# All-optical coherent quantum-noise cancellation in cascaded optomechanical
systems

Jakob Schweer  Institute for Gravitational Physics, and Max Planck Institute
for Gravitational Physics (Albert Einstein Institute), Leibniz Universität
Hannover, Callinstraße 38, 30167 Hannover, Germany  Daniel Steinmeyer
Institute for Gravitational Physics, and Max Planck Institute for
Gravitational Physics (Albert Einstein Institute), Leibniz Universität
Hannover, Callinstraße 38, 30167 Hannover, Germany  Klemens Hammerer
Institute for Theoretical Physics, Leibniz Universität Hannover, Appelstraße
2, 30167 Hannover, Germany  Institute for Gravitational Physics, and Max
Planck Institute for Gravitational Physics (Albert Einstein Institute),
Leibniz Universität Hannover, Callinstraße 38, 30167 Hannover, Germany
Michèle Heurs  Institute for Gravitational Physics, and Max Planck Institute
for Gravitational Physics (Albert Einstein Institute), Leibniz Universität
Hannover, Callinstraße 38, 30167 Hannover, Germany

###### Abstract

Coherent quantum noise cancellation (CQNC) can be used in optomechanical
sensors to surpass the standard quantum limit (SQL). In this paper, we
investigate an optomechanical force sensor that uses the CQNC strategy by
cascading the optomechanical system with an all-optical effective negative
mass oscillator. Specifically, we analyze matching conditions, losses and
compare the two possible arrangements in which either the optomechanical or
the negative mass system couples first to light. While both of these orderings
yield a sub-SQL performance, we find that placing the effective negative mass
oscillator before the optomechanical sensor will always be advantageous for
realistic parameters. The modular design of the cascaded scheme allows for
better control of the sub-systems by avoiding undesirable coupling between
system components, while maintaining similar performance to the integrated
configuration proposed earlier. We conclude our work with a case study of a
micro-optomechanical implementation.

##  I Introduction

Achieving force measurements at the quantum limit has been a significant focus
for several decades [1, 2] and has fueled the development of optomechanics [3,
4, 5]. Optomechanical sensors exploit the interaction of a light field with
the motion of a mechanical oscillator to measure its displacement with high
precision. Force measurements based on these schemes are subject to shot noise
and quantum radiation pressure backaction noise [6, 7]. Shot noise is caused
by the uncertainty in the number of photons over time and can be decreased
relative to the signal by increasing the intensity of the optical field.
Contrary, the backaction noise arises from the fluctuation in the radiation
pressure of the optical field, which will increase with its intensity. The
trade-off between these competing processes then sets a lower bound to the
precision of the measurement, which is called the standard quantum limit (SQL)
[7, 8, 9].

The SQL is not a fundamental limit, and many different approaches have been
suggested to achieve measurements with sub-SQL accuracy. These approaches
include frequency-dependent squeezing [10, 11, 12], variational measurements
[13, 14, 15], dual mechanical resonators [16, 17, 18, 19] and optical spring
effects [20, 21, 22]. In essence, these ideas go beyond the SQL by measuring a
quantum non-demolition (QND) variable of the probe, that is, a variable that
commutes with itself for different moments in time. In a QND measurement, the
backaction is transmitted to the canonically conjugate observable and thus
avoided. A more general approach to QND measurements is gained by introducing
another system that acts like a reference frame with an effective negative
mass [23]. By measuring with respect to this reference system, a QND
measurement is realized. When the reference system is a harmonic oscillator,
an effective negative mass amounts to a negative eigenfrequency. This idea was
first experimentally utilized in demonstrating Einstein-Podolsky-Rosen (EPR)
states of two atomic spin oscillators of positive and negative mass [24].
Based on this, back action cancellation was demonstrated by Wasilewski et al.
[25] in the context of magnetometry. Extending this idea, several proposals
have been made in a hybrid setting of a mechanical oscillator and atomic spin
ensembles [26, 23], and the evasion of backaction noise in these spin
ensembles experimentally verified in [27]. Independently, Tsang and Caves [28,
29] developed this idea in a more general context, called quantum-mechanics-
free subsystems. In the context of optomechanics, the main idea is to
introduce an ‘anti-noise’ path to the dynamics of the optomechanical sensor
upon coupling to an ancillary resonator that acts as an effective negative
mass. This way, the backaction noise can be cancelled coherently, and sub-SQL
force sensing is achieved for all measurement frequencies. Appropriately, this
approach is called coherent quantum noise cancellation (CQNC). Details and
experimental feasibility of this all-optical effective negative mass
oscillator were discussed in more detail by Wimmer et al. [30].

Within the area of CQNC force sensing, many other possible negative mass
oscillators and setups have been considered. Other setups include the use of
ultra-cold atoms inside a separate cavity [31, 32], hybrid optomechanical
cavity, i.e. implementing an atomic ensemble inside the optomechanical sensor
[33, 34] and employing Bose-Einstein condensates [35]. Even a new all-optical
setup was suggested using two detuned optical modes inside the force sensor
[36]. These approaches can be categorized into integrated setups, where the
effective negative mass is introduced directly into the optomechanical force
sensor, and cascaded setups [37, 38], where the effective negative mass
oscillator is a separate system. Recently, Zeuthen et al. [39] considered a
broad class of effective negative mass oscillators in a cascaded setting and
even considered a possible coupling between the positive and negative mass
oscillator in a parallel topology.

Inspired by this, we want to discuss a cascaded version of the original all-
optical setup [28, 30]. Instead of implementing the ‘anti-noise’ path directly
into the optomechanical sensor, an all-optical effective negative mass
oscillator is built as a separate system. The backaction is then cancelled by
coupling the force sensor to the effective negative mass oscillator via a
strong coherent field. This approach will give more freedom in the
experimental design and simplify reaching the challenging conditions for a
CQNC experiment. The main challenge before cancelling quantum backaction noise
is to measure the backaction noise. Due to the modular nature of the cascaded
approach, this can be tackled entirely separate from the effective negative
mass oscillator. We will see that under some modifications to the matching
conditions, our cascaded setup recovers the ideal CQNC performance described
in [30], and the additional degrees of freedom by expanding the dimension of
the system lead to novel phenomena for sub-SQL force sensing. This includes
the recovery of ideal CQNC around an off-resonant frequency and possible CQNC
performance in the low- or high-frequency regime, even for unmatched CQNC
conditions.

This paper is organized as follows. In Sec. II, we describe the model of our
cascaded CQNC scheme and derive the quantum Langevin equations of motion. In
Sec. III, we discuss force sensing in optomechanical sensors and derive the
optimal parameters for ideal CQNC. In Sec. IV, we analyze possible deviations
from the ideal conditions and their impact on the performance of coherent
quantum noise reduction. Then, in Sec. V, a case study is provided. Finally,
we summarize our findings in Sec. VI.

Figure 1: (a) Sketch of the cascaded setup. The optomechanical sensor (green)
consists of a mechanical oscillator inside an optical cavity, depicted in a
membrane-in-the-middle (MiM) setup. The effective negative mass oscillator
(blue) comprises two optical modes, coupled via a beam-splitter and down-
conversion process. A non-linear (PPKTP) crystal exemplifies the coupling for
the down-conversion and a wave plate for the beam-splitter coupling. (b)
Simplified depiction of the possible arrangements of the cascaded scheme.

##  II Model

Fig. 1(a) illustrates a possible, schematic realization of our setup. We refer
to [40] for details on the experimental implementation of all-optical CQNC. An
optomechanical sensor (OMS), subject to an external force and radiation
pressure noise, is connected to an effective negative mass oscillator (NMO) by
a coherent light field. The force is then measured by detecting outgoing light
after the second system. The order of sub-systems can be chosen freely, and
the two possible arrangements are depicted in Fig. 1(b). In the first case,
the light travels through the NMO, followed by the OMS. We will refer to this
case as $\mathrm{NMO}\mapsto\mathrm{OMS}$. In the second case, the order is
reversed; hence the light will travel through the OMS first, and we will refer
to this case as $\mathrm{OMS}\mapsto\mathrm{NMO}$. The effects of these
different arrangements on the performance of the backaction cancellation will
be discussed later.

The OMS is modelled by an optical cavity with resonance frequency
$\omega_{\mathrm{om}}$, containing a damped mechanical positive mass
oscillator (PMO) with resonance frequency $\omega_{\mathrm{m}}$ and linewidth
$\gamma_{\mathrm{m}}$, coupled to the cavity field via radiation pressure
interaction and subjected to an external force $\mathrm{F}$. Following the
standard treatment of these force sensors [5], we move to a rotating frame
with respect to the frequency $\omega_{\mathrm{\scriptscriptstyle{L}}}$ of the
strong driving laser field and arrive at the linearized Hamiltonian

$\displaystyle\begin{split}H_{\mathrm{\scriptscriptstyle{OMS}}}=\Delta_{\mathrm{om}}c^{\dagger}_{\mathrm{om}}c_{\mathrm{om}}+\frac{\omega_{\mathrm{m}}}{2}(x_{\mathrm{m}}^{2}+p_{\mathrm{m}}^{2})+\frac{gx_{\mathrm{m}}}{\sqrt{2}}(c_{\mathrm{om}}+c^{\dagger}_{\mathrm{om}}).\end{split}$
(1)

Here,
$\Delta_{\mathrm{om}}=\omega_{\mathrm{om}}-\omega_{\mathrm{\scriptscriptstyle{L}}}$
is the detuning of the optomechanical cavity to the incoming field,
$c_{\mathrm{om}}$ ($c_{\mathrm{om}}^{\dagger}$) are the annihilation
(creation) operators of the optical mode and
$x_{\mathrm{m}}=X/x_{\mathrm{\scriptscriptstyle{ZPF}}}$,
$p_{\mathrm{m}}=Px_{\mathrm{\scriptscriptstyle{ZPF}}}/\hbar$ are the position
and momentum operators of the mechanical oscillator normalized to the zero-
point fluctuation
$x_{\mathrm{\scriptscriptstyle{ZPF}}}=\sqrt{\hbar/m\omega_{\mathrm{m}}}$, such
that $[x_{\mathrm{m}},p_{\mathrm{m}}]=i$. The last term in Eq. (1) describes
the radiation pressure interaction of the cavity mode and the mechanical
oscillator. Its strength is given by
$g=\sqrt{2}\omega_{\mathrm{c}}x_{\mathrm{\scriptscriptstyle{ZPF}}}\alpha_{\mathrm{c}}/L$,
where $L$ is the cavity length and $\alpha_{\mathrm{c}}\propto\sqrt{P}$ is the
field amplitude of the cavity mode, proportional to the input power $P$.
Introducing dimensionless amplitude and phase quadratures
$c_{\mathrm{om}}=(x_{\mathrm{om}}+ip_{\mathrm{om}})/\sqrt{2}$, the Hamiltonian
(1) implies the quantum Langevin equations (QLEs)

$\displaystyle\dot{x}_{\mathrm{m}}$
$\displaystyle=\omega_{\mathrm{m}}p_{\mathrm{m}},$ (2a)
$\displaystyle\dot{p}_{\mathrm{m}}$
$\displaystyle=-\omega_{\mathrm{m}}x_{\mathrm{m}}-\gamma_{\mathrm{m}}p_{\mathrm{m}}-gx_{\mathrm{om}}+\sqrt{\gamma_{\mathrm{m}}}F,$
(2b) $\displaystyle\dot{x}_{\mathrm{om}}$
$\displaystyle=-\frac{\kappa_{\mathrm{om}}}{2}x_{\mathrm{om}}+\Delta_{\mathrm{om}}p_{\mathrm{om}}+\sqrt{\kappa_{\mathrm{om}}}x_{\mathrm{om}}^{\mathrm{in}},$
(2c) $\displaystyle\dot{p}_{\mathrm{om}}$
$\displaystyle=-\frac{\kappa_{\mathrm{om}}}{2}p_{\mathrm{om}}-\Delta_{\mathrm{om}}x_{\mathrm{om}}-gx_{\mathrm{m}}+\sqrt{\kappa_{\mathrm{om}}}p_{\mathrm{om}}^{\mathrm{in}}.$
(2d)

Here, $\kappa_{\mathrm{om}}$ is the decay rate of the cavity mode and
$c_{\mathrm{om}}^{\mathrm{in}}=(x_{\mathrm{om}}^{\mathrm{in}}+ip_{\mathrm{om}}^{\mathrm{in}})/\sqrt{2}$
is its vacuum input noise. The noise process fulfills $\langle
c_{\mathrm{om}}^{\mathrm{in}}(t)c_{\mathrm{om}}^{\mathrm{in}\dagger}(\tau)\rangle=\delta(t-\tau)$.
In Eq. (2) we have defined the scaled force operator $F=\mathrm{F}/\sqrt{\hbar
m\gamma_{\mathrm{m}}\omega_{\mathrm{m}}}$ with dimension $\sqrt{\mathrm{Hz}}$.
It consists of the to-be-detected force signal
$F_{\mathrm{\scriptscriptstyle{sig}}}$ acting on the mechanical oscillator and
Brownian thermal noise $F_{\mathrm{\scriptscriptstyle{th}}}$ of the
oscillator. The scaled thermal noise satisfies $\langle
F_{\mathrm{\scriptscriptstyle{th}}}(t)F_{\mathrm{\scriptscriptstyle{th}}}(\tau)\rangle=n_{\mathrm{\scriptscriptstyle{th}}}\delta(t-\tau)$,
where $n_{\mathrm{\scriptscriptstyle{th}}}=k_{B}T/\hbar\omega_{\mathrm{m}}$ is
the average phonon number of the mechanical oscillator.

The NMO consists of two optical modes $c_{\mathrm{c}}$ and $a$, with resonance
frequencies $\omega_{\mathrm{c}}$ and $\omega_{\mathrm{a}}$, coupled with a
beam-splitter and down-conversion process. In analogy to [30, 40], we refer to
$\omega_{\mathrm{c}}$ as the meter cavity and $\omega_{\mathrm{a}}$ as the
ancilla cavity. The Hamiltonian of this system, see Appendix A for details, is
given by

$H_{\mathrm{\scriptscriptstyle{NMO}}}=\Delta_{\mathrm{c}}c^{\dagger}_{\mathrm{c}}c_{\mathrm{c}}+\Delta_{\mathrm{a}}a^{\dagger}a+g_{\mathrm{\scriptscriptstyle{BS}}}(ac^{\dagger}_{\mathrm{c}}+a^{\dagger}c_{\mathrm{c}})+g_{\mathrm{\scriptscriptstyle{DC}}}(ac_{\mathrm{c}}+a^{\dagger}c^{\dagger}_{\mathrm{c}}),$
(3)

with the detunings
$\Delta_{\mathrm{c},\mathrm{a}}=\omega_{\mathrm{c},\mathrm{a}}-\omega_{\mathrm{\scriptscriptstyle{L}}}$,
the beam-splitter coupling strength $g_{\mathrm{\scriptscriptstyle{BS}}}$ and
the coupling strength of the down-conversion process
$g_{\mathrm{\scriptscriptstyle{DC}}}$. As above, we introduce amplitude and
phase quadratures associated with the meter and ancilla cavity. Then the
Hamiltonian (3) implies the following QLEs for the NMO,

$\displaystyle\dot{x}_{\mathrm{c}}$
$\displaystyle=-\frac{\kappa_{\mathrm{c}}}{2}x_{\mathrm{c}}+\Delta_{\mathrm{c}}p_{\mathrm{c}}+(g_{\mathrm{\scriptscriptstyle{BS}}}-g_{\mathrm{\scriptscriptstyle{DC}}})p_{\mathrm{a}}+\sqrt{\kappa_{\mathrm{c}}}x_{\mathrm{c}}^{\mathrm{in}},$
(4a) $\displaystyle\dot{p}_{\mathrm{c}}$
$\displaystyle=-\frac{\kappa_{\mathrm{c}}}{2}x_{\mathrm{c}}-\Delta_{\mathrm{c}}x_{\mathrm{c}}-(g_{\mathrm{\scriptscriptstyle{BS}}}+g_{\mathrm{\scriptscriptstyle{DC}}})x_{\mathrm{a}}+\sqrt{\kappa_{\mathrm{c}}}p_{\mathrm{c}}^{\mathrm{in}},$
(4b) $\displaystyle\dot{x}_{\mathrm{a}}$
$\displaystyle=-\frac{\kappa_{\mathrm{a}}}{2}x_{\mathrm{a}}+\Delta_{\mathrm{a}}p_{\mathrm{a}}+(g_{\mathrm{\scriptscriptstyle{BS}}}-g_{\mathrm{\scriptscriptstyle{DC}}})p_{\mathrm{c}}+\sqrt{\kappa_{\mathrm{a}}}x_{\mathrm{a}}^{\mathrm{in}},$
(4c) $\displaystyle\dot{p}_{\mathrm{a}}$
$\displaystyle=-\frac{\kappa_{\mathrm{a}}}{2}p_{\mathrm{a}}-\Delta_{\mathrm{a}}x_{\mathrm{a}}-(g_{\mathrm{\scriptscriptstyle{BS}}}+g_{\mathrm{\scriptscriptstyle{DC}}})x_{\mathrm{c}}+\sqrt{\kappa_{\mathrm{a}}}p_{\mathrm{a}}^{\mathrm{in}},$
(4d)

with the cavity linewidth $\kappa_{\mathrm{c}}$ and $\kappa_{\mathrm{a}}$ and
the input noise processes $\mathrm{a}^{\mathrm{in}}$ and
$c_{\mathrm{c}}^{\mathrm{in}}$. Under the condition
$g_{\mathrm{\scriptscriptstyle{BS}}}-g_{\mathrm{\scriptscriptstyle{DC}}}=0$,
the QLEs (4) generate similar interaction as in Eqs. (2). Additionally,
driving the meter cavity on resonance $\Delta_{\mathrm{c}}=0$, results in
$\Delta_{\mathrm{a}}=\omega_{\mathrm{a}}-\omega_{\mathrm{c}}$. Thus the
detuning between the meter and ancilla cavity can be used to generate an
effective negative mass oscillator.

##  III Force sensing and ideal CQNC

To solve the dynamics of both systems, we turn to the frequency domain.
Introducing the Fourier domain operators

$O(\omega)=\frac{1}{\sqrt{2\pi}}\int\mathrm{d}t\,O(t)e^{i\omega t},$

equations (2)–(4) can be solved using the standard input-output formalism [41]

$\displaystyle x^{\mathrm{out}}=\sqrt{\kappa}x-x^{\mathrm{in}},$ (5)
$\displaystyle p^{\mathrm{out}}=\sqrt{\kappa}p-p^{\mathrm{in}}.$ (6)

We consider the systems separately, first the OMS. On resonance,
$\Delta_{\mathrm{om}}=0$, the output quadratures read

$\displaystyle\begin{split}x^{\mathrm{out}}_{\mathrm{om}}&=e^{i\phi}x^{\mathrm{in}}_{\mathrm{om}},\\\
p^{\mathrm{out}}_{\mathrm{om}}&=e^{i\phi}p^{\mathrm{in}}_{\mathrm{om}}-\chi_{\mathrm{m}}g^{2}\kappa_{\mathrm{om}}\chi_{\mathrm{om}}^{2}x^{\mathrm{in}}_{\mathrm{om}}\\\
&\quad+\chi_{\mathrm{m}}\sqrt{\kappa_{\mathrm{om}}}g\chi_{\mathrm{om}}\sqrt{\gamma_{\mathrm{m}}}F,\end{split}$
(7)

where
$e^{i\phi}=(\frac{\kappa_{\mathrm{om}}}{2}-i\omega)/(\frac{\kappa_{\mathrm{om}}}{2}+i\omega)$.
We have defined the susceptibilities for the optomechanical cavity and the
mechanical oscillator, as

$\displaystyle\chi_{\mathrm{om}}(\omega)$
$\displaystyle=[i\omega+\frac{\kappa_{\mathrm{om}}}{2}]^{-1},$ (8)
$\displaystyle\chi_{\mathrm{m}}(\omega)$
$\displaystyle=\omega_{\mathrm{m}}[(\omega^{2}-\omega^{2}_{\mathrm{m}})-i\gamma_{\mathrm{m}}\omega]^{-1}.$
(9)

The mechanical oscillator is susceptible to the external force $F$, which
contains the force signal $F_{\mathrm{\scriptscriptstyle{sig}}}$ and thermal
noise $F_{\mathrm{\scriptscriptstyle{th}}}$. By measuring with light, the
force signal can be estimated via a phase measurement, which also introduces
additional noise due to the radiation pressure. From the measured phase
$p^{\mathrm{out}}_{\mathrm{om}}$ in Eq. (7) we can give an unbiased estimator
$\hat{F}$ of the force $F$, as

$\displaystyle\hat{F}=\frac{1}{\sqrt{\gamma_{\mathrm{m}}}\chi_{\mathrm{m}}\,g\sqrt{\kappa_{\mathrm{om}}}\chi_{\mathrm{om}}}p_{\mathrm{om}}^{\mathrm{out}}=F+F_{\mathrm{\scriptscriptstyle{add}}},$
(10)

where the additional force noise, added by the measurement light, is defined
as

$\displaystyle\begin{split}F_{\mathrm{\scriptscriptstyle{add}}}&=F_{\mathrm{\scriptscriptstyle{th}}}+\frac{e^{i\phi}}{\sqrt{\gamma_{\mathrm{m}}}\chi_{\mathrm{m}}\,g\sqrt{\kappa_{\mathrm{om}}}\chi_{\mathrm{om}}}p_{\mathrm{om}}^{\mathrm{in}}\\\
&\quad-\frac{g\sqrt{\kappa_{\mathrm{om}}}\chi_{\mathrm{om}}}{\sqrt{\gamma_{\mathrm{m}}}}x_{\mathrm{om}}^{\mathrm{in}}.\end{split}$
(11)

To characterize the sensitivity of the force measurement, we use the (power)
spectral density of the added noise defined by

$S_{F}(\omega)\delta(\omega-\omega^{\prime})=\frac{1}{2}\left(\Bigl{\langle}F_{\mathrm{\scriptscriptstyle{add}}}(\omega)F_{\mathrm{\scriptscriptstyle{add}}}(-\omega^{\prime})\Bigr{\rangle}+\mathrm{c}.\mathrm{c}.\right).$
(12)

Assuming uncorrelated amplitude and phase quadratures, the added noise
spectral density is

$S_{F}=\frac{k_{B}T}{\hbar\omega_{\mathrm{m}}}+\frac{1}{2G_{\mathrm{om}}\gamma_{\mathrm{m}}\lvert\chi_{\mathrm{m}}\rvert^{2}}+\frac{G_{\mathrm{om}}}{2\gamma_{\mathrm{m}}},$
(13)

where we defined the frequency dependent measurement strength as

$\displaystyle\begin{split}G_{\mathrm{om}}(\omega)&=g^{2}\kappa_{\mathrm{om}}\lvert\chi_{\mathrm{om}}(\omega)\rvert^{2}=\Gamma_{\mathrm{om}}\;\frac{(\frac{\kappa_{\mathrm{om}}}{2})^{2}}{\omega^{2}+(\frac{\kappa_{\mathrm{om}}}{2})^{2}},\end{split}$
(14)

with Lorentzian shape and maximum
$\Gamma_{\mathrm{om}}=\frac{4\,g^{2}}{\kappa_{\mathrm{om}}}$. In this form,
the noise spectral density (13) is dimensionless. To arrive at a force noise
spectral density in units of $\mathrm{N}^{2}/\mathrm{Hz}$, one has to re-scale
it, such that $S_{F}(\omega)=\hbar
m\gamma_{\mathrm{m}}\omega_{\mathrm{m}}S_{F}(\omega)$ for a given
optomechanical force sensor [30]. The terms in Eq. (13) are thermal noise due
to Brownian motion of the mechanical oscillator (first term), shot noise in
the phase quadrature (second term) and backaction noise from the amplitude
quadrature (third term). The thermal noise adds a flat background to the force
sensitivity, which is independent of the measurement rate $G_{\mathrm{om}}$
and frequency. Throughout this paper, we will assume that the thermal noise is
either dominated by backaction noise [30] or is suppressed by cooling of the
mechanical mode and therefore neglect this term.

The shot noise term scales inversely proportional to the measurement rate
$G_{\mathrm{om}}$ and thus inversely proportional to the power, as
$G_{\mathrm{om}}\propto P$. Additionally, the backaction noise is proportional
to $G_{\mathrm{om}}$. This implies that an optimal power value, which
minimizes Eq. (13), exists for each frequency. Minimizing Eq. (13) with
respect to $G_{\mathrm{om}}$ for all frequencies gives the achievable lower
bound

$S_{F}(\omega)\geq\frac{1}{\gamma_{m}\lvert\chi_{\mathrm{m}}(\omega)\rvert}\equiv
S_{\mathrm{\scriptscriptstyle{SQL}}}(\omega),$ (15)

which is the standard quantum limit (SQL). The optimal measurement rate
required is

$G_{\mathrm{\scriptscriptstyle{SQL}}}(\omega)=\frac{1}{\lvert\chi_{\mathrm{m}}(\omega)\rvert}.$
(16)

Next, we consider the NMO. On resonance, $\Delta_{\mathrm{c}}=0$, and for
$g_{\mathrm{\scriptscriptstyle{BS}}}=g_{\mathrm{\scriptscriptstyle{DC}}}=\frac{1}{2}g_{\mathrm{a}}$
the output quadratures read

$\displaystyle\begin{split}x^{\mathrm{out}}_{\mathrm{c}}&=e^{i\theta}x^{\mathrm{in}}_{\mathrm{c}},\\\
p^{\mathrm{out}}_{\mathrm{c}}&=e^{i\theta}p^{\mathrm{in}}_{\mathrm{c}}-\chi_{\mathrm{a}}g_{\mathrm{a}}^{2}\kappa_{\mathrm{c}}\chi_{\mathrm{c}}^{2}x^{\mathrm{in}}_{\mathrm{c}}\\\
&\quad+\chi_{\mathrm{a}}\sqrt{\kappa_{\mathrm{c}}}g_{\mathrm{a}}\chi_{\mathrm{c}}\sqrt{\kappa_{\mathrm{a}}}\left(\frac{\kappa_{\mathrm{a}}/2+i\omega}{\Delta_{\mathrm{a}}}x_{\mathrm{a}}^{\mathrm{in}}+p_{\mathrm{a}}^{\mathrm{in}}\right),\end{split}$
(17)

where
$e^{i\theta}=(\frac{\kappa_{\mathrm{c}}}{2}-i\omega)/(\frac{\kappa_{\mathrm{c}}}{2}+i\omega)$.
We have defined susceptibilities of the meter cavity $\chi_{\mathrm{c}}$ and
the ancilla cavity $\chi_{\mathrm{a}}$ as

$\displaystyle\chi_{\mathrm{c}}$
$\displaystyle=[i\omega+\frac{\kappa_{\mathrm{c}}}{2}]^{-1},$ (18)
$\displaystyle\chi_{\mathrm{a}}$
$\displaystyle=\Delta_{\mathrm{a}}[(\omega^{2}-\Delta_{\mathrm{a}}^{2}-\frac{\kappa_{\mathrm{a}}^{2}}{4})-i\kappa_{\mathrm{a}}\omega]^{-1}.$
(19)

The aim of our setup is to couple the two systems in a manner that the
backaction noise in the force spectrum will cancel, hence allowing for a sub-
SQL performance. In our dual-cavity setup, this is done by cascading the two
systems and matching the parameters of the NMO accordingly.

As seen in Fig. 1(b), the whole scheme has two possible arrangements. Thus, to
cascade the systems, we choose
$x_{\mathrm{c}}^{\mathrm{out}}=x_{\mathrm{om}}^{\mathrm{in}}$ and
$p_{\mathrm{c}}^{\mathrm{out}}=p_{\mathrm{om}}^{\mathrm{in}}$ for the case
$\mathrm{NMO}\mapsto\mathrm{OMS}$ and
$x_{\mathrm{om}}^{\mathrm{out}}=x_{\mathrm{c}}^{\mathrm{in}}$ and
$p_{\mathrm{om}}^{\mathrm{out}}=p_{\mathrm{c}}^{\mathrm{in}}$ for
$\mathrm{OMS}\mapsto\mathrm{NMO}$. For ideal CQNC, the order will not matter.
We will discuss cases that depend on the order further below. After cascading
the two systems, we can again identify the additional force noise as in Eq.
(10) and derive the added noise spectral density

$\displaystyle\begin{split}S_{F}&=\frac{1}{2G_{\mathrm{om}}\gamma_{\mathrm{m}}|\chi_{\mathrm{m}}|^{2}}\\\
&\quad+\frac{G_{\mathrm{a}}^{2}\lvert\chi_{\mathrm{a}}\rvert^{2}+G_{\mathrm{om}}^{2}\lvert\chi_{\mathrm{m}}\rvert^{2}+2\,G_{\mathrm{a}}G_{\mathrm{om}}\Re(\chi_{\mathrm{m}}\chi_{\mathrm{a}}^{*})}{2G_{\mathrm{om}}\gamma_{\mathrm{m}}|\chi_{\mathrm{m}}|^{2}}\\\
&\quad+\frac{G_{\mathrm{a}}\kappa_{\mathrm{a}}|\chi_{\mathrm{a}}|^{2}}{2G_{\mathrm{om}}\gamma_{\mathrm{m}}|\chi_{\mathrm{m}}|^{2}}\left(\frac{\omega^{2}+\kappa_{\mathrm{a}}^{2}/4}{\Delta_{\mathrm{a}}^{2}}+1\right),\end{split}$
(20)

see Appendix B for details. Analogous to the measurement strength of the OMS
in Eq. (14), we have defined the frequency dependent measurement strength of
the NMO,

$\displaystyle\begin{split}G_{\mathrm{a}}(\omega)&=g_{\mathrm{a}}^{2}\kappa_{\mathrm{c}}\lvert\chi_{\mathrm{c}}(\omega)\rvert^{2}=\Gamma_{\mathrm{a}}\;\frac{(\frac{\kappa_{\mathrm{c}}}{2})^{2}}{\omega^{2}+(\frac{\kappa_{\mathrm{c}}}{2})^{2}},\end{split}$
(21)

with $\Gamma_{\mathrm{a}}=\frac{4\,g_{\mathrm{a}}^{2}}{\kappa_{\mathrm{c}}}$.
The terms in Eq. (20) are shot-noise (first term), backaction noise (second
term) and shot noise from the ancilla cavity (last term). The backaction noise
is then cancelled if the conditions are such that

$\displaystyle g_{\mathrm{\scriptscriptstyle{BS}}}$
$\displaystyle=g_{\mathrm{\scriptscriptstyle{DC}}}=\frac{1}{2}g_{\mathrm{a}},$
(22) $\displaystyle G_{\mathrm{om}}(\omega)$
$\displaystyle=G_{\mathrm{a}}(\omega),$ (23)
$\displaystyle\chi_{\mathrm{m}}(\omega)$
$\displaystyle=-\chi_{\mathrm{a}}(\omega),$ (24)

for all $\omega$. This means that the ancilla cavity should couple to the
light with the same strength as the PMO, but the response to the force signal
should be opposite to the PMO, hence it behaves as an effective negative mass.
Considering the explicit form of Eqs. (9) and (19), condition (24) entails
further restrictions:

  1. 1.

The detuning of the ancilla cavity to the meter cavity is

$\Delta_{\mathrm{a}}=-\omega_{\mathrm{m}},$ (24a)

which effectively moves the ancilla cavity to the negative mass frame.

  2. 2.

The linewidth of the ancilla cavity $\kappa_{\mathrm{a}}$ should match the
damping rate of the mechanical oscillator

$\kappa_{\mathrm{a}}=\gamma_{\mathrm{m}},$ (24b)

to mimic the oscillating behaviour of the PMO.

  3. 3.

The susceptibilities $\chi_{\mathrm{m}}$ and $\chi_{\mathrm{a}}$ differ by a
factor $\kappa_{\mathrm{a}}^{2}/4$. To alleviate this, the detuning
$\lvert\Delta_{\mathrm{a}}\rvert\gg\kappa_{\mathrm{a}}$, and together with
forgoing points this implies the resolved sideband limit of the ancilla cavity

$\omega_{\mathrm{m}}\gg\kappa_{\mathrm{a}},$ (24c)

and a large quality factor of the mechanical oscillator,

$Q_{\mathrm{m}}=\frac{\omega_{\mathrm{m}}}{\gamma_{\mathrm{m}}}\gg 1.$ (24d)

These conditions are similar to the integrated setup [30], but instead of the
coupling strength $g_{\mathrm{a}}$ and $g$, the measurement strengths
$G_{\mathrm{a}}$ and $G_{\mathrm{om}}$ need to match.

Assuming conditions (22)–(24) are met, the backaction term in Eq. (20) will
vanish, and we arrive at

$S_{F}=\frac{1}{2G_{\mathrm{om}}\gamma_{\mathrm{m}}|\chi_{\mathrm{m}}|^{2}}+\frac{1}{2}\left(\frac{\omega^{2}+\gamma_{\mathrm{m}}^{2}/4}{\omega_{\mathrm{m}}^{2}}+1\right),$
(25)

which contains only shot-noise contributions of the measured phase quadrature
and the ancilla cavity. The contribution of the OMS is sometimes referred to
as the fundamental quantum limit (FQL)[9], energetic quantum limit [42] or the
quantum Cramér-Rao bound [43, 44, 45]. In the limit of large measurement
strength, we arrive at the lower bound

$\displaystyle
S_{F}(\omega)\geq\frac{1}{2}\left(\frac{\omega^{2}+\gamma_{\mathrm{m}}^{2}/4}{\omega_{\mathrm{m}}^{2}}+1\right)\equiv
S_{\mathrm{\scriptscriptstyle{CQNC}}}(\omega).$ (26)

Combining Eqs. (15) and (26) we find

$\displaystyle
S_{\mathrm{\scriptscriptstyle{CQNC}}}=S_{\mathrm{\scriptscriptstyle{SQL}}}\times\frac{1}{2Q_{\mathrm{m}}}\left(\frac{\omega^{2}+\gamma_{\mathrm{m}}^{2}/4+\omega_{\mathrm{m}}^{2}}{\sqrt{(\omega^{2}-\omega_{\mathrm{m}}^{2})^{2}+\gamma_{\mathrm{m}}^{2}\omega^{2}}}\right).$
(27)

Thus, for $Q_{\mathrm{m}}\gg 1$, we can summarize

$\displaystyle
S_{\mathrm{\scriptscriptstyle{CQNC}}}=S_{\mathrm{\scriptscriptstyle{SQL}}}\times\begin{cases}\quad
1&\text{on resonance }\omega=\omega_{\mathrm{m}}\\\
1/(2Q_{\mathrm{m}})&\text{off resonance }\omega\,\begin{subarray}{c}\ll\\\
\gg\end{subarray}\,\omega_{\mathrm{m}}.\end{cases}$ (28)

In conclusion, under the additional condition that
$G_{\mathrm{om}}=G_{\mathrm{a}}$, the cascaded setup reproduces the same
findings of [30], leading to an enhancement in performance up to a factor of
$2Q_{\mathrm{m}}$ off-resonance and SQL performance on resonance.

##  IV Imperfect CQNC

Conditions (22)–(24) are the ideal case for a perfect cancellation of
backaction noise and will not be satisfied in an actual experiment. Therefore,
we will discuss possible imperfections and their impact on the performance of
our cascaded scheme. These imperfections include mismatches to the parameters
in Eqs. (22)–(24), and possible losses. Another degree of freedom of our setup
is the order in which the light passes through the sub-systems (i.e.
$\mathrm{NMO}\mapsto\mathrm{OMS}$ or $\mathrm{OMS}\mapsto\mathrm{NMO}$), but
this will only affect the force sensing performance for imperfections that
directly affect the force signal. Hence, we split our discussion into order
dependent and independent categories. Data shown in the figures of this
section refer to an OMS given by the parameters in Table 1.

Table 1: Parameters of the optomechanical sensor used in Figs. 2 to 5 Parameter | Norm. value | Value
---|---|---
$\omega_{\mathrm{m}}$ | mechanical resonance frequency | 1 | 500 kHz
$\gamma_{\mathrm{m}}$ | mechanical linewidth | $10^{-3}\,\omega_{\mathrm{m}}$ | 500 Hz
$Q_{\mathrm{m}}$ | mechanical quality factor | $\frac{\omega_{\mathrm{m}}}{\gamma_{\mathrm{m}}}$ | 1000
$\kappa_{\mathrm{om}}$ | optomechanical cavity linewidth | $10\,\omega_{\mathrm{m}}$ | 5 MHz

###  IV.1 Order-independent imperfections

The parameters discussed in this subsection will impede the cancellation of
backaction noise and, as a result, limit the CQNC performance but will not
affect the force signal. Hence, the possible CQNC performance in the face of
these imperfections will not depend on the system order.

####  IV.1.1 Non-ideal ancilla cavity linewidth
$\kappa_{\mathrm{a}}\neq\gamma_{\mathrm{m}}$

The strictest requirement for an all-optical CQNC setup is to match the
ancilla cavity linewidth to the damping rate of the mechanical oscillator.
Assuming all conditions for ideal CQNC are matched, except
$\kappa_{\mathrm{a}}\neq\gamma_{\mathrm{m}}$. Since the measurement strengths,
$G_{\mathrm{om}}=G_{\mathrm{a}}$ are matched for all frequencies, and we
assume no propagation losses between the system, this effectively reduces to
the integrated CQNC setup [30]. The spectral density of added noise (20) in
this case becomes

$\displaystyle\begin{split}S_{F}&=\frac{1}{2G_{\mathrm{om}}\gamma_{\mathrm{m}}|\chi_{\mathrm{m}}|^{2}}+\frac{G_{\mathrm{om}}}{2\gamma_{\mathrm{m}}}\left\lvert\frac{\chi_{\mathrm{m}}+\chi_{\mathrm{a}}}{\chi_{m}}\right\rvert^{2}\\\
&\quad+\frac{\kappa_{\mathrm{a}}\lvert\chi_{\mathrm{a}}\rvert^{2}}{2\gamma_{\mathrm{m}}\lvert\chi_{\mathrm{m}}\rvert^{2}}\left(\frac{\omega^{2}+\kappa_{\mathrm{a}}^{2}/4}{\omega_{\mathrm{m}}^{2}}+1\right).\end{split}$
(29)

For an optimal $G_{\mathrm{om}}$, we find the minimal spectral density for the
added noise,

$\displaystyle
S_{F}=\frac{\lvert\chi_{\mathrm{m}}+\chi_{\mathrm{a}}\rvert}{\gamma_{\mathrm{m}}\lvert\chi_{\mathrm{m}}\rvert}+\frac{\kappa_{\mathrm{a}}\lvert\chi_{\mathrm{a}}\rvert^{2}}{2\gamma_{\mathrm{m}}\lvert\chi_{\mathrm{m}}\rvert^{2}}\left(\frac{\omega^{2}+\kappa_{\mathrm{a}}^{2}/4}{\omega_{\mathrm{m}}^{2}}+1\right)$
(30)

This is composed of measurement shot and backaction noise (first term) and
noise introduced by the ancilla cavity (second term). The second term will
dominate the first one for frequencies off-resonance, setting a bound to the
achievable performance. The ratio between the spectral density (30) and the
SQL is

$\displaystyle S_{F}=\frac{\kappa_{\mathrm{a}}}{2\,\omega_{\mathrm{m}}}\times
S_{\mathrm{\scriptscriptstyle{SQL}}},$ (31)

for $\kappa_{\mathrm{a}}<\omega_{\mathrm{m}}$. For
$\kappa_{\mathrm{a}}\gg\omega_{\mathrm{m}}$, the effect of CQNC will vanish
for low frequencies, converging to the SQL, while for large frequencies, the
added noise is larger than the SQL. This is illustrated in Fig. 2.

Figure 2: Force noise for a mismatch between ancilla cavity linewidth
$\kappa_{\mathrm{a}}$ and damping rate of the mechanical oscillator
$\gamma_{\mathrm{m}}$. For $\kappa_{\mathrm{a}}<\omega_{\mathrm{m}}$ an
improvement of $\kappa_{\mathrm{a}}/2\omega_{\mathrm{m}}$ can be achieved off-
resonance (solid green). For $\kappa_{\mathrm{a}}\gg\omega_{\mathrm{m}}$, the
effect of CQNC is completely cancelled for low frequencies, and the
sensitivity is worse than the SQL for high frequencies (red). The shaded areas
mark the bounds for sub-SQL sensitivity, from below the fundamental limit
given by Eq. (26), from above the SQL given by Eq. (15). Parameters are given
in Table 1

####  IV.1.2 Unequal measurement strengths $G_{\mathrm{om}}\neq
G_{\mathrm{a}}$

Next, we consider a mismatch of the measurement strengths $G_{\mathrm{a}}\neq
G_{\mathrm{om}}$, while matching the other CQNC conditions. This entails
unmatched cavity linewidth $\kappa_{\mathrm{c}}\neq\kappa_{\mathrm{om}}$ and
unmatched couplings $g_{\mathrm{a}}\neq g$. Introducing parameters for the
linewidth mismatch $\kappa_{\mathrm{c}}=\epsilon\,\kappa_{\mathrm{om}}$ and
coupling mismatch $g_{\mathrm{a}}=\sqrt{\delta}g$, we find for the spectral
density

$\displaystyle\begin{split}S_{F}&=\frac{1}{2G_{\mathrm{om}}\gamma_{\mathrm{m}}|\chi_{\mathrm{m}}|^{2}}+\frac{G_{\mathrm{om}}}{2\gamma_{\mathrm{m}}}\left\lvert
1-\delta\epsilon\frac{\lvert\chi_{\mathrm{c}}\rvert^{2}}{\lvert\chi_{\mathrm{om}}\rvert^{2}}\right\rvert^{2}\\\
&\quad+\delta\epsilon\frac{\lvert\chi_{\mathrm{c}}\rvert^{2}}{\lvert\chi_{\mathrm{om}}\rvert^{2}}\frac{1}{2}\left(\frac{\omega^{2}+\gamma_{\mathrm{m}}^{2}/4}{\omega_{\mathrm{m}}^{2}}+1\right).\end{split}$
(32)

For suitable couplings $g$ and cavity linewidth $\kappa$, we can find a
frequency where the backaction term in Eq. (32) will vanish, and ideal CQNC is
possible. This is the case when the Lorentzians $G_{\mathrm{a}}$ and
$G_{\mathrm{om}}$ are such that they will intersect at a frequency $\omega\neq
0$. We find that

$\displaystyle\omega_{*}=\pm\sqrt{\frac{\delta\,\epsilon-\epsilon^{2}}{1-\delta\,\epsilon}}\frac{\kappa_{\mathrm{om}}}{2}$
(33)

is a real-valued frequency for the following parameters:

$\displaystyle g_{\mathrm{a}}=g$ $\displaystyle\quad\Rightarrow\;$
$\displaystyle\kappa_{\mathrm{c}}$ $\displaystyle<\kappa_{\mathrm{om}}$
$\displaystyle\mathrm{or}$ $\displaystyle\kappa_{\mathrm{c}}$
$\displaystyle>\kappa_{\mathrm{om}},$ (34a) $\displaystyle g_{\mathrm{a}}<g$
$\displaystyle\quad\Rightarrow\;$ $\displaystyle\epsilon$
$\displaystyle>\frac{1}{\delta}$ $\displaystyle\mathrm{or}$
$\displaystyle\epsilon$ $\displaystyle<\delta,$ (34b) $\displaystyle
g_{\mathrm{a}}>g$ $\displaystyle\quad\Rightarrow\;$ $\displaystyle\epsilon$
$\displaystyle>\delta$ $\displaystyle\mathrm{or}$ $\displaystyle\epsilon$
$\displaystyle<\frac{1}{\delta}.$ (34c)

Consequently, a cavity linewidth mismatch can compensate for every possible
matching condition of the couplings, and ideal CQNC can be achieved at
$\omega_{*}$.

For non-vanishing backaction, we can again minimize the spectral density (32)
with an optimal $G_{\mathrm{om}}$. Turning to the low-frequency limit
($\kappa_{\mathrm{c},\mathrm{om}}\gg\omega$), the measurement strengths become
frequency independent,
$G_{\mathrm{om},\mathrm{a}}\to\Gamma_{\mathrm{om},\mathrm{a}}$, and the ratio
$\lvert\chi_{\mathrm{c}}\rvert^{2}/\lvert\chi_{\mathrm{om}}\rvert^{2}\to
1/\epsilon^{2}$. The minimal noise spectral density is then

$\displaystyle S_{F}=|1-\frac{\delta}{\epsilon}|\times
S_{\mathrm{\scriptscriptstyle{SQL}}}+\left(\frac{\delta}{\epsilon}\right)\times
S_{\mathrm{\scriptscriptstyle{CQNC}}}.$ (35)

Ideal CQNC can be recovered for $\epsilon=\delta$, which means
$\Gamma_{\mathrm{om}}=\Gamma_{\mathrm{a}}$. Hence, as long as the rate at
which the backaction information leaks out of the system is matched, ideal
CQNC is possible.

For the converse case ($\kappa_{\mathrm{c},\mathrm{om}}\ll\omega$) the cavity
susceptibilities
$\lvert\chi_{\mathrm{c}}\rvert^{2}\approx\lvert\chi_{\mathrm{om}}\rvert^{2}$,
effectively cancelling in Eq. (32). The optimal spectral density becomes

$\displaystyle S_{F}=|1-\delta\,\epsilon|\times
S_{\mathrm{\scriptscriptstyle{SQL}}}+\left(\delta\,\epsilon\right)\times
S_{\mathrm{\scriptscriptstyle{CQNC}}}.$ (36)

In this case, ideal CQNC can be recovered for $\delta=1/\epsilon$, which
entails $g_{\mathrm{a}}^{2}\kappa_{c}=g^{2}\kappa_{\mathrm{om}}$. We have
depicted our main findings in Fig. 3.

We also considered a combination of the imperfections discussed in this
section. If, for example, $G_{\mathrm{om}}\neq G_{\mathrm{a}}$ and
additionally $\kappa_{\mathrm{a}}\neq\gamma_{\mathrm{m}}$, the noise spectral
density will be a combination of Eqs. (29) and 32. In this case, the
cancellation of backaction noise is possible for the cases discussed above,
but the ancilla cavity noise floor is higher because of the linewidth mismatch
$\gamma_{\mathrm{m}}\neq\kappa_{\mathrm{a}}$. Thus, our findings will remain
the same, but the achievable performance off-resonance is given by the noise
spectral density (31).

Figure 3: Force noise for imperfect matching of measurement strength. For
mismatched coupling strength compensated by linewidth mismatch, perfect noise
cancellation can be recovered at low frequencies (solid green for
$\epsilon=\delta=0.9$). For matched linewidth but mismatched coupling
strength, noise cancellation is limited, but sub-SQL performance is possible
(dashed red for $\delta=0.9$). In the case of matched coupling strength but
mismatched linewidth, we find a frequency (33) where perfect noise
cancellation is possible (dash-dotted purple line, with $\epsilon=0.9$). The
shaded areas mark the bounds for sub-SQL sensitivity, from below the
fundamental limit given by Eq. (26) and from above the SQL given by Eq. (15).
Parameters are given in Table 1.

###  IV.2 Order-dependent imperfections

The parameters discussed in this subsection, namely losses, not only hamper
the cancellation of backaction noise but also affect the force signal
directly.

####  IV.2.1 Losses

We first consider propagation losses, which occur between the first and the
second system. The propagation losses are modelled via mixing the output
signal of the first system with vacuum in a beam-splitter-like interaction.
This leads to a modified output signal [8],

$\displaystyle
x_{\mathrm{out}}^{\prime}=\sqrt{\eta}\,x_{\mathrm{out}}+\sqrt{1-\eta}\,x_{\mathrm{\scriptscriptstyle{vac}}},$
(37)

where $x_{\mathrm{\scriptscriptstyle{vac}}}$ represents the vacuum field and
$\eta\in[0,1]$ is the efficiency of the process. Due to this additional noise,
information about the backaction interaction of the first system is lost to
the vacuum, hence perfect cancellation of backaction noise is not possible. As
before, we can find an optimal coupling strength to minimize the additional
noise. For the system order $\mathrm{NMO}\mapsto\mathrm{OMS}$, we achieve a
minimal spectral density off-resonance

$\displaystyle S_{F}=\sqrt{1-\eta}\times
S_{\mathrm{\scriptscriptstyle{SQL}}}.$ (38)

In the opposite order $\mathrm{OMS}\mapsto\mathrm{NMO}$, additionally to the
loss of backaction information, some force signal will be lost due to
propagation losses. Hence, the added noise will increase for this topology. We
find

$\displaystyle S_{F}=\frac{\sqrt{1-\eta}}{\eta}\times
S_{\mathrm{\scriptscriptstyle{SQL}}},$ (39)

for the minimal spectral density off-resonance. The spectral density is
increased by $1/\eta$ compared to the case $\mathrm{NMO}\mapsto\mathrm{OMS}$
and hence directly proportional to the lost force signal. Losses after the
second system constitute the detection efficiency and can be modelled
similarly. Since this will not affect the cancellation of backaction noise, we
will omit detection losses for now.

Apart from propagation losses, we take intracavity losses into account.
Introducing a Markovian bath for each cavity, with coupling rates
$\kappa_{\mathrm{c}}^{\mathrm{\scriptscriptstyle{bath}}}$ and
$\kappa_{\mathrm{om}}^{\mathrm{\scriptscriptstyle{bath}}}$, the intracavity
losses can be described in terms of the escape efficiencies

$\displaystyle\eta_{\mathrm{om},\mathrm{c}}^{\mathrm{\scriptscriptstyle{esc}}}=\frac{\kappa^{\mathrm{in}}_{\mathrm{om},\mathrm{c}}}{\kappa^{\mathrm{in}}_{\mathrm{om},\mathrm{c}}+\kappa^{\mathrm{\scriptscriptstyle{bath}}}_{\mathrm{om},\mathrm{c}}}=\frac{\kappa^{\mathrm{in}}_{\mathrm{om},\mathrm{c}}}{\kappa_{\mathrm{om},\mathrm{c}}}.$
(40)

Similar to propagation losses, introducing intracavity losses will always
impede the cancellation of backaction noise, and depending on the order of the
systems, the available force signal information will differ. For the case
$\mathrm{NMO}\mapsto\mathrm{OMS}$, with optimal measurement strength, we find
the minimal spectral density

$\displaystyle
S_{F}=\frac{\sqrt{\eta^{\mathrm{\scriptscriptstyle{esc}}}_{c}+\eta^{\mathrm{\scriptscriptstyle{esc}}}_{\mathrm{om}}-2\eta^{\mathrm{\scriptscriptstyle{esc}}}_{c}\eta^{\mathrm{\scriptscriptstyle{esc}}}_{\mathrm{om}}}}{\eta^{\mathrm{\scriptscriptstyle{esc}}}_{\mathrm{om}}}\times
S_{\mathrm{\scriptscriptstyle{SQL}}}.$ (41)

This encompasses both cases with propagation loss, for
$\eta^{\mathrm{\scriptscriptstyle{esc}}}_{\mathrm{om}}\to 1$ we retrieve Eq.
(38) and for $\eta^{\mathrm{\scriptscriptstyle{esc}}}_{\mathrm{c}}\to 1$ Eq.
(39) respectively. Thus, for the configuration
$\mathrm{NMO}\mapsto\mathrm{OMS}$, the intracavity loss can be handled
similarly to propagation loss.

For the case $\mathrm{OMS}\mapsto\mathrm{NMO}$, we lose additional force
signal due to intrinsic loss in the meter cavity. Moreover, the signal also
picks up additional information of the phase quadrature. We arrive at the
minimal spectral density,

$\displaystyle S_{F}$
$\displaystyle=\frac{\sqrt{\eta^{\mathrm{\scriptscriptstyle{esc}}}_{\mathrm{c}}+\eta^{\mathrm{\scriptscriptstyle{esc}}}_{\mathrm{om}}-2\eta^{\mathrm{\scriptscriptstyle{esc}}}_{\mathrm{c}}\eta^{\mathrm{\scriptscriptstyle{esc}}}_{\mathrm{om}}}}{\eta^{\mathrm{\scriptscriptstyle{esc}}}_{\mathrm{om}}\,|1-\eta^{\mathrm{\scriptscriptstyle{esc}}}_{\mathrm{c}}\kappa_{\mathrm{c}}\chi_{\mathrm{c}}|^{2}}\times
S_{\mathrm{\scriptscriptstyle{SQL}}}.$ (42)

The term
$|1-\eta^{\mathrm{\scriptscriptstyle{esc}}}_{\mathrm{c}}\kappa_{\mathrm{c}}\chi_{\mathrm{c}}|^{2}$
describes the meter cavity’s phase and noise contribution. Due to its
dependence on the meter cavity susceptibility $\chi_{\mathrm{c}}$, this
difference is frequency-dependent and will vanish for frequencies
$\omega>\kappa_{c}$. For low frequencies, it will be at a maximal value of
$|1-2\eta^{\mathrm{\scriptscriptstyle{esc}}}_{\mathrm{c}}|^{2}$, making
intracavity losses extra punishing for configuration
$\mathrm{OMS}\mapsto\mathrm{NMO}$. We see that introducing losses is
detrimental to the possible noise reduction. As losses will never be
avoidable, the system order $\mathrm{NMO}\mapsto\mathrm{OMS}$ should always be
preferable since higher levels of noise reduction are achieved.

####  IV.2.2 Relative mismatch of $g_{\mathrm{\scriptscriptstyle{BS}}}$ and
$g_{\mathrm{\scriptscriptstyle{DC}}}$

Figure 4: Flow chart between the mode of the ancilla and the meter cavity. The
solid line shows the original backaction flow, and the dashed line shows the
noise introduced by the relative mismatch $g_{\mathrm{r}}$ of the beam-
splitter and down-conversion coupling.

In addition to losses, a relative mismatch between the beam-splitter coupling
$g_{\mathrm{\scriptscriptstyle{BS}}}$ and down-conversion coupling
$g_{\mathrm{\scriptscriptstyle{DC}}}$ will also affect the noise cancellation
depending on the system order. So far, we assumed
$g_{\mathrm{\scriptscriptstyle{BS}}}=g_{\mathrm{\scriptscriptstyle{DC}}}=1/2\,g_{\mathrm{a}}$,
in order to mimic the backaction interaction of the OMS. We will now fix
$g_{\mathrm{\scriptscriptstyle{BS}}}+g_{\mathrm{\scriptscriptstyle{DC}}}=g$
and introduce a mismatch between the beam-splitter and down-conversion
couplings

$\displaystyle\frac{g_{\mathrm{\scriptscriptstyle{BS}}}-g_{\mathrm{\scriptscriptstyle{DC}}}}{g_{\mathrm{\scriptscriptstyle{BS}}}+g_{\mathrm{\scriptscriptstyle{DC}}}}=g_{\mathrm{r}}\neq
0.$ (43)

As shown in Fig. 4, the relative mismatch $g_{\mathrm{r}}$ allows the phase
quadratures to couple back into the amplitude quadrature and thus deviate from
the backaction interaction of the OMS. This introduces a noise path and will
limit the cancellation of backaction noise. It also affects the force noise
differently for the different system orders. For the case
$\mathrm{OMS}\mapsto\mathrm{NMO}$, the force signal is imprinted on the output
phase quadrature of the OMS, and with the introduced mismatch, it is possible
for the signal to couple to the amplitude quadrature. Contrary, for
$\mathrm{NMO}\mapsto\mathrm{OMS}$, the force signal will remain fully in the
output phase quadrature. Thus, this results in different spectral noise
densities for our phase measurement. For general mismatches, this will not
reduce to a simple expression. The resulting spectral densities were
calculated numerically and are shown in Fig. 5. The CQNC performance is
limited for low frequencies, but sub-SQL levels are still possible. Contrary
to losses, the order $\mathrm{OMS}\mapsto\mathrm{NMO}$ seems advantageous for
a relative mismatch of the couplings. CQNC will vanish entirely in the high-
frequency limit, and no sub-SQL performance is possible.

Figure 5: Force noise for relative mismatch of beam-splitter and down-
conversion coupling. (a) Plotted for $g_{\mathrm{r}}=0.2$. (b) Plotted for
$g_{\mathrm{r}}=-0.2$. The relative mismatch introduces additional noise by
modifying the effective backaction term of the NMO. Perfect noise cancellation
is not possible, but sub-SQL levels are achievable for low frequencies. For
high frequency, no noise reduction is possible. Traversing through the OMS
first seems advantageous for noise cancellation. The shaded areas mark the
bounds for sub-SQL sensitivity, from below the fundamental limit given by Eq.
(26) and from above the SQL given by Eq. (15). Parameters are given in Table
1.

##  V case study

After discussing ideal CQNC and the most relevant deviations from the ideal
parameters, we now turn to a realistic situation one would expect in an actual
experiment. For an integrated setup, reasonable parameters have been discussed
in [30], which were revised in [40] for a cascaded setup, and two reasonable
sets of parameters were given. From there, we found a new set of parameters
which achieve broadband noise reduction for frequencies below the mechanical
resonance of the oscillator. Losses are of particular interest in our case
study, as they influence the noise reduction depending on the system order.
Our set of parameters is shown in Table 2.

Table 2: Proposed set of parameters Parameter | Norm. value | Value/$2\pi$
---|---|---
$\omega_{\mathrm{m}}$ | mechanical resonance frequency | 1 | 500 kHz
$\gamma_{\mathrm{m}}$ | mechanical linewidth | $10^{-8}\,\omega_{\mathrm{m}}$ | 5 mHz
$\kappa_{\mathrm{c}}$ | meter cavity linewidth | $4\,\omega_{\mathrm{m}}$ | 2 MHz
$\Delta_{\mathrm{a}}$ | ancilla cavity detuning | $-0.99\,\omega_{m}$ | -495 kHz
$\kappa_{\mathrm{a}}$ | ancilla cavity linewidth | $\frac{2}{5}\,\omega_{\mathrm{m}}$ | 200 kHz
$g_{\mathrm{\scriptscriptstyle{BS}}}$ | beam-splitter coupling strength | $1.01\,\frac{g}{2}$ | 253 kHz
$g_{\mathrm{\scriptscriptstyle{DC}}}$ | down-conversion coupling strength | $0.97\,\frac{g}{2}$ | 243 kHz
$\eta^{\mathrm{\scriptscriptstyle{esc}}}_{\mathrm{c}}$ | escape efficiency NMO | $0.9$ |
$\kappa_{\mathrm{om}}$ | optomechanical cavity linewidth | $0.99\,\kappa_{\mathrm{c}}$ | 1.98 MHz
$g$ | optomechanical coupling strength | $\omega_{m}$ | 500 kHz
$\eta^{\mathrm{\scriptscriptstyle{esc}}}_{\mathrm{om}}$ | escape efficiency OMS | $0.9$ |
$\eta_{\mathrm{\scriptscriptstyle{prop}}}$ | propagation efficiency | $0.97$ |
$\eta_{\det}$ | detection efficiency | $0.97$ |

The OMS must be limited by quantum backaction noise to measure the possible
cancellation of backaction noise. For this, the quantum backaction noise in
Eq. (13) must be much larger than the thermal noise. In the low-frequency
limit $(\kappa_{\mathrm{om}}\gg\omega)$, this can be expressed in terms of the
quantum cooperativity, as

$\displaystyle
C_{\mathrm{q}}=\frac{\Gamma_{\mathrm{om}}}{\gamma_{\mathrm{m}}}\frac{\hbar\,\omega_{\mathrm{m}}}{k_{B}T}=\frac{4g^{2}\hbar}{\kappa_{\mathrm{om}}k_{B}T}\,Q_{\mathrm{m}}\gg
1.$ (44)

Modern silicon-nitride membranes have exceeded quality factors of
$Q_{\mathrm{m}}=10^{8}$ [46], thus the OMS would be quantum backaction limited
for a temperature $T=4\text{ K}$, a temperature achievable with cryogenics.
For higher temperatures, the quality factor must be increased to elevate the
backaction effects over the thermal noise floor, and similarly, lower
temperatures allow for a lower quality factor. In order to account for this
and compare all OMS of frequency $\omega_{\mathrm{m}}$, once they can resolve
the quantum backaction, we normalize our force noise by the quality factor
$Q_{\mathrm{m}}$.

Matching most parameters, such as ancilla cavity detuning
$\Delta_{\mathrm{a}}$ and the cavity linewidth $\kappa_{\mathrm{c}}$ and
$\kappa_{\mathrm{om}}$ should not be a problem; we assume them to be closely
matched. More delicate to match are the coupling strengths. A down-conversion
coupling of $g_{\mathrm{\scriptscriptstyle{DC}}}=2\pi\times 250\text{ kHz}$
and a beam-splitter coupling of $g_{\mathrm{\scriptscriptstyle{BS}}}\geq
2\pi\times 235\text{ kHz}$ were readily achieved [40], thus we set the
optomechanical coupling strength to $g=2\pi\times 500\text{ kHz}$.
Optomechanical coupling strengths of $g=2\pi\times 440\text{ kHz}$ have been
reported in micro-mechanical setups [27] and higher couplings in the order of
MHz should be possible [47]. Hence, our assumed coupling strength should be
reasonable. If these levels cannot be reached for the optomechanical coupling
strength, one could still compensate for this mismatch by the cavity
linewidths, as described in Eq. (30) and increase the performance for low
frequencies.

For a negative mass oscillator, where the two modes are not spatially
separated, as depicted in Fig. 1(a), the escape efficiency will also dictate
the achievable linewidth of the ancilla cavity. An escape efficiency of 90%
should be achievable [48], which, with a meter cavity linewidth of
$\kappa_{\mathrm{c}}=2\text{ MHz}$, makes an ancilla cavity linewidth of
$\kappa_{\mathrm{a}}=200\text{ kHz}$ possible. For the OMS, similar escape
efficiencies should be achievable. Detection efficiencies over $97\%$ were
already realized [49]. Similarly, propagation losses between the systems
should not be an issue. We assume $3\%$ losses from both propagation and
detection.

The achievable sensitivities for the parameters in Table 2 are shown in Fig.
6. In the low-frequency regime, the configuration
$\mathrm{NMO}\mapsto\mathrm{OMS}$ shows a reduction of 20% below the SQL and
almost comparable results to the integrated setup. No sub-SQL sensitivity can
be achieved for the other system order $\mathrm{OMS}\mapsto\mathrm{NMO}$. This
is not surprising, as we saw in subsection IV.2 that this configuration
suffers additional penalties from losses. We see that instead of matching the
parameters (22)–(24), the limiting factor for noise reduction in a realistic
case will be losses. Additionally, as losses will never be entirely avoidable,
choosing the right system ordering, $\mathrm{NMO}\mapsto\mathrm{OMS}$, is of
utmost importance.

Figure 6: Force noise normalised to $Q_{\mathrm{m}}$ for the parameters given
by Table 2 and temperature $T=4\text{ K}$. For low frequencies, sub-SQL
performance is possible for the integrated setup (solid green) and the case
$\mathrm{NMO}\mapsto\mathrm{OMS}$ (dashed blue). No sub-SQL levels are
possible for the case $\mathrm{OMS}\mapsto\mathrm{NMO}$ (dash-dotted orange).
The shaded area shows levels above the SQL.

##  VI Conclusion

In this work, we discussed a cascaded version of the all-optical coherent
quantum noise cancellation setup proposed by Tsang and Caves [28, 30]. Instead
of introducing the anti-noise path directly into the optomechanical cavity, we
considered an all-optical effective negative mass oscillator as a standalone
system and removed the backaction noise of the positive mass oscillator by
coupling both systems coherently via a strong drive field. Under the
conditions (22)–(24), we then rediscovered the perfect cancellation of
backaction noise. Afterwards, we discussed deviations from the ideal
conditions, including losses and the influence of the system order. We saw
that for mismatched measurement strengths, by choosing the cavity linewidth
and coupling strength in a specific way, CQNC can be recovered in the high- or
low-frequency regime, or even at a specific frequency
$\omega_{*}\neq\omega_{\mathrm{m}}$. For losses and a relative mismatch of
beam-splitter and down-conversion coupling, the system order will also affect
the noise cancellation performance. Finally, we discussed the performance of
our setup for a set of realistic parameters and showed that a quantum noise
reduction of 20% below the SQL is possible for the order
$\mathrm{NMO}\mapsto\mathrm{OMS}$ in the low-frequency regime.

We thank Jonas Junker and Bernd Schulte for fruitful discussions regarding the
experimental setup. This research was funded by the Deutsche
Forschungsgemeinschaft (Excellence Cluster QuantumFrontiers (EXC 2123 Project
ID 390837967), SFB 1227 (DQ-mat, project A05), GRK 1991) and the Quantum- and
Nano-Metrology (QUANOMET) initiative from VW-Vorab (ZN3294).

##  Appendix A Details on the Hamiltonian
$H_{\mathrm{\scriptscriptstyle{NMO}}}$

Here, we give some more details on the Hamiltonian
$H_{\mathrm{\scriptscriptstyle{NMO}}}$ in Eq. (3). The effective negative mass
oscillator consists of two optical modes with different frequencies
$\omega_{\mathrm{c}}$ and $\omega_{\mathrm{a}}$, coupled by a beam-splitter
and down-conversion process. The whole Hamiltonian reads

$\displaystyle
H_{\mathrm{\scriptscriptstyle{NMO}}}=H_{0}+H_{\mathrm{drive}}+H_{\mathrm{\scriptscriptstyle{DC}}}+H_{\mathrm{\scriptscriptstyle{BS}}},$
(45)

where the first term represents the free Hamiltonian of the optical modes,

$\displaystyle
H_{0}=\omega_{\mathrm{c}}c^{\dagger}_{\mathrm{c}}c_{\mathrm{c}}+\omega_{\mathrm{a}}a^{\dagger}a.$
(46)

The second term describes the laser drive, which drives the meter cavity mode
$c_{\mathrm{c}}$, and is given by

$\displaystyle
H_{\mathrm{drive}}=iE(e^{-i\omega_{\mathrm{\scriptscriptstyle{L}}}t}c^{\dagger}_{\mathrm{c}}-e^{i\omega_{\mathrm{\scriptscriptstyle{L}}}t}c_{\mathrm{c}}).$
(47)

Here, $\omega_{\mathrm{\scriptscriptstyle{L}}}$ denotes the laser frequency
and $E$ describes the laser field amplitude, which is given by $\lvert
E\rvert=\sqrt{\kappa_{\mathrm{c}}P/\hbar\omega_{\mathrm{\scriptscriptstyle{L}}}}$,
with the laser power $P$ and cavity linewidth $\kappa_{\mathrm{c}}$. The third
term describes a two-mode squeezing process. In this, a pump field impinges on
a non-linear crystal, and a pump photon of frequency $\omega_{\mathrm{P}}$ is
converted into two photons of lower frequency. In the usual treatment of such
processes, the pump is assumed to be a strong coherent field. In a rotating
frame with respect to the pump frequency and linearized pump field [50], the
Hamiltonian is then

$\displaystyle
H_{\mathrm{\scriptscriptstyle{DC}}}=g_{\mathrm{\scriptscriptstyle{DC}}}(ac_{\mathrm{c}}+a^{\dagger}c^{\dagger}_{\mathrm{c}}).$
(48)

The coupling strength of the down-conversion process is given by
$g_{\mathrm{\scriptscriptstyle{DC}}}=\Gamma l\frac{c}{L}$, where $l$ is the
length of the crystal, $L$ is the cavity length, $c$ the speed of light and
$\Gamma$ the gain parameter. We refer to [30, 51] for more details on the gain
parameter. The last term describes the beam-splitter interaction. It is given
by

$\displaystyle
H_{\mathrm{\scriptscriptstyle{BS}}}=g_{\mathrm{\scriptscriptstyle{BS}}}(a^{\dagger}c_{\mathrm{c}}+c^{\dagger}_{\mathrm{c}}a),$
(49)

where $g_{\mathrm{\scriptscriptstyle{BS}}}$ denotes the coupling strength of
this process. For a generic beam-splitter, the strength is defined by
$g_{\mathrm{\scriptscriptstyle{BS}}}=rc/L$ with $r$ the reflectivity of the
beam-splitter. Alternatively, for a setup considered in Fig. 1(a), where the
two modes are not spatially separated, the beam-splitter interaction can be
achieved with a wave plate [40]. The strength is then given by

$\displaystyle
g_{\mathrm{\scriptscriptstyle{BS}}}=\frac{c}{L}\frac{\theta}{2}\sin 2\tau,$
(50)

with $\theta$ the wave plate angle and $\tau$ the delay. Moving to a rotating
frame with respect to the laser frequency
$\omega_{\mathrm{\scriptscriptstyle{L}}}$ and assuming a strong driving field,
the Hamiltonian can be linearized, and we arrive at Eq. (3).

##  Appendix B Calculation of noise spectral densities

We consider a general linear quantum system consisting of $n$ system
variables, $k$ in- and outputs and $m$ bath variables. The input-output
relations can be written as

$\displaystyle\mathbf{x}_{\mathrm{out}}=K_{\mathrm{in}}^{\top}\mathbf{x}-\mathbf{x}_{\mathrm{in}}$
(51)

with a vector $\mathbf{x}$ containing the $n$ system variables, and
$\mathbf{x}_{\mathrm{in}}$ and $\mathbf{x}_{\mathrm{out}}$ vectors for the $k$
inputs and outputs. The whole system is governed by the equations of motion

$\displaystyle\dot{\mathbf{x}}(t)=M_{\mathrm{\scriptscriptstyle{sys}}}\mathbf{x}(t)+K_{\mathrm{in}}\mathbf{x}_{\mathrm{in}}(t)+K_{\mathrm{\scriptscriptstyle{bath}}}\mathbf{x}_{\mathrm{\scriptscriptstyle{bath}}}(t),$
(52)

where we have introduced the system matrix
$M_{\mathrm{\scriptscriptstyle{sys}}}$ and input matrices $K_{\mathrm{in}}$
and $K_{\mathrm{\scriptscriptstyle{bath}}}$ for the input and bath
quadratures. The equations of motion (52) can be solved in the Fourier domain,
where $\dot{\mathbf{x}}(t)=i\omega\mathbf{x}(\omega)$. It follows,

$\displaystyle\mathbf{x}=(i\omega\mathbb{1}-M_{\mathrm{\scriptscriptstyle{sys}}})^{-1}\,(K_{\mathrm{in}}\mathbf{x}_{\mathrm{in}}+K_{\mathrm{\scriptscriptstyle{bath}}}\mathbf{x}_{\mathrm{\scriptscriptstyle{bath}}}).$
(53)

Together with Eq. (51), we derive the output quadratures as

$\displaystyle\begin{split}\mathbf{x}_{\mathrm{out}}&=K_{\mathrm{in}}^{\top}\mathbf{x}-\mathbf{x}_{\mathrm{in}}\\\
&=\left(K_{\mathrm{in}}^{\top}(i\omega\mathbb{1}-M_{\mathrm{\scriptscriptstyle{sys}}})^{-1}K_{\mathrm{in}}-\mathbb{1}\right)\mathbf{x}_{\mathrm{in}}\\\
&\quad+K_{\mathrm{in}}^{\top}(i\omega\mathbb{1}-M_{\mathrm{\scriptscriptstyle{sys}}})^{-1}K_{\mathrm{\scriptscriptstyle{bath}}}\mathbf{x}_{\mathrm{\scriptscriptstyle{bath}}}\\\
&=T_{\mathrm{in}}\mathbf{x}_{\mathrm{in}}+T_{\mathrm{\scriptscriptstyle{bath}}}\mathbf{x}_{\mathrm{\scriptscriptstyle{bath}}}\\\
&=\mathcal{T}\,\tilde{\mathbf{x}}_{\mathrm{in}},\end{split}$ (54)

where

$\displaystyle\tilde{\mathbf{x}}_{\mathrm{in}}$
$\displaystyle=\bigl{(}\begin{smallmatrix}\mathbf{x}_{\mathrm{in}}\\\
\mathbf{x}_{\mathrm{\scriptscriptstyle{bath}}}\end{smallmatrix}\bigr{)},$ (55)
$\displaystyle\mathcal{T}$
$\displaystyle=\left(T_{\mathrm{in}},T_{\mathrm{\scriptscriptstyle{bath}}}\right).$
(56)

From this, we can calculate the (symmetrized) spectral density matrix as

$\displaystyle\begin{split}\delta(\omega-\omega^{\prime})S_{\mathrm{out}}(\omega)&=\frac{1}{2}\langle\mathbf{x}_{\mathrm{out}}(\omega)\mathbf{x}^{\dagger}_{\mathrm{out}}(\omega^{\prime})\rangle+\mathrm{c}.\mathrm{c}.\\\
&=\frac{1}{2}\langle\mathcal{T}(\omega)\tilde{\mathbf{x}}_{\mathrm{in}}\tilde{\mathbf{x}}^{\dagger}_{\mathrm{in}}\mathcal{T}^{\dagger}(-\omega^{\prime})\rangle+\mathrm{c}.\mathrm{c}.\\\
&=\frac{1}{2}\langle\mathcal{T}(\omega)S_{\mathrm{in}}\mathcal{T}^{\dagger}(-\omega^{\prime})\rangle+\mathrm{c}.\mathrm{c}.,\end{split}$
(57)

with $S_{\mathrm{in}}$ the input spectral density matrix. Every sub-system in
our setup has four system variables. Hence the system matrices
$M_{\mathrm{\scriptscriptstyle{sys}}}$ and bath input matrices
$K_{\mathrm{\scriptscriptstyle{bath}}}$ are all $4\times 4$-dimensional. The
in- and output variables are the two quadratures of the laser light, making
the input matrices $K_{\mathrm{in}}$ $4\times 2$-dimensional.

To model losses, the output quadratures are mixed with vacuum noise via a
beam-splitter interaction, which are then

$\displaystyle\mathbf{x}_{\mathrm{out}}^{\prime}=\underbrace{\begin{pmatrix}1&0&0&0\\\
0&1&0&0\end{pmatrix}\eta_{4\times 4}\begin{pmatrix}\mathcal{T}&0\\\
0&\mathbb{1}\end{pmatrix}}_{\mathcal{T}_{\mathrm{\scriptscriptstyle{loss}}}}\begin{pmatrix}\tilde{\mathbf{x}}_{\mathrm{in}}\\\
\mathbf{x}_{\mathrm{\scriptscriptstyle{vac}}}\end{pmatrix}.$ (58)

The second matrix mixes the cavity output with vacuum, and the first matrix is
the partial trace over the lost output port of the beam-splitter.

Finally, we need to cascade the two sub-systems. For this choose
$\mathbf{x}_{\mathrm{in},2}=\mathbf{x}^{\prime}_{\mathrm{out},1}$, where the
subscripts $1$ and $2$ stand for the first and second system. The total output
quadratures are then given by

$\displaystyle\begin{split}\mathbf{x}^{\prime}_{\mathrm{out},2}&=\mathcal{T}^{\mathrm{\scriptscriptstyle{loss}}}_{2}\scalebox{0.85}{\mbox{$\displaystyle\begin{pmatrix}\mathcal{T}^{\mathrm{\scriptscriptstyle{loss}}}_{1}&0&0\\\
0&\mathbb{1}_{4\times 4}&0\\\ 0&0&\mathbb{1}_{2\times
2}\end{pmatrix}\begin{pmatrix}\mathbf{x}_{\mathrm{in}}\\\
\mathbf{x}_{\mathrm{\scriptscriptstyle{bath}},1}\\\
\mathbf{x}_{\mathrm{\scriptscriptstyle{vac}},1}\\\
\mathbf{x}_{\mathrm{\scriptscriptstyle{bath}},2}\\\
\mathbf{x}_{\mathrm{\scriptscriptstyle{vac}},2}\end{pmatrix}$}}\\\
&=\mathcal{T}_{\mathrm{\scriptscriptstyle{total}}}\mathbf{x}_{\mathrm{in},\mathrm{\scriptscriptstyle{total}}}.\end{split}$
(59)

The equations of motion (2) for the optomechanical sensor imply the following
matrices,

$\displaystyle
M_{\mathrm{\scriptscriptstyle{sys}}}^{\mathrm{\scriptscriptstyle{OMS}}}$
$\displaystyle=\scalebox{0.75}{\mbox{$\displaystyle\begin{pmatrix}-\frac{\kappa_{\mathrm{om}}}{2}&\Delta_{\mathrm{om}}&0&0\\\
-\Delta_{\mathrm{om}}&-\frac{\kappa_{\mathrm{om}}}{2}&-g&0\\\
0&0&0&\omega_{\mathrm{m}}\\\
-g&0&-\omega_{\mathrm{m}}&-\gamma_{\mathrm{m}}\end{pmatrix}$}},$ (60a)
$\displaystyle K^{\mathrm{\scriptscriptstyle{OMS}}}_{\mathrm{in}}$
$\displaystyle=\scalebox{0.75}{\mbox{$\displaystyle\begin{pmatrix}\sqrt{\kappa^{\mathrm{in}}_{\mathrm{om}}}&0\\\
0&\sqrt{\kappa^{\mathrm{in}}_{\mathrm{om}}}\\\ 0&0\\\ 0&0\end{pmatrix}$}},$
(60b) $\displaystyle
K^{\mathrm{\scriptscriptstyle{OMS}}}_{\mathrm{\scriptscriptstyle{bath}}}$
$\displaystyle=\scalebox{0.75}{\mbox{$\displaystyle\begin{pmatrix}\sqrt{\kappa^{\mathrm{\scriptscriptstyle{bath}}}_{\mathrm{om}}}&0\\\
0&\sqrt{\kappa^{\mathrm{\scriptscriptstyle{bath}}}_{\mathrm{om}}}\\\ 0&0\\\
0&\sqrt{\gamma_{\mathrm{m}}}\end{pmatrix}$}}.$ (60c)

Similarly, the equations of motion (4) for the effective negative mass
oscillator implies

$\displaystyle
M_{\mathrm{\scriptscriptstyle{sys}}}^{\mathrm{\scriptscriptstyle{NMO}}}$
$\displaystyle=\scalebox{0.75}{\mbox{$\displaystyle\begin{pmatrix}-\frac{\kappa_{\mathrm{c}}}{2}&\Delta_{\mathrm{c}}&0&(g_{\mathrm{\scriptscriptstyle{BS}}}-g_{\mathrm{\scriptscriptstyle{DC}}})\\\
-\Delta_{\mathrm{c}}&-\frac{\kappa_{\mathrm{c}}}{2}&-(g_{\mathrm{\scriptscriptstyle{BS}}}+g_{\mathrm{\scriptscriptstyle{DC}}})&0\\\
0&(g_{\mathrm{\scriptscriptstyle{BS}}}-g_{\mathrm{\scriptscriptstyle{DC}}})&-\frac{\kappa_{\mathrm{a}}}{2}&\Delta_{\mathrm{a}}\\\
-(g_{\mathrm{\scriptscriptstyle{BS}}}+g_{\mathrm{\scriptscriptstyle{DC}}})&0&-\Delta_{\mathrm{a}}&-\frac{\kappa_{\mathrm{a}}}{2}\end{pmatrix}$}},$
(61a) $\displaystyle K^{\mathrm{\scriptscriptstyle{NMO}}}_{\mathrm{in}}$
$\displaystyle=\scalebox{0.75}{\mbox{$\displaystyle\begin{pmatrix}\sqrt{\kappa^{\mathrm{in}}_{\mathrm{c}}}&0\\\
0&\sqrt{\kappa^{\mathrm{in}}_{\mathrm{c}}}\\\ 0&0\\\ 0&0\end{pmatrix}$}},$
(61b) $\displaystyle
K^{\mathrm{\scriptscriptstyle{OMS}}}_{\mathrm{\scriptscriptstyle{bath}}}$
$\displaystyle=\scalebox{0.75}{\mbox{$\displaystyle\begin{pmatrix}\sqrt{\kappa^{\mathrm{\scriptscriptstyle{bath}}}_{\mathrm{c}}}&0\\\
0&\sqrt{\kappa^{\mathrm{\scriptscriptstyle{bath}}}_{\mathrm{c}}}\\\
\sqrt{\kappa_{\mathrm{a}}}&0\\\ 0&\sqrt{\kappa_{\mathrm{a}}}\end{pmatrix}$}}.$
(61c)

From these expressions, we calculate the total transfer matrix in Eq. (59) and
together with the input spectral density,

$\displaystyle
S_{\mathrm{in}}=\scalebox{0.85}{\mbox{$\displaystyle\begin{cases}\frac{1}{2}\,\mathrm{diag}\left(1,1,1,1,1,1,1,1,0,2S_{F}\right)&\text{for
}\mathrm{NMO}\mapsto\mathrm{OMS}\\\
\frac{1}{2}\,\mathrm{diag}\left(1,1,1,1,0,2S_{F},1,1,1,1\right)&\text{for
}\mathrm{OMS}\mapsto\mathrm{NMO}\end{cases}$}},$ (62)

we obtain the output spectral density with Eq. (57). The spectral density of
the added noise is then estimated from the phase component
$S^{pp}_{\mathrm{out}}$, by dividing it by the coefficient of $S_{F}$.

## References

  * Braginsky  _et al._ [1980] V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, Science 209, 547 (1980).
  * Braginsky and Khalili [1995] V. B. Braginsky and F. Y. Khalili,  _Quantum measurement_ (Cambridge University Press, 1995).
  * Meystre [2013] P. Meystre, Annalen der Physik 525, 215 (2013).
  * Chen [2013] Y. Chen, Journal of Physics B: Atomic, Molecular and Optical Physics 46, 104001 (2013).
  * Aspelmeyer  _et al._ [2014] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Reviews of Modern Physics 86, 1391 (2014).
  * Caves [1980] C. M. Caves, Phys. Rev. Lett. 45, 75 (1980).
  * Clerk  _et al._ [2010] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Reviews of Modern Physics 82, 1155 (2010).
  * Bowen and Milburn [2020] W. P. Bowen and G. J. Milburn,  _Quantum Optomechanics_ (CRC Press, 2020).
  * Danilishin  _et al._ [2019] S. L. Danilishin, F. Y. Khalili, and H. Miao, Living Reviews in Relativity 2019 22:1 22, 1 (2019).
  * Unruh [1983] W. G. Unruh, in  _Quantum Optics, Experimental Gravity, and Measurement Theory_, edited by P. Meystre and M. O. Scully (Springer US, 1983) pp. 647–660.
  * Bondurant and Shapiro [1984] R. S. Bondurant and J. H. Shapiro, Phys. Rev. D 30, 2548 (1984).
  * Jaekel and Reynaud [1990] M. T. Jaekel and S. Reynaud, Europhys. Lett. 13, 301 (1990).
  * Vyatchanin and Zubova [1995] S. Vyatchanin and E. Zubova, Physics Letters A 201, 269 (1995).
  * Kimble  _et al._ [2001] H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, Physical Review D 65, 022002 (2001).
  * Khalili [2010] F. Y. Khalili, Physical Review D - Particles, Fields, Gravitation and Cosmology 81, 122002 (2010).
  * Woolley and Clerk [2013] M. J. Woolley and A. A. Clerk, Physical Review A - Atomic, Molecular, and Optical Physics 87, 063846 (2013).
  * Briant  _et al._ [2003] T. Briant, M. Cerdonio, L. Conti, A. Heidmann, A. Lobo, and M. Pinard, Physical Review D 67, 102005 (2003).
  * Caniard  _et al._ [2007] T. Caniard, P. Verlot, T. Briant, P. F. Cohadon, and A. Heidmann, Physical Review Letters 99, 110801 (2007).
  * de Lépinay  _et al._ [2021] L. M. de Lépinay, C. F. Ockeloen-Korppi, M. J. Woolley, and M. A. Sillanpää, Science 372, 625 (2021).
  * Buonanno and Chen [2001] A. Buonanno and Y. Chen, Physical Review D 64, 042006 (2001).
  * Verlot  _et al._ [2010] P. Verlot, A. Tavernarakis, T. Briant, P. F. Cohadon, and A. Heidmann, Physical Review Letters 104, 133602 (2010).
  * Chen  _et al._ [2011] Y. Chen, S. L. Danilishin, F. Y. Khalili, and H. Müller-Ebhardt, General Relativity and Gravitation 43, 671 (2011).
  * Polzik and Hammerer [2015] E. S. Polzik and K. Hammerer, Annalen der Physik 527, A15 (2015).
  * Julsgaard  _et al._ [2001] B. Julsgaard, A. Kozhekin, and E. S. Polzik, Nature 2001 413:6854 413, 400 (2001).
  * Wasilewski  _et al._ [2010] W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, Physical Review Letters 104, 133601 (2010).
  * Hammerer  _et al._ [2009] K. Hammerer, M. Aspelmeyer, E. S. Polzik, and P. Zoller, Physical Review Letters 102, 020501 (2009).
  * Møller  _et al._ [2017] C. B. Møller, R. A. Thomas, G. Vasilakis, E. Zeuthen, Y. Tsaturyan, M. Balabas, K. Jensen, A. Schliesser, K. Hammerer, and E. S. Polzik, Nature 547, 191 (2017).
  * Tsang and Caves [2010] M. Tsang and C. M. Caves, Physical Review Letters 105, 123601 (2010).
  * Tsang and Caves [2012] M. Tsang and C. M. Caves, Physical Review X 2, 031016 (2012).
  * Wimmer  _et al._ [2014] M. H. Wimmer, D. Steinmeyer, K. Hammerer, and M. Heurs, Physical Review A - Atomic, Molecular, and Optical Physics 89, 053836 (2014).
  * Bariani  _et al._ [2015] F. Bariani, H. Seok, S. Singh, M. Vengalattore, and P. Meystre, Physical Review A - Atomic, Molecular, and Optical Physics 92, 043817 (2015).
  * Gebremariam  _et al._ [2019] T. Gebremariam, Y. X. Zeng, M. Mazaheri, and C. Li, Science China Physics, Mechanics and Astronomy 2019 63:1 63, 1 (2019).
  * Motazedifard  _et al._ [2016] A. Motazedifard, F. Bemani, M. H. Naderi, R. Roknizadeh, and D. Vitali, New Journal of Physics 18, 073040 (2016).
  * Singh  _et al._ [2022] S. K. Singh, M. Mazaheri, J.-X. Peng, and M. Asjad, (2022), arXiv:2201.10805 .
  * Zhang  _et al._ [2013] K. Zhang, P. Meystre, and W. Zhang, Physical Review A - Atomic, Molecular, and Optical Physics 88, 043632 (2013).
  * Yan and Jing [2021] J. Yan and J. Jing, Annalen der Physik 533, 2100119 (2021).
  * Carmichael [1993] H. J. Carmichael, Physical Review Letters 70, 2273 (1993).
  * Gardiner [1993] C. W. Gardiner, Physical Review Letters 70, 2269 (1993).
  * Zeuthen  _et al._ [2022] E. Zeuthen, E. S. Polzik, and F. Y. Khalili, PRX Quantum 3, 020362 (2022).
  * Steinmeyer [2019] D. Steinmeyer,  _Subsystems for all-optical coherent quantum-noise cancellation_ , Ph.D. thesis, Leibniz Universität Hannover (2019).
  * Walls and Milburn [2007] D. F. Walls and G. J. Milburn,  _Quantum optics_, 2nd ed. (Springer, 2007).
  * Braginsky  _et al._ [2001] V. B. Braginsky, M. L. Gorodetsky, F. Y. Khalili, and K. S. Thorne, AIP Conference Proceedings 523, 180 (2001).
  * Tsang  _et al._ [2011] M. Tsang, H. M. Wiseman, and C. M. Caves, Physical Review Letters 106, 090401 (2011).
  * Miao  _et al._ [2017] H. Miao, R. X. Adhikari, Y. Ma, B. Pang, and Y. Chen, Physical Review Letters 119, 050801 (2017).
  * Khalili and Zeuthen [2021] F. Y. Khalili and E. Zeuthen, Physical Review A 103, 043721 (2021).
  * Mason  _et al._ [2019] D. Mason, J. Chen, M. Rossi, Y. Tsaturyan, and A. Schliesser, Nature Physics 2019 15:8 15, 745 (2019).
  * Norte  _et al._ [2016] R. A. Norte, J. P. Moura, and S. Gröblacher, Physical Review Letters 116, 147202 (2016).
  * [48] J. Junker, (private communication).
  * Vahlbruch  _et al._ [2016] H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, Physical Review Letters 117, 110801 (2016).
  * Ou  _et al._ [1992] Z. Y. Ou, S. F. Pereira, and H. J. Kimble, Applied Physics B 1992 55:3 55, 265 (1992).
  * Byer and Herbst [1977] R. L. Byer and R. L. Herbst,  _Parametric oscillation and mixing_ (Springer, Berlin, Heidelberg, 1977) pp. 81–137.