Fock-State Lattices in Quantum Systems

• Fock-state lattices are synthetic graphs in Hilbert space where basis states act as lattice sites connected via Hamiltonian or Liouvillian-induced transitions.
• They map complex quantum systems, including cavity QED and many-body interactions, to tight-binding models on finite, semi-infinite, or higher-dimensional synthetic graphs.
• FSLs enable analysis of topological transport, dynamic localization, and interference effects, offering practical insights for experimental and theoretical quantum research.

Fock-state lattices (FSLs) are synthetic lattices in Hilbert space whose sites are basis states of a quantum system and whose bonds are the Hamiltonian-induced or Liouvillian-induced transitions between those states. In the quantum-optical formulation, a many-body Fock state such as $|n_1,n_2,\dots,n_d\rangle$ is treated as a lattice site, each bosonic mode provides a lattice dimension, and matrix elements of creation and annihilation operators generate site-dependent hopping amplitudes, typically with square-root bosonic factors. This viewpoint recasts models from cavity and circuit QED, waveguide photonics, and many-body lattice physics as tight-binding problems on finite, semi-infinite, or higher-dimensional synthetic graphs, and it has been used to analyze topological transport, dynamic localization, artificial gauge fields, dark states, non-Hermitian effects, and open-system dynamics (Saugmann et al., 2022, Deng et al., 2022, Naves et al., 27 Mar 2025).

1. State-space definition and basic construction

A general FSL can be defined from a basis $|\{n\}\rangle$ and the matrix elements of a Hamiltonian $\hat H$. In this representation, the on-site energy of site $\{n\}$ is

$$\varepsilon_{\{n\}}=\langle \{n\}|\hat H|\{n\}\rangle,$$

and a bond between $\{m\}$ and $\{n\}$ exists whenever

$$J_{\{nm\}}=\langle \{n\}|\hat H|\{m\}\rangle \neq 0.$$

Accordingly,

$$\hat H=\sum_{\{n\}} \varepsilon_{\{n\}}\,|\{n\}\rangle\!\langle\{n\}|+\sum_{\{n\}\neq \{m\}} J_{\{nm\}}\,|\{n\}\rangle\!\langle\{m\}|.$$

In quantum-optical settings, the natural basis is a bare Fock basis, so the lattice coordinates are occupation numbers and internal-state labels; continuous symmetries such as excitation-number conservation reduce the effective dimension of the reachable graph (Ferraro et al., 10 Apr 2026, Saugmann et al., 2022).

This synthetic-lattice interpretation is not restricted to bosonic few-mode models. In disordered interacting spinless-fermion systems at fixed filling $\nu=N_e/N$, each occupation-number basis state

$|\{n\}\rangle$0

is a Fock-space lattice site, with total site count

$|\{n\}\rangle$1

The diagonal energy of a site is

$|\{n\}\rangle$2

while off-diagonal hopping connects configurations differing by a single nearest-neighbour fermion hop in real space (Welsh et al., 2018). This broader usage makes clear that an FSL is defined by basis connectivity rather than by a particular physical platform.

A recurring misconception is to identify an FSL with a lattice in real space. In the superconducting-circuit literature, the opposite statement is explicit: FSL sites are not positions in real space but photon-number basis states of a multimode quantum system, and the Hilbert space itself acts like a spatial lattice (Zhang et al., 2024). This distinction is central to the entire framework.

2. Semi-infinite bosonic lattices and the Glauber-Fock paradigm

A particularly clean FSL is the Glauber-Fock lattice, a semi-infinite tight-binding chain with site index $|\{n\}\rangle$3 and nearest-neighbor hopping rates

$|\{n\}\rangle$4

Its sites represent bosonic number states $|\{n\}\rangle$5, and the $|\{n\}\rangle$6 hopping law is exactly the bosonic ladder-operator matrix element $|\{n\}\rangle$7 (Longhi et al., 2013).

In photonic implementations, the same structure appears in a semi-infinite waveguide array with coupled-mode equation

$|\{n\}\rangle$8

Here every excited waveguide represents a Fock state, and propagation along $|\{n\}\rangle$9 reproduces the coefficients of displaced Fock states: $\hat H$0 Launching waveguide $\hat H$1 therefore realizes the classical analogue of the displaced number state $\hat H$2 (Keil et al., 2011).

Under a homogeneous time-dependent force $\hat H$3, the driven Glauber-Fock lattice obeys

$\hat H$4

After the gauge transformation $\hat H$5, $\hat H$6, the dynamics is governed by the complex quantity

$\hat H$7

The exact revival condition is

$\hat H$8

For a sinusoidal drive $\hat H$9, this becomes

$\{n\}$0

A notable result is that this dynamic-localization condition, familiar from homogeneous infinite lattices, remains exactly valid in the semi-infinite inhomogeneous Glauber-Fock case; numerical comparison showed that with $\{n\}$1 and $\{n\}$2, revival fails in a homogeneous truncated chain but is exact in the Glauber-Fock lattice (Longhi et al., 2013). This establishes that the boundary at $\{n\}$3 need not spoil self-imaging when it is intrinsic to the Fock-space structure.

The non-Hermitian extension reinforces the same point from a different direction. In a semi-infinite non-Hermitian Glauber-Fock lattice with unidirectional coupling,

$\{n\}$4

the exceptional-point limit $\{n\}$5 yields a continuous family of stationary states with coherent-state coefficients

$\{n\}$6

At $\{n\}$7, this reduces to the left-edge state $\{n\}$8, while finite truncations of these profiles propagate almost diffraction-free over long distances (Yuce et al., 2020). The semi-infinite FSL thus supports exact solvability in both Hermitian and non-Hermitian regimes.

3. Topological transport and pumping in Jaynes-Cummings-type FSLs

The multimode Jaynes-Cummings (JC) family provides a second major route to FSLs. In a two-mode JC model with time-dependent couplings $\{n\}$9 and $$\varepsilon_{\{n\}}=\langle \{n\}|\hat H|\{n\}\rangle,$$0,

$$\varepsilon_{\{n\}}=\langle \{n\}|\hat H|\{n\}\rangle,$$1

the conserved excitation number confines the dynamics to a fixed-$$\varepsilon_{\{n\}}=\langle \{n\}|\hat H|\{n\}\rangle,$$2 sector. In that sector, the basis states form a one-dimensional synthetic lattice with matrix elements

$$\varepsilon_{\{n\}}=\langle \{n\}|\hat H|\{n\}\rangle,$$3

$$\varepsilon_{\{n\}}=\langle \{n\}|\hat H|\{n\}\rangle,$$4

Bright and dark modes,

$$\varepsilon_{\{n\}}=\langle \{n\}|\hat H|\{n\}\rangle,$$5

reduce the dynamics to an instantaneous zero-energy dark state that expands as a binomial wavepacket on the FSL (Wu et al., 2 Jun 2026).

For power-law coupling profiles

$$\varepsilon_{\{n\}}=\langle \{n\}|\hat H|\{n\}\rangle,$$6

the global adiabatic criterion

$$\varepsilon_{\{n\}}=\langle \{n\}|\hat H|\{n\}\rangle,$$7

shows that fast transfer is controlled not by a constant gap alone but by the mean and variance of the nonadiabatic factor $$\varepsilon_{\{n\}}=\langle \{n\}|\hat H|\{n\}\rangle,$$8. For $$\varepsilon_{\{n\}}=\langle \{n\}|\hat H|\{n\}\rangle,$$9, $\{m\}$0 is constant and $\{m\}$1; within this family, sinusoidal coupling is therefore globally optimal. The same work constructs a constant-gap family with $\{m\}$2 for all $\{m\}$3 and shows that constant gap alone is neither necessary nor sufficient for fast, high-fidelity transfer. Incorporating experimental decoherence parameters, the predicted optimal duration is

$\{m\}$4

so that for $\{m\}$5, $\{m\}$6, compared with $\{m\}$7 used in the experiment; the predicted transferred photon number is $\{m\}$8 (Wu et al., 2 Jun 2026).

A closely related but physically distinct construction uses a Rydberg superatom coupled to microwave and optical cavities. Restricting to a fixed total excitation number $\{m\}$9, the basis states $\{n\}$0 form a cross-linked one-dimensional FSL with Hamiltonian

$\{n\}$1

where

$\{n\}$2

This is mapped to an extended SSH chain with a zero-energy defect state that moves across the lattice as the ratio $\{n\}$3 is varied. The energy gap to the nearest bulk states is

$\{n\}$4

and adiabatic following pumps the state from $\{n\}$5 to $\{n\}$6. With realistic dissipation, about $\{n\}$7 optical photons are produced from $\{n\}$8 initial microwave photons, and under $\{n\}$9 coupling disorder the average optical output remains above $$J_{\{nm\}}=\langle \{n\}|\hat H|\{m\}\rangle \neq 0.$$0 (Song et al., 10 Apr 2025).

More broadly, superconducting-circuit experiments have realized one- and two-dimensional FSLs in which topological transport of zero-energy states, strain induced pseudo-Landau levels, valley Hall effect, and Haldane chiral edge currents were demonstrated (Deng et al., 2022). This suggests that FSL topology is not confined to one-dimensional SSH analogies but extends naturally to higher-dimensional synthetic lattices.

4. Interference, dark subspaces, and flux-induced localization

Because FSLs often contain multiple transition paths between the same pair of synthetic sites, destructive interference is structurally generic. In $$J_{\{nm\}}=\langle \{n\}|\hat H|\{m\}\rangle \neq 0.$$1-mode JC models, the Hamiltonian in a fixed-$$J_{\{nm\}}=\langle \{n\}|\hat H|\{m\}\rangle \neq 0.$$2 excitation sector acquires an arrowhead block form,

$$J_{\{nm\}}=\langle \{n\}|\hat H|\{m\}\rangle \neq 0.$$3

and dark states are exactly the lower-state superpositions in the null space of the coupling matrix,

$$J_{\{nm\}}=\langle \{n\}|\hat H|\{m\}\rangle \neq 0.$$4

For a general $$J_{\{nm\}}=\langle \{n\}|\hat H|\{m\}\rangle \neq 0.$$5-mode JC model under equal detunings, the number of orthogonal dark states in the $$J_{\{nm\}}=\langle \{n\}|\hat H|\{m\}\rangle \neq 0.$$6-excitation subspace is

$$J_{\{nm\}}=\langle \{n\}|\hat H|\{m\}\rangle \neq 0.$$7

When the dark-subspace dimension exceeds one, the subspace is unique but the choice of dark-state basis is not (Zhao et al., 2024).

This null-space mechanism has direct higher-dimensional consequences. In superconducting quantum circuits, four transmon-type qutrits were used to construct 2D, pseudo-3D, and genuine 3D FSLs from photon-number configurations such as $$J_{\{nm\}}=\langle \{n\}|\hat H|\{m\}\rangle \neq 0.$$8, $$J_{\{nm\}}=\langle \{n\}|\hat H|\{m\}\rangle \neq 0.$$9, $$\hat H=\sum_{\{n\}} \varepsilon_{\{n\}}\,|\{n\}\rangle\!\langle\{n\}|+\sum_{\{n\}\neq \{m\}} J_{\{nm\}}\,|\{n\}\rangle\!\langle\{m\}|.$$0, and $$\hat H=\sum_{\{n\}} \varepsilon_{\{n\}}\,|\{n\}\rangle\!\langle\{n\}|+\sum_{\{n\}\neq \{m\}} J_{\{nm\}}\,|\{n\}\rangle\!\langle\{m\}|.$$1. Artificial gauge fields were introduced by Floquet engineering, and the gauge-invariant flux through a loop was defined by the accumulated Peierls phase. In the 2D plaquette, $$\hat H=\sum_{\{n\}} \varepsilon_{\{n\}}\,|\{n\}\rangle\!\langle\{n\}|+\sum_{\{n\}\neq \{m\}} J_{\{nm\}}\,|\{n\}\rangle\!\langle\{m\}|.$$2 produces a free quantum walk, whereas $$\hat H=\sum_{\{n\}} \varepsilon_{\{n\}}\,|\{n\}\rangle\!\langle\{n\}|+\sum_{\{n\}\neq \{m\}} J_{\{nm\}}\,|\{n\}\rangle\!\langle\{m\}|.$$3 causes perfect destructive interference and Aharonov-Bohm caging. The same mechanism was extended to a pseudo-3D structure with two perpendicular plaquettes and to a 3D octahedral FSL, where a coherent superposition state,

$$\hat H=\sum_{\{n\}} \varepsilon_{\{n\}}\,|\{n\}\rangle\!\langle\{n\}|+\sum_{\{n\}\neq \{m\}} J_{\{nm\}}\,|\{n\}\rangle\!\langle\{m\}|.$$4

was localized within the equatorial $$\hat H=\sum_{\{n\}} \varepsilon_{\{n\}}\,|\{n\}\rangle\!\langle\{n\}|+\sum_{\{n\}\neq \{m\}} J_{\{nm\}}\,|\{n\}\rangle\!\langle\{m\}|.$$5 plane by applying $$\hat H=\sum_{\{n\}} \varepsilon_{\{n\}}\,|\{n\}\rangle\!\langle\{n\}|+\sum_{\{n\}\neq \{m\}} J_{\{nm\}}\,|\{n\}\rangle\!\langle\{m\}|.$$6 fluxes on the longitudinal plaquettes (Zhang et al., 2024). Unlike Anderson localization, this caging does not rely on disorder; it is produced by flux-controlled interference.

Finite FSLs can also encode spin-dependent circulation. In a system of three cavities coupled to one two-level atom, periodic modulation of the cavity frequencies with relative phase shift $$\hat H=\sum_{\{n\}} \varepsilon_{\{n\}}\,|\{n\}\rangle\!\langle\{n\}|+\sum_{\{n\}\neq \{m\}} J_{\{nm\}}\,|\{n\}\rangle\!\langle\{m\}|.$$7 yields the effective Hamiltonian

$$\hat H=\sum_{\{n\}} \varepsilon_{\{n\}}\,|\{n\}\rangle\!\langle\{n\}|+\sum_{\{n\}\neq \{m\}} J_{\{nm\}}\,|\{n\}\rangle\!\langle\{m\}|.$$8

which realizes a finite spin-orbit-coupled FSL. The atomic state selects the chirality of photon circulation, and an atomic superposition routes the cavity field into a superposition of distinct photon-number configurations, enabling NOON states, entangled coherent states, and micro-macro entangled states (Wang et al., 2016). In the condensed-matter language adopted for FSLs, this is a finite triangular lattice with synthetic gauge structure.

5. Algebraic and graph-theoretic formulations

An algebraic formulation makes the geometry of an FSL explicit before any particular Hamiltonian is chosen. Starting from a Lie algebra $$\hat H=\sum_{\{n\}} \varepsilon_{\{n\}}\,|\{n\}\rangle\!\langle\{n\}|+\sum_{\{n\}\neq \{m\}} J_{\{nm\}}\,|\{n\}\rangle\!\langle\{m\}|.$$9, diagonal Cartan generators label the lattice sites, and off-diagonal root generators define the lattice bonds. If the rank of the algebra is $\nu=N_e/N$0, then

$\nu=N_e/N$1

This identifies $\nu=N_e/N$2 with a semi-infinite one-dimensional chain, $\nu=N_e/N$3 with a finite one-dimensional chain, $\nu=N_e/N$4 with a two-dimensional triangular lattice, $\nu=N_e/N$5 with a two-dimensional square lattice with diagonal bonds, and $\nu=N_e/N$6 with parity-separated one-dimensional chains. The same framework also associates each algebra with a Lie phase space of coherent states, and in many cases both the phase space and the FSL carry nontrivial curvature (Ferraro et al., 10 Apr 2026).

This viewpoint clarifies why some FSLs support perfect revivals, synthetic fluxes, or chirality. It also exposes a limit of algebraic reconstruction: not every integrable Hamiltonian admits a finite Lie algebra reproducing the same FSL structure. In the cited analysis, this failure is explicit for Hamiltonians nonlinear in the generators, and for systems combining bosonic and fermionic degrees of freedom the appropriate structure may instead be a Lie superalgebra; the Jaynes-Cummings model and Tavis-Cummings model are cited as cases requiring a superalgebraic description, while the quantum Rabi model and Lipkin-Meshkov-Glick model do not admit the relevant finite closure (Ferraro et al., 10 Apr 2026).

A complementary graph-theoretic language orders multi-photon FSL states by a pseudo-energy operator,

$\nu=N_e/N$7

which assigns a unique integer label to each $\nu=N_e/N$8-photon, $\nu=N_e/N$9-mode configuration. In this representation, tunneling operators $|\{n\}\rangle$00 act as pseudo-energy ladder operators, Fock graphs are encoded by an adjacency matrix $|\{n\}\rangle$01, and for $|\{n\}\rangle$02 and $|\{n\}\rangle$03 nearest-neighbor couplings in real space induce nonlocal couplings in Fock space, giving rise to all-optical dark states and nontrivial co-tunneling pathways (Tschernig et al., 2020).

The same algebraic infrastructure has been extended to discrete-time quantum walks on FSLs. There the step operator

$|\{n\}\rangle$04

uses displacement operators rather than ordinary nearest-neighbour shifts. The resulting walks exhibit ballistic spreading, coin-walker entanglement, symmetry-induced interference patterns, and algebra-dependent anomalies including super-ballistic spreading in $|\{n\}\rangle$05 and localization effects in $|\{n\}\rangle$06 and $|\{n\}\rangle$07 (Ferraro et al., 10 Apr 2026). This suggests that FSLs are not merely static synthetic graphs but a natural setting for state-space transport protocols.

6. Non-Hermitian, many-body, and open-system extensions

FSLs also appear in non-Hermitian and open-system settings, where the lattice interpretation becomes dynamical rather than purely spectral. A solvable semi-infinite FSL-based SSH model,

$|\{n\}\rangle$08

maps exactly to the Jaynes-Cummings Hamiltonian under a displacement transformation. Its intercell coupling

$|\{n\}\rangle$09

creates a site-dependent semi-infinite SSH chain with a stable zero mode that exists either as a left-edge state for $|\{n\}\rangle$10 or as a domain-wall-bound state near $|\{n\}\rangle$11 for $|\{n\}\rangle$12. With balanced gain and loss,

$|\{n\}\rangle$13

the same system exhibits a non-Hermitian bound effect: any state overlapping with the bound state stabilizes to the domain wall after renormalization, and the minimal stabilization time occurs near the exceptional point (Mi et al., 21 Jun 2025).

Open quantum systems lead to a doubled construction, the Liouville Fock-state lattice (LFSL). Vectorizing the Lindblad master equation,

$|\{n\}\rangle$14

maps $|\{n\}\rangle$15 to a superstate

$|\{n\}\rangle$16

in the doubled Hilbert space $|\{n\}\rangle$17. The Liouvillian matrix then defines a synthetic lattice whose diagonal sites represent populations, off-diagonal sites represent coherences, and whose non-Hermitian structure yields directed transport, sources, sinks, decaying sectors, and steady-state manifolds. In the SIC-POVM representation the dynamics acquires a classical-looking form,

$|\{n\}\rangle$18

although negative off-diagonal rates remain possible. A central result is that geometrical frustration in LFSLs can produce infinitely degenerate steady-state manifolds (Naves et al., 27 Mar 2025).

Finally, there are adjacent frameworks that are FSL-like in structure without using the modern terminology. One example is the construction of unconventional Fock algebras in which operators $|\{n\}\rangle$19 satisfy

$|\{n\}\rangle$20

For $|\{n\}\rangle$21 single-particle labels, the resulting tensor-product occupation basis has dimension $|\{n\}\rangle$22, and the root operators act as raising and lowering operators on a finite occupation-number graph. The formalism was developed to model fractionally charged quasiparticles, parafermions, and Majorana modes, rather than as a general FSL theory, but it supplies an explicit operator-level realization of a finite occupation-number lattice with unconventional exchange statistics (Cobanera, 2014).

Taken together, these developments show that FSLs are not a single model class but a general representation principle. Their common content is the replacement of real-space geometry by basis-state connectivity; their diversity lies in the underlying algebra, conservation laws, interference structure, and openness of the dynamics.