Electrooptomechanical Equivalent Circuits for Quantum Transduction

With use of 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 with linear response theory based on the Heisenberg-Langevin equations. While these equations are 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 electrooptomechanical 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 for both static (dc) and harmonically varying (ac) bias 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.

Figure 1

Example electrooptomechanical transducer: A serial $RLC$ circuit with intrinsic resistance $R_{LC}$ is loaded by a semi-infinite transmission (TX) line of characteristic impedance $Z_{\mathrm{TX}}$. The circuit is capacitively coupled to a mechanical mode of intrinsic linewidth ${\gamma }_m$ with coupling strength $G$. Analogously, the optical mode has an intrinsic loss rate ${\kappa }_0$ and a read-out rate ${\kappa }_{\mathrm{ext}}$ and couples to the mechanical mode with strength $G_{\mathrm{OM}}$.

Figure 2

Dc-biased electromechanical interface based on capacitive coupling. (a) A mechanically modulated coupling capacitor, $C_c(x)$, where $x$ is a suitable mechanical position coordinate and $\delta Q$ is an added charge. (b) Equivalent circuit for the linearized dynamics of the system in (a) in which the mechanical mode is represented by a serial $RLC$ circuit parallel to the unmodulated coupling capacitor ${\overline{C}}_c$. The charge $\delta Q_m$ and the magnetic flux ${\varphi }_m$ play the roles of position and momentum, respectively. The equivalent mechanical resistance $R_m$, inductance $L_m$, and capacitance $C_m$ correspond to damping rate, mass, and (inverse) spring constant, respectively.

Figure 3

Ac-biased electromechanical interface. (a) Thévenin-equivalent circuit considered for the ac-biased system. An ideal voltage source $\delta V(\omega )$ in series with an impedance $Z(\omega )$ drives the coupled membrane-capacitor system. (b) EM equivalent circuit. The mechanical motion is replaced by the central loop current $I_m(\mathrm{\Omega })$. Through capacitors of capacitance ${\overline{C}}_c$, the mechanical current is connected to loop currents $I_{e,+}$ and $I_{e,-}$ representing the upper and lower sidebands of the electrical current, which are driven by voltage sources $V_{e,+}(\mathrm{\Omega })=\delta V({\omega }_d+\mathrm{\Omega })$ and $V_{e,-}(\mathrm{\Omega })=\delta V^{\ast }({\omega }_d-\mathrm{\Omega })$. (c) The spectral mapping between the electrical and mechanical frequency scales. The electrical spectrum is folded around the carrier frequency ${\omega }_d$ [with a gain profile determined by $Z(\omega )$] such that the upper and lower sidebands at ${\omega }_d\pm \mathrm{\Omega }$ are mapped into a single mechanical frequency $\mathrm{\Omega }$.

Figure 4

$N$-port network. Each port is associated with an incoming and an outgoing traveling field, $V_{\mathrm{in},i}$ and $V_{\mathrm{out},i}$. The outgoing fields are linked to the incoming fields by the scattering matrix $\mathbf{S}(\omega )$.

Figure 5

Mapping between (a) a resistor $R$ with Johnson voltage noise $V_R$ and (b) a semi-infinite lossless transmission line with characteristic impedance $Z_{\mathrm{TX}}=R$ and incoming signal $2V_{\mathrm{in}}=V_R$.

Figure 6

Ac-biased electromechanical example circuit. (a) $RLC$ circuit (ohmic resistance $R_{LC}$) coupled to a single transmission line of characteristic impedance $Z_{\mathrm{TX}}$ and, at the same time, capacitively coupled to a mechanical mode (intrinsic damping rate ${\gamma }_m$) by means of an ac bias of frequency ${\omega }_d$ (not shown) resulting in an EM coupling strength $G$. (b) The electrical part of the example system in (a) that determines the Thévenin-equivalent quantities $Z(\omega )$ and $\delta V(\omega )$ [cf. Fig. 3]. This subcircuit does not include the coupling capacitor. The voltage sources $V_{e,\mathrm{in}}^{\mathrm{(TX)}}$ and $V_{e,\mathrm{in}}^{(LC)}$ represent the transmission-line input and the Johnson noise of the resistor with resistance $R_{LC}$, respectively. (c) EM equivalent circuit for the system depicted in (a). The characteristic impedance $Z_{\mathrm{TX}}$ of the transmission line loads the upper/lower sideband loop by approximately $\pm ({\omega }_d/\mathrm{\Omega })Z_{\mathrm{TX}}$ [Eqs. (39) and (40), for $\mathrm{\Omega }\ll {\omega }_d$] (the inductor and the ohmic loss resistor are represented by $Z_{e,\pm }^{\prime }$). In the quantum limit, the interaction of the electrical sideband fields $V_{e,\pm }$ with the mechanical system can lead to squeezing of the outgoing transmission-line signal.

Figure 7

Squeezing of a traveling electromagnetic field produced by a cold microwave $LC$ resonator ($n_{\mathrm{TX}}\approx 0$, ${\overline{\omega }}_{LC}=2\pi \times 1\mathrm{GHz}$) coupled to a mechanical mode (${\omega }_{m,V}=2\pi \times 1\mathrm{MHz}$) at an ambient temperature $T_m=4\mathrm{K}$. Plotted is the homodyne power spectral density (PSD) $S(\mathrm{\Omega })$ of ${\hat{X}}_{\theta }(\mathrm{\Omega })$ for a local oscillator phase $\theta \approx 0.18\pi$, which maximizes the squeezing obtainable for a single spectral component (at $\mathrm{\Omega }\approx 0.99{\omega }_{m,V}$ for these parameters). The shot-noise (SN) limit at 1 (in the normalization used here) is indicated by the dashed line. Other system parameters used are $Z_{\mathrm{TX}}=50\mathrm{\Omega }$, $Q_{LC}={\overline{\omega }}_{LC}L/Z_{\mathrm{TX}}=100$, ${\omega }_d={\overline{\omega }}_{LC}-2\pi \times 5\mathrm{MHz}$, ${\gamma }_m=2\pi \times 0.1\mathrm{Hz}$, and $G/\sqrt{m}=3.2\times {10}^{11}\mathrm{V}/(\mathrm{m}\sqrt{\mathrm{kg}})$.

Figure 8

Electrooptomechanical equivalent circuit for a mechanical mode acting as an intermediary between an arbitrary linear electrical circuit and a single optical mode. Each of the electrical or optical sidebands is represented by a loop current $I_{e,\pm }$ or $I_{o,\pm }$ and is coupled capacitively to the mechanical loop current $I_m$ via ${\overline{C}}_c$ or ${\overline{C}}_{\mathrm{opt}}$. The effective voltage sources $V_{e,\pm },V_{o,\pm },V_m$ represent electrical, optical, and mechanical noise or signal inputs. Using standard circuit rules to determine the current in an external loop, we may determine the output at the corresponding sideband.

Figure 9

Reduced electrooptomechanical equivalent circuit in which electrical and optical modes have been adiabatically eliminated. It consists of an effective mechanical loop of shifted resonance frequency loaded by a resistive element for each sideband coupling. The resistances are positive for the upper sidebands, $R_{\mathrm{EM/OM},+}$, and negative for the lower sidebands, $-R_{\mathrm{EM/OM},-}$, leading to amplification effects. Each sideband coupling drives the effective mechanical loop with a Thévenin voltage representing the noise and signal ports of that subsystem. The read-out of the system corresponds to the signal dissipated in the various ports.