initial state

$\Psi_{\text{I},1}=\mathcal{N}_{\text{I},1}^{-\frac{1}{2}}\prod_{i=1}^{N}\left(\sum_{m=1}^{N}c_{m}{z_{i}^{m}}\right)\prod_{i}e^{-|z_{i}|^{2}/2},c_{m}=1.$
(S11)

and project to different angular momentum channels

$\Psi_{\text{I},1}=\mathcal{N}_{\text{I},1}^{-\frac{1}{2}}(\sum_{l}\psi_{l})\prod_{i}e^{-|z_{i}|^{2}/2}.$
(S12)

We obtain that $\psi_{l}=\sum_{\lambda}\prod_{i}d_{\lambda_{i}}m_{\lambda}$.

For example for $N=3$,

$\displaystyle\psi_{6}$
$\displaystyle=d_{2}^{3}m_{222}+d_{3}d_{2}d_{1}m_{321}+d_{3}^{2}d_{0}m_{330}$
(S13) $\displaystyle\psi_{7}$
$\displaystyle=d_{3}^{2}d_{1}m_{331}+d_{3}d_{2}^{2}m_{322}$ (S14)
$\displaystyle\psi_{8}$ $\displaystyle=d_{3}^{2}d_{2}m_{332}$ (S15)
$\displaystyle\psi_{9}$ $\displaystyle=d_{3}^{3}m_{333}.$ (S16)

Though the initial state could couple to angular momentum less than 6, we can
safely ignore them since they have zero overlap with the final state. Finally,
combining the equations

$\displaystyle\braket{\psi_{6}}{J_{\lambda(6)}}$
$\displaystyle=(-\eta)^{3}O_{c}\braket{J_{\lambda(6)}}{J_{\lambda(6)}}$ (S17)
$\displaystyle\braket{\psi_{7}}{J_{\lambda(7)}}$
$\displaystyle=(-\eta)^{2}O_{c}\braket{J_{\lambda(7)}}{J_{\lambda(7)}}$ (S18)
$\displaystyle\braket{\psi_{8}}{J_{\lambda(8)}}$
$\displaystyle=(-\eta)O_{c}\braket{J_{\lambda(8)}}{J_{\lambda(8)}}$ (S19)
$\displaystyle\braket{\psi_{9}}{J_{\lambda(9)}}$
$\displaystyle=O_{c}\braket{J_{\lambda(9)}}{J_{\lambda(9)}}$ (S20)

and with the condition $d_{3}=1$, we can solve for $O_{d}$, and $d_{i}$. We
solve for $O_{d}$ first from Eq. (S20), then solve for $d_{2}$ from Eq. (S19),
etc. The solution is guaranteed because they are linear in each step. We
obtain
$O_{d}=-(9/220),d_{0}=-((41991\eta^{3})/332750),d_{1}=((3303\eta^{2})/6050),d_{2}=-((81\eta)/55)$.
Note, $O_{d}$ differ the overlap $\braket{\Psi_{I,1}}{\Phi_{1,\eta}}$ by a
normalization factor $(\mathcal{N}_{1}\mathcal{N}_{1,\eta})^{1/2}$.

##  IV Generalization to two quasi-hole state in the main text

The same analysis could be performed for the two quasihole state. We need to
replace $\lambda_{i}$ by $\lambda_{i}+1$. Once we express
$\Phi_{1,\eta}(\\{z_{i}\\})$, the state with one quasi-hole, in terms of Jack
Polynomials, we could immediately rewrite the state with two quasi-holes,
$\Phi_{2,\eta}(\\{z_{i}\\})$ also in terms of Jack Polynomials, since
$\Phi_{2,\eta}(\\{z_{i}\\})\sim(\prod_{i}z_{i})\Phi_{2,\eta}(\\{z_{i}\\})$.

The normalization constants $N(\lambda)$ changes but everything else,
including the total number of basis and the coefficients in front of the basis
remains the same. Likewise, generalization to cones, e.g., replace
$\lambda_{i}$ by $3\lambda_{i}$ also only changes the normalization factors.
Using Jack polynomial, we are able to calculate the coefficients for up to
$N=8$. The coefficients required to generate the one-quasi-hole and two-quasi-
hole state for $N$ up to 4 is summarized in table S1.

Table S1:  The coefficients $d_{i}$ ($c_{i}$) are for the wave functions with one quasi hole exicitation at $\eta$ (one quasi hole excitation at origin and another at $\eta$). $O_{d}$ ($O_{c}$) is the overlap of the initial wave function and the final state.  $N$ | $O_{d}$ | $d_{0}$ | $d_{1}$ | $d_{2}$ | $d_{3}$ | $O_{c}$ | $c_{1}$ | $c_{2}$ | $c_{3}$ | $c_{4}$
---|---|---|---|---|---|---|---|---|---|---
2 | $-\frac{2}{7}$ | $\frac{4\eta^{2}}{49}$ | $-\frac{8\eta}{7}$ |  |  | $3/10$ | $-4\eta^{2}/25$ | $-9\eta/10$ |  |
3 | $-\frac{9}{220}$ | $-\frac{41991\eta^{3}}{332750}$ | $\frac{3303\eta^{2}}{6050}$ | $-\frac{81\eta}{55}$ |  | $12/257$ | $-138478\eta^{3}/16974593$ | $2928\eta^{2}/66049$ | $-264\eta/257$ |
4 | $\frac{8}{2377}$ | $\frac{3571758875792\eta^{4}}{861946858349307}$ | $-\frac{41724784088\eta^{3}}{120873209697}$ | $\frac{20331502\eta^{2}}{16950387}$ | $-\frac{13504\eta}{7131}$ | $\frac{125}{28931}$ | $\frac{-88255919558988767\eta^{4}}{1401147243843246242}$ | $\frac{1768215130300\eta^{3}}{24215326878491}$ | $\frac{132909825\eta^{2}}{837002761}$ | $\frac{-35075\eta}{28931}$

##  V solutions to the master equation

Here, we use dissipative dynamics on a cone as an example. Discussions can be
straightforwardly generalized to a flat plane. We consider a cavity supporting
states with angular momenta $3,6,$ and $9$. To simplify notations, the decay
rate of the $p$-orbital has been taken as a constant in these angular momentum
channels. As an example, we set $g/\Omega=4.3/1.5$. We perform a time-
dependent calculation using the master equation and find that an initial pure
state of 2 photons decays into a density matrix that is a mixture of the
vacuum, the one-body dark state $d_{l}^{\dagger}\ket{vac}$ with angular
momentum $l=3,6,$ and $9$, and a two-body quasihole state
$D^{\dagger}\ket{vac}=\frac{1}{\sqrt{6.2}}d_{6}^{\dagger}d_{6}^{\dagger}\ket{vac}-\sqrt{\frac{2.1}{3.1}}d_{3}^{\dagger}d_{9}^{\dagger}\ket{vac}$.
Here $\lambda=\Omega/\sqrt{g^{2}+\Omega^{2}}=0.329$, and
$d_{l}^{\dagger}=\lambda
a_{l}^{\dagger}-\frac{g\lambda}{\Omega}r_{l}^{\dagger}$. To be more explicit,
the density matrix of the final steady state reads
$c_{vac}\ket{vac}\bra{vac}+c_{3}d_{3}^{\dagger}\ket{vac}\bra{vac}d_{3}+c_{9}d_{9}^{\dagger}\ket{vac}\bra{vac}d_{9}+c_{6}d_{6}^{\dagger}\ket{vac}\bra{vac}d_{6}+c_{tb}D^{\dagger}\ket{vac}\bra{vac}D$.
Because the initial two photons have angular momentum 6, $c_{3}=c_{9}$. There
are no off diagonal terms in the final density matrix.

For convenience, we define anti-dark state
$\tilde{d}_{l}^{\dagger}=\frac{g\lambda}{\Omega}a_{l}^{\dagger}+\lambda
r_{l}^{\dagger}$. We rewrite the initial photonic state
$(a_{6}^{\dagger})^{2}\ket{vac}$ as a superposition of states that are
proportional to
$(\tilde{d}_{6}^{\dagger})^{2},\tilde{d}_{6}^{\dagger}d_{6}^{\dagger},(d_{6}^{\dagger})^{2}$.
The first part decays completely. The second part decays to a one-body dark
state of angular momentum 6 with probability of
$2\lambda^{2}(1-\lambda^{2})=0.19$, which is the majority of $c_{6}$. The last
part decays to the two-body quasihole state with the probability of
$c_{tb}=\lambda^{4}/3.1=0.38\%$ and an anti quasihole state. The anti
quaishole states decays into the one-body dark states with $l=3,6,9$. The
coefficients $c_{3},c_{9},c_{6}$ depend on the interaction strength and the
decay rate. For example, for $\gamma=2.3\Omega$ and $U_{0}=74\Omega$ [Fig.
S2(a)], $c_{3}=0.0008$ and $c_{6}=0.2001$; for $\gamma=2.3\Omega$ and
$U_{0}=111\Omega$ [Fig. S2(b)], $c_{3}=0.0004$, and $c_{6}=0.201$; for
$\gamma=1.15\Omega$ and $U_{0}=111\Omega$ [Fig. S2(c)], $c_{3}=0.007$, and
$c_{6}=0.190$.

The lines in Fig. S2 show the time-dependent coefficient, $c(t)$, of the
density matrix. The blue lines denote $c_{tb}(t)$. Since it is the coefficient
for the dark state, $c_{tb}(t)$ stays as a constant regardless of the
interaction strength and the decay rate. The red, green, black lines denote
$c_{3}$, $c_{6}$, and $c_{vac}$, respectively. Changing the interaction
strength [Fig. S2(a)] or the decay rate [Fig. S2(c)] leads to changes in
$c_{3}$, $c_{6}$, and $c_{vac}$.

Figure S2:  The black, green, red, and blue lines shows the probability $c(t)$
for the diagonal density matrix elements $\ket{vac}\bra{vac}$,
$d_{6}^{\dagger}\ket{vac}\bra{vac}d_{6}$,
$d_{3}^{\dagger}\ket{vac}\bra{vac}d_{3}$, $D^{\dagger}\ket{vac}\bra{vac}D$ as
a function of time. $d_{9}^{\dagger}\ket{vac}\bra{vac}d_{9}$ is the same as
$d_{3}^{\dagger}\ket{vac}\bra{vac}d_{3}$. Panel (a) is for $\gamma=2.3\Omega$
and $U_{0}=74\Omega$. Panel (b) is for $\gamma=2.3\Omega$ and
$U_{0}=111\Omega$. Panel (c) is for $\gamma=1.15\Omega$ and $U_{0}=111\Omega$.
Changing the interaction energy or decay rate do not change the two-body dark
state at all but they affects the one-body states and vacuum state. From (a)
to (b), changing $U$ does not affect the decay to the one-body sector much,
which is mainly governed by the decay rate in the $p$-orbitals.