Constant-Excitation Fock-State Spaces

• Constant–excitation Fock–state spaces are defined as subspaces of Hilbert space with fixed excitation numbers (e.g., photons, particles), fundamental for describing specific quantum systems.
• They enable detailed operator analysis and variational optimization, facilitating efficient state preparation, simulation in quantum optics, and control in condensed matter physics.
• Advanced techniques, including lattice mappings and sampling theory, are employed to analyze spectral properties, optimize quantum interference, and generate robust Fock and squeezed states.

Constant–excitation Fock–state spaces comprise the subspaces of Hilbert space (typically Fock space, the canonical state space for bosonic or fermionic quantum systems) consisting of all states with a fixed excitation number. The excitation number may correspond to total particle number, photon number, total spin, or energy quanta. Such subspaces are ubiquitous in quantum optics, condensed matter physics, and quantum information, playing essential roles in describing exactly N-particle, N-photon, or N-excitation phenomena. The structure, analysis, and manipulation of these spaces involve both analytic aspects (function theory, operator theory, representation theory) and deep connections to lattice models, sampling theory, and optimal quantum state control.

1. Mathematical Characterizations and Operator Theory

A Fock space is, in its canonical form, the Hilbert space of entire functions $f$ on $\mathbb{C}^n$ with Gaussian weights, or an infinite direct sum $\mathcal{F} = \bigoplus_{k=0}^\infty \mathcal{F}_k$ where $\mathcal{F}_k$ contains all states of excitation number $k$. The constant–excitation Fock–state space for some $k$ is then simply $\mathcal{F}_k$. In analytic terminology, this can correspond to spaces of homogeneous polynomials of degree $k$.

Operator-theoretic features hinge on excitation number. For instance, the differentiation operator $D$, when acting on generalized Fock spaces $F(m, p)$, mimics the quantum annihilation operator—lowering excitation number by one. Its spectral properties are strongly dependent on the growth rate of the weight: for critical weights ($m = 1$) its spectrum is precisely the closed unit disk in the complex plane, revealing deep algebraic and analytic symmetries (Mengestie, 2016). The boundedness and compactness of such operators are strictly contingent on excitation constraints and weight decay properties.

A recent result formalizes the maximal Fock space $F^\infty_\alpha$ as a Lipschitz space with a specific metric $d_\alpha$, specializing the classical Bloch space criterion to the Fock setting (Bao et al., 1 Apr 2025):

$$|f(z) - f(w)| \leq C d_\alpha(z, w) \text{ for all } z, w \in \mathbb{C}^n, \quad f \in F^\infty_\alpha$$

where $d_\alpha(z,w)$ captures the geometry imposed by fixed excitation via the Gaussian measure.

2. Fock–State Lattice Construction and Constant–Excitation Subspaces

Fock–state lattices (FSLs) represent interaction Hamiltonians (e.g., Jaynes–Cummings or multi-mode boson models) in a basis indexed by excitation number, mapping quantum dynamics onto synthetic lattice models where lattice sites correspond to occupation number configurations with fixed total excitation (Saugmann et al., 2022). These lattices can have nontrivial geometry and topological properties:

• For multimode boson systems, fixed total excitation yields a triangular or hexagonal lattice geometry in occupation space.
• In the central spin model, the FSL is equivalent to a finite SSH chain with topologically protected edge states; spatially alternating tunneling strengths manifest exponential localization characteristic of edge modes.

Symmetries (continuous U(1) vs. discrete $\mathbb{Z}_2$) serve to block-diagonalize the state space, with constant–excitation subspaces corresponding to invariant lattice sectors. The interplay of synthetic gauge fields (via phase-engineered couplings $e^{i\phi_{ij}}$) produces fractal spectra or chiral propagation patterns of excitations.

3. Variational Principles and Optimization Algorithms

Efficient computation in constant–excitation Fock–state spaces leverages variational principles tailored for excited states, notably the functional (Zhao et al., 2015):

$$\Omega(\Psi) = \frac{\langle\Psi|(\omega-H)|\Psi\rangle}{\langle\Psi|(\omega-H)^2|\Psi\rangle}$$

where $H$ is the Hamiltonian and $\omega$ is positioned between desired eigenvalues. This construct ensures the global minimum targets the eigenstate just above $\omega$ in energy. With suitable ansatzes (e.g., Jastrow Antisymmetric Geminal Power, multi-Slater expansions), direct Monte Carlo evaluation of $\Omega(\Psi)$ permits efficient Fock-space sampling restricted to fixed excitation sectors. Optimization parallels the linear method, solving generalized eigenvalue problems in the space spanned by the ansatz and its parameter derivatives. The cost of matrix element evaluation scales polynomially with system size, making these techniques practical for large constant–excitation subspaces.

4. Quantum Interference, Dark States, and Topological Phenomena

Quantum interference in constant–excitation FSLs underpins phenomena such as dark states—superpositions decoupled from atomic excitation, and thus protected from radiative decay (Zhao et al., 29 Sep 2024). Using the arrowhead-matrix method in multimode Jaynes–Cummings models, the number of orthogonal dark states in the $n$-excitation subspace for $N$ modes is

$C_{N+n-2}^{N-2} = \frac{(N+n-2)!}{(N-2)! n!}$

These states are encoded in the null space of the coupling matrix between atomic and photonic degrees of freedom and can be explicitly constructed using bright/dark-mode decompositions. Topological edge states and flat-band phenomena are realized as spatially localized dark states in models mapped to SSH and related lattices (Saugmann et al., 2022). Such states play roles not only in quantum simulation and coherent control but also as loss-resilient channels in photonic networks.

5. Sampling, Trace Formulas, and Signal Reconstruction

Sampling theory in generalized Fock spaces yields explicit bounds for reconstructing state norms from partial data—providing quantitative control over sampling constants in terms of the relative density of the sampling set (Konate et al., 2021). For a dominating set $E$ and $f$ in a weighted Fock space $F^p_\phi$, one obtains

$$\|f\|_{L^p_\phi(E)} \ge \left(\frac{\gamma}{c}\right)^{L} \|f\|_{L^p_\phi(\mathbb{C})}$$

where the exponent $L$ and constant $c$ depend on coverage, measure-doubling, and dimension parameters. Such results underpin reliable state reconstruction and stability results for Toeplitz operator invertibility.

Trace formulas for the evaluation of physical quantities (e.g., partition functions, Gibbs states) exploit constant–excitation subspaces as finite-dimensional approximants to infinite-dimensional Fock spaces (Wick et al., 2020). Explicitly,

$$\operatorname{Tr}(X) = \lim_{n \to \infty} \int_{\mathbb{C}^n} [X K_x^{(n)}(z) K_x^{(n)}(z)] d\lambda_n(x)$$

where $K_x^{(n)}$ is the reproducing kernel for the $n$-particle sector.

6. Generation of Large Fock States and Squeezed States

Modern nonlinear nanophotonic architectures allow deterministic generation and stabilization of large-number Fock states and highly squeezed states via bound states in the continuum (BICs) (Rivera et al., 2022). Engineered cavities with Kerr nonlinearity can be tuned such that radiation loss vanishes precisely when the cavity contains $n_0$ photons; injection of a coherent state then results in natural evolution toward an exactly $n_0$-photon Fock state ($|\Delta n| \ll 1$). Squeezing protocols realize noise reduction in photon number variance by over 90% (up to 10 dB below shot noise) even in the presence of moderate loss.

7. Open Quantum Systems: Liouville Fock State Lattices

Vectorization of the Lindblad master equation leads to Liouville Fock State Lattices (LFSLs)—synthetic lattices in a doubled Hilbert space (density-matrix formalism) (Naves et al., 27 Mar 2025). Each “site” corresponds to a component $|n, m\rangle\rangle$ of the density matrix, and non-Hermitian Liouvillian dynamics introduce population drift, sources, and sinks reminiscent of classical stochastic processes. In constant-excitation sectors, open-system dynamics can exhibit infinite degeneracy in steady states due to geometric frustration. Alternative representations (Bloch and SIC-POVM) yield positive semidefinite probability-like distributions, allowing these systems to function as simulators for classical nonequilibrium phenomena and anomalous transport.

8. Operator Structure, Allowed Mappings, and Quantum Channels

Weighted superposition operators $S_{(u,\psi)}$—where $u, \psi$ are entire analytic symbols—are severely restricted when mapping between Fock spaces (Mengestie, 19 Jun 2025). For a transformation $S_{(u,\psi)}$ to be bounded and Lipschitz between $\mathcal{F}_p$ and $\mathcal{F}_q$, $\psi$ must be affine and $u$ either constant or of controlled exponential quadratic form with $|a_2| < 1/2$. No nontrivial compact weighted superposition operator exists. This suggests that practically only “gentle” (linear or trivial) operations preserve constant–excitation Fock–state structure, providing a functional-analytic analogue to quantum channel constraints for number-conserving operations.

Summary Table: Constant–Excitation Fock–State Spaces—Concepts and Roles

Aspect	Description	Example Reference
Operator-theoretic structure	Excitation-number–preserving subspaces, spectral analysis of D	(Mengestie, 2016, Bao et al., 1 Apr 2025)
FSL mappings & topological modes	Synthetic lattices, dark states, topological edge modes	(Saugmann et al., 2022, Zhao et al., 29 Sep 2024)
Variational excited-state methods	Direct targeting of excited subspaces, efficient Monte Carlo	(Zhao et al., 2015)
Sampling and reconstruction	Polynomial bounds on sampling constants, trace formulas	(Konate et al., 2021, Wick et al., 2020)
Nonlinear quantum state control	Deterministic generation/squeezing of Fock states via BIC	(Rivera et al., 2022)
Open-system LFSLs & simulators	Non-Hermitian dynamics, classical simulation analogues	(Naves et al., 27 Mar 2025)
Operator mapping restrictions	Superposition operators, boundedness, Lipschitz continuity	(Mengestie, 19 Jun 2025)

Constant–excitation Fock–state spaces serve as rigorous frameworks for quantum state preparation, excited-state computation, physical simulation, and topological state engineering. Their mathematical characterizations—via operator theory, analytic function spaces, synthetic lattices, and variational optimization—underpin much of modern quantum theory and quantum technology research.