Electrooptomechanical Equivalent Circuits for Quantum Transduction

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 (OM) 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. CONTENTS

Sensitive optomechanical transduction of electric and magnetic signals to the optical domain

“…The biasq is the charge induced on C for a DC bias or half the peak charge induced for an AC bias. The electromechanical coupling scales with the mechanical motion's modulation of [16]. Here, C m is a shorthand notation for the derivative of the membrane-capacitor C m with respect to the membrane displacement x from its equilibrium, and we see that it alone characterizes how much the membrane-capacitor implementation affects the electromechanical coupling.…”

“…Here, all inductors includes a parasitic series resistance and L e includes a parasitic parallel capacitance to account for its self-resonance. Note that during transduction the membrane-capacitor corresponds to an equivalent circuit of a capacitor connected in parallel with another series LC resonator in [16], but without the drive bias it is a capacitor.…”

“…To derive an expression for G em in terms of simple and measurable quantities, we take the following two steps: first, we approximate the electrostatically-induced frequency shift of the membrane ∆Ω m [16] as…”

Microwave-to-Optical Transduction Using a Mechanical Supermode for Coupling Piezoelectric and Optomechanical Resonators

“…It results as a limiting case of a more general equivalent circuit derived in Ref. [43] and summarized in Appendix B.…”

“…In Section III, we apply the equivalent circuit analysis of optoelectromechanical systems proposed in Ref. [43] to establish formulas for key metrics based on physical parameters that characterize the component elements. After laying down the theoretical groundwork for transduction efficiency η, added noise N , and conversion bandwidth ∆ω, two optimization scenarios will be addressed: maximizing η (Section IV) and minimizing N (Section V).…”