图 1  周期性驱动非平衡量子输运和其中几何性质的示意图　(a) 非平衡量子系统示意图. 量子系统由一个包含多个能级的系统来表示, 它可以与多个热库相连. 热库温度($ {T}_{\mathrm{h}} $和$ {T}_{\mathrm{c}} $)和系统参数$(\lambda)$都被含时地驱动. 由此, 可以产生系统与热库间的热量交换($ {Q}_{\mathrm{h}} $和$ {Q}_{\mathrm{c}} $)以及系统的功输出(W). (b) 此非平衡量子系统在参数空间($\boldsymbol{\varLambda }\equiv \left(T, \lambda \right) $)中的几何性质. 曲线坐标系表现出非均匀的热力学距离, 而各点的箭头表示几何联络. 几何联络在几何上对应平行移动一个微小参数时带来的和乐(holonomy)角. 热力学距离定义了一个具有度规的黎曼曲面

Fig. 1.  A scheme of periodically driven nonequilibrium quantum transport and its geometry. (a) A diagrammatic nonequilibrium quantum system. The middle quantum system is illustrated by a multi-level system, which is coupled with several thermal reservoirs. The temperature of reservoirs ($ {T}_{\mathrm{h}} $ and $ {T}_{\mathrm{c}} $) and the mechanical parameter of the system ($ \lambda $) are simultaneously and periodically modulated. The heat exchange ($ {Q}_{\mathrm{h}} $ and $ {Q}_{\mathrm{c}} $) and work output $(W)$ are thus generated. (b) The geometry of this nonequilibrium quantum system in the space of parameters $(\boldsymbol{\varLambda }\equiv $$ \left(T, \lambda \right) )$. The curvilinear coordinate is adopted to show the inhomogeneous thermodynamic distance and the local vectors are for the geometric connection, as derived in the main text. Geometrically, the geometric connection is the holonomy angle during an infinitesimal parallel transport and the thermodynamic distance between neighboring points defines a Riemannian space with endowed metric.