Non-reciprocal population dynamics in a quantum trimer.

C A Downing,D Zueco

doi:10.1098/rspa.2021.0507

Abstract

We study a quantum trimer of coupled two-level systems beyond the single-excitation sector, where the coherent coupling constants are ornamented by a complex phase. Accounting for losses and gain in an open quantum systems approach, we show how the mean populations of the states in the system crucially depend on the accumulated phase in the trimer. Namely, for non-trivial accumulated phases, the population dynamics and the steady states display remarkable non-reciprocal behaviour in both the singly and doubly excited manifolds. Furthermore, while the directionality of the resultant chiral current is primarily determined by the accumulated phase in the loop, the sign of the flow may also change depending on the coupling strength and the amount of gain in the system. This directionality paves the way for experimental studies of chiral currents at the nanoscale, where the phases of the complex hopping parameters are modulated by magnetic or synthetic magnetic fields.

Highlights

Reciprocity in the animal kingdom is manifested by the evolution of reciprocal altruism: ‘you scratch my back, and I will scratch yours’ [1]
Inspired by the landmark experiment of Roushan et al [23], who modelled their photonic system as harmonic oscillators, in this work we study a trimer of two-level systems (2LSs) in order to probe the whole energy ladder, including the effects of saturation due to the strong interactions
We have considered a trimer of 2LSs in an open quantum systems approach, where both the magnitude and phase of the coherent coupling constants are important

Summary

Introduction

Reciprocity in the animal kingdom is manifested by the evolution of reciprocal altruism: ‘you scratch my back, and I will scratch yours’ [1]. The triply excited state is characterized by Hσ3†σ2†σ1†|0 = ω8σ3†σ2†σ1†|0 , and is associated with the maximal eigenvalue ω8 = 3ω0 (pink lines) These two extreme rungs of the energy ladder are the same in the coupled and uncoupled regimes (left and right in figure 1b), because they are associated with the wholly unoccupied state and the wholly occupied state. In a two-site dimer, with Hamiltonian Hdi = ω0(σ1†σ1 + σ2†σ2) + g(eiθ σ1†σ2 + h.c.), the single-excitation eigenfrequencies are unaffected by the phase θ12 They read ω± = ω0 ± g, such that the energy ladder of the dimer is formed by {2ω0, ω+, ω−, 0} [36,65]. Upon assuming weak coupling to the environment and Markovian behaviour, and after discarding fast-oscillating (non-resonant) terms, the quantum master equation of the trimer eigenfrequencies ⁄ w0 royalsocietypublishing.org/journal/rspa Proc. Becoming sensitive to the gauge-independent phase φ, and the loss and gain in the open quantum system, which can be controlled through the parameters γn and Pn, respectively

Chiral steady states

Non-reciprocal dynamics

Discussion