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Manipulating anyons in quantum Hall droplets of light using dissipations

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 π [1][2][3][4]. 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 [5,6]. 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 [7][8][9]. Synthetic Landau levels have been realized in twisted cavities to access quantum Hall physics in both a flat plane or a cone [8,[10][11][12]. In the latter case, only a fraction of the lowest Landau levels is occupied, for instance, z 3m j , where z j = (x j + iy j ) is the complex coordinate of the jth particle and m is an integer. Strong interactions between photons have been introduced by hybridizing photons with atoms [13][14][15][16][17][18][19]. Furthermore, two photons uploaded to a twisted cavity have been prompted to the quantum Hall regime [9]. 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 [20]. 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 ∼ a †N m |0 , where a † 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 ∼ ( m c m a † m ) N |0 , 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 [21]. 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.
We define |vac as the state with all atoms occupying the s orbital and no photon in the cavity. The time evolution of the density matrix, ρ, of this open system is determined by a master equation [22,23] 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 † m (r † 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).
z m e −|z| 2 /2 is the wavefunction with an angular momentum m in the LLL, and a † z is the operator for photons at position z. The magnetic length m has been set to 1. p † m and r † m 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 [24] and the interactions between two atoms at the s and p orbitals are negligible, In Eq. (3), γ describes the decay of atoms in the shortlived p orbital, and Ω pump denotes the strength of an additional field that continuously pump photons to the cavity. Again, γ has been treated as a constant for simplicity. We first consider a vanishing pump, Ω m = 0. The initial state, |Ψ I = ( m c m a † m ) N |vac / √ N !, can be rewritten in the coordinate space as , {z i } is a shorthand notation for {z 1 , z 2 , ...z N }, and N 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, where λ = Ω √ Ω 2 +g 2 and d † z = λa † z − gλ Ω r † z is the creation operator for a dark state at z (Supplemental Materials). Since the p orbital is not involved, |Ψ o F as a superposition of eigenstates of H 0 survives the dissipative dynamics.
When N > 1, a N -polariton zero-energy dark states, |D l , is labeled by l, the angular momentum that is conserved in the dissipative dynamics. |D l 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 with zero-energies, whereũ l = D l |Ψ I [25], and φ l ({z i }) = 0 when z j = z k for any j = k (Supplemental Materials). The above discussions show that, if an appropriate |Ψ F 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, where the first anyon is located at z = 0 and the other at z = η. All results can be immediately generalized to a cone by replacing z i by z 3 i . Expanding Φ({z i }) in terms of η, we see that it is a superposition of multiple zero-energy states, where Apparently, each φ 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 Φ is a superposition of multiple states with different angular momenta, l = N 2 , N 2 + 1, ..., N 2 + N , when η = 0. Interestingly, each φ l ({z i }) is precisely a Jack polynomial, a powerful tool that has been extensively studied in quantum Hall physics (Supplemental Materials).
To access |Φ at long times in the dissipative dynamics, we required that the allowed single-particle states in the LLL are m = 1, 2, . . . , N . As such, the initial state is written as which depends on N parameters. Apparently, Ψ I is also a superposition of multiple states with angular momenta N , ..., N 2 + N , 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 ψ l<N 2 ({z i }) can only be turned into bright states and eventually be dissipated away. When N 2 ≤ l ≤ N 2 + N , φ l ({z i }) is only zero energy eigenstate for each l, provided that the available single-particle states are constrained between m min = 1 and m max = N + 1. The cutoffs, m min = 1 and m max = N + 1, can be provided by either the finite size of the mirror or purposely limiting available angular momentum channels [23], as shown in Fig. 1. Therefore, ψ l ({z i }), which has a fixed angular momentum l, must decay to φ l ({z i }) in long times.
We denote the overlap between ψ l ({z i }) and φ l as µ l , To ensure that Ψ I ({z i }) in Eq. (16) eventually dissipates to Φ({z i }) as shown in Eq. (12), it is required that 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 = −4η 2 /25, c 2 = −9η/10. A few other examples of the explicit expressions of c m are given in the supplemental materials. In fact, ψ 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 i.e., the dissipative dynamics turns initial state into desired quantum Hall states. Here the λ 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 δ m , or the creation of a single quasi-hole state at location η (Supplemental Materials).
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 ) 2 |vac , 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 Ω pump . The steady state is a superposition of states with the number of photons centered around N [26]. 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 [27]. 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, where ρ 1,η is the density of the state with one hole located at z = η and ρ 2,η is that of the state with an additional hole at the origin [28]. 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) shows the statistical phase factor over π 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 → ∞ can be extrapolated [ Fig. 3(c)]. Considering the cone leads to smaller required N [29] [Fig 3(d)]. For example, φ 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 φ 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 [20,[30][31][32][33]. 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 [20]. To this end, we consider the quasihole 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 [20]. 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, |Φ = 1 √ 2 (| ↑ |Φ η +| ↓ |Φ η ), 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 m mE δ a † m a m , where E δ is a constant energy. After tilting LLL for a certain time t, each φ l in Eq. (S5) acquires a dynamical phase, e −iE δ lt . This phase can then be absorbed by η such that η → η = ηe −iωt . Thus, as time goes by, one hole encircle the other with an angular frequency, ω = E δ , 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 |Φ = 1 √ 2 (| ↑ |Φ η +| ↓ |Φ η ) At this time, the coherence of the quantum spin-valve is given by ↓ | ↑ = 1 2 F ≡ 1 2 Φ η |Φ η . In other words, the overlap between two quasi-hole states, F, is encoded in the quantum spin-valve. In particular, when η and η are close to each other, i.e., t is small, the Berry connection can be obtained from F. In this particular braiding scheme, only the amplitude of η changes, and A = Aθ, where θ = arg η,θ is the unit vector in the angular direction, and A = i Φ η |∇ θ |Φ η /|η|. It is straightforward to see that A = i(1 − F)/(arg η − arg η ). Once A is obtained for each θ, a loop integral of 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.
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. 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.

THE ONE-BODY DARK STATE
The spatial wave function in first quantization form reads Ψ I ({z i }) = N i=1 ( m c m ϕ 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 where λ = Ω √ Ω 2 +g 2 . Apparently, the dark state, |D m , of a polariton is immune to the dissipation as it does not contain the p orbital. The two bright states of polaritons, |B m,± , eventually decay and will not contribute to the final density matrix when t 1/γ. If we use |Ψ 1 I to denote the initial state of a single particle, all bright states vanish at long times, and

JACK POLYNOMIALS AND QUANTUM HALL STATES
The wave function of the quasihole state carrying a single quasihole is written as where η denotes the location of quasihole. When η is finite, this state is a superposition of multiple angular momentum states. Expanding Eq. (S3) in power series of η, we obtain where l=6 (−η) N 2 −l φ l (z 1 , z 2 , z 3 ) e −(|z1| 2 +|z2| 2 +|z3| 2 )/2 , and It turns out that φ l ({z i }) corresponds to a Jack polynomial J α λ ({z i }) [34]. A Jack polynomial [35], which has N arguments, {z i } ≡ {z 1 , z 2 , ..., z N }, depends on a number of indices. One of them is denoted by α. For relevant states in our case, α = −2. For the Moore-Read and other quantum Hall states, α takes different values. λ, a short-hand notation for a set of indices, {λ 1 , λ 2 , ..., λ N }, where λ 1 ≥ λ 2 ≥ λ 3 ≥ ... ≥ λ 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 λ i can be written as .., λ N } specifies one of the monomials in the Jack polynomial, where n k is the number of times for the integer k to appear in {λ 1 , λ 2 , ..., λ N }. For instance, m 420 = (z 4 1 z 2 2 z 0 3 + z 4 2 z 2 3 z 0 1 + z 4 3 z 2 1 z 0 2 ). Each of these three integers, 4, 2, 0, shows up in λ = {4, 2, 0} once and any other integer does not exist. As such, ∞ k=0 (n k !) = 1 in this example. All other monomials in a Jack polynomial can be derived from m λ by squeezings.
The explicit expression of a Jack Polynomial can be written in terms of a determinant, where λ (n) ≡ λ. A squeezing operation, which allows two particles to exchange a certain amount of angular momentum, changes m λ (n) to another monomial, such as m λ (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., λ (n) = {4, 2, 0} has become λ (n−1) = {4, 1, 1}. Repeating the squeezing operations to λ (n) , or other states obtained from previous squeezings, all states in the first row of the above equation are obtained. Fig. S1 shows an example of how to obtain all states from squeezing m 420 . These m λ (i) are ordered based on λ (1) < λ (2) < ... < λ (n) . λ (j) < λ (k) means that there exist a i * such that λ for any i ≤ 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 i2 )/2. As shown in Fig. S1, λ The final next-to-diagonal line is defined as Finally, the normalization factor, 1/N J α λ is chosen such that J α λ = m λ (n) + n−1 j=1 c j m λ (j) , i.e., the coefficient in front of m λ (n) is one.
As an example, the explicit expression of J 2 420 is written as, Compare the above equation to φ 6 that we previously obtained, we find that φ 6 = J 2 420 . Similarly, we could conclude that φ 7 = J 520 , φ 8 = J 530 , φ 9 = J 531 .
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 ψ 6 |J λ(6) = (−η) 3 O c J λ(6) |J λ(6) (S17) ψ 7 |J λ(7) = (−η) 2 O c J λ(7) |J λ(7) (S18)   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.