Two-State Quantum Geometric Tensor

• Two-State Quantum Geometric Tensor is a framework that encodes both the quantum metric and Berry curvature in two-level systems, elucidating intrinsic geometry and dynamical responses.
• It extends to non-Hermitian settings by employing left-right formulations, thereby uncovering unique signatures at exceptional points and during topological transitions.
• Experimental protocols, including polarization tomography and generalized force methods, enable direct reconstruction of its components in systems like microcavity polaritons.

The two-state quantum geometric tensor (QGT) encodes the full local geometry of quantum states in a two-level (band) quantum system, encompassing both quantum metric and Berry curvature. This tensor generalizes to both Hermitian and non-Hermitian systems, describing not only the intrinsic geometric characteristics of quantum bands but also underpinning measurable dynamical and topological phenomena, including anomalous transport and phase transitions. Both the real (metric) and imaginary (curvature) components of the QGT emerge in semiclassical wave-packet dynamics and can be directly reconstructed from experimental data, particularly in optical and exciton-polariton systems.

1. Definitions and Structural Decomposition

For a two-level system with a smoothly parameterized eigenstate $|\psi(\boldsymbol\zeta)\rangle$, the Hermitian QGT is defined as

$$Q_{\mu\nu} = \langle \partial_\mu \psi | \partial_\nu \psi \rangle - \langle \partial_\mu \psi | \psi \rangle \langle \psi | \partial_\nu \psi \rangle,$$

with $\mu,\nu$ labeling parameters. Decomposing into real and imaginary parts yields the quantum metric

$g_{\mu\nu} = \mathrm{Re}\, Q_{\mu\nu}$

and Berry curvature

$\Omega_{\mu\nu} = -2\,\mathrm{Im}\, Q_{\mu\nu}.$

Under a Bloch-sphere (pseudospin) parametrization, explicit expressions for a two-band model (with eigenstates parameterized by angles $(\theta,\phi)$) are

$$g_{ij} = \frac{1}{4}\left[\partial_i \theta \, \partial_j \theta + \sin^2 \theta \, \partial_i \phi \, \partial_j \phi \right],$$

$$\Omega_{xy} = \frac{1}{2} \sin\theta\, (\partial_x\phi \, \partial_y\theta - \partial_y\phi \, \partial_x\theta).$$

These structures extend naturally to more general Hamiltonians and form the backbone for band-geometry analysis (Hu et al., 2023, Gianfrate et al., 2019).

In non-Hermitian systems (where $H \neq H^\dagger$), the eigenstates split into right ($|u_n^R\rangle$) and left ($$Q_{\mu\nu} = \langle \partial_\mu \psi | \partial_\nu \psi \rangle - \langle \partial_\mu \psi | \psi \rangle \langle \psi | \partial_\nu \psi \rangle,$$0) biorthogonal bases, satisfying $$Q_{\mu\nu} = \langle \partial_\mu \psi | \partial_\nu \psi \rangle - \langle \partial_\mu \psi | \psi \rangle \langle \psi | \partial_\nu \psi \rangle,$$1. Two widely adopted generalizations of the QGT exist:

• Right–Right (RR) QGT:

$$Q_{\mu\nu} = \langle \partial_\mu \psi | \partial_\nu \psi \rangle - \langle \partial_\mu \psi | \psi \rangle \langle \psi | \partial_\nu \psi \rangle,$$2

• Left–Right (LR) QGT:

$$Q_{\mu\nu} = \langle \partial_\mu \psi | \partial_\nu \psi \rangle - \langle \partial_\mu \psi | \psi \rangle \langle \psi | \partial_\nu \psi \rangle,$$3

For pseudo-Hermitian cases with real spectra, these structures are further constrained and the usual Hermitian-like decomposition applies (Huang et al., 21 Sep 2025, Hu et al., 2024).

2. Explicit Formulas for Two-State Hamiltonians

Consider a generic two-level Hamiltonian

$$Q_{\mu\nu} = \langle \partial_\mu \psi | \partial_\nu \psi \rangle - \langle \partial_\mu \psi | \psi \rangle \langle \psi | \partial_\nu \psi \rangle,$$4

with $$Q_{\mu\nu} = \langle \partial_\mu \psi | \partial_\nu \psi \rangle - \langle \partial_\mu \psi | \psi \rangle \langle \psi | \partial_\nu \psi \rangle,$$5 taking real or complex values for Hermitian or non-Hermitian systems, respectively. Introducing the (possibly complex) normalized Bloch vector $$Q_{\mu\nu} = \langle \partial_\mu \psi | \partial_\nu \psi \rangle - \langle \partial_\mu \psi | \psi \rangle \langle \psi | \partial_\nu \psi \rangle,$$6, the RR QGT components for a given band are:

$$Q_{\mu\nu} = \langle \partial_\mu \psi | \partial_\nu \psi \rangle - \langle \partial_\mu \psi | \psi \rangle \langle \psi | \partial_\nu \psi \rangle,$$7

$$Q_{\mu\nu} = \langle \partial_\mu \psi | \partial_\nu \psi \rangle - \langle \partial_\mu \psi | \psi \rangle \langle \psi | \partial_\nu \psi \rangle,$$8

The LR QGT is constructed analogously but with left–right derivatives, leading to potentially complex metric and curvature. First-order perturbation relations yield (Huang et al., 21 Sep 2025):

$$Q_{\mu\nu} = \langle \partial_\mu \psi | \partial_\nu \psi \rangle - \langle \partial_\mu \psi | \psi \rangle \langle \psi | \partial_\nu \psi \rangle,$$9

3. Measurement Protocols and Experimental Realization

In planar microcavity exciton-polariton systems, the full QGT can be experimentally reconstructed through polarization tomography. The procedure involves:

1. Measuring emitted intensities in six polarization bases to reconstruct the Stokes vector and, by extension, the band pseudospin texture and $\mu,\nu$0 maps.
2. Cleaning the polarization data and numerically deriving $\mu,\nu$1, $\mu,\nu$2.
3. Computing the local $\mu,\nu$3 and $\mu,\nu$4 on a momentum grid via finite-difference approximation (Gianfrate et al., 2019).

For non-Hermitian or pseudo-Hermitian two-band models, direct extraction of the full QGT—including both LR and RR components—was demonstrated through two experimentally viable protocols (Huang et al., 21 Sep 2025):

• Energy-Fluctuation Method: Measurement of the generalized expectation value of the squared energy fluctuation operator between time-evolved right and left states yields $\mu,\nu$5, enabling simultaneous access to metric and curvature.
• Generalized Force Method: Measurement of the generalized force operator between time-evolved states, $\mu,\nu$6, enables separate extraction of the real (metric) and imaginary (curvature) components.

Both schemes leverage adiabatic perturbation ramps and controlled-swap circuits to achieve high-fidelity access to the QGT without full tomography.

4. Semiclassical Wave-Packet Dynamics

The QGT components directly determine observable features of semiclassical wave-packet dynamics under external forces. For a wave packet centered at $\mu,\nu$7 in a two-band non-Hermitian system under force $\mu,\nu$8:

$\mu,\nu$9

$g_{\mu\nu} = \mathrm{Re}\, Q_{\mu\nu}$0

Here $g_{\mu\nu} = \mathrm{Re}\, Q_{\mu\nu}$1 and $g_{\mu\nu} = \mathrm{Re}\, Q_{\mu\nu}$2 are respectively the RR and LR Berry connections. Notably:

• RR Berry curvature $g_{\mu\nu} = \mathrm{Re}\, Q_{\mu\nu}$3 induces an anomalous Hall-like drift.
• The difference $g_{\mu\nu} = \mathrm{Re}\, Q_{\mu\nu}$4 gives a field-induced correction to the group velocity, unique to non-Hermitian systems.
• Both RR and LR QGT contributions, including nonadiabatic (field-induced) corrections, affect the dynamics, particularly near degeneracies and exceptional points (Hu et al., 2023, Hu et al., 2024).

5. Special Limits: Exceptional Points, Gap-Closing, and Topological Transitions

In two-band non-Hermitian systems, the QGT structure exhibits distinctive behaviors at exceptional points (EPs) and upon the opening of spectral gaps:

• At EPs ($g_{\mu\nu} = \mathrm{Re}\, Q_{\mu\nu}$5), both RR and LR tensors diverge with distinct critical exponents ($g_{\mu\nu} = \mathrm{Re}\, Q_{\mu\nu}$6, $g_{\mu\nu} = \mathrm{Re}\, Q_{\mu\nu}$7, $g_{\mu\nu} = \mathrm{Re}\, Q_{\mu\nu}$8), reflecting the non-Hermitian topology (Hu et al., 2023).
• In the gapless regime, the RR Berry curvature can maintain a finite texture in parameter space, yielding observable anomalous Hall drifts even without real magnetic fields.
• With gap opening, the LR Berry curvature evolves to a smooth topological Dirac-like profile, supporting quantized Chern numbers and associated phase transitions in topological models (Huang et al., 21 Sep 2025).
• In pseudo-Hermitian systems, quantum metric and curvature can be measured dynamically and track transitions in the topological invariants, such as the Chern number (Huang et al., 21 Sep 2025).

6. Connection to Degenerate (Non-Abelian) Two-State Systems

For a two-fold–degenerate two-level manifold (non-Abelian case), the QGT generalizes to a matrix-valued tensor

$g_{\mu\nu} = \mathrm{Re}\, Q_{\mu\nu}$9

with associated Hermitian (metric) and anti-Hermitian (curvature) decompositions. Traces and determinants of these objects relate directly to the unit Bloch vector structure, and algebraic identities connect the determinant of the quantum metric to the absolute value of the Berry curvature component, coinciding with the Abelian case in the nondegenerate limit (Ding et al., 2023).

7. Applications and Extensions

The two-state QGT is directly relevant in:

• Topological photonics and polaritonics: governs anomalous Hall effect, orbital magnetic susceptibility, and superfluidity in flat bands (Gianfrate et al., 2019).
• Non-Hermitian systems: reveals unique geometric and dynamical signatures, e.g., novel Hall effects without Berry phase quantization.
• Measurement schemes: both polarization-resolved photoluminescence (in microcavities) and quantum simulation protocols (energy fluctuation and generalized force approaches) enable direct mapping of the full QGT (Huang et al., 21 Sep 2025).
• Detection and quantification of topological phase transitions across a range of two-level and degenerate systems.

The QGT thus forms a unifying geometric framework for analyzing, measuring, and predicting the band structure, dynamics, and topology of quantum systems, both Hermitian and non-Hermitian. These results have been rigorously demonstrated in platforms including microcavity polariton systems (Gianfrate et al., 2019), non-Hermitian hybrid photonic systems (Hu et al., 2023), and theoretically via dynamical protocols in pseudo-Hermitian and two-fold–degenerate models (Hu et al., 2024, Ding et al., 2023, Huang et al., 21 Sep 2025).