Reaction-diffusion dynamics of the weakly dissipative Fermi gas

Abstract

We study the one-dimensional Fermi gas subject to dissipative reactions. The dynamics is governed by the quantum master equation, where the Hamiltonian describes coherent motion of the particles, while dissipation accounts for irreversible reactions. For lattice one-dimensional fermionic systems, emergent critical behavior has been found in the dynamics in the reaction-limited regime of weak dissipation. Here, we address the question whether such features are present also in a gas in continuum space. We do this in the weakly dissipative regime by applying the time-dependent generalized Gibbs ensemble method. We show that for two body $2A\to \emptyset$ and three $3A\to \emptyset$ body annihilation, as well as for coagulation $A+A\to A$, the density features an asymptotic algebraic decay in time akin to the lattice problem. In all the cases, we find that upon increasing the temperature of the initial state the density decay accelerates, but the asymptotic algebraic decay exponents are not affected. We eventually consider the competition between branching $A\to A+A$ and the decay processes $A\to \emptyset$ and $2A\to \emptyset$. We find a second-order absorbing-state phase transition in the mean-field directed percolation universality class. This analysis shows that emergent behavior observed in lattice quantum reaction-diffusion systems is present also in continuum space, where it may be probed using ultra-cold atomic physics.

Figures (12)

• Figure 1: Time dependent generalised Gibbs ensemble (tGGE). Time evolution of the density matrix in the reaction-limited regime of weak dissipation $\widetilde{\Gamma}_{\nu} n^{z-2}/\Omega \ll 1$. Reactions occur on a much longer time-scale $\sim (n^z \widetilde{\Gamma}_{\nu})^{-1}$ than that of coherent motion $\sim n^{-2} \Omega^{-1}$, such that in-between consecutive reactions the system locally relaxes to the stationary state of the Hamiltonian. This local stationary state is a generalized Gibbs ensemble, which is for the here considered non-interacting Fermi gas a gaussian state. The time-dependence $\rho_{\mathrm{GGE}}(t)$ of the GGE accounts for the breaking of the conservation laws due to the reactions.
• Figure 2: Momentum occupation function at various times $\tau$ for binary annihilation. (a) $C_q$ is plotted as a function of $q$ for increasing $\tau$ (top to bottom) for a thermal initial state with $T=10$ and initial density $n_0=1$. In panel (b) we plot the same times for a different temperature $T=0.01$ and the same $n_0$ value. The curves correspond to the numeric solution of Eq. \ref{['eq:rate_alpha']}. The occupation function becomes gaussian in time with a peak at $q=0$.
• Figure 3: Reaction-limited binary annihilation in the continuum limit. (a) Log-log plot of the decay of the density $n(\tau)$ as a function of the rescaled time $\tau$ for increasing values of the temperature $T$ (from top to bottom).The solution of Eq. \ref{['eq:rate_alpha']} is found with fourth-order Runge-Kutta integrator with the parameters $d\tau=10^{-5}$ (time step of the numerical integrator) $dk:=2\pi/L=0.1$ (momentum grid step), $a=0.1$ (lattice spacing setting the maximum wavevector $|k|<k_{M}=\pi/a$). For the analytic curves (black-dashed lines) we used Eq. \ref{['n_ann']}, while the coloured curves are the numerical solution of Eq. \ref{['eq:rate_alpha']}. (b) Effective exponent $\delta(\tau)$, see Eq. \ref{['def:delta']}, as a function of $\tau$ with $b=2$. Increasing the temperature from $T=0.01$ (green) to $T=100$ (red) accelerates the convergence to the power law exponent $\delta=-1/2$. In both panels the initial density is set to $n_0=1$.
• Figure 4: Reaction-limited binary annihilation on the lattice. Log-log plot of the decay of the density $n(\tau)$ as a function of the rescaled time $\tau$ for increasing values of the temperature (from top to bottom) and initial filling $n_0=0.1$. (a) Numerical solution of Eq. \ref{['eq:rate-l']} for $\theta=0$ with parameters $a=0.1$, $dk=0.1$. (b) Numerical solution of Eq. \ref{['eq:rate-l']} for $\theta=\pi/4$. The dashed-dotted lines represent the asymptotic behavior at long times computed from Eqs. \ref{['eq:initial_filling_lattice']} and \ref{['eq:asymp_lattice']}. For high (bottom-purple line) and low temperature (top green line) the asymptotics approaches \ref{['eq:l_as_th']} and \ref{['eq:l_as_0']}, respectively.
• Figure 5: Momentum occupation function for various values of time $\tau$ for three-body annihilation. (a) $C_q$ is plotted as a function of $q$ for increasing $\tau$ (top to bottom) for a thermal initial state with $T=0.1$ and $n_0=1$. In panel (b) we plot the same times for a different temperature $T=10$ and the same $n_0=1$. The curves correspond to the numeric solution of Eq. \ref{['eq:3ann_kin_eq']}. The occupation function is non gaussian in time and it develops a double peak profile.