  We propose a geometrical engine undergoing an adiabatic (Thouless) pumping
process for a small system connected to external isothermal reservoirs with the
control of electrochemical potentials of the reservoirs and one parameter in
the system Hamiltonian. Thanks to the geometrical nature of this process, the
entropy production is characterized by the geometric metric tensor which is
connected to the Fisher information and the Hessian of the density matrix in a
nonequilibrium steady state. The existence of an inequality between the
thermodynamic length and entropy production is established. We also establish
that the work done on this system is characterized by a vector potential and is
equivalent to the thermodynamic flux. To characterize the engine, we the
introduce effective efficiency as the relation between the work and entropy
production. Through the theoretical analysis of the quantum master equation for
the Anderson model of a quantum dot within the wide-band approximation, we
illustrate the explicit values of the work, thermodynamic length, and effective
efficiency of the engine as functions of the phase difference of the externally
controlled electrochemical potentials.
