Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.16628

Quantum Simulation of Two-Dimensional Free Dirac Hamiltonian in Multimode Circuit QED

Jiwon Kang, Jiuk Lee, Eliya Blumenthal, Shay Hacohen-Gourgy, Eunseong Kim

quant-phSubmitted Sep 15, 2026• Announced Sep 16, 2026• 0 citations
arXivHTMLPDF

Abstract

Two-dimensional massive Dirac systems feature a gapped Dirac-cone dispersion central to a broad range of phenomena in condensed-matter and topological physics. While quantum simulations have demonstrated one-dimensional Dirac dynamics and two-dimensional massless Weyl dynamics, programmable simulation of massive two-dimensional Dirac dynamics remains experimentally unexplored. Here, we realize a programmable two-dimensional free Dirac Hamiltonian in circuit QED, with independently tunable spin-momentum couplings and mass, using a single Rabi-driven qubit coupled to two modes of a multimode cavity. Using this Hamiltonian, we observe rotational Zitterbewegung of a two-dimensional massive Dirac particle and its dependence on the effective mass. Time-dependent master-equation simulations incorporating measured decoherence, corrections beyond the rotating-wave approximation (RWA), and anharmonicity of the transmon reproduce the observed dynamics. Our results establish multimode circuit QED as a compact, programmable platform for higher-dimensional relativistic quantum dynamics and provide a foundation for exploring dynamical and topological phenomena in gapped Dirac systems.

Comments
8 pages, 4 figures; Supplemental Material included

Related papers

Analog Simulation of Massive Relativistic Quantum Fields in 2 + 1 Dimensions

Yansheng Zhang, Feiyang Wang, Paul H. C. Wong +8 more

Quantum field theories provide fundamental models of complex interacting systems, from high-energy physics and cosmology to condensed matter. However, solving these models in non-perturbative and dynamical regimes is often challenging, particularly in more than one spatial dimension. Analog simulation using tunable synthetic quantum systems can both verify existing theoretical predictions and lead to new physical insights. Here, we realize analog simulation of massive relativistic quantum fields in $2+1$ dimensions (two spatial dimensions and time), using two coherently coupled spin components in a uniform two-dimensional atomic Bose--Einstein condensate. Specifically, we encode the paradigmatic sine-Gordon model in the long-wavelength modes of the field describing the relative phase, $φ$, of the two components. In the perturbative regime, collective field excitations exhibit a relativistic dispersion with a tunable mass gap. We also observe explicitly non-perturbative phenomena, including the existence of topological domain walls across which $φ$ rapidly winds by $2π$. Our work opens possibilities for studies of cosmologically relevant phenomena including preheating, dynamics of topological defects, and relativistic false-vacuum decay.

cond-mat.quant-gasCross-listedarXiv:2603.08840•Mar 11, 2026•0 citations

Multimode Strong-Coupling Processes in Circuit QED Lattices

Won Chan Lee, Ali Fahimniya, Kellen O'Brien +5 more

Circuit QED systems provide an ideal platform for exploring the strong-coupling regime of multimode cavity QED. Here we present two new phenomena from multimode strong coupling: a circuit Lagrangian analysis which captures beyond tight-binding effects of strong photon-photon coupling and experimental observation of strong wave-mixing resonances in the qubit response. Our circuit analysis reveals qualitatively new features such as emergent band gaps, lifted degeneracies, broadened flat bands, and frequency-dependent hopping. Within the multimode photon environment, strong qubit-photon coupling in turn gives rise to multiphoton processes involving multiple normal modes. We demonstrate a strong four-wave-mixing process involving excitation of a qubit and simultaneous frequency conversion between modes. Notably, this wave-mixing process is dominated by localized flat-band modes of the photonic lattice, which exhibit the strongest coupling to the transmon qubit.

quant-pharXiv:2605.00983•May 5, 2026•0 citations

Dirac particle dynamics of a superconducting circuit

Elisha Svetitsky, Nadav Y. Katz

The core concept of quantum simulation is the mapping of an inaccessible quantum system onto a controllable one by identifying analogous dynamics. We map the Dirac equation of relativistic quantum mechanics in $3+1$ dimensions onto a multilevel superconducting Josephson circuit. Resonant drives determine the particle mass and momentum, and the quantum state represents the internal spinor dynamics which are cast in the language of multilevel quantum optics. Zitterbewegung and wave-packet dynamics are reproduced. The degeneracy of the Dirac spectrum corresponds to a degeneracy of bright and dark states within the system, and particle spin and helicity are employed to interpret the multilevel dynamics. We also simulate the Schwinger mechanism of electron-positron pair production by introducing an analogous electric field as a doubly degenerate Landau-Zener problem. All proposed measurements can be performed well within typical decoherence times. This work opens an avenue for experimental study of the Dirac equation and provides a tool for control of complex dynamics in multilevel systems.

arXiv:1710.10797•Physical Review A•Apr 3, 2019•3 citations

Experimental simulation of Dirac equation in superconducting qubits

Yu Zhang, Jianwen Xu, Zhuang Ma +4 more

The Dirac equation elucidates fundamental physical principles and predicts significant physical phenomena. Its simulation further offers explanations for subsequent physics observations. We map the 3+1-dimensional Dirac equation onto the first excited manifold of four qubits to simulate the internal four-component spinor dynamics. Additionally, we simulate two intriguing and challenging-to-implement physical phenomena: the Zitterbewegung effect and the Sauter-Schwinger effect. Compared to previous simulations, our system provides a larger parameter space, allowing for a more comprehensive exploration of physical properties. This work enhances our understanding of spinor dynamics. The Dirac equation, one of the most fundamental equations in relativistic quantum mechanics, has influenced diverse research fields. The authors demonstrate an analogue simulation of the 3+1-dimensional Dirac equation by applying parametric modulations in superconducting circuits.

Communications Physics•Jun 12, 2025•3 citations

Oscillatory multiband dynamics of free particles: The ubiquity ofzitterbewegungeffects

R. Winkler, Ulrich Zülicke, Jens Bolte

In the Dirac theory for the motion of free relativistic electrons, highly oscillatory components appear in the time evolution of physical observables such as position, velocity, and spin angular momentum. This effect is known as zitterbewegung. We present a theoretical analysis of rather different Hamiltonians with gapped and/or spin-split energy spectrum (including the Rashba, Luttinger, and Kane Hamiltonians) that exhibit analogs of zitterbewegung as a common feature. We find that the amplitude of oscillations of the Heisenberg velocity operator $\mathbf{v}(t)$ generally equals the uncertainty for a simultaneous measurement of two linearly independent components of $\mathbf{v}$. It is also shown that many features of zitterbewegung are shared by the simple and well-known Landau Hamiltonian, describing the dynamics of two-dimensional (2D) electron systems in the presence of a magnetic field perpendicular to the plane. Finally, we also discuss the oscillatory dynamics of 2D electrons arising from the interplay of Rashba spin splitting and a perpendicular magnetic field.

arXiv:cond-mat/0609005•Physical Review B•May 10, 2007•70 citations

Heisenberg-Langevin Formalism For Open Circuit-QED Systems

Moein Malekakhlagh

We present a Heisenberg-Langevin formalism to study the effective dynamics of a superconducting qubit coupled to an open multimode resonator, without resorting to the rotating wave, two level, Born or Markov approximations. Our effective equations are derived by eliminating resonator degrees of freedom while encoding their effect in the Green's function of the electromagnetic background. We account for the openness of the resonator exactly by employing a spectral representation for the Green's function in terms of a set of non-Hermitian modes. A well-behaved time domain perturbation theory is derived to systematically account for the nonlinearity of weakly nonlinear qubits like transmon. We apply this method to the problem of spontaneous emission, capturing accurately the non-Markovian features of the qubit dynamics, valid for any qubit-resonator coupling strength. Any discrete-level quantum system coupled to the electromagnetic continuum is subject to radiative decay and renormalization of its energy levels. When inside a cavity, these quantities can be strongly modified with respect to vacuum. Generally, this modification can be captured by including only the closest resonant cavity mode. In circuit-QED architecture, with substantial coupling strengths, it is however found that such rates are strongly influenced by far off-resonant modes. A multimode calculation over the infinite set of cavity modes leads to divergences unless an artificial cutoff is imposed. Previous studies have not pointed out what the source of this divergence is. We show that unless the effect of $A^2$ is accounted for up to all orders exactly, any multimode calculations of circuit-QED quantities is bound to diverge. Subsequently, we present the calculation of finite radiative corrections to qubit properties that is free of an artificially introduced high frequency cut-off.

arXiv:1711.05699•Nov 11, 2017•0 citations
Two-dimensional massive Dirac systems feature a gapped Dirac-cone dispersion central to a broad range of phenomena in condensed-matter and topological physics. While quantum simulations have demonstrated one-dimensional Dirac dynamics and two-dimensional massless Weyl dynamics, programmable simulation of massive two-dimensional Dirac dynamics remains experimentally unexplored. Here, we realize a programmable two-dimensional free Dirac Hamiltonian in circuit QED, with independently tunable spin-momentum couplings and mass, using a single Rabi-driven qubit coupled to two modes of a multimode cavity. Using this Hamiltonian, we observe rotational Zitterbewegung of a two-dimensional massive Dirac particle and its dependence on the effective mass. Time-dependent master-equation simulations incorporating measured decoherence, corrections beyond the rotating-wave approximation (RWA), and anharmonicity of the transmon reproduce the observed dynamics. Our results establish multimode circuit QED as a compact, programmable platform for higher-dimensional relativistic quantum dynamics and provide a foundation for exploring dynamical and topological phenomena in gapped Dirac systems.
Quantum field theories provide fundamental models of complex interacting systems, from high-energy physics and cosmology to condensed matter. However, solving these models in non-perturbative and dynamical regimes is often challenging, particularly in more than one spatial dimension. Analog simulation using tunable synthetic quantum systems can both verify existing theoretical predictions and lead to new physical insights. Here, we realize analog simulation of massive relativistic quantum fields in 2+12+12+1 dimensions (two spatial dimensions and time), using two coherently coupled spin components in a uniform two-dimensional atomic Bose--Einstein condensate. Specifically, we encode the paradigmatic sine-Gordon model in the long-wavelength modes of the field describing the relative phase, φφφ, of the two components. In the perturbative regime, collective field excitations exhibit a relativistic dispersion with a tunable mass gap. We also observe explicitly non-perturbative phenomena, including the existence of topological domain walls across which φφφ rapidly winds by 2π2π2π. Our work opens possibilities for studies of cosmologically relevant phenomena including preheating, dynamics of topological defects, and relativistic false-vacuum decay.
Circuit QED systems provide an ideal platform for exploring the strong-coupling regime of multimode cavity QED. Here we present two new phenomena from multimode strong coupling: a circuit Lagrangian analysis which captures beyond tight-binding effects of strong photon-photon coupling and experimental observation of strong wave-mixing resonances in the qubit response. Our circuit analysis reveals qualitatively new features such as emergent band gaps, lifted degeneracies, broadened flat bands, and frequency-dependent hopping. Within the multimode photon environment, strong qubit-photon coupling in turn gives rise to multiphoton processes involving multiple normal modes. We demonstrate a strong four-wave-mixing process involving excitation of a qubit and simultaneous frequency conversion between modes. Notably, this wave-mixing process is dominated by localized flat-band modes of the photonic lattice, which exhibit the strongest coupling to the transmon qubit.
The core concept of quantum simulation is the mapping of an inaccessible quantum system onto a controllable one by identifying analogous dynamics. We map the Dirac equation of relativistic quantum mechanics in 3+13+13+1 dimensions onto a multilevel superconducting Josephson circuit. Resonant drives determine the particle mass and momentum, and the quantum state represents the internal spinor dynamics which are cast in the language of multilevel quantum optics. Zitterbewegung and wave-packet dynamics are reproduced. The degeneracy of the Dirac spectrum corresponds to a degeneracy of bright and dark states within the system, and particle spin and helicity are employed to interpret the multilevel dynamics. We also simulate the Schwinger mechanism of electron-positron pair production by introducing an analogous electric field as a doubly degenerate Landau-Zener problem. All proposed measurements can be performed well within typical decoherence times. This work opens an avenue for experimental study of the Dirac equation and provides a tool for control of complex dynamics in multilevel systems.
The Dirac equation elucidates fundamental physical principles and predicts significant physical phenomena. Its simulation further offers explanations for subsequent physics observations. We map the 3+1-dimensional Dirac equation onto the first excited manifold of four qubits to simulate the internal four-component spinor dynamics. Additionally, we simulate two intriguing and challenging-to-implement physical phenomena: the Zitterbewegung effect and the Sauter-Schwinger effect. Compared to previous simulations, our system provides a larger parameter space, allowing for a more comprehensive exploration of physical properties. This work enhances our understanding of spinor dynamics. The Dirac equation, one of the most fundamental equations in relativistic quantum mechanics, has influenced diverse research fields. The authors demonstrate an analogue simulation of the 3+1-dimensional Dirac equation by applying parametric modulations in superconducting circuits.
In the Dirac theory for the motion of free relativistic electrons, highly oscillatory components appear in the time evolution of physical observables such as position, velocity, and spin angular momentum. This effect is known as zitterbewegung. We present a theoretical analysis of rather different Hamiltonians with gapped and/or spin-split energy spectrum (including the Rashba, Luttinger, and Kane Hamiltonians) that exhibit analogs of zitterbewegung as a common feature. We find that the amplitude of oscillations of the Heisenberg velocity operator v(t)\mathbf{v}(t)v(t) generally equals the uncertainty for a simultaneous measurement of two linearly independent components of v\mathbf{v}v. It is also shown that many features of zitterbewegung are shared by the simple and well-known Landau Hamiltonian, describing the dynamics of two-dimensional (2D) electron systems in the presence of a magnetic field perpendicular to the plane. Finally, we also discuss the oscillatory dynamics of 2D electrons arising from the interplay of Rashba spin splitting and a perpendicular magnetic field.
We present a Heisenberg-Langevin formalism to study the effective dynamics of a superconducting qubit coupled to an open multimode resonator, without resorting to the rotating wave, two level, Born or Markov approximations. Our effective equations are derived by eliminating resonator degrees of freedom while encoding their effect in the Green's function of the electromagnetic background. We account for the openness of the resonator exactly by employing a spectral representation for the Green's function in terms of a set of non-Hermitian modes. A well-behaved time domain perturbation theory is derived to systematically account for the nonlinearity of weakly nonlinear qubits like transmon. We apply this method to the problem of spontaneous emission, capturing accurately the non-Markovian features of the qubit dynamics, valid for any qubit-resonator coupling strength. Any discrete-level quantum system coupled to the electromagnetic continuum is subject to radiative decay and renormalization of its energy levels. When inside a cavity, these quantities can be strongly modified with respect to vacuum. Generally, this modification can be captured by including only the closest resonant cavity mode. In circuit-QED architecture, with substantial coupling strengths, it is however found that such rates are strongly influenced by far off-resonant modes. A multimode calculation over the infinite set of cavity modes leads to divergences unless an artificial cutoff is imposed. Previous studies have not pointed out what the source of this divergence is. We show that unless the effect of A2A^2A2 is accounted for up to all orders exactly, any multimode calculations of circuit-QED quantities is bound to diverge. Subsequently, we present the calculation of finite radiative corrections to qubit properties that is free of an artificially introduced high frequency cut-off.