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 2 A → ∅ and three 3 A → ∅ body annihilation, as well as for coagulation A + A → 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 → A + A and the decay processes A → ∅ and 2 A → ∅ . 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.
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 2 A → ∅ and three 3 A → ∅ body annihilation, as well as for coagulation A + A → 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 → A + A and the decay processes A → ∅ and 2 A → ∅ . 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.
Abstract We investigate quantum reaction–diffusion (RD) systems in one-dimension with bosonic particles that coherently hop in a lattice, and when brought in range react dissipatively. Such reactions involve binary annihilation ( A + A → ∅ ) and coagulation ( A + A → A ) of particles at distance d. We consider the reaction-limited regime, where dissipative reactions take place at a rate that is small compared to that of coherent hopping. In classical RD systems, this regime is correctly captured by the mean-field approximation. In quantum RD systems, for noninteracting fermionic systems, the reaction-limited regime recently attracted considerable attention because it has been shown to give universal power law decay beyond mean field for the density of particles as a function of time. Here, we address the question whether such universal behavior is present also in the case of the noninteracting Bose gas. We show that beyond mean-field density decay for bosons is possible only for reactions that allow for destructive interference of different decay channels. Furthermore, we study an absorbing-state phase transition induced by the competition between branching A → A + A , decay A → ∅ and coagulation A + A → A . We find a stationary phase-diagram, where a first and a second-order transition line meet at a bicritical point which is described by tricritical directed percolation. These results show that quantum statistics significantly impact on both the stationary and the dynamical universal behavior of quantum RD systems.
We consider the quantum nonequilibrium dynamics of systems where fermionic particles coherently hop on a one-dimensional lattice and are subject to dissipative processes analogous to those of classical reaction-diffusion models. Particles can either annihilate in pairs, A+A→0, or coagulate upon contact, A+A→A, and possibly also branch, A→A+A. In classical settings, the interplay between these processes and particle diffusion leads to critical dynamics as well as to absorbing-state phase transitions. Here, we analyze the impact of coherent hopping and of quantum superposition, focusing on the so-called reaction-limited regime. Here, spatial density fluctuations are quickly smoothed out due to fast hopping, which for classical systems is described by a mean-field approach. By exploiting the time-dependent generalized Gibbs ensemble method, we demonstrate that quantum coherence and destructive interference play a crucial role in these systems and are responsible for the emergence of locally protected dark states and collective behavior beyond mean field. This can manifest both at stationarity and during the relaxation dynamics. Our analytical results highlight fundamental differences between classical nonequilibrium dynamics and their quantum counterpart and show that quantum effects indeed change collective universal behavior.
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→∅, triplets 3A→∅, and quadruplets 4A→∅ of neighboring 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→∅, while the mean-field power-law prediction just acquires a logarithmic correction for 3A→∅ and is exact for 4A→∅. 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→∅ 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.
We consider the quantum reaction-diffusion dynamics in d spatial dimensions of a Fermi gas subject to binary annihilation reactions $A+A\ensuremath{\rightarrow}\ensuremath{\emptyset}$. These systems display collective nonequilibrium long-time behavior, which is signalled by an algebraic decay of the particle density. Building on the Keldysh formalism, we devise a field theoretical approach for the reaction-limited regime, where annihilation reactions are scarce. Combining a perturbative expansion of the dissipative interaction with Euler-hydrodynamic scaling limit, we derive a description in terms of a large-scale universal kinetic equation. Our approach shows how the time-dependent generalized Gibbs ensemble assumption, which is often employed for treating low-dimensional nonequilibrium dissipative systems, emerges from systematic diagrammatics. It also allows us to exactly compute---for arbitrary spatial dimension---the decay exponent of the particle density. The latter is based on the large-scale description of the quantum dynamics and it differs from the mean-field prediction even in dimension larger than one. We moreover consider spatially inhomogeneous setups involving an external potential. In confined systems the density decay is accelerated towards the mean-field algebraic behavior, while for deconfined scenarios the power-law decay is replaced by a slower nonalgebraic decay.
We discuss many-body fermionic and bosonic systems subject to dissipative particle losses in arbitrary spatial dimensions d d , within the Keldysh path-integral formulation of the quantum master equation. This open quantum dynamics represents a generalisation of classical reaction-diffusion dynamics to the quantum realm. We first show how initial conditions can be introduced in the Keldysh path integral via boundary terms. We then study binary annihilation reactions A+A\to \emptyset A + A → ∅ , for which we derive a Boltzmann-like kinetic equation. The ensuing algebraic decay in time for the particle density depends on the particle statistics. In order to model possible experimental implementations with cold atoms, for fermions in d=1 d = 1 we further discuss inhomogeneous cases involving the presence of a trapping potential. In this context, we quantify the irreversibility of the dynamics studying the time evolution of the system entropy for different quenches of the trapping potential. We find that the system entropy features algebraic decay for confining quenches, while it saturates in deconfined scenarios.
We consider an extension of classical stochastic reaction-diffusion (RD) dynamics to open quantum systems. We study a class of models of hard-core particles on a one-dimensional lattice whose dynamics is generated by a quantum master operator. Particle hopping is coherent while reactions, such as pair annihilation or pair coalescence, are dissipative. These are quantum open generalizations of the A+A→⌀ and A+A→A classical RD models. We characterize the relaxation of the state towards the stationary regime via a decomposition of the system Hilbert space into transient and recurrent subspaces. We provide a complete classification of the structure of the recurrent subspace (and the nonequilibrium steady states) in terms of the dark states associated to the quantum master operator and its general spectral properties. We also show that, in one dimension, relaxation towards these absorbing dark states is slower than that predicted by a mean-field analysis due to fluctuation effects, in analogy with what occurs in classical RD systems. Numerical simulations of small systems suggest that the decay of the density in one dimension, in both the open quantum A+A→⌀ and A+A→A systems, behaves asymptotically as t-b with 1/2<b<1.