Geometric criticality in the driven Jaynes-Cummings model

TL;DR

This work analyzes geometric criticality in the driven Jaynes-Cummings model by computing the quantum geometric tensor and Berry curvature for its eigenstates as the drive parameters $(\eta,\phi)$ vary. The driven JCM is solved to yield a discrete spectrum with $E_0=0$ and $E_{n,\pm}=\pm\sqrt{n}\,\Omega\,(1-\eta^2)^{3/4}$ and corresponding squeezed-displaced eigenstates, enabling a QGT analysis in the critical region of the photon-blockade breakdown transition. A perturbative QGT expression shows divergent quantum metric components $\mathcal{G}_{\eta\eta}$, $\mathcal{G}_{\phi\phi}$ and Berry-curvature components $\mathcal{B}_{\eta\phi}$, $\mathcal{B}_{\phi\eta}$ near the critical point, with brighter states exhibiting much stronger divergences than the dark state; Bures metric divergences are also enhanced and dominated by the photonic sector. The results are experimentally accessible in circuit QED via adiabatic, counterdiabatic, or excited-state preparation techniques, offering a geometric lens on criticality in light-matter quantum systems and informing control protocols in the presence of quantum phase transitions.

Abstract

When the photonic mode in the Jaynes-Cummings model is driven by an external classical field, the system can undergo the photon-blockade breakdown phase transition at a critical point. Such a phase transition has been detailedly investigated, but the critical properties of the eigenstates remain largely unexplored so far. We here study the geometric criticality associated with these eigenstates. The amplitude and phase of the drive serve as the control parameter of the governing Hamiltonian. We find the quantum metric and Berry curvature tensors for each eigenstate display divergent behaviors in the critical region. More importantly, the divergence associated with bright eigenstates is much more pronounced than that for the unique dark state. Our theoretical results can be experimentally confirmed in circuit quantum electrodynamics systems, where the driven Jaynes-Cummings model has been realized.

Figures (2)

• Figure 1: The quantum metric $\mathcal{G}_{\eta\eta}$ (a), $\mathcal{G}_{\phi\phi}$ (c) and the Berry curvature $\mathcal{B}_{\eta \phi}$ (b), $\mathcal{B}_{\phi \eta}$ (d) with respect to $\eta$ for $\phi=0$. The eigenstates $|\psi_i\rangle \ (i=0,1,2,3,4,5)$ correspond to $n = 0,1,2,3,4,5$, respectively. The quantum metric $\mathcal{G}_{\eta \phi}$, $\mathcal{G}_{\phi \eta}$ and the Berry curvature $\mathcal{B}_{\eta\eta}$, $\mathcal{B}_{\phi\phi}$ are equal to zero, and not shown here.
• Figure 2: The Bures metric $g_{\eta\eta}$ (a) and $g_{\phi\phi}$ (b) as a function of $\eta$. Here $|\psi_i\rangle$, $|\psi_{iq}\rangle$ and $|\psi_{ir}\rangle$ ($i = 0,1,2$) represent the eigenstates of the composite system, the two-level subsystem and the quantum field mode, respectively. The $g_{\eta\phi}$ and $g_{\phi\eta}$ are equal to zero, and not shown here.