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

In the physical world, many systems possess a hidden potential for symmetry that they ultimately choose not to use. Consider a magnet: the laws of physics treat all directions equally, yet a magnet spontaneously picks a specific north and south, breaking that perfect balance. This phenomenon, known as spontaneous symmetry breaking, is a fundamental principle that explains everything from how magnets work to why particles have mass. When a system breaks this symmetry, it does not just settle into a new state; it also gives rise to ripples, or collective vibrations, that move through the material. In the realm of superfluids—fluids that flow without friction—these ripples come in two distinct flavors. One type involves the fluid sloshing back and forth in density, like a wave moving across a pond. The other type involves the fluid pulsing in strength, expanding and contracting like a breathing lung, without changing its overall density. The first is a well-understood wave, while the second, a more elusive vibration of the system's very strength, has been notoriously difficult to isolate and study.

A team of researchers has now successfully isolated and visualized these two distinct vibrations in a controlled laboratory setting, using a quantum simulator built from superconducting circuits. By creating a miniature, two-dimensional grid of artificial atoms, they were able to excite and watch the behavior of these modes with a precision that is impossible in larger, natural systems. The experiment focused on a specific state of matter where the artificial atoms are arranged in a grid, with exactly one particle sitting on each spot on average. In this state, the system behaves like a superfluid. The researchers found that by carefully tweaking the connections between the atoms, they could make the entire grid pulse in strength, creating the elusive "breathing" vibration. Conversely, by applying a specific pattern of local pushes and pulls, they could generate the density waves. Crucially, they discovered that in their small, finite system, the breathing vibration does not fade away quickly. Instead, it remains a sharp, clear signal for a long time, a behavior that stands in stark contrast to what happens in the infinite, macroscopic world.

The researchers achieved this by constructing a lattice of transmon qubits, a type of superconducting circuit that can act like an artificial atom. They programmed these circuits to mimic the Bose-Hubbard model, a theoretical framework describing how particles interact when they hop between sites on a grid. In their setup, the strength of the interaction between neighboring sites could be tuned, as could the energy cost of having a particle sit on a specific site. The team worked in a regime where the grid was filled with exactly one particle per site on average, a condition that allows for the emergence of superfluidity. To see what was happening, they did not measure the abstract "order parameter" that physicists usually use to describe superfluids. Instead, they looked directly at the number of particles on each site. They found that when the system vibrated in the "breathing" mode, the number of sites holding two particles at once would rise and fall in sync with the number of empty sites, while the total number of particles remained constant. This oscillation between having two particles and having none, while the average stays the same, is the signature of the amplitude mode.

To trigger this specific vibration, the researchers applied a uniform modulation to the connections between all the sites simultaneously. This global push and pull on the links caused the entire grid to expand and contract in its internal structure. They detected the resulting oscillation by measuring the fraction of sites that held two particles, a quantity they call the doublon fraction. The data showed a clear, rhythmic rise and fall in this fraction, confirming that the system was vibrating in the amplitude mode. In a separate set of experiments, they excited the density wave mode by applying a spatially varying potential, essentially pushing the particles in a wave-like pattern across the grid. This caused the local density of particles to rise and fall as the wave traveled through the lattice. The two modes responded to different types of inputs and did not interfere with one another, confirming that they are distinct physical phenomena with different symmetries.

A key discovery of this work is how the size of the system affects the lifetime of these vibrations. In a theoretical, infinitely large system, the "breathing" mode is expected to decay rapidly. It does this by breaking apart into pairs of lower-energy density waves, a process that blurs the signal and makes it difficult to observe. However, in the researchers' mesoscopic system, which consisted of grids ranging from 4x4 to 7x7 sites, this decay channel was blocked. Because the system was finite, the possible lower-energy states were not a continuous stream but a discrete set of steps. This discreteness prevented the breathing mode from easily breaking apart, allowing it to persist as a coherent, long-lived excitation. The researchers observed this by ramping the connections between the sites and watching the system evolve over time. They saw the doublon fraction oscillate for hundreds of nanoseconds, a duration that would be impossible in a larger system where the signal would have washed out almost immediately.

The team also mapped out how these vibrations change as the system moves from a state where particles are stuck in place to the superfluid state. They varied the strength of the hopping between sites and measured the frequency of the vibrations. They found that the frequency of the breathing mode reached a minimum at a specific point, which corresponds to the transition between the insulating and superfluid phases. The density wave mode, on the other hand, showed a different behavior, with its frequency dropping as the system became more superfluid. By comparing their experimental results with numerical simulations, they confirmed that their observations matched theoretical predictions for these finite-sized systems. The simulations showed that as the system size increases, the discrete peaks of the breathing mode begin to merge, eventually forming the broad, indistinct feature expected in the thermodynamic limit.

This work demonstrates that finite quantum simulators can act as powerful tools to probe physical phenomena that are often obscured in the large-scale limit. By working with a system small enough to control but large enough to exhibit collective behavior, the researchers were able to isolate the Higgs-like amplitude mode and observe it clearly. They showed that the finite size of the system, often considered a limitation in experiments, actually provides a unique advantage here by creating spectral gaps that protect the excitation from decaying. The ability to selectively excite and detect these modes without needing to measure the abstract order parameter directly opens new avenues for studying the dynamics of symmetry breaking. The results provide a clear, experimental picture of how collective excitations behave in a controlled environment, bridging the gap between microscopic quantum mechanics and macroscopic physical laws.