© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Advanced Quantum Technologies

Hofstadter Butterfly Variants in Inhomogeneous Magnetic Fields: Quantum Simulation with Superconducting Circuits

Xin Liu, Jun‐Yao Li, Guoqiang Zhang, Lijiong Shen, Y Kang, Wei Feng

Published Mar 1, 2026• 0 citations

Abstract

ABSTRACT Quantum simulation offers an effective approach to explore novel phenomena in complex quantum systems, such as the famous Hofstadter butterfly that are inaccessible in conventional natural systems. We investigate variants of the Hofstadter butterfly arising in lattices subject to inhomogeneous magnetic fields. Using a superconducting‐circuit platform, we implement a zigzag spin chain and a two‐leg spin ladder, where spatially modulated interaction phases produce position‐dependent magnetic fluxes through the plaquettes. By periodically modulating the resonance frequency of superconducting qubits, we engineer the desired synthetic magnetic field. The resulting single‐excitation dynamics map onto a tight‐binding model of a charged particle moving in a non‐uniform magnetic field, enabling the exploration of fractal energy spectra beyond the uniform‐field case. We numerically simulate the system dynamics using experimentally realistic parameters and reveal variants of the Hofstadter butterfly through a spectroscopic method. The simulation results demonstrate a high degree of agreement with theoretical predictions. Our proposal provides a promising route for investigating exotic fractal energy spectra via quantum simulation.

DOI: 10.1002/qute.202500943

Related papers

Quantum simulation of Hofstadter butterfly with synthetic gauge fields on two-dimensional superconducting-qubit lattices

Wei Feng, Dexi Shao, Guo-Qiang Zhang +3 more

Motivated by recent realizations of two-dimensional (2D) superconducting-qubit lattices, we propose a protocol to simulate Hofstadter butterfly with synthetic gauge fields in superconducting circuits. Based on the existing 2D superconducting-qubit lattices, we construct a generalized Hofstadter model on zigzag lattices, which has a fractal energy spectrum similar to the original Hofstadter butterfly. By periodically modulating the resonant frequencies of qubits, we engineer a synthetic gauge field to mimic the generalized Hofstadter Hamiltonian. A spectroscopic method is used to demonstrate the Hofstadter butterfly from the time evolutions of experimental observables. We numerically simulate the dynamics of the system with realistic parameters, and the results show a butterfly spectrum clearly. Our proposal provides a promising way to realize the Hofstadter butterfly on the latest 2D superconducting-qubit lattices and will stimulate the quantum simulation of novel properties induced by magnetic fields in superconducting circuits.

arXiv:2306.11245•Frontiers of Physics•Jul 1, 2023•4 citations

Hofstadter butterfly in a cavity-induced dynamic synthetic magnetic field

Elvia Colella, Farokh Mivehvar, Francesco Piazza +1 more

Energy bands of electrons in a square lattice potential threaded by a uniform magnetic field exhibit a fractal structure known as the Hofstadter butterfly. Here we study a Fermi gas in a 2D optical lattice within a linear cavity with a tilt along the cavity axis. The hopping along the cavity axis is only induced by resonant Raman scattering of transverse pump light into a standing-wave-cavity mode. Choosing a suitable pump geometry allows us to realize the Hofstadter-Harper model with a cavity-induced dynamical synthetic magnetic field, which appears at the onset of the superradiant phase transition. The dynamical nature of this cavity-induced synthetic magnetic field arises from the delicate interplay between collective superradiant scattering and the underlying fractal band structure. Using a sixth-order expansion of the free energy as a function of the order parameter and by numerical simulations, we show that at low magnetic fluxes the superradiant ordering phase transition is first order, while it becomes second order for higher flux. The dynamic nature of the magnetic field induces a nontrivial deformation of the Hofstadter butterfly in the superradiant phase. At strong pump far above the self-ordering threshold, we recover the Hofstadter butterfly one would obtain in a static magnetic field.

arXiv:1909.05975•Physical review. B./Physical review. B•Dec 23, 2019•20 citations

Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms

Dieter Jaksch, P. Zoller

We investigate the dynamics of neutral atoms in a 2D optical lattice which traps two distinct internal states of the atoms in different columns. Two Raman lasers are used to coherently transfer atoms from one internal state to the other, thereby causing hopping between the different columns. By adjusting the laser parameters appropriately we can induce a non-vanishing phase of particles moving along a closed path on the lattice. This phase is proportional to the enclosed area and we thus simulate a magnetic flux through the lattice. This set-up is described by a Hamiltonian identical to the one for electrons on a lattice subject to a magnetic field and thus allows us to study this equivalent situation under very well defined controllable conditions. We consider the limiting case of huge magnetic fields—which is not experimentally accessible for electrons in metals—where a fractal band structure, the Hofstadter butterfly, characterizes the system.

arXiv:quant-ph/0304038•New Journal of Physics•May 30, 2003•777 citations

Light-Induced Hofstadter's Butterfly Spectrum in Optical Lattices

Jing-Min Hou

We propose a scheme to create an effective magnetic field, which can be perceived by cold neutral atoms in a two-dimensional optical lattice, with a laser field with a space-dependent phase and a conventional laser field acting on Λ-type three-level atoms. When the dimensionless parameter α, being the ratio of flux through a lattice cell to one flux quantum, is rational, the energy spectrum shows a fractal band structure, which is so-called Hofstadter's butterfly.

Communications in Theoretical Physics•Mar 1, 2009•1 citation

Coherence of interacting bosons in optical lattices in synthetic magnetic fields with a large number of subbands

B. Grygiel, K. Patucha, T. A. Zaleski

We study the behavior of interacting ultracold bosons in optical lattices in synthetic magnetic fields with wide range of in-cell fluxes $\ensuremath{\alpha}=p/q$. The problem is similar to the one of an electron moving in a tight-binding scheme in the magnetic field and becomes difficult to tackle for a growing number of magnetic subbands, $q$. To overcome this, we focus on the interplay of the width, shape, and number of the subbands on the formation of the coherent state of cold bosons. Using the quantum rotor approach, which goes beyond the mean-field approximation, we are able to pinpoint the elements of the band structure, which are the most significant in a proper theoretical description of the synthetic magnetic field in a bosonic lattice system. As a result, we propose a method of reconstruction of the Hofstadter butterfly spectrum by replacing the magnetic subbands with renormalized bands of a square lattice. This allows us to effectively investigate the properties of the studied system for a wide range of magnetic fluxes and their impact on the Mott-insulator--superfluid transition.

Physical Review A•May 9, 2016•6 citations

Spectroscopic signatures of localization with interacting photons in superconducting qubits

P. Roushan, C. Neill, Jirawat Tangpanitanon +23 more

Quantized eigenenergies and their associated wave functions provide extensive information for predicting the physics of quantum many-body systems. Using a chain of nine superconducting qubits, we implement a technique for resolving the energy levels of interacting photons. We benchmark this method by capturing the main features of the intricate energy spectrum predicted for two-dimensional electrons in a magnetic field-the Hofstadter butterfly. We introduce disorder to study the statistics of the energy levels of the system as it undergoes the transition from a thermalized to a localized phase. Our work introduces a many-body spectroscopy technique to study quantum phases of matter.

arXiv:1709.07108•Science•Nov 30, 2017•459 citations

ABSTRACT Quantum simulation offers an effective approach to explore novel phenomena in complex quantum systems, such as the famous Hofstadter butterfly that are inaccessible in conventional natural systems. We investigate variants of the Hofstadter butterfly arising in lattices subject to inhomogeneous magnetic fields. Using a superconducting‐circuit platform, we implement a zigzag spin chain and a two‐leg spin ladder, where spatially modulated interaction phases produce position‐dependent magnetic fluxes through the plaquettes. By periodically modulating the resonance frequency of superconducting qubits, we engineer the desired synthetic magnetic field. The resulting single‐excitation dynamics map onto a tight‐binding model of a charged particle moving in a non‐uniform magnetic field, enabling the exploration of fractal energy spectra beyond the uniform‐field case. We numerically simulate the system dynamics using experimentally realistic parameters and reveal variants of the Hofstadter butterfly through a spectroscopic method. The simulation results demonstrate a high degree of agreement with theoretical predictions. Our proposal provides a promising route for investigating exotic fractal energy spectra via quantum simulation.

Motivated by recent realizations of two-dimensional (2D) superconducting-qubit lattices, we propose a protocol to simulate Hofstadter butterfly with synthetic gauge fields in superconducting circuits. Based on the existing 2D superconducting-qubit lattices, we construct a generalized Hofstadter model on zigzag lattices, which has a fractal energy spectrum similar to the original Hofstadter butterfly. By periodically modulating the resonant frequencies of qubits, we engineer a synthetic gauge field to mimic the generalized Hofstadter Hamiltonian. A spectroscopic method is used to demonstrate the Hofstadter butterfly from the time evolutions of experimental observables. We numerically simulate the dynamics of the system with realistic parameters, and the results show a butterfly spectrum clearly. Our proposal provides a promising way to realize the Hofstadter butterfly on the latest 2D superconducting-qubit lattices and will stimulate the quantum simulation of novel properties induced by magnetic fields in superconducting circuits.

Energy bands of electrons in a square lattice potential threaded by a uniform magnetic field exhibit a fractal structure known as the Hofstadter butterfly. Here we study a Fermi gas in a 2D optical lattice within a linear cavity with a tilt along the cavity axis. The hopping along the cavity axis is only induced by resonant Raman scattering of transverse pump light into a standing-wave-cavity mode. Choosing a suitable pump geometry allows us to realize the Hofstadter-Harper model with a cavity-induced dynamical synthetic magnetic field, which appears at the onset of the superradiant phase transition. The dynamical nature of this cavity-induced synthetic magnetic field arises from the delicate interplay between collective superradiant scattering and the underlying fractal band structure. Using a sixth-order expansion of the free energy as a function of the order parameter and by numerical simulations, we show that at low magnetic fluxes the superradiant ordering phase transition is first order, while it becomes second order for higher flux. The dynamic nature of the magnetic field induces a nontrivial deformation of the Hofstadter butterfly in the superradiant phase. At strong pump far above the self-ordering threshold, we recover the Hofstadter butterfly one would obtain in a static magnetic field.

We investigate the dynamics of neutral atoms in a 2D optical lattice which traps two distinct internal states of the atoms in different columns. Two Raman lasers are used to coherently transfer atoms from one internal state to the other, thereby causing hopping between the different columns. By adjusting the laser parameters appropriately we can induce a non-vanishing phase of particles moving along a closed path on the lattice. This phase is proportional to the enclosed area and we thus simulate a magnetic flux through the lattice. This set-up is described by a Hamiltonian identical to the one for electrons on a lattice subject to a magnetic field and thus allows us to study this equivalent situation under very well defined controllable conditions. We consider the limiting case of huge magnetic fields—which is not experimentally accessible for electrons in metals—where a fractal band structure, the Hofstadter butterfly, characterizes the system.

We study the behavior of interacting ultracold bosons in optical lattices in synthetic magnetic fields with wide range of in-cell fluxes $\ensuremath{\alpha}=p/q$. The problem is similar to the one of an electron moving in a tight-binding scheme in the magnetic field and becomes difficult to tackle for a growing number of magnetic subbands,

q

. To overcome this, we focus on the interplay of the width, shape, and number of the subbands on the formation of the coherent state of cold bosons. Using the quantum rotor approach, which goes beyond the mean-field approximation, we are able to pinpoint the elements of the band structure, which are the most significant in a proper theoretical description of the synthetic magnetic field in a bosonic lattice system. As a result, we propose a method of reconstruction of the Hofstadter butterfly spectrum by replacing the magnetic subbands with renormalized bands of a square lattice. This allows us to effectively investigate the properties of the studied system for a wide range of magnetic fluxes and their impact on the Mott-insulator--superfluid transition.

Quantized eigenenergies and their associated wave functions provide extensive information for predicting the physics of quantum many-body systems. Using a chain of nine superconducting qubits, we implement a technique for resolving the energy levels of interacting photons. We benchmark this method by capturing the main features of the intricate energy spectrum predicted for two-dimensional electrons in a magnetic field-the Hofstadter butterfly. We introduce disorder to study the statistics of the energy levels of the system as it undergoes the transition from a thermalized to a localized phase. Our work introduces a many-body spectroscopy technique to study quantum phases of matter.