Dissipation in fermionic two-body continuous-time quantum walk under the steepest entropy ascent formalism

TL;DR

This work investigates dissipation in a two-walker CTQW of spinless fermions on a ring using the thermodynamically consistent steepest entropy ascent (SEA) formalism. The authors derive both the single-component and two-component SEA equations, employing local-perception operators to preserve no-signaling in the composite system, and apply them to a ring with tunable Hubbard-like interactions. Across four interaction regimes, SEA drives greater probability spreading and entropy production than unitary dynamics, with the extent of these effects modulated by the interaction terms; the Loschmidt echo decays more rapidly under SEA, yet becomes more unitary-like as interactions strengthen, signaling nuanced thermalization behavior. The results underscore SEA as a practical, environment-agnostic framework for modeling nonlinear dissipation and thermalization in many-body quantum systems, with potential implications for quantum information processing and experimental platforms such as superconducting qubits.

Abstract

Quantum walks play a crucial role in quantum algorithms and computational problems. Many-body quantum walks can reveal and exploit quantum correlations that are unavailable for single-walker cases. Studying quantum walks under noise and dissipation, particularly in multi-walker systems, has significant implications. In this context, we use a thermodynamically consistent formalism of dissipation modeling, namely the steepest entropy ascent (SEA) formalism. We analyze two spinless fermionic continuous-time walkers on a 1D graph with tunable Hubbard and extended Hubbard-like interactions. By contrasting SEA-driven dynamics with unitary evolution, we systematically investigate how interaction strengths modulate thermalization and entropy production. Our findings highlight the relevance of SEA formalism in modeling nonlinear dissipation in many-body quantum systems and its implications for quantum thermalization.

Figures (11)

• Figure 1: A schematic of the two-walker model on a ring graph of $N$ vertices indexed from $0$ to $N-1$. The two-walker wave-function is an element of the joint Hilbert space $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$. Initially, walkers $A$ and $B$ are localized in distinct regions, allowing the composite state to be written as a product state. As correlations develop during evolution, the system is described by a composite antisymmetric density matrix ($\rho_a$). The reduced density matrix $\rho_{J\newline}$ represents the ${J\newline}^\text{th}$ walker.
• Figure 2: Joint probability distribution (JPD) of two-walker evolution on a ring with 11 nodes (indexed from zero). The walkers evolve without interaction ($\alpha_i=0$$\forall$$i$). Panels (a) and (b) show the initial JPD, while panels (c) and (d) depict the JPD at $t/\tau=30$. SEA evolution is shown in (a) and (c), and unitary evolution in (b) and (d). Color bars next to each panel indicate the corresponding probability values.
• Figure 3: Joint probability distribution (JPD) of two-walker evolution (see Table \ref{['table:cases']}) on a ring with 11 nodes (indexed from zero). The walkers experience strong, full interaction ( $\alpha_i = 10$$\forall$$i$). Panels (a) and (b) show the initial JPD, while panels (c) and (d) depict the JPD at $t/\tau=30$. SEA evolution is shown in (a) and (c), and unitary evolution in (b) and (d). Color bars next to each panel indicate the corresponding probability values.
• Figure 4: Marginal probability of walker A on a ring graph with 11 sites, evolving under a Hamiltonian with no interaction. Panels (a)--(c) depict unitary evolution, while panels (d)--(f) show SEA evolution. Panels (a) and (e) compare the initial evolution of the marginal probability distribution under unitary and SEA dynamics, respectively. Panels (b) and (f) show the late-time evolution for the same cases. The large color bars on the left of each panel indicate probability values, while the smaller ones on the left of panels (b) and (f) correspond to zoomed-in probability scales. The x-axis represents time $(t/\tau)$, and the y-axis denotes the site number. Each of the smaller panels ((a), (b)) and ((e), (f)) are zoomed portions of panels (c) and (d), respectively, with the zoomed-in sections marked by rectangles.
• Figure 5: Marginal probability of walker A on a ring graph with 11 sites, evolving under the Full Interaction Hamiltonian with $\alpha_i = 10$ for all $i$s (see Table \ref{['table:cases']}). Panels (a)--(c) correspond to unitary evolution, while panels (d)--(f) depict SEA evolution. Panels (a) and (e) show the initial evolution of the marginal probability distribution for unitary and SEA dynamics, respectively, while panels (b) and (f) illustrate the late-time evolution. The large color bars on the left of each panel represent probability values, and the smaller ones on the left of panels (b) and (f) show corresponding zoomed-in probability scales. The x-axis represents time $(t/\tau)$, and the y-axis denotes the site number. Each of the smaller panels ((a), (b)) and ((e), (f)) are zoomed portions of panels (c) and (d), respectively, with the zoomed-in sections marked by rectangles.