Geometric control theory for quantum back-action evasion

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.

Geometric control theory for quantum back-action evasion

http://crossmark.crossref.org/dialog/?doi=10.1140/epjqt/s40507-016-0053-5&domain=pdf

Web End = Yu Yokotera* and Naoki Yamamoto

http://orcid.org/0000-0003-3125-4188

Web End = *Correspondence: mailto:[email protected]

Web End [email protected] Department of Applied Physics and Physico-Informatics, Keio University, Hiyoshi 3-14-1, Kohoku, Yokohama, 223-8522, Japan

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 eld of gravitational wave detection []. In fact, a variety of linear sensors composed of opto-mechanical oscillators have been proposed [], and several experimental implementations of those systems in various scales have been reported [].

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) [, ], 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 eld of gravitational wave detection, e.g., the variational measurement technique [] or the quantum locking scheme []. Moreover, towards more accurate detection, recently we nd some high-level approaches to design a BAE sensor, based on those specic BAE methods. For instance, Ref. [] provides a systematic comparison of several BAE methods and gives an optimal solution. Also

2016 Yokotera and Yamamoto. 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.

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 2 of 22

systems and control theoretical methods have been developed to synthesize a BAE sensor for a specic opto-mechanical system [, ]; in particular, the synthesis is conducted by connecting an auxiliary system to a given plant system by direct-interaction [] or coherent feedback [].

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 congurations. The key tool used here is the geometric control theory [], 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 rst 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 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 , we provide the general theory for designing a coherent feedback controller achieving BAE, and demonstrate an example for an opto-mechanical system. In Section , we discuss the case of direct interaction scheme, also based on the geometric control theory. Finally, in Section , 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 nd an approximated BAE controller and show how much the designed controller can suppress the noise.

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

Notation For a matrix A = (aij), A , A, and A represent the transpose, Hermitian conjugate, and element-wise complex conjugate of A, respectively; i.e., A = (aji), A = (aji), and

A = (aij) = (A) . (a) and (a) denote the real and imaginary parts of a complex num

ber a. O and In denote the zero matrix and the nn identity matrix. Ker A and Im A denote

the kernel and the image of a matrix A, i.e., Ker A = {x|Ax = } and Im A = {y|y = Ax, x}.

2 Preliminaries2.1 Geometric control theory for disturbance decoupling

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

dx(t)dt = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t), ()

where x(t) X :=

Rn is a vector of system variables, u(t) U :=

Rm and y(t) Y :=

Rl 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

Y(s) = (s)U(s), (s) = C(sI A)B + D,

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

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 3 of 22

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

Denition Let A : X X be a linear map. Then, a subspace V X is said to be

A-invariant, if AV V.

Denition Given a linear map A : X X and a subspace Im B X , a subspace V X

is said to be (A, B)-invariant, if AV V Im B.

Denition Given a linear map A : X X and a subspace Ker C X , a subspace V X

is said to be (C, A)-invariant, if A(V Ker C) V.

Denition Assume that V is (C, A)-invariant, V is (A, B)-invariant, and V V. Then,

(V, V) is said to be a (C, A, B)-pair.

From Denitions and , we have the following two lemmas.

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

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

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

dx(t)dt = Ax(t) + Bu(t) + Ed(t), y(t) = Cx(t), z(t) = Hx(t),

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 . 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 aect at all on z(t).a This control goal is called the disturbance decoupling. Here we describe a specic 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 conguration is illustrated in Figure ; that is, the system modication is carried out by combining an auxiliary system (controller) with the original system (plant), so that the whole closed-loop system satises the disturbance decoupling

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 4 of 22

condition. The controller with variable xK XK :=

Rnk is assumed to take the following

form:

dxK(t)

dt = AKxK(t) + BKy(t), u(t) = CKxK(t) + DKy(t),

where AK : XK XK, BK : Y XK, CK : XK U, and DK : Y U are real matrices.

Then, the closed-loop system dened in the augmented space XE := X XK is given by

d dt

x xK

=

A + BDKC BCK

BKC AK

x xK

+

E O

d, z =

H O x xK

. ()

The control goal is to design (AK, BK, CK, DK) so that, in Eq. (), the disturbance signal d(t) dose not appear in the output z(t): see the endnote in Page . Here, let us dene

AE =

A + BDKC BCK

BKC AK

, ()

B = Im B, C = Ker C, E = Im E, and H = Ker H. Then, the following theorem gives the solv

ability condition for the disturbance decoupling problem.

Theorem For the closed-loop system (), the disturbance decoupling problem via the dynamical feedback controller has a solution if and only if there exists a (C, A, B)-pair (V, V)

satisfying

E V V H. ()

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 satised.

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

V(C, E) V(B, H),

where V(B, H) is the maximum element of (A, B)-invariant subspaces contained in H,

and V(C, E) is the minimum element of (C, A)-invariant subspaces containing E. These

subspaces can be computed by the algorithms given in Appendix A.

Once the solvability condition described above is satised, then we can explicitly construct the controller matrices (AK, BK, CK, DK). The following intersection and projection subspaces play a key role for this purpose; that is, for a subspace VE XE = X XK, let

us dene

VI :=

x X

x

O

VE , VP := x X

x xK

VE, xK XK .

Then, the following theorem is obtained:

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 5 of 22

Theorem Suppose that (V, V) is a (C, A, B)-pair. Then, there exist F F(V), G G(V), and DK : Y U such that Ker F V and Im G V hold, where F = F DKC,

G = G BDK.

Moreover, there exists XK with dim XK = dim V dim V, and AE has an invariant sub

space VE XE such that V = VI and V = VP. Also, (AK, BK, CK) satises

CKN = F, BK = NG, AKN = N(A + BF + GC), ()

where N : V XK is a linear map satisfying Ker N = V.

In fact, under the condition given in Theorem , let us dene the following augmented subspace VE XE:

VE :=

x Nx

x

V .

Then, V = VI and V = VP hold, and we have

AE

x Nx

=

A + BDKC BCK

BKC AK

x Nx

=

(A + BF)x N(A + BF)x

VE,

implying that VE is actually AE-invariant. Now suppose that Theorem holds, and let

us take the (C, A, B)-pair (V, V) satisfying Eq. (). Then, together with the above result

(AEVE VE), we have Im[E O] VE Ker[H O]. This implies that d(t) must be con

tained in the unobservable subspace with respect to z(t), and thus the disturbance decoupling is realized.

2.2 Linear quantum systems

Here we describe a general linear quantum system composed of n bosonic subsystems. The jth mode can be modeled as a harmonic oscillator with the canonical conjugate pairs (or quadratures) qj and pj satisfying the canonical commutation relation (CCR) qj pk

pk qj = ijk. Let us dene the vector of quadratures as x = [q, p, . . . , qn, pn] . Then, the

CCRs are summarized as

xx xx

= i n, n = diag{ , . . . , }, =

.

Note that n is a nn block diagonal matrix. The linear quantum system is an open sys

tem coupled to m environment elds via the interaction Hamiltonianint = i

mj=(Ljj

Ljj), wherej(t) is the eld annihilation operator satisfyingj(t)k(t ) k(t )j(t) =

jk(t t ). Also Lj is given by Lj = c jx with cj

Cn. In addition, the system is driven by the Hamiltonian = x Rx/ with R = R

Rnn. Then, the Heisenberg equation of x is

given by

dx(t)

dt = Ax(t) +

m

j=

Bjj(t), ()

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 6 of 22

wherej(t) is dened by

j =

Qj Pj

=

(j +j)/

(j j)/i

.

The matrices are given by A = n(R +

mj= C j Cj/) and Bj = nC j with Cj =

[ (cj), (cj)]

Rn. Also, the instantaneous change of the eld operatorj(t) via the system-eld coupling is given by

outj(t) = Cjx(t) +j(t). ()

To summarize, the linear quantum system is characterized by the dynamics () and the output (), which are exactly of the same form as those in Eq. () (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 []:

A n + nA +

m

j=

Bj B j = O, Bj = nC j . ()

2.3 Weak signal sensing, SQL, and BAE

The opto-mechanical oscillator illustrated in Figure is a linear quantum system, which serves as a sensor for a very weak signal. Let q and p be the oscillators position and

momentum operators, and = (q + ip)/ represents the annihilation operator of the

cavity mode. The system Hamiltonian is given by = m(q + p)/ g q q; that is, the

oscillators free evolution with resonant frequency m plus the linearized radiation pressure interaction between the oscillator and the cavity eld with coupling strength g. The system couples to an external probe eld (thus m = ) via the coupling operator L = , with the coupling constant between the cavity and probe elds. The corresponding matrix R and vector c are then given by

R =

m g

m

g

, c =

i

.

The oscillator is driven by an unknown force f(t) with coupling constant ; f(t) is the very weak signal we would like to detect. Then the vector of system variables x = [q, p, q, p]

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 7 of 22

satises

dx

dt = Ax + B + bf,out = Cx +,

where

A =

m

m g

/ g /

, B = C =

,

()

b =

, = [ Q, P] ,out =

Qout, Pout

.

Note that we are in the rotating frame at the frequency of the probe eld. These equations indicate that the information about f can be extracted by measuring Pout by a homodyne detector. Actually the measurement output in the Laplace domain is given by

Pout(s) = f (s)f(s) + Q(s) Q(s) + P(s)P(s), ()

where f , Q, and P are transfer functions given by

f (s) = g

m

(s + m)(s + /),

Q(s) = g

m

(s + m)(s + /) ,

P(s) = s

/ s + /.

Thus, Pout certainly contains f. Note however that it is subjected to two noises. The rst one, Q, is the back-action noise, which is due to the interaction between the oscillator and the cavity. The second one, P, is the shot noise, which inevitably appears. Now, the normalized output is given by

y(s) = Pout(s)

f (s) = f(s) +

Q(s)

f (s) Q(s) +

P(s)

f (s) P(s),

and the normalized noise power spectral density of y in the Fourier domain (s = i) is calculated as follows:

S() = |y f|

=

Q

f

| Q| +

P

f

|P|

| Q| |P| | m| m = SSQL( ).

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., | Q| |P| /. Hence, the

essential reason why SQL appears is that Pout contains both the back-action noise Q and the shot noise P. Therefore, toward the high-precision detection of f, we need BAE; that

| Q|| P|

| f |

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 8 of 22

is, the system structure should be modied 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 [, ]; i.e., for the modied (controlled) sensor, the transfer function from the back-action noise to the measurement output must satisfy

Q(s) = , s. ()

Equivalently, Pout contains only the shot noise P; hence, in this case the signal to noise ratio can be further improved by injecting a P-squeezed (meaning |P| < /) probe

eld into the system.

3 Coherent feedback control for back-action evasion3.1 Coherent and measurement-based feedback control

There are two schemes for controlling a quantum system via feedback. The rst one is the measurement-based feedback [] illustrated in Figure (a). In this scheme, we measure the output elds and feed the measurement results back to control the plant system. On the other hand, in the coherent feedback scheme [, ] shown in Figure (b), the feedback loop dose not contain any measurement component and the plant system is controlled by another quantum system. Recently we nd several works comparing the performance of these two schemes [, ]. In particular, it was shown in [] that there are some control tasks that cannot be achieved by any measurement-based feedback but can be done by a coherent one. More specically, those tasks are realizing BAE measurement, generating a quantum non-demolished variable, and generating a decoherence-free subsystem; in our case, of course, the rst 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 ., the geometric control theory for disturbance decoupling problem is formulated for the controlled system with special structure (); in particular, the coecient matrix of the disturbance d(t) is of the form [E , O] and that of the state vector in the output z(t) is [H, O]. Here we consider a class of coherent feedback conguration such that the whole closed-loop system dynamics has this structure, in order for the geometric control theory to be directly applicable.

First, for the plant system given by Eqs. () and (), we assume that the system couples to all the probe elds in the same way; i.e.,

Bj = B, j. ()

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 9 of 22

This immediately leads to Cj = C, j. Next, as the controller, we take the following special

linear quantum system with (m ) input-output elds:

dxK

dt = AK xK +

m

j=

BKj,outj = CK xK +j (j = , , . . . , m ), ()

where the matrices (AK, BK, CK) satisfy the physical realizability condition (). Note that, corresponding to the plant structure, we assumed that the controller couples to all the elds in the same way, specied by CK. Here we emphasize that the number of channels, m, should be as small as possible from a viewpoint of implementation; hence in this paper let us consider the case m = . Now, we consider the coherent feedback connection illustrated in Figure , i.e.,

= Sout, = Sout, = Tout, = Tout,

where Sj and Tj are unitary matrices representing the scattering process of the elds;

recall that the scattering processout = ei with

R the phase shift can be represented

in the quadrature form as

Qout

Pout

= S(

)

Q

P

=

cos sin sin cos

Q

P

.

Combining the above equations, we nd that the whole closed-loop system with the augmented variable xE = [x , x K] is given by

dxE

dt = AE xE + BE + bEf,out = CE xE + DE, ()

where

AE =

A + B{TS + TS(TS + I)}C B{T + T(I + ST)}CK

BK{(I + ST)S + S}C AK + BKSTCK

,

BE =

B(I + TS + TSTS)

BK(I + ST)S

,

CE =

(TSTS + TS + I)C T(ST + I)CK

,

DE = TSTS, bE =

b O

.

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 10 of 22

Therefore, the desired system structure of the form () is realized if we take

ST = I. ()

In addition, it is required that the back-action noise Q dose not appear directly in Pout, which can be realized by taking

DE = TS = I. ()

Here we set Sj and Tj to be the /-phase shifter (see Figure ) to satisfy the above conditions () and ();

Sj = Tj = S =

(j = , ). ()

As a consequence, we end up with

AE =

A BC BSCKBKSC AK BKCK

, bE =

b O

, BE =

B

O

,

()

This is certainly of the form () with DK = I. Hence, we can now directly apply the geometric control theory to design a coherent feedback controller achieving BAE; that is, our aim is to nd (AK, BK, CK) such that, for the closed-loop system (), the back-action noise Q (the rst element of) does not appear in the measurement output Pout (the second element ofout). Note that those matrices must satisfy the physical realizability condition (), and thus they cannot be freely chosen. We need to take into account this additional constraint when applying the geometric control theory to determine the controller matrices.

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

Here we apply the coherent feedback scheme elaborated in Section . to the opto-mechanical system studied in Section .. The goal is, as mentioned before, to determine the controller matrices (AK, BK, CK) such that the closed-loop system achieves BAE. Here,

CE =

C O , DE = I.

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we provide a step-by-step procedure to solve this problem; the relationships of the class of controllers determined in each step is depicted in Figure .(i) First, to apply the geometric control theory developed above, we need to modify the plant system so that it is a input-output linear quantum system; here we consider the plant composed of a mechanical oscillator and a -ports optical cavity, shown in Figure . As assumed before, those ports have the same coupling constant . In this case the matrix A given in Eq. () is replaced by

A =

m

m g

/ g /

.

Now we focus only on the back-action noise Q and the measurement output Pout; hence the closed-loop system () and (), which ignores the shot noise term in the dynamical equation, is given by

dxE

dt =

A BC BSCK

BKSC AK BKCK

xE +

E O

Q +

b O

f,

H O xE + P,

where B = B, C = C, and b are given in Eq. (), and

E =

, H =

Pout =

.

This system is certainly of the form (), where now DK = I.(ii) In the next step we apply Theorem to check if there exists a feedback controller such that the above closed-loop system achieves BAE; recall that the necessary and sucient condition is Eq. (), i.e., E V V H, where now

E = Im E = span

, H = Ker H = span

,

,

.

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To check if this solvability condition is satised, we use Corollary ; from E C = Im E

Ker C = and H B = Ker H Im B =

R, the algorithms given in Appendix A yield

V(C, E) = E, V(B, H) = H, ()

implying that the condition in Corollary , i.e., V(C, E) V(B, H), is satised. Thus, we

now see that the BAE problem is solvable, as long as there is no constraint on the controller parameters.

Next we aim to determine the controller matrices (AK, BK, CK), using Theorem . First we set V = V(C, E) = E and V = V(B, H) = H; note that (V, V) is a (C, A, B)-pair. Then,

from Theorem , there exists a feedback controller with dimension dim XK = dim V

dim V = . Moreover, noting again that DK = I, there exist matrices F F(V), G G(V), and N such that

Ker F = Ker(F + C) V, Im G = Im(G + B) V, Ker N = V.

These conditions lead to

F =

f f f

g f

g

g

g g g

, G =

, N =

n n n

n n n

,

where fij, gij, and nij are free parameters. Then the controller matrices (AK, BK, CK) can be identied by Eq. () with the above matrices (F, G, N); specically, by substituting CK

SCK, BK BKS, and AK AK BKCK in Eq. (), we have

SCKN = F + C, BKS = N(G + B), (AK BKCK)N = N(A + BF + GC),

which yield

AK = N(A + BF + GC + GF)N+, BK = NG , CK = FN+, ()

where N+ is the right inverse to N, i.e., NN+ = I.(iii) Note again that the controller () has to satisfy the physical realizability condition (), which is now AK + A K + BK B K = O and BK = C K . These constraints are represented in terms of the parameters as follows:

f = g, f = g, nn nn = , fn = fn f,

f + = g

n,

+ f

!n + mn = f,

mn

()

+ f

!n = f,

where n = nn nn and n = nn nn. This is one of our main results; the linear controller () achieving BAE for the opto-mechanical oscillator can be fully

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 13 of 22

parametrized by Eq. () satisfying the condition (). We emphasize that this full parametrization of the controller can be obtained thanks to the general problem formulation based on the geometric control theory.(iv) In practice, of course, we need to determine a concrete set of parameters to construct the controller. Especially here let us consider a 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 []. Now the condition for the system (AK, BK, CK) to be passive is given by AK = AK and BK = BK; the general result of this fact is given in Theorem in Appendix B. From these conditions, the system parameters are imposed to satisfy, in addition to Eq. (), the following equalities:

f = g

, f = , n = n, n = n. ()

There is still some freedom in determining nij, which however corresponds to simply the phase shift at the input-output ports of the controller, as indicated from Eq. (). Thus, the passive controller achieving BAE in this example is unique up to the phase shift. Here particularly we chose n = and n = . Then the controller matrices () satisfying Eqs. () and () are determined as

AK =

As illustrated in Figure , the controller specied by these matrices can be realized as a single-mode, -inputs and -outputs optical cavity with decay rate g/ and detuning m. In other words, if we take the cavity with the following Hamiltonian and the coupling operator ( = (q + ip)/ is the cavity mode)

K = =

g

g

g m m g

, CK = B K =

.

, LK = K = K (q + ip), ()

then to satisfy the BAE condition the controller parameters ( , K) must satisfy

= m, K = g/. ()

To summarize, the above-designed sensing system composed of the opto-mechanical oscillator (plant) and the optical cavity (controller), which are combined via coherent feedback, satises the BAE condition. Hence, it can work as a high-precision detector of the force f below the SQL, particularly when the P-squeezed probe input eld is used; this fact will be demonstrated in Section .

4 Direct interaction scheme

In this section, we study another control scheme for achieving BAE. As illustrated in Figure (a), the controller in this case is directly connected to the plant, not through a coher-

q + p

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 14 of 22

ent feedback; hence this scheme is called the direct interaction. The controller is characterized by the following two Hamiltonians:

K =

x KRK xK,int =

x RxK + x KRx

, ()

where xK = [q , p , . . . , q nk, p nk] is the vector of controller variables with nk the number of modes of the controller.K is the controllers self Hamiltonian with RK

Rnknk .

Alsoint with R

Rnnk , R

Rnkn represents the coupling between the plant and the controller. Note that, for the HamiltoniansK andint to be Hermitian, the matrices must satisfy RK = R K and R = R; these are the physical realizability conditions in the scenario of direct interaction. In particular, here we consider a plant system interacting with a single probe eld, with coupling matrices B = B and C = C. Then, the whole dynamics of the augmented system with variable xE = [x , x K] is given by

dxE

dt = AE xE + BE + bEf,out = CE xE +, ()

where

AE =

A nR nkR nkRK

, BE =

B

O

, CE =

C

O

, bE =

b

O

. ()

Note that BE, CE, and bE are the same matrices as those in Eq. (). Also, comparing the matrices () and (), we have that DK = O, which thus leads to F = F and G = G in Theorem . Now, again for the opto-mechanical system illustrated in Figure , let us aim to design the direct interaction controller, so that the whole system () achieves BAE; that is, the problem is to determine the matrices (RK, R, R) so that the back-action noise Q does not appear in the measurement output Pout. For this purpose, we go through the same procedure as that taken in Section ..

(i) Because of the structure of the matrices BE and CE, the system is already of the form (), where the geometric control theory is directly applicable.(ii) Because we now focus on the same plant system as that in Section ., the same conclusion is obtained; that is, the BAE problem is solvable as long as there is no constraint on the controller matrices (RK, R, R).

The controller matrices can be determined in a similar way to Section . as follows. First, because the (C, A, B)-pair (V, V) is the same as before, it follows that dim XK = ,

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 15 of 22

i.e., nk = . Then, from Theorem with the fact that F = F and G = G, we nd that the direct interaction controller can be parameterized as follows:

RK = N(A + BF + GC)N+, R = BFN+, R = NGC. ()

The matrices F, G, and N satisfy Ker F V, Im G V, and Ker N = V, which lead to

F =

f f f

g f

, G =

g

g

g g g

,

()

N =

n n n

n n n

,

where fij, gij, and nij are free parameters.(iii) The controller matrices have to satisfy the physical realizability conditions RK = R K

and R = R; these constraints impose the parameters to satisfy

f = g, f = g, nn nn = , fn = fn f,

f = g

n,

f

!n + mn = f,

mn

()

f

!n = f,

where n = nn nn and n = nn nn. Equations (), (), and () provide the full parametrization of the direct interaction controller.(iv) To specify a set of parameters, as in the case of Section ., let us aim to design a passive controller. From Theorem in Appendix B, RK and R = R satisfy the condition

RK = RK and R = R, which lead to the same equalities given in Eq. (). Then, setting the parameters to be n = and n = , we can determine the matrices RK and R as follows:

RK =

m

m

, R = R =

g g

.

The controller specied by these matrices can be physically implemented as illustrated in Figure (b); that is, it is a single-mode detuned cavity with HamiltonianK = m,

which couples to the plant through a beam-splitter (BS) represented byint = g( +

).

Remark We can employ an active controller, as proposed in []. In this case the interaction Hamiltonian is given byint = gB( +) + gD( +), while the systems self-Hamiltonian is the same as above;K = m. That is, the controller couples to

the plant through a non-degenerate optical parametric amplication process in addition

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 16 of 22

to the BS interaction. To satisfy the BAE condition, the parameters must satisfy gB +gD = g.

Note that this direct interaction controller can be specied, in the full-parameterization (), (), and (), by

f = f = f = , n = n = , n = n = .

5 Approximate back-action evasion

We have demonstrated in Sections . and that the BAE condition can be achieved by engineering an appropriate auxiliary system and connecting it to the plant. However, in a practical situation, it cannot be expected to realize such perfect BAE due to several experimental imperfections. Hence, in a realistic setup, we should modify our strategy for engineering a sensor so that it would accomplish approximate BAE. Then, looking back into Section . where the BAE condition, Q(s) = , s, was obtained, we are naturally

led to consider the following optimization problem to design an auxiliary system achieving the approximate BAE:

min """"

()

where denotes a valid norm of a complex function. In particular, in the eld of robust

control theory, the following H norm and the H norm are often used []:

=

$AE xE + BE + bE(fth + f),out = CE xE +, ()

Q(s)

f (s)

"""",

# (i) d,

(i) .

That is, the H or H control theory provides a general procedure for synthesizing a feedback controller that minimizes the above norm. In this paper, we take the H norm, mainly owing to the broadband noise-reduction nature of the H controller. Then, rather than pursuing an optimal quantum H controller based on the quantum H control theory [, ], here we take the following geometric-control-theoretical approach to solve the problem (). That is, rst we apply the method developed in Section or to the idealized system and obtain the controller achieving BAE; then, in the practical setup containing some unwanted noise, we make a local modication of the controller parameters obtained in the rst step, to minimize the cost Q(s)/ f (s) .

As a demonstration, here we consider the coherent feedback control for the opto-mechanical system studied in Section ., which is now subjected to the thermal noise fth.

Following the above-described policy, we employ the coherent feedback controller constructed for the idealized system that ignores fth, leading to the controller given by Eqs. ()

and (), illustrated in Figure . The closed-loop system with variable xE = [x , x K] ,

which now takes into account the realistic imperfections, then obeys the following dynamics:

dxE

dt =

= max

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 17 of 22

where

$AE =

m

m g

/ K

g / K

K

K

.

BE, CE, and bE are the same matrices given in Eq. (). fth is the thermal noise satisfying fth(t)fth(t ) n(t t ), where n is the mean phonon number at thermal equilibrium [, ]. Note that the damping eect appears in the (, ) component of

$AE due to the stochastic nature of fth. Also, again, K and are the decay rate and the detuning of the controller cavity, respectively. In the idealized setting where fth is negligible, the perfect

BAE is achieved by choosing the parameters satisfying Eq. (). The measurement output of this closed-loop system is, in the Laplace domain, represented by

Pout(s) =

$ f

$ P P(s).

The normalized noise power spectral density of y(s) = Pout(s)/

$ f (s) is calculated as

fth(s) + f(s)

+ $ Q Q(s) +

$S() =

y

(i) f(i)

=

|fth| +

$ Q

$ f

| Q| +

$ P

$ f

|P|

. ()

The coecient of the back-action noise is given by

$ Q(s)

$ f (s) =

{K (s + s + m) + gm(s + )}

gm {(s + /)(s + ) + Ks}

. ()

Our goal is to nd the optimal parameters (K, ) that minimize the H norm of the transfer function,

$ Q/

$ f .

The system parameters are taken as follows []: m/ = . MHz, / = . MHz, / = . kHz, g/ = . MHz, n . , and the eective mass is . kg.

We then have Figure , showing

$ Q/

$ f as a function of K and . This gure shows that there exists a unique pair of (optK, opt) that minimizes the norm, and they are given by optK/ = . MHz and opt/ = . MHz, which are actually close to the ideal

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 18 of 22

values (). Figure shows the value of Eq. () with these optimal parameters (optK, opt),

where the noise oor |fth| is subtracted. The solid black line represents the SQL, which

is now given by

$SSQL() = |(

m) i |

m . ()

Then the dot-dashed blue and dotted green lines indicate that, in the low frequency range, the coherent feedback controller can suppress the noise below the SQL, while, by denition, the noise power of the autonomous (i.e., uncontrolled) plant system is above the SQL. Moreover, this eect can be enhanced by injecting a P-squeezed probe eld (meaning | Q| = er/ and |P| = er/) into the system. In fact the dashed red line in the

gure illustrates the case r = (about dB squeezing), showing the signicant reduction of the noise power.

6 Conclusion

The main contribution of this paper lies in that it rst 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 []. 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 []. 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 we have provided a simple approach based on the geometric control theory for solving this problem, the H or H 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.

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 19 of 22

Appendix A: Algorithms for computing V and V

The set of (A, B)-invariant subspaces has a unique maximum element contained in a given subspace H X . This space, denoted by V(B, H), can be computed by the following

algorithm:

V-algorithm:(Step ) V := H,(Step ) Vi := H A(Vi B) (i = , , . . .), (Step ) V(B, H) = Vi (if Vi = Vi in Step ).

Similarly, the set of (C, A)-invariant subspaces has a unique minimum element containing a given subspace E X , and this space, denoted by V(C, E), can be computed by the

following algorithm:

V-algorithm:(Step ) V := E,(Step ) Vi := E A(Vi C) (i = , , . . .), (Step ) V(C, E) = Vi (if Vi = Vi in Step ).

Appendix B: Passivity condition of linear quantum systems

This appendix provides the passivity condition of a general linear quantum system, which have been given in []. First note that the system dynamics () and (), which can be represented as

dx

dt = Ax + B,out = Cx + D, ()

with = [, . . . ,m] , has the following equivalent expression:

d dt

= A

+ B

, out

out

=

C

+ D

, ()

where = [, . . . ,n] and = [, . . . ,m] are vectors of annihilation operators. By def

inition, = [, . . . ,n] . The coecient matrices are of the form

A =

A A+ A + A

, B =

B B+ B + B

,

()

As in the case of (), these matrices have to satisfy the physical realizability condition; see [, ]. The passivity condition of this system is dened as follows:

C =

C C+ C + C

, D =

D D+ D + D

.

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 20 of 22

Denition The system () is said to be passive if the matrices satisfy A+ = O and B+ = O, in addition to the physical realizability condition.

Note that a passive system is constituted only with annihilation operator variables; a typical optical realization of the passive system is an empty optical cavity. Moreover,

D+ = O is already satised and B+ = O leads to C+ = O. This is the reason why it is sucient to consider the constraints only on A+ and B+. Then the goal here is to represent the conditions A+ = B+ = O in terms of the coecient matrices of Eq. (). For this purpose, let us introduce the permutation matrix Pn as follows; for a column vector z = [z, z, . . . , zn] , Pn is dened through Pnz = [z, z, . . . , zn, z, z, . . . , zn] . Note that Pn satises PnP n = P nPn = In. Then, the coecient matrices of the above two system representations are connected by

A = P n APn, B = P n BPm, C = P m CPn, D = P m DPm,

where

A =

A + A + A+ + A + i(A A A+ + A +)

i(A A + A+ A +) A + A (A+ + A +)

B, C, and D have the same forms as above. Then, we have the following theorem, providing the passivity condition in the quadrature form:

Theorem The system () is passive if and only if, in addition to the physical realizability condition (), the following equalities hold:

nA n = A, nB m = B.

Proof Let us rst dene n = In = [O, In; In, O], which leads to P n nPn = n. Then,

we can prove

nA n = A n A n = A.

The condition in the right hand side is equivalent to A+ + A + = O and A+ A + = O, which thus leads to A+ = O. Also, from a similar calculation we obtain nB m = B B+ = O.

Let us next consider the passivity condition of the direct interaction controller discussed in Section . The setup is that, for a given linear quantum system, we add an auxiliary component with variable xK, which is characterized by the Hamiltonians (). The point

is that these Hamiltonians have the following equivalent representations in terms of the vector of annihilation operators andK:

K =

K K

K

K

.

K K RK

,

()

int =

%

R

+ K K R

&

.

Yokotera and Yamamoto EPJ Quantum Technology (2016) 3:15 Page 21 of 22

The matrices RK, R, and R are of the same forms as those in Eq. (). Note that they have to satisfy the physical realizability conditions RK = RK and R = R. Now we can dene the passivity property of the direct interaction controller; that is, if the Hamiltonians () does not contain any quadratic term such asK, andK,, then the direct

interaction controller is passive. The formal denition is given as follows:

Denition The direct interaction controller constructed by Hamiltonians () is said to be passive if, in addition to the physical realizability conditions RK = RK and R = R, the matrices satisfy RK+ = O and R+ = O.

Through almost the same way shown above, we obtain the following result:

Theorem The direct interaction controller constructed by Hamiltonians () is passive if and only if, in addition to the physical realizability conditions RK = R K and R = R, the following equalities hold:

nkRK nk = RK, nkR n = R.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to the writing of this paper. All authors read and approved the nal manuscript.

Acknowledgements

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

Endnote

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

Received: 13 September 2016 Accepted: 18 October 2016

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