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Modelling the electrical response of multi-level quantum systems at finite frequency has been typically performed in the context of two incomplete paradigms: (i) input-output theory, which is valid at any frequency but neglects dynamic losses, and (ii) semiclassical theory, which captures well dynamic dissipation effects but is only accurate at low frequencies. Here, we develop a unifying theory, valid for arbitrary frequencies, that captures both the quantum behaviour and the non-unitary effects introduced by relaxation and dephasing. The theory allows a multi-level system to be described by a universal small-signal equivalent circuit model, a resonant RLC circuit, whose topology only depends on the number of energy levels. We apply our model to a double quantum-dot charge qubit and a Majorana qubit, showing the capability to continuously describe the systems from adiabatic to resonant and from coherent to incoherent, suggesting new and realistic experiments for improved quantum state readout. Our model will facilitate the design of hybrid quantum-classical circuits and the simulation of qubit control and quantum state readout.

Semiconductor quantum dots operated dynamically are the basis of many quantum technologies such as quantum sensors and computers. Hence, modelling their electrical properties at microwave frequencies becomes essential to simulate their performance in larger electronic circuits. Here, we develop a self-consistent quantum master equation formalism to obtain the admittance of a quantum dot tunnel-coupled to a charge reservoir under the effect of a coherent photon bath. We find a general expression for the admittance that captures the well-known semiclassical (thermal) limit, along with the transition to lifetime and power broadening regimes due to the increased coupling to the reservoir and amplitude of the photonic drive, respectively. Furthermore, we describe two new photon-mediated regimes: Floquet broadening, determined by the dressing of the QD states, and broadening determined by photon loss in the system. Our results provide a method to simulate the high-frequency behaviour of QDs in a wide range of limits, describe past experiments, and propose novel explorations of QD-photon interactions.

Silicon offers the enticing opportunity to integrate hybrid quantum-classical computing systems on a single platform. For qubit control and readout, high-frequency signals are required. Therefore, devices that can facilitate its generation are needed. Here, we present a quantum dot-based radiofrequency multiplier operated at cryogenic temperatures. The device is based on the non-linear capacitance-voltage characteristics of quantum dot systems arising from their low-dimensional density of states. We implement the multiplier in a multi-gate silicon nanowire transistor using two complementary device configurations: a single quantum dot coupled to a charge reservoir and a coupled double quantum dot. We study the harmonic voltage conversion as a function of energy detuning, multiplication factor and harmonic phase noise and find near ideal performance up to a multiplication factor of 10. Our results demonstrate a method for high-frequency conversion that could be readily integrated into silicon-based quantum computing systems and be applied to other semiconductors., Comment: 17 pages, 16 figures