Observation of disorder-induced superfluidity

Technical Summary: Observation of Disorder-Induced Superfluidity

Problem Statement
The interplay between disorder and interactions in quantum many-body systems typically leads to localization, suppressing the particle mobility required for long-range phase coherence. In the standard Bose-Hubbard model, increasing disorder drives a transition from a superfluid (SF) to a Mott insulator (MI) and subsequently to a Bose glass (BG), a gapless insulating phase. While theoretical proposals and numerical simulations have suggested the existence of a "re-entrant" superfluid phase induced by disorder—where resonant tunneling between sites with specific detunings could restore global coherence—definitive experimental evidence for this phenomenon has remained elusive. The primary challenges have been the difficulty in synthesizing materials with tunable disorder and the inability of existing quantum simulators to resolve the necessary dynamical quantities to distinguish between glassy insulators and true superfluids.

Methodology
The authors utilized a superconducting quantum processor developed by Google Quantum AI, configured as a two-dimensional array of coupled nonlinear resonators (transmons). By restricting the dynamics to the lowest three energy levels ($|0\rangle, |1\rangle, |2\rangle$), the system effectively simulates qutrits, allowing for the study of strongly interacting bosons described by the Bose-Hubbard model:
$$H = -J \sum_{\langle i,j \rangle} (a^\dagger_i a_j + \text{h.c.}) + \frac{U}{2} \sum_i n_i(n_i-1) - \sum_i \mu_i n_i$$
Key experimental capabilities included:

• Tunable Parameters: Independent control of nearest-neighbor hopping ($J$) and on-site disorder strength ($W$), where $\mu_i$ is drawn from a uniform distribution of width $W$.
• State Preparation: Adiabatic ramping from a product state ($|1\rangle^{\otimes N}$) to the target Hamiltonian parameters.
• Compressibility Measurement: To distinguish between ergodic (superfluid) and non-ergodic (glassy) states, the authors employed two distinct state preparation protocols analogous to classical glass experiments: Zero-Field Cooled (ZFC) and Field-Cooled (FC). Compressibility ($\kappa$) was measured by applying a static potential gradient and observing the density response.
• Correlation and Condensate Fraction: Spatially resolved two-point correlators ($C_{ij} = \langle a^\dagger_i a_j \rangle$) were measured using a protocol involving local microwave rotations (Givens rotations) and number measurements to construct the single-particle density matrix (SPDM). The largest eigenvalue of the SPDM provided the condensate fraction ($n_0$).
• Dynamical Structure Factor: The excitation spectrum was probed via Bragg spectroscopy, applying a spatiotemporal potential modulation to measure the system's response and identify collective modes.

Key Contributions and Results

1. Mapping the Phase Diagram: The study successfully mapped the phase diagram of interacting bosons in a disordered landscape. It identified the Mott insulator (gapped, incompressible), the Bose glass (gapless, incompressible, non-ergodic), and the superfluid phase.
2. Evidence for Disorder-Induced Superfluidity: The central finding is the observation of a superfluid phase emerging at intermediate to strong disorder strengths ($W \sim U$) and low hopping ($J$).
   • Compressibility: The system exhibited a transition to a compressible state as disorder increased. Crucially, in the high-disorder regime, the compressibility values differed between the ZFC and FC protocols ($\Delta \kappa \neq 0$), signaling non-ergodic glassy behavior in the surrounding phases, while the superfluid region showed distinct characteristics.
   • Condensate Fraction: Measurements of the condensate fraction revealed a non-monotonic dependence on disorder. A significant condensate fraction emerged and peaked when the disorder strength was comparable to the on-site interaction ($W \approx U$), indicating the formation of a macroscopic coherent state despite the disorder.
   • Resonant Tunneling Mechanism: The data supports the mechanism where disorder induces resonances between sites with mutual detuning $U$. These resonances facilitate the formation of particle-hole pairs (doublons and holons), which act as mobile carriers that restore global phase coherence, effectively creating "puddles" of superfluidity that percolate into a global state.
3. Dynamical Signature: The dynamical structure factor measurements revealed a linearly dispersing phonon mode (Bogoliubov mode) in the disordered superfluid phase. This mode, characteristic of superfluidity, appeared even when disorder was introduced to a clean Mott insulator, although the speed of sound was suppressed relative to the clean case due to reduced superfluid density.

Significance
The paper provides strong experimental evidence for the existence of disorder-induced superfluidity in the Bose-Hubbard model, a phenomenon that had been theoretically proposed but not experimentally verified. The results demonstrate that disorder does not always lead to localization; under specific conditions involving multi-level systems and resonant tunneling, it can enhance mobility and induce long-range coherence. This work bridges the gap between theoretical predictions of re-entrant superfluid phases and experimental reality, offering insights into the dirty superfluid–insulator transition relevant to systems such as thin films, granular superconductors, and potentially high-$T_c$ cuprates, where superconductivity emerges from doped Mott insulators. The study also highlights the utility of superconducting quantum processors in simulating complex many-body physics with tunable disorder and measuring dynamical quantities that are difficult to access in other platforms.