Generalized thermodynamic closure in ultrafast phonon dynamics

Abstract

Driven-dissipative dynamics underlie a wide range of nonequilibrium phenomena in quantum materials, yet reduced descriptions beyond the quasi-equilibrium picture remain difficult to establish. Here, we experimentally demonstrate that a resonantly driven phonon mode admits a generalized thermodynamic description in which coherence and energy jointly organize the nonequilibrium evolution. Beyond a threshold driving field strength, we observe a delayed ultrafast response of a coherently driven phonon mode. Combined with experimentally constrained Lindblad dynamics, we show that this delay reflects the finite-time spreading of excitations across many phonon levels. At the same time, the full density-matrix trajectories for three driving conditions collapse onto a common surface defined by energy and coherence. Our results establish a coherence-extended thermodynamic regime for driven phonons and provide a framework for broader state engineering in driven-dissipative bosonic excitations.

Figures (4)

• Figure 1: Experimental realization of a driven-dissipative phonon system far from equilibrium. (a) The differential transmission spectrum, $\Delta T /T$, of MAPI at 100 K via the THz pump pulse centered at 1 THz. The spectra at the probe energy below 1.65 eV are magnified to better display the sub-band-gap response (see main text). (b) The temporal profile of the pump THz pulse. (c) The dynamics of the differential transmission at the probe energy of 1.6 eV. The 1.6 eV response reaches its maximum after $\sim 3$ ps from the maximum of the THz pump. (d) The differential transmission spectrum at the pump-probe delay time, $t_{\rm pr}$, of 3 ps. (e) The transient spectra of the suppressed sub-gap transition at $t_{\rm pr}=$ 3 ps at various peak field strengths of the drive THz pulses. (f) The threshold behavior of the suppression of sub-gap transition upon the drive peak field strengths.
• Figure 2: Failure of alternative interpretations for the observed electronic response. (a) (a, b) The expected differential transmission spectra (black) obtained from the temperature-dependent, steady-state absorption curves (Fig. S1(c)). $\Delta T_{\rm thermal}/T$ ($\Delta T_{\rm PT}/T$)is estimated from $T_{120K}-T_{100K}$ ($T_{180K}-T_{100K}$), respectively. Each of the estimated curves is compare with the pump-probe differential spectrum $\Delta T_{\rm exp}/T$ (blue) shown in Fig. 1(e). (c) The expected dynamic response (blue) of THz-induced electronic interband transition, calculated from optical Bloch equations (Methods) based on the experimental THz field profile (black). (d) Temperature dependence of the differential transmission spectra. Burgundy curves are dominated by the Franz Keldysh effect of coexisting tetragonal crystallites near the critical temperature (136 to 160 K). Solid and dashed green curves (100 and 112 K) show no discernible Franz Keldysh effect, with a distinctive spectral feature. The dotted green curve (120 K) shows a mixture of Franz Keldysh effect and purely phonon-induced suppression of sub-gap transitions (see text).
• Figure 3: Modeling the driven phonon as an open quantum system. (a) Effective participation number ($N_{\rm eff}(t)$) from the experimentally constrained Lindblad simulation (solid curves) compared with the measured responses (open circles) for three drive conditions: 1 THz, 183 kV/cm (red); 0.8 THz, 189 kV/cm (blue); and 1 THz, 107 kV/cm (green) (b) Simulated population dynamics of individual phonon levels (middle) and population distributions at selected pump-probe delays (bottom; -5, 3, and 20 ps) for the 1 THz, 183 kV/cm drive condition; the corresponding experimental THz waveform is shown above (top). (c) Comparison of simulations with Hilbert-space truncation to $N=$ 3 (red) and $N=$20 (blue), together with the measured dynamics (black circles) (d) Comparison of scalar observables extracted from the simulation: entropy production ($\Sigma$, green), trace distance from the equal-energy Gibbs state ($D_{\rm trace}$, red), total coherence ($C_{\rm total}$, blue), energy expectation value ($\langle H \rangle$, black solid), and von Neumann entropy ($S_{\rm vN}$, dark gray dashed), together with the measured dynamics (gray circles).
• Figure 4: Coherence-extended thermodynamic closure. (a) Three-dimensional representation of the phonon energy expectation value ($\langle H \rangle$), the total coherence ($C_{\rm total}$), and the von Neumann entropy ($S_{\rm vN}$), extracted from the full density matrix ($\rho(t)$) obtained from the experimentally constrained Lindblad simulation. The coherence-extended thermodynamic surface, $S_{\rm vN}= S_{\rm vN}(\langle H \rangle, C_{\rm total})$, is shown as a semi-transparent light-blue layer. (b) Hysteresis in the energy-entropy plane, shown together with the equilibrium thermodynamic relation, $S_{\rm vN}=S_{\rm vN}(\langle H \rangle)$, obtained from the equal-energy Gibbs reference state (dotted black). (c) Hysteresis in the energy-coherence plane. In all panels, solid curves denote the excitation trajectories up to the maximum of the energy (yellow triangle) and dashed curves the subsequent relaxation trajectories for the three experimental drive conditions (red: 1 THz with 183 kV/cm, blue: 0.8 THz with 189 kV/cm, green: 1 THz with 107 kV/cm).