Induced Self-Stabilization in Fractional Quantum Hall States of Light

Recent progress in nanoscale quantum optics and superconducting qubits has made the creation of strongly correlated, and even topologically ordered, states of photons a real possibility. Many of these states are gapped and exhibit anyon excitations, which could be used for a robust form of quantum information processing. However, while numerous qubit array proposals exist to engineer the Hamiltonian for these systems, the question of how to stabilize the many-body ground state of these photonic quantum simulators against photon losses remains largely unanswered. We here propose a simple mechanism that achieves this goal for Abelian and non-Abelian fractional quantum Hall states of light. Our construction uses a uniform two-photon drive field to couple the qubits of the primary lattice with an auxiliary “shadow” lattice, composed of qubits with a much faster loss rate than the qubits of the primary quantum simulator itself. This coupling causes hole states created by photon losses to be rapidly refilled, and the system’s many-body gap prevents further photons from being added once the strongly correlated ground state is reached. The fractional quantum Hall state (with a small, transient population of quasihole excitations) is thus the most stable state of the system, and all other configurations will relax toward it over time. The physics described here could be implemented in a circuit QED architecture, and the device parameters needed for our scheme to succeed are in reach of current technology. We also propose a simple six-qubit device, which could easily be built in the near future, that can act as a proof of principle for our scheme.

Popular Summary

The emerging field of quantum simulation involves engineering collections of simple quantum systems to produce exotic and complex states. Microwave photons trapped in a superconducting circuit are a particularly promising platform for quantum simulation experiments of this type, but the fact that photons can escape from the circuit and become lost to the environment poses a serious fundamental challenge. We propose a method to guard against the loss of photons in superconducting circuits, which overcomes the last remaining theoretical obstacle to engineering fractional quantum Hall states of light.

We consider a quantum simulator in which photons in a two-dimensional quantum device array are tuned to act as if they were charged particles moving in a magnetic field. The many-photon ground states of such a system are fractional quantum Hall states, which are highly exotic states that contain fractional charges—emergent particles that behave as if they were half of a photon. By coupling an auxiliary “shadow” lattice of intentionally bad quantum bits—qubits with very fast loss rates—to the primary circuit, we demonstrate analytically and numerically that photons lost in the primary lattice can be passively refilled. We assume that photons are lost at a uniform rate from each site in the primary lattice. The shadow lattice acts as a local energy pump, with the requirement that the energies of the two lattices match. This interplay between the two lattices stabilizes the many-photon ground state indefinitely without any active intervention from an external observer; the refilling stops when a fractional quantum Hall state is achieved. We also propose a simple circuit of only six quantum bits that could use this principle as a form of passive quantum error correction.

Since the device parameters needed for our scheme to succeed have already been realized with current technology, we anticipate that our findings will be verified experimentally.

Figure 1

(a) Simple example of dissipative protection of a quantum state. We consider a nonlinear resonator $P$, coupled to an auxilliary “shadow” spin $S$, and we wish to hold the resonator in the one-photon state $|1_P⟩$ against a loss rate ${\mathrm{\Gamma }}_P$. We do so by coupling the resonator and shadow spin with a two-photon drive $\mathrm{\Omega }$, which resonantly adds a photon to the resonator and excites the spin or removes a photon from the resonator and returns the spin to its ground state. The nonlinearity $\mathrm{\Delta }\gg \mathrm{\Omega }$ suppresses the $|1⟩\to |2⟩$ transition, and if the shadow spin has a fast relaxation rate ${\mathrm{\Gamma }}_S\gg {\mathrm{\Gamma }}_P$, then $|1,\downarrow ⟩$ is the most stable state of the system and all other configurations will rapidly relax toward it, as described in the Introduction. (b) Many-body generalization of this mechanism to protect a lattice fractional quantum Hall system (blue) from losses. The primary quantum Hall lattice is coupled to a shadow lattice (red) through a two-photon drive field, so that when a many-body hole excitation is created by a photon loss on the primary lattice (rate ${\mathrm{\Gamma }}_P$), it is resonantly mixed with a particle excitation on the shadow lattice (rate $\mathrm{\Omega }$), which rapidly decays (rate ${\mathrm{\Gamma }}_S$), returning the primary lattice to its many-body ground state. If ${\mathrm{\Gamma }}_S\gg {\mathrm{\Gamma }}_P$ and $\mathrm{\Omega }\gg {\mathrm{\Gamma }}_P$, holes will be refilled much more quickly than they are created, and the fractional quantum Hall ground state becomes the most stable state of the system. The many-body gap of the topological ground state suppresses further particle addition once the fractional quantum Hall state is reached.

Figure 2

Generic level diagram of lowest Landau level bosons interacting through a repulsive contact interaction, governed by the Hamiltonian $H_P$ [Eq. (1)]. If the contact interaction is a ($k+1$)-body term, the system is gapless and degenerate until $N=N^{*}\equiv k{N}_{\varphi }/2$, at which point the exact ground state is a Read-Rezayi state of level $k$ [41]. For all $N<N^{*}$, particles can always be added at a lowest Landau level energy of exactly $-\tilde{\mu }$, but for $N>N^{*}$, the lowest energy state is at least the gap energy $\mathrm{\Delta }$ (which scales as the smaller of the hopping $J$ and the interaction energy $U_{k+1}$) above $-\tilde{\mu }N$. This is simply the statement that the fractional quantum Hall state accommodates exactly $N^{*}$ particles in its ground state and additional particles added then must go above the many-body gap. That hole states are massively degenerate and isoenergetic reflects the fact that the anyonic quasiholes are dispersionless and noninteracting, and thus do not fractionalize on their own after being created locally through a single-photon loss. Noninteracting topological defects are not a generic feature of most anyon systems, and the lack of interactions allows us to induce self-stabilization of the correlated states through the shadow lattice construction.

Figure 3

Commensurability and incompressibility of the stationary state, computed in the refilling rate approximation for a $6\times 4$ lattice with periodic boundary conditions, $\varphi =1/3$, ${\mathrm{\Gamma }}_R/{\mathrm{\Gamma }}_P\simeq 50$, and ${\mathrm{\Gamma }}_P=0.00067J$. (a) The three curves initialize the system’s density matrix in the empty state $N=0$ (blue, bottom curve), the Laughlin state (purple, center curve), and the Laughlin state with a quasiparticle excitation (gold, top curve), and time is evolved by numerically integrating the Lindblad equation (8). In all cases, the system relaxes to a Laughlin state with a negligible density of excitations, as discussed in the text. Identical behavior is observed in all cases studied. (b) Signatures of incompressibility in this system. We plot the average number fluctuations $(⟨N^2⟩-{⟨N⟩}^2)/N^{*}$ for the same parameters and initial conditions, demonstrating that number fluctuations are highly suppressed by the combination of the energy gap $\mathrm{\Delta }$ and the rapid refilling rate ${\mathrm{\Gamma }}_R$. The small asymptotic value of $(⟨N^2⟩-{⟨N⟩}^2)/N^{*}\sim 0.05$ stands in contrast to the case of a gapless system subject to a coherent drive field, where $(⟨N^2⟩-{⟨N⟩}^2)/⟨N⟩$ would be $O(1)$.

Figure 4

Scaling of the equilibrium density versus the ratio of the refilling rate ${\mathrm{\Gamma }}_R$ to the primary loss rate ${\mathrm{\Gamma }}_P$. The data (blue dots) are taken from simulations of $4\times 4$, $5\times 5$, and $6\times 4$ lattices with $N_{\varphi }=6$, 8, or 10 and hard-core two-body interactions, for various choices of $\mathrm{\Omega }\sim {\mathrm{\Gamma }}_S$ [with the refilling rate ${\mathrm{\Gamma }}_R$ estimated from Eq. (4) with $\tilde{\mu }={\epsilon }_S$], and ${\mathrm{\Gamma }}_P$. Values of $N/N^{*}>1$ are due to incoherent quasiparticle addition above the Laughlin state, which occurs at a rate ${\mathrm{\Gamma }}_E\ll {\mathrm{\Gamma }}_P$, as described in the text. The blue curve $N/N^{*}=1.008-0.66{\mathrm{\Gamma }}_P/{\mathrm{\Gamma }}_R$ is a numerical fit to the data, and the coefficient of 0.66, rather than a number close to unity, stems from geometric considerations (the finite size of quasiholes means that multiple sites contribute to their refilling, while a single site contributes to their creation for a given loss event) and the contribution from incoherent quasiparticle addition rates. The incoherent addition rate increases with increasing $\mathrm{\Omega }$ and ${\mathrm{\Gamma }}_S$, and thus generically grows with increasing ${\mathrm{\Gamma }}_R$ for a given ${\mathrm{\Gamma }}_P$.

Figure 5

Long-time stabilization of (a) Moore-Read and (b) Laughlin states. (a) Using the refilling rate approximation [Eqs. (7) and (8)], we study the dynamics of a $6\times 4$ lattice with periodic boundary conditions, $\varphi =1/4$, and a hard-core three-body interaction. In this limit, the exact ground state at $N=N^{*}=6$ is a Moore-Read state [49], which is threefold degenerate on the torus. In the figure, we plot the logarithm of the ordered eigenvalues ${\rho }_{\alpha }$ describing the equilibrium occupation of the lowest Landau level states (other states not shown) for ${\mathrm{\Gamma }}_R=200{\mathrm{\Gamma }}_P$ and ${\mathrm{\Gamma }}_R\ll \mathrm{\Delta }$. Each level corresponds to a fixed particle number $N$, with $N=6$ being the Moore-Read state. The equilibrium occupations (probability of finding the system in that state when a measurement is made) of the LLL states can be modeled by an induced chemical potential and temperature ${\mu }_{\text{ind}}/T_{\text{ind}}\simeq \log ({\mathrm{\Gamma }}_R/{\mathrm{\Gamma }}_P)$. The final occupation of the three Moore-Read states is 99% in this case. In such a small system, equilibration was rapid, but as remarked in the text, the exponential suppression of fractionalization in large systems leaves the equilibration rate an open question. (b) Long-time configuration of a $5\times 5$ lattice with $N_{\varphi }=10$ and hard-core two-body interactions, with ${\mathrm{\Gamma }}_R=100{\mathrm{\Gamma }}_P$. The occupation of the two Laughlin states at $N=N^{*}=5$ is 95%.

Figure 6

A simple circuit to demonstrate passive error correction. (a) Ring of three primary superconducting transmon qubits ($P_1,P_2,P_3$, outlined in blue), coupled capacitively to three shadow qubits ($S_1,S_2,S_3$, outlined in red). The small Josephson couplings that link the primary qubits are biased by a time-dependent flux $\mathrm{\Phi }(t)$, which implements the two-photon drive field. Combined with the internal capacitances of the junctions, the resulting interaction is $-J{\sigma }_{iP}^x{\sigma }_{jP}^x$ in the rotating frame. As described in the text, the shadow lattice qubits rapidly correct single bit-flip errors, protecting both of the degenerate ground states of the primary lattice Hamiltonian. (b) Demonstration of state protection for this circuit, with $g=0.05J$ and ${\mathrm{\Gamma }}_S=0.1J$ (${\mathrm{\Gamma }}_R=0.05J$). We initialize the system in the ground state $|111⟩$ and numerically integrate the Lindblad equations for the full system’s evolution. For ${\mathrm{\Gamma }}_P=\{10^{-3},5\times 10^{-4},2.5\times 10^{-4}\}J$, the observed decay rates for the ground states $|000⟩$ and $|111⟩$ are $\{4.6\times 10^{-5},1.3\times 10^{-5},3.8\times 10^{-6}\}$, demonstrating the effectiveness of the error correction.