First-Principles Quantum Kinetic Theory

• First-Principles Quantum Kinetic Theory is a rigorous ab initio method that models nonequilibrium quantum many-body dynamics via density matrix and master equation approaches.
• It systematically incorporates photon absorption, electron–phonon scattering, and radiative recombination to predict both transient and steady-state photocurrents.
• The framework reveals direct links between nonlinear optical responses and quantum geometric quantities like Berry curvature and quantum metric, enhancing material design.

First-principles quantum kinetic theory provides a rigorous ab initio framework for describing the nonequilibrium dynamics of quantum many-body systems, explicitly linking microscopic Hamiltonians and quantum geometry to emergent macroscopic nonlinear responses. In the context of the photogalvanic effect—dc current generation induced by optical illumination in non-centrosymmetric materials—this theory systematically incorporates all relevant quantum scattering channels (photon absorption, electron–phonon interactions, radiative recombination) and enables the calculation of both transient and steady-state photocurrents. The formalism utilizes the density matrix approach and quantum master equations, reproducing conventional shift and injection current results in the phonon-free limit and predicting new phonon-mediated contributions that reconcile theoretical predictions with experimental measurements. The ab initio framework reveals direct connections between nonlinear photogalvanic responses and fundamental quantum geometric quantities such as Berry curvature and the quantum metric, establishing a geometric foundation for the understanding and prediction of nonlinear optical phenomena in real materials (Yu et al., 3 Jan 2026).

1. Quantum Kinetic Framework: Liouville–von Neumann Hierarchy and Master Equation

The starting point is the many-body Hamiltonian for electrons coupled to bosonic modes (photons and phonons),

$H_{\text{tot}} = H_0^{e} + H_0^{b} + H'$

where

• $H_0^{e} = \sum_{n} \epsilon_n\, c_n^\dagger c_n$ describes the unperturbed electronic bands,
• $H_0^{b} = \sum_m \hbar \omega_m\, a_m^\dagger a_m$ the free photon/phonon bath,
• $$H' = \sum_{m,ij} \lambda_{ij}^m c_i^\dagger c_j (a_m + a_{-m}^\dagger)$$ is the electron–boson interaction.

Tracing out bosonic degrees of freedom in the Born–Markov approximation yields the quantum master equation (QME) for the reduced electronic density matrix $\rho(t)$: $$\frac{d\rho}{dt} = -\frac{i}{\hbar}\big[H_0^{e},\rho\big] + \left.\frac{d\rho}{dt}\right|_{\text{light}} + \left.\frac{d\rho}{dt}\right|_{e\text{–ph}} + \left.\frac{d\rho}{dt}\right|_{\text{rec}}$$ The rightmost terms encode, respectively, the coherent light–matter coupling, electron–phonon scattering, and radiative (electron–hole) recombination.

Expanding $\rho$ in powers of the optical field ($\mathbf{A}(t)$), one obtains a hierarchy: $$\rho(t) = \rho^{(0)} + \rho^{(1)}(t) + \rho^{(2)}(t) + \dots$$ with equations of motion: $$\frac{d}{dt}\rho^{(0)} = -\frac{i}{\hbar}[H_0^{e},\rho^{(0)}], \quad \frac{d}{dt}\rho^{(1)} + \frac{i}{\hbar}[H_0^{e},\rho^{(1)}] = -\frac{i}{\hbar}[H'(t),\rho^{(0)}], \ldots$$ The integral solution for each order involves time‐ordered convolutions with the Liouvillian evolution operator, systematically including relaxation and decoherence.

2. Photon- and Phonon-Mediated Scattering: Collision Integrals and Population Dynamics

Within the Born–Markov framework, scattering by photon emission/absorption and electron–phonon interaction yields bosonic collision integrals: $$\left.\frac{d\rho_{12}}{dt}\right|_c = \sum_{345} \Big[ (I-\rho)_{13}\,\rho_{45}\,P^c_{3245} - \rho_{32}(I-\rho)_{45}\,P^c_{5431} \Big] + (1\leftrightarrow 2)^\dagger$$ with

$$P^c_{1234} = \frac{\pi}{\hbar}\sum_{m\in c,\pm} N_m^{\mp}\,\lambda_{13}^{\pm m} (\lambda_{24}^{\pm m})^*$$

Here $N_m$ is the boson occupation factor. Projecting to diagonal elements yields the standard quantum Boltzmann (rate) equations for band populations: $$\left.\frac{df_i}{dt}\right|_c = \sum_j \left[(1-f_i)f_j\,W^c_{i\leftarrow j} - f_i(1-f_j)\,W^c_{j\leftarrow i}\right]$$ with

$$W^c_{i\leftarrow j} = \frac{2\pi}{\hbar} \sum_{m,\pm}|\lambda_{ij}^{m}|^2(N_m+\frac{1}{2}\mp\frac{1}{2})\,\delta(\epsilon_i-\epsilon_j\mp\hbar\omega_m)$$

Intraband (phonon-mediated, same band) and interband (phonon- or photon-mediated, conduction/valence) relaxation drive the system toward thermal or ground-state occupation.

3. Nonlinear Photogalvanic Response: Shift Current, Injection Current, and Transient Dynamics

Shift Current

For linearly polarized optical excitation, the steady-state dc (shift) current is obtained from off-diagonal density-matrix elements at second order: $$J^{\text{shift}}_a = -e\,\text{Tr}[\rho^{(2)} v_a] \equiv \sigma_{abc}E_b(\omega)E_c(-\omega)$$ The shift current tensor in velocity gauge is

$$\sigma_{abc} = \frac{\pi e^3}{2\hbar^2 V} \sum_{cvk} f_v(k)\left[ r^b_{vc}(k)\,r^c_{cv;a}(k) + r^c_{vc}(k)\,r^b_{cv;a}(k) \right] \delta(\epsilon_c(k)-\epsilon_v(k)-\hbar\omega)$$

where $r^b_{vc} = \langle u_v|i\partial_{k_b}|u_c\rangle$ and $r^c_{cv;a}$ denotes the covariant derivative. The shift vector $$R^a_{cv} = \partial_{k_a} \arg r^b_{cv} - (A^a_{cc} - A^a_{vv})$$ measures real-space charge shift.

Inclusion of explicit electron–phonon scattering yields a phonon-mediated second-order shift current: $$J^{sh}_{ph} = \frac{2e\pi}{\hbar V} \sum_{k,k',s,s',\nu} S^\nu_{k's'\leftarrow ks}\,|g^\nu_{ks;k's'}|^2\,\delta(\epsilon_{ks} - \epsilon_{k's'} + \hbar\omega_{q\nu})\left[ (N_{q\nu}+1)(1-f_{ks})f_{k's'} - N_{q\nu}(1-f_{k's'})f_{ks} \right]$$ where $S^\nu$ is the phonon-shift vector. For small phonon momentum $q$, $S^\nu \sim q \times \Omega$ with $\Omega$ the Berry curvature.

Injection Current

Under circularly polarized light, the nonequilibrium band occupation asymmetry in momentum space results in the steady-state injection current: $$J^{inj}_a = i\frac{\pi e^3}{2\hbar V}|A_0|^2 \sum_{ks\neq s'} f^k_{ss'}\left[\tau_{ks} v^a_{ss} - \tau_{ks'} v^a_{s's'}\right] (v^b_{s's}v^c_{ss'} - v^c_{s's}v^b_{ss'})\delta(\epsilon_s(k)-\epsilon_{s'}(k)-\hbar\omega)$$ with state-dependent relaxation times $\tau_{ks}$ (typically derived from Fan–Migdal self-energy) and occupation imbalance $f^k_{ss'}$. In the two-band scenario, the injection conductivity reduces to

$$\sigma^{inj}_{ab} = \frac{\pi e^2}{2\hbar} \oint_{\epsilon_c-\epsilon_v = \hbar\omega} dk\,\Omega_{11}^b(k)\left[ \tau_{k1} v^a_{11}(k) - \tau_{k2} v^a_{22}(k) \right]$$

where $\Omega_{11}^b$ is the Berry curvature of band 1, and phonon-corrected $\tau_{k1}^{-1}$ also acquires quantum metric contributions.

Transient Photogalvanic Response

The full time dependence of the QME yields a convolution form for transient shift current components: $$J^{sh}_x(t) = \int_{-\infty}^t K_x(t-t') I(t')\,dt',\quad x\in\{\text{exc},\;\text{ph}\}$$ with excitation and phonon kernel functions $K_x$ decaying on different relaxation timescales ($\tau_{e-ph}, \tau_{rec}$), predicting both unipolar and bipolar current waveforms as seen in THz emission experiments.

4. Limiting Behavior and the Physical Role of Phonons

In the absence of phonon-mediated scattering ($\tau \to \infty$), the quantum master equation reduces to the standard perturbative expansion, yielding the traditional shift current formula and a non-saturating (“runaway”) injection current ($dJ^{inj}/dt \propto \beta E^2$) with no steady state. Finite electron–phonon coupling introduces relaxation times, ensuring that the injection current saturates in accordance with experiment.

Crucially, the phonon-mediated shift current can be comparable in magnitude to the photo-excitation shift current for materials like BaTiO$_3$ at high photon energies. This supplementation corrects the underestimation of photogalvanic current found in phonon-free (purely electronic) ab initio predictions and is essential for quantitative agreement with experiment.

5. Quantum Geometric Structure and Material Implications

The theory yields transparent expressions for both phonon-mediated shift currents, self-consistent (finite-lifetime) injection currents, and their full time-dependent dynamics. These closed-form expressions explicitly involve Berry curvature and quantum metric, establishing that nonlinear photocurrent phenomena in noncentrosymmetric solids are fundamentally governed by quantum geometry.

For instance, in the shift current response, the Berry connection and its derivatives directly appear in the nonlinear conductivity tensor. In phonon-mediated processes, the quantum metric tensor enters through its influence on the scattering rates and the lifetimes. The injection current, too, is dictated by the Berry curvature and the state-dependent relaxation governed by quantum metric corrections.

Quantitative application to BaTiO$_3$ demonstrates that accounting for both phonon-mediated current and realistic lifetimes eliminates the major discrepancy between theoretical and experimental photogalvanic responses. The framework predicts new features, such as the tunable appearance of unipolar versus bipolar THz currents controlled by photon frequency, and enables reliable ab initio prediction of nonlinear currents in a broad range of materials.

6. Perspective and Broader Significance

The first-principles quantum kinetic theory developed here establishes a predictive and flexible platform for calculating nonlinear optical responses in complex materials (Yu et al., 3 Jan 2026). All bosonic scattering mechanisms are incorporated on equal footing, and the approach maintains full consistency with quantum geometry concepts. Neither simplified relaxation-time nor phenomenological scattering models are necessary; instead, the entire response, including relaxation and recombination, is rigorously captured within a unified master-equation formalism.

This geometric, first-principles quantum kinetics is poised to become the reference computational paradigm for ab initio nonlinear optics in solids, guiding future experimental and theoretical exploration of ultrafast and nonequilibrium quantum phenomena in quantum materials.