On Fermionic walkers interacting with a correlated structured environment

Abstract

We study the large-time behaviour of a sample $\mathcal {S}$ consisting of an ensemble of fermionic walkers on a graph interacting with a structured infinite reservoir of fermions $\mathcal {E}$ through an exchange of particles in preferred states. We describe the asymptotic state of $\mathcal {S}$ in terms the initial state of $\mathcal {E}$, with especially simple formulae in the limit of small coupling strength. We also study the particle fluxes into the different parts of the reservoir.

Appendix: Comments on the statistics

Following [9], we have made the choice of considering different species of fermions for the sample $\mathcal {S}$ and the environment $\mathcal {E}$. Considering the same species for both components of the system would have amounted to imposing the anticommutation relation $\{a^\sharp (\psi ),{\tilde{b}}(\varphi )\} = 0$ for all $\psi \in \mathcal {H}_\mathcal {S}$ and $\varphi \in \mathcal {H}_\mathcal {E}$ instead of the commutation relation $[a^\sharp (\psi ),b(\varphi )] = 0$. This is realized on the Fock space $\varGamma ^{-}_{}(\mathcal {H}_\mathcal {E}) \otimes \varGamma ^{-}_{}(\mathcal {H}_\mathcal {S})$ by setting . In this case, one finds with the same techniques formulae such as

$$\begin{aligned} {\tilde{K}}_\alpha ^* a^*(\psi ) {\tilde{K}}_\alpha = a^*((\varvec{1}+ (\cos \alpha - 1)P)\psi ) +\mathrm {i}\sin \alpha \, {\tilde{b}}^*(\iota \psi ), \end{aligned}$$

leading to the same formulae as in Lemma 2.2. Therefore, the asymptotics of the state in the sample $\mathcal {S}$ and the fluxes are same.

With this choice of statistics, one may alternatively view $\tilde{K}_\alpha$ as arising from the second quantization of a one-body operator on $\mathcal {H}_\mathcal {E}\oplus \mathcal {H}_\mathcal {S}$:

$$\begin{aligned} {\tilde{K}}_\alpha = \mathcal {U}\varGamma _{}\!\,\,(\varvec{1}+ (\cos \alpha -1)(\iota ^*\iota + \iota \iota ^*) - \mathrm {i}\sin \alpha \,(\iota ^* + \iota )) \mathcal {U}^*, \end{aligned}$$

where $\mathcal {U} : \varGamma ^{-}_{}(\mathcal {H}_\mathcal {E}\oplus \mathcal {H}_\mathcal {S}) \rightarrow \varGamma ^{-}_{}(\mathcal {H}_\mathcal {E}) \otimes \varGamma ^{-}_{}(\mathcal {H}_\mathcal {S})$ is the usual fermionic exponential map; see, for example, [5, §5.1]. The dynamics implemented by the unitary $\varGamma _{}\!\,\,((S \otimes U \oplus W)\mathrm {e}^{-\mathrm {i}\alpha (\iota +\iota ^*)})$ gives rise to a quasi-free dynamics and the corresponding one-particle Møller operator

$$\begin{aligned} \varOmega _+ =\mathop {\hbox {s-lim}}\limits _{t \rightarrow \infty }(S \otimes U \oplus W)^{t}((S \otimes U \oplus W)\mathrm {e}^{-\mathrm {i}\alpha (\iota +\iota ^*)})^{-t} \end{aligned}$$

exists and satisfies

$$\begin{aligned} \varOmega _+ (0 \oplus \varvec{1}) = \mathrm {i}\sin \alpha \sum _{t'=0}^{\infty } (S\otimes U)^{t'+1}\iota W^* ((\varvec{1}+ (\cos \alpha -1)\iota ^*\iota W^*)^{t'}. \end{aligned}$$

In particular, one quickly recovers

$$\begin{aligned} (0 \oplus \varvec{1}) \varOmega _+^* (\varSigma \oplus \varXi ) \varOmega _+ (0 \oplus \varvec{1}) = 0 \oplus \varDelta \end{aligned}$$

for all $\varXi \in \mathcal {B}(\mathcal {H}_\mathcal {S})$, showing that—at least when the initial state in the sample is a giqf state associated with a density $\varXi$ invariant for the free dynamics—the limiting state is the same as in the case previously considered. This reduction to a one-body problem also suggests the same behaviour for Bose statistics.