Nonlinear Time-Synthetic Mesh Lattices

• The paper presents a novel optical thermodynamic framework that predicts equilibrium modal distributions and measurable radiation pressure in synthetic mesh lattices.
• It demonstrates how time multiplexing and dynamic modulation enable programmable connectivity and Kerr-induced nonlinearity in engineered photonic systems.
• The results validate applications in ultrafast optical diagnostics, coherent state preparation, and controlled nonequilibrium thermodynamic processes.

Nonlinear time-synthetic mesh lattices are engineered photonic systems that exploit synthetic dimensions—specifically, time multiplexing—to realize mesh networks with tunable nonlinearity, controllable topology, and tailored bandstructure. These structures generalize the discrete nonlinear Schrödinger lattice framework to settings where the spatial degrees of freedom are replaced or supplemented by discrete time bins or delay-line loops, and the effective lattice connectivity is programmed through external modulation. The interplay of Kerr-type nonlinearities, mode–mode scattering, and driven–dissipative processes in such architectures places them at the forefront of both classical and quantum photonic many-body thermodynamics. The optical thermodynamic framework developed for large-scale, highly multimode nonlinear lattices extends naturally to these platforms, providing an analytic approach to their equilibrium and non-equilibrium physics.

1. Foundations: Synthetic Mesh Lattices and Nonlinearity

Time-synthetic mesh lattices operate by mapping spatial sites onto discrete time bins, typically realized through a sequence of fiber loops or dynamically modulated cavities interconnected via fiber couplers or electro-optic modulators. Each "site" represents a temporal slot, and inter-site hopping is determined by programmable delays or modulations. Nonlinearity is introduced by Kerr materials (e.g., silica fibers, integrated nonlinear waveguides) so that the evolution of the field amplitude in each time bin is governed by a discrete nonlinear Schrödinger (DNLS) equation with vertices corresponding to time bins:

$$i\frac{d\Psi_n}{dz} + \sum_{m} K_{nm}\Psi_m + \gamma |\Psi_n|^2 \Psi_n = 0$$

where $K_{nm}$ encodes mesh connectivity (set by the modulation scheme), and $\gamma$ characterizes the Kerr nonlinearity. In the weakly nonlinear regime, this framework supports statistical-mechanical analysis, extending the optical thermodynamic paradigm established for spatially multimode photonic arrays (Efremidis et al., 2022, Selim et al., 2022).

A distinctive feature of time-synthetic lattices is the ease of engineering both bandstructure and coupling topology dynamically. This approach enables the realization of topologically nontrivial bands, Floquet synthetic gauge fields, or higher-dimensional phenomenology in lower-dimensional physical settings.

2. Thermodynamic Description: State Variables and Equations of State

The macroscopic state of a nonlinear time-synthetic mesh lattice is captured by:

• Total power (optical "particle number") $P=\sum_n |c_n|^2$, an extensive invariant under Hamiltonian evolution.
• Internal energy $U = \sum_n E_n |c_n|^2$ where $E_n$ is the site-dependent (or mode-dependent) propagation constant, encoding the bandstructure.
• System size (number of synthetic time bins or modes) $M$.
• Entropy $S = \sum_n \ln n_n$ (in classical regime), which is maximized in thermal equilibrium.
• Temperature $T$, the Lagrange multiplier enforcing internal energy conservation.
• Chemical potential $\mu$, enforcing power conservation.

Under ergodic (long-time) evolution and sufficient nonlinearity to induce mode mixing, the modal power distribution follows the Rayleigh–Jeans law (Efremidis et al., 2022, Selim et al., 2022, Wu et al., 2019):

$n_n = \frac{T}{E_n - \mu}$

subject to $P = \sum_n n_n$ and $U = \sum_n E_n n_n$, yielding the fundamental equation of state

$U - \mu P = M T$

This "optical thermodynamic" description enables the prediction of equilibrium modal distributions, entropy production, and macroscopic observables for arbitrary synthetic lattice topologies.

3. Optical Thermodynamic Pressure and Nonlinear Mesh Dynamics

In nonlinear mesh lattices, the pressure conjugate to the number of time bins $M$ emerges as a central intensive variable. This optical thermodynamic pressure $p$ is defined thermodynamically as:

$$p = T \left(\frac{\partial S}{\partial M}\right)_{U,P}$$

Its explicit evaluation splits into two parts (Ren et al., 10 Apr 2024):

$p = Q p^{EM}_T + p_{entr}$

where $p^{EM}_T$ is the average electromagnetic radiation pressure exerted by the multimode field (directly measurable at the boundary), $Q$ is a geometry-dependent prefactor, and $p_{entr}$ is an entropic contribution arising from the change in mode density. For homogeneous mesh lattices, $p$ can be written analytically in terms of $T, \mu, M$ (Efremidis et al., 2022, Ren et al., 10 Apr 2024):

$p = T \ln\left(\frac{T}{E_M - \mu}\right)$

The physical implication is that variation of the synthetic lattice size (adding or subtracting time bins) or modification of coupling modulates a measurable radiation pressure, which can be repulsive or attractive depending on the internal energy and modal distribution. In time-synthetic systems, this provides an all-optical means to probe nonequilibrium thermomechanical phenomena, analogous to mechanical pressure in conventional lattices.

4. Entropy, Statistical Ensembles, and Nonlinear Equilibration

The entropy of nonlinear time-synthetic mesh lattices is directly computed from the distribution of mode occupancies:

$$S = \sum_{n=1}^M \ln n_n = M \ln T - \sum_n \ln(E_n - \mu)$$

In the dense-mode, large-$M$ limit, the entropy becomes extensive ($S \propto M$), and closed-form, Sackur–Tetrode–type expressions appear (Wu et al., 2020, Selim et al., 2022):

$$S(E,N,L) = L \ln\left(\frac{4 L k^2 N}{4k^2 N^2 - E^2}\right)$$

where $L$ represents the synthetic "volume" (number of time bins, or equivalently the mesh length), $N$ the total optical power, and $E$ the net internal energy. Thermodynamic intensive variables (temperature $T$, chemical potential $\mu$) follow from derivatives of $S$.

Upon initialization far from equilibrium (e.g., a short pulse addressing a subset of time bins), the system relaxes under nonlinear mode mixing to the Rayleigh–Jeans equilibrium, maximizing $S$ under the constraints of conserved $P$ and $U$. Experimental evidence in large-scale photonic meshes and fiber-based time-synthetic lattices confirms that the predicted equilibrium modal distributions and thermodynamic pressure are realized on accessible timescales (Ferraro et al., 2022, Kirsch et al., 15 Nov 2025).

5. Thermodynamic Processes: Isentropic Evolution and Joule–Thomson Effects

Optical thermodynamics in nonlinear mesh lattices supports direct analogues of adiabatic, isentropic, and free expansion processes:

• Isentropic mesh deformation: Adiabatic modulation of coupling coefficients or lattice size leads to rescaling of $T, \mu, U$ such that certain ratios (e.g., $\mu/T$) are conserved (Efremidis et al., 2021). The modal distribution shape is preserved, but the overall bandwidth and the thermodynamic variables rescale in proportion.
• Joule–Thomson expansion in time-synthetic meshes: Sudden addition of time bins (expansion of the synthetic dimension) at fixed total power and energy implements an isenthalpic process. As predicted by the thermodynamic equations, such expansion yields a drop in the optical temperature ($T$), and, at high power, induces condensation into the ground mode (full coherence), a phenomenon recently observed in large-scale mesh experiments (Kirsch et al., 15 Nov 2025).

This formalism generalizes to nonequilibrium protocols (e.g., time-dependent drive, fluctuating mesh connectivity), enabling the design of all-optical engines, refrigerators, and thermodynamic cycles (including Otto and Carnot analogues).

6. Numerical Validation and Practical Significance

Analytic predictions for equilibrium distributions, thermodynamic pressure, and entropy growth in nonlinear time-synthetic mesh lattices have been quantitatively validated against direct time-domain numerical simulations (e.g., full integration of the DNLS with large $M$ on complex connectivity graphs) (Efremidis et al., 2022, Ren et al., 10 Apr 2024, Selim et al., 2022). The agreement between ensemble averages and simulation outcomes for modal distributions, pressure vs. mesh size, and entropy production confirms the applicability of the thermodynamic framework to realistic settings.

Practical implications include:

• Ultrafast optical thermometric diagnostics: All-optical determination of $T$ and $\mu$ from mode-resolved intensity measurements.
• Radiation-pressure engineering: Tunable optical actuation via programmed mesh dynamics and nonequilibrium energy injection.
• Light-by-light management and beam combining: Harnessing isothermal and isentropic lattice expansions for coherent state preparation, as in photon-gas JT condensation (Kirsch et al., 15 Nov 2025).
• Platform for open-system and stochastic thermodynamics: The modular, programmable nature of time-mesh lattices enables studies of fluctuation theorems, work/heat statistics, and nonthermal steady states (Ramesh et al., 26 Jul 2024).

7. Outlook and Extensions

Nonlinear time-synthetic mesh lattices exemplify the synthesis of quantum optics, nonlinear dynamics, and statistical thermodynamics in programmable photonic systems. Future work will further develop nonequilibrium extensions—exploring transport, fluctuation theorems, open quantum engine protocols—and leverage the synthetic-lattice paradigm to access higher-dimensional and topologically nontrivial models. The analytic tractability of the thermodynamic approach obviates the need for direct electromagnetic field simulation or stress-tensor evaluation in the dense-mode regime, providing a highly efficient toolset for device design and phenomenological exploration (Kirsch et al., 15 Nov 2025, Selim et al., 2022, Ren et al., 10 Apr 2024).