### Geometric control theory for disturbance decoupling

Let us consider the following *classical* linear time-invariant system:

$$\begin{aligned} \frac{dx(t)}{dt}=Ax(t)+Bu(t),\qquad y(t)=Cx(t)+Du(t), \end{aligned}$$

(1)

where \(x(t) \in \mathcal{X}:=\mathbb{R}^{n}\) is a vector of system variables, \(u(t) \in \mathcal{U}:=\mathbb{R}^{m}\) and \(y(t) \in \mathcal{Y}:=\mathbb{R}^{l}\) are vectors of input and output, respectively. *A*, *B*, *C*, and *D* are real matrices. In the Laplace domain, the input-output relation is represented by

$$\begin{aligned} Y(s)=\Xi (s)U(s), \qquad \Xi (s)=C(sI-A)^{-1}B+D, \end{aligned}$$

where \(U(s)\) and \(Y(s)\) are the Laplace transforms of \(u(t)\) and \(y(t)\), respectively. \(\Xi (s)\) is called the *transfer function*. In this subsection, we assume \(D=0\).

Now we describe the geometric control theory, for the disturbance decoupling problem [24, 25]. The following *invariant subspaces* play a key role in the theory.

### Definition 1

Let \(A: \mathcal{X} \rightarrow \mathcal{X}\) be a linear map. Then, a subspace \(\mathcal{V} \subseteq \mathcal{X}\) is said to be *A*-*invariant*, if \(A\mathcal{V} \subseteq \mathcal{V}\).

### Definition 2

Given a linear map \(A: \mathcal{X} \rightarrow \mathcal{X}\) and a subspace \(\operatorname{Im} B \subseteq \mathcal{X}\), a subspace \(\mathcal{V} \subseteq \mathcal{X}\) is said to be \((A, B)\)-*invariant*, if \(A\mathcal{V} \subseteq \mathcal{V} \oplus \operatorname{Im} B\).

### Definition 3

Given a linear map \(A: \mathcal{X} \rightarrow \mathcal{X}\) and a subspace \(\operatorname {Ker}C \subseteq \mathcal{X}\), a subspace \(\mathcal{V}\subseteq \mathcal{X}\) is said to be \((C, A)\)-*invariant*, if \(A(\mathcal{V} \cap \operatorname {Ker}C) \subseteq \mathcal{V}\).

### Definition 4

Assume that \(\mathcal{V}_{1}\) is \((C, A)\)-invariant, \(\mathcal{V}_{2}\) is \((A, B)\)-invariant, and \(\mathcal{V}_{1} \subseteq \mathcal{V}_{2}\). Then, \((\mathcal{V}_{1}, \mathcal{V}_{2})\) is said to be a \((C,A,B)\)-pair.

From Definitions 2 and 3, we have the following two lemmas.

### Lemma 1

\(\mathcal{V} \subseteq \mathcal{X}\)
*is*
\((A, B)\)-*invariant if and only if there exists a matrix*
*F*
*such that*
\(F \in \mathcal{F}(\mathcal{V}):= \{ F: \mathcal{X} \rightarrow \mathcal{U} |(A+BF)\mathcal{V} \subseteq \mathcal{V} \}\).

### Lemma 2

\(\mathcal{V} \subseteq \mathcal{X}\)
*is*
\((C, A)\)-*invariant if and only if there exists a matrix*
*G*
*such that*
\(G \in \mathcal{G}(\mathcal{V}):= \{ G: \mathcal{Y} \rightarrow \mathcal{X} |(A+GC)\mathcal{V} \subseteq \mathcal{V} \}\).

The disturbance decoupling problem is described as follows. The system of interest is represented, in an extended form of Eq. (1), as

$$\begin{aligned} \frac{dx(t)}{dt}=Ax(t)+Bu(t)+Ed(t),\qquad y(t)=Cx(t),\qquad z(t)=Hx(t), \end{aligned}$$

where \(d(t)\) is the disturbance and \(z(t)\) is the output to be regulated. *E* and *H* are real matrices. The other output \(y(t)\) may be used for constructing a feedback controller; see Figure 1. The disturbance \(d(t)\) can degrade the control performance evaluated on \(z(t)\). Thus it is desirable if we can modify the system structure by some means so that eventually \(d(t)\) dose not affect at all on \(z(t)\).^{Footnote 1} This control goal is called the disturbance decoupling. Here we describe a specific feedback control method to achieve this goal; note that, as shown later, the direct-interaction method for linear quantum systems can also be described within this framework. The controller configuration is illustrated in Figure 1; that is, the system modification is carried out by combining an auxiliary system (controller) with the original system (plant), so that the whole closed-loop system satisfies the disturbance decoupling condition. The controller with variable \({x}_{{K}} \in \mathcal{X}_{{ K}} :=\mathbb{R}^{n_{k}}\) is assumed to take the following form:

$$\begin{aligned} \frac{dx_{{ K}}(t)}{dt} =A_{K}x_{{ K}}(t) +B_{K}y(t),\qquad u(t)=C_{K}x_{{ K}}(t) +D_{K}y(t), \end{aligned}$$

where \(A_{K}: \mathcal{X}_{{ K}} \rightarrow \mathcal{X}_{{ K}}\), \(B_{K}: \mathcal{Y} \rightarrow \mathcal{X}_{{ K}}\), \(C_{K}: \mathcal{X}_{{ K}} \rightarrow \mathcal{U}\), and \(D_{K}: \mathcal{Y} \rightarrow \mathcal{U}\) are real matrices. Then, the closed-loop system defined in the augmented space \(\mathcal{X}_{{ E}}:= \mathcal{X} \oplus \mathcal{X}_{{ K}}\) is given by

$$\begin{aligned} \frac{d}{dt}\left [\textstyle\begin{array}{@{}c@{}} x\\ x_{{ K}} \end{array}\displaystyle \right ] = \left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} A+BD_{K}C & BC_{K}\\ B_{K}C & A_{K} \end{array}\displaystyle \right ] \left [ \textstyle\begin{array}{@{}c@{}} x\\ x_{{ K}} \end{array}\displaystyle \right ] +\left [ \textstyle\begin{array}{@{}c@{}} E\\ O \end{array}\displaystyle \right ] d, \qquad z=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} H & O \end{array}\displaystyle \right ] \left [ \textstyle\begin{array}{@{}c@{}} x\\ x_{{ K}} \end{array}\displaystyle \right ]. \end{aligned}$$

(2)

The control goal is to design \((A_{K}, B_{K}, C_{K}, D_{K})\) so that, in Eq. (2), the disturbance signal \(d(t)\) dose not appear in the output \(z(t)\): see the endnote in Page 21. Here, let us define

$$\begin{aligned} A_{{ E}}=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} A+BD_{K}C & BC_{K}\\ B_{K}C & A_{K} \end{array}\displaystyle \right ], \end{aligned}$$

(3)

\(\mathcal{B}= \operatorname{Im} B\), \(\mathcal{C}= \operatorname {Ker}C\), \(\mathcal{E}= \operatorname{Im} E\), and \(\mathcal{H}= \operatorname {Ker}H\). Then, the following theorem gives the solvability condition for the disturbance decoupling problem.

### Theorem 1

*For the closed*-*loop system* (2), *the disturbance decoupling problem via the dynamical feedback controller has a solution if and only if there exists a*
\((C,A,B)\)-*pair*
\((\mathcal{V}_{1},\mathcal{V}_{2})\)
*satisfying*

$$\begin{aligned} \mathcal{E} \subseteq \mathcal{V}_{1} \subseteq \mathcal{V}_{2} \subseteq \mathcal{H}. \end{aligned}$$

(4)

Note that this condition does not depend on the controller matrices to be designed. The following corollary can be used to check if the solvability condition is satisfied.

### Corollary 1

*For the closed*-*loop system* (2), *the disturbance decoupling problem via the dynamical feedback controller has a solution if and only if*

$$\mathcal{V}_{*} (\mathcal{C}, \mathcal{E}) \subseteq \mathcal{V}^{*}(\mathcal{B}, \mathcal{H}), $$

*where*
\(\mathcal{V}^{*} (\mathcal{B}, \mathcal{H})\)
*is the maximum element of*
\((A, B)\)-*invariant subspaces contained in*
\(\mathcal{H}\), *and*
\(\mathcal{V}_{*} (\mathcal{C}, \mathcal{E})\)
*is the minimum element of*
\((C, A)\)-*invariant subspaces containing*
\(\mathcal{E}\). *These subspaces can be computed by the algorithms given in Appendix *
A.

Once the solvability condition described above is satisfied, then we can explicitly construct the controller matrices \((A_{K}, B_{K}, C_{K}, D_{K})\). The following intersection and projection subspaces play a key role for this purpose; that is, for a subspace \(\mathcal{V}_{{ E}} \subseteq \mathcal{X}_{{ E}}= \mathcal{X} \oplus \mathcal{X}_{{ K}}\), let us define

$$\begin{aligned} \mathcal{V}_{{ I}}:= & \left \{ x \in \mathcal{X}\bigg| \left [ \textstyle\begin{array}{@{}c@{}} x\\ O \end{array}\displaystyle \right ] \in \mathcal{V}_{{ E}} \right \}, \qquad \mathcal{V}_{{ P}}:= \left \{ x \in \mathcal{X}\bigg| \left [ \textstyle\begin{array}{@{}c@{}} x\\ x_{{ K}} \end{array}\displaystyle \right ] \in \mathcal{V}_{{ E}}, \exists {x}_{{ K}} \in \mathcal{X}_{{ K}} \right \}. \end{aligned}$$

Then, the following theorem is obtained:

### Theorem 2

*Suppose that*
\((\mathcal{V}_{1}, \mathcal{V}_{2})\)
*is a*
\((C, A, B)\)-*pair*. *Then*, *there exist*
\(F \in \mathcal{F}(\mathcal{V}_{2})\), \(G \in \mathcal{G}(\mathcal{V}_{1})\), *and*
\(D_{K}: \mathcal{Y} \rightarrow \mathcal{U}\)
*such that*
\(\operatorname {Ker}F_{0} \supseteq \mathcal{V}_{1}\)
*and*
\(\operatorname{Im} G_{0} \subseteq \mathcal{V}_{2}\)
*hold*, *where*
\(F_{0}=F-D_{K}C\), \(G_{0}=G-BD_{K}\).

*Moreover*, *there exists*
\(\mathcal{X}_{{ K}}\)
*with*
\(\operatorname{dim} \mathcal{X}_{{ K}}=\operatorname{dim} \mathcal{V}_{2} - \operatorname{dim} \mathcal{V}_{1}\), *and*
\(A_{{ E}}\)
*has an invariant subspace*
\(\mathcal{V}_{{ E}} \subseteq \mathcal{X}_{{ E}}\)
*such that*
\(\mathcal{V}_{1}=\mathcal{V}_{{ I}}\)
*and*
\(\mathcal{V}_{2}=\mathcal{V}_{{ P}}\). *Also*, \((A_{K}, B_{K}, C_{K})\)
*satisfies*

$$\begin{aligned} C_{K}N=F_{0},\qquad B_{K}=-NG_{0},\qquad A_{ K}N=N(A+BF_{0}+GC), \end{aligned}$$

(5)

*where*
\(N: \mathcal{V}_{2} \rightarrow \mathcal{X}_{{ K}}\)
*is a linear map satisfying*
\(\operatorname {Ker}N=\mathcal{V}_{1}\).

In fact, under the condition given in Theorem 2, let us define the following augmented subspace \(\mathcal{V}_{{ E}} \subseteq \mathcal{X}_{{ E}}\):

$$\begin{aligned} \mathcal{V}_{{ E}}:= \left \{ \left [ \textstyle\begin{array}{@{}c@{}} x\\ Nx \end{array}\displaystyle \right ] \bigg| x \in \mathcal{V}_{2} \right \}. \end{aligned}$$

Then, \(\mathcal{V}_{1}=\mathcal{V}_{{ I}}\) and \(\mathcal{V}_{2}=\mathcal{V}_{{ P}}\) hold, and we have

$$A_{{ E}}\left [ \textstyle\begin{array}{@{}c@{}} x\\ Nx \end{array}\displaystyle \right ] =\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} A+BD_{ K}C & BC_{ K}\\ B_{ K}C & A_{ K} \end{array}\displaystyle \right ] \left [ \textstyle\begin{array}{@{}c@{}} x\\ Nx \end{array}\displaystyle \right ] =\left [ \textstyle\begin{array}{@{}c@{}} (A+BF)x\\ N(A+BF)x \end{array}\displaystyle \right ] \in \mathcal{V}_{{ E}}, $$

implying that \(\mathcal{V}_{{ E}}\) is actually \(A_{{ E}}\)-invariant. Now suppose that Theorem 1 holds, and let us take the \((C,A,B)\)-pair \((\mathcal{V}_{1}, \mathcal{V}_{2})\) satisfying Eq. (4). Then, together with the above result (\(A_{{ E}}\mathcal{V}_{{ E}} \subseteq \mathcal{V}_{{ E}}\)), we have \(\operatorname{Im} [E^{\top }~O]^{\top } \subseteq \mathcal{V}_{{ E}} \subseteq \operatorname {Ker}[H~O]\). This implies that \(d(t)\) must be contained in the unobservable subspace with respect to \(z(t)\), and thus the disturbance decoupling is realized.

### Linear quantum systems

Here we describe a general linear quantum system composed of *n* bosonic subsystems. The *j*th mode can be modeled as a harmonic oscillator with the canonical conjugate pairs (or quadratures) \(\hat{q}_{j}\) and \(\hat{p}_{j}\) satisfying the canonical commutation relation (CCR) \(\hat{q}_{j}\hat{p}_{k}-\hat{p}_{k}\hat{q}_{j}=i \delta _{jk}\). Let us define the vector of quadratures as \(\hat{x}=[\hat{q}_{1}, \hat{p}_{1}, \ldots, \hat{q}_{n}, \hat{p}_{n}]^{\top }\). Then, the CCRs are summarized as

$$\begin{aligned} \hat{x}\hat{x}^{\top }-\bigl(\hat{x}\hat{x}^{\top } \bigr)^{\top } = i \Sigma _{n},\qquad \Sigma _{n} = \operatorname{diag}\{\Sigma, \ldots, \Sigma \},\qquad \Sigma =\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} 0 & 1 \\ -1 & 0 \end{array}\displaystyle \right ]. \end{aligned}$$

Note that \(\Sigma _{n}\) is a \(2n \times 2n\) block diagonal matrix. The linear quantum system is an open system coupled to *m* environment fields via the interaction Hamiltonian \(\hat{H}_{\mathrm{int}}=i\sum_{j=1}^{m} (\hat{L}_{j} \hat{A}_{j}^{*}-\hat{L}_{j}^{*} \hat{A}_{j})\), where \(\hat{A}_{j}(t)\) is the field annihilation operator satisfying \(\hat{A}_{j}(t)\hat{A}_{k}^{*}(t')-\hat{A}_{k}^{*}(t')\hat{A}_{j}(t)=\delta _{jk}\delta (t-t')\). Also \(\hat{L}_{j}\) is given by \(\hat{L}_{j}=c_{j}^{\top }\hat{x}\) with \(c_{j}\in {\mathbb{C}}^{2n}\). In addition, the system is driven by the Hamiltonian \(\hat{H}=\hat{x}^{\top }R\hat{x}/2\) with \(R=R^{\top }\in \mathbb{R}^{2n \times 2n}\). Then, the Heisenberg equation of *x̂* is given by

$$\begin{aligned} \frac{d\hat{x}(t)}{dt}=A\hat{x}(t)+\sum_{j=1}^{m} B_{j}\hat{W}_{j}(t), \end{aligned}$$

(6)

where \(\hat{W}_{j}(t)\) is defined by

$$\hat{W}_{j} = \left [ \textstyle\begin{array}{@{}c@{}} \hat{Q}_{j} \\ \hat{P}_{j} \end{array}\displaystyle \right ] = \left [ \textstyle\begin{array}{@{}c@{}} (\hat{A}_{j}+\hat{A}_{j}^{*})/\sqrt{2} \\ (\hat{A}_{j}-\hat{A}_{j}^{*})/\sqrt{2} i \end{array}\displaystyle \right ]. $$

The matrices are given by \(A=\Sigma _{n}(R + \sum_{j=1}^{m} C_{j}^{\top }\Sigma C_{j}/2)\) and \(B_{j}=\Sigma _{n} C_{j}^{\top }\Sigma \) with \(C_{j}=\sqrt{2}[\Re (c_{j}), \Im (c_{j})]^{\top }\in \mathbb{R}^{2 \times 2n}\). Also, the instantaneous change of the field operator \(\hat{W}_{j}(t)\) via the system-field coupling is given by

$$\begin{aligned} \hat{W}_{j}^{\mathrm{out}}(t)=C_{j}\hat{x}(t)+ \hat{W}_{j}(t). \end{aligned}$$

(7)

To summarize, the linear quantum system is characterized by the dynamics (6) and the output (7), which are exactly of the same form as those in Eq. (1) (\(l=m\) in this case). However note that the system matrices have to satisfy the above-described special structure, which is equivalently converted to the following *physical realizability condition* [29]:

$$ A\Sigma _{n}+\Sigma _{n} A^{\top } + \sum_{j=1}^{m} B_{j} \Sigma B_{j}^{\top }=O,\qquad B_{j}=\Sigma _{n} C_{j}^{\top }\Sigma. $$

(8)

### Weak signal sensing, SQL, and BAE

The opto-mechanical oscillator illustrated in Figure 2 is a linear quantum system, which serves as a sensor for a very weak signal. Let \(\hat{q}_{1}\) and \(\hat{p}_{1}\) be the oscillator’s position and momentum operators, and \(\hat{a}_{2}=(\hat{q}_{2}+i \hat{p}_{2})/\sqrt{2}\) represents the annihilation operator of the cavity mode. The system Hamiltonian is given by \(\hat{H}=\omega _{ m}(\hat{q}_{1}^{2}+\hat{p}_{1}^{2})/2 -g\hat{q}_{1}\hat{q}_{2}\); that is, the oscillator’s free evolution with resonant frequency \(\omega _{ m}\) plus the linearized radiation pressure interaction between the oscillator and the cavity field with coupling strength *g*. The system couples to an external probe field (thus \(m=1\)) via the coupling operator \(\hat{L}_{1}=\sqrt{\kappa }\hat{a}_{2}\), with *κ* the coupling constant between the cavity and probe fields. The corresponding matrix *R* and vector \(c_{1}\) are then given by

$$\begin{aligned} R=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \omega _{ m} &0&-g&0\\ 0&\omega _{ m} &0&0\\ -g&0&0&0\\ 0&0&0&0 \end{array}\displaystyle \right ],\qquad c_{1}=\sqrt{\frac{\kappa }{2}} \left [ \textstyle\begin{array}{@{}c@{}} 0 \\ 0 \\ 1 \\ i \end{array}\displaystyle \right ]. \end{aligned}$$

The oscillator is driven by an unknown force \(\hat{f}(t)\) with coupling constant *γ*; \(\hat{f}(t)\) is the very weak signal we would like to detect. Then the vector of system variables \(\hat{x}=[\hat{q}_{1}, \hat{p}_{1}, \hat{q}_{2}, \hat{p}_{2}]^{\top }\) satisfies

$$ \frac{d\hat{x}}{dt} =A\hat{x}+B_{1}\hat{W}_{1}+b \hat{f},\qquad \hat{W}_{1}^{\mathrm{out}} =C_{1}\hat{x}+ \hat{W}_{1}, $$

where

$$\begin{aligned} \begin{aligned} &{A=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0&\omega _{ m}&0&0\\ -\omega _{ m} &0 &g&0\\ 0&0&-\kappa /2 &0\\ g&0&0&-\kappa /2 \end{array}\displaystyle \right ],\qquad B_{1}=-C_{1}^{\top } =-\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} 0 & 0 \\ 0 & 0 \\ \sqrt{\kappa } & 0 \\ 0 & \sqrt{\kappa } \end{array}\displaystyle \right ],} \\ &{b=\sqrt{\gamma }\left [ \textstyle\begin{array}{@{}c@{}} 0 \\ 1 \\ 0 \\ 0 \end{array}\displaystyle \right ],\qquad \hat{W}_{1} =[\hat{Q}_{1}, \hat{P}_{1}]^{\top },\qquad \hat{W}_{1}^{\mathrm{out}} = \bigl[\hat{Q}_{1}^{\mathrm{out}}, \hat{P}_{1}^{\mathrm{out}} \bigr]^{\top }.} \end{aligned} \end{aligned}$$

(9)

Note that we are in the rotating frame at the frequency of the probe field. These equations indicate that the information about *f̂* can be extracted by measuring \(\hat{P}_{1}^{\mathrm{out}}\) by a homodyne detector. Actually the measurement output in the Laplace domain is given by

$$\begin{aligned} \hat{P}_{1}^{\mathrm{out}}(s) =\Xi _{f}(s) \hat{f}(s)+\Xi _{Q}(s) \hat{Q}_{1}(s) +\Xi _{P}(s) \hat{P}_{1}(s), \end{aligned}$$

(10)

where \(\Xi _{f}\), \(\Xi _{Q}\), and \(\Xi _{P}\) are transfer functions given by

$$\begin{aligned} \Xi _{f}(s)=\frac{g\omega _{ m}\sqrt{\gamma \kappa }}{(s^{2}+\omega _{ m}^{2})(s+\kappa /2)},\qquad \Xi _{Q}(s)=-\frac{g^{2} \omega _{ m}\kappa }{(s^{2}+\omega _{ m}^{2})(s+\kappa /2)^{2}},\qquad \Xi _{P}(s)= \frac{s-\kappa /2}{s+\kappa /2}. \end{aligned}$$

Thus, \(\hat{P}_{1}^{\mathrm{out}}\) certainly contains *f̂*. Note however that it is subjected to two noises. The first one, \(\hat{Q}_{1}\), is the back-action noise, which is due to the interaction between the oscillator and the cavity. The second one, \(\hat{P}_{1}\), is the shot noise, which inevitably appears. Now, the normalized output is given by

$$y_{1}(s) =\frac{\hat{P}_{1}^{\mathrm{out}}(s)}{\Xi _{f}(s)} =\hat{f}(s) + \frac{\Xi _{Q}(s)}{\Xi _{f}(s)} \hat{Q}_{1}(s) + \frac{\Xi _{P}(s)}{\Xi _{f}(s)}\hat{P}_{1}(s), $$

and the normalized noise power spectral density of \(y_{1}\) in the Fourier domain \((s=i\omega )\) is calculated as follows:

$$\begin{aligned} S(\omega ) &=\bigl\langle \vert y_{1}-\hat{f}\vert ^{2} \bigr\rangle =\biggl\vert \frac{\Xi _{Q}}{\Xi _{f}} \biggr\vert ^{2} \bigl\langle \vert \hat{Q}_{1}\vert ^{2} \bigr\rangle + \biggl\vert \frac{\Xi _{P}}{\Xi _{f}} \biggr\vert ^{2} \bigl\langle \vert \hat{P}_{1}\vert ^{2} \bigr\rangle \\ & \geq 2\sqrt{\frac{\vert \Xi _{Q}\vert ^{2}\vert \Xi _{P}\vert ^{2}}{\vert \Xi _{f}\vert ^{4}} \bigl\langle \vert \hat{Q}_{1} \vert ^{2} \bigr\rangle \bigl\langle \vert \hat{P}_{1} \vert ^{2} \bigr\rangle } \geq \frac{\vert \omega ^{2}-\omega _{ m}^{2} \vert }{\gamma \omega _{ m}} =S_{\mathrm{SQL}}( \omega ). \end{aligned}$$

The lower bound is called the SQL. Note that the last inequality is due to the Heisenberg uncertainty relation of the normalized noise power, i.e., \(\langle \vert \hat{Q}_{1}\vert ^{2} \rangle \langle \vert \hat{P}_{1}\vert ^{2} \rangle \geq 1/4\). Hence, the essential reason why SQL appears is that \(\hat{P}_{1}^{\mathrm{out}}\) contains both the back-action noise \(\hat{Q}_{1}\) and the shot noise \(\hat{P}_{1}\). Therefore, toward the high-precision detection of *f̂*, we need BAE; that is, the system structure should be modified by some means so that the back-action noise is completely evaded in the output signal (note that the shot noise can never be evaded). The condition for BAE can be expressed in terms of the transfer function as follows [22, 23]; i.e., for the modified (controlled) sensor, the transfer function from the back-action noise to the measurement output must satisfy

$$\begin{aligned} \Xi _{Q}(s)=0,\quad \forall s . \end{aligned}$$

(11)

Equivalently, \(\hat{P}_{1}^{\mathrm{out}}\) contains only the shot noise \(\hat{P}_{1}\); hence, in this case the signal to noise ratio can be further improved by injecting a \(\hat{P}_{1}\)-squeezed (meaning \(\langle \vert \hat{P}_{1}\vert ^{2} \rangle < 1/2\)) probe field into the system.