Theory of multi-dimensional quantum capacitance and its application to spin and charge discrimination in quantum-dot arrays

Quantum states of a few-particle system capacitively coupled to a metal gate can be discriminated by measuring the quantum capacitance (QC), which can be identified with the second derivative of the system energy with respect to the gate voltage. If more than one gates are capacitively coupled with the system (e.g., in a quantum-dot array), the theory must be generalized to account for the dependence of the energy on all the applied gate voltages. With this aim, we have introduced the concept of QC matrix. The matrix formalism allows us to determine the dependence of the QC on both the working point and direction of the voltage oscillations in the parameter space, and to identify the optimal combination of gate voltages that maximizes the outcome of a QC measurement. From the application to a quantum-dot array, we predict novel measurable features that are specific of the multi-gate approach, such as QC plateaus and voltage-tunable heights of the QC peaks. We show how such features depend on the quantum tunneling processes involved in the transitions between different charge stability regions. Altogether, our work provides a procedure for optimizing the discrimination between states with different particle numbers and/or total spins, which can be realized experimentally.