# Geometric control theory for quantum back-action evasion

- Yu Yokotera
^{1}Email authorView ORCID ID profile and - Naoki Yamamoto
^{1}

**3**:15

https://doi.org/10.1140/epjqt/s40507-016-0053-5

© Yokotera and Yamamoto 2016

**Received: **13 September 2016

**Accepted: **18 October 2016

**Published: **28 October 2016

## Abstract

Engineering a sensor system for detecting an extremely tiny signal such as the gravitational-wave force is a very important subject in quantum physics. A major obstacle to this goal is that, in a simple detection setup, the measurement noise is lower bounded by the so-called standard quantum limit (SQL), which is originated from the intrinsic mechanical back-action noise. Hence, the sensor system has to be carefully engineered so that it evades the back-action noise and eventually beats the SQL. In this paper, based on the well-developed geometric control theory for classical disturbance decoupling problem, we provide a general method for designing an auxiliary (coherent feedback or direct interaction) controller for the sensor system to achieve the above-mentioned goal. This general theory is applied to a typical opto-mechanical sensor system. Also, we demonstrate a controller design for a practical situation where several experimental imperfections are present.

## Keywords

## 1 Introduction

Detecting a very weak signal which is almost inaccessible within the classical (i.e., non-quantum) regime is one of the most important subjects in quantum information science. A strong motivation to devise such an ultra-precise sensor stems from the field of gravitational wave detection [1–5]. In fact, a variety of linear sensors composed of opto-mechanical oscillators have been proposed [6–9], and several experimental implementations of those systems in various scales have been reported [10–14].

It is well known that in general a linear sensor is subjected to two types of fundamental noises, i.e., the *back-action noise* and the *shot noise*. As a consequence, the measurement noise is lower bounded by the *standard quantum limit* (*SQL*) [1, 2], which is mainly due to the presence of back-action noise. Hence, high-precision detection of a weak signal requires us to devise a sensor that evades the back-action noise and eventually beats the SQL; i.e., we need to have a sensor achieving *back-action evasion* (*BAE*). In fact, many BAE methods have been developed especially in the field of gravitational wave detection, e.g., the variational measurement technique [15–17] or the quantum locking scheme [18–20]. Moreover, towards more accurate detection, recently we find some high-level approaches to design a BAE sensor, based on those specific BAE methods. For instance, Ref. [21] provides a systematic comparison of several BAE methods and gives an optimal solution. Also systems and control theoretical methods have been developed to synthesize a BAE sensor for a specific opto-mechanical system [22, 23]; in particular, the synthesis is conducted by connecting an auxiliary system to a given plant system by *direct-interaction* [22] or *coherent feedback* [23].

Along this research direction, therefore, in this paper we set the goal to develop a general systems and control theory for engineering a sensor achieving BAE, for both the coherent feedback and the direct-interaction configurations. The key tool used here is the *geometric control theory* [24–28], which had been developed a long time ago. This is indeed a beautiful theory providing a variety of controller design methods for various purposes such as the non-interacting control and the disturbance decoupling problem, but, to our best knowledge, it has not been applied to problems in quantum physics. Actually in this paper we first demonstrate that the general synthesis problem of a BAE sensor can be formulated and solved within the framework of geometric control theory, particularly the above-mentioned disturbance decoupling problem.

This paper is organized as follows. Section 2 is devoted to some preliminaries including a review of the geometric control theory, the general model of linear quantum systems, and the idea of BAE. Then, in Section 3, we provide the general theory for designing a coherent feedback controller achieving BAE, and demonstrate an example for an opto-mechanical system. In Section 4, we discuss the case of direct interaction scheme, also based on the geometric control theory. Finally, in Section 5, for a realistic opto-mechanical system subjected to a thermal environment (the perfect BAE is impossible in this case), we provide a convenient method to find an approximated BAE controller and show how much the designed controller can suppress the noise.

A part of the results in Section 3.3 in this paper will appear in Proceedings of the 55th IEEE Conference on Decision and Control.

### Notation

For a matrix \(A=(a_{ij})\), \(A^{\top }\), \(A^{\dagger }\), and \(A^{\sharp }\) represent the transpose, Hermitian conjugate, and element-wise complex conjugate of *A*, respectively; i.e., \(A^{\top }=(a_{ji})\), \(A^{\dagger }=(a_{ji}^{*})\), and \(A^{\sharp }=(a_{ij}^{*})=(A^{\dagger })^{\top }\). \(\Re (a)\) and \(\Im (a)\) denote the real and imaginary parts of a complex number *a*. *O* and \(I_{n}\) denote the zero matrix and the \(n \times n\) identity matrix. Ker*A* and Im*A* denote the kernel and the image of a matrix *A*, i.e., \(\operatorname {Ker}A=\{x |Ax=0\}\) and \(\operatorname{Im} A=\{y |y=Ax, \forall x\}\).

## 2 Preliminaries

### 2.1 Geometric control theory for disturbance decoupling

*classical*linear time-invariant system:

*A*,

*B*,

*C*, and

*D*are real matrices. In the Laplace domain, the input-output relation is represented by

*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*
\(G \in \mathcal{G}(\mathcal{V}):= \{ G: \mathcal{Y} \rightarrow \mathcal{X} |(A+GC)\mathcal{V} \subseteq \mathcal{V} \}\).

*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 \(z(t)\).

^{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:

### 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*

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*

*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.

### 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*

*where*\(N: \mathcal{V}_{2} \rightarrow \mathcal{X}_{{ K}}\)

*is a linear map satisfying*\(\operatorname {Ker}N=\mathcal{V}_{1}\).

### 2.2 Linear quantum systems

*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

*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

*physical realizability condition*[29]:

### 2.3 Weak signal sensing, SQL, and BAE

*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

*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

*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

*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

*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

## 3 Coherent feedback control for back-action evasion

### 3.1 Coherent and measurement-based feedback control

*measurement-based feedback*[30–33] illustrated in Figure 3(a). In this scheme, we measure the output fields and feed the measurement results back to control the plant system. On the other hand, in the

*coherent feedback*scheme [29, 34–37] shown in Figure 3(b), the feedback loop dose not contain any measurement component and the plant system is controlled by another quantum system. Recently we find several works comparing the performance of these two schemes [34, 38–41]. In particular, it was shown in [23] that there are some control tasks that cannot be achieved by any measurement-based feedback but can be done by a coherent one. More specifically, those tasks are realizing BAE measurement, generating a quantum non-demolished variable, and generating a decoherence-free subsystem; in our case, of course, the first one is crucial. Hence, here we aim to develop a theory for designing a coherent feedback controller such that the whole controlled system accomplishes BAE.

### 3.2 Coherent feedback for BAE

As discussed in Section 2.1, the geometric control theory for disturbance decoupling problem is formulated for the controlled system with special structure (2); in particular, the coefficient matrix of the disturbance \(d(t)\) is of the form \([E^{\top }, O]^{\top }\) and that of the state vector in the output \(z(t)\) is \([H, O]\). Here we consider a class of coherent feedback configuration such that the whole closed-loop system dynamics has this structure, in order for the geometric control theory to be directly applicable.

*j*. Next, as the controller, we take the following special linear quantum system with \((m-1)\) input-output fields:

*m*, should be as small as possible from a viewpoint of implementation; hence in this paper let us consider the case \(m=3\). Now, we consider the coherent feedback connection illustrated in Figure 4, i.e.,

### 3.3 Coherent feedback realization of BAE in the opto-mechanical system

*κ*. In this case the matrix

*A*given in Eq. (9) is replaced by

*b*are given in Eq. (9), and

*N*such that

*N*, i.e., \(NN^{+}=I_{2}\).

*passive system*; this is a static quantum system such as an empty optical cavity. The main reason for choosing a passive system rather than a non-passive (or

*active*) one such as an optical parametric oscillator is that, due to the external pumping energy, the latter could become fragile and also its physical implementation must be more involved compared to a passive system [42]. Now the condition for the system \((A_{ K}, B_{ K}, C_{ K})\) to be passive is given by \(\Sigma A_{ K}\Sigma =-A_{ K}\) and \(\Sigma B_{ K}\Sigma =-B_{ K}\); the general result of this fact is given in Theorem 3 in Appendix B. From these conditions, the system parameters are imposed to satisfy, in addition to Eq. (21), the following equalities:

*f̂*below the SQL, particularly when the \(\hat{P}_{1}\)-squeezed probe input field is used; this fact will be demonstrated in Section 5.

## 4 Direct interaction scheme

*direct interaction*. The controller is characterized by the following two Hamiltonians:

(i) Because of the structure of the matrices \(B_{{ E}}\) and \(C_{{ E}}\), the system is already of the form (2), where the geometric control theory is directly applicable.

(ii) Because we now focus on the same plant system as that in Section 3.3, the same conclusion is obtained; that is, the BAE problem is solvable as long as there is no constraint on the controller matrices \((R_{ K}, R_{1}, R_{2})\).

*F*,

*G*, and

*N*satisfy \(\operatorname {Ker}F \supseteq \mathcal{V}_{1}\), \(\operatorname{Im} G \subseteq \mathcal{V}_{2}\), and \(\operatorname {Ker}N=\mathcal{V}_{1}\), which lead to

### Remark

## 5 Approximate back-action evasion

*approximate BAE*. Then, looking back into Section 2.3 where the BAE condition, \(\Xi _{Q}(s)=0\), ∀

*s*, was obtained, we are naturally led to consider the following optimization problem to design an auxiliary system achieving the approximate BAE:

*n̄*is the mean phonon number at thermal equilibrium [44, 45]. Note that the damping effect appears in the \((2, 2)\) component of \(\widetilde{A}_{{ E}}\) due to the stochastic nature of \(\hat{f}_{\mathrm{th}}\). Also, again, \(\kappa _{ K}\) and Δ are the decay rate and the detuning of the controller cavity, respectively. In the idealized setting where \(\hat{f}_{\mathrm{th}}\) is negligible, the perfect BAE is achieved by choosing the parameters satisfying Eq. (24). The measurement output of this closed-loop system is, in the Laplace domain, represented by

## 6 Conclusion

The main contribution of this paper lies in that it first provides the general theory for constructing a back-action evading sensor for linear quantum systems, based on the well-developed classical geometric control theory. The power of the theory has been demonstrated by showing that, for the typical opto-mechanical oscillator, a full parametrization of the auxiliary coherent-feedback and direct interaction controller achieving BAE was derived, which contains the result of [22]. Note that, although we have studied a simple example for the purpose of demonstration, the real advantage of the theory developed in this paper will appear when dealing with more complicated multi-mode systems such as an opto-mechanical system containing a membrane [46–49]. Another contribution of this paper is to provide a general procedure for designing an approximate BAE sensor under realistic imperfections; that is, an optimal approximate BAE system can be obtained by solving the minimization problem of the transfer function from the back-action noise to the measurement output. While in Section 5 we have provided a simple approach based on the geometric control theory for solving this problem, the \(H_{2}\) or \(H_{\infty }\) control theory could be employed for systematic design of an approximate BAE controller even for the above-mentioned complicated system. This is also an important future research direction of this work.

This condition is satisfied if the transfer function from \(d(s)\) to \(z(s)\) is zero for all *s*, for the modified system. Or equivalently, the controllable subspace with respect to \(d(t)\) is contained in the unobservable subspace with respect to \(z(t)\).

## Declarations

### Acknowledgements

This work was supported in part by JSPS Grant-in-Aid No. 15K06151.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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