Single-Excitation Friedrichs Model

• The single-excitation Friedrichs model is a paradigm that describes a discrete quantum state linearly coupled to a continuum, foundational for studying resonance and decay dynamics.
• It employs an exact resolvent method and pole equations to classify bound, virtual, and resonance states within a rigorously defined rigged-Hilbert-space framework.
• Extensions of the model support simulations of non-Markovian decay and open-system engineering in fields like quantum optics and many-body theory.

The single-excitation Friedrichs model is a fundamental paradigm for the analysis of quantum systems with a single discrete state (the "bare" or "excited" level) linearly coupled to a continuum of states. It provides an analytically tractable setting for the study of resonance phenomena, virtual states, bound-state formation, and non-Markovian decay dynamics, and serves as the archetype for open system models in quantum optics, many-body theory, and quantum statistical mechanics. Central features include exact expressions for the resolvent, explicit secular (pole) equations, and a rigorous framework for generalized completeness—allowing for bound, virtual, and resonance states and higher-order poles in rigged-Hilbert-space formalism (Xiao et al., 2016, Gadella et al., 2011).

1. Mathematical Structure and Hamiltonian

The canonical single-excitation Friedrichs model is defined on a Hilbert space

$$\mathcal{H} = \mathrm{span}\{|1\rangle\} \oplus L^2(0,\infty),$$

where $|1\rangle$ denotes the bare discrete (excited) level of energy $\omega_0\in\mathbb{R}$ and $|\omega\rangle$, $\omega>0$, are continuum eigenstates. The Hamiltonian is decomposed as

$H = H_0 + V,$

with

$$H_0 = \omega_0 |1\rangle\langle 1| + \int_0^\infty \omega\,|\omega\rangle\langle\omega|\,d\omega,$$

$$V = \lambda\int_0^\infty [g(\omega)|\omega\rangle\langle 1| + g^*(\omega)|1\rangle\langle\omega|]\,d\omega,$$

where $g(\omega)$ is the form factor (real for $\omega>0$), and $\lambda\in\mathbb{R}$ is a real, tunable coupling constant (Xiao et al., 2016, Gadella et al., 2011). The model generalizes to multiple discrete levels and multiple continua by extending to $N$-level systems and multiple cut structures in the complex plane (Zhang et al., 19 Dec 2025, Xiao et al., 2016).

2. Resolvent, Self-Energy, and Analytic Continuation

The full resolvent $G(z) = (z - H)^{-1}$, projected onto the discrete subspace, yields

$$G_{PP}(z) = P G(z) P = \frac{1}{z - \omega_0 - \Sigma(z)} P,$$

where the self-energy

$$\Sigma(z) = \lambda^2 \int_0^\infty \frac{|g(\omega')|^2}{z - \omega'}\,d\omega'$$

encodes all effects of the continuum and the energy-dependent decay processes (Xiao et al., 2016, Lonigro, 2021, Gadella et al., 2011). For $z$ above the continuum threshold ($\Im z \neq 0$, or $z > 0$), $\Sigma(z)$ develops an imaginary part due to dissipation:

$$\Sigma^\pm(E) = \lambda^2\,\mathrm{P.V.}\int_0^\infty \frac{|g(\omega')|^2}{E - \omega'}\,d\omega' \mp i \pi \lambda^2 |g(E)|^2, \qquad E>0.$$

The function $\Sigma(z)$ is then analytically continued from the physical (first) sheet to unphysical sheets—crucial for classifying all pole (state) types (Xiao et al., 2016, Facchi et al., 2019).

3. Pole Equation and Classification of States

All nontrivial spectral features arise as isolated poles $z_p$ of the resolvent, governed by the transcendental equation

$z_p - \omega_0 - \Sigma^{(\mathrm{II})}(z_p) = 0,$

where $\Sigma^{(\mathrm{II})}$ denotes analytic continuation to the appropriate Riemann sheet (second or higher) (Xiao et al., 2016, Lonigro, 2021). The classification is as follows:

• Bound State: $z_p \in \mathbb{R}$, $z_p < 0$ on the first (physical) sheet.
• Virtual State: $z_p\in\mathbb{R}$, $z_p<0$ on the unphysical (second) sheet.
• Resonance (Gamow State): $z_p \in \mathbb{C}\setminus\mathbb{R}$ on the second or higher sheet, $\Im z_p < 0$ (lower half-plane).
• Higher-Order Poles: Multiple simple poles may coalesce into a double or triple pole as parameters ($\omega_0, \lambda$, form-factor singularities) are tuned, corresponding to exceptional points or Jordan block structures.

Notably, when the bare state is below the continuum threshold ($\omega_0 < 0$), tuning on the coupling $\lambda$ generates both a bound-state pole (on the first sheet) and an accompanying virtual-state pole (on the second sheet), as established by analytic continuation and the argument principle (Xiao et al., 2016).

4. Spectral and Dynamical Consequences

The survival amplitude $a(t) = \langle 1|e^{-i H t}|1\rangle$ admits a representation dominated at intermediate times by the residue at the resonance pole:

$$a(t) = Z e^{-i E_R t - \Gamma t/2} + \mathrm{background},$$

where $E_R = \Re z_R$ and $\Gamma = -2\Im z_R$ are the position and width of the resonance. The background integral produces quadratic short-time (Zeno) behavior and long-time power-law corrections (Gadella et al., 2011, Lonigro, 2021). For models with $N$ levels or multiple continua, the long-time asymptotics are set by the number of bound states: pure decay ($M=0$), fractional decay to a constant ($M=1$), or persistent oscillations ($M\geq2$) in the survival probability (Zhang et al., 19 Dec 2025).

The model supports bound states in the continuum (BIC) when the spectral density vanishes on-shell, $J(E_b) = 0$ inside the continuum—arising for certain symmetry-protected or interference conditions (Zhang et al., 19 Dec 2025).

5. Completeness Relations and Rigged Hilbert Space

The single-excitation Friedrichs model admits a generalized completeness relation,

$$1 = \sum_b |z_b\rangle \langle z_b| + \int_C dE\, |\Psi(E)\rangle\langle\tilde{\Psi}(E)| + \sum_v |z_v\rangle\langle\tilde{z}_v| + \sum_R |z_R\rangle\langle\tilde{z}_R| + \dots,$$

where terms include contributions from bound-state poles (first sheet), virtual-state poles (second sheet), resonant (Gamow) poles, and scattering states along the contour $C$ which has been deformed to enclose all lower-half-plane second-sheet (resonance/virtual) poles (Xiao et al., 2016). If a pole is of $n$-th order, it contributes a chain of generalized Gamow vectors $|z^{(r)}\rangle$, satisfying

$$H|z^{(r)}\rangle = z|z^{(r)}\rangle + |z^{(r-1)}\rangle, \quad r=2,\ldots,n,$$

integrated into the completeness relation as

$$1 = \int_C dE\,|\Psi(E)\rangle\langle\tilde{\Psi}(E)| + \sum_{r=1}^n |z^{(r)}\rangle\langle\tilde{z}^{(n-r+1)}| +\ldots.$$

This construction is rigorously justified in the rigged-Hilbert-space framework as detailed in the appendix of (Xiao et al., 2016).

6. Generalizations and Physical Interpretation

The single-excitation Friedrichs model extends naturally to include multiple discrete levels (multi-level Friedrichs), multiple continua, and models with structured or singular atom-field coupling. In multi-continuum cases, the spectrum acquires a Riemann surface structure with multiple sheets, generating multiple virtual and resonance poles for each threshold (cut), with pole trajectories crossing from one sheet to another as couplings are varied (Xiao et al., 2016). The completeness and pole-structure analyses generalize accordingly.

Form-factor singularities (e.g., $g(\omega)\sim 1/(\omega+\rho^2)^{1/2}$) generate additional second-sheet poles associated with dynamical "molecular" states, distinguishable from seed poles deriving from discrete levels ("quark" states in hadronic language) (Xiao et al., 2016). The dynamical merger of poles and formation of higher-order (exceptional) points is described by discriminant conditions on the pole equations.

In open-system quantum optics and quantum statistical applications, the Friedrichs model provides a paradigm for exact non-Markovian decay, memory effects, temperature-dependent relaxation, and the direct engineering of spectral and decay profiles via form-factor (spectral density) design (Gadella et al., 2011, Teretenkov, 2020, Lonigro, 2021).

7. Numerical Realizations and Applications

Modern numerical implementations recast the single-excitation Friedrichs model as a Volterra-type integro-differential equation for the discrete state amplitude, incorporating memory kernels derived from the spectral density. Efficient algorithms deploy high-order temporal collocation (Gauss-Legendre) and sum-of-exponential history compression for nearly linear computational cost over long times (Hoskins et al., 2021). These schemes support simulations of spontaneous emission, photon-pulse driving, and multi-mode decay signatures, robustly capturing phenomena such as algebraic decay tails, mode trapping, and collective effects in extended systems.

Applications range from analytic modeling of spontaneous emission and resonance enhancement to open-system engineering in photonic crystals, atomic chains in waveguides, and rigorous resolution of the direct/inverse spectral problem for single-quanta quantum dynamics (Zhang et al., 19 Dec 2025, Facchi et al., 2019, Hoskins et al., 2021).