Selective Control and Imaging of Higgs and Goldstone Modes with a Quantum Simulator

AI summary

Overview

Research area: Quantum simulation of strongly correlated many-body physics—specifically, the Bose–Hubbard model realized with superconducting transmon qubits—and the spectroscopy of collective (Higgs and Goldstone) modes associated with a superfluid order parameter.

Technical level: Advanced. The paper assumes familiarity with spontaneous symmetry breaking, the Bose–Hubbard model, and circuit-QED spectroscopy, although the central physics question is stated clearly enough for non-specialists.

Scope (one sentence): The authors use a two-dimensional transmon lattice at unit filling as an analog quantum simulator to selectively excite, detect, and image the amplitude (Higgs) and phase (Goldstone) modes of a lattice superfluid, and to track how those modes behave as the system size is varied.

What This Paper Is About

When a system spontaneously breaks a symmetry—as a superfluid does—its order parameter can fluctuate in two distinct ways: in amplitude and in phase. These fluctuations appear as the Higgs and Goldstone modes, but the Higgs mode is notoriously hard to probe because it couples weakly to conventional electromagnetic probes and, in two dimensions, can decay into pairs of lower-energy Goldstone modes, smearing its spectral signature.

The authors' goal is to show that a finite-sized (mesoscopic) quantum simulator can circumvent this problem: by exciting each mode with a drive of the correct spatial symmetry and reading out site-resolved occupation numbers, they isolate sharp, long-lived Higgs and Goldstone excitations, and they study how both modes evolve as the lattice grows.

Key Contributions

1. Selective excitation protocols for each mode. A spatially uniform modulation of the hopping strength J excites the Higgs mode, while a modulation of the local chemical potential μᵢ with a spatial profile chosen from the momentum q = kπ/L excites the k = 1 Goldstone mode. The two drives couple to modes of different symmetry and do not cross-excite one another.

2. Local observables that distinguish the modes without measuring the order parameter. The Higgs mode is detected through the doublon fraction (the fraction of sites in |2⟩), which tracks density variance; the Goldstone mode is detected through oscillations of the local density ⟨nᵢ⟩. Both are extracted from qutrit-resolved site-by-site readout.

3. Two complementary spectroscopic techniques. "Vacuum Rabi spectroscopy"—a diabatic ramp of J that produces a Landau–Zener superposition of ground and excited states, followed by free decay—and driven spectroscopy, in which the response is demodulated at the drive frequency. The two methods agree quantitatively except for the Higgs mode in the largest system at small J/U, which the authors attribute to nonlinear drive effects.

4. Experimental finite-size scaling. Repeating the measurements on 4×4, 5×5, and 7×7 lattices (the 7×7 with one qubit removed due to poor swapping behavior) traces how the mode frequencies approach the thermodynamic limit, and a tensor-network calculation of the kinetic spectral function for a 4×4 lattice shows the discrete peak structure that replaces the broad continuum.

Main Findings

• A sharp, long-lived Higgs resonance. For J/U = 0.09 and ⟨U⟩/2π = −200.0 ± 0.3 MHz, the doublon fraction oscillates predominantly at a single frequency, with the maximum frequency component at ω/|U| = 0.314, in agreement with the theoretically calculated value. The authors interpret the finite-size-induced spectral gaps as the reason the Higgs mode appears coherent rather than overdamped.

• Spatial imaging of the amplitude mode. Snapshots of the site-resolved number variance ⟨nᵢ²⟩ − ⟨nᵢ⟩² over one period of the doublon oscillation resemble drum-head modes; residual spatial structure is attributed to differences in individual transmon properties such as readout fidelities, coherence times, non-linearities, or coupling strengths.

• Ramp rate affects amplitude, not frequency. Shortening the ramp time increases the response amplitude—confirming that a more diabatic ramp populates excited modes more strongly—while the peak frequencies and widths remain independent of ramp rate, since the ramp only sets state-preparation weights.

• Mode-selective response across the transition. On the 5×5 lattice, the Higgs drive produces a strong response in both the superfluid and insulating regimes, whereas the Goldstone drive responds strongly only in the superfluid regime. This contrast is presented as a signature of superfluidity and spontaneous symmetry breaking: in the Mott regime the order parameter magnitude vanishes and its phase is ill-defined, but an amplitude mode is still permitted.

• A minimum near the crossover. The driven Higgs response shows a minimum near J/U = 0.08, a signature of proximity to the Mott–superfluid transition. The Higgs frequency reaches a minimum in the crossover regime, while the Goldstone frequency falls monotonically with increasing J and approaches a constant above that point; in the superfluid phase the Goldstone frequency is related to the speed of sound c by ω = cπ/L.

• The two branches cross without an avoided crossing. For all three system sizes the Goldstone and Higgs frequencies cross as J is varied, with no avoided crossing resolved within the experimental range and resolution—consistent with the modes belonging to different spatial symmetry sectors.

• Consistency with finite-size scaling expectations. The critical ratio g* at which each mode reaches its minimum frequency is expected to approach g_T as g*(L) − g_T ∝ L^(−1/ν), with 1/ν ≈ 1.49 and g_T ≈ 0.06. The data are consistent with this prediction, though the authors state that confirming the critical exponent would require finer coupling resolution and a larger range of system sizes.

• Discrete peak comb in simulation. A Lorentzian-broadened (η = 0.05) tensor-network calculation of the kinetic spectral function for a 4×4 lattice shows multiple clearly separated peaks; smaller systems give more widely spaced peaks, and the experiment probes the first and most heavily weighted branch, with the others accessible at higher drive frequency.

• No true transition in a finite system. With fixed particle number, the global U(1) symmetry cannot be broken; the transition becomes a crossover and the state is typically a superposition of symmetry-breaking configurations. Beyond J/U ≫ g_T—outside the probed region—the two-excitation-per-site description in terms of doublons and holons would break down.

Methodology in Plain English

The team treats microwave photons in a lattice of coupled superconducting transmons as bosons, with hopping set by tunable couplers and on-site repulsion set by the transmon nonlinearity. Working at one particle per site and keeping J/U small lets each transmon be treated as a three-level system (qutrit), so occupations of 0, 1, and 2 particles per site can be read out directly.

To calibrate, they iteratively perform single-photon swap spectroscopy between neighboring qubits, adjusting coupler and qubit flux biases until the swap rate matches the target coupling; calibrations are required to converge within 0.02 MHz for a modulation amplitude δJ = 0.3 MHz, and couplings up to 20 MHz are used. The work uses subsets of a 105-qubit device, with lattice sizes of 3×3, 4×4, 5×5, and 7×7 (3×3 data appear only in the appendices). Average anharmonicities of the four grids at their idle points are U/2π = −200.4 ± 0.8, −200.1 ± 0.7, −200.0 ± 0.8, and −199.8 ± 1.0 MHz, and U/2π = −200 MHz is treated as constant across coupling strengths.

Two measurement strategies are used. In the first, the hopping is ramped from zero to its final value fast enough to leave the system in a superposition of the ground and Higgs states; after a variable hold time, the doublon fraction is measured and Fourier analyzed. In the second, the hopping (for Higgs) or the qubit detunings (for Goldstone, with a cos(yq) spatial profile) are modulated sinusoidally, and the response is obtained by a continuous Fourier transform at the drive frequency. Peak frequencies are extracted from Lorentzian fits within ±50 MHz of the maximum response, searching only above 20 MHz to avoid low-frequency noise.

All data are post-selected on total photon count, with shot counts increased at long times to keep shot noise below a target. Because post-selection survival drops steeply—about 67% at 0 ns and 5% at 200 ns out of 50,000 total shots—the authors analyze two mechanisms for a rising doublon fraction: T₁ loss acting differently on sites with different decay rates, and frequency noise driving transitions between singlon and doublon–holon configurations within the number-conserving sector.

Why This Matters

Impact on research. The work demonstrates that a finite quantum simulator can access physics that is hard to resolve in the thermodynamic limit: in an infinite two-dimensional system the Higgs mode decays into a continuum of Goldstone pairs, but in a mesoscopic lattice that continuum becomes a discrete comb of resolvable peaks. It also provides an experimental handle on finite-size scaling of collective mode frequencies, a tool more familiar from numerics than from experiment, and offers a mode-selective drive-and-detect toolkit that does not require direct measurement of the order parameter.

Potential applications (the paper itself reports a quantum-simulation experiment, not device applications; these are areas the techniques could serve):

• Benchmarking and calibrating coupler-mediated lattices of superconducting qubits, including the analog pairwise swap-spectroscopy procedure described here.
• Characterizing how defects, readout infidelity, and inhomogeneity perturb collective behavior in multi-qubit arrays.
• Providing analog intuition for amplitude-mode physics in condensed-matter systems such as superconductors and lattice superfluids, where the amplitude mode resists conventional electromagnetic probes.
• Testing numerical methods—such as tensor-network and infinite matrix-product-state calculations—against experimental spectra in the intermediate-size regime where exact simulation is still feasible.

Industry relevance. For quantum computing hardware groups, the results quantify how device-level imperfections (coherence times, readout fidelity, anharmonicity spread, failing qubits) propagate into many-body observables, and they demonstrate qutrit-resolution readout and high-fidelity analog control at couplings up to 20 MHz on a 105-qubit device—capabilities that underpin analog quantum simulation and calibration at scale.

Future Directions

• Improving coupling resolution and extending the range of system sizes to actually confirm the predicted critical exponent 1/ν ≈ 1.49 rather than merely show consistency.
• Accessing the higher-lying branches of the Higgs spectral comb, which the authors note could be probed by increasing the drive frequency.
• Resolving whether any avoided crossing appears between the Higgs and Goldstone branches at finer resolution, since none is observed within the current experimental range.
• Extending the approach to stronger J/U, where the restriction to at most two excitations per site fails and the doublon–holon picture of amplitude and phase modes would need to be replaced.

Target Audience

Physicists and quantum-information researchers interested in analog quantum simulation with superconducting circuits; condensed-matter theorists working on spontaneous symmetry breaking, superfluidity, and the Higgs mode in lattice systems; and hardware-oriented readers who want a concrete example of mode-selective control, qutrit readout, and finite-size scaling on a multi-qubit transmon lattice. Readers without background in many-body physics will find the introduction and figures accessible but the spectroscopy details demanding.

Authors’ abstract

We realize the mesoscopic Bose-Hubbard model at unit-filling on a two-dimensional array of transmon qubits and selectively excite collective modes associated with amplitude or phase fluctuations of the superfluid order parameter. We identify these modes respectively as the Higgs and Goldstone modes using complementary local observables, extracted from measuring the number of particles on each site. Modulation of the coupling between sites excites the Higgs mode, which we detect via the density of doubly occupied sites. In contrast, modulation of the local chemical potential with an appropriately chosen spatial profile excites the lowest-frequency Goldstone mode, which we detect via changes in the local density. The two modes have different symmetries and do not couple to one another. By varying the system geometry, we characterize the impact of finite system size on the evolution of both modes. In the thermodynamic limit, decay into a continuum of lower-energy excitations can obscure the Higgs response. In our mesoscopic system, however, the finite system size leads to spectral gaps and allows us to detect the Higgs mode as a coherent, long-lived excitation. Our work demonstrates how finite quantum simulators can isolate and probe physical phenomena that become difficult to resolve in the thermodynamic, large-scale limit.