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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo> <mml:mtext>G</mml:mtext> <mml:mo>⟩</mml:mo></mml:mrow> </mml:math> and the excited state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo> <mml:mtext>E</mml:mtext> <mml:mo>⟩</mml:mo></mml:mrow> </mml:math> , and a new nonunitary operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> N(g) for an asymmetric transition between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo> <mml:mtext>G</mml:mtext> <mml:mo>⟩</mml:mo></mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo> <mml:mtext>E</mml:mtext> <mml:mo>⟩</mml:mo></mml:mrow> </mml:math> . The non-Hermiticity parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo> <mml:mtext>G</mml:mtext> <mml:mo>⟩</mml:mo></mml:mrow> </mml:math> and the excited state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo> <mml:mtext>E</mml:mtext> <mml:mo>⟩</mml:mo></mml:mrow> </mml:math> both in one and two dimensions.