Spin-1 models in the ultrastrong-coupling regime of circuit QED

We propose a superconducting circuit platform for simulating spin-1 models. To this purpose we consider a chain of $N$ ultrastrongly coupled qubit-resonator systems interacting through a grounded superconducting quantum interference device (SQUID). The anharmonic spectrum of the qubit-resonator system and the selection rules imposed by the global parity symmetry allow us to activate well controlled two-body quantum gates via ac pulses applied to the SQUID. We show that our proposal has the same simulation time for any number of spin-1 interacting particles. This scheme may be implemented within the state-of-the-art circuit QED in the ultrastrong coupling regime.

Figure 1

Top: chain of quantum Rabi systems, given by transmon-qubit (green box) ultrastrongly coupled to transmission lines (gray solid line). The ${\mathrm{QRS}}_j$ and ${\mathrm{QRS}}_{j+1}$ are coupled through grounded ${\mathrm{SQUID}}_{j,j+1}$ (box with crosses). The chain is composed by interleaved species of QRSs (A and B). Bottom: the spatial function $\chi \left(x\right)$ of the voltage, that define a $\lambda /4$-resonator for the edges, and $\lambda /2$-resonators for the bulk.

Figure 2

Energy-level diagram for two adjacent QRSs. The blue solid arrows match the ${\epsilon }_1^{\ell }-{\epsilon }_o^{\ell }$ transition, and the red dashed arrows match the ${\epsilon }_2^{\ell }-{\epsilon }_1^{\ell }$ transition. ${\nu }_n$ are the necessary frequencies to adjust the resonance condition for different transitions.

Figure 3

Energy diagram of Hamiltonian (7). Blue continuous line indicate the states with parity $+1$ and red dashed lines states with parity $-1$, the vertical dashed lines indicate the coupling strength for the ${\mathrm{QRS}}_A$ and for the ${\mathrm{QRS}}_B$ used in our numerical calculations.

Figure 4

Diagram of the digital quantum simulation of the anisotropic Heisenberg model, for (a) adjacent sites and (b) the entire chain.

Figure 5

Fidelities as a function of time for the simulation of (a) the anisotropic Heisenberg model and (b) the Ising model. In both figures blue circles stand for a lattice size of $N=2$, red crosses $N=3$, and yellow triangles $N=4$ spin-1 particles.

Figure 6

Diagram of the circuit that models two adjacent transmission lines coupled through a grounded SQUID.