Dynamical Quantum Geometric Tensor

• DQGT is a gauge-invariant tensor extending the conventional quantum geometric tensor by incorporating time-dependent modulations, non-adiabatic corrections, and open-system effects.
• It is experimentally accessible through modulation protocols in platforms like NV centers and ultracold atoms, enabling precise measurements of quantum metrics and Berry curvatures.
• DQGT informs optimal quantum control strategies and serves as a diagnostic tool for phase transitions and dynamical behavior in many-body, non-Hermitian, and open quantum systems.

The dynamical quantum geometric tensor (DQGT) is a gauge-invariant, tensorial structure that encodes how quantum geometric properties of pure or mixed quantum states respond under time-dependent controls, non-equilibrium evolution, and open-system dynamics. It generalizes the conventional quantum geometric tensor—comprising the quantum metric and Berry curvature—by incorporating dynamic couplings, non-adiabatic corrections, and the effects of parameter modulations or sudden quenches. The DQGT directly governs dynamical observables such as Rabi frequencies in modulated systems, non-adiabatic rates in quantum control, intrinsic noise in transport, and the phase and trajectory corrections in semiclassically evolving quantum states. Its measurement protocols, analytic structure, and predictive role have been established across solid-state qubits, many-body systems, non-Hermitian and open quantum systems, and bosonic excitations, forming a unifying framework for quantum geometry in dynamic and non-equilibrium settings.

1. Mathematical Structure and Core Definitions

The DQGT generalizes the quantum geometric tensor to time-dependent or modulated settings. For a family of Hamiltonians $H(\vec{\lambda}(t))$ with instantaneous eigenstates $|n(\vec{\lambda}(t))\rangle$ and eigenvalues $E_n(\vec{\lambda}(t))$, the standard quantum geometric tensor (QGT) is

$$\chi^{(n)}_{\mu\nu} = \langle \partial_\mu n | (1 - |n\rangle\langle n|) | \partial_\nu n \rangle,$$

where $\mu,\nu$ index control parameters. Its decomposition yields the Fubini–Study metric $g_{\mu\nu} = \operatorname{Re}\chi_{\mu\nu}$ and Berry curvature $F_{\mu\nu} = -2\operatorname{Im}\chi_{\mu\nu}$.

In dynamical protocols, the relevant generalization is

$$g^{(n)}_{\mu\nu}(\vec{\lambda}) = \operatorname{Re} \sum_{m \neq n} \frac{ \langle m | \partial_\mu H | n \rangle \langle n | \partial_\nu H | m \rangle } { [E_n - E_m]^4 },$$

where the fourth-order denominator reflects the squared non-adiabatic rate for finite-time protocols (Chen, 2022, Li et al., 23 Dec 2025). This DQGT metric governs the instantaneous rate and accumulated probability of non-adiabatic transitions, and introduces a Riemannian metric on the path traversed in parameter space.

For nonequilibrium or quenched systems, the time-dependent DQGT is constructed by promoting the quantum geometric tensor to the Heisenberg picture or parameterizing with $(k,t)$ for lattice systems (Rattacaso et al., 2019, Tang et al., 27 Jan 2026):

$$Q_{\mu\nu}(k, t) = \langle \partial_\mu \psi_k(t) | \partial_\nu \psi_k(t) \rangle - \langle \partial_\mu \psi_k(t) | \psi_k(t)\rangle \langle \psi_k(t) | \partial_\nu \psi_k(t)\rangle,$$

where the arguments can be momentum, time, or additional control parameters.

2. Dynamical Probing and Measurement Protocols

The DQGT is directly accessible by dynamical modulation protocols. In the paradigmatic case of a two-level NV center (single qubit), small parametric modulations of control parameters (e.g., microwave drive amplitude or phase) induce Rabi oscillations between instantaneous eigenstates (Yu et al., 2018). The Rabi frequency for transitions between states $|n(\vec{\lambda}(t))\rangle$0 and $|n(\vec{\lambda}(t))\rangle$1 is

$|n(\vec{\lambda}(t))\rangle$2

and its squared sum over $|n(\vec{\lambda}(t))\rangle$3 reconstructs the diagonal DQGT metric. Off-diagonal elements and the Berry curvature are accessed via simultaneous or phase-offset modulations in multiple parameters, with phase-sensitive detection isolating the real and imaginary parts of $|n(\vec{\lambda}(t))\rangle$4.

For multilevel or pseudo-Hermitian systems, general protocols utilize either energy fluctuation observables or generalized force operators, leveraging distinct time-evolved left and right (or biorthogonal) eigenstates (Huang et al., 21 Sep 2025). The full DQGT is reconstructed by measuring appropriately designed generalized expectation values and dividing by known ramp parameters.

Experiments on NV centers demonstrate $|n(\vec{\lambda}(t))\rangle$5-level measurement of all components $|n(\vec{\lambda}(t))\rangle$6 and $|n(\vec{\lambda}(t))\rangle$7, including the tracking of topological transitions via measurement of the Berry curvature and extracted Chern numbers as parametric biases are swept (Yu et al., 2018).

3. Physical Interpretation: Dynamical Effects and Observables

The DQGT governs a wide range of dynamical effects:

• Non-adiabatic corrections: The DQGT provides a quadratic-in-velocity correction to the accumulated phase in finite-duration protocols, in contrast to the usual Berry phase, and determines algebraic (rather than exponential) scaling of residual transition probability or phase error beyond the Landau–Zener regime (Bleu et al., 2016, Chen, 2022).
• Optimal quantum control: The metric structure of the DQGT defines a geodesic problem: driving parameters at constant "DQGT speed" minimizes the non-adiabatic transition probability and leads to optimal protocols for rapid adiabatic passage, such as OSTIRAP in multilevel systems, where transfer efficiency >98% can be achieved in times an order of magnitude faster than conventional STIRAP (Li et al., 23 Dec 2025).
• Wavepacket dynamics and positional shifts: In semiclassical wavepacket theory, the DQGT appears as a second-order correction to the anomalous Hall velocity and geometric phase, both in Hermitian and non-Hermitian (open) systems, and modifies the effective trajectory under external perturbations (Bleu et al., 2016, Hu et al., 2024).

In periodically driven or Floquet systems, Berry curvature and metric fluctuations contribute directly to observable quantities such as Hall current noise, where the frequency-dependent DQGT enters the Kubo formula for noise spectra—even in time-reversal-invariant systems (Wei et al., 2023).

4. DQGT in Open, Non-Hermitian, and Driven Quantum Systems

Extension of the DQGT framework to non-Hermitian and open quantum systems requires care with regards to left–right eigenvectors and the biorthogonal formalism (Hu et al., 2024, Huang et al., 21 Sep 2025, Cariñena et al., 2017). Two distinct QGTs arise:

• Right–right (RR) QGT: Defined solely in terms of right eigenstates, its real part provides a genuine quantum metric and its imaginary part an effective Berry curvature controlling anomalous drift and positional shifts.
• Left–right (LR) QGT: Built from biorthogonal left and right eigenstates, generally complex-valued, and dictating geometric phase corrections unique to open systems.

The DQGT in this regime incorporates additional dissipative corrections, and the mixing of real and imaginary parts introduces Hall-type effects inaccessible in Hermitian systems. Open-system Lindblad dynamics carries the quantum metric and curvature tensors along the flow, capturing deformations of quantum geometry under Markovian evolution (Cariñena et al., 2017).

5. Nonequilibrium, Quenched, and Many-Body DQGT

In many-body and out-of-equilibrium quantum systems, the DQGT serves as both a dynamic probe of spreading and coherence and a tool for diagnosing phase transitions (Rattacaso et al., 2019, Tang et al., 27 Jan 2026). After a quantum quench, the DQGT evaluated in $|n(\vec{\lambda}(t))\rangle$8-space encodes:

• Wavepacket spreading: The spatial DQGT metric grows quadratically in time (ballistic scaling), reflecting the group velocity variance.
• Energy fluctuation: The temporal DQGT metric quantifies the variance of the final Hamiltonian in the evolved state, setting characteristic evolution timescales.
• Covariances and non-adiabatic curvatures: The off-diagonal DQGT component correlates position and energy, and its imaginary part reflects non-adiabatic-induced Berry curvature in the extended parameter space.

Importantly, the DQGT preserves equilibrium phase diagrams under global quenches: singularities linked to quantum criticality persist after time evolution, with equilibration of the tensor modulo exponentially suppressed fluctuations in large systems (Rattacaso et al., 2019, Tang et al., 27 Jan 2026).

6. Experimental Platforms and Applications

Dynamical measurement and utilization of the DQGT now span a broad set of platforms:

Platform	DQGT Role	Key Reference
NV centers, transmons, dots	Direct parametric modulation	(Yu et al., 2018)
Ultracold atoms	Lattice depth, Raman-coupling	(Yu et al., 2018)
Bosonic/phononic systems	DSF-based full QGT extraction	(Wu et al., 20 Jan 2026)
Optically active polaritons	Trajectory/phase corrections	(Bleu et al., 2016)
Many-body spin chains	Quench-induced DQGT dynamics	(Rattacaso et al., 2019, Tang et al., 27 Jan 2026)
Non-Hermitian microcavities	Non-Hermitian DQGT dynamics	(Hu et al., 2024)

DQGT protocols have enabled high-precision mapping of quantum metric hotspots and Berry-curvature fluxes in real solids and synthesized systems, as well as the identification of topological invariants (Chern numbers, $|n(\vec{\lambda}(t))\rangle$9 invariants). In quantum control and computation, DQGT-guided protocols deliver order-of-magnitude improvements in fidelity and speed and can be robust against amplitude and detuning fluctuations (Li et al., 23 Dec 2025).

7. Theoretical Extensions and Outlook

Current research generalizes the DQGT framework into higher-rank tensors (capturing tensor monopoles in multi-parameter spaces), many-body entanglement diagnostics (via the connection to quantum Fisher information and susceptibilities), and complex, non-Hermitian, or Floquet-engineered bandstructures. The DQGT is further anticipated to clarify geometric noise effects, nonlinear response, and drive-induced topological transitions in interacting systems (Yu et al., 2018, Wei et al., 2023, Hu et al., 2024). Open challenges include computational handling for large, multi-parameter systems, robust protocols for extracting non-Abelian DQGTs in degenerate manifolds, and further elucidation of DQGT-driven phenomena in dissipative and strongly correlated regimes.