Quantum simulation offers a powerful approach to study many-body physics intractable for classical computers. Analog quantum simulation directly realizes model Hamiltonians by engineering the platform Hamiltonians, enabling controlled studies of real quantum system dynamics. Platforms such as trapped ions, atoms in optical lattices, superconducting qubits, and nuclear spins have been used. Superconducting quantum simulation excels in exploring localization and thermalization in non-equilibrium systems. The 1D Aubry-André-Harper (AAH) model is a key model for studying localization and topological states, both theoretically and experimentally. A generalization of the AAH model incorporates next-nearest-neighbor hopping, resulting in a GAAH model with both on-site and off-diagonal quasi-periodic modulations. The GAAH model exhibits intriguing localization and topological properties, including a critical phase with multifractal wave functions and topological adiabatic pumping. Experimental realizations have been achieved in photonic crystals and cold atom systems, but mostly at the single-particle or mean-field level. This experiment utilizes a superconducting processor with tunable coupling architecture to simulate the GAAH model across a wide parameter range. By adjusting qubit frequencies and couplers, we experimentally observe the dynamics of the extended, localized, and critical phases and investigate the phase transitions through non-equilibrium many-body dynamics. We analyze spin transport for single- and multi-excitation initial states and quantify the spreading of initial states using participation entropies.

The study builds upon extensive theoretical and experimental work on the Aubry-André-Harper (AAH) model and its generalizations. Previous work demonstrated the AAH model's ability to exhibit localization and topological properties. Generalizations of the model, incorporating next-nearest-neighbor hopping, have been shown to possess richer phase diagrams, including critical phases characterized by multifractal wave functions. Experimental realizations in various platforms, such as photonic crystals and cold atoms, have provided insights into single-particle or mean-field dynamics. However, the study of many-body dynamics in the GAAH model using a fully controllable quantum processor was lacking, creating the motivation for the presented work.

The experiment uses a 10-transmon superconducting qubit processor with 9 tunable couplers. The Hamiltonian of the system is given by: H = Σᵢ¹⁰ ħᵢ + Σᵢ¹⁰ ħᵢ,ᵢ+₁, where ħᵢ represents the tunable local potential on qubit i, and ħᵢ,ᵢ+₁ represents the nearest-neighbor coupling strength between qubits i and i+1. The coupling strength Jᵢ,ᵢ+₁ consists of direct coupling and superexchange interaction via the coupler, enabling individual adjustment by controlling coupler frequencies. Quasi-periodic modulations are implemented: Jᵢ,ᵢ+₁ = J(1 + cos(2π(i+1/2)α + δ)) and ħᵢ = V cos(2πiα + δ), where J and V represent off-diagonal and on-site modulation amplitudes, α is the irrational frequency, and δ is a global phase offset. Localization properties are characterized by the inverse participation ratio (IPR). The phase diagram, consisting of extended, localized, and critical phases, is determined using IPR analysis. Spin transport is studied by initializing the system in single-excitation (|1000000000⟩) and Néel states (|1010101010⟩) and measuring on-site populations over time. The participation entropy S_q^PE(t) = -1/(q-1) log Σᵢ pᵢ(t)^q quantifies how fast the initial state spreads over the Hilbert space, where pᵢ(t) is the probability of finding the system in the i-th computational basis state at time t. The second-order participation entropy (q=2) is the focus, and its time evolution is measured for various parameters to characterize the phase transitions. The experimental setup involves careful calibration of hopping coupling using a joint probability measurement during qubit-qubit swapping. The on-site potential calibration uses vacuum Rabi oscillations, with staggered frequencies to mitigate AC Stark effects. For spin transport measurements, 5000 repeated single-shot measurements are performed at each time point. Participation entropy calculations utilize post-selection within the half-filled sector due to U(1) symmetry.

The experiment successfully simulated the GAAH model's three phases (extended, localized, and critical) using a superconducting qubit processor. In the extended phase, spin excitations spread ballistically. In the localized phase, spin remains confined to the initially excited qubits. The critical phase exhibits intermediate spin transport behavior, with oscillations around adjacent sites. These observations held for both single- and multi-excitation initial states. The time evolution of participation entropies provided further characterization of phase transitions. In the extended phase, participation entropy remained high. In the localized phase, it oscillated around a low value. The critical phase showed an intermediate plateau. Analyzing the late-time averaged participation entropy along paths in the µ-V plane confirmed the phase transitions, showing good agreement between experimental data and numerical simulations. Finite-size effects were observed, with the minimum participation entropy shifting slightly from the theoretical transition point for the experimental system size (L=10). The scaling behavior of the participation entropy with system size confirmed the nature of the phase transitions.

The experimental results directly demonstrate the existence of three distinct phases and the phase transitions in the GAAH model, validating theoretical predictions. The use of a superconducting qubit processor with tunable coupling allowed for a precise and controlled study of many-body dynamics in this model. The observation of intermediate transport behavior and participation entropy in the critical phase highlights the unique characteristics of this phase, which cannot be captured by the traditional Anderson model, where any non-zero disorder would lead to localization. The agreement between experimental data and numerical simulations further reinforces the accuracy of the simulation and the effectiveness of the experimental methods. The work opens up exciting possibilities for exploring the critical phase further. Future studies could focus on larger systems to further investigate finite-size effects and explore the impact of many-body interactions on the phase diagram.

This study successfully simulated the generalized Aubry-André-Harper model's three phases using a superconducting quantum processor, directly observing and characterizing phase transitions through spin transport and participation entropy measurements. The tunable coupling architecture of the processor proved highly effective for generating the required Hamiltonian. The results agree well with theoretical predictions and demonstrate the power of superconducting quantum processors for exploring complex quantum many-body systems. Future research could focus on investigating the impact of many-body interactions on the critical phase and on exploring other models using similar tunable architectures.

The experiment is limited by the finite size of the superconducting qubit chain (10 qubits). Finite-size effects were observed, particularly a slight shift in the minimum participation entropy from the theoretical transition point. While decoherence effects were considered in simulations, more investigation could be done to explore the interplay between decoherence and the observed dynamics. The experimental accessible range of coupling strengths is limited by hardware constraints.