Floqeut topological phases in PT symmetric quantum walks with gain and loss

摘要

A quantum walk, that is, a synthetic quantum system mainly impremented by photonic systems, whose dynamics is described by a time-evolution operator, provides potential applications for quantum computing and information. It is further interesting that the quantum walk possesses novel topological phases akin to those of Floquet topological insulators, which are topological insulators driven by a time-periodic field[1,2]. Recently, a one-dimensional quantum walk dynamics associated with gain and loss is experimentally implemented by optical fiber loops [3]. The experiment shows that the energy of the system is kept to be real as a manifestation of PT symmetry (combined parity and time-reversal symmetry) in spite of the open quantum system.In this work, we theoretically study Floquet topological phases driven by the PT symmetric non-unitary time evolution. We have found the presence of combined parity and chiral symmetry, in addition to PT symmetry[4]. The presence of these symmetries allows us to study Floquet topological phases in open quantum systems by using a procedure for ordinal quantum walks belonging to class BDI[5]. We have confirmed that the number of edge states originating from Floquet topological phases and topological numbers satisfy the bulk-edge correspondence, while a modification due to the imaginary energy is required due to breaking of PT symmetry. We have also found that the probability corresponding to edge states exponentially increases due to the imaginary energy with increasing time as shown in Fig.1. This provides a way to observe huge probabilities originating from edge states in actual experimental setups. We also mention the experimental setup and result confirming the above theoretical predictions.