Average Excitation Number (AEN)

• Average Excitation Number (AEN) is defined as the expectation value of the excitation-number operator, measuring mean excitations like photons or electrons in quantum systems.
• AEN is applied in quantum optics, nuclear physics, and electronic structure analysis to assess phenomena such as energy inflow, collective motion, and electron correlations.
• Recent studies show that relying solely on the average can mask high-excitation events and correlations, urging more robust statistical and metrological frameworks.

Average Excitation Number (AEN) is a foundational concept spanning quantum optics, quantum information, nuclear physics, statistical analysis of excitations, interferometric metrology, and electronic structure theory. It quantifies the mean number of excitations—such as photons, quasiparticles, or electronic transitions—within a quantum system or ensemble, providing a statistical or dynamical measure of the system's excitation landscape. The definition and relevance of AEN depend closely on the physical context and the structure of the underlying quantum states or spectra. Recent research emphasizes AEN's nuanced role: while it naturally captures excitation-level information in quantum optical and spectroscopic scenarios, overly simplistic use of the average alone as a resource quantifier may obscure the influence of high-number events or collective correlations, necessitating a more granular analysis.

1. Formal Definition and Quantification of AEN

In quantum systems, the Average Excitation Number is defined as the expectation value of an excitation-number operator. For photonic systems, this typically corresponds to $\langle a^\dagger a \rangle$, where $a^\dagger$ and $a$ are the creation and annihilation operators. In atomic, nuclear, or molecular settings, the AEN may quantify collective electronic, vibrational, or nucleonic excitations.

Mathematically, for a system described by a density matrix $\rho$ and excitation number operator $N$, the AEN takes the form: $\mathrm{AEN} = \mathrm{Tr}[\rho\,N]$ Depending on application, this may represent photon number, electron transitions, seniority excitations in configuration interaction expansions, or statistical averages over quantum level populations.

In interferometry and metrology, the average particle number $\langle n \rangle$ is commonly used; however, as demonstrated in (Branford et al., 2021), the full distribution over particle number is often more revealing of resource cost and sensitivity.

2. AEN in Quantum Optical and Statistical Physics Contexts

The AEN emerges as a primary diagnostic in quantum optics—for example, in cavity quantum electrodynamics (QED). In the context of the anti-rotating term in the Rabi Hamiltonian combined with Lindblad-type dephasing, the evolution of the mean photon number obeys a linear growth law: $$\langle n(t) \rangle = \langle n(0) \rangle + \frac{2\gamma g^2}{\omega^2 + \Omega^2 + \gamma^2} t$$ with $g$ (coupling strength), $\omega$ (cavity frequency), $\Omega$ (atomic transition frequency), and $\gamma$ (aggregate dephasing rate) controlling the rate (Dodonov, 2012). This demonstrates that non-unitary effects (decay, decoherence) combined with terms neglected through rotating wave approximations can mediate continuous energy inflow, driving up the AEN beyond equilibrium expectations.

In nuclear physics and collective excitation analysis, statistical treatment of spectra reveals regimes of order and chaos, reflected in the distribution of nearest-neighbor spacings (NNSDs) and parametrized by the level repulsion density $g(s)$ (1711.01848): $$p(s) = \frac{1}{\mathcal{N}} g(s) \, \exp\left[-\int_0^s g(s') \,\mathrm{d}s'\right]$$ Transitions between Poisson and Wigner-Dyson statistics indicate changes in the effective AEN—the number of collective degrees of freedom actively participating in nuclear dynamics.

3. Limitations and Subtleties in AEN as a Resource Metric

Recent quantum metrology research (Branford et al., 2021) exposes critical limitations of using AEN—or average number alone—as a cost or sensitivity metric. For variable-number probe states (e.g., vacuum–Fock superpositions), low values of $\langle n \rangle$ can coexist with extremely high quantum Fisher information (QFI), as the metrological advantage is concentrated in rare, high-photon-number events: $$\mathcal{J}(\psi) = \langle n \rangle \left[N - \frac{\langle n \rangle}{2}\right]$$ This decoupling implies that the average exposure or "damage" per trial, judged via AEN, may severely underestimate the true risk or resource demand in schemes where the tails of the number distribution dominate informative outcomes (e.g. N00N state interferometry).

Bayesian and frequentist approaches both confirm that the metrological figure of merit scales with higher moments or specific components in the number distribution, not simply the mean. This necessitates an expanded framework for resource accounting beyond AEN.

4. AEN in Quantum Many-Body and Electronic Structure Methods

In electronic structure theory, AEN provides a measure of electronic correlation and excitation complexity. The Hierarchy Configuration Interaction (hCI) method (Kossoski et al., 2022) introduces a hierarchy parameter $h = (e + s/2)/2$ (Editor’s term) that combines excitation degree $e$ and seniority number $s$, implicitly controlling the average excitation character:

• Low $h$ encompasses determinants with minimal excitations or seniority.
• Higher $h$ includes configurations important for both dynamic and static correlations. While explicit formulae for AEN are not provided, the model achieves improved accuracy for bond-breaking and dissociation processes by balancing the inclusion of determinants across excitation and seniority space.

In ppRPA (particle–particle random phase approximation) and active-space methods (Li et al., 2023), AEN can be calculated as the mean over computed excitation eigenvalues for a subset of relevant states: $\mathrm{AEN} = \frac{1}{M} \sum_{m=1}^{M} \omega_m$ where $\omega_m$ are eigenvalues of the truncated ppRPA matrix. Accurate evaluation of AEN reflects the quality of the low-energy excitation spectrum and the multi-configurational character of molecular excited states.

5. Analytical and Computational Strategies for Determining AEN

Computational approaches to AEN depend on context:

• In quantum optics, analytical differential equations yield closed-form expressions for $\langle n(t) \rangle$.
• In nuclear spectral analysis, statistical fitting to NNSD using linear level repulsion or Brody parameters provides empirical AEN-related metrics.
• In electronic structure, CI wave function expansions or ppRPA eigenvalue statistics are used to obtain average excitation energies and configuration weights.

Recent advances employ neural network wave functions (e.g., using Ferminet) with effective core potentials (ECPs) to calculate electronic excited states at high precision (Liu et al., 2023). Orthogonality-enforced QMC loss functions enable accurate resolution of multiple excited states, enhancing the effective AEN within a chosen energy window and yielding results competitive with experimental benchmarks.

6. AEN in Non-Markovian Open Quantum Systems and Real-World Control

Non-Markovianity introduces memory effects in open quantum systems, fundamentally altering excitation dynamics. The steady-state AEN in a quantum oscillator coupled to a bath can escalate rapidly with increased non-Markovian coupling (small spectral width $\gamma$) (Wang et al., 2024), and detuning between system and bath frequencies can stabilize the AEN at its initial value over extended periods. Analytical solution of the system’s master equation yields: $$\langle a^\dagger(t)a(t)\rangle = |A(t)|^2 \langle a^\dagger(0)a(0)\rangle + \sum_j |B_j(t)|^2 \langle b_j^\dagger(0)b_j(0)\rangle$$ where $A(t)$ and $B_j(t)$ are expansion coefficients determined by system-bath interactions. Pulse-control strategies leveraging leakage elimination operators (LEO) can tune dynamical evolution of the AEN, either prolonging relaxation or accelerating return to equilibrium. The emergence of the quantum Mpemba effect in this setting—faster cooling for initially hotter states upon kick pulses—demonstrates the nontrivial impact of dynamical control on excitation statistics.

7. Broader Implications and Applications

The AEN acts as an essential measure in:

• Assessing photon generation and energy inflow mechanisms in cavities under non-unitary evolution.
• Quantifying nuclear collective motion and transitions between order and chaos in many-body systems.
• Designing quantum metrology protocols with tailored resource profiles.
• Evaluating electron correlation and excitation complexity in ab initio and neural network–based electronic structure theory.
• Diagnosing and controlling dissipation and decoherence in realistic quantum environments.

However, careful interpretation is required—especially in quantum metrology, where naïve use of AEN as a sole resource or damage metric may misrepresent actual operational risks or advantages. A plausible implication is the need for resource metrics incorporating full excitation number distributions, complementing AEN with higher statistical moments and event-specific contributions.

In sum, the concept of Average Excitation Number remains indispensable yet should be contextualized with a comprehensive statistical and dynamical analysis appropriate to each application domain.