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Indian Journal of Pure & Applied Physics

Quantum Simulation of One-Dimensional Crystal Band Structure with a Tunable Superconducting Electric Circuit

Nikolay Berezkin

Published Jan 1, 2026• 0 citations
Publisher

Abstract

This paper presents a concept for a quantum simulator based on a superconducting qubit coupled to a readout resonator, designed to experimentally emulate the band structure of a one-dimensional crystal. The superconducting qubit, with its intrinsic periodic potential from the Josephson effect, serves as a direct analog of an electron in a periodic lattice. The Hamiltonian of the system is analyzed in two complementary regimes: the weak-coupling regime, corresponding to nearly free electrons, and the strong-coupling regime, corresponding to electrons in the tight-binding limit. We show that the system can be continuously swept between these regimes using a flux-tunable symmetric DC-SQUID to vary the effective Josephson energy. To probe the resulting band structure, we propose an experimental method based on two-tone spectroscopy, which maps the qubit's flux-dependent transition spectrum via the dispersive shift of a coupled resonator. Numerical simulations confirm the feasibility of this approach, visualizing the transition from a broad band dispersion in the weak-coupling limit to exponentially narrow bands in the strong-coupling limit. This platform demonstrates how tunable superconducting circuits can be used as versatile quantum simulators for fundamental solid-state physics phenomena.

DOI
10.56042/ijpap.v64i4.27155

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This paper presents a concept for a quantum simulator based on a superconducting qubit coupled to a readout resonator, designed to experimentally emulate the band structure of a one-dimensional crystal. The superconducting qubit, with its intrinsic periodic potential from the Josephson effect, serves as a direct analog of an electron in a periodic lattice. The Hamiltonian of the system is analyzed in two complementary regimes: the weak-coupling regime, corresponding to nearly free electrons, and the strong-coupling regime, corresponding to electrons in the tight-binding limit. We show that the system can be continuously swept between these regimes using a flux-tunable symmetric DC-SQUID to vary the effective Josephson energy. To probe the resulting band structure, we propose an experimental method based on two-tone spectroscopy, which maps the qubit's flux-dependent transition spectrum via the dispersive shift of a coupled resonator. Numerical simulations confirm the feasibility of this approach, visualizing the transition from a broad band dispersion in the weak-coupling limit to exponentially narrow bands in the strong-coupling limit. This platform demonstrates how tunable superconducting circuits can be used as versatile quantum simulators for fundamental solid-state physics phenomena.
Metamaterial resonant structures made from arrays of superconducting lumped-circuit elements can exhibit microwave-mode spectra with left-handed dispersion for the standing-wave resonances, resulting in a high density of modes in the same frequency range where superconducting qubits are typically operated, as well as a bandgap at lower frequencies that extends down to dc. Using this regime for multimode circuit quantum electrodynamics, we perform a series of measurements of such a superconducting metamaterial resonator coupled to a flux-tunable transmon qubit. Through microwave measurements of the metamaterial, we observe the coupling of the qubit to each of the modes that it passes through. Using a separate readout resonator, we probe the qubit dispersively and characterize the qubit energy relaxation as a function of frequency, which is strongly affected by the Purcell effect in the presence of the dense mode spectrum. Additionally, we investigate the ac Stark shift of the qubit as the photon number in the various metamaterial modes is varied. Through numerical simulations, we explore designs based on this scheme with enhanced coupling so that the coupling energy between the qubit and the metamaterial modes can exceed the intermode spacing in future devices with achievable circuit parameters. The ability to tailor the dense mode spectrum through the choice of circuit parameters and manipulate the photonic state of the metamaterial through interactions with qubits makes this a promising platform for analog quantum simulation with microwave photons and quantum memories.
Arrays of coupled superconducting qubits are analog quantum simulators able to emulate a wide range of tight-binding models in parameter regimes that are difficult to access or adjust in natural materials. In this work, we use a superconducting qubit array to emulate a tight-binding model on the rhombic lattice, which features flat bands. Enabled by broad adjustability of the dispersion of the energy bands and of on-site disorder, we examine regimes where flat-band localization and Anderson localization compete. We observe disorder-induced localization for dispersive bands and disorder-induced delocalization for flat bands. Remarkably, we find a sudden transition between the two regimes and, in its vicinity, the semblance of quantum critical scaling.
Analog quantum simulators can be used to study quantum correlation in novel many-body systems by emulating the Hamiltonian of these systems. One essential question in quantum simulation is to probe the properties of an emulated many-body system. Here we present a circuit QED scheme for probing such properties by measuring the spectrum of a superconducting resonator coupled to a quantum simulator. We first study a general framework of this approach and show that the spectrum of the resonator is directly related to the correlation function of the coupling operator between the resonator and the simulator. We then apply this scheme to a simulator of the transverse field Ising model implemented with superconducting qubits, where the resonance peaks in the resonator spectrum correspond to the frequencies of the elementary excitations. The effects of resonator damping, qubit decoherence, and resonator backaction are also discussed. This setup can be used to probe a broad range of many-body models.
Controlling the energy spectrum of quantum-coherent superconducting circuits, i.e., the energies of excited states, the circuit anharmonicity, and the states' charge dispersion, is essential for designing performant qubits. This control is usually achieved by adjusting the circuit's geometry. In situ control is traditionally obtained via an external magnetic field, in the case of tunnel Josephson junctions. More recently, semiconductor-weak-links-based Josephson junctions have emerged as an alternative building block with the advantage of tunability via the electric-field effect. Gate-tunable Josephson junctions have been succesfully integrated in superconducting circuits using, for instance, semiconducting nanowires or two-dimensional electron gases. In this Letter we demonstrate, in a graphene superconducting circuit, a large gate tunability of qubit properties: frequency, anharmonicity, and charge dispersion. We rationalize these features using a model considering the transmission of Cooper pairs through Andreev bound states. Noticeably, we show that the high transmission of Cooper pairs in such weak link strongly suppresses the charge dispersion. Our Letter illustrates the potential for graphene-based qubits as versatile building blocks in advanced quantum circuits.
We study a hybrid quantum system consisting of spin ensembles and superconducting flux qubits, where each spin ensemble is realized using the nitrogen-vacancy centers in a diamond crystal and the nearest-neighbor spin ensembles are effectively coupled via a flux qubit. We show that the coupling strengths between flux qubits and spin ensembles can reach the strong and even ultrastrong coupling regimes by either engineering the hybrid structure in advance or tuning the excitation frequencies of spin ensembles via external magnetic fields. When extending the hybrid structure to an array with equal coupling strengths, we find that in the strong-coupling regime, the hybrid array is reduced to a tight-binding model of a one-dimensional bosonic lattice. In the ultrastrong-coupling regime, it exhibits quasiparticle excitations separated from the ground state by an energy gap. Moreover, these quasiparticle excitations and the ground state are stable under a certain condition that is tunable via the external magnetic field. This may provide an experimentally accessible method to probe the instability of the system.
Tunable Josephson harmonics open new avenues for qubit design. We demonstrate a superconducting circuit element consisting of a tunnel junction in series with a SQUID loop, yielding a Josephson potential whose harmonic content is strongly tunable by magnetic flux. Through spectroscopy of the first four qubit transitions, together with an effective single-mode model renormalized by the internal mode, we resolve a second harmonic with an amplitude up to ∼10%\sim10\%∼10% of the fundamental. We identify a flux sweet spot where the dispersive shift vanishes, achieved by balancing the dispersive couplings to the internal and qubit modes. This highly tunable element provides a route toward protected qubits and customizable nonlinear microwave devices.