Excess energy and countercurrents after a quantum kick

TL;DR

This work analyzes the excess energy imparted to a quantum many-body system when its external potential suddenly translates with velocity $v$, showing that $\Delta E(t)=\mathbf{v}\cdot\langle\mathbf{P}\rangle(t)$ and deriving the framework using Galilean boosts (RF1 vs RF2). It provides a comprehensive classification of long-time behavior for bound, emitting, and extended systems, including countercurrents in metals and nonlinear $v^3$ countercurrents in insulators, and connects the mechanical kick to an ultrafast electric-field pulse within linear and nonlinear response theory. The authors validate the predictions with first-principles rt-TDDFT simulations on Al (metal) and Diamond (insulator), observing Drude-like current in the metal and nonlinear countercurrents in the insulator, and discuss implications for stopping power and ultrafast phenomena. The results offer a unified energetics perspective on non-equilibrium quantum dynamics with potential relevance to ultrafast experiments, electronic stopping, and cold-atom lattice systems.

Abstract

A quantum system of interacting particles under the effect of a static external potential is hereby described as kicked when that potential suddenly starts moving with a constant velocity v. If initially in a stationary state, the excess energy at any time after the kick equals $v \langle P \rangle (t)$, with P being the total momentum of the system. If the system is finite and remains bound, the long time average of the excess energy tends to $Mv^2$, with M the system's total mass, or a related expression if there is particle emission. $Mv^2$ is twice what expected from an infinitely smooth onset of motion, and any monotonic onset is expected to increase the average energy to a value within both limits. In a macroscopic system, a particle flow emerges countering the potential's motion when electrons stay partially behind. For charged particles the described kinetic kick is equivalent to the kick given by the infinitely short electric-field pulse $E = \frac{m}{q} v δ(t)$ to the system at rest, useful as a formal limit in ultrafast phenomena. A linear-response analysis of low-v countercurrents in kicked metals shows that the coefficient of the linear term in v is the Drude weight. Non-linear in v countercurrents are expected for insulators through the electron-hole excitations induced by the kick, going as $v^3$ at low v for centrosymmetric ones. First-principles calculations for simple solids are used to ratify those predictions, although the findings apply more generally to systems such as Mott insulators or cold lattices of bosons or fermions.

Figures (3)

• Figure 1: Kicked bulk aluminum. Upper panel: excess energy per unit cell $\Delta E$ versus time $t$ after a sudden onset of motion with $v=1.37$Å/fs (black line). The red line gives the theoretical reference of $\Delta E=Mv^2$, $M$ being the mass of the three electrons per cell. The difference between the red line and the average of the black line indicates the presence of a countercurrent [Eqs. \ref{['eq:countercurrent']} and \ref{['eq:counteral']}]. Middle: Magnitude of average momentum (or of current density) $\langle \mathbf{P}\rangle'/m_e = \frac{\Omega}{e}\mathbf{j}'$ versus velocity $v$. Lower: Fourier transform of $\Delta E(t)$ versus frequency $\omega$ for $v= 0.27$ Å/fs.
• Figure 2: Kicked diamond. Upper two panels: excess energy per unit cell $\Delta E$ versus time $t$ after a sudden onset of motion with $v=7.38$ Å/fs (inset zoom-in as indicated) and $v=13.13$ Å/fs (black lines). Red lines: theoretical reference without countercurrent of $\Delta E=Mv^2$, with $M=8 m_e$ for the eight valence electrons per cell. Lower: Black dots: magnitude of the average momentum per unit cell (or current density) $\langle \mathbf{P}\rangle'/m_e = \frac{\Omega}{e}\mathbf{j}'$ versus velocity $v$ from rt-TDDFT results. Red line: linear component at $v\rightarrow 0$, due to the non-local pseudopotential stengel2025pseudopotentials. Blue dots: $\langle \mathbf{P}\rangle'/m_e$ after removing that linear term. Blue line: fit of the latter as $\langle \mathbf{P}\rangle'/m_e = a v^3 + b v^5$.
• Figure 3: Comparison of atomic-orbital and plane-wave results for the evolution of $\Delta E$ for diamond as kicked with the three velocities indicated. The red straight line is the $Mv^2$ reference, and the black (blue) line is as obtained with an atomic-orbital (plane-wave) basis.