Propagation and Localization of Collective Excitations on a 24-Qubit Superconducting Processor

Superconducting circuits have emerged as a powerful platform of quantum simulation, especially for emulating the dynamics of quantum many-body systems, because of their tunable interaction, long coherence time, and high-precision control. Here in experiments, we construct a Bose-Hubbard ladder with a ladder array of 20 qubits on a 24-qubit superconducting processor. We investigate theoretically and demonstrate experimentally the dynamics of single- and double-excitation states with distinct behaviors, indicating the uniqueness of the Bose-Hubbard ladder. We observe the linear propagation of photons in the single-excitation case, satisfying the Lieb-Robinson bounds. The double-excitation state, initially placed at the edge, localizes; while placed in the bulk, it splits into two single-excitation modes spreading linearly toward two boundaries, respectively. Remarkably, these phenomena, studied both theoretically and numerically as unique properties of the Bose-Hubbard ladder, are represented coherently by pairs of controllable qubits in experiments. Our results show that collective excitations, as a single mode, are not free. This work paves the way to simulation of exotic logic particles by subtly encoding physical qubits and exploration of rich physics by superconducting circuits.

Figure 1

Experimental setup. (a) Circuit diagram of the device. There are 24 superconducting transmon qubits ($Q_{1A}\text{−}{Q}_{10A}$, $Q_{1B}\text{−}{Q}_{10B}$ and $Q_a\text{−}{Q}_d$) constituting a ladder [41, 42, 43, 44, 45]. The nearest-neighbor hopping is realized by the capacitive coupling between qubits, and the negative anharmonicity $U$ of each qubit gives the on-site attractive interaction. The frequency of each qubit is tunable by individual microwave driving through the flux-bias line. Readout resonators are separated into four groups, and the resonators in each group share a microwave transmission line for multiplexed readout. More experimental details of our system are presented in the Supplemental Material [46]. In our experiments, the middle 20 qubits ($Q_{1A}\text{−}{Q}_{10A}$, $Q_{1B}\text{−}{Q}_{10B}$) are biased to the working frequency 4.8 GHz. The other four qubits ($Q_a\text{−}{Q}_d$) are always idled with frequencies 3.5, $<3.5$, 5.4, and $<3\mathrm{GHz}$, respectively. Thus, ${\mathrm{Q}}_a\text{−}{\mathrm{Q}}_d$ are far away from the working frequency, and thus, a 20-qubit ladder is employed. (b) The image of the corresponding Bose-Hubbard ladder with 6 rungs. The red ball represents the photon (the excitation of the qubit), which can tunnel to the nearest-neighbor sites (red arrows). We label two qubits, which share the same rung (black box), as a unit cell $C_j$.

Figure 2

Time evolutions of single-excitation states up to 100 ns. Time evolutions of the probability distributions of single excitations $P_j^{\mathrm{se}}$ starting with one initial single-excitation state $|10{⟩}_1$ and $|10{⟩}_6$ at (a) the leftmost cell $C_1$ and (b) the central cell $C_6$, and the states $|10{⟩}_1\otimes |01{⟩}_{10}$ of two initial single excitations at (c) $C_1$ and $C_{10}$, respectively, are shown. For (b), $C_1$ is at the idle frequency to be turned off to make $C_6$ central. (d)–(f) are for the corresponding numerical results for comparisons with (a)–(c). In the numerical simulation, decoherence is not considered, and the experimental parameters are used. (g) Time evolutions of the probability distributions of (a) for $C_2–C_6$. The scattering points are experimental data, while the solid curves are for the corresponding Gaussian fitting analysis. (h) The linear fitting analysis of (g) between the time of the first peak of the probability distribution and the cell distance. The experimental data (blue cross) are obtained from the Gaussian fitting curves, while the theoretical ones (orange stars) are obtained from the numerical results. We calculate the experimental group velocity as $v_g^{\mathrm{Ex}}=112.4\mathrm{cells}/\mu \mathrm{s}$, and theoretic one as $v_g^{\mathrm{Th}}=123.5\mathrm{cells}/\mu \mathrm{s}$.

Figure 3

Time evolutions of double-excitation states up to 100 ns. Time evolutions of the probability distributions of double excitations $P_j^{11}$ with initial double-excitation states $|11{⟩}_1$ and $|11{⟩}_6$ at: (a),(b) $C_1$ and (c),(d) $C_6$, respectively, are plotted. In (a) and (c), experimental results are shown; in (b) and (d), corresponding numerical results are shown. Similarly, $C_1$ is at the idle frequency to be turned off for the central localized initial state in (c),(d). The (e) experimental and (f) theoretical evolutions of the probability distributions of single excitations $P_j^{\mathrm{se}}$ for the central localized initial state in (c),(d) are shown, respectively.

Figure 4

Numerical results of time evolutions up to 100 ns of the double-excitation states of 7 cells for different regimes of interaction strength with initial double-excitation states at $C_1$. Here, we choose $J/2\pi =12\mathrm{MHz}$, and $U/J=1$, 5, 10, 20, respectively.