Programmable spectral symmetries in an anisotropic quantum Rabi simulator

Technical Summary: Programmable Spectral Symmetries in an Anisotropic Quantum Rabi Simulator

Problem Statement
The Quantum Rabi Model (QRM) describes the fundamental interaction between a two-level system and a single bosonic mode. While the isotropic QRM, where rotating and counterrotating processes are locked together, has been extensively studied experimentally, the broader symmetry landscape remains largely unexplored. Specifically, the independent control of rotating ($g_1$) and counterrotating ($g_2$) couplings, and the resulting anisotropy, dictates the model's conserved quantities, selection rules, and level crossings. Existing experiments have largely focused on the isotropic limit or the Jaynes-Cummings (JCM) limit, leaving open critical questions regarding how anisotropy reorganizes the spectrum, how symmetry-protected crossings emerge or vanish, and whether hidden symmetries can be dynamically detected within a single programmable device.

Methodology
The authors realize a programmable anisotropic quantum Rabi model (AQRM) using a superconducting quantum processor. The experimental platform consists of a flip-chip architecture with a central bus resonator and 24 frequency-tunable transmon qubits. The system utilizes one transmon as the system qubit ($Q$) coupled to the resonator ($R$) via a tunable coupler ($C$).

The core innovation lies in the engineering of the light-matter interaction Hamiltonian:
$$\hat{H} = \frac{\Omega}{2} \hat{\sigma}_z + \omega \hat{a}^\dagger \hat{a} + g_1 (\hat{H}_r + \lambda \hat{H}_{cr}) + \frac{\varepsilon}{2} \hat{\sigma}_x$$
where $\hat{H}_r$ and $\hat{H}_{cr}$ represent rotating and counterrotating interactions, respectively. The anisotropy ratio $\lambda = g_2/g_1$ and the transverse bias $\varepsilon$ are independently controlled.

The control mechanism combines longitudinal qubit frequency modulation ($\tilde{\omega}_q$) with a multi-tone transverse drive ($E$). In the rotating frame, the carrier tone sets the bias $\varepsilon$, while the transverse modulation renormalizes the effective coupling strengths of the two field quadratures via a Bessel factor $J_0(2\beta)$. This allows continuous tuning of $\lambda$ from 0 (JCM limit) to values $>1$ (anti-JCM limit). To access the full parameter space, including $\lambda > 1$, the authors employ a duality mapping: $\hat{H}(\omega, \Omega, g_1, g_2, \varepsilon) = \hat{\sigma}_x \hat{H}(\omega, -\Omega, g_2, g_1, \varepsilon) \hat{\sigma}_x$. This allows the simulation of high-anisotropy regimes by implementing the dual Hamiltonian with opposite qubit frequency and applying $\pi$ pulses.

Experimental characterization involves time-resolved measurements of qubit excitation probability and cavity photon population, adiabatic state preparation for ground-state tracking, and joint qubit-resonator tomography to reconstruct Wigner functions and extract parity.

Key Contributions and Results

1. Continuous Anisotropy Tuning and Symmetry Breaking:
   The authors demonstrate independent control of $g_1$ and $g_2$, enabling continuous tuning of $\lambda$. They observe that for $\lambda=0$ (JCM), the total excitation number is conserved ($U(1)$ symmetry). As $\lambda$ increases, counterrotating processes activate, breaking $U(1)$ symmetry and mixing excitation manifolds. The extracted total photon number confirms the transition from bounded JCM oscillations to the photon growth characteristic of the deep-strong-coupling (DSC) regime.

2. Spectral Reconstruction and Incomplete Revivals:
   In the DSC regime ($g/\omega \gtrsim 1$), the study investigates collapse-revival dynamics. In the isotropic case ($\lambda=1$), the system exhibits complete revivals due to equally spaced energy levels in the effective harmonic oscillator picture. The authors show that anisotropy ($\lambda \neq 1$) disrupts this equal spacing, leading to incomplete revivals. Even near degeneracy, the anisotropy leaves a dynamical fingerprint, characterized by lifted minima in photon-number fluctuations ($\delta n$), directly linking microscopic coupling imbalance to spectral interference reconstruction.

3. Ground-State Parity Switch:
   The authors identify a unique feature of the AQRM: an exact crossing between the two lowest energy levels of opposite parity. In the isotropic QRM, such crossings are generally avoided or restricted to specific points. In the AQRM, as the coupling strength $g_0$ increases, an exact crossing occurs at a predicted critical point $g_{0c} = (1+\lambda)g_{1c}$. Using adiabatic state preparation and joint tomography, the authors resolve this crossing, observing a ground-state parity switch where the system transitions from an even-parity ground state to an odd-parity ground state. This phenomenon has no analogue in the isotropic model.

4. Hidden Symmetry and Anisotropic Displacement:
   For biased models ($\varepsilon \neq 0$), the $Z_2$ parity symmetry is generally broken. However, the authors observe that exact level crossings reemerge at discrete commensurate biases due to hidden symmetries. In the isotropic case, these occur at $\varepsilon/\omega = k$ (integer $k$). The study demonstrates that in the anisotropic case, these "hidden symmetry points" shift according to the condition $\varepsilon_h/\omega = k \frac{2\sqrt{\lambda}}{1+\lambda}$. By tracking the peak of the transfer probability between displaced Fock states, the authors experimentally verify this anisotropic displacement, confirming the presence of generalized hidden symmetries.

Significance
The paper establishes a controllable route to engineering nonperturbative light-matter Hamiltonians where symmetry, spectrum, and dynamics can be programmed independently. By moving beyond the isotropic limit, the authors demonstrate that anisotropy is not merely a perturbation but a fundamental control parameter that reconstructs the spectral landscape. The ability to program spectral symmetries allows for the observation of phenomena such as ground-state parity switching and anisotropic hidden symmetry tunnelling, which are inaccessible in standard QRM implementations. This work transforms the AQRM from a theoretical interpolation into an experimentally programmable symmetry landscape, providing a platform for exploring quantum criticality, nonclassical state generation, and complex Floquet-engineered Hamiltonians.