Engineering three-body interaction and Pfaffian states in circuit QED systems

We demonstrate a scheme to engineer the three-body interaction in circuit-QED systems by tuning a fluxonium qubit. Connecting such qubits in a square lattice and controlling the tunneling dynamics in the form of a synthesized magnetic field for the photon-like excitations of the system allows the implementation of a parent Hamiltonian whose ground state is the Pfaffian wave function. Furthermore, we show that the addition of the next-nearest-neighbor tunneling stabilizes the ground state, recovering the expected topological degeneracy even for small lattices. Finally, we discuss the implementation of these ideas with the current technology.

Figure 1

(a) Two-body and (b) three-body interactions emerge as two- and three-excitation nonlinearity in an anharmonic oscillator. (c) Circuit model to implement Eq. (1). Each site is composed of an anharmonic oscillator with the three-body interaction. The qubits are detuned from each other according to a staggered pattern (blue/red/green/black) and are coupled by externally modulated SQUIDs.

Figure 2

(a) Three-body $\left(U_3\right)$ and (b) two-body $\left(U_2\right)$ interaction strength as function of the external bias flux and the shunted inductance. The dashed line on (b) shows where the two-body interaction vanishes. (c) Shows the three-body interaction strength when $U_2$ is optimized to be less than $0.0005E_J$. All the plots are for $E_c/E_J=0.05.$

Figure 3

Dynamics of different excitation numbers when two qubits are inductively coupled to each other, as shown in (a): (b) one excitation, (c) two excitations, (d) three excitations. The black line shows the total population in (b) and (c) and $P_{21}+P_{12}$ in (d).

Figure 4

(a) First thirteen eigenvalues for 4 particles on a 4× 4 lattice with torus boundary conditions, i.e., a total of 16 plaquettes, $\alpha =0.25$, and $U_3=\infty$. The index $n$ labels eigenvalues. Different shapes show different tunneling schemes: circles for only nearest-neighbor tunneling (gap/$J\simeq 0.04)$, triangles for the nearest- and the next-nearest-neighbor tunneling (gap/$J\simeq 0.12)$, and squares for the long-range tunneling (gap/$J\simeq 0.12)$. The gaps are denoted by arrows. (b) The order parameter as a function of $U_3$ and $U_2$, characterizing the relative size of the gap.