Strongly coupled Josephson-junction array for simulation of frustrated one-dimensional spin models

We study the capacitance-coupled Josephson-junction array beyond the small-coupling limit. We find that, when the scale of the system is large, its Hamiltonian can be obtained without the small-coupling approximation and the system can be used to simulate strongly frustrated one-dimensional Ising spin problems. To engineer the system Hamiltonian for an ideal theoretical model, we apply a dynamical-decoupling technique to eliminate undesirable couplings in the system. Using a six-site junction array as an example, we numerically evaluate the system to show that it exhibits important characteristics of the frustrated spin model.

Figure 1

The capacitance-coupled Josephson-junction array. $C_c$ is the coupling capacitance between neighboring junctions. $V_{gi}$ is the bias voltage and $C_{gi}$ is the bias capacitance of the Josephson junction at site $i$. The total capacitance and Josephson energy are $C_{ji}$ and ${\mathcal{E}}_J$.

Figure 2

The relationship between $\lambda$ and $\beta$. The dashed red line corresponds to $\lambda =1/2$.

Figure 3

The dynamical decoupling control scheme to eliminate interactions between spins that are three sites apart. The ellipses in groups of three along the vertical direction represent simultaneous $\pi$ or $3\pi$ pulses applied to the qubits.

Figure 4

The finite-size scaling of the second-order derivative of ground-state energy with $\lambda \doteq 0$ (solid lines) and $\lambda =0.2$ (dashed lines)

Figure 5

(a) Fine structures and small dips in the $d{E}_g^2/d{B}^2$ curve with the parameters $\lambda =0.7$ and $N=60$. (b) The phase diagram of the ANNNI model. (c) The spin-spin correlation function $c_s^2\left(d\right)$ vs the spin separation $d$ at point A ($\lambda =0.7$ and $B=0.2$) in the phase diagram. (d) $c_s^2\left(d\right)$ at point B ($\lambda =0.7$ and $B=0.3$) in the phase diagram. (e) $c_s^2\left(d\right)$ at point C ($\lambda =0.7$ and $B=0.6$) in the phase diagram.

Figure 6

(a) The probability of the ground state ${|\psi \rangle }_g$ of the six-site Josephson-junction array being the ferromagnetic state. (b) The probability of ${|\psi \rangle }_g$ being the paramagnetic state.

Figure 7

The fidelity of the system state to the state of the corresponding ANNNI model for different numbers of dynamical decoupling control sequences.

Figure 8

(a) Phase-transition process of the ANNNI model with $\lambda =0.4$. (b) Phase-transition process of the Josephson-junction array system with and without the dynamical-decoupling control sequence. The black dashed curve is the result without dynamical-decoupling control sequences. The solid blue and dotted red curves are the results with one and four control sequences in unit time.