Electron transport in solids is a fundamental topic in condensed matter physics. Bloch and Zener predicted, approximately 80 years ago, that under a constant force, electrons in a crystal lattice would oscillate and localize, exhibiting Bloch oscillations (BOs) and Wannier-Stark localization (WSL) respectively. These phenomena are intrinsically quantum mechanical, showcasing the wave-like nature of electrons. However, direct observation in bulk materials is challenging due to short coherence times. Experimental observation of BOs and WSL was first achieved in semiconductor superlattices in the 1990s, but relaxation times remained a limiting factor. Advances in quantum technology offer alternative platforms for simulating BOs and WSL, such as cold atoms and photonic waveguide arrays. These artificial systems offer significantly longer coherence times compared to semiconductor superlattices. Superconducting circuits, with their scalability, long coherence times, and precise control, have emerged as a powerful platform for quantum simulation experiments, including quantum many-body dynamics, quantum chemistry, and quantum algorithms. This work leverages a 1D array of superconducting qubits to study the essential transport properties of spin and energy in BOs and WSL, focusing particularly on energy transport, a feature missing from previous simulations, which necessitates multi-qubit simultaneous readout capabilities for acquiring nearest-neighbor two-site correlations. The experiment uses a 5-qubit superconducting processor to investigate BOs and WSL in a spin system, modeled using an isotropic XY chain. By precisely tuning qubit frequencies, a linear potential is created. The researchers anticipated observing suppressed spin propagation and localized oscillations under this potential, confirming the BO and WSL characteristics. Furthermore, they aimed to quantify the WSL length and study its relationship to the potential gradient, and analyze the impact of the linear potential on thermal transport in the system.

Numerous studies have explored Bloch oscillations and Wannier-Stark localization in various systems. Early experimental work focused on semiconductor superlattices, demonstrating the existence of these phenomena but hampered by short coherence times. The advent of quantum simulators provided opportunities to study these effects in systems with extended coherence, such as cold atom experiments. Studies using cold atoms in optical lattices have shown clear evidence of Bloch oscillations and their dependence on interaction strength and lattice parameters. Photonic waveguide arrays have also been utilized, providing an alternative platform to investigate the linear and nonlinear regimes of optical Bloch oscillations. However, many of these previous studies lacked the ability to directly measure energy transport. This work addresses this gap by utilizing a superconducting quantum processor with multi-qubit readout capabilities, enabling a comprehensive investigation of both spin and energy transport in the context of BOs and WSL.

The experiment utilized a 5-qubit superconducting processor arranged in a 1D chain, featuring high-precision simultaneous readouts and full control over each qubit. The system's Hamiltonian was described by a 1D Bose-Hubbard model, which, under specific conditions (large and staggered on-site interaction, U), reduced to an effective isotropic XY model. This simplification allowed for a two-dimensional Fock space truncation at each qubit, effectively representing the system as a spin-1/2 chain. A linear potential was implemented by linearly varying the qubit frequencies along the chain. Two main experimental sequences were used: one to study spin transport and another for thermal transport. For spin transport, the leftmost qubit was initially excited to |1⟩, and the subsequent time evolution of the system was monitored by measuring the probability distribution of the |1⟩ state at each qubit. For thermal transport, the initial state was prepared to create a kinetic energy gradient between the edges of the chain, using a combination of X/2 gates. The time evolution of the kinetic energy density at both edges was then measured through simultaneous two-qubit readout. The WSL length was extracted by fitting the time-evolution of the maximum photon occupancy probability at the furthest qubit using a Gaussian function. The relationship between WSL length and the potential gradient was analyzed. The numerical simulations involved solving the Lindblad master equation, incorporating experimentally calibrated decoherence and dephasing parameters. Experimental data were processed and analyzed by averaging multiple single-shot measurements, and errors were estimated using standard statistical methods.

The experiments demonstrated the characteristic features of Bloch oscillations and Wannier-Stark localization. When a linear potential was applied, spin transport was significantly suppressed, with the spin exhibiting oscillations near its initial position. The observed oscillation frequency was consistent with theoretical predictions and was much slower than the decoherence time of the qubits. A novel method was employed to extract the WSL length from the maximum probability of a photon propagating to the opposite end of the chain in the presence of the linear potential. This approach successfully revealed an inverse relationship between the WSL length and the potential gradient, in agreement with theoretical expectations. The numerical simulations accurately reproduced the experimental observations, validating the theoretical model and experimental techniques. Furthermore, the simultaneous readout capabilities of the superconducting processor allowed for the investigation of thermal transport. The results showed that the linear potential also suppressed thermal transport, preventing the free exchange of kinetic energy between the ends of the chain, mirroring the behavior of spin transport. This observation suggests a connection between spin and energy transport in this system, consistent with the Wiedemann-Franz law observed in classical systems.

The findings of this study provide strong experimental evidence for Bloch oscillations and Wannier-Stark localization in a superconducting quantum processor. The ability to directly observe and quantify these phenomena, particularly the inverse relationship between WSL length and potential gradient, confirms the theoretical understanding of these quantum transport effects. The successful demonstration of suppressed thermal transport further enriches the picture and reveals a link between energy and spin transport under the influence of a linear potential. The experimental approach offers a significant advantage over previous studies with other quantum simulators due to the simultaneous readout capability of the superconducting qubits, allowing for the measurement of energy transport directly. This work establishes the superconducting quantum processor as a versatile platform for exploring out-of-equilibrium quantum phenomena, opening avenues for future investigations.

This paper presents the first experimental observation of Bloch oscillations and Wannier-Stark localization in a superconducting quantum processor, confirming theoretical predictions. The observed inverse correlation between WSL length and potential gradient, combined with the evidence of suppressed thermal transport, highlights the power of this platform for simulating complex quantum phenomena. Future research could explore more complex scenarios, such as the impact of disorder potentials, engineered noise, and interactions, potentially leading to the observation of Stark many-body localization. Scaling the system to include a larger number of qubits with improved coherence would further enhance the possibilities for simulating a wide range of out-of-equilibrium many-body problems.

The current study is limited to a 5-qubit system. While this allows for the observation of BOs and WSL, scaling up to larger systems would be beneficial for investigating the interplay between interactions and localization. The experimental techniques, while precise, are still subject to inherent noise and decoherence effects in the superconducting qubits. Further improvements in qubit coherence and control could lead to more accurate measurements and the exploration of longer time scales. Finally, the current study primarily focuses on a 1D system, and future work could explore higher dimensional systems to further expand the scope of this research.