Observation of critical phase transition in a generalized Aubry-André-Harper model with superconducting circuits

The study aims to simulate and probe phase transitions and non-equilibrium many-body dynamics in the one-dimensional generalized Aubry-André-Harper (GAAH) model using a superconducting quantum processor. The research question is how quasi-periodic modulations in both on-site potentials and off-diagonal hopping couplings affect localization properties and dynamical behavior, and how to experimentally distinguish the extended, localized, and critical phases. The work leverages analog quantum simulation with precise control of qubit frequencies and tunable couplers to implement the GAAH Hamiltonian and to measure dynamical signatures such as spin transport and participation entropies. This is important because the GAAH model exhibits rich localization and topological phenomena, including a critical phase with multifractal wave functions, and prior experimental investigations have largely been limited to single-particle dynamics. The superconducting platform allows multi-qubit state preparation, control, and readout, enabling exploration of many-body dynamics and phase transitions.

The Aubry-André-Harper (AAH) model is a canonical platform for studying localization and topology in 1D systems, originally derived from a 2D quantum Hall system with nearest-neighbor hopping. Inclusion of next-nearest-neighbor hopping leads to a generalized AAH (GAAH) model with both on-site and off-diagonal quasi-periodic modulations, yielding extended, localized, and a critical phase with multifractal properties, and enabling topological adiabatic pumping. Experimentally, GAAH realizations have been demonstrated in photonic lattices and cold atom systems, but observations were mostly at the single-particle or mean-field level. Superconducting circuits offer flexible, precise control suitable for realizing off-diagonal quasi-periodic modulation, enabling studies beyond single-particle physics. The work builds on advances in superconducting quantum simulation that have explored localization, thermalization, and quantum walks, and on tunable coupler architectures that permit precise control of inter-qubit coupling strengths.

Device and Hamiltonian: A linear array of L=10 transmon qubits (Q1–Q10) is interconnected by nine tunable couplers (C1–C9), enabling accurate control of nearest-neighbor couplings. The effective Hamiltonian is H = sum_j h_j + sum_j h_{j,j+1}, comprising on-site energies and nearest-neighbor hoppings. The coupling J_{j,j+1} consists of direct qubit-qubit coupling and a superexchange interaction via the tunable coupler, J_{j,j+1} = J^{d}_{j,j+1} + J^{se}_{j,j+1}, with the latter tunable via fast flux bias; the accessible coupling range is approximately −30.0 to +4.8 MHz (−15.0 to +3.8 MHz used experimentally). Qubit Z-pulses tune the local on-site potentials relative to a reference frequency near 4.36 GHz. Implementation of GAAH modulations: Quasi-periodic modulations are programmed as follows: off-diagonal coupling J_{j,j+1} = λ(1 + μ cos(2π (j+1/2) α + δ)) and on-site potential h_j = λ V cos(2π j α + δ), where μ and V are modulation amplitudes for hopping and on-site terms, respectively; λ/(2π) = 4 MHz; α = (√5 − 1)/2 is the irrational frequency (golden ratio); δ is a global phase offset. μ and V are set via Z-control of couplers and qubits, respectively. Calibration: For each μ, all couplings are calibrated using swap oscillation experiments at qubit resonance (~4.36 GHz), with fine-tuning to within ~0.1 MHz. On-site potentials are calibrated via two vacuum Rabi oscillation experiments per target qubit using staggered frequency configurations of non-target qubits (±80 MHz from reference) to mitigate AC Stark effects; the final Z amplitude is the average over the two measurements. Readout correction uses single-qubit readout fidelity matrices. Initial states and measurement protocols: Spin transport is probed with (i) a single-excitation initial state |1000000000⟩ and (ii) a half-filled Néel state |1010101010⟩. After programming μ and V, the system evolves for t ∈ [0, 500] ns. At each time point and for each δ, on-site populations P_i(t) are measured simultaneously over 5000 single-shot repetitions; experimental data are averaged over 5 choices of δ. Numerical simulations are performed for comparison, including decoherence where indicated. Participation entropy measurements: Full 2^10 bitstring probabilities are recorded with readout error mitigation (product of single-qubit fidelity matrices). Due to U(1) symmetry, analysis post-selects the half-filled sector (N = C(10,5) = 252). The second-order dynamical participation entropy is S_2^{PE}(t) = −log Σ_i p_i(t)^2 (negative log of the IPR). Ten far-from-equilibrium initial states are used and averaged: M=10 states of the form |ψ_i(0)⟩ and their global spin-flipped partners (constructed symmetrically about the chain center). Late-time averages are taken over t ∈ [350, 450] ns. For global μ–V scans, numerical calculations neglect decoherence in this time window since it is much shorter than the average T1 ≈ 22.3 μs. Analysis paths: Phase transitions are probed by varying parameters along three paths in the μ–V plane: I) fixed μ=0.5, varying V (extended to localized); II) fixed V=1, varying μ (extended to critical); III) fixed V=3, varying μ (localized to critical). Finite-size analysis compares L=10 data to numerics at larger sizes (L=14,18,22) via rescaled S^{PE}(L) = (log N_{L=10}/log N_L) S^{PE}(L).

- Realization of the GAAH model on a 10-qubit superconducting processor with tunable couplers, implementing simultaneous quasi-periodic on-site and off-diagonal modulations. - Spin transport observations: • Extended phase (e.g., μ=0.5, V=0.5 or 1.0): ballistic, lightcone-like propagation with reflections at boundaries; populations spread across the chain. • Localized phase (e.g., μ=0.5, V=4.0): excitation remains near the initially excited sites; single-excitation case stays localized at Q1; in Néel initial state, populations remain near initial pattern (∼1 on initially excited qubits, ∼0 otherwise). • Critical phase (e.g., μ=2.0, V=0.5 or 1.0): intermediate transport; excitations oscillate around initial sites with nonzero probability reaching distant sites; for Néel initial state, long-time populations hover near 0.5 but show bias depending on initial bit (initially excited sites >0.5, others <0.5). - Dynamical participation entropy S_2^{PE}(t) distinguishes phases: • After an initial rise, S_2^{PE}(t) plateaus at the highest values in the extended phase, lowest in the localized phase, and intermediate in the critical phase; oscillation amplitudes are stronger in localized and critical phases than in the extended phase for the same system size and δ-averaging. • Parameter sweeps: For fixed μ=0.5, increasing V from 1 to 4 strongly suppresses S_2^{PE}(t) growth, indicating extended-to-localized transition. For fixed V=1, increasing μ slows S_2 growth near μ≈1, then S_2 slightly rises again in the critical regime, consistent with multifractality. - Phase transition characterization via averaged late-time participation entropy S^{PE} (averaged over t ∈ [350,450] ns and 10 initial states): • Path I (μ=0.5, vary V): S^{PE} decreases monotonically with V, consistent with extended→localized transition. • Path II (V=1, vary μ): S^{PE} exhibits a minimum near μ≈1.25 for L=10 (shifted from the theoretical μ=1 due to finite-size effects). Larger-system numerics (L=14,18,22) show the minimum at μ=1 after rescaling, confirming finite-size origin. • Path III (V=3, vary μ): S^{PE} decreases slightly for μ≤1, then increases across the localized→critical transition at μ=1.5, approaching an intermediate value; the slope at μ=1.5 grows with system size, indicating sharpening in the thermodynamic limit. - Agreement between experiment and numerics: Time-resolved P_i(t) and S_2^{PE}(t) curves match simulations that include decoherence where appropriate; μ–V plane mapping via numerically computed S^{PE} reproduces the phase diagram similar to eigenstate-averaged IPR, with no mobility edges. - Experimental parameters and statistics: Evolution times 0–500 ns; 5000 single-shot measurements per time point; averages over 5 values of δ; average T1 ≈ 22.3 μs; coupling range used −15.0 to +3.8 MHz; half-filled sector size N=252.

The results demonstrate that a tunable-coupler superconducting platform can faithfully emulate the GAAH model and capture its distinctive dynamical signatures across extended, localized, and critical phases. By measuring site-resolved spin transport and the time evolution of participation entropies, the study establishes experimentally accessible indicators of phase identity and transitions without resorting to full spectroscopy. The critical phase exhibits intermediate transport and multifractal-like participation entropy plateaus, bridging behaviors of extended and localized regimes. Mapping transitions via averaged late-time S^{PE} along controlled parameter paths aligns with theoretical boundaries and finite-size-corrected numerics, validating the methodology. The platform’s simultaneous control of couplings and on-site energies significantly broadens the class of Hamiltonians accessible to analog quantum simulation, paving the way for studies of topology (e.g., adiabatic pumping), coherent time reversal (Loschmidt echo) and scrambling (OTOCs), and interaction-induced phenomena beyond single-particle physics.

This work realizes the generalized Aubry-André-Harper model on a 10-qubit superconducting processor with tunable couplers, implementing both on-site and off-diagonal quasi-periodic modulations. It experimentally observes distinct spin-transport behaviors and quantifies dynamical localization via participation entropies, identifying extended, localized, and critical phases and their transitions. The approach showcases the power of superconducting circuits with tunable couplings for simulating complex lattice models and characterizing their non-equilibrium dynamics. Future directions include extending to larger system sizes and incorporating controlled interactions to explore the many-body critical phase that is delocalized yet nonthermal, and studying transitions among ergodic, many-body localized, and many-body critical phases. The platform also enables investigations of topological effects and dynamical probes such as Loschmidt echo and OTOCs via engineered time reversal.

- Finite system size (L=10) leads to noticeable finite-size effects, e.g., the minimum of averaged late-time participation entropy along V=1 occurs at μ≈1.25 instead of the theoretical μ=1; larger-size numerics recover μ=1. - Limited tunable coupling range (approximately −15.0 to +3.8 MHz used) and residual ZZ interactions constrain accessible parameter regimes and may introduce systematic deviations. - Decoherence is present though mitigated by short evolution times; some simulations neglect decoherence for late-time averaging because the window (350–450 ns) is much shorter than average T1 (~22.3 μs), which may still introduce small discrepancies. - Measurements average over a finite number of global phase offsets (five δ values), and post-selection to the half-filled sector relies on readout error mitigation; remaining readout and calibration errors can affect quantitative precision. - Observations are primarily dynamical in the single-particle or non-interacting limit; many-body interaction effects and true thermodynamic-limit behavior remain to be explored.