11.1 Introduction

Microwave cavity optomechanics is the part of the field of nano- and micro-electromechanics (NEMS and MEMS) that investigates the interaction between mechanical motion and electrical energy stored in microwave resonant circuits. It is one of the most promising strategies for observing and exploiting quantum behavior of macroscopic mechanical oscillators. The topic is the electrical analog of cavity optomechanics but has it roots in circuit quantum electrodynamics (cQED), the search for quantum effects in NEMS devices, and the effort to detect gravitational waves. The name microwave cavity optomechanics is a misnomer as the electrically resonant structures are not literally cavities but resonant circuits. The misnomer “cavity” persists to strengthen the analogy to cavity optomechanics and to resolve an ambiguity; the terms “resonator” and “oscillator” apply equally well to both the mechanical and electrical degrees of freedom but “cavity” is understood to refer only to the electrical degree of freedom.

11.1.1 Relationship to Cavity Optomechanics

The reemergence of interest in radiation pressure in optical cavities occurred because advances in microfabrication enabled the creation of high finesse optical cavities incorporating light and floppy mechanical oscillators [1–3]. This advent of cavity optomechanics sparked a new and vigorous effort to observe the quantum motion of a mechanical oscillator and the quantum nature of radiation pressure [4, 5]. A few years earlier, experiments with superconducting circuits had beautifully demonstrated the quantum nature of microwave light [6]. These experiments exploited superconducting microwave circuits to create artificial atoms (qubits) and cavities (superconducting resonators) and achieved coherent coupling between a single microwave photon and a qubit. It was natural to consider if the same sort of superconducting circuits could be used to create a strong interaction between microwave light and mechanical motion, revealing the quantum nature of both mechanical motion and microwave radiation pressure. Indeed, a superconducting resonant circuit with a mechanically compliant element (whose motion alters the resonance frequency of that circuit) is described by the same optomechanical Hamiltonian as a Fabry-Perot cavity with a movable mirror.

Working with microwave light and superconducting circuits has a number of appealing features. The superconducting circuitry provides a highly flexible, engineerable platform for creating particular optomechanical interactions, including interaction between mechanical oscillators and superconducting qubits. Although all experiments must be performed at low temperatures, typically below 1 K, these experimental techniques are also immediately compatible with low temperatures. At low temperatures, the random thermal forces that decohere the quantum states of mechanical oscillators are greatly reduced. More practically, microwave cavity optomechanics is free from the experimental nuisances associated with aligning and stabilizing an optical cavity. The microwave cavity, which is formed from a lithographically patterned metal film, is absolutely passively stable and rigid except for the mechanically compliant element of interest.

Against these appealing features of microwave, rather than optical, light one must weigh two consequences of the lower frequency, lower energy photons. First, the scale for optomechanical forces is proportional to the photon energy, and is therefore 4–5 orders of magnitude smaller for microwave photons compared to optical photons. Second, the lower photon energy precludes using the quantum efficient photodetection technology of optics. Nevertheless, the experiments that have demonstrated interaction between micro- and nano-mechanical oscillators and microfabricated electrical circuits have been at the forefront of optomechanical physics. Initial work focused on coupling nanomechanical oscillators to superconducting cavities. These results progressed from observing the thermomechanical motion of a mechanical oscillator cooled below 40 mK [7], to detecting the dynamical backaction of microwave fields on mechanical oscillators [8], allowing them to be cooled by radiation pressure. This radiation pressure cooling has been used to bring a mechanical oscillator close to its motional ground state [9].

Recent work has focused on overcoming the poor quantum efficiency of the microwave measurement and the weak interaction between microwaves and mechanical motion. The poor measurement efficiency has been tackled either by creating quantum limited microwave measurement tools [10] or by implementing a backaction evading measurement of just one quadrature of mechanical motion [11]. The weak interaction can be enhanced by exploiting the ability to confine microwave photons into tiny volumes, much smaller than a cubic wavelength, using lumped element resonant circuits [12] or advanced nanofabriction [13]. In a parallel development, mechanical oscillators have been coupled to superconducting qubits either dispersively or directly. Respectively, these experiments demonstrated that it is possible to infer the state of the qubit through a modification of the mechanical oscillator’s response [14], and to prepare the mechanical oscillator in a pure quantum state containing zero or one phonon through interaction with the qubit [15].

11.1.2 Emergence from Resonant Gravitational Wave Detectors and Quantum NEMS

Experimental work in microwave cavity optomechanics emerged from quantum nano-electromechanical systems [16], but was preceded by superconducting resonant-mass gravitational-wave detectors. Much of the early work in building gravitational wave detectors focused on meter-sized mechanical oscillators into which a gravitational wave would deposit energy (Ref. [17] has a good review of the experimental efforts). Energy deposited in the oscillator would be detected if it exceeded \(k_B(T+T_N)\) where \(k_BT\) is the thermal energy available to the oscillator and \(T_N\) is the noise added by the displacement measurement expressed as an apparent increase in temperature [18]. In order to detect the deflection of the mechanical oscillator various schemes were proposed and implemented. These schemes include transducing motion into a changing capacitance or inductance [19], which would then result in a changing current or voltage. A related scheme envisioned coupling motion to a nanometer-sized gap between electrodes, modulating a tunneling current [20].

Rather than detect directly the current or voltage created by a time varying capacitance, several groups pursue schemes where mechanical motion alters the resonance frequency of a literal microwave cavity [21, 22] (also see Ref. [23] and references therein). The microwave cavities are typically cylindrical Helmholtz resonators distorted so that bottom and top surfaces are brought close together forming something like a parallel plate capacitor. Strain in the cavity itself alters the size of the gap between the capacitor plates. In related work, the coupling of motion to microwave energy is studied in sapphire cylinders, which are simultaneously high-\(Q\) microwave cavities and acoustical oscillators [24].

But for matters of scale, these gravitational wave detectors contain the same physics as the microwave optomechanical structures that incorporate nano- or micro-mechanical oscillators. The meter-scale gravitational-wave detectors achieve excellent energy sensitivity [17], display dynamical backaction damping of the mechanical motion [24], and have even been used in a proof-of-principle demonstration of a backaction evading scheme [25].

Although work with resonant-mass gravitational-wave detectors continues, the dominant strategy for detecting gravitational waves now uses kilometer-sized optical interferometers. The theory of resonant-mass gravitational-wave detectors has found a new application in quantum NEMS devices. Describing the performance of NEMS devices in the language of quantum measurement occurred when micro- and nano- electromechanics had progressed to the point that it was plausible to imagine observing the quantum motion of these much smaller, higher-frequency mechanical oscillators. Initially, much of the effort was devoted to coupling nano-mechanical oscillators to ultrasensitive nano-electronic transistors, such as single-electron transistors (SETs) [26, 27], quantum point contacts (QPCs) [28], atomic point contacts (APCs) [29] and superconducting quantum interference devices (SQUIDs) [30]. Most of the insights gained from this work were in fact about the physics of the transistors, as both the imprecision and backaction of the displacement measurement are determined by the physics of the transistor itself [31].

In order to more directly pursue the goals of observing quantum behavior in macroscopic mechanical motion, nanomechanical oscillators have been embedded in microwave cavities [7]. The backaction on the mechanical oscillator is then determined by the the quality of the microwave source, which can be quantum-limited inside an ultralow temperature dilution refrigerator cryostat. That is, a random backaction force associated with the measurement arises from the amplitude or phase fluctuations of the microwave tone that excites the cavity. If the microwave tone is in a pure coherent state, the amplitude and phase fluctuations are associated only with the fundamental quantum noise of that state.

Detecting the motion of a nanomechanical oscillator via parametric, or dispersive, coupling to a microwave cavity recapitulated both the development of the resonant-mass gravitational-wave detectors and superconducting qubits. In fact, it was the development of cQED that inspired the development of microwave cavity optomechanics measurements of nanomechanical oscillators. In detecting the state of a superconducting qubit, particularly the Cooper-pair box, a readout of the qubit state using a microwave resonator provided dramatic advantages [6] over direct measurements with an SET [32]. Because the best measurements of nanomechanical motion were accomplished at that time with an SET [26, 27], it was natural to ask if the same benefits of cavity readout could be realized for detection of nanomechanical motion.

11.2 Superconducting Resonant Circuits

A true microwave cavity, meaning an empty volume enclosed by metal, is not well suited measuring nanomechanical motion. While cavities of the type described in [23] have a volume of approximately, \(1~\mathrm {cm^{3}}\) set by the wavelength of microwave signals, a nanomechanical beam oscillator has a much smaller size, perhaps \(10^{-10}~ \mathrm {cm^{3}}\) (Fig. 11.1). It is clear that only a tiny fraction of the energy stored in the cavity would interact with the oscillator. Instead, microwave cavity optomechanical structures use transmission line resonators or lumped element circuits comprising a separate inductive and capacitive element because they can have mode volumes much less than one cubic wavelength. A transmission line resonator has only one dimension on the order of a wavelength, while a lumped element resonator can be much smaller than a wavelength in all dimensions. For example, the cross-sectional dimension for a typical transmission line resonator used in these experiments is 5 \(\upmu \)m, limited by lithography or other constraints. As such, these circuits or transmission lines have mode volumes only moderately larger than the size of the mechanical oscillator itself.

11.2.1 Circuit Design

In order for these small volume “cavities” to have any detectable resonance—let alone a high quality factor \(Q\)—they must be built from superconducting metals.Footnote 1 Consequently, the experiments only operate at or below liquid helium temperatures. Even lower temperatures are required if one would like use the interaction of the mechanical oscillator with the microwave field to cool the mechanical oscillator to its ground state. For this type of cooling, the microwave cavity must itself be in a pure quantum state [9]; therefore, the cavity should be in equilibrium with a microwave environment at temperature \(T \ll \hbar \omega _r/k_B\), where \(\omega _r\) is the cavity resonance frequency. An ultralow temperature cryostat with \(T<200\) mK is usually required to meet this condition because practical aspects of microwave technology are much simpler at frequencies \(\omega /2 \pi < 26\) GHz.

Fig. 11.1
figure 1

A Nanomechanical oscillator embedded in a resonant circuit. In this device, a 150 \(\upmu \)m long aluminum (gold false color) string is freely suspended over a hole etched into the silicon substrate (green false color). Motion of the string in the plane of the figure alters the capacitance between the two aluminum electrodes. Not visible in the figure is the inductor that completes the resonant circuit by connecting the two electrodes

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The physical and technical constraints of microwave cavity optomechanics are similar to those of cQED and a type of astrophysical detector known as a microwave kinetic inductance detector (MKID). The first cavity optomechanical structures [7] looked rather like cQED devices or MKID structures (Fig. 11.2, c.f. Ref. [33]).

Fig. 11.2
figure 2

Images of chips with microwave cavity optomechanical structures formed by patterning superconducting aluminum films. a The two meandering lines are coplanar waveguide (CPW) transmission line cavities. At the bottom of the meandering line the center of the coplanar waveguide structure is shorted to the adjacent ground planes, while the top is nearly an electrical open. The horizontal transmission line (called the feedline) passes near the electrically open ends of the cavities. Microwave power passing through the feedline excites the cavities and they resonate when the excitation frequency has a wavelength one-quarter of the cavity length. In this device, the two cavities have different lengths and consequently different resonance frequencies, allowing them to be studied separately. The wire-bonds visible in the image suppress the antisymmetric mode of the CPW feedline [34]. b This image shows a lumped element superconducting resonant circuit. The meandering wire forms the inductor, while the two ends of that wire fabricated close to one another form the capacitor. While this structures resonates at about the same frequency as in subfigure a, it is substantially smaller as indicated by the scale bars

Full size image

In contrast to the cQED experiments in which the interaction between a qubit and a cavity is strong even with no photons in the cavity, the limiting factor in microwave optomechanics is the weak interaction between the motion of an oscillator and a cavity’s resonance frequency. To increase this small coupling the microwave cavities have evolved into \(LC\) resonant circuits. These structures comprise separate inductive and capacitive elements and are more compact than a transmission line resonator with the same resonance frequency (Fig. 11.2b). Because they confine electromagnetic fields into even smaller volumes than their transmission line cousins, lumped element circuits can enhance the interaction between mechanical motion and electrical energy [12].

11.2.2 Circuit Analysis

Whether the resonant structure is a lumped element circuit or a transmission line cavity, it is useful to model the circuit as composed of separate inductor and capacitor elements. In particular, the interaction of a mechanical oscillator with the circuit is readily described as a capacitor whose capacitance varies with oscillator position. Although the experimental apparatus for studying these optomechanical structures is a full microwave network, when good microwave practice is followed one need only analyze electrical circuit models to describe the microwave resonance. In essence, good practice ensures that the left and right moving waves carrying microwave energy in a cable or transmission line do not couple to each other except at the ports as labeled in Fig. 11.3. Indeed, the simple one port network shown in Fig. 11.3a, is sufficient to understand the two-port microwave resonators used in the majority of microwave cavity optomechanics experiments [7–11]. Simple transformations extend the analysis of the one port network to either of the two types of two-port networks used, those represented by Fig. 11.3b and used in Refs.  [7, 8, 10], or those represented by Fig. 11.3c and used in [9, 11].

Fig. 11.3
figure 3

Circuit diagrams for a lumped element model of a microwave cavity optomechanical structure. a The basic one-port network model of a resonant circuit. b The two-port model that describes the capacitively coupled feedline geometry of Fig. 11.2. c A two port network that describes the half wave resonators used for example in Ref.  [35]

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Whether fabricated as lumped element circuits or as a transmission line resonators, all of the microwave cavities used in recent experiments [7–12] can be modeled as parallel \(LC\) circuits. The circuit shown in Fig. 11.2b is fabricated as a parallel \(LC\) circuit, while the quarter-wave transmission line cavities in Fig. 11.2a can be modeled as a parallel \(LC\) circuit. From standard microwave network analysis an electrical short at one end of a transmission line transforms to an open a quarter wavelength distant from the short. The lumped element model for such a quarter-wave resonator is then a parallel \(LC\) circuit because that circuit is also open on resonance. Finally, the transmission through a half-wave resonator, as implemented in  [9, 11], can also be modeled as a parallel \(LC\) circuit, as it is essentially two quarter-wave resonators back-to-back. Loss in any of the resonators is modeled by including the \(R\) circuit element in Fig. 11.3a. A small capacitor \(C_c\) isolates the resonant circuit from power lost through to port 1 and dissipated in the source impedance \(Z_0\).

For the circuit shown in Fig. 11.3a, straightforward circuit analysis yields the frequency-dependent impedance \(Z_{\mathrm {in}}(\omega )\) at port 1. One can relate the amplitude and phase of the voltage wave reflected from port 1 to the incident wave through the voltage reflection coefficient \(\varGamma =(Z_{\mathrm {in}}(\omega )-Z_0)/(Z_{\mathrm {in}}(\omega )+Z_0)\), which can be written in terms of four dimensionless parameters as \(N/D\) with

$$\begin{aligned} N&=i(\gamma _I - \gamma _E \varepsilon ) +\gamma _I \gamma _E -2z(1-\gamma _I \gamma _E -i(\gamma _I+2\gamma _E-3\gamma _E \varepsilon )/2)\nonumber \\&\quad +z^2(3i\gamma _E(1-\varepsilon ) +\gamma _I \gamma _E -1) + iz^3\gamma _E(1-\varepsilon ) \end{aligned}$$
(11.1)
$$\begin{aligned} D&= i(\gamma _I + \gamma _E \varepsilon ) -\gamma _I \gamma _E -2z(1+\gamma _I \gamma _E-i(\gamma _I-2\gamma _E+3\gamma _E \varepsilon )/2) \nonumber \\&\quad + z^2(-3i\gamma _E(1-\varepsilon ) -\gamma _I \gamma _E -1) - iz^3\gamma _E(1-\varepsilon ) \end{aligned}$$
(11.2)

where \(z=(\omega -\omega _r) / \omega _r\), \(\omega _r=\sqrt{1/(L(C+C_c))}\), \(\gamma _I=\omega _r L/R\), \(\gamma _E=\omega _r C_c Z_0\), and \(\varepsilon =C_c/(C+C_c)\). While this expression is simpler when written as function of \((\omega / \omega _r)\), in this form each dimensionless parameter is small for a high-\(Q\) circuit probed near resonance. This exact expression captures more than is necessary to model a high-\(Q\) resonant circuit probed near its resonance frequency. From Eq. 11.1, one can find a simple approximate expression that describes the frequency dependence near a high-\(Q\) resonance in a systematic way

$$\begin{aligned} \varGamma (z) = \frac{i( \gamma _I - \gamma _E \varepsilon ) - 2z}{i (\gamma _I + \gamma _E \varepsilon ) - 2z}. \end{aligned}$$
(11.3)

Although this expression accurately captures the magnitude of \(\varGamma \) and its phase dependence, it differs from the exact expression by an overall phase shift even in the high-\(Q\) limit.Footnote 2 This deviation is of little consequence because the overall phase shift is usually not under experimental control. Ignoring the overall phase shift, Eq. 11.3 can be fit to a measured response extracting three parameters \(\omega _r\), \(\gamma _I\), and \(\gamma _E \varepsilon \). The rate of energy lost to dissipation in the resonant circuit itself is evidently \(\omega _r\gamma _I=\kappa _I\). The rate at which energy is lost to port 1 is likewise \(\omega _r\gamma _E \varepsilon =\kappa _E\).

Examining Eq. 11.3 one recognizes the reflection coefficient of a series \(RLC\) circuit probed near resonance. Namely, at resonance for a lossless resonator \(\varGamma =-1\), the reflection from a short circuit. Even though the resonator itself is modeled as a parallel \(RLC\) circuit, when excited through \(C_c\) its behavior is that of a series \(RLC\) circuit with transformed values of \(R\), \(L\), and \(C\). This transformation to a series circuit will simplify the analysis of electromechanics in Sect. 11.3.

From the reflection coefficient of Fig. 11.3a, the two-port parameters of networks B and C are determined. The dynamics of resonant circuits are the same in all three examples, only the interference of the excitation signal with the reflected signal changes. For Fig. 11.3b, the S-parameters for the network are \(s_{11}=s_{22}=(\varGamma -1)/(\varGamma +3)\) and \(s_{21}=s_{12}=(2\varGamma +2)/(\varGamma +3)\). For Fig. 11.3c, the expressions are a little more complex as they must explicitly contain the impedance of the coupling capacitor \(Z_c=1/i \omega C_c\). They are \(s_{11}=s_{22}=(\varGamma -1)[(Z_0+Z_c)^2 + 4Z_0 Z_c]/D_G\), and \(s_{21}=s_{12}=2 Z_0[(\varGamma +1) Z_0 + (\varGamma -1) Z_c]/D_G\), where \(D_G = (Z_0 + Z_c)[(\varGamma +3) Z_0 + (\varGamma -1)Z_c].\)

Equation 11.3 is simple enough to allow compact analysis of a microwave optomechanical circuit, yet general enough to capture the behavior of the microwave networks used in recent experiments. In fact, approximating Eq. 11.1 with Eq. 11.3 is a version of a standard approximation for resonant systems where one replaces the full frequency dependence with a simpler Lorentzian form. In essence the frequency dependant impedance has been replaced with a response proportional to \(1/(2z Q+i)\), a function whose magnitude is a Lorentzian. This simple form is used to model most resonant phenomena in cavity optomechanics, as both the cavities and mechanical oscillators are high-\(Q\) structures probed near resonance. Furthermore, the quantum equations of motion for a driven harmonic oscillator coupled to a simple environment are described in this form. (See for example Appendix E of Ref. [36]). In that sense, the high-\(Q\) and close to resonance approximation made in deriving Eq. 11.3 yield the same response function as the Markov approximation of the quantum input–output theory for cavities.

11.2.3 Electromechanical Coupling

The capacitors and inductors that determine the resonance frequency of an electrical circuit are structures that store either electrical or magnetic energy. The inductance and capacitance depend only on the geometry of the metal that makes up the circuit and the permeability or permittivity of the insulating regions that contain the magnetic and electrical fields. Any change in that geometry should change the stored electrical energy and there will be an associated stress on the capacitor or inductor itself. If one of the elements is mechanically compliant, there will be an interaction between the motion of that object and the electrical energy stored in the circuit. Of course, microphones, speakers, relays and other electromechanical devices exploit exactly this type of interaction between motion and electricity. In the case of microwave cavity electromechanical structures mechanical motion alters the resonance frequency of the electrical circuit by a changing capacitance.Footnote 3 The mechanically compliant structure is itself a harmonic oscillator and in most cases the mechanical resonance frequency is far below the electrical resonance frequency. (Reference [15] is an important exception where the circuit and mechanical oscillator are nearly resonant with each other.) In these cases, the mechanical oscillator does not respond substantially to forces oscillating at the microwave frequency of the circuit resonance, but only to lower frequency variations in the microwave energy stored in the circuit.

In general, these electrical forces can reshape the very object whose motion defines the mechanical oscillator and one should include the electrical energy, as well as the elastic energy, when calculating the shapes of the resonant modes and their frequencies. In the analysis that follows, I will assume that the electrical forces are much smaller than the elastic forces of the mechanical oscillators, both because it is a great simplification and because it is accurate for the electromechanical experiments with superconducting circuits.

In this limit, each mode of a mechanically resonant structure can be modeled as an independent single degree of freedom, even though the mode shape is a full three-dimensional function of space. In the cavity optomechanical structures the motion of this mode changes the shape of a metal structure, thereby altering the capacitance \(C\) of this metal object to nearby metal structures. Referring to the coordinate of this single degree of freedom as \(x\), the electro-mechanical coupling is determined by the function \(C(x)\). As a matter of principle this function could be calculated by presuming that all metal objects are held at different fixed potentials and finding the change in electrical energy as function of \(x\). In practice, the detailed geometry of the micro- or nano- fabricated mechanical oscillator is not known well enough for this to be a fruitful exercise. Estimates of \(C(x)\) where the mechanical oscillator is treated as a thin wire [7] or a flat plate [12] translated by its motion towards or away from a nearby counter-electrode are accurate within a factor of two.

11.3 Electromechanics

Armed with the function \(C(x)\), one can find the dynamics of the coupled electrical and mechanical system. In full generality this calculation is difficult because the interaction between motion and electricity is not linear. Nevertheless, microwave cavity optomechanical experiments have all been performed in a limit where the interaction can be linearized and the essential physics captured by a simple model.

11.3.1 Electromechanical Model

As the model for electromechanical effects, I assume a series \(RLC\) circuit driven with an ac-voltage source of source impedance \(Z_0\), amplitude \(V_0\), and angular frequency \(\omega _L\), as \(V(t)=(V_0/2) (e^{i \omega _L t}) + c.c.\), where \(c.c.\) is the complex conjugate (Fig. 11.4). If I identify \(R\sqrt{C/L} \rightarrow \gamma _I\), \(Z_0\sqrt{C/L} \rightarrow \gamma _E \varepsilon \), and \(\omega \sqrt{LC}-1 \rightarrow z\), this circuit has the same behavior close to resonance as Fig. 11.3a and has a reflection coefficient given by Eq. 11.3. The capacitor will be the electromechanical element; I model the mechanical coordinate \(x\) as a mass \(m\) subject to a spring force with constant \(k_s\). For a distributed mechanical oscillator such as a beam or drumhead, there is not a unique definition of the mass \(m\), spring constant \(k_s\), or coordinate \(x\). For microwave cavity optomechanical structures, in which no particular point on the distributed mechanical oscillator has special significance, the most sensible convention uses the total mass of the compliant structure as \(m\). With this choice, \(x\) is the RMS deviation of the distributed oscillator away from its equilibrium position, and \(k_s\) is determined from the angular resonance frequency \(\varOmega _{\mathrm{M}}\) as \(k_s=m\varOmega _{\mathrm{M}}^2\).

Fig. 11.4
figure 4

A simple electromechanical circuit. The capacitor is assumed to be a mechanically compliant element whose capacitance depends on a coordinate \(x\). That coordinate experiences a spring force with constant \(k_s\)

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11.3.2 Dynamics of Electromechanical Systems

Figure 11.4 is a simple enough model that it can be solved with a few approximations; yet, detailed enough to contain almost all of the phenomena observed in the recent experiments [7–12]. In order to solve for the dynamics of this electromechanical circuit, I will use a Lagrangian description. For electromechanical circuits, a Lagrangian description is preferable to standard ac circuit analysis because the electrical and mechanical degrees of freedom are treated equivalently in deriving the coupled equations of motion. Because the energy of the circuit \(U=q^2/2C(x) + (L/2) \dot{q}^2\) is easily written in terms the charge \(q\) and its time derivative \(\dot{q}=dq/dt\), I use \(q\) as the generalized coordinate of the circuit. The Lagrangian that describes the model is then \(\fancyscript{L} =[(1/2)L\dot{q}^2 + (1/2)m\dot{x}^2] - [(1/2)q^2/C(x) + (1/2)k_s x^2 - qV]\), where \(V\) is a generalized force acting on \(q\). Rather than include dissipation into the Lagrangian formalism, I will introduce the dissipative forces directly into the equations of motion. The equations of motion with dissipation are

$$\begin{aligned} L\ddot{q}&= -\frac{q}{C(x)} + V(t) -\dot{q}(R+Z_0)\end{aligned}$$
(11.4)
$$\begin{aligned} m\ddot{x}&= -k_s x +\frac{q^2}{2 C(x)^2}\frac{\partial C(x)}{\partial x} -\varGamma _M m\dot{x}, \end{aligned}$$
(11.5)

where the dissipative mechanical force is \(-\varGamma _M m \dot{x}\).

Even numerically, these equations are too general to solve because the function \(C(x)\) is unspecified. To make progress, one approximates the coupling between \(x\) and \(q^2\) to linear order by expanding \(1/C(x)\approx 1/C(\bar{x})-(1/C(\bar{x})^2)(\partial C(\bar{x})/\partial {x})(x-\bar{x})\), where \(\bar{x}\) is the static value of the coordinate \(x\). In order to determine \(\bar{x}\) one must know the function \(C(x)\) and specify the applied voltage \(V(t)\). Equations 11.4 and 11.5 can then be solved for the static values of \(q^2\) and \(x\). If the applied voltage is sufficiently small that the change in \(C(x)\) due to the application of \(V\) can be approximated linearly then one can find \(\bar{x}=-\bar{F}/k_s\), where \(\bar{F}=(\partial C/\partial x)(|V_0|^2/4)/[(1-\omega _L LC)^2 + (\omega _L (R + Z_0))^2].\) In experimental practice, the technical limitations of micro- or nano- fabrication imply that quantities such as \(C(x)\), \(k_s\), and \(m\) that depend on the full 3-dimensional shape of an object are not known with better than 10 % accuracy. One still can proceed with the linearization without knowing \(\bar{x}\); it will have some value that will determine \((1/C(\bar{x}))\partial C(\bar{x})/\partial x\) and that quantity can be determined experimentally. From the Lagrangian with linearized coupling we can write more tractable equations of motion as

$$\begin{aligned} L\ddot{q}&= V -\frac{q}{C}\left( 1-\frac{1}{C}\frac{\partial C}{\partial x} x\right) -\dot{q}(R+Z_0)\end{aligned}$$
(11.6)
$$\begin{aligned} m\ddot{x}&= -k_s (x+\bar{x}) + \frac{q^2}{2C^2}\frac{\partial C}{\partial x} -\varGamma _M m\dot{x}, \end{aligned}$$
(11.7)

where \(C = C(\bar{x})\) and \(x\) now refers to displacement from \(\bar{x}\) (i.e. \(x-\bar{x} \rightarrow x\)). Although the coupling no longer contains explicit \(x\) dependence, these equations of motion are themselves non-linear through the \(xq\) and \(q^2\) terms. They can be solved either numerically or using approximate methods.

Let us find approximate solutions to Eqs. 11.6 and 11.7. In the presence of the drive \(V(t)\), I will seek solutions of the form \(q(t)=(1/2)(q_0 + q_1(t))e^{i \omega _L t} + c.c.\) and \(x=x(t)\). In deriving linear equations of the motion for \(q_1\) and \(x\), I assume a strong drive \(|q_1|/|q_0| \ll 1\) and a slowly varying response \(|\dot{q_1}(t)|/|\omega _L q_1(t)| \ll 1\) and work to first order in both small quantities. In addition, I ignore terms oscillating near \(2\omega _L\) as neither the circuit nor the oscillator will respond so far from resonance. The linear equations of motion are

$$\begin{aligned} {-}G\frac{q_0q_1^*(t)+q_0^*q_1(t)}{2C\omega _{\mathrm{opt}}} + F_{\mathrm {ext}}=(\ddot{x} + \varGamma _m \dot{x}+\varOmega _{\mathrm{M}}^2 x)m \end{aligned}$$
(11.8)
$$\begin{aligned} 2G q_0\omega _{\mathrm{opt}}x(t)=q_1(t)(\omega _L^2-\omega _{\mathrm{opt}}^2 - i \omega _L \kappa ) + \dot{q_1}(t)(2 i \omega _L + \kappa ), \end{aligned}$$
(11.9)
$$\begin{aligned} 2G q_0^*\omega _{\mathrm{opt}}x(t)=q_1^*(t)(\omega _L^2-\omega _{\mathrm{opt}}^2 + i \omega _L \kappa ) + \dot{q_1}^*(t)(-2 i \omega _L + \kappa ) \end{aligned}$$
(11.10)

where the \(k_s \bar{x}\) has vanished by construction, \(\kappa =(\kappa _I + \kappa _E)=(R+Z_0)/L\) is the total rate at which energy is lost from the circuit and I have introduced a force \(F_{\mathrm {ext}}\) applied to the mechanical oscillator by an external agent. In addition, I define the static resonance frequency \(\omega _{\mathrm{opt}}=(LC(\bar{x}))^{-1/2}\) and the electromechanical coupling \(G=-\omega _{\mathrm{opt}}(1/2C)(\partial C / \partial x)\). These equations are remarkable; they show that fluctuations in the charge act as a force that drives the oscillator and fluctuations in the position of the oscillator act as a voltage source that creates charge fluctuations on the capacitor. Implicitly the LHS of Eq. 11.8 contains the position of the oscillator. One could arrange such a condition by measuring the position of the oscillator and applying a force based on that measurement. Indeed literal feedback of this type is commonplace, but here the coupled electromechanics has reduced to a kind of endogenous feedback.

One can justify the assumptions made in deriving the equations of motion by noting the parameters used in the microwave cavity experiments to date [7–12]. Mechanical resonance frequencies range between \(\varOmega _{\mathrm{M}}/(2\pi )\approx 100\) kHz to 10 MHz, while typical cavity resonances are in the range 5 GHz \(<\omega _{\mathrm{opt}}/(2 \pi )<8\) GHz. The cavity decay rates are in the range 200 kHz \(< \kappa /(2 \pi )< \) 2 MHz, while the mechanical dissipation rates \(\varGamma _M< 1\) kHz are much smaller. For these parameters the mechanical degree of freedom is always slow compared to the electrical degree of freedom (this is not true for [15]) and both the microwave cavity and mechanical oscillator have \(Q>1{,}000\). When discussing quantum effects in these systems (Sect. 11.4.2) it will also become clear that a strong drive is required for substantial coupling between electricity and motion.

These coupled equations can be readily solved via Fourier transform (or less formally by ansatz \(x(t)\rightarrow x(\omega )e^{i \omega t}\) and \(q_1(t)\rightarrow q_1(\omega )e^{i \omega t})\) for the purely mechanical susceptibility \(x(\omega )/F_{\mathrm {ext}}(\omega )\) or for the electromechanical susceptibility \(q_1(\omega )/F_{\mathrm {ext}}(\omega )\). The procedure requires a little more care than is usually necessary for linear equations of motion because \(q_1(t)\) is complex. (I write both Eq. 11.9 and it complex conjugate Eq. 11.10 to emphasize that \(q_1(t)\) is complex.) Furthermore \(q_1^*(t)=q_1^*(\omega )e^{- i \omega t}\), thus; the frequency components rotating as \(e^{i \omega t}\) are \(q_1(\omega )\), \(x(\omega )\), and \(q_1^*(-\omega )\). The frequency domain equations are

$$\begin{aligned} x(\omega )&= \chi _M(\omega )\left[ F_{\mathrm {ext}}(\omega )-\frac{G(q_0q_1^*(-\omega ) + q_0^*q_1(\omega ))}{2C\omega _{\mathrm{opt}}}\right] \end{aligned}$$
(11.11)
$$\begin{aligned} q_1(\omega )&= \chi _c(\omega )q_0 G x(\omega ) \end{aligned}$$
(11.12)
$$\begin{aligned} q_1^*(-\omega )&= \chi _c^*(-\omega )q_0^* G x(\omega ) \end{aligned}$$
(11.13)

where I have defined the bare mechanical susceptibility \(\chi _M^{-1}= m(\varOmega _{\mathrm{M}}^2-\omega ^2 + i \varGamma _M \omega )\), the circuit susceptibility \(\chi _c(\omega )= [1-(\varDelta /2\omega _L)]/[(\varDelta ^{\prime } -\omega ) + i (\kappa /2)(1-\omega /\omega _L)]\). I have written the circuit susceptibility in terms of the detuning of the drive from circuit resonance \(\varDelta =\omega _L - \omega _{\mathrm{opt}}\) and defined \(\varDelta ^{\prime }=\varDelta -(\varDelta ^2/2\omega _L)\).

Fig. 11.5
figure 5

A simple block diagram description of the effect of electromechanics on the susceptibility of a mechanical oscillator. As in the text, \(Y(\omega )=A(\chi _c(\omega )+\chi _c^*(-\omega )).\)

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These coupled equations constitute solutions to the electromechanical circuit linearized about a strong oscillatory drive. For example, they can be used to find the charge “sidebands” about a drive frequency \(\omega _L\) generated by an oscillatory \(F_{\mathrm {ext}}\). In addition, they determine the mechanical susceptibility in the presence of electromechanical effects

$$\begin{aligned} \frac{x(\omega )}{F_{\mathrm {ext}}}=\widetilde{\chi _M}=\frac{\chi _M}{1+\chi _M A(\chi _c(\omega )+\chi _c^*(-\omega ))}, \end{aligned}$$
(11.14)

where \(A=G^2|q_0|^2/(2C\omega _{\mathrm{opt}})\). Written in this form, the description of electromechanical effects as endogenous feedback is evident (Fig. 11.5). The open loop response of the mechanical oscillator \(\chi _M\) and the closed loop response is \(\widetilde{\chi _M}\). In the recent literature, it is not control theory which has served as the intellectual touchstone for these electro- or opto-mechanical effects but rather quantum field theory. For example, the term \(Y(\omega )=A(\chi _c(\omega )+\chi _c^*(-\omega ))\) is known as the self-energy and Eq. 11.14 is a version of Dyson’s equation [37]. This choice is natural when these equations of motion are derived directly from the quantum Hamiltonian and perturbation theory. Nevertheless, a classical feedback control offers a helpful alternative perspective. Most importantly, any effect contained within such a description is unambiguously classical. In addition, control theory has developed many powerful calculational techniques, all of which can be used once the electromechanical interaction is described in this language. For example, with these techniques it is simple to decide if the electromechanical circuit is stable as a function of \(A\).

One of the most studied effects of electro- and opto- mechanics is the ability to damp, amplify, stiffen, or soften the motion of the mechanical oscillator by the radiation pressure. Such effects are easily calculated from Eq. 11.14. First I rewrite the closed loop response in a different form \(\widetilde{\chi _M}(\omega )^{-1}=\chi _M(\omega )^{-1}+Y(\omega )\). For a high \(Q\) mechanical oscillator studied near resonance, I approximate \(\chi _M(\omega )^{-1}\approx 2\varOmega _M m[\varOmega _M-\omega +i{\varGamma /2}]\). It is then easy to see that for a wide range of conditions \(\widetilde{\chi _M}\) has the same form as \(\chi _M\) but with a new resonance frequency and linewidth determined by the real and imaginary parts of \(Y(\omega =\varOmega _M)/(2\varOmega _M m)\). Following through with this calculation, the additional “optical” damping and the “optical” frequency shift are

$$\begin{aligned} \varGamma _{\mathrm{opt}}&= \frac{2A}{\varOmega _M m}\left[ \frac{\kappa }{4(\varDelta +\varOmega _M)^2+\kappa ^2}-\frac{\kappa }{4(\varDelta -\varOmega _M)^2+\kappa ^2} \right] \end{aligned}$$
(11.15)
$$\begin{aligned} \delta \varOmega _{\mathrm{M},\mathrm{opt}}&= \frac{2A}{\varOmega _M m}\left[ \frac{\varDelta -\varOmega _M}{4(\varDelta -\varOmega _M)^2+\kappa ^2} +\frac{\varDelta +\varOmega _M}{4(\varDelta +\varOmega _M)^2 +\kappa ^2} \right] , \end{aligned}$$
(11.16)

where I have assumed \(\varOmega _M\ll \omega _{\mathrm{opt}}\) in order to write a simpler form for \(\chi _c(\varOmega _{\mathrm{M}}) \approx [(\varDelta -\varOmega _{\mathrm{M}})+(i \kappa /2)]^{-1}\) and arrive at the form appearing in Refs. [8, 37]. In those papers, the prefactor \(2A/\varOmega _M m\) is written in a quantum form \(4nG^2 x_{\mathrm{ZPF}}^2\), with an explicit dependence on the zeropoint motion of the oscillator \(x_{\mathrm{ZPF}}^2=\hbar /(2 \varOmega _M m)\) and on the number of photons in the cavity \(n=E/\hbar \omega _{\mathrm{opt}}\). By recognizing \(E=q_0^2/2C\) as the average energy stored in the circuit, it is clear that (\(4nG^2 x_{\mathrm{ZPF}}^2=2A/\varOmega _M m\)) the prefactors are indeed the same and that \(x_{\mathrm{ZPF}}^2\) appears if the energy stored in the circuit is expressed in units of energy quanta \(n=E/\hbar \omega _{\mathrm{opt}}\).

11.4 Quantum Effects in Electromechanics

By following the procedure of canonical quantization, the quantum optomechanical Hamiltonian can be derived from the classical circuit Lagrangian. Although the quantum description is widely used, the majority of microwave cavity optomechanical experiments are still contained within classical physics. With the recent demonstration of many photon strong coupling [12], microwave optomechanics is poised to escape a classical description.

11.4.1 The Cavity Optomechanical Hamiltonian

In the body of theory associated with cavity optomechanics, quantum equations of motion reminiscent of Eqs. 11.6 and 11.7 are usually derived by starting from a quantum cavity optomechanical Hamiltonian [37, 38]. One can show that the quantum version of the electromechanical circuit model is indeed this Hamiltonian. Returning to the Lagrangian with linearized coupling, one can recast this into a classical Hamiltonian using the formal definition, \(H=\dot{q}\varPhi + \dot{x}p - \fancyscript{L}\), where the canonical momenta are \(p\equiv \partial \fancyscript{L}/\partial \dot{x}=m \dot{x}\) and \(\varPhi \equiv \partial \fancyscript{L}/ \partial \dot{q}=L\dot{q}\). Writing \(H\) as a function of \(q\), \(x\), \(p\), and \(\varPhi \) gives

$$\begin{aligned} H=\frac{p^2}{2m}+ \frac{k_s x^2}{2}+ \frac{\varPhi ^2}{2L} + \frac{q^2}{2 C}\left( 1-\frac{1}{C}\frac{\partial C}{\partial x} x\right) - qV. \end{aligned}$$
(11.17)

Having written the classical Hamiltonian in terms of system coordinates and their canonical momenta, one simply writes the quantum Hamiltonian by replacing each coordinate or momentum by its operator (e.g. \(q\rightarrow \hat{q})\) and requires that they obey canonical commutations relations \([\hat{x},\hat{p}]=i\hbar \), and \([\hat{q},\hat{\varPhi }]=i\hbar \). By defining the creation and annihilation operators for the two harmonic oscillators

$$ \hat{a}=\hat{q}\sqrt{Z_c/2\hbar } + i\hat{\varPhi }\sqrt{1/2Z_c\hbar }$$
$$\hat{b}=\hat{x}\sqrt{Z_M /2\hbar } + i\hat{p}\sqrt{1/2Z_M\hbar }, $$

one arrives at the celebrated optomechanical Hamiltonian

$$\begin{aligned} \hat{H}&=\hbar \omega _{\mathrm{opt}}\left( \hat{a}^\dag \hat{a}+\frac{1}{2}\right) + \hbar \varOmega _{\mathrm{M}}\left( \hat{b}^\dag \hat{b}+\frac{1}{2}\right) + \hbar Gx_{\mathrm{ZPF}}(\hat{b}+ \hat{b}^{\dag })\hat{a}^\dag \hat{a}\nonumber \\&\quad -(\hat{a}+ \hat{a}^\dag ) \sqrt{\frac{\hbar }{2Z_c}}V + \frac{\hbar G}{2}\left[ \hat{a}\hat{a}+ \hat{a}^\dag \hat{a}^\dag +1\right] x_{\mathrm{ZPF}}(\hat{b}+ \hat{b}^{\dag }), \end{aligned}$$
(11.18)

where \(Z_c = \sqrt{L/C}\) and \(Z_M = \sqrt{k_s m}\). The terms in square brackets of Eq. 11.18 are not usually included as part of the optomechanical Hamiltonian. Indeed there is a good physical argument to ignore them for the cavity optomechanical systems for which \(\varOmega _{\mathrm{M}}\ll \omega _{\mathrm{opt}}\). The force associated with these terms oscillates at about twice \(\omega _{\mathrm{opt}}\), where the susceptibility of the mechanical oscillator is likely to be vanishingly small. Nevertheless, I have included them because it is possible [15] to build mechanical structures that responds at \(2\omega _{\mathrm{opt}}\) when \(2\omega _{\mathrm{opt}}\) is a microwave rather than optical frequency.

11.4.2 Observing Quantum Effects in Microwave Cavity Optomechanics

The quantum Hamiltonian of Eq. 11.18 has been studied in most of its possible limits [37, 38] and extended to include additional physical phenomena. In this work, the formalism of quantum optics and the descriptive language of optics has been dominant. Rather than recreate those calculations, I instead focus on the purely classical equations of motion. This choice is partly to acknowledge that the quantization of the cavity optomechanical Hamiltonian is an act more of aspiration than necessity. In addition, solving the classical equations of motion in the language of electromechanics, rather than quantum optics, establishes intuition for those more familiar with circuits.

From Eq. 11.18, one can understand why true quantum effects are elusive in electro-mechanics. The quantity \(Gx_{\mathrm{ZPF}}< 100\) Hz [12], which sets the rate for single photon-phonon processes in these circuits, is slower than the rate at which dissipative and decohering processes destroy the purity of quantum states of the cavity or oscillator (\(\kappa > 100\) kHz). At the moment, strong coupling between photons and phonons requires the application of a strong cavity drive [12, 39], through the \(qV\) term in Eq. 11.17. In the presence of a strong drive, the effective Hamiltonian is bilinear in the photon \(\hat{a}\) and phonon \(\hat{b}\) operators. The phonon is just another boson; such linear interactions are analogous to the beam splitter and parametric down conversion interactions in quantum optics [39]. As a consequence of a theorem of quantum information [40], a system of harmonic oscillators: with only linear interactions, prepared in a Gaussian state, and measured with linear amplifiers has an average behavior that is classical, while the fluctuations away from the average remain Gaussian. In quantum optics, profoundly quantum behavior can be observed even for these linear interactions when the measurement process is nonlinear (e.g. photon detection). In electromechanics where there exists no readily available photon-counting technology, the predictions of quantum and classical equations of motion are very difficult to distinguish. In the linear measurement of microwave fields that have interacted with an electromechanical circuit, quantum effects are manifest primarily as an irreducible Gaussian noise process, identified as quantum noise. Indeed, many of the prominent results of this field constitute the measurement of a noisy quantity, such as the residual motion of a mechanical oscillator [9] upon cooling, or the added noise of measurement [10], which is then compared to a quantum-limited value. To date, there is one marvelous counter example [15] to the relative weakness of quantum effects in electro- or opto-mechanical structures. In this case, the mechanical oscillator inherits its nonlinearity through a strong interaction with a superconducting qubit with which it is nearly resonant. For the situation where low frequency mechanical motion alters the resonance of a high frequency cavity, the optomechanical nonlinearity is currently too weak for a single photon and phonon to influence one another. I am nevertheless optimistic that this may be overcome in the future. In the three years from the inception of the effort to measure and manipulate nanomechanical oscillators with microwave cavities, the coupling strength between motion and microwave electricity has increased more than 10,000 times (Fig. 11.6). Furthermore, the linewidth of microfabricated microwave cavities are decreasing as the loss processes in those structures are better understood [41–43].

Fig. 11.6
figure 6

The increase in \(G\) versus publication date. The cited works are Regal et al. [7], Teufel et al. [8, 10, 12, 44], Rocheleau et al. [9], Hertzberg et al. [11], Sulkko et al. [13]. A reasonable target for single photon strong coupling is about \(10^{7}\) kHz/nm

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In the meantime, the level of quantum interaction available is still quite useful for measurement and control. Returning to the feedback description of electromechanics, the performance of the endogenous electromechanical feedback can be compared to an active feedback scheme. For any feedback scheme, the quality of the oscillator control will be limited by any stochastic component in the feedback force. In an active feedback scheme, the dominant stochastic force may well arise from error in the position measurement. For electromechanial feedback, no explicit measurement is required. The stochastic force will arise from the random components in the applied voltage. For microwave signals applied inside a dilution refrigerator, these stochastic forces can be quantum-limited. From one point of view, the cooling of a mechanical oscillator by radiation pressure is simply an exchange between the thermal state which characterizes the mechanical oscillator and the coherent state of the microwave field. Upon cooling the mechanical oscillator acquires the zero-entropy coherent state of the microwave field while the microwave field acquires the thermal state of the oscillator, carrying it away.

This point of view also demonstrates that it should be possible to use microwave cavity optomechanical structures to prepare mechanical oscillators in profoundly quantum non-Gaussian states before the single photon strong coupling regime is achieved. Non-Gaussian states of the microwave field are routinely created using superconducting qubits [45]. These in-principle can be transferred to the state of the mechanical oscillator using the already achieved many photon strong coupling [12, 39]. Whether using this strategy or direct coupling to a qubit [15], mechanical oscillators may soon be prepared in a quantum superposition of two places at once (so-called Schrodinger cat states), using already extant electromechanical technology.