# Electric circuit model of microwave optomechanics

Xin Zhou  Corresponding Author<EMAIL_ADDRESS>CNRS, Univ. Lille, Centrale
Lille, Univ. Polytechnique Hauts-de-France, IEMN UMR8520, Av. Henri Poincaré,
Villeneuve d’Ascq 59650, France  Dylan Cattiaux  Univ. Grenoble Alpes,
Institut Néel - CNRS UPR2940, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex
9, France  Didier Theron  CNRS, Univ. Lille, Centrale Lille, Univ.
Polytechnique Hauts-de-France, IEMN UMR8520, Av. Henri Poincaré, Villeneuve
d’Ascq 59650, France  Eddy Collin  Univ. Grenoble Alpes, Institut Néel - CNRS
UPR2940, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France

###### Abstract

We report on the generic classical electric circuit modeling that describes
standard single-tone microwave optomechanics. Based on a parallel $RLC$
circuit in which a mechanical oscillator acts as a movable capacitor,
derivations of analytical expressions are presented, including key features
such as the back-action force, the input-output expressions, and the spectral
densities associated, all in the classical regime. These expressions coincide
with the standard quantum treatment performed in optomechanics when the
occupation number of both cavity and mechanical oscillator are large. Besides,
the derived analytics transposes optics into electronics terms, which is
mandatory for quantitative measurement and design purposes. Finally, the
direct comparison between the standard quantum treatment and the classical
model addresses the bounds between quantum and classical regimes, highlighting
the features which are truly quantum, and those which are not.

††preprint: APS/123-QED

##  I Introduction

In the last decades, great scientific successes have been achieved in cavity
optomechanics, which uses laser photons to explore the interaction between
optical fields and mechanical motion (Aspelmeyer et al., 2014a). Cavity
optomechanics allows to cool down suspended micro-mirrors and to excite cold
atom clouds through the radiation-pressure effect, which enables the
investigation of mechanical systems in the quantum regime (Arcizet et al.,
2006; Brennecke et al., 2008; Schliesser et al., 2009a). Optical forces also
offer a method to enhance the resolution of nano-mechanical sensors through
the optical spring effect (Yu et al., 2016). Optomechancial platforms are thus
both model systems containing rich physics to be explored (Aspelmeyer et al.,
2014a), but also unique sensors able to detect extremely tiny
forces/displacements, especially in the quantum regime as foreseen in the 80’s
(Caves et al., 1980). Besides, the amazing sensitivity of optomechanics for
displacement detection lead recently to the tour de force detection of the
long-thought gravitational waves (Abbott et al., 2016).

Inspired by the achievements of circuit quantum electrodynamics (QED),
researchers eventually started to use microwave photons confined in a
superconducting resonator to probe micro/nano-mechanical oscillators, leading
to the new experimental field of microwave optomechanics (Regal et al., 2008).
It inherits the abundant physical and technological properties emerging from
cavity optomechanics, and benefits from the capabilities of microwave circuit
designs. Especially, low temperature experiments performed in this wavelength
range allow the use of quantum electronics components such as Josephson
parametric amplifiers (Teufel et al., 2009). Building on microwave
optomechanical schemes, sideband cooling of mechanical motion down to the
quantum ground state and entanglement of massive mechanical oscillators have
been recently achieved (Teufel et al., 2011; Ockeloen-Korppi et al., 2018).
Moreover, microwave optomechanical platforms with cavity-enhanced sensitivity
have been built, squeezing the classical thermal fluctuations of the
mechanical element (Pernpeintner et al., 2014; Faust et al., 2012; Huber et
al., 2019). The latter clearly demonstrates that optomechanics is not only
reserved for frontiers experiments in quantum mechanics, but represents also a
new resource for classical devices with novel applications.

Within circuit QED, Hamiltonian formulations adapted to various quantum
circuits have been developed using the mathematical toolbox of quantum
mechanics. This has been achieved by quantizing (i.e. promoting to operators)
variables of electrical engineering (e.g. voltage, current) and building the
corresponding generic quantum circuit theory (Yurke and Denker, 1984; Vion et
al., 2002; Cottet, 2002). Signal propagation in coplanar waveguides (CPW) is
thus described by propagating bosonic modes (i.e. photons, similarly to laser
light), and RLC resonators by localized modes. Input-output theory and quantum
noise formalisms (Gardiner et al., 2004; Gardiner and Zoller, 2015) are then
required to describe driving fields and detected signals, in the framework of
the quantum version of the fluctuation-dissipation theorem (applied here to
superconducting circuits) (Göppl et al., 2008; Clerk et al., 2010). Among the
properties of interest, quantum electric circuit theory enables to model non-
linear features and dissipation, two crucial properties at the core of device
performance and novel circuit development. For instance, quantum-limited
Josephson amplifiers are described by modeling Josephson junctions as tunable
nonlinear inductors (Vijay et al., 2009; Zhou et al., 2014).

For microwave optomechanics, although its basic principle and relevant
applications are not restricted to quantum mechanics, today the full classical
circuit model has not yet been presented, despite useful pioneering
discussions (Aspelmeyer et al., 2014b). Therefore, the theoretical framework
in use for all experiments, even when purely classical in essence, still
relies on the Hamiltonian deduced from quantum optics (Aspelmeyer et al.,
2014a). This means both that the physical description is over-complexified,
losing from sight the real (classical) nature of most properties; and also
that the connection between circuit parameters and optomechanical properties
is not clearly identified, while it is a need for design and optimization.

In this paper, we present the generic classical electric circuit modeling of
microwave optomechanics. First in part II.1, we review the different RLC
equivalent circuits at stake and then describe the framework of our formalism
in section II.2. In the following part II.3, we derive the cavity back-action
force on the mechanics with its two components: the dynamic one (proportional
to motion amplitude) that modifies the mechanical susceptibility, and the
stochastic one (from the cavity current noise) that limits the mechanical mode
fluctuations. In part II.4, we then work out the corresponding input-output
expressions with classical spectral densities. In the last part III of the
paper, we compare the classical description to the conventional quantum
Hamiltonian method and discuss the key ingredients of the models: namely
sideband asymmetry, the Heisenberg uncertainty, and the motion optimal
detection limit for which we define a (relative) standard classical limit
(SCL), in analogy with the (ultimate) standard quantum limit (SQL). The major
merit of this electric RLC circuit model lies in the fact that it gives a
straightforward connection between experimentally relevant quantities and
electric elements of the devices. It also offers a clear understanding of most
features, demonstrating their classical nature (with no need to invoke quantum
mechanics).

##  II Modeling of the classical electric analogue

###  II.1 Generic microwave optomechanical circuit

In the electromechanical version of optomechanics, the mechanical element of
the circuit is a movable capacitor which is part of an RLC resonator. In
practice, this resonant circuit can be realized in many ways: a quarter-wave
coplanar waveguide (CPW) element (Regal et al., 2008), a microfabricated
superconducting inductor-capacitor meander (Teufel et al., 2009), or even a
parallel plate capacitor shunted by a spiral inductor (Teufel et al., 2011).
Any resonator can be described by equivalent lumped RLC elements near a
resonance; see e.g. Ref. (Göppl et al., 2008) for the case of CPW resonators,
providing analytic expressions. For complex geometries however, finite element
analysis is required (like e.g. in Ref. (Zhou et al., 2019)). In most
experiments, the coupling to the outside is capacitive (Regal et al., 2008;
Teufel et al., 2009, 2011; Göppl et al., 2008; Zhou et al., 2019): the
resonator is almost isolated from the outside world, while the electromagnetic
field from input/output waveguides is allowed to “leak in-and-out” through
weakly coupled ports. This defines the microwave cavity element, in which the
motion $x(t)$ of the mechanical object modulates the effective $C$.

In Fig. 1 we thus show the three standard microwave setups. In (a) a two-port
scheme, in which a lumped RLC parallel circuit couples to distinct input and
output ports through different capacitors $C_{c1}$ and $C_{c2}$, yielding
effective coupling rates $\kappa_{1}$ and $\kappa_{2}$ respectively; they
quantify how energy decays over time from the resonator to each port
(Hertzberg et al., 2010; Lei et al., 2016). The total coupling rate to the
outside is thus defined as $\kappa_{ex}=\kappa_{1}+\kappa_{2}$. The internal
damping rate of the circuit is modeled through $R_{in}$, leading to a
$\kappa_{in}$ decay rate (measuring the decay toward internal degrees of
freedom). The total decay rate of the microwave mode is then
$\kappa_{t}=\kappa_{ex}+\kappa_{in}$.

Figure 1:  The three equivalent electric circuits for microwave optomechanics
with different input and output designs. In each, a mechanical element $x(t)$
capacitively couples to a parallel RLC circuit. (a) two-port, (b) bi-
directional and (c) single-port (reflection design, a specific component is
then required to separate incoming ant outcoming waves). The defined electric
elements are discussed in the text, and the outside world, as seen from the
circuit, is depicted only in (a).

The ports are realized by coaxial cables of characteristic impedance
$\text{Z}_{0}$ connected to adapted elements, i.e. the outside impedance seen
from the port is also $Z_{0}$ [see Fig. 1 (a) input and output]: the voltage
source $V_{\text{rf}}(t)=V_{p}\cos(\omega_{p}t)$ on the left has an output
load of $Z_{0}$, while the detection of the $V_{\text{out}}$ voltage on the
right is realized by an amplifier of input load $Z_{0}$ as well. $\omega_{p}$
is the (angular) microwave drive frequency, while $V_{p}$ is the applied
amplitude. In Fig. 1 (b) we show the electric schematic of a bi-directional
coupling: the RLC resonator couples evanescently to a nearby transmission line
with an effective capacitance $Cc$ (Palomaki et al., 2013; Teufel et al.,
2011; Zhou et al., 2019). This is strictly equivalent to scheme (a) when
imposing $C_{c1}=C_{c2}=Cc/2$ (and thus
$\kappa_{2}=\kappa_{1}=\kappa_{ex}/2$). At last, the third scheme is shown in
Fig. 1 (c). Only one port is connected to the device, requiring thus the use
of a specific nonreciprocal component (e.g. like a circulator) to separate the
drive signal from the response (reflection mode) (Massel et al., 2011). This
is again equivalent to scheme (a) with $C_{c1}=C_{c}$ and no $C_{c2}$ (i.e.
$\kappa_{2}=0$, $\kappa_{ex}=\kappa_{1}$). The problem at hand is solved below
in terms of generalized fluxes $\phi(t)$ = $\int^{t}V(t^{\prime})dt^{\prime}$
(Clerk et al., 2010); the dynamics equation will be written for the $\phi$
corresponding to the RLC node, see Fig. 1 (c). Incoming and outcoming
traveling waves (equivalent to the laser signals in conventional
optomechanics) are thus defined as $\phi_{in}$ (the tone pump, orange) and
$\phi_{out}$ (the response, green) in Fig. 1.

###  II.2 Dynamics equation and solution

We shall consider in the following a single port configuration [Fig. 1 (c)],
the extension to the other models being straightforward from what has been
said above. Whenever necessary, this correspondence will be explicitly
discussed.

The circuits shown in Fig. 1 are a combination of transmission lines (the
coaxial cables) and lumped elements ($RLC$, $Z_{0}$ impedances, and source).
The first step of the modeling is thus to get rid of the coaxial elements, in
order to model an ideal lumped circuit. To start with, we consider the source
$V_{\text{rf}}$ which generates the incoming wave $\phi_{in}$. In schemes (a)
and (c) of Fig. 1, the drive port is terminated by an (almost) open circuit
since the coupling capacitance is very small ($C_{c}\omega\,Z_{0}\ll 1$). The
incoming wave is thus almost fully reflected, and the standing wave voltage on
the input capacitor is $V_{d}\approx 2V_{\text{rf}}$ (Pozar, 2009). On the
other hand for scheme (b), the transmission line is almost unperturbed by the
coupling element $C_{c}$, and the incoming wave travels toward the output port
(almost) preserving its magnitude; on the coupling capacitor we have
$V_{d}\approx V_{\text{rf}}$.

Applying Norton’s theorem, we transform the series voltage source input
circuit into a parallel RC, which drives a total current $I_{d}$ across it.
This is shown in Fig. 2 (a), with finally the total loaded RLC resonator in
Fig. 2 (b). The effective components of the Norton drive circuit are defined
from the real and imaginary parts of the complex admittance
$Y_{c}(\omega)=[Z_{0}+1/(iC_{c}\omega)]^{-1}$, in the limit
$C_{c}\omega\,Z_{0}\ll 1$ (weak coupling):

$\displaystyle\Re[Y_{c}(\omega)]$ $\displaystyle=$
$\displaystyle\frac{1}{R_{ex}}\approx(\omega_{c}C_{c})^{2}\,Z_{0},$ (1)
$\displaystyle\Im[Y_{c}(\omega)]$ $\displaystyle\approx$ $\displaystyle
i\omega_{c}C_{c},$ (2)

with the approximation $\omega\approx\omega_{c}$ ($i$ is the imaginary unit).
The current $I_{d}$ flowing into the resonator then writes:

$I_{d}\approx i\omega_{c}C_{c}\,V_{d}.$ (3)

The detected voltage is calculated from the current flowing through the
amplifier’s impedance $Z_{0}$. For circuits Fig. 1 (a) and (c), this simply
leads to:

$V_{\text{out}}\approx-\omega_{c}^{2}C_{c}Z_{0}\,\phi,$ (4)

assuming again $\omega\approx\omega_{c}$. For circuit (b), the evanescent
coupling leads to a loading composed of two impedances $Z_{0}$ in parallel
(half of the signal is fed back to the voltage source):

$V_{\text{out}}\approx-\omega_{c}^{2}C_{c}\frac{Z_{0}}{2}\,\phi.$ (5)

From our definitions of $\kappa_{ex}$, the subtlety of these different
writings shall obviously be accounted for in our final expressions (see
discussion of Section II.4).

Figure 2:  (a) Circuit schematics converting the series voltage source $V_{d}$
(impedance $Z_{0}$) that loads the capacitor $C_{c}$ into a parallel current
source $I_{d}$; the equivalent resistance $R_{ex}$ is defined from $Z_{0}$ and
$C_{c}$ (see text). (b) Equivalent parallel $RLC$ circuit with current source
for the one-port optomechanical circuit [replacing Fig. 1 (c)]. The mechanical
contribution to the capacitance is explicitly defined as $C_{g}(x)$.

The classical dynamics equation that describes this problem writes:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\frac{d}{dt}\left(\left(C_{c}+C_{k}+C_{g}[x]\right)\frac{d\phi}{dt}\right)$
(6) $\displaystyle+$
$\displaystyle\left(\frac{1}{R_{ex}}+\frac{1}{R_{in}}\right)\frac{d\phi}{dt}+\frac{1}{L}\phi$
$\displaystyle=$ $\displaystyle I_{d}+I_{\text{noise}}.$

In Eq. (6), the fraction of the capacitance modulated by the mechanics is
defined as $C_{g}(x)$ [see Fig. 2 (b)]. The Johnson-Nyquist electric noise (at
temperature $T_{c}$) seen by the cavity is modeled as a source
$I_{\text{noise}}$, in parallel with the imposed drive circuit generating
$I_{d}$ (Clerk et al., 2010).

In the following, we will consider small motion. We therefore write
$C_{g}(x)\approx C_{g}(0)+x(t)\times dC_{g}(0)/dx$, defining the total
(static) capacitance $C_{t}=C_{c}+C_{k}+C_{g}(0)$. $C_{g}(0)$ corresponds to
the contribution of the mobile element when at rest, while $C_{c}$ comes from
the slight “leakage” of the cavity mode into the coaxial lines. The rates are
then defined from the electronic components:

$\displaystyle\kappa_{ex}$ $\displaystyle=$
$\displaystyle\frac{1}{R_{ex}C_{t}},$ (7) $\displaystyle\kappa_{in}$
$\displaystyle=$ $\displaystyle\frac{1}{R_{in}C_{t}}.$ (8)

In the two-port case, one simply defines $R_{1}$ and $R_{2}$ leading to
$1/R_{ex}=1/R_{1}+1/R_{2}$, and similarly $1/R_{t}=1/R_{ex}+1/R_{in}$; we
write the corresponding quality factors $Q_{i}=\omega_{c}/\kappa_{i}$ (with
$i=in,ex,t$). Besides, the microwave resonance of the loaded $RLC$ circuit is
given by $\omega_{c}=1/\sqrt{LC_{t}}$.

Introducing the coupling strength $G=-d\omega_{c}/dx=-d\omega_{c}/dC_{g}\times
dC_{g}/dx$, Eq. (6) can be re-written in the more compact form:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\left(1+\frac{2G}{\omega_{c}}{x}\right)\frac{d^{2}\phi}{dt^{2}}$
(9) $\displaystyle+$
$\displaystyle\left({\kappa_{t}}+\frac{2G}{\omega_{c}}\frac{dx}{dt}\right)\frac{d\phi}{dt}+{\omega_{c}}^{2}\phi$
$\displaystyle=$
$\displaystyle\frac{I_{d}}{C_{t}}+\frac{I_{\text{noise}}}{C_{t}},$

The drive current writes
$I_{d}(t)=\frac{1}{2}I_{p}e^{-i\omega_{p}t}+\text{c.c.}$ with $\omega_{p}$ the
frequency at which the microwave pumping is applied and $I_{p}$ its (complex)
amplitude. We introduce the frequency detuning $\Delta=\omega_{p}-\omega_{c}$.
From Eq. (3), $I_{p}$ is derived from the input voltage drive amplitude
$V_{p}$ not (Author’s note). The mechanical displacement is written as
$x(t)=\frac{1}{2}x_{0}(t)e^{-i\Omega_{m}t}+\text{c.c.}$ with $x_{0}(t)$ the
(complex) motion amplitude translated in frequency around $\Omega_{m}$, the
mechanical resonance frequency of the movable element. This amplitude is a
stochastic variable: the Brownian motion of the moving element thermalized at
temperature $T_{m}$(see Fig. 3).

The terms where motion $x(t)$ multiplies flux $\phi(t)$ in Eq. (9) then
generate harmonics at $\omega_{n}=\omega_{p}+n\,\Omega_{m}$, with
$n\in\mathbb{Z}$: this phenomenon is known as nonlinear mixing. We can thus
find an exact solution using the ansatz:

$\phi(t)=\\!\\!\\!\sum_{n=-\infty}^{+\infty}\\!\\!\\!\phi_{n}(t)=\\!\\!\\!\sum_{n=-\infty}^{+\infty}\\!\\!\frac{1}{2}\mu_{n}(t)e^{-i\left(\omega_{p}+n\Omega_{m}\right)t}+\text{c.c.}$
(10)

which, when injected in Eq. (9) generates a system of coupled equations for
the $\mu_{n}$ (complex) amplitudes. In order to match the decomposition, the
white noise component is thus naturally written as
$I_{\text{noise}}=\sum_{n}\frac{1}{2}\delta
I_{n}(t)e^{-i\left(\omega_{p}+n\Omega_{m}\right)t}+\text{c.c.}$ with $\delta
I_{n}(t)$ the (complex) amplitude translated at frenquency
$\omega_{p}+n\,\Omega_{m}$.

Figure 3:  Spectrum landscape: here, schematic example of “blue sideband”
pumping (pump tone in red, $\Delta$ around $+\Omega_{m}$), in the resolved
sideband limit ($\Omega_{m}\gg\kappa_{t}$, with $\kappa_{t}$ the width of the
cavity response, here in violet). On the left, the Brownian mechanical peak
around $\Omega_{m}$. In dashed gray, the mixing mechanism and the two main
response peaks (around $\omega_{l}$ and $\omega_{h}$, see text).

In practice, we are interested only in schemes where $\Delta\approx
0,+\Omega_{m}$ and $-\Omega_{m}$. The resonant feature brought in by the $RLC$
element implies that only the spectral terms the closest to $\omega_{c}$ in
Eq. (10) will be relevant when $\kappa_{t}\ll\Omega_{m}$; this is known as the
resolved sideband limit. From the full comb, we thus keep only $n=-1,0,+1$
which we rename in ’l’ (lower), ’p’ (pump) and ’h’ (higher) respectively for
clarity. Considering $\Omega_{m}\ll\omega_{c}$, we also justify the assumption
$\omega\approx\omega_{c}$ used in Eqs. (1 \- 5).

Eq. (9) is solved by mimicking a rotating wave approximation, a method
initially developed for resonances in atom optics and nuclear magnetic
resonance (Abragam, 2011). For our classical treatment, it simply means that
we are concerned by the dynamics of each component of Eq. (10) only around its
main frequency $\omega_{n}$ (what is called a “rotating frame”, as opposed to
the “laboratory frame” for the full signal), assuming time-variations of $x$
and $\mu_{n}$ to be slow (valid for high-$Q$ microwave and mechanical
resonances). Reminding
$\omega_{l}\approx\omega_{p}\approx\omega_{h}\approx\omega_{c}$, the flux
amplitudes inside the cavity acting on the mechanical oscillator are:

$\displaystyle\begin{split}&\mu_{p}(t)=+\frac{i}{2}\left(\frac{I_{p}}{\omega_{c}C_{t}}+\frac{\delta
I_{p}(t)}{\omega_{c}C_{t}}\right)\chi_{p},\\\
&\mu_{l}(t)=+\frac{i}{2}\left(G\,x_{0}^{*}(t){\mu_{p}}(t)+\frac{\delta
I_{l}(t)}{\omega_{c}C_{t}}\right)\chi_{l},\\\
&\mu_{h}(t)=+\frac{i}{2}\left(G\,x_{0}(t){\mu_{p}}(t)+\frac{\delta
I_{h}(t)}{\omega_{c}C_{t}}\right)\chi_{h},\end{split}$ (11)

having defined:

$\displaystyle\chi_{p}=\frac{1}{-i\Delta+\frac{\kappa_{t}}{2}},$ (12)
$\displaystyle\chi_{l}=\frac{1}{-i\left(\Delta-\Omega_{m}\right)+\frac{\kappa_{t}}{2}},$
(13)
$\displaystyle\chi_{h}=\frac{1}{-i\left(\Delta+\Omega_{m}\right)+\frac{\kappa_{t}}{2}},$
(14)

the cavity susceptibilities associated to each spectral component.

When $\Delta\approx 0$, the pump tone ’p’ is resonant with the cavity; the
scheme is symmetric and the two ’l’ and ’h’ amplitudes are equivalent. We
shall call this the “green” pumping scheme in the following. When
$\Delta\approx-\Omega_{m}$, the ’h’ component is resonant with the cavity and
the ’l’ one is greatly suppressed. This is known as the “red sideband” pumping
scheme. When $\Delta\approx+\Omega_{m}$, the situation is reversed and ’l’ is
resonant with the cavity and ’h’ suppressed. This is the “blue” scheme
schematized in Fig. 3. The meaning of the naming colors will be clarified in
Section II.4.

Figure 4:  Schematic frequency dependence of the ${\mu}_{l}$ (left) and
${\mu}_{h}$ (right) components, under “blue sideband” pumping. Each curve
corresponds to a different ratio of ${\kappa_{t}}/2\Omega_{m}$ (see legend:
not in the sideband resolved limit), in the small $\mu_{p}$ drive limit (see
text).

The two satellite signals $\mu_{l}$ and $\mu_{h}$ at $\omega_{p}\pm\Omega_{m}$
generated by the pump tone $\mu_{p}$ are schematized in Fig. 4 for an
arbitrary Brownian noise $x$, in the small drive limit. They correspond to
energy up-converted from the pump ’p’ (for ’h’), or down-converted (for ’l’).
When the drive $\mu_{p}$ (i.e. $I_{p}$ or $V_{p}$) becomes large enough, back-
action of the cavity onto the mechanical element has to be taken into account.
This is derived in the next Section.

###  II.3 Classical back-action of cavity onto mechanics

The voltage bias on the mechanical element is responsible for a force
$F_{\text{ba}}$, defined as the gradient of the electromagnetic energy:

$F_{\text{ba}}(t)=+\frac{d}{dx}\left[\frac{1}{2}C_{g}(x)\left(\frac{d\phi(t)}{dt}\right)^{2}\right],$
(15)

using our notations (see Fig. 2). This is the so-called back-action of the
cavity onto the mechanical element. The sign definition corresponds to the
fixed electrode acting upon the mobile part. We are interested only in the
component of this force that can drive the mechanics; we therefore define
$F_{\text{ba}}(t)=\frac{1}{2}F_{0}(t)e^{-i\Omega_{m}t}+\text{c.c.}$ with
$F_{0}(t)$ the (complex) force amplitude acting in the frame rotating at the
mechanical frequency $\Omega_{m}$. Re-writing $\phi$ in terms of the defined
flux amplitudes, and keeping only the lowest order for the spatial derivative
(small motion limit), we get:

$F_{0}(t)\approx+C_{t}\omega_{c}G\left[\mu_{p}(t)\mu_{l}^{*}(t)+\mu_{p}^{*}(t)\mu_{h}(t)\right].$
(16)

From Eqs. (11), we immediately see that $F_{0}(t)$ will depend on both the
motion amplitude $x_{0}(t)$ and the current noise components of the cavity
$\delta I_{n}(t)$. Eq. (16) is thus recasted in:

$\displaystyle F_{0}(t)$ $\displaystyle\approx$
$\displaystyle+i\frac{G^{2}}{\omega_{c}}\left(\frac{C_{t}\omega_{c}^{2}\left|\mu_{p}\right|^{2}}{2}\right)\,\Big{[}\chi_{h}-\chi_{l}^{*}\Big{]}\,x_{0}(t)$
(17) $\displaystyle+$ $\displaystyle\\!\\!\\!i\frac{G}{2}\Big{[}\delta
I_{h}(t)\,\mu_{p}^{*}\,\chi_{h}-\delta
I_{l}^{*}(t)\,\mu_{p}\,\chi_{l}^{*}\Big{]},$

with the first term the so-called dynamic component (proportional to $x_{0}$),
and the last term the stochastic component that is fed back from the cavity
onto the mechanical degree of freedom. In Eq. (17), the term $\delta I_{p}$
has been dropped at lowest order and
$\mu_{p}\approx\frac{i}{2}\frac{I_{p}}{\omega_{c}C_{t}}\chi_{p}$ is now time-
independent; only the noise current at the two sidebands is relevant.

The governing equation for the mechanical motion is expressed in the rotating
frame as (neglecting again the slow dynamics):

$-2i\,\dot{x}_{0}(t)-i\Gamma_{m}x_{0}(t)=\frac{L_{0}(t)+F_{0}(t)}{m\Omega_{m}},$
(18)

with $\Gamma_{m}$ the mechanical damping rate, and
$Q_{m}=\Omega_{m}/\Gamma_{m}$ the mechanical quality factor. $m$ is the mass
of the moving element. We write $L(t)$ the Langevin force (at temperature
$T_{m}$), with $L_{0}(t)$ the component acting in the frame rotating at
$\Omega_{m}$. Injecting Eq. (17) into Eq. (18) and taking the Fourier
transform, the solution can be written in the simple usual form:

$x_{0}(\omega)=\chi_{m}(\omega)\left[L_{0}(\omega)+\delta
F_{0}(\omega)\right],$ (19)

where we have defined:

$\displaystyle\chi_{m}(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{2m\Omega_{m}\left(-\omega-i\frac{\Gamma_{m}}{2}\right)+\Sigma},$
(20) $\displaystyle\Sigma$ $\displaystyle=$
$\displaystyle-i\frac{G^{2}}{\omega_{c}}\left(\frac{C_{t}\omega_{c}^{2}\left|\mu_{p}\right|^{2}}{2}\right)\,\Big{[}\chi_{h}-\chi_{l}^{*}\Big{]},$
(21) $\displaystyle\delta F_{0}(\omega)$ $\displaystyle=$
$\displaystyle+i\frac{G}{2}\Big{[}\delta
I_{h}(\omega)\,\mu_{p}^{*}\,\chi_{h}-\delta
I_{l}^{*}(\omega)\,\mu_{p}\,\chi_{l}^{*}\Big{]}.$ (22)

Eq. (20) is the mechanical susceptibility of the moving element. The
mechanical linear response is thus modified by the interaction with the
microwave field through the term $\Sigma$ [Eq. (21)]; it is usually
(abusively) referred to as the optical “self-energy”, see e.g. Ref.
(Aspelmeyer et al., 2014a). Matching the expressions of this Review, note the
difference in the definition of susceptibilities between mechanical and
optical fields [ Eqs. (12-14)]: an $i$ factor has been incorporated in
between. The last Eq. (22) corresponds to the stochastic component of the
back-action: noise originating from the Johnson-Nyquist current that adds up
with the Langevin force.

In the following Section we shall discuss the spectra associated to Eq. (19)
with their specific properties. The link between injected power, Brownian
motion and measured spectrum of the voltage $V_{\text{out}}$ is finally
presented.

###  II.4  Spectral properties and input-output relationships

Taking real and imaginary parts of $\Sigma$, we see from Eq. (20) that the
optomechanical interaction is responsible for a frequency shift
$\delta\Omega_{m}$ and an additional damping term $\Gamma_{\text{opt}}$:

$\displaystyle\delta\Omega_{m}$ $\displaystyle=$ $\displaystyle
G^{2}\frac{1}{\omega_{c}\,(2m\Omega_{m})}\left(\frac{C_{t}\omega_{c}^{2}\left|\mu_{p}\right|^{2}}{2}\right)\times$
(23)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left[\frac{\Delta+\Omega_{m}}{(\Delta+\Omega_{m})^{2}+(\frac{\kappa_{t}}{2})^{2}}+\frac{\Delta-\Omega_{m}}{(\Delta-\Omega_{m})^{2}+(\frac{\kappa_{t}}{2})^{2}}\right]\\!,$
$\displaystyle\Gamma_{\text{opt}}$ $\displaystyle=$ $\displaystyle
G^{2}\frac{1}{\omega_{c}\,(2m\Omega_{m})}\left(\frac{C_{t}\omega_{c}^{2}\left|\mu_{p}\right|^{2}}{2}\right)\times$
(24)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left[\frac{\kappa_{t}}{(\Delta+\Omega_{m})^{2}+(\frac{\kappa_{t}}{2})^{2}}-\frac{\kappa_{t}}{(\Delta-\Omega_{m})^{2}+(\frac{\kappa_{t}}{2})^{2}}\right]\\!.$

The former expression above is referred to as the optical spring and the
latter the optical damping effects (Aspelmeyer et al., 2014a). Physically,
these effects originate in the radiation pressure exerted on the movable
capacitor by the electromagnetic field confined inside it.

The spectrum of the stochastic back-action term [Eq. (22)] writes:

$\displaystyle S_{\delta F_{0}}(\omega)$ $\displaystyle=$
$\displaystyle\frac{G^{2}}{\omega_{c}^{2}}\left(\frac{C_{t}\omega_{c}^{2}\left|\mu_{p}\right|^{2}}{2}\right)\frac{R_{t}S_{\delta
I_{n}}}{2}\times$ (25)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left[\frac{\kappa_{t}}{(\Delta+\Omega_{m})^{2}+(\frac{\kappa_{t}}{2})^{2}}+\frac{\kappa_{t}}{(\Delta-\Omega_{m})^{2}+(\frac{\kappa_{t}}{2})^{2}}\right]\\!,$

as a function of the white current noise spectrum $S_{\delta I_{n}}=S_{\delta
I_{l}}(\omega)=S_{\delta I_{h}}(\omega)$, with ’l’ and ’h’ components
uncorrelated; therefore $S_{\delta F_{0}}$ is also white. As a result, the
displacement spectrum deduced from Eq. (19) is:

$S_{x_{0}}(\omega)=\left|\chi_{m}(\omega)\right|^{2}\left[S_{L_{0}}+S_{\delta
F_{0}}\right],$ (26)

with $S_{L_{0}}$ the white force spectrum associated to Brownian motion. This
result is a Lorentzian peak (depicted in Fig. 3 on the left) with full-width
$\Gamma_{m}+\Gamma_{\text{opt}}$ and position $\delta\Omega_{m}$ (in the frame
rotating at $\Omega_{m}$); the total area is proportional to the total white
noise force felt by the mechanics, namely $S_{{\cal
L}_{0}}=S_{L_{0}}+S_{\delta F_{0}}$. Equivalently, Eq. (26) writes with the
original spectra in the laboratory frame:

$S_{x}(\omega)=\left[\left|\chi_{m}(\omega-\Omega_{m})\right|^{2}+\left|\chi_{m}(\omega+\Omega_{m})\right|^{2}\right]S_{\cal
L},$ (27)

defined for $\omega$ ranging from $-\infty$ to $+\infty$; the classical
spectrum is even with two identical peaks $S^{-}_{x}(\omega)$,
$S^{+}_{x}(\omega)$ located at $\omega\approx\pm\Omega_{m}$. From the rotating
wave transform, we have $S_{L_{0}}/4=S_{L}=2k_{B}T_{m}\,m\Gamma_{m}$ and
$S_{\delta I_{n}}/4=S_{I_{\text{noise}}}=2k_{B}T_{c}/R_{t}$ (with $k_{B}$
Boltzmann’s constant) (Clerk et al., 2010; Aspelmeyer et al., 2014a); by
construction, each fluctuating current $\delta I_{n}(t)$ is defined over a
bandwidth of order $\Omega_{m}$ (while $I_{\text{noise}}$ covers ${\rm
I\\!R}$).

Figure 5:  Illustration (arbitrary units) of the dynamical back-action
properties, in the “blue sideband” pumping case, for different
$\kappa_{t}/2\Omega_{m}$ ratios (resolved sideband, see legend). Left: optical
spring $\delta\Omega_{m}$ versus normalized detuning $\Delta/\Omega_{m}$.
Right: optical damping $\Gamma_{\text{opt}}$. The amplitudes of both effects
scale as $g^{2}\propto\left|\mu_{p}\right|^{2}$: When the total damping
$\Gamma_{m}+\Gamma_{\text{opt}}$ reaches $0$, the system starts to self-
oscillate (see text).

The stochastic force acting on the mechanics can thus be recasted in the
simple form:

$\frac{S_{{\cal L}_{0}}}{4}=S_{{\cal
L}}=2k_{B}\,m\left(\Gamma_{m}+\Gamma_{\text{opt}}\right)T_{\text{eff}},$ (28)

with:

$\displaystyle T_{\text{eff}}$ $\displaystyle=$
$\displaystyle\frac{T_{m}\,\Gamma_{m}+T_{c}\,\Gamma_{\text{opt}}^{\prime}}{\Gamma_{m}+\Gamma_{\text{opt}}},$
(29) $\displaystyle\Gamma_{\text{opt}}^{\prime}$ $\displaystyle=$
$\displaystyle
G^{2}\frac{1}{\omega_{c}\,(2m\Omega_{m})}\left(\frac{C_{t}\omega_{c}^{2}\left|\mu_{p}\right|^{2}}{2}\right)\frac{\Omega_{m}}{\omega_{c}}\times$
(30)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left[\frac{\kappa_{t}}{(\Delta+\Omega_{m})^{2}+(\frac{\kappa_{t}}{2})^{2}}+\frac{\kappa_{t}}{(\Delta-\Omega_{m})^{2}+(\frac{\kappa_{t}}{2})^{2}}\right]\\!.$

The term $T_{\text{eff}}$ in Eq. (28) is thus interpreted as an effective
temperature for the mechanical mode, created by the combination of the force
fluctuations around $\Omega_{m}$ (in the radiofrequency domain) and the
current fluctuations around $\omega_{c}$ (in the microwave domain), both
derived within the same physical framework: the fluctuation-dissipation
theorem. Note the similarity between Eq. (24) and Eq. (30). Besides, we see
that the magnitudes of optical spring, optical damping, and back-action noise
are all governed by a single parameter:

$g=G\sqrt{\frac{1}{\omega_{c}\,(2m\Omega_{m})}\left(\frac{C_{t}\omega_{c}^{2}\left|\mu_{p}\right|^{2}}{2}\right)}.$
(31)

Replacing $g$ in the above expressions, they are formally equivalent to the
optomechanics results (Aspelmeyer et al., 2014a); this fact shall be discussed
in the following Section.

Depending on the scheme used, Eqs. (23,24,30) behave very differently. When
$\Delta\approx+\Omega_{m}$, the second term in the brackets of these
expressions dominate and $\delta\Omega_{m}\approx
g^{2}\,(\Delta-\Omega_{m})/(\kappa_{t}/2)^{2}$,
$\Gamma_{\text{opt}}\approx-g^{2}\,4/\kappa_{t}$ and
$\Gamma_{\text{opt}}^{\prime}\approx
g^{2}\,\frac{\Omega_{m}}{\omega_{c}}4/\kappa_{t}$ (resolved sideband limit).
The optical damping is negative (the Lorentz mechanical peak has a smaller
effective width $\Gamma_{\text{eff}}=\Gamma_{m}-|\Gamma_{\text{opt}}|$),
therefore the mechanical response is enhanced (the mechanical $Q$ factor
grows), and the effective temperature $T_{\text{eff}}$ is increased: energy is
pumped into the mechanical mode, a mechanism called Stokes scattering in
optics (Aspelmeyer et al., 2010). We adopt the language used in this community
and call this scheme the “blue sideband” pumping. When
$\Gamma_{m}+\Gamma_{\text{opt}}=0$ the system reaches an instability and
starts to self-oscillate (Aspelmeyer et al., 2014a; Marquardt et al., 2006;
Carmon et al., 2005). The properties of this scheme are illustrated in Fig. 5.
When $\Delta\approx-\Omega_{m}$, the situation is reversed and
$\delta\Omega_{m}\approx g^{2}\,(\Delta+\Omega_{m})/(\kappa_{t}/2)^{2}$,
$\Gamma_{\text{opt}}\approx+g^{2}\,4/\kappa_{t}$ and
$\Gamma_{\text{opt}}^{\prime}\approx
g^{2}\,\frac{\Omega_{m}}{\omega_{c}}4/\kappa_{t}$. The optical damping is now
positive ($\Gamma_{\text{eff}}=\Gamma_{m}+|\Gamma_{\text{opt}}|$ is larger),
the mechanical response is damped (the mechanical $Q$ factor decreases) and
the temperature $T_{\text{eff}}$ is reduced: energy is pumped out of the
mechanical mode, a mechanism called Anti-Stokes scattering in optics
(Aspelmeyer et al., 2010). This scheme, known as sideband cooling is also
referred to as “red sideband” pumping. At last, when $\Delta\approx 0$ the
situation is symmetric: no energy is pumped in or out, and
$\delta\Omega_{m}\approx g^{2}\,(2\Delta)/(\Omega_{m})^{2}$,
$\Gamma_{\text{opt}}\approx 0$ and $\Gamma_{\text{opt}}^{\prime}\approx
g^{2}\,\frac{\Omega_{m}}{\omega_{c}}2\kappa_{t}/(\Omega_{m})^{2}$
($\Gamma_{\text{eff}}=\Gamma_{m}$ unchanged). In order to distinguish it from
the two other schemes, we name it “green sideband” pumping. Note that this
scheme has the smallest back-action contribution; it is thus also referred to
as the optimal scheme in optics (Aspelmeyer et al., 2014a).

The final step of the modeling requires to link the input (the $V_{\text{rf}}$
source) to the measured spectrum of the output voltage
$S_{V_{\text{out}}}(\omega)$. From Eq. (4), and including the voltage noise on
the detector $V_{\text{noise}}(t)$, we have:

$\displaystyle S_{V_{\text{out}}}(\omega)$ $\displaystyle=$ $\displaystyle
S_{V_{\text{noise}}}+\omega_{c}^{4}(C_{c}Z_{0})^{2}\times$ (32)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{n=-\infty}^{+\infty}\left[\frac{S_{\mu_{n}}(\omega-\omega_{n})}{4}+\frac{S_{\mu_{n}}(\omega+\omega_{n})}{4}\right]$
$\displaystyle-\omega_{c}^{2}(C_{c}Z_{0})\times$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{n=-\infty}^{+\infty}\left[\frac{S_{\mu_{n},\delta
V_{n}}(\omega-\omega_{n})}{4}+\frac{S_{\mu_{n},\delta
V_{n}}(\omega+\omega_{n})}{4}\right]\\!,$

reminding $\omega_{n}=\omega_{p}+n\Omega_{m}$, with $S_{\mu_{n}}(\omega)$ the
spectrum of the $n^{\text{th}}$ component of the flux $\phi(t)$ decomposition,
Eq. (10), and $S_{\mu_{n},\delta V_{n}}(\omega)$ the cross-correlations
between flux and voltage noise components (with
$V_{\text{noise}}(t)=\sum_{n}\frac{1}{2}\delta
V_{n}(t)e^{-i\left(\omega_{p}+n\Omega_{m}\right)t}+\text{c.c.}$, same
decomposition as for the current $I_{\text{noise}}$). By construction, from
the admittance $Y_{c}$ introduced in Section II.2, we have:

$V_{\text{noise}}\approx+i\omega_{c}C_{c}Z_{0}R_{ex}\,I_{\text{noise}},$ (33)

which defines the output voltage noise from the cavity current noise. We have:

$\displaystyle S_{\mu_{p},\delta V_{p}}(\omega)$ $\displaystyle=$
$\displaystyle+\frac{2(\omega_{c}C_{c}Z_{0})}{\omega_{c}C_{t}}\left[\chi_{p}+\chi_{p}^{*}\right]\frac{R_{ex}S_{\delta
I_{n}}}{4},$ (34) $\displaystyle S_{\mu_{l},\delta V_{l}}(\omega)$
$\displaystyle=$
$\displaystyle+\frac{2(\omega_{c}C_{c}Z_{0})}{\omega_{c}C_{t}}\left[\chi_{l}+\chi_{l}^{*}\right]\frac{R_{ex}S_{\delta
I_{n}}}{4}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left|\mu_{p}\right|^{2}G^{2}(\omega_{c}C_{c}Z_{0})\left[i(\chi_{l}^{2}\chi_{m}^{*}-{\chi_{l}^{*}}^{2}\chi_{m})\right]\frac{R_{ex}S_{\delta
I_{n}}}{4},$ $\displaystyle S_{\mu_{h},\delta V_{h}}(\omega)$ $\displaystyle=$
$\displaystyle+\frac{2(\omega_{c}C_{c}Z_{0})}{\omega_{c}C_{t}}\left[\chi_{h}+\chi_{h}^{*}\right]\frac{R_{ex}S_{\delta
I_{n}}}{4}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left|\mu_{p}\right|^{2}G^{2}(\omega_{c}C_{c}Z_{0})\left[i(\chi_{h}^{2}\chi_{m}-{\chi_{h}^{*}}^{2}\chi_{m}^{*})\right]\frac{R_{ex}S_{\delta
I_{n}}}{4},$

using the properties of the Johnson-Nyquist current. On the right-hand-side,
the first term involves only the microwave cavity; for Eqs. (II.4,II.4) the
last term involves the mechanics. These terms are nonzero since the same
current noise generating the detection background also drives the cavity, and
is fed back to the mechanics from Eq. (22).

The total spectrum $S_{V_{\text{out}}}(\omega)$ is composed of identical combs
around $\omega\approx\pm\omega_{p}$. What is measured by any classical
apparatus (say, a spectrum analyzer) is the power spectral density [in
Watt/(radian/second)]:

$\displaystyle S_{\text{PSD}}(\omega)$ $\displaystyle=$
$\displaystyle\frac{2S_{V_{\text{out}}}(\omega>0)}{Z_{0}}=2R_{ex}S_{I_{\text{noise}}}$
(37)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\frac{\omega_{c}^{4}(C_{c}Z_{0})^{2}}{2Z_{0}}\times\sum_{n=^{\prime}p^{\prime},^{\prime}l^{\prime},^{\prime}h^{\prime}}S_{\mu_{n}}(\omega-\omega_{n})$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\frac{\omega_{c}^{2}(C_{c}Z_{0})}{2Z_{0}}\times\sum_{n=^{\prime}p^{\prime},^{\prime}l^{\prime},^{\prime}h^{\prime}}S_{\mu_{n},\delta
V_{n}}(\omega-\omega_{n}),$

with all power folded in the $\omega>0$ range. Similarly to the cavity itself,
we define a temperature for the detection port as
$R_{ex}S_{I_{\text{noise}}}=R_{ex}S_{\delta I_{n}}/4=k_{B}T_{ex}$, ensuring
that the background noise in Eq. (37) reduces to $2k_{B}T_{ex}$, as it should.
In the sum of Eq. (32), only the ’p’, ’l’ and ’h’ terms have been kept: the
measured spectrum is composed of 3 peaks (see Figs. 3 and 4), defined from
Eqs. (11):

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{\mu_{p}}(\omega)$
$\displaystyle\\!\\!=$
$\displaystyle\\!\left|\mu_{p}\right|^{2}\,2\pi\delta_{0}(\omega)+\frac{R_{t}S_{\delta
I_{n}}}{4}\frac{\kappa_{t}}{C_{t}\omega_{c}^{2}}\left|\chi_{p}\right|^{2}\\!\\!,$
(38) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{\mu_{l}}(\omega)$
$\displaystyle\\!\\!=$
$\displaystyle\\!\left|\mu_{p}\right|^{2}G^{2}\frac{S_{x_{0}}(\omega)}{4}\left|\chi_{l}\right|^{2}+\frac{R_{t}S_{\delta
I_{n}}}{4}\frac{\kappa_{t}}{C_{t}\omega_{c}^{2}}\left|\chi_{l}\right|^{2}$
(39)
$\displaystyle\\!\\!\\!\\!\\!+\frac{\left|\mu_{p}\right|^{2}G^{2}\kappa_{t}}{2\omega_{c}}\left[i(\chi_{m}^{*}\chi_{l}-\chi_{m}\chi_{l}^{*})\right]\frac{R_{t}S_{\delta
I_{n}}}{4}\left|\chi_{l}\right|^{2}\\!,$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{\mu_{h}}(\omega)$
$\displaystyle\\!\\!=$
$\displaystyle\\!\\!\left|\mu_{p}\right|^{2}G^{2}\frac{S_{x_{0}}(\omega)}{4}\left|\chi_{h}\right|^{2}+\frac{R_{t}S_{\delta
I_{n}}}{4}\frac{\kappa_{t}}{C_{t}\omega_{c}^{2}}\left|\chi_{h}\right|^{2}$
(40)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\frac{\left|\mu_{p}\right|^{2}G^{2}\kappa_{t}}{2\omega_{c}}\left[i(\chi_{m}\chi_{h}-\chi_{m}^{*}\chi_{h}^{*})\right]\frac{R_{t}S_{\delta
I_{n}}}{4}\left|\chi_{h}\right|^{2}\\!,$

applying again the properties of the Johnson-Nyquist current ($\delta_{0}$ is
the Dirac function). The second terms in each expressions correspond to the
cavity alone, being driven by the current noise. Eq. (38) is due to the pump
tone signal; Eqs. (39,40) include the two sidebands, proportional to the
mechanical motion spectrum and
$\left|\mu_{p}\right|^{2}=\frac{\left|I_{p}\right|^{2}}{4(\omega_{c}C_{t})^{2}}\left|\chi_{p}\right|^{2}$.
The last terms in Eqs. (39,40) correspond to cross-correlations between the
cavity noise current and the motion.

We should now clarify the energy flow in this system. The power injected
$P_{in}$ by the traveling wave $\phi_{in}$ (Fig. 1) is by definition:

$P_{in}=\frac{1}{2}\frac{\left|V_{p}\right|^{2}}{Z_{0}},$ (41)

and the energy $E_{c}$ stored in the microwave resonator writes:

$E_{c}=\frac{C_{t}\omega_{c}^{2}\left|\mu_{p}\right|^{2}}{2}=P_{in}\,\kappa_{ex}\left|\chi_{p}\right|^{2},$
(42)

where we made use of Eqs. (1,3,7). The pump power $P_{pump}$ measured in the
output spectrum at $\omega_{p}$ is then:

$P_{pump}=\frac{\omega_{c}^{4}(C_{c}Z_{0})^{2}}{2Z_{0}}\left|\mu_{p}\right|^{2}=E_{c}\,\kappa_{ex}.$
(43)

Replacing Eq. (42) in Eq. (43) leads to
$P_{pump}=P_{in}\,\kappa_{ex}^{2}\left|\chi_{p}\right|^{2}$; the ratio
$P_{pump}/P_{in}$ is thus a straightforward calibration of the quantity
$\kappa_{ex}$. The power spectral density can thus be recasted in the compact
form:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!S_{\text{PSD}}(\omega)=2k_{B}T_{ex}+P_{pump}\,2\pi\delta_{0}(\omega-\omega_{p})$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\kappa_{ex}\,k_{B}(T_{c}-T_{ex})\,\kappa_{t}\left|\chi_{p}\right|^{2}\Pi(\omega-\omega_{p})$
$\displaystyle+\kappa_{ex}\,g^{2}\,\frac{S^{-}_{x}(\omega-\omega_{p})}{\bar{x}^{2}}\left|\chi_{l}\right|^{2}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\kappa_{ex}\,k_{B}(T_{c}-T_{ex})\,\kappa_{t}\left|\chi_{l}\right|^{2}\Pi(\omega-\omega_{l})$
$\displaystyle+\kappa_{ex}\,g^{2}\,\frac{S^{+}_{x}(\omega-\omega_{p})}{\bar{x}^{2}}\left|\chi_{h}\right|^{2}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\kappa_{ex}\,k_{B}(T_{c}-T_{ex})\,\kappa_{t}\left|\chi_{h}\right|^{2}\Pi(\omega-\omega_{h}),$
(44)

having defined $\bar{x}^{2}=\frac{1}{\omega_{c}\,(2m\Omega_{m})}$ (in
meter2/Joule). $\Pi(\omega)$ denotes the door function (here, of width
$\sim\Omega_{m}$), reminding that each cavity component is defined around a
precise angular frequency $\omega_{n}$. Note that to detect the cavity as a
peak or a dip, one requires $T_{c}\neq T_{ex}$; $T_{ex}$ also defines the
background noise level that ultimately limits a measurement. Eq. (31) then
reads $g=G\,\bar{x}\sqrt{E_{c}}$ (in radian/second).

The last terms of Eqs. (II.4,39) and (II.4,40), which correspond to the cross-
correlations involving the mechanics, affect the measurement by mimicking an
extra force noise $\delta F_{ex\,l},\delta F_{ex\,h}$ which depends on the
sideband (Aspelmeyer et al., 2014a; Weinstein et al., 2014):

$\displaystyle S^{-}_{x}(\omega)$ $\displaystyle=$ $\displaystyle
S_{x}(\omega)+|\chi_{m}(\omega-\Omega_{m})|^{2}S_{\delta F_{ex\,l}},$ (45)
$\displaystyle S^{+}_{x}(\omega)$ $\displaystyle=$ $\displaystyle
S_{x}(\omega)+|\chi_{m}(\omega+\Omega_{m})|^{2}S_{\delta F_{ex\,h}}.$ (46)

These shall not be confused with the true back-action force noise $S_{\delta
F}$: $\delta F_{ex\,l},\delta F_{ex\,h}$ do not actually affect the mechanical
degree of freedom. Their relevance is discussed in Section III.1, on the basis
of the 3 standard measuring schemes (with i.e. $\Delta=0,\pm\Omega_{m}$).
Notwithstanding this fact, the two sidebands are thus the image of the two
peaks of the mechanical spectrum, translated around $\omega_{p}$ (one being
thus at $\omega_{p}-\Omega_{m}=\omega_{l}$ and the other at
$\omega_{p}+\Omega_{m}=\omega_{h}$), with an amplitude proportional to $g^{2}$
and modulated by the susceptibilities
$\left|\chi_{l}\right|^{2},\left|\chi_{h}\right|^{2}$. Integrating the peaks,
we obtain an area proportional to the observed variance of the displacement
$\frac{1}{2\pi}\int S^{\pm}_{x}d\omega={\sigma^{\pm}_{x}}^{2}$, including thus
the cross-correlation contribution.

The above applies to the reflection setup, Fig. 2 (c). For the two-port one
Fig. 2 (a), one should replace $\kappa_{ex}\rightarrow\kappa_{1}$ in Eq. (42)
and $\kappa_{ex}\rightarrow\kappa_{2}$ in Eqs. (43,II.4). In the case of a bi-
directional arrangement Fig. 2 (b), one should replace
$\kappa_{ex}\rightarrow\kappa_{ex}/2$ in all expressions. Up to this point, we
relied only on classical mechanics, and all optomechanical properties (at
fixed $\Delta$) depend only on $g$ (tuned experimentally through $P_{in}$) and
$T_{m},T_{c},T_{ex}$. We shall now explicitly link our results to the quantum
formalism.

##  III  Discussion

###  III.1  Sideband asymmetry

Cross-correlations between the cavity current noise and the mechanics, Eqs.
(39,40) and cross-correlations between the detection background and the cavity
noise Eqs. (II.4,II.4) can be recast into apparent stochastic force components
that depend on the sideband, Eqs. (45,46) for the ’l’ and ’h’ ones
respectively.

For the “blue” pumping scheme, only the ’l’ sideband is measurable in the
sideband-resolved limit. Injecting $\Delta=+\Omega_{m}$ in the above mentioned
equations, we obtain:

$S_{\delta
F_{ex\,l}}=2m\Gamma_{\text{eff}}\,k_{B}\left(+2T_{c}-T_{ex}\right)\frac{\Omega_{m}}{\omega_{c}},$
(47)

with $T_{c}$ and $T_{ex}$ the temperatures of the cavity and the detection
port respectively, as introduced in the preceding Section. Similarly for the
“red” scheme, with $\Delta=-\Omega_{m}$ and looking at the ’h’ sideband we
have:

$S_{\delta
F_{ex\,h}}=2m\Gamma_{\text{eff}}\,k_{B}\left(+T_{ex}-2T_{c}\right)\frac{\Omega_{m}}{\omega_{c}}.$
(48)

In both expressions, $\Gamma_{\text{eff}}=\Gamma_{m}+\Gamma_{\text{opt}}$ but
$\Gamma_{\text{opt}}$ is different: negative for the “blue” scheme, and
positive for the “red” one. However, for low drive powers
$\Gamma_{\text{opt}}\approx 0$ and $\Gamma_{\text{eff}}\approx\Gamma_{m}$. In
this case, a very simple result emerges: the two apparent force noises are
opposite, a result referred to in the literature as sideband asymmetry
(Aspelmeyer et al., 2014a; Weinstein et al., 2014).

In the case of a “green” pumping scheme, $\Delta=0$ and both sidebands can be
measured at the same time. The resulting expressions for the cross-correlation
apparent stochastic force components are:

$\displaystyle S_{\delta F_{ex\,l}}$ $\displaystyle=$ $\displaystyle
2m\Gamma_{m}\,k_{B}\left(+T_{ex}\right)\frac{\Omega_{m}}{\omega_{c}},$
$\displaystyle S_{\delta F_{ex\,h}}$ $\displaystyle=$ $\displaystyle
2m\Gamma_{m}\,k_{B}\left(-T_{ex}\right)\frac{\Omega_{m}}{\omega_{c}},$ (49)

again in the sideband-resolved limit. Eqs. (49) are very similar to Eqs.
(47,48): again the two forces are opposite, but this time they depend only on
$T_{ex}$.

Let us consider the case of an ideally thermalized system were
$T_{c}=T_{ex}=T_{m}$. Then in the limit $\Gamma_{\text{opt}}\approx 0$,
sideband asymmetry measured by comparing the ’l’ peak in “blue” pumping Eq.
(47) with the ’h’ peak in “red” Eq. (48) gives strictly the same result as the
direct comparison of the two sidebands Eqs. (49) observed with a “green”
scheme. Besides, the sideband asymmetry effect simply renormalizes the
observed mechanical temperature by $T_{m}\rightarrow
T_{m}(1+\Omega_{m}/\omega_{c})$ on the ’l’ side, and by $T_{m}\rightarrow
T_{m}(1-\Omega_{m}/\omega_{c})$ on the ’h’ side; since
$\Omega_{m}/\omega_{c}\ll 1$, this effect can be safely neglected in this
case. One needs to artificially create a situation where $T_{m}\ll T_{ex}$ to
make the sideband asymmetry detectable (e.g. by sideband cooling the
mechanical mode, and injecting noise through the microwave port) (Weinstein et
al., 2014). As soon as $T\rightarrow 0~{}$K, the classical picture breaks down
and all features should be interpreted in the framework of quantum mechanics;
including sideband asymmetry. The direct link between the two theories shall
be discussed in the next Section.

###  III.2  The quantum limit and Heisenberg uncertainty

Quantum mechanics tells us that energy comes in quanta, each carrying $\hbar$
times the frequency corresponding to the concerned mode. To link the classical
writing to the quantum expressions, we thus have to introduce the following
populations:

$\displaystyle n_{c}$ $\displaystyle=$
$\displaystyle\frac{E_{c}}{\hbar\omega_{c}},$ (50) $\displaystyle n_{c}^{th}$
$\displaystyle=$ $\displaystyle\frac{k_{B}T_{c}}{\hbar\omega_{c}},$ (51)
$\displaystyle n_{ex}^{th}$ $\displaystyle=$
$\displaystyle\frac{k_{B}T_{ex}}{\hbar\omega_{c}},$ (52) $\displaystyle
n_{m}^{th}$ $\displaystyle=$
$\displaystyle\frac{k_{B}T_{m}}{\hbar\Omega_{m}},$ (53)

with respectively the (coherent) cavity population, the cavity thermal
population, the external microwave port (thermal) population and the
mechanical mode thermal population.

Let us consider first the case of a mechanical oscillator driven only by the
Langevin force; we do not consider yet the back-action stochastic drive,
neither the sideband asymmetry apparent contributions discussed in Section
III.1. Then $\frac{1}{2\pi}\int
S^{\pm}_{x}d\omega={\sigma^{\pm}_{x}}^{2}={\sigma_{x}}^{2}$, the two detected
sidebands are equivalent, and we have:

$\displaystyle{\sigma_{x}}^{2}$ $\displaystyle=$
$\displaystyle\left\langle{\delta
x}^{2}\right\rangle\frac{\Gamma_{m}}{\Gamma_{\text{eff}}},$ (54)
$\displaystyle\left\langle{\delta x}^{2}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{k_{B}T_{m}}{2m{\Omega_{m}}^{2}}=x_{zpf}^{2}\,n_{m}^{th},$
(55)

with $\left\langle{\delta x}^{2}\right\rangle$ the half-variance of the motion
(computed on one sideband only), having defined
$x_{zpf}=\sqrt{\hbar/(2m\Omega_{m})}$ the zero point fluctuations. The
$\Gamma_{m}/\Gamma_{\text{eff}}$ factor comes from the dynamical part of the
back-action, causing optical damping/anti-damping, with
$\Gamma_{\text{eff}}=\Gamma_{m}+\Gamma_{\text{opt}}$ for “blue” and “red”
pumping schemes, and $\Gamma_{\text{eff}}=\Gamma_{m}$ for “green”. From this
definition we have $x_{zpf}^{2}=\hbar\omega_{c}\,\bar{x}^{2}$, and we can
recast $g=g_{0}\sqrt{n_{c}}$ with $g_{0}=G\,x_{zpf}$ (in Rad/s). Eq. (55)
coincides exactly with the quantum-mechanical high-temperature limit; when
$T\rightarrow 0~{}$K, quantum mechanics predicts that $n_{m}^{th}\rightarrow
1/2$ (Aspelmeyer et al., 2014a).

Any measurement comes with an acquisition imprecision. In the literature, one
finds a discussion focused on the phase of the optical “green” readout
(Aspelmeyer et al., 2014a). An equivalent discussion can be performed on the
amplitude of the signal; this is what we will present in the following. A
position fluctuation $\delta x$ transduces into a cavity frequency shift
$\delta\omega_{c}=\delta
x\,G\,(\kappa_{t}/2)/{\left|-i\omega+(\kappa_{t}/2)\right|}$, taking into
account the finite response time of the microwave mode. This shift
$\delta\omega_{c}$ will in turn modify the output signal energy $\delta
E/E\approx 4{\left(\delta\omega_{c}\right)}^{2}/{\kappa_{t}}^{2}$. This
variation can be expressed in terms of detection noise quanta $\delta
E/(\hbar\omega_{c})=n_{det}$ with $\delta
E/E=n_{det}/\left(n_{c}\kappa_{c}\,t\right)$, measured during a time $t$. For
an ideal quantum detector the measurement is shot-noise limited with
$n_{det}=1$ (Aspelmeyer et al., 2014a; Clerk et al., 2010); in the classical
case, $n_{det}\,\hbar\omega_{c}$ corresponds to the noise background (in
Joules) affecting the detection, arising from the whole amplification chain
(with obviously $n_{det}\gg 1$). The imprecision in position resulting from
the finite $n_{det}$ can be interpreted as the integral of a flat noise
$S^{imp}_{x}$ (over a bandwidth $t^{-1}$) (Schliesser et al., 2009b):

$S^{imp}_{x}=\frac{k^{2}_{t}n_{det}}{16\,G^{2}n_{c}\kappa_{ex}}\left(1+4\frac{\omega^{2}}{\kappa_{t}^{2}}\right),$
(56)

where $\omega$ is set by the measuring scheme, “blue”, “red” or “green”.

Focusing on the “green” scheme, Eq. (25) leads to $S_{\delta
F}=\hbar^{2}G^{2}n_{c}n_{c}^{th}\,2\kappa_{t}/\Omega_{m}^{2}$, and in Eq. (56)
one should consider $\omega=\pm\Omega_{m}$ (for each sideband). This leads to
the result:

$S^{imp}_{x}\,S_{\delta
F}=\frac{\hbar^{2}}{4}\frac{\kappa_{t}}{\kappa_{ex}}\left(2n_{c}^{th}\right)n_{det}.$
(57)

At $T\rightarrow 0~{}$K, quantum mechanics predicts that
$n_{c}^{th}\rightarrow 1/2$. Thus for a shot-noise limited detection
($n_{det}=1$), we recover from Eq. (57) the famous Heisenberg limit
$S^{imp}_{x}S_{\delta F}\geq\hbar^{2}/4$, reached only for an overcoupled
cavity $\kappa_{t}\approx\kappa_{ex}$.

It is enlightening to evaluate the minimal mechanical displacement which can
be detected with such a microwave optomechanical scheme. The imprecision noise
Eq. (56) can be taken into account by adding it up with $S_{x}^{\pm}$ in Eq.
(44). Subtracting the $2k_{B}T_{ex}$ background and integrating each sideband
over a bandwidth $\Delta\omega$ large enough to cover the peaks, we are led to
define a signal component ${\cal S}_{ig}$, and a noise component ${\cal
N}_{oise}$, for each sideband $i=$’l’ or ’h’:

$\displaystyle{\cal S}_{ig}$ $\displaystyle=$
$\displaystyle\hbar\omega_{c}|\chi_{i}|^{2}\kappa_{ex}\,G^{2}n_{c}\left[\left\langle{\delta
x}^{2}\right\rangle\frac{\Gamma_{m}}{\Gamma_{\text{eff}}}\right.$ (58)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\left.+\frac{S_{\delta
F_{ex\,i}}}{(2m\Omega_{m})^{2}\,\Gamma_{\text{eff}}}\right],$
$\displaystyle{\cal N}_{oise}$ $\displaystyle=$
$\displaystyle\hbar\omega_{c}|\chi_{i}|^{2}\kappa_{ex}\Bigg{[}\Bigg{.}$ (59)
$\displaystyle\\!\\!\\!\\!\\!\Bigg{.}\kappa_{t}\frac{\Delta\omega}{2\pi}\left(\frac{n_{det}\,\kappa_{t}}{16\,\kappa_{ex}}\left(1+4\frac{\omega^{2}}{\kappa_{t}^{2}}\right)+n_{c}^{th}-n_{ex}^{th}\right)\Bigg{.}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\Bigg{.}+\frac{G^{4}n_{c}^{2}\,x_{zpf}^{4}\kappa_{t}}{\Gamma_{m}}n_{c}^{th}\left(|\chi_{l}|^{2}+|\chi_{h}|^{2}\right)\frac{\Gamma_{m}}{\Gamma_{\text{eff}}}\,\Bigg{]},$

where ${\cal N}_{oise}$ contains both imprecision (former term, with also
cavity noise $n_{c}^{th}-n_{ex}^{th}$) and back-action (latter). We now wrote
explicitly the sideband asymmetry contribution $S_{\delta F_{ex\,i}}$ in Eq.
(58).

Considering again the “green” scheme, we have $\left\langle{\delta
x}^{2}\right\rangle\Gamma_{m}/\Gamma_{\text{eff}}=x_{zpf}^{2}n_{m}^{th}$ and
$|\chi_{l}|^{2}=|\chi_{h}|^{2}$. Eqs. (49) demonstrate that the extra term in
Eq. (58) modifies the measured peaks from Eq. (55) into Eq. (58) by
substituting $n_{m}^{th}\rightarrow n_{m}^{th}+n_{ex}^{th}$ on the ’l’ side,
and $n_{m}^{th}\rightarrow n_{m}^{th}-n_{ex}^{th}$ on the ’h’ side; this is
sideband asymmetry in the quantum mechanics language. The difference
${\sigma^{-}_{x}}^{2}-{\sigma^{+}_{x}}^{2}$ is then proportional to
$2n_{ex}^{th}$, which tends towards 1 in quantum mechanics at $T\rightarrow
0~{}$K; a similar result can be obtained comparing the ’l’ and ’h’ peaks
obtained in “blue” and “red” pumping schemes respectively (Weinstein et al.,
2014). However from $({\sigma^{-}_{x}}^{2}+{\sigma^{+}_{x}}^{2})/2$, the
“green” scheme leads to a quantity insensitive to sideband asymmetry, which is
directly the image of the mechanical motion. Discussing now on this quantity,
our signal is then ${\cal S}_{ig}\propto\left\langle{\delta
x}^{2}\right\rangle$.

Figure 6:  Illustration of inverse signal-to-noise ratio $\left({\cal
S}_{ig}/{\cal N}_{oise}\right)^{-1}$ in the “green ” pumping scheme, as a
function of photon number $n_{c}$ inside the cavity. Driven at optimal
$n_{c}$, the best signal-to-noise ratio can be obtained for the detection of
$\left\langle{{\delta}x}^{2}\right\rangle$ (so-called standard classical
limit, see text). Curve computed with , $\kappa_{t}=\kappa_{ex}$,
$\Delta\omega=6\,\Gamma_{m}$, and
$\Gamma_{m}\Omega_{m}^{2}/(\kappa_{t}g_{0}^{2})=1.6\,10^{8}$ for
$n_{det}=100$, $n_{c}^{th}=n_{ex}^{th}=6.\,10^{3}$ and
$n_{m}^{th}=6.\,10^{5}$. At the quantum limit, the optimum point (the minimum
of the curve) reaches $\left({\cal S}_{ig}/{\cal N}_{oise}\right)^{-1}\approx
2$ (SQL, see text).

Carrying out the substitutions $\Gamma_{\text{eff}}=\Gamma_{m}$,
$\omega=\pm\Omega_{m}$ and $|\chi_{l}|^{2}+|\chi_{h}|^{2}\approx
2/\Omega_{m}^{2}$ valid for “green” pumping in Eqs. (58,59), we can define a
signal-to-noise ratio ${\cal S}_{ig}/{\cal N}_{oise}$ that illustrates our
sensitivity to the quantity $\left\langle{\delta x}^{2}\right\rangle$. This is
represented in Fig. 6 as a function of drive power with the parameter $n_{c}$;
similar plots can be found in Refs. (Aspelmeyer et al., 2014a; Clerk et al.,
2010; Teufel et al., 2009). On the left, the sensitivity is lost $\propto
1/n_{c}$ because of our finite detection noise $n_{det}$. On the right, the
measurement is dominated by back-action $\propto n_{c}$ arising from
$n_{c}^{th}$. There is an optimum defined by $d({\cal S}_{ig}/{\cal
N}_{oise})/d\,n_{c}=0$. This point verifies
$n_{c}\approx\frac{1}{4\sqrt{\pi}}\sqrt{\frac{\Delta\omega}{\Gamma_{m}}}\sqrt{\frac{\kappa_{t}}{\kappa_{ex}}}\sqrt{\frac{n_{det}}{n_{c}^{th}}}\,\frac{\Gamma_{m}\Omega_{m}^{2}}{\kappa_{t}g_{0}^{2}}$,
with ${\cal S}_{ig}/{\cal
N}_{oise}\approx\frac{n_{m}^{th}}{\sqrt{n_{det}2n_{c}^{th}}}\sqrt{2\pi}\sqrt{\frac{\kappa_{ex}}{\kappa_{t}}}\sqrt{\frac{\Gamma_{m}}{\Delta\omega}}$
(resolved sideband limit, assuming $n_{ex}^{th}=n_{c}^{th}$). At the
$T\rightarrow 0~{}$K quantum limit $n_{c}^{th},n_{m}^{th}\rightarrow 1/2$ with
a shot-noise limited detector $n_{det}=1$, we reach at best ${\cal
S}_{ig}/{\cal N}_{oise}\approx 1/2$ (for $\kappa_{t}\approx\kappa_{ex}$,
$\Delta\omega\approx 6\,\Gamma_{m}$): the signal is about half the total
detected noise (Clerk et al., 2010). This is the ultimate physical limit
called standard quantum limit (SQL). In contrast, the classical optimum which
we shall call standard classical limit (SCL) is relative and depends both on
$n_{det}$ (quality of classical detector) and $n_{c}^{th}$ (Johnson-Nyquist
noise of the cavity). The main optomechanical results applying to the “green”
pumping scheme are compared in Tab. 1 in the classical and quantum regimes.
The key point revealed by the classical modeling is that all features have a
classical analogue; only the $T\rightarrow 0~{}$K quantities are a true
signature of quantumness, which highlights the importance of calibrations in
all conducted experiments.

Table 1: $S^{imp}_{x}\,S_{\delta F}$ product, signal ${\cal S}_{ig}$, noise ${\cal N}_{oise}$ and signal-to-noise ratio in the quantum and classical limits (the latter are given at the optimal $n_{c}$ for the “green” pumping scheme; $\delta E$ energy detection resolution and $T_{c}$ cavity temperature, see text).  Quantity | Quantum limit | Classical limit
---|---|---
$S^{imp}_{x}\,S_{\delta F}$ | $\frac{\kappa_{t}}{\kappa_{ex}}\hbar^{2}/4$ | $\frac{1}{2}\frac{\kappa_{t}}{\kappa_{ex}}k_{B}T_{c}\,\delta E/(\omega_{c}^{2})$
${\cal S}_{ig}$ | $\propto x_{zpf}^{2}/{2}$ | $\propto k_{B}T_{m}/(2\,m\Omega_{m}^{2})$
${\cal N}_{oise}$ | $\propto x_{zpf}^{2}$ | $\propto\sqrt{\delta E\,k_{B}T_{c}}/(\sqrt{2}\,m\Omega_{m}\omega_{c})$
${\cal S}_{ig}/{\cal N}_{oise}$ | 1/2 | $k_{B}T_{m}\omega_{c}/(\sqrt{2}\sqrt{\delta E\,k_{B}T_{c}}\,\Omega_{m})$

A similar reasoning can be performed for the “blue” pumping scheme. While the
measurement is limited by the parametric instability for $n_{c}$, the ${\cal
S}_{ig}/{\cal N}_{oise}$ can be recast into $\sim
T_{m}\omega_{c}/\left(T_{c}\Omega_{m}\right)\gg 1$ close to it. This makes it
a very practical technique to perform thermometry Zhou et al. (2019).

##  IV Conclusion

In summary, we have presented the generic classical electric circuit model
which is analogous to the standard optomechanics quantum treatment. The
developed analytics provides the bridge between circuit parameters and quantum
optics quantities, a mandatory link for design and optimization. The two
approaches are strictly equivalent, provided temperatures are high enough for
both the mechanical and the electromagnetic degrees of freedom.

Besides, a thorough comparison of the two models gives a profound
understanding of what measured properties are fundamentally quantum. To match
the mathematics of the two computation methods, we introduce populations by
means of an energy per quanta proportional to the mode resonance frequency:
the early Planck postulate. Sideband asymmetry is derived in classical terms,
and we distinguish the temperatures of the mechanical mode $T_{m}$ from the
one of the microwave mode $T_{c}$ and the microwave (detection) port $T_{ex}$.
Considering the measurement protocol in itself, we derive the resolution limit
of a classical experiment performed with the optimal optomechanical scheme. We
obtain the classical (and relative) analogue of the (absolute) standard
quantum limit (SQL) fixed by the Heisenberg principle in quantum mechanics; we
shall name it the standard classical limit (SCL). Only the
$T_{m},T_{c},T_{ex}\rightarrow 0~{}$K measured quantities appear to be
specific to quantum mechanics, since all features present a classical
analogous counterpart.

###### Acknowledgements.

We would like to acknowledge support from the STaRS-MOC project No. 181386
from Region Hauts-de-France, the ERC generator project No. 201050 from ISITE
and the ERC CoG grant ULT-NEMS No. 647917. The research leading to these
results has received funding from the European Union’s Horizon 2020 Research
and Innovation Programme, under grant agreement No. 824109, the European
Microkelvin Platform (EMP).

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