Proposal of a quantum version of active particles via a nonunitary quantum walk

Abstract

The main aim of the present paper is to define an active particle in a quantum framework as a minimal model of quantum active matter and investigate the differences and similarities of quantum and classical active matter. Although the field of active matter has been expanding, most research has been conducted on classical systems. Here, we propose a truly deterministic quantum active-particle model with a nonunitary quantum walk as the minimal model of quantum active matter. We aim to reproduce results obtained previously with classical active Brownian particles; that is, a Brownian particle, with finite energy take-up, becomes active and climbs up a potential wall. We realize such a system with nonunitary quantum walks. We introduce new internal states, the ground state and the excited state, and a new nonunitary operator $N(g)$ for an asymmetric transition between the two states. The non-Hermiticity parameter $g$ promotes the transition to the excited state; hence, the particle takes up energy from the environment. For our quantum active particle, we successfully observe that the movement of the quantum walker becomes more active in a nontrivial manner as we increase the non-Hermiticity parameter $g$, which is similar to the classical active Brownian particle. We also observe three unique features of quantum walks, namely, ballistic propagation of peaks in one dimension, the walker staying on the constant energy plane in two dimensions, and oscillations originating from the resonant transition between the ground state and the excited state both in one and two dimensions.

Figures (9)

• Figure 1: Schematic views of active matter. (a) A component that takes up energy from the environment, stores it internally, converts the stored internal energy to the kinetic energy and moves, and (b) an interacting collection of such components.
• Figure 2: Time evolution of our quantum active particle in one dimension. There are four internal states, namely, $(\ket{\mathrm{L}}\oplus\ket{\mathrm{R}})\otimes(\ket{\mathrm{G}}\oplus\ket{\mathrm{E}})$ at each site. The parameters $\varepsilon$, $w$ and $g$ are all real. We have the coin operator (②), the shift operator (③), and the new operator for the asymmetric transition between the ground state $\ket{\mathrm{G}}$ and the excited state $\ket{\mathrm{E}}$, which describes the energy take-up (①). We use different $\theta$ values for the ground state $\ket{\mathrm{G}}$ and the excited state $\ket{\mathrm{E}}$ to realize a system without momentum conservation.
• Figure 3: Normalized probability distributions $\tilde{P}(x,T=100)$ of the ground state [(a)], excited state [(b)] and the sum of the both states [(c), (d)] after 100 time steps of evolution for $g=0$ [(a), (b), (c)] and $g=1$ [(d)]. The system size is $L_x=401$ with $-200\le x\le200$; the probability outside the plotting range is significantly small.
• Figure 4: The time-step dependence of the standard deviation $\Delta x$ for $w=0$ [(a)] and $w=0.25$ [(b), (c), (d)]. (a) and (b) are computed for $g=0$, (c) is computed for $g=0.5$, and (d) is computed for $g=1$. The red triangles and blue circles indicate the standard deviation with respect to the excited and ground states, respectively. The green plus symbols indicate the standard deviation normalized by the total probability.
• Figure 5: Density plot of the time evolution of the quantum walker focusing on the ground state [(c), (d)], the excited state [(e), (f)] and the sum of the two states [(a), (b)] for $g=0$ [(a), (c), (e)] and $g=1$ [(b), (d), (f)]. The color indicates the probability of the walker at each site and time step.