Analog Circuit-QED Simulator of Quantum Spin Dynamics Through the Extended Bose-Hubbard Model

TL;DR

The paper tackles the challenge of simulating quantum spin dynamics of the Heisenberg model by mapping spins to bosons through the Dyson-Maleev transformation, yielding a Hermitian extended Bose-Hubbard ($EBH$) Hamiltonian that is equivalent to the Holstein-Primakoff encoding for spin-1/2. It then details a circuit-QED implementation based on a Josephson-junction array that realizes the $EBH$ model under a controlled parameter regime, providing explicit relations between spin couplings $J_{jk}$ and circuit parameters. Numerical validation using exact diagonalization confirms that the microwave-photon dynamics in the $EBH$ simulator reproduce the target spin dynamics across 1D and 2D geometries, including dimerized chains, spinon propagation, and disordered lattices. The work demonstrates a scalable, experimentally feasible platform for probing complex quantum spin dynamics in a highly tunable bosonic setting, with potential extensions to higher spins, longer-range couplings, and open-system dynamics.

Abstract

We propose and validate a framework for analog simulation of the Heisenberg spin model using a circuit quantum electrodynamics (circuit-QED) platform. Our method involves the Dyson-Maleev transformation, for which we develop a procedure to circumvent its inherent non-Hermiticity, yielding the extended Bose-Hubbard (EBH) Hamiltonian. We demonstrate the equivalence of this approach to the Holstein-Primakoff encoding for spin-1/2 systems. For the experimental realization of this EBH model, we design a scalable circuit-QED architecture based on an engineered Josephson junction array. Numerical simulations confirm that the microwave photon dynamics in this simulator accurately reproduces the original spin dynamics. Our work establishes an experimentally accessible method for investigating complex quantum spin dynamics in a highly controllable bosonic setting.

Figures (6)

• Figure 1: Design of the analog circuit-QED-based simulator for a one-dimensional Heisenberg model with (a) open and (b) periodic boundary conditions. The simulator is the Josephson-junction array that consists of $N$ Josephson junctions with Josephson energies $E_{J,j}$, each coupled in parallel to a capacitance $C_j$, $j=1,\ldots, N$, and connected to ground. Flux variables $\phi_j$ indicated by red dots play the role of nonlinear microwave oscillators. These nonlinear oscillators interact via capacitive couplings $C'_{jk}$ and nonlinear inductive couplings $E'_{J,jk}$. Adapted from Ref. RamosJJA.
• Figure 2: Design of the analog circuit-QED-based simulator for the Heisenberg model of $N$ spins on a rectangular lattice with open boundary conditions. The spins are enumerated row-wise from the top left to the bottom right. Other parameters are the same as in Fig. \ref{['fig:1D']}
• Figure 3: Comparison of the dynamics of the circuit-QED simulator (solid line), Eq. (\ref{['eq:H_JJA_simplified']}), and the dimerized antiferromagnetic Heisenberg chain with open boundary conditions (dotted line), Eq. (\ref{['eq:H1']}). The initial state is the domain wall state. The number of sites is $N=14$. The coupling constant is $J=2\pi\hbar\times 40$ MHz. The parameters of the circuit-QED simulator are presented in Table \ref{['tab1']}. (a) Magnetization flow through the center for dimerization strength $\delta = 0,0.1,0.2$, and $0.3$. (b) Deviations of the spin expectation values from their initial values for sites in the middle ($j = 6, 7, 8, 9$) and at the edges ($j = 1, 14$) of the chain. The dimerization coefficient is $\delta=0$.
• Figure 4: Comparison of the dynamics of the circuit-QED simulator (solid line), Eq. (\ref{['eq:H_JJA_simplified']}), and the antiferromagnetic Heisenberg chain with periodic boundary conditions (dotted line), Eq. (\ref{['eq:H2']}). The number of sites is $N=14$. The correlation function is shown for sites $j=1,2,3,4$. The coupling constant is $J=2\pi\hbar\times 40$ MHz. The parameters of the circuit-QED simulator are presented in Table \ref{['tab2']}.
• Figure 5: Comparison of the dynamics of the circuit-QED simulator (solid line), Eq. (\ref{['eq:H_JJA_simplified']}), and the spatially anisotropic Heisenberg model (dotted line), Eq. (\ref{['eq:H3']}) on a $5\times 3$ lattice with open boundary conditions. The quantum Fisher information density is shown for different anisotropy parameters $\eta = 0, 0.1, 0.2, 0.3$. The coupling constant is $J=2\pi\hbar\times 40$ MHz. The parameters of the circuit-QED simulator are presented in Table \ref{['tab3']}.