The effect of discrete-time evolution on thermalisation in a lattice

Abstract

Particles subject to weak contact interactions in a finite-size lattice tend to thermalise. The Hamiltonian evolution ensures energy conservation and the final temperature is fully determined by the initial conditions. In this work we show that equilibration processes are radically different in lattices subject to discrete-step unitary evolution, which have been implemented in photonic circuits, coin polarisation walkers and time-multiplexed pulses in fibres. Using numerical simulations, we show that weak nonlinearities lead to an equilibrium state of equal modal occupation across all the bands, i.e, a thermal state with infinite temperature and chemical potential. This state is reached no matter the initial distribution of modes and independent of the bands dispersions and gaps. We show that by engineering the temporal periodicity inherent to these systems, equilibration can be accelerated through resonant Floquet channels. Our results demonstrate that discrete time plays a crucial role in wave dynamics, fundamentally altering thermalisation processes within weak wave turbulence theory.

Figures (4)

• Figure 1: (a) One dimensional lattice subject to a cascade of amplitude splitters (red diamonds) and phase shifters (purple units). The space-time unit cell of the lattice is marked in dashed lines, with a two-step temporal periodicity and two-site spatial periodicity. The orange and blue circles highlight the sites of the two natural sublattices of the system. (b) The dispersion relation for the case with parameters $(t^2,\phi_0) = (0.75,\pi/3)$ and N=40. The Brillouin zone is highlighted in the pink shaded region, while copies of the Brillouin zones above and below in the $\omega$-direction are presented for clarity.
• Figure 2: Wave-action spectra at different time steps for two sets of parameters: (a) $(t^2, \phi_0) = (0.3,\pi/8)$ and (b) $(t^2, \phi_0) = (0.75,\pi/3)$. The spectra are written as a function of the eigenfrequencies, namely $n_{\omega_{k}}^{m\pm} = n_k^{m\pm} + n_{-k}^{m\pm}$. The grey shaded region show the initial spectra, the blue circles show the pre-thermal state, and the green dots refer to the full thermal state. Red and orange curves represents the Rayleigh-Jeans distributions (\ref{['Rayleigh-Jeans']}) and (\ref{['Rayleigh-Jeans2']}), respectively, whose best-fit values are shown in the legend.
• Figure 3: Evolution of power $P$, linear energy $E_{lin}$, and $T/\mu$ in either band for the sets of parameters $(t^2, \phi_0) = (0.3,\pi/8)$ in panels a), c), e) and $(t^2, \phi_0) = (0.75,\pi/3)$ in panels b), d), f). Green points mark the upper band quantity and purple the lower band quantity. The grey region indicates the pre-thermal state within the upper band and the blue region the thermal state.
• Figure 4: The plots in (a) and (b) shows the dispersion relation for the cases $(t^2,\phi_0 )= (0.3,\pi/8)$ and $(t^2,\phi_0) = (0.75,\pi/3)$, respectively. The latter case allows for 4-wave Umklapp processes along the $\omega$-direction; an example of such process is sketched in (b) using red and blue dots where $(k_1,k_2,k_3,k_4) = (-17\pi/20,2\pi/20,-10\pi/20,-5\pi/20)$. (c) The system's parameter space of $(t^2,\phi_0)$ showing regions where the 4-wave Umklapp processes are absent (white) and present (red). By moving further into the red region ($t^2\rightarrow1$) full thermalisation timescales continually become shorter (See supplementary for details).