Quantum reaction-limited reaction-diffusion dynamics of annihilation processes

We investigate the quantum reaction-diffusion dynamics of fermionic particles which coherently hop in a one-dimensional lattice and undergo annihilation reactions. The latter are modelled as dissipative processes which involve losses of pairs $2A \to \emptyset$, triplets $3A \to \emptyset$, and quadruplets $4A \to \emptyset$ of neighbouring particles. When considering classical particles, the corresponding decay of their density in time follows an asymptotic power-law behavior. The associated exponent in one dimension is different from the mean-field prediction whenever diffusive mixing is not too strong and spatial correlations are relevant. This specifically applies to $2A\to \emptyset$, while the mean-field power-law prediction just acquires a logarithmic correction for $3A \to \emptyset$ and is exact for $4A \to \emptyset$. A mean-field approach is also valid, for all the three processes, when the diffusive mixing is strong, i.e., in the so-called reaction-limited regime. Here, we show that the picture is different for quantum systems. We consider the quantum reaction-limited regime and we show that for all the three processes power-law behavior beyond mean field is present as a consequence of quantum coherences, which are not related to space dimensionality. The decay in $3A\to \emptyset$ is further, highly intricate, since the power-law behavior therein only appears within an intermediate time window, while at long times the density decay is not power-law. Our results show that emergent critical behavior in quantum dynamics has a markedly different origin, based on quantum coherences, to that applying to classical critical phenomena, which is, instead, solely determined by the relevance of spatial correlations.