Electro-optomechanical equivalent circuits for quantum transduction

TL;DR

This paper introduces an equivalent circuit formalism for electro-optomechanical transducers, enabling intuitive analysis, design optimization, and quantum characterization of hybrid systems bridging microwave and optical frequencies.

Contribution

It develops a unified impedance-based circuit model for electro-optomechanical transducers, applicable to static and dynamic regimes, facilitating better understanding and optimization of quantum transduction devices.

Findings

Provides a complete quantum mechanical scattering matrix for transducers.

Unifies the transducer description within electrical engineering circuit analysis.

Applicable beyond the resolved-sideband regime, for arbitrary linear circuits.

Abstract

Using the techniques of optomechanics, a high-$Q$ mechanical oscillator may serve as a link between electromagnetic modes of vastly different frequencies. This approach has successfully been exploited for the frequency conversion of classical signals and has the potential of performing quantum state transfer between superconducting circuitry and a traveling optical signal. Such transducers are often operated in a linear regime, where the hybrid system can be described using linear response theory based on the Heisenberg-Langevin equations. While mathematically straightforward to solve, this approach yields little intuition about the dynamics of the hybrid system to aid the optimization of the transducer. As an analysis and design tool for such electro-optomechanical transducers, we introduce an equivalent circuit formalism, where the entire transducer is represented by an electrical circuit. Thereby we integrate the transduction functionality of optomechanical systems into the toolbox of electrical engineering allowing the use of its well-established design techniques. This unifying impedance description can be applied both for static (DC) and harmonically varying (AC) drive fields, accommodates arbitrary linear circuits, and is not restricted to the resolved-sideband regime. Furthermore, by establishing the quantized input-output formalism for the equivalent circuit, we obtain the scattering matrix for linear transducers using circuit analysis, and thereby have a complete quantum mechanical characterization of the transducer. Hence, this mapping of the entire transducer to the language of electrical engineering both sheds light on how the transducer performs and can at the same time be used to optimize its performance by aiding the design of a suitable electrical circuit.