Time Derivatives of Excitation Coefficients

• Time derivatives of excitation coefficients are the rates at which quantum state amplitudes evolve, crucial for analyzing transitions and relaxation in dynamic systems.
• They are derived using frameworks such as Dyson-like equations, time-dependent perturbation theory, and path-ordered exponentials to capture nonadiabatic effects.
• Accurate computation of these derivatives underpins the evaluation of oscillator strengths, energy transfer, and conservation laws in both closed and open quantum systems.

The time derivatives of excitation coefficients refer to the dynamical rates of change of quantum amplitudes or occupation probabilities associated with excited states under time-dependent perturbations. These derivatives govern transition probabilities, energy relaxation, population transfer, and response properties in quantum, electronic, or field-theoretic systems. A comprehensive understanding of their analytical structure, computation, and physical implications is central to fields including many-body quantum theory, time-dependent density functional theory (TDDFT), quantum statistical mechanics, and quantum kinetic theory.

1. Formal Definitions and Mathematical Framework

Time derivatives of excitation coefficients typically arise in the expansion of a time-dependent quantum state $|\Psi(t)\rangle$ in a basis of energy eigenstates $\{|n\rangle\}$. Writing

$$|\Psi(t)\rangle = \sum_n c_n(t) e^{-iE_nt/\hbar} |n\rangle,$$

the excitation coefficients $c_n(t)$ encode the probability amplitudes for occupancy of the $n$th state, and their time derivatives $dc_n/dt$ control the evolution between states.

In time-dependent perturbation theory and response formulations, these derivatives manifest in different physical contexts:

• TDDFT Linear Response: The evolution of Kohn–Sham (KS) excitation amplitudes is governed by integro-differential equations coupling single-particle transitions via the frequency-dependent Hartree–exchange–correlation kernel $f_{\text{xc}}(\omega)$ (0807.0091).
• Landau Problem: In a charged particle subject to a time-dependent electric field in a magnetic background, the time derivatives of excitation coefficients are governed by nonadiabatic transition operators expressed as path-ordered exponentials of canonical variables (Chee, 2012).
• Open Quantum System Dynamics: The coefficients' rates are governed by non-Hermitian and stochastic terms in the presence of an environment, e.g., as in stochastic Schrödinger or Lindblad equations (Hofmann et al., 2012).
• Kinetic Theories and Diffusion: The time derivative of nonequilibrium distribution functions (which generalize excitation coefficients to Fermi systems or quantum gases) is governed by kinetic or nonlinear diffusion equations, with solutions characterizing relaxation dynamics (Lukyanov, 14 Oct 2024).

2. Derivation and Core Theoretical Structures

The explicit time-dependence of excitation coefficients is controlled by several distinct mathematical mechanisms:

A. Dyson-Like Equations in TDDFT

In time-dependent density functional theory, the density response function

$$\chi(\omega) = \chi_s(\omega) + \chi_s(\omega) [v + f_{\text{xc}}(\omega)] \chi(\omega)$$

encapsulates the frequency-domain evolution of excitation coefficients (0807.0091). The poles of $\chi(\omega)$ yield excitation energies $\omega_q$, while residues at these poles relate to oscillator strengths and (via inverse Fourier transformation) to the time derivatives of excitation amplitudes. Critically, the frequency dependence of $f_{\text{xc}}(\omega)$ induces nontrivial time evolution not captured by adiabatic kernels.

B. Quantum Adiabatic Theorem and Nonadiabatic Transitions

Exact factorization of quantum time-evolution operators in explicitly time-dependent Hamiltonians facilitates analytic tracking of excitation coefficients. In the Landau problem with a general time-dependent electric field, separation into geometric, dynamical, and nonadiabatic factors isolates the explicit time-integral dependence of coefficients responsible for quantum transitions across Landau levels:

$$u(t) = -\frac{q}{mc} \int_0^t e^{-i\omega s} E^*(s) ds$$

with $|u(t)|^2$ setting leading-order transition probabilities (Chee, 2012).

C. Higher Time Derivatives in Effective Theories

Integrating out quantum fluctuations (moments) in canonical systems (e.g., anharmonic oscillators) via semiclassical/adiabatic expansions yields effective equations for expectation values with an infinite series of higher time derivatives:

$$\ddot{q} + \omega^2 q + U'(q)/m + \hbar f(q, \dot{q}, \ddot{q}, \ldots) = 0$$

The time derivatives of excitation coefficients are then connected to quantum back-reaction terms, which modify the structure and rates of quantum transitions (Bojowald et al., 2012).

D. Representational Effects and Unitary Transformations

In nonstationary (time-dependent) external fields, the choice of representation (e.g., Dirac vs. Foldy–Wouthuysen) crucially affects the correct extraction of excitation coefficients' time derivatives. Under a time-dependent unitary transformation, the operator $i \hbar \partial/\partial t$ acquires an extra term:

$$H' = U H U^{-1} - i\hbar U \frac{\partial U^{-1}}{\partial t}$$

This correction must be included to obtain correct time evolution and transition amplitudes (Silenko, 2014).

E. Diffusive Relaxation and Effective Kinetics

For Fermi systems, time derivatives of excitation coefficients correspond to the time evolution of the momentum-distribution deviation $\delta f(p, t)$, which relaxes according to nonlinear diffusion:

$$\frac{\partial f(p)}{\partial t} = - K_{p,0} m \left\{ p (1 - 2f(p)) \frac{\partial f(p)}{\partial p} + 3f(p)(1 - f(p)) \right\} + D_{p,0} \left\{ \frac{\partial^2 f(p)}{\partial p^2} + \frac{2}{p} \frac{\partial f(p)}{\partial p} \right\}$$

leading to a phenomenological relaxation $d\delta f/dt = -\delta f/\tau_{\text{eq}}$ (Lukyanov, 14 Oct 2024).

3. Physical Interpretations and Impact on Dynamics

Excitation Coefficient Dynamics and Transition Probabilities

The time derivatives of excitation coefficients govern transition rates between quantum states, the structure of absorption/emission spectra, and the temporal evolution of populations or coherences. For example, the Lehmann representation for a frequency-dependent polarizability,

$$\alpha(\omega) = \sum_q \frac{f_q}{\omega^2 - \omega_q^2 + i\delta},$$

links the positions and residues of poles (encoding excitation energies and oscillator strengths) to the precise time derivatives of the underlying excitation coefficients (0807.0091, Dar et al., 2023).

Nonadiabatic Effects and Kernel Frequency Dependence

The inclusion of explicit frequency dependence in kernels, such as in the time-dependent exact-exchange or dressed TDDFT formalisms, ensures proper conservation laws (e.g., the $f$-sum rule) and accurate dynamical evolution. At high frequencies, as $f_{\text{xc}}(\omega)$ approaches constant values and $\chi_s(\omega) \propto 1/\omega^2$, the correct cancellation guarantees physically meaningful time derivatives and excitation dynamics (0807.0091).

The modulus of transition matrix elements, and their dynamical evolution, are strongly driven by time-varying external fields, bath interactions, and intrinsic nonadiabatic couplings, with the time derivatives sensitive to both the parameters of driving (field strength, detuning, symmetry-breaking) and the microscopic interaction structure.

Open System Dynamics, Dissipation, and Relaxation

In open quantum systems, environmental coupling introduces stochastic and dissipative contributions to excitation coefficient evolution, often captured by master equations or stochastic Schrödinger equations with bath operators. The temporal derivative of site excitation probability, or amplitude, then determines energy transfer and relaxation time scales, which in turn are sensitive to detuning, coupling topology, and system–bath interactions (Hofmann et al., 2012).

In kinetic and nonequilibrium settings, effective relaxation times extracted from the time dependence of distribution function deviations ($\delta f$) summarize the net effect of many-body collisions and diffusive processes on the net decay of excitation coefficients (Lukyanov, 14 Oct 2024).

4. Computational Strategies and Implementation

The practical computation of time derivatives of excitation coefficients is context-dependent:

Linear Response and Matrix Equations

Numerically, frequency-dependent kernels are discretized and the Dyson-like equation for the response function is solved at each frequency, with excitation energies and oscillator strengths extracted from the matrix equation's poles and residues. Basis sets such as cubic splines facilitate efficient representation and numerical inversion in spherical (e.g., atomic) systems (0807.0091).

Path-Ordered Exponentials and Diagrammatic Expansions

For Landau-level problems, explicit evaluation of path-ordered exponentials yields analytic expressions for transition amplitudes, which can be expanded perturbatively in small parameters (e.g., adiabatic limits), revealing dependence on time integrals of the driving electric field (Chee, 2012).

Time-dependent perturbation theory for many-body systems employs diagrammatic techniques (Wick's theorem, Hugenholtz diagrams) to precompute time-independent contractions, separating the overall transition probability into time-dependent coefficients (propagators, field interaction amplitudes) and precomputed static structures (Bayne et al., 2017).

Energy Methods and Regularization in PDEs

In analysis of hyperbolic PDEs with low-regularity coefficients, time derivatives of excitation coefficients arise in energy estimates where commutator terms and regularized coefficients are controlled using paradifferential calculus. Corrector terms in the energy functional are designed to cancel potentially divergent contributions from nonsmooth coefficients (Colombini et al., 2013, Colombini et al., 2020).

Phenomenological Relaxation and Statistical Models

Relaxation dynamics in kinetic models are quantified by integrating deviations over time and matching areas to ideal exponential decays, extracting effective timescales for the decay of excitation amplitudes or occupation probabilities (Lukyanov, 14 Oct 2024).

5. Special Issues: Conservation Laws, Representations, and Non-Markovianity

Conservation Laws and Sum Rules

Frequency dependence of exchange–correlation kernels is essential to satisfy sum rules (e.g., $f$-sum rule) in TDDFT, ensuring that the dynamical response—tied to the time derivatives of excitation coefficients—remains physically consistent across the frequency domain (0807.0091, Dar et al., 2023).

Role of Representation and Gauge Choices

In nonstationary fields, the proper computation of excitation coefficients' time derivatives requires careful selection of representation. The Foldy–Wouthuysen representation eliminates extra time-derivative terms associated with explicitly time-dependent unitary transformations, resulting in direct correspondence between operator action and physical time evolution (Silenko, 2014).

Non-Markovian Effects and Beyond Golden Rule

Including the effect of coherent, memory-retaining dynamics (beyond simple exponential relaxation), such as those governed by Bessel functions in stimulated emission (Rabi oscillations), shows that the time derivatives of excitation coefficients can be oscillatory, precluding thermodynamic equilibrium as postulated in traditional rate-equation (Pauli) models (Islam et al., 2017).

6. Advanced Applications and Implications

Double Excitation and Frequency-Dependent Kernels

Dressed TDDFT and similar approaches exploit explicit frequency dependence in kernels to access double-excitation character and redistribute oscillator strengths among states, directly affecting the time derivatives of excitation coefficients. Accurate prediction of excited-to-excited-state couplings and quadratic response in dynamics simulations hinges on these corrections (Dar et al., 2023, Baranova et al., 5 Jun 2025).

Quantum Gravity, Cosmology, and Effective Theories

Formalisms integrating higher time derivatives, as in quantum cosmology or gravity, show that time derivatives of excitation coefficients encapsulate back-reaction effects and may produce non-local-in-time dynamics and corrections to classical evolution, with phenomenological significance for early-universe and strong-gravity scenarios (Bojowald et al., 2012).

Energy-Transfer and Open-System Measurement

For energy transfer in molecular assemblies, the time derivatives of excitation coefficients provide a quantitative window into transfer times, efficiency reduction by defects, and the measurement process via norm decay in stochastic quantum-jump algorithms (Hofmann et al., 2012).

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The paper and computation of time derivatives of excitation coefficients intersect the foundations of non-equilibrium statistical mechanics, quantum kinetic theory, advanced response theory, and field-theoretic analysis. Their explicit time evolution, structural corrections via kernel frequency dependence, and dependence on representation and environmental coupling remain subjects of methodological and conceptual significance in quantum science.