Quantum functionalities via feedback amplification
Rion Shimazu Naoki Yamamoto



[]shark98t@keio.jp

[]yamamoto@appi.keio.ac.jp
Department of Applied Physics and Physico-Informatics, 
Keio University, Hiyoshi 3-14-1, Kohoku, Yokohama 223-8522, Japan



Feedback amplification is a key technique for synthesizing various important 
functionalities, especially in electronic circuits involving op-amps. 
This paper presents a quantum version of this methodology, where the general 
phase-preserving quantum amplifier and coherent (i.e., measurement-free) feedback 
are employed to construct various type of systems having useful functionalities: 
quantum versions of differentiator, integrator, self-oscillator, and active filters. 
The class of active filters includes the Butterworth filter, which can be used to 
enhance the capacity of an optical quantum communication channel, and 
the non-reciprocal amplifier, which enables measurement of a superconducting qubits system 
as well as protection of it by separating input from output fields. 
A particularly detailed investigation is performed on the active phase-cancelling filter 
for realizing a broadband gravitational-wave detector; 
that is, the feedback amplification method is used to construct an active filter that 
compensates the phase delay of the signal and eventually recovers the sensitivity 
in the high frequency regime. 


Quantum functionalities via feedback amplification
    Naoki Yamamoto
    December 30, 2023
==================================================













§ INTRODUCTION





 



The amplifier is an essential component in modern technological systems, and it is 
usually involved in those systems in some feedback form. 
Let us consider a classical amplification process y=Gu where u and 
y are input and output signals, and G>1 is the gain of the amplifier. 
Then by feeding a fraction of the output back to the input through the controller 
K, as depicted in Fig. <ref>, the input-output relation is 
modified to 


    y=G^( fb) u,      
         G^( fb)=G/1+GK=1/1/G + K.


Then by making the gain G large, we find y=(1/K)u; hence if K is a passive 
device with gain K<1, the entire system works as a robust amplifier which is 
insensitive to the parameter change in G. 
The importance of this feedback amplification technique <cit.> 
is not limited to realizing of such a robust amplifier. 
That is, by combining high-gain amplifiers (op-amps in the electrical circuits) 
with several passive devices such as resistors and capacitors, one can devise a 
variety of functional systems; e.g., integrator, active filters, switches, and self-oscillators 
<cit.>. 



This paper develops the quantum version of feedback amplification theory, which is 
expected to be of particular importance to make the existing quantum technological 
devices robust and further to engineer systems with functionalities. 
In fact this idea has been implicitly employed in some specific systems 
<cit.>. An explicit research direction was addressed in 
<cit.>, showing 
a general quantum analogue to the above-described robust amplification method; 
more precisely, it is shown that a coherent (i.e., measurement-free) feedback 
control <cit.> of a high-gain phase-preserving amplifier 
<cit.> and a passive device 
(e.g., a beam splitter) yields a robust phase-preserving amplifier. 



This paper begins with Sec. II to introduce the models of the quantum phase-preserving amplifier 
and some linear passive systems. Then, using those models, 
we extend the quantum feedback amplification scheme presented in <cit.> from 
the Fourier domain to the Laplace domain (Sec. III), together with developing a basic stability test 
method (Sec. IV). 
We then apply the theory to construct systems having several useful functionalities: 
quantum versions of differentiator and integrator (Sec. V), self-oscillator 
(Sec. VI), and active filters (Sec. VII). 
As for the quantum integrator, it will be proven applicable for improving the 
detection efficiency of an itinerant field. 
The ability to synthesize a quantum self-oscillator might also be useful for several purposes 
as in the classical case, such as analogue quantum memory and frequency converter 
<cit.>, though in this paper we do not provide a concrete example. 
Active filtering is a typical application of feedback amplification, which in our 
case includes the quantum version of Butterworth filter <cit.> and 
non-reciprocal amplifier; 
the former is used to realize the steep roll-off characteristic in frequency, 
which enables the enhancement of the capacity of a quantum communication channel 
<cit.>; the latter enables precise measurement of a superconducting qubits system 
while protecting it from the unwanted backward field generated in the amplification process 
<cit.>. 



In particular, in Sec. VIII we show a detailed investigation on the quantum phase-cancellation 
filter applied to the gravitational-wave detection problem; 
this is an active filter that can compensate the delayed phase of an incoming signal 
for the purpose of enhancing the detection bandwidth. 
The quantum phase-cancellation filters proposed in the literature 
<cit.> are based on an opto-mechanical 
implementation, but it requires an extremely low environmental temperature. 
The proposed phase-cancellation filter based on the feedback amplification method, on the 
other hand, can be all-optically implemented in the room temperature. 
We demonstrate a numerical simulation to show how much this filter can broaden the 
bandwidth of a typical gravitational-wave detector in a practical setting. 











§ PRELIMINARIES






 §.§ Phase preserving linear amplifier










In this paper we consider a general phase preserving linear amplifier 
<cit.>. 
A typical realization of this system is given by the non-degenerate parametric 
amplifier (NDPA) <cit.>. 
In optics case, as depicted in Fig. <ref>, the NDPA is an optical cavity having 
two orthogonally polarized fields with modes a_1 and a_2, which are created and 
coupled with each other at the pumped non-linear crystal (the green box in 
Fig. <ref>) inside the cavity. 
Also, the mode a_1 (a_2) couples with an input field b_1 (b_2) at the mirror 
with transmissibity proportional to γ. 
The Hamiltonian of the NDPA is given by


    H_NDPA=   ħω_1a_1^† a_1+ħω_2a_2^† a_2
       +iħλ (a_1^† a_2^† e^-2iω_pt - a_1a_2e^2iω_pt),


with ω_k the resonant frequencies of a_k, λ∈ℝ the coupling 
strength between a_1 and a_2, and 2ω_p the pump frequency. 
Here we assume that ω_1=ω_2=ω_p. 
Then, in the rotating frame at frequency ω_p, the dynamics of the NDPA is given 
by the following Langevin equation <cit.>:


    [ [   ȧ_1; ȧ_2^†;       ]]
           = [ [ -γ/2    λ;    λ -γ/2;      ]]
                [ [   a_1; a_2^†;       ]]
               -√(γ)[ [   b_1; b_2^†;       ]].


Note that the canonical commutation relation of input fields is given by 
[b(t), b^†(t')]=δ(t-t'), with δ(t-t') the Dirac delta function. 
The output equations are given by


    b̃_1=√(γ)a_1 + b_1,   b̃_2^†=√(γ)a_2^† + b_2^†.


From Eqs. (<ref>) and (<ref>), 
the input-output relation of the NDPA is represented as 


    [ [     b̃_1(s); b̃_2^†(s^∗);             ]]
           = 1/(s+γ/2)^2-λ^2
          ×[ [ s^2 -λ^2 - γ^2/4              -γλ;              -γλ s^2 -λ^2 - γ^2/4;                  ]]            
                [ [     b_1(s); b_2^†(s^∗);            ]].


The operator b(s) is related to b(t) via the Laplace transformation 
<cit.>: 


    b(s)=∫_0^∞ e^-stb(t)dt,   
            b^†(s)=[b(s)]^†=∫_0^∞ e^-s^∗ tb^†(t)dt.


From Eq. (<ref>), γ>2λ if and only if 
the amplifier is stable (i.e., every solution of the characteristic polynomial 
(s+γ/2)^2-λ^2=0 has negative real part). 
The output mode b̃_1 at s=0 is given by 


    b̃_1(0) 
            = -γ^2 + 4λ^2/γ^2 - 4λ^2 b_1(0) 
              + -4γλ/γ^2 - 4λ^2 b_2^†(0),


which diverges as γ→ 2λ+0. 
Hence, in this parameter limit, the signal with s satisfying |s|≪γ 
is largely amplified. 



In this paper we consider the general phase-preserving linear amplifier with 
the following input-output relation: 


    [ [     b̃_1(s); b̃_2^†(s^∗);             ]]
            =     G(s)
                [ [     b_1(s); b_2^†(s^∗);            ]],
    
            G(s)=   [ [ G_11(s) G_12(s); G_21(s) G_22(s);         ]].


The condition on the transfer function matrix G(s) is represented in the 
Fourier domain as follows. 
The Fourier transformation of the field operators are defined as


    b(iω)=∫_-∞^∞ e^-iω tb(t)dt,
       b^†(iω)=[b(iω)]^†
                =∫_-∞^∞ e^iω tb^†(t)dt,


which satisfy [b(iω),b^†(iω')]=2πδ(ω-ω'). 
This commutation relation requires G(s) to satisfy 


    |G_11(iω)|^2 - |G_12(iω)|^2 
        = 
           |G_22(iω)|^2 -|G_21(iω)|^2 = 1, 
        G_21(iω)G_11^*(iω) - G_22(iω)G_12^*(iω)=0,  
          ∀ω,


where G_ij^*(iω)=[G_ij(iω)]^∗ is the complex conjugate of 
G_ij(iω). 






 §.§ Passive systems





 



The general form of passive linear system from the inputs (b_3, b_4) to 
the outputs (b̃_3, b̃_4) in the Laplace domain is represented as 


    [ [ b̃_3^†(s^∗); b̃_4^†(s^∗);             ]]
            =   K(s) 
                [ [ b_3^†(s^∗); b_4^†(s^∗);            ]],
    
            K(s)=   [ [ K_11(s) K_12(s); K_21(s) K_22(s);         ]],


where the creation-mode representation is used to simplify the notation. 
The transfer function K(s) satisfies |K_11(iω)|^2 + |K_12(iω)|^2 = 1, 
|K_21(iω)|^2 + |K_22(iω)|^2 = 1, and 
K_21(iω)K_11^*(iω) + K_22(iω)K_12^*(iω)=0, 
∀ω. 
These conditions are derived from unitarity of the response function matrix of 
passive quantum system:

    K^-1(iω) = K^† (iω) =
             [ [ K_11^∗ (iω) K_21^∗ (iω); K_12^∗ (iω) K_22^∗ (iω);             ]].


A typical passive device is a single-mode optical cavity having two input-output 
ports, depicted in Fig. <ref>(a). 
The dynamics of the cavity is given by 


    ȧ_3^† =  (- κ_1+κ_2/2 + iΔ)a_3^†
        - √(κ_1)b_3^† - √(κ_2)b_4^†,


where a_3 is the cavity mode, κ_i is the coupling strength between a_3 and 
the input itinerant field b_i, and Δ is the detuning. 
Also the output equations are given by 


    b̃_3^†=√(κ_1)a_3^† + b_3^†,   b̃_4^†=√(κ_2)a_3^† + b_4^†.


Then the transfer function matrix K(s) is given by


    K   (s) =  1/s+(κ_1+κ_2)/2-iΔ
       ×[ [ s + (κ_2 - κ_1)/2 -iΔ            -√(κ_1κ_2);            -√(κ_1κ_2) s + (κ_1 - κ_2)/2 -iΔ;                       ]].


In the special case κ_1=κ_2=κ and Δ=0, it is 


    K(s)       
            =  1/s+κ[ [  s -κ; -κ  s;    ]].


Hence the relation between b_3 and b̃_4 is given by 


    b̃_4^†(s^∗) 
             = -κ/s+κ b_3^†(s^∗)
             + s /s+κ b_4^†(s^∗).


That is, in the domain |s|≫κ, the cavity works as an integrator for 
the transmitting field from b_3 to b̃_4. 
Also it works as a low-pass filter with bandwidth κ; that is, the frequency 
components of b_3 with |s|=|iω|≪κ can only pass through the cavity, 
and hence this cavity is called the mode-cleaning cavity (MCC). 




In this paper we also work on the case where b̃_4 is the reflected field of b_3, 
as shown in Fig. <ref>(b); 
in this case the transfer function is given by 


    K(s) =  1/s+(κ_1+κ_2)/2-iΔ
          ×[ [            -√(κ_1κ_2) s + (κ_1 - κ_2)/2 -iΔ; s + (κ_2 - κ_1)/2 -iΔ            -√(κ_1κ_2);                       ]].


Again in the special case κ_1=κ_2=κ and Δ=0, it is 


    K(s)       
            =  1/s+κ[ [ -κ  s;  s -κ;    ]].


Hence the relation between b_3 and b̃_4 is given by 


    b̃_4^†(s^∗) 
             = s/s+κ b_3^†(s^∗)
             + -κ/s+κ b_4^†(s^∗).


That is, at around s = 0, the cavity works as a differentiator for the 
reflected field from b_3 to b̃_4. 
Also it works as a high-pass filter with bandwidth κ; that is, the optical 
components of b_3 in the domain |s|=|iω|≫κ can only pass through 
the cavity. 
We also call this cavity as a MCC. 










§ QUANTUM FEEDBACK AMPLIFICATION





 


In this paper we consider the general feedback-connected system shown in 
Fig. <ref>, composed of the high-gain quantum phase-preserving 
amplifier G and a passive system K. 
The feedback structure is made by 


    b̃_2=b_3,   
          b_2=b̃_4,


which are of course the same as b̃_2^†=b_3^† and 
b_2^†=b̃_4^†. 
The entire system has the inputs (b_1, b_4^†) and the outputs 
(b̃_1, b̃_3^†). 
From Eqs. (<ref>) and (<ref>), 
the input-output relation of this system is given by


    [ [     b̃_1(s); b̃_3^†(s^∗);             ]]
              = G^( fb)(s)
                [ [     b_1(s); b_4^†(s^∗);            ]],
    
          
             G^( fb)(s) =
             [ [ G_11^( fb)(s) G_12^( fb)(s); G_21^( fb)(s) G_22^( fb)(s);               ]],


where 

    G_11^( fb)
        =   G_11 - K_21[G]/1-K_21G_22,
    
    G_12^( fb)
        =   G_12 K_22/1-K_21G_22,
    
    G_21^( fb)
        =   G_21 K_11/1-K_21G_22,
    
    G_22^( fb)
        =   K_12 + G_22[K]/1-K_21G_22,


with [G]= G_11G_22-G_12G_21 and 
[K]= K_11K_22-K_12K_21. 
These matrix entries satisfy 
|G_11^( fb)(iω)|^2 - |G_12^( fb)(iω)|^2 =1, ∀ω, etc, 
meaning that it also functions as a phase-preserving amplifier. 



It was shown in <cit.> that 
|G_11^( fb)(iω)| ≈ 1/|K_21(iω)| holds in the high-gain 
amplification limit |G_11(iω)| →∞; 
because the characteristic change in the passive transfer function K(s) is usually 
very small, this realizes the robust quantum amplification, which is the quantum 
analogue to the classical feedback amplification technique mentioned in the first 
paragraph in Sec. <ref>. 
We now extend this idea to the Laplace domain. 
The point to derive the result is that, from Eq. (<ref>), we have 


    [G(iω)]
            =G_11(iω)G_22(iω)
               -G_12(iω)
                  G_22(iω)G_12^∗(iω)/G_11^∗(iω)
          
        =(|G_11(iω)|^2
            -|G_12(iω)|^2)G_22(iω)/G_11^∗(iω)
        =G_22(iω)/G_11^∗(iω),


and thus 


    [G(iω)]/G_22(iω)
             =1/G_11^∗(iω)→ 0,


in the high-gain limit |G_11(iω)| →∞. 
Also again from Eq. (<ref>), |G_11(iω)|=|G_22(iω)| and 
|G_12(iω)|=|G_21(iω)| hold. Then in the same limit, 
Eq. (<ref>) leads to 


    1-|G_12(iω)/G_11(iω)|^2
            =1/|G_11(iω)|^2→ 0  ⟹  |G_12(iω)/G_22(iω)|→ 1,
    
       1-|G_21(iω)/G_22(iω)|^2
            =1/|G_22(iω)|^2→ 0  ⟹  |G_21(iω)/G_22(iω)|→ 1.


These are equivalent to 


    G_12(iω)/G_22(iω)→ e^iθ(ω),   G_21(iω)/G_22(iω)→ e^iφ(ω),


where θ(ω) and φ(ω) are certain real functions of ω.



We now extend the above result and assume that 


    [G(s)]/G_22(s)→ 0,   G_12(s)/G_22(s)→ 1,   G_21(s)/G_22(s)→ 1


hold in the domain 𝒟={  s∈ℂ  ;  |G_11(iω))|→∞}. 
Moreover, we assume G_11(s) = G_22(s) for all s∈ℂ. 
These conditions are indeed satisfied in the case of NDPA shown in 
Sec. <ref>, for s satisfying |s|≪γ, 
where the high-gain limit is realized by taking γ→ 2λ+0. 
Under the above assumptions, the transfer function matrix of the entire 
closed-loop system can be approximated by 


    G^( fb)(s)= -1/K_21(s)[ [       1 K_22(s); K_11(s)  [K(s)];         ]],


in the domain 𝒟. 
Hence, we now have a quantum system that, as will be proven later, generates 
several interesting and robust functionalities available in the feedback 
amplification setting. 



The proof of Eq. (<ref>) is as follows:


    G_11^(fb)
          =   G_11-K_21[G]/1-K_21G_22
          =1 - K_21( [G]/G_22)/(1/G_22)-K_21
    →    -1/K_21,
    
          G_12^(fb)
              =   G_12K_22/1-K_21G_22
              =(G_12/G_22)K_22/(1/G_22)-K_21→  -K_22/K_21,
    
         G_21^(fb)
         =   G_21K_11/1-K_21G_22
         =(G_21/G_22)K_11/(1/G_22)-K_21→ 
             -K_11/K_21.
    
          G_22^(fb)
            =   K_12+G_22[K]/1-K_21G_22
            =(K_12/G_22) + [K]/(1/G_22)-K_21
    →    - [K]/K_21.


We again emphasize that Eq. (<ref>) is the system only depending on 
the passive component K, meaning that G^(fb) is robust against the 
characteristic change in G. 
Note that, for a general phase preserving amplifier which does not necessarily 
satisfy Eq. (<ref>) and 
G_11(s) = G_22(s) ∀ s∈ℂ, the resulting closed-loop system 
in the high-gain limit may still contain some components of G. 
Hence, following the convention of the classical feedback amplification theory, 
we call G satisfying these assumptions the ideal quantum op-amp. 










§ STABILITY ANALYSIS METHOD







From an engineering viewpoint, it is important to guarantee the stability of 
the entire controlled system before activating it (more precisely, before closing the loop for control). 
In the classical case the seminal Nyquist method <cit.>
is often used for this purpose. Here we show the quantum version of this method, 
particularly for the quantum 
feedback-controlled system with transfer function matrix (<ref>); 
note that, hence, the stability must be guaranteed for the system with finite 
amplification gain. 


Let us represent the matrix entries of G(s) and K(s) as 
G_ij(s)=g_ij(s)/g(s) and K_ij(s)=k_ij(s)/k(s), respectively, where 
g(s), g_ij(s), k(s), and k_ij(s) are the polynomial functions. 
Then, it is easy to see that G^( fb)(s) has the following form: 


    G^( fb)(s)
             = 1/g^2(s) k^2(s) {1-K_21(s)G_22(s)}[ [ ⋆ ⋆; ⋆ ⋆;   ]],


where for simplicity the matrix entries, the polynomial functions denoted by 
⋆, are not shown. 
Here we assume that the original systems G(s) and K(s) are stable; 
then because k(s) and g(s) are stable polynomial functions (meaning that 
the zeros of k(s) and g(s) lie in the left side plane in ℂ), 
the stability of the closed-loop system is completely characterized by the 
zeros of 1-K_21(s)G_22(s). 

    
We can now apply the classical Nyquist method to test the stability of this 
closed-loop system. 
As in the classical case let us define the open-loop transfer function:


    L(s) = -K_21(s)G_22(s).


Then, from the Nyquist theorem, the simplest stability criterion is as follows. 
If the point -1 lies outside the Nyquist plot (i.e., the trajectory of 
L(iω) for ω∈ (-∞, +∞) in the complex plane), then 
the closed-loop system is stable; 
otherwise, it is unstable. 
The point is that this stability test can be carried out for an open-loop system 
illustrated in Fig. <ref>, which is constructed via simply 
cascading the amplifier and the controller. 
In fact the input-output relation of this open-loop system is given by 


    [ [     b̃_1(s); b̃_3^†(s^∗); b̃_4^†(s^∗);             ]]
            =   G^(open)(s)
                [ [     b_1(s); b_2^†(s^∗); b_4^†(s^∗);            ]],
    
           G^(open)(s)=   [ [        G_11(s)        G_12(s)              0; K_11(s)G_21(s) K_11(s)G_22(s)        K_12(s); K_21(s)G_21(s) K_21(s)G_22(s)        K_22(s);                ]].


Therefore, the Nyquist plot can be obtained by setting b_1 and b_4 to 
the vacuum fields and injecting the coherent field |α e^iω⟩ 
in the b_2 port with several frequencies ω. 
In fact, measuring the amplitude of the output field b̃_4 gives us 
the Nyquist plot in the form 
L(iω)=-b̃_4^†(-iω)/α^*. 
Note that the measurement result of b̃_4 must be probabilistic, and 
hence the Nyquist plot constitutes a “band" with variance 
|Δb̃_4(iω)|^2, meaning that the stability margin 
should be taken into account. 









§ FUNCTIONALITIES 1: QUANTUM PID




We now start describing several functionalities realized by the developed 
quantum feedback amplification method. 
The first functionality is the quantum PID <cit.>. 
That is, we show that, via the proper choice of the system K, the ideal closed-loop system 
(<ref>) functions as a differentiator (D) or integrator (I) on the input itinerant field b_1; 
hence together with the proportional component (P), which simply attenuates or amplifies the 
amplitude of the input, now P, I, and D components are provided to us. 
These three are clearly the most basic components involved in almost all electrical circuits and 
used for constructing several useful systems such as a PID feedback controller and an analogue 
computer. 
In fact, there have been proposed a classical PID control for quantum systems 
<cit.>; the fully-quantum PID controller constructed using 
feedback amplification may have some advantages over the classical one, but we leave 
this analysis for future work. 
Instead, we will show a simple application of the quantum integrator at the end of this section. 






 §.§ Differentiator










Let us take the symmetric cavity (<ref>) as the controller K(s). 
In this case, the transfer function of the ideal closed-loop system 
(<ref>) is given by 

    G^( fb)(s)       
            =  1/κ[ [ s + κ     s;     s  s -κ;       ]].


Hence from Eq. (<ref>), the output b̃_3^†(s) is given by 


    b̃_3^†(s^∗) 
             = s/κ b_1(s)
             + s-κ/κ b_4^†(s^∗),


or in the time-domain it is 


    b̃_3^†(t) 
             = 1/κd/dt b_1(t)
             + 1/κd/dt b_4^†(t) - b_4^†(t),


meaning that the closed-loop system works as a differentiator for the 
itinerant field b_1(t). 



As discussed in Sec. III, the approximation is valid only in a specific 
s-region such that the high-gain limit is effective. 
To show a concrete example of this region, 
we study a feedback controlled system composed 
of the optical NDPA and the control cavity, depicted in 
Fig. <ref>(a). 
Recall that, in the case of NDPA, the high-gain limit is achieved in the 
regime |s|≪γ≈ 2λ, which becomes wider as λ 
increases. 
Actually this can be seen in Fig. <ref>(b), 
showing the gain plot of the transfer function (<ref>) of this optical 
system with parameters γ=2.01λ and its high-gain limit 
|iω/κ|; 
that is, the frequency range such that this controlled system effectively 
approximates the ideal differentiator becomes wider as λ gets bigger. 








Note that, as in the classical case, the differentiator itself is an unstable 
system, and thus this system should be used together with other components 
such that the entire system is stable. 
This instability can be readily seen using the method addressed in 
Sec. <ref>; 
the open-loop transfer function in this case is 


    L(s)=κ(s^2-γ^2/4-λ^2)/(s+κ)(s^2+γ s+γ^2/4-λ^2),


and the Nyquist's plot is given by Figs. <ref>(a) and 
(b), showing that the point -1 lies inside the trajectory of L(iω) 
and thus the system is unstable. 
Note that the actual Nyquist's plot fluctuates along the curve shown in the 
figure, with variance |Δb̃_4(iω)|^2. 








 §.§ Integrator










Next we take the high-pass filtering cavity (<ref>) as the controller, 
where in this case the reflected field is fed back to the amplifier, as 
shown in Fig. <ref>(a) in optics case. 
Then the transfer function matrix (<ref>) of the ideal closed-loop 
system is given by 


    G^( fb)(s)       
            =  1/s[ [ - s - κ       κ;       κ    s -κ;         ]].


Hence from Eq. (<ref>) the output b̃_3^†(s^∗) 
is connected to the input b_1(s) as


    b̃_3^†(s^∗) 
             = κ/s b_1(s)
             + s-κ/s b_4^†(s^∗),


or in the time-domain it is 


    b̃_3^†(t) 
             = κ∫_0^t b_1(τ)dτ 
             + b_4^†(t)
             - κ∫_0^t b_4^†(τ)dτ .


This means that in a specific s-region where the high-gain limit of the 
amplifier is effective (|s|≪γ in the NDPA case, as discussed in 
Sec. <ref>), the closed-loop system works as an 
integrator for the itinerant field b_1(t). 
Note again that the feedback controlled system can approximate the 
integrator in the low-frequency regime, while the 
standard non-controlled cavity (<ref>) 
can do the same task in a high-frequency regime |iω|≫κ; 
in this sense the integrator shown here is the functionality realized only via the 
feedback amplification method. 



Now, unlike the differentiator, the integrator forms a circulating field 
in the feedback loop between the amplifier and the controller cavity. 
Therefore, we regard this loop as another cavity with mode a_4, 
as depicted in Fig. <ref>(b) in optics case
and Fig. <ref>(c) in a microwave system case, 
which we call the loop cavity. 
In fact, for the model depicted in Fig. <ref>(a) where 
b_2, b̃_2, b_3, and b̃_4 are treated as the itinerant 
fields, they violate the Ito rule such as dB_2(t) dB_2^†(t)=dt 
with B_2(t)=∫_0^t b_2(τ)dτ <cit.>. 
In what follows we show that this modified model still maintains the 
functionality of integration. 
More precisely, we will show that the transfer function from b_1 to b̃_3^† 
in Fig. <ref>(b) or Fig. <ref>(c) 
approaches to κ/s in the same high-gain limit. 


First, the Hamiltonian of the system is given by 


    H_fbamp
           =    ∑_k=1^4ħω_ka^†_ka_k
           +iħλ(a_1^† a_2^† e^-2iω_pt-a_1a_2e^2iω_pt)
    
        +ħ g_24(a_2^† a_4+a_2a_4^†)
                      +ħ g_34(a_3^† a_4+a_3a_4^†),


where ω_3 and ω_4 are the resonant frequencies of a_3 and a_4, 
respectively. 
g_24 (g_34) describes the coupling strength 
between a_2 and a_4 (a_3 and a_4), which are given by 


    g_24=√(cγ/L_4),    
             g_34=√(cκ/L_4),


with L_4 the round trip length of the loop cavity and c=3×10^8 m/s 
the speed of light. 
Here we assume ω_k=ω_p (k=1, …, 4). 
Together with the coupling to the external fields, we find that, in the rotating frame at 
frequency ω_p, the dynamical equations are given by


    ȧ_1= -γ/2a_1+λ a_2^†-√(γ)b_1,   ȧ_2^† = λ a_1+ig_24a_4^†,
    
       ȧ_3^† = -κ/2a_3^†+ig_34a_4^†-√(κ)b_4^†,   ȧ_4^† = ig_24a_2^†+ig_34a_3^†,
    
       b̃_1 = √(γ)a_1 + b_1,   b̃_3^† = √(κ)a_3^† + b_4^†.


The input-output equation of this system in the Laplace domain is of the form 


    [ [     b̃_1(s); b̃_3^†(s^∗);             ]]
             = [ [ G_11^( fb')(s) G_12^( fb')(s); G_21^( fb')(s) G_22^( fb')(s);                ]]
               [ [     b_1(s); b_4^†(s^∗);            ]],


where particularly G_21^( fb')(s) is given by


    G_21^( fb')(s)
            =α_0/s^4+β_3 s^3+ β_2 s^2+β_1 s +β_0,


with 


    α_0=   √(γκ)λ g_24g_34,   β_0=γκ g_24^2/4-λ^2g_34^2, 
    
    β_1=   (γ g_24^2+κ g_24^2+γ g_34^2-κλ^2)/2,
    
    β_2=   γκ/4-λ^2+g_24^2+g_34^2,   β_3=(γ+κ)/2.


In the limit γ→2λ+0, together with 
Eq. (<ref>), the above coefficients are approximated as 


    α_0≈    2cκλ^2/L_4,   β_0≈ 0, 
    
    β_1≈    2cλ(κ+λ)/L_4-κλ^2/2, 
    
    β_2≈   κλ/2-λ^2+c(κ+2λ)/L_4,   β_3≈κ/2+λ.


Furthermore, we assume |s|≪γ, so that the higher-order term of s can 
be neglected. 
Then the transfer function (<ref>) can be approximated by


    G_21^( fb')(s)
           ≈   α_0/β_1s≈2cκλ^2/L_4/{2cλ(κ+λ)/L_4-κλ^2/2 } s
    
            =   κ/(1+κ/λ- L_4κ/4c)s.


Thus, if κ≪λ and L_4κ/(4c)≪1, 
this system becomes the integrator which we wish to obtain:


    G_21^( fb')(s)≈κ/s.





In Fig. <ref>(d), the solid black line shows  
the ideal gain plot of the integrator, (i. e., |κ/iω|), 
while the dotted lines represent 
|G_21^( fb')(iω)| in Eq. (<ref>) 
with parameters γ=2.01λ, c/L_4=10^3κ, and 
λ=nκ (n=3, 5, 7, 9). 
Clearly, |G_21^( fb')(iω)| well approximates |κ/iω| 
in a specific s region where the high-gain limit of the NDPA is effective, 
which is now given by |s|≪γ≈ 2λ. 
Hence, λ should be relatively large to guarantee that the integrator 
works in a wider region in s; this can be actually seen in the figure, 
although making λ large does not make a big difference in the 
parameter regime considered here. 
However, the approximation is not valid in the frequency domain where 
ω / κ < 10^-2. 
This is because, if s=0, the transfer function in Eq. (<ref>) is 
G_21^(fb')(0)=α_0/β_0, whereas the transfer function of the integrator 
goes towards infinity if s→0. 
Therefore, the approximation is valid in the relatively low frequency domain where s^4, s^3, s^2 
can be ignored and where |β_1 s| ≫ |β_0| (⇔ |s| ≫ |β_0/β_1|). 


The stability of the modified model depicted in 
Figs. <ref>(b) and <ref>(c) 
cannot be investigated via the 
stability test discussed in Sec. <ref>, 
which can only be applied to the case where the feedback loop does not 
form a cavity. 
Therefore, instead of the Nyquist's theorem, we use the 
Routh-Hurwitz method <cit.>. 
In our case, the system is stable if and only if every root of the 
characteristic polynomial function in the denominator in 
Eq. (<ref>) has a negative real part; 
the Routh-Hurwirz method systematically leads to the stability condition 
as follows: 


    β_3>0,  β_2β_3-β_1/β_3>0,  β_3^2β_0/β_1-β_2β_3+β_1>0.


Note that β_3>0 is already satisfied. 








 §.§ Application to qubit detection



Here we give an application of the integrator, which can be used in a 
stand-alone fashion unlike the differentiator. 
The system of interest (not the feedback controlled system) is a 
qubit that is dissipatively coupled to the external itinerant field 
b_0(t), such as a transmon qubit coupled to a superconducting 
resonator; 
the Langevin equation of the system variable σ_x(t) is given by 


    d/dtσ_x(t) = -Γ/2σ_x(t)
                     + √(Γ)σ_z(t) {b_0(t)+b^†_0(t)},


where Γ is the strength of the dissipative coupling 
<cit.>. 
The output field is given by b_1(t)=√(Γ)σ_-(t) + b_0(t); 
the quadrature q_1(t)={b_1(t)+b_1^†(t)}/√(2) thus follows 


    q_1(t) = √(Γ/2)σ_x(t) 
                    + b_0(t)+b^†_0(t)/√(2).


Now the field state is set to the vacuum |0⟩_F. 
Then we find 


    _F⟨0|q_1(t)|0⟩_F 
           = √(Γ/2)e^-Γ t/2σ_x,


where for a system-field operator X, _F⟨0|X|0⟩_F 
represents an operator living in the system Hilbert space. 
This means that, for a very short time interval Γ t≪ 1, 
the above operator is approximated as 
_F⟨0|q_1(t)|0⟩_F ≈√(Γ/2)σ_x, 
which thus takes ±√(Γ/2) when measuring it. 
In other words, to measure the qubit state we need a high-speed detector. 



Using the integrator changes this condition. 
Let us place the ideal integrator having the input-output relation 
(<ref>) along the output field of the qubit. 
That is, the output b_1(t) is taken as the input to the 
integrator, and we measure its output b̃_3(t). 
The quadrature 
q̃_3(t)={b̃_3(t)+b̃_3^†(t)}/√(2) then 
satisfies 


    _F⟨0|q̃_3(t)|0⟩_F 
           = _F⟨0|(κ∫_0^t b_1(τ)+b_1^†(τ)/√(2) dτ)|0⟩_F
    
          
           = κ√(2/Γ)(1-e^-Γ t/2)σ_x,


where b_4 is set to be a vacuum field. 
Therefore, in the long time limit Γ t ≫ 1, this output 
itinerant field becomes 


    _F⟨0|q̃_3(t)|0⟩_F
              ≈κ√(2/Γ)σ_x.


Hence the measurement result is ±κ√(2/Γ), 
which equals to ±√(Γ/2) by choosing the integrator 
parameter as κ=Γ/2. 
This means that, even if the given detector is slow, the integrator assist 
it to capture the same amount of measurement signal as that obtained by a fast detector.












§ FUNCTIONALITIES 2: SELF OSCILLATION




Self-oscillation is also an important functionality realized with the 
feedback amplification method, which is indeed widely used in a variety 
of engineering scenes. To realize a sustained oscillation, of course, 
some nonlinearities such as a voltage saturation are necessary to be involved, 
but here we only focus on the linear part; a possible practical application of 
the proposed method which combines a nonlinear component will be presented in a 
future work.


Let us consider the cavity (<ref>) with 
κ_1 = κ_2 = κ. 
Then the transfer function of the ideal closed-loop system, 
Eq. (<ref>), is given by 


    G^( fb)(s)       
            =  1/s-iΔ[ [   s + κ-iΔ         -κ;          κ s - κ - iΔ;            ]].


Therefore the output b̃_3 is given by 


    b̃_3^†(s^∗) 
             = κ/ s-iΔ b_1(s)
               +  s-κ-iΔ/ s-iΔ b^†_4(s^∗).


Because the pole is on the imaginary axis, this represents a self-oscillation 
of b̃_3. 
In fact, if both b_1 and b_4 are set to the vacuum and 
b̃_3(0)≠ 0, then in the time domain b̃_3 satisfies 


    b̃_3(t)=e^-iΔ tb̃_3(0),


hence it oscillates with frequency -Δ (in the rotating-frame). 
Also, the spectral broadening of this oscillation can be seen 
from 


    1/2π∫_-∞^∞b̃_3^†(-iω) b̃_3(-iω')dω' 
              = κ^2/ (ω - Δ)^2 .


In practice, the cavity parameter κ_1-κ_2 is set to a small positive number, 
which makes the system oscillating almost with a fixed frequency yet with growing amplitude; 
but the amplitude is saturated by some dissipative nonlinearities, and as a result 
a sustained oscillation called the quantum limit cycle can be realized. 
As in the classical case, there may be several applications of this functionality, e.g., 
a quantum memory <cit.> and synchronization for spectroscopy <cit.>; 
also see Ref. <cit.> for the general semi-classical method for analyzing the quantum 
limit cycle and synchronization. 


Now we show the dynamics of the entire closed loop system in nearly 
the ideal amplification limit. In the optics case, the realization of 
the self-oscillator is very similar to the integrator shown in 
Fig. <ref>(a). 
The only difference between the self-oscillator and the integrator is 
that the detuning of the MCC is not zero for the case of self-oscillator, 
while it is zero for the integrator. 
Also as in the integrator, the self-oscillator forms a loop cavity between 
the NDPA and MCC, and thus it should be modeled as the system shown in 
Fig. <ref>(b) or Fig. <ref>(c) 
with non-zero detuning Δ in the mode a_3. 
Then the entire dynamical equation of the self-oscillator are the same as those in 
Eqs. (<ref>) except that iΔ a_3^† is added 
to the right hand side of the equation that has a term ȧ_3^† in the left hand side. Here, 
if b_1(t)=b_4^†(t)=0 ∀ t, the mean dynamics is 


    d/dt[
    [   a_1; a_2^†; a_3^†; a_4^† ]]=A_osc[
    [   a_1; a_2^†; a_3^†; a_4^† ]],  b̃_3^†=√(κ)a_3^†,


where


    A_osc=[
    [    -γ/2       λ       0       0;       λ       0       0   ig_24;       0       0 -κ/2+iΔ   ig_34;       0   ig_24   ig_34       0 ]].


Figure <ref> shows the mean time evolution of the quadratures of 
b̃_3(t): 


    q̃_3(t)=b̃_3(t)+b̃_3^†(t)/√(2),
             p̃_3(t)=b̃_3(t)-b̃_3^†(t)/√(2)i.


The parameters and the initial conditions are set as follows: 
Δ=1, λ=0.01Δ, γ=2.01λ, and 
c/L_4=0.1Δ. 
Also κ is set to κ=0.1Δ (black lines in the figure) or 
κ=0.01Δ (light blue lines). 
The initial condition is a_1(0)=a_2^†(0)=a_3^†(0)
=a_4^†(0)=1/√(2). 
Now the ideal oscillation (<ref>) is represented 
in terms of the quadratures as 


    [
    [ q̃_3(t); p̃_3(t) ]]=
    [
    [  cos(Δ t)  sin(Δ t); -sin(Δ t) cos(Δ t ) ]]
    [
    [ q̃_3(0); p̃_3(0) ]].


With the initial values mentioned above, these are given by 
q̃_3(t)=√(κ)cos(Δ t) and p̃_3(t)
=-√(κ)sin(Δ t), meaning that there is a π/2 phase difference 
between the two quadratures. 
Figure <ref> shows that the case κ=0.01Δ 
closely reproduces this ideal oscillation. 
In particular, the smaller value of κ leads to the slower attenuation 
of the oscillation. 
This is simply because the smaller κ is, the less amount of photons 
leaks out from the MCC with detuning Δ. 
Thus, by setting κ smaller, we can preserve the coherence of the 
light field oscillating with frequency Δ in the MCC. 
However, making κ smaller also reduces the amount of photons coming 
from the NDPA into the MCC, and thus the amplitude of oscillation is limited. 
Conversely, a large value of κ allows a flow of large amount of 
photons from NDPA to MCC, resulting in a large amplitude of the oscillation 
as demonstrated in the case κ=0.1Δ in Fig. <ref>. 
Therefore, there is a tradeoff between the coherence time and the amplitude 
of the self-oscillation. 















§ FUNCTIONALITIES 3: ACTIVE FILTERS




As we have seen in Sec. <ref>, the 2-input 2-output cavity 
works as a low-pass or high-pass filter with bandwidth κ and maximal gain 1. 
Here we show that, by the feedback amplification method, several types of filter with 
tunable bandwidth, gain, and phase, i.e., the quantum version of active filters, can 
be engineered. 




 §.§ High-Q active filter




First we show a simple first-order active filter. 
As in the quantum integrator, the controller is chosen as the high-pass filtering cavity 
(<ref>) with zero-detuning Δ=0, which in this case is set to be asymmetric 
(i.e., κ_1 ≠κ_2). Then, the closed-loop system (<ref>) realized in the high-gain amplification limit 
is given by 


    [ [     b̃_1(s); b̃_3^†(s^∗);             ]]
            = G^ (fb)(s)
                [ [     b_1(s); b_4^†(s^∗);            ]], 
    
                      
           G^( fb)(s)       
            =  1/s+(κ_2-κ_1)/2
          ×[ [ - s - (κ_1+κ_2)/2         √(κ_1κ_2);         √(κ_1κ_2) s - (κ_1 + κ_2)/2;                   ]].


Here we focus on the output b̃^†_3(s^∗):  


    b̃^†_3(s^∗) 
             = √(κ_1κ_2)/s+(κ_2-κ_1)/2b_1(s)
               + s-(κ_1+κ_2)/2 /s+(κ_2-κ_1)/2b^†_4(s^∗),


with κ_2-κ_1>0. 
Hence, this system functions as a low-pass filter for b_1(s) with bandwidth 
(κ_2-κ_1)/2. 
In contrast to the standard low-pass filter (<ref>) with bandwidth κ, 
the bandwidth of this active filter can be made very small by making κ_1 and 
κ_2 close to each other. 
As a result, the Q-factor can be largely enhanced from Q=ω_0/2κ to 
Q'=ω_0/(κ_2-κ_1). 
For instance for a coherent light field with frequency ω_0=3× 10^14 Hz, 
an optical cavity κ=3× 10^6 leads to Q=5× 10^7, 
while the active filter with κ_1=κ and κ_2=1.01κ leads to 
Q'=1× 10^10. 
Note that this active filter also functions as an amplifier with gain 
2√(κ_1κ_2)/(κ_2-κ_1), which becomes large if Q-factor 
increases. 
Importantly, in this case the idler noise mode b_4 is also amplified with gain 
(κ_2+κ_1)/(κ_2-κ_1) at s=0. 
This means that basically the filtering makes sense only for an input field with amplitude 
much bigger than (κ_2+κ_1)/(κ_2-κ_1). 
Also we remark that, as discussed in the case of integrator, the feedback loop now constructs 
another cavity, which should be taken into account for more precise modeling of the filter; 
we will give such a detailed investigation in 
Sec. <ref> 
for another type of active filter discussed in the next subsection. 








 §.§ Phase-cancellation filter




The functionality provided by an active filter is not only modifying the gain profile, 
but changing the phase of an input filed. 
In fact Miao et al. proposed a quantum active filter that can effectively 
change the phase and thereby enhance the bandwidth of the gravitational 
wave detector or, in a wider sense, any cavity-based quantum sensor 
<cit.>. 
A rough description of their idea is as follows. 
When a gravitational wave propagates through the interferometer (arm cavity), 
then it must pick up a phase 
ϕ_arm(Ω)=-2Ω L_arm/c, where Ω, 
L_arm, and c are the gravitational-wave frequency, the length of the cavity, 
and c the speed of light, respectively. 
Note that only in Secs. <ref> and <ref> 
we use the conventional Ω rather than ω to recall that 
this is the gravitational-wave frequency. 
This extra phase eventually limits the bandwidth of the detector; 
hence constructing an auxiliary intra-cavity filter with the transfer function 
e^-iϕ_arm(Ω)=e^2iΩ L_arm/c will compensate this 
extra phase and thus may recover the bandwidth. 



Here we show that the feedback amplification method can be employed to realize such a 
phase-cancelling filter in a fully optical setting. 
We again use the closed-loop system (<ref>) and now consider 
the output b̃_1: 


    b̃_1(s)
             =     G_11^(fb)(s)b_1(s)+G_12^(fb)(s)b_4^†(s^∗)
    
    
             =    - s + (κ_1+κ_2)/2 /s+(κ_2-κ_1)/2b_1(s)
               + √(κ_1κ_2)/s+(κ_2-κ_1)/2b^†_4(s^∗).


Let us then set κ_2=0:


    b̃_1(s) = 
         	G_11^(fb)(s)b_1(s) =
             - s + κ_1/2 /s -κ_1/2b_1(s).


In the frequency domain s=iΩ this equation reduces to 


    b̃_1(iΩ)=G_11^(fb)(iΩ)b_1(iΩ)
                =-iΩ+κ_1/2/iΩ-κ_1/2b_1(iΩ).


Then by setting Ω≪κ_1 and κ_1=2c/L_arm, we actually 
find that G_11^(fb) approximates our target filter: 


    G_11^(fb)(iΩ)
                  =    -Ω^2+κ_1^2/4+iΩκ_1/Ω^2+κ_1^2/4≈κ_1^2/4+iΩκ_1/κ_1^2/4
    ≈     e^4iΩ/κ_1
                  = e^2iΩ L_arm/c
                  = e^-iϕ_arm(Ω).


This phase-cancelling filter might be realizable in practice by carefully devising the 
controller cavity so that the optical loss κ_2 is very small. 
Note that in the literature works <cit.> 
an opto-mechanical oscillator was employed to realize the same filter where in that case 
κ_2 represents the magnitude of the thermal bath added on the oscillator; 
hence κ_2 ≈ 0 requires the oscillator to be in an ultra-low temperature 
environment. 



Lastly note that the system (<ref>) is clearly unstable; particularly 
the system (<ref>) represents a phase-lead filter that 
violates the causality. 
Similar to the case of integrator, therefore, in a practical setting such a phase-cancellation 
filter must be incorporated in a bigger system that is totally stable. 
In Sec. <ref> we give a detailed study to see 
how much the phase-cancellation filter (<ref>) could compensate the phase 
delay and thereby enhance the bandwidth of the stabilized gravitational-wave detector 
in a practical setup. 








 §.§ Butterworth filter






Let us move back to the problem of modifying the gain profile via an active filter. 
A particularly useful bandpass filter, which is often used in classical electrical circuits, 
is the Butterworth filter. 
The transfer function of the n-th order classical Butterworth filter is given by 
T_n(s)=g/B_n(s) where g is a constant and the followings are examples of polynomials 
B_n(s): 


    B_1(s)=   s+1,   
          B_2(s)=s^2+√(2)s+1,
    
          B_3(s)=   (s+1)(s^2+s+1).


The gain of the filter is given by 


    |T_n(iω)| = g/√((ω/ω_B)^2n+1),


which has the steep roll-off characteristic of frequency, particularly for large n, at the 
cut-off frequency ω_B. 


A quantum version of Butterworth filter has actually been employed in the literature; 
in Ref. <cit.>, a fourth-order quantum Butterworth filter was applied to 
enhance the channel capacity of a linear time-invariant bosonic channel. 
However, its physical realization has not been discussed. 
Here we show that, in the simple case n=2, the feedback amplification technique can be 
used to realize the quantum Butterworth filter. 



The controller K is chosen as the cascaded cavities, an optical case of which 
is depicted in Fig. <ref>. 
The left cavity with mode c_1 has two inputs (b_3, b_4) and two outputs (b'_3, b'_4), 
and the right one with mode c_2 has two inputs (b”_3, b”_4) and two outputs 
(b̃_3, b̃_4). 
We assume that the detuning of the left and right cavities are Δ and -Δ, 
respectively. 
A phase shifter e^iπ(=-1) is placed in the path from b'_4 to b”_3. 
Then the two input and output fields are connected as follows: 


    b”_3^†(s^∗) = -b'_4^†(s^∗),    
         b”_4^†(s^∗) = b'_3^†(s^∗).


The input-output relations of each cavities are given by 


    [[ b'_3^†(s^∗); b'_4^†(s^∗) ]]
    	   =K_l(s)
            [[ b_3^†(s^∗); b_4^†(s^∗) ]],   
    [[ b̃_3^†(s^∗); b̃_4^†(s^∗) ]]
              =K_r(s)
           [[ b”_3^†(s^∗); b”_4^†(s^∗) ]],


where K_l(s) is given by the left hand side of Eq. (<ref>) and

    K_r(s) =  1/s+(κ_1+κ_2)/2+iΔ
          ×[ [ s + (κ_1 - κ_2)/2 + iΔ             -√(κ_1κ_2);             -√(κ_1κ_2) s - (κ_1 - κ_2)/2 + iΔ;                        ]].


The transfer function of the controller is thus given by 


    K(s)=K_r(s)
            [
    	[  0 -1;  1  0 ]]
    	K_l(s).


Here we set Δ=(κ_1+κ_2)/2. 
The feedback configuration is depicted in Fig. <ref>, where K(s) is now given 
by Eq. (<ref>). 
Then from Eqs. (<ref>) and (<ref>), we find that 
the output b̃_3^† of the closed-loop system composed of this controller and 
a high-gain amplifier G is given by


    b̃_3^†(s)
         =     G_21^(fb)(s)b_1(s)+G_22^(fb)(s)b_4^†(s^∗)
    
    
         =    -{κ_1-κ_2+i(κ_1+κ_2) }√(κ_1κ_2)/s^2 + (κ_2-κ_1)s + (κ_2-κ_1)^2/2b_1(s)
    
        - s^2 - (κ_2+κ_1)s + (κ_2+κ_1)^2/2/s^2 + (κ_2-κ_1)s + (κ_2-κ_1)^2/2b^†_4(s^∗).








The transfer function from b_1 to b̃_3^† has a form of the second order 
Butterworth filter with cut-off frequency ω_B=(κ_2-κ_1)/√(2) and 
maximal gain g=√(2κ_1κ_2(κ_1^2+κ_2^2))/ω_B^2. 
Also, it is easy to see that the transfer function from b_4^† to b̃_1 has the 
same form of second order Butterworth filter as above. 
Note that, as mentioned in Sec. <ref>, the amplitude of the input field 
should be much bigger than that of the amplified idler vacuum noise. 



Figure <ref> shows the gain plot of the second-order Butterworth filter 
developed above. 
In this figure, the black solid line shows the gain plot of the ideal transfer function 
G_21^(fb)(iω)=-K_11(iω)/K_21(iω), which corresponds to 
γ→ 2λ+0, while the dotted lines show the gain plot of 
G_21^(fb)(iω)
=G_21(iω)K_11(iω)/{1-K_21(iω)G_22(iω) } 
with γ=2.01λ and several parameters λ=mκ_1 (m=1, 3, 5, 7). 
The other parameters are fixed to κ_2=1.5κ_1 and 
Δ=(κ_1+κ_2)/2. 
Now, as mentioned before, |G_11(iω)|≫1 holds in the frequency range 
ω≪γ=2.01λ. 
Therefore, making λ bigger results in broadening the frequency range where the 
approximation is valid, and in fact Fig. <ref> shows that the dotted 
line approaches to the ideal solid line as λ gets larger. 










 §.§ Non-reciprocal amplifier








The last topic in this section is a proposal of a non-reciprocal (directional) 
amplifier, constructed via the feedback amplification method. 
Non-reciprocal amplifiers are particularly important in the field of superconducting 
circuit based quantum technologies <cit.>. 
In the microwave regime the phase-preserving amplifier obeys the same equation 
(<ref>), but the configuration differs from the optical 
case; the input b_1 and the corresponding output 
b̃_1 propagate along the same transmission line yet with opposite 
direction as shown in Fig. <ref>(c). 
If the purpose of the use of amplifier is to detect a small signal b_1 generated from a 
source system, e.g., a superconducting qubit, then the propagating direction of the reflected 
output field b̃_1 must be changed to protect the source system from the backward 
field (if the output b̃_1 is the amplified vacuum noise field) or to measure it 
(if the output b̃_1 is the amplified signal). 
In fact there have been a number of theoretical and experimental proposals of 
active non-reciprocal amplifier <cit.>. 
Our scheme is similar to <cit.>, but it has a clear concept of using the 
feedback amplification to realize a robust non-reciprocal amplifier as described below. 



Before describing the result, we note that the assumptions in the coherent feedback theory 
(e.g., see <cit.>) are not always satisfied in superconducting devices, while 
there is certainly the case where the theory is valid and was experimentally demonstrated 
<cit.>. 
Here we leave this problem, i.e., the analysis for experimental implementability of the method 
in the proposed superconducting circuit, for the future work. 




The proposed microwave non-reciprocal amplifier has a form of coherent feedback shown in 
Fig. <ref>(a), which takes into account the above-mentioned fact 
that the input and reflected output fields propagate along the same transmission line. 
This whole system has three inputs (b_1, b_3, b_4) and three outputs 
(b̃_2, b̃_3, b̃_4); 
particularly b_3 is the input signal and b̃_3 is the amplified signal to be detected, 
while b_1 and b_4 are the vacuum field. 
G and G are both phase-preserving amplifiers, and K is a passive system. 
As mentioned above, the source system generating b_3 may be contaminated due 
to the backward field b̃_4, which is not necessarily vacuum. 
Hence b̃_4 should be sufficiently suppressed. 


Each system component has the following input-output relations:


    [[       b̃_1(s); b̃̃̃_1^†(s^∗) ]]=   [[ G_11(s) G_12(s); G_21(s) G_22(s) ]]
          [[  b̃̃̃_3(s); b_1^†(s^∗) ]],
    
    [[ b̃_2^†(s^∗);   b̃̃̃_2(s) ]]=   [[ G_11(s) G_12(s); G_21(s) G_22(s) ]]
         [[ b̃̃̃_1^†(s^∗);        b_2(s) ]],
    
    [[   b̃_3(s); b̃̃̃_3(s) ]]=   [[ K_11(s) K_12(s); K_21(s) K_22(s) ]]
           [[    b_3(s); b̃̃̃_2(s) ]],
    
    [
            [ b̃_4(s);  b_2(s) ]]
        =   [
            [ K_11^∗(s) K_21^∗(s); K_12^∗(s) K_22^∗(s) ]]
        [
            [  b_4(s); b̃_1(s) ]].


Combining these equations, the input-output relation of the whole feedback 
controlled system is represented as follows; 


    [ [ b̃_2^†(s^∗);     b̃_3(s);     b̃_4(s);             ]]
               = G^ (fb)(s)
                [ [ b_1^†(s^∗);     b_3(s);     b_4(s);            ]],  
    
             G^(fb)(s)=   [[ G_11^(fb)(s) G_12^(fb)(s) G_13^(fb)(s); G_21^(fb)(s) G_22^(fb)(s) G_23^(fb)(s); G_31^(fb)(s) G_32^(fb)(s) G_33^(fb)(s) ]].


Here we found that, in a s-domain such that both G(s) and G(s) have a large gain, 
the transfer function matrix G^(fb)(s) converges to 


    G^ (fb)(s)
          
       →[
            [                         -1/K_22(s)                   -K_21(s)/K_22(s)                                  0;                   -K_12(s)/K_22(s)                     [K(s)]/K_22(s)                                  0;                                  0                                  0 K_11^∗(s) + [K^†(s)]/1 + K_22^∗(s) ]],


where

    |
            
                K_11^∗(iω) + [K^†(iω)]/
                1 + K_22^∗(iω)
            |
        =
        1,
          ∀ω

holds. 
The proof of Eq. (<ref>) is given in 
Appendix <ref>, and Eq. (<ref>) can be 
proven by using the unitary property of K(iω) (i.e., 
|K_11(iω)|^2 + |K_12(iω)|^2 = 1, 
|K_21(iω)|^2 + |K_22(iω)|^2 = 1, and 
K_21(iω)K_11^*(iω) + K_22(iω)K_12^*(iω)=0). 
The point of this result is that, because b_4 is vacuum, the backward field mode b̃_4 
propagating towards the input port (see Fig. <ref>(a)) is also a vacuum 
field in this high-gain limit. 
This means that the noise of the backward field at the input port is suppressed to the vacuum 
in the feedback configuration. 
Therefore, because the output signal b̃_3 contains the input signal b_3 with amplification 
gain [K]/K_22 that only depends on the passive component, this feedback-controlled 
system functions as a robust non-reciprocal amplifier or more broadly a robust non-reciprocal active 
filter. 


Let us consider an example. 
If K is given by a beam splitter with power transmissivity T: 


    K(s)=[[    √(T) -√(1-T);  √(1-T)    √(T) ]],


then we have 


    G^(fb)(s)=
              [[   -1/√(T) -√(1/T-1)         0;  √(1/T-1)    1/√(T)         0;         0         0         1 ]].


Hence, the input signal b_3 is amplified with amplification gain 1/√(T). 
Importantly, this non-reciprocal amplification is robust against the characteristic 
changes in the original amplifiers (G, G) because the gain 
1/√(T) is a tunable yet static quantity. 


Lastly we provide a way for intuitively understanding the mechanism of non-reciprocity of the 
proposed amplifier. 
For this purpose, let us reconsider the classical feedback amplifier shown in Fig. <ref>. 
In this case, the input to the amplifier G, which is u-Ky, converges to zero when G→∞ 
due to the boundedness of the output y. 
This means that the signal injected to G effectively vanishes in the high-gain limit; as a result, 
the signal flows only along the line illustrated by the orange thick line in the left figure of 
Fig. <ref>(b). 
This view implies that the proposed non-reciprocal amplifier has a similar characteristic. 
That is, the signal fields injected to the amplifiers G and G effectively vanish (more 
precisely, suppressed to the vacuum field), and the signal propagates only through the passive system 
K as illustrated by the orange thick line in the right figure of Fig. <ref>(b). 


This is indeed a useful property, because the signal does not pass through the circulator; 
in fact, microwave circulators are in general noisy, and thus, developing a non-reciprocal amplifier 
without circulators is what has been pursued recently 
<cit.>. 
Therefore, it would be interesting to conduct an experiment to see how much the signals injected to the 
amplifiers G and G are suppressed. 











§ APPLICATION TO GRAVITATIONAL-WAVE DETECTION




As noted before, any functionality realized via the feedback amplification method 
should be evaluated in such a way that it is incorporated in a concrete setup 
with particular engineering purpose, to see its actual performance under practical 
constraints. 
Here we study the phase-cancellation filter discussed in Sec. <ref>, 
and see how much it might broaden the bandwidth of the typical gravitational-wave 
detector. 






 §.§ Basics of gravitational-wave detector







The most basic schematic of the gravitational-wave detector, particularly the one 
that uses a laser interferometer <cit.>, 
is shown in Fig. <ref>. 
The input laser with frequency ω_0 is injected to the arm cavities through the 
power recycling mirror (PRM). 
Each arm cavity is composed of two mirrors: the input test mass (ITM) and the end 
test mass (ETM). 
A tidal force of gravitational-wave F_GW with frequency Ω induces 
a pendulum motion of ETMs. 
Then the arm cavities create the signal light fields with frequency ω_0 + Ω, 
which are combined at the center half mirror and leak to outside through the signal 
recycling mirror (SRM); 
this output field is denoted as d_out. 
Note that a vacuum field d_in unavoidably enters into the system at SRM. 



The Hamiltonian of the entire system, in the rotating frame at frequency ω_0, 
is given by <cit.> 


    H=P^2/2M+ħΔ_d d^† d
            -ħ G_arm(d+d^†)X-F_GWX.


(X,P) are the differential (position, momentum) operators of ETMs, and they satisfy 
[X(t),P(t)]=iħ. 
M is the mass of ETMs. 
d is the sideband mode of the interferometer field, with detuning Δ_d, 
which satisfies [d(t),d^†(t)]=1. 
G_arm represents the coupling strength between X and d, and it is given by 
G_arm=√(2P_armω_0/(ħ cL_arm)) with 
P_arm the arm cavity power <cit.>. 
Then the dynamics of the system is given by


    Ẋ= 1/MP,   Ṗ= ħ G_arm(d+d^†)+F_GW,
    
       ḋ= -(iΔ_d+γ_IFO/2)d
            	+iG_armX-√(γ_IFO)d_in,


where γ_IFO is the coupling between d and d_in. 
Also, the output equation of the system is given by 


    d_out = √(γ_IFO)d + d_in.


Note that 
[d_in(t),d_in^†(t')]=[d_out(t),d_out^†(t')]
=δ(t-t'). 
The input-output relation in the Laplace domain, in terms of the quadratures 
Q_d^in,out=(d_in,out+d_in,out^†)/√(2) and 
P_d^in,out=(d_in,out-d_in,out^†)/√(2)i, is obtained as 


    [[ Q_d^out(s); P_d^out(s) ]]
           =J(s)[[   F_GW(s); Q_d^in(s); P_d^in(s) ]],


with


    J(s)=[
    	[ J_11(s) J_12(s) J_13(s); J_21(s) J_22(s) J_23(s) ]]
    	=
       [
    	[                                  0                s-γ_IFO/2/s+γ_IFO/2                                  0;     √(2γ_IFO)G_arm/Ms^2(s+γ_IFO/2) -2ħ G_arm^2γ_IFO/Ms^2(s+γ_IFO/2)^2                s-γ_IFO/2/s+γ_IFO/2 ]],


where Δ_d=0 is assumed. 
The gravitational-wave strain signal h, which is defined as 
F_GW(t)=ML_armḧ(t), can be detected by homodyne measuring 
P_d^out. 
The quantum noise operator is then defined as 


    F_N(s)=   P_d^out(s)/ML_arms^2J_21(s)-h(s)
    
    
                   =    Ξ_Q(s)Q_d^in(s)+Ξ_P(s)P_d^in(s),


where


    Ξ_Q(s)=-√(2γ_IFO)ħ G_arm/ML_arms^2(s+γ_IFO/2), 
    
    Ξ_P(s)=s-γ_IFO/2/√(2γ_IFO)G_armL_arm.


Hence F_N(s) is composed of the radiation pressure noise 
Ξ_Q(s)Q_d^in(s) and the shot noise Ξ_P(s)P_d^in(s), which are 
dominant in the low and high frequency range, respectively. 
The magnitude of F_N(iΩ) is quantified by the spectral density S(Ω), 
which is generally defined by <cit.>


    2π    S(Ω)δ(Ω-Ω')  
    
    
               =   ⟨ F_N(iΩ)F_N^†(iΩ') 
                      + F_N^†(iΩ')F_N(iΩ) ⟩/2.


It is now calculated as 


    S(Ω)= ( |Ξ_Q(iΩ) |^2 + |Ξ_P(iΩ) |^2 )/2,


which is lower bounded by the standard quantum limit (SQL) 
<cit.>: 
S_SQL(Ω)=|Ξ_QΞ_P | = ħ/(ML_arm^2Ω^2). 
Note that, however, there have been several proposals to beat SQL, which can be now even 
experimentally observed in the real LIGO system <cit.>. 
Figure <ref> shows the sensitivity √(S(Ω)) in the following typical setup 
<cit.>: M=40 kg, L_arm=4 km, 
P_arm=800 kW, 
ω_0=2π c/λ_laser, 
λ_laser=1064 nm, 
Δ_d=0, 
γ_IFO=2π×200 Hz. 
Also recall that Ω is the gravitational-wave frequency.













 §.§ Effect of the phase-cancellation filter



As seen above, the detection sensitivity (roughly the inverse of the noise magnitude) is 
limited by the quantum noise. 
Especially, the following equality holds <cit.>, meaning that there is a tradeoff 
between the bandwidth and the peak sensitivity: 


    ∫_0^∞1/|Ξ_P(iΩ) |^2  dΩ 
              = 2 π G_arm^2 L_arm^2.


In fact, because the integral does not depend on the bandwidth of the cavity, 
γ_IFO, a broad-band enhancement of the sensitivity is not allowed. 



As described in Sec. <ref>, the above tradeoff is attributed to the 
frequency-dependent propagation phase 
ϕ_arm(Ω)=-2Ω L_arm/c. 
The idea proposed in Ref. <cit.> is to construct the phase-cancellation filter 
with transfer function e^-iϕ_arm(Ω)=e^2iΩ L_arm/c 
to compensate ϕ_arm(Ω). 
Now, unlike the optomechanics-based scheme proposed in Ref. <cit.>, we can 
construct the same filter (<ref>) in all-optics setup, using 
the feedback amplification method. 
Figure <ref>(b) shows ϕ_arm(Ω) and 


    ϕ_G
              = arg[G_11^(fb)(iΩ)]
              = arg[ G_11 - K_21 det[G]/1-K_21G_22],


where G is the transfer function of NDPA and K is given by 
Eq. (<ref>) with Δ=κ_2=0; 
this approximates -ϕ_arm in the high-gain limit, as proven in 
Eq. (<ref>). 
The parameters are set as 
L_arm=4 km, λ=3×10^6 Hz, γ=2.01λ, and 
κ_1=2c/L_arm. 
We can see from Fig. <ref>(b) that the filter certainly 
achieves the desired phase cancellation in the frequency range where 
Ω≪κ_1=2c/L_arm≈2π× 2.39×10^4 Hz 
is satisfied. 








Now recall that the filter is realized as the feedback-controlled system shown in 
Fig. <ref>(a). 
That is, as discussed in the case of integrator in Sec. <ref>, the feedback 
loop between the amplifier (G) and the cavity (K) forms a loop cavity with mode a_4. 
The total Hamiltonian of the filter is given by Eq. (<ref>), and 
we again assume ω_k=ω_p (k=1, …, 4). 
Then in the rotating frame at frequency ω_p, the dynamics and output equation of the filter are given by


    ȧ_1= -γ/2a_1+λ a_2^†-√(γ)b_in,   ȧ_2^† = λ a_1+ig_24a_4^†,
    
       ȧ_3^† = ig_34a_4^†,   ȧ_4^† = ig_24a_2^†+ig_34a_3^†,
    
        b_out = √(γ)a_1 + b_in,


with g_24=√(cγ/L_4) and g_34=√(cκ_1/L_4). 
This equation is the same as Eq. (<ref>) except that control cavity K 
couples to only the loop cavity. 
The input-output relation of this system in the Laplace domain is represented as


    b_out(s)    = Z(s) b_in(s), 
    
    
          Z(s)     =s^4+α_3s^3+α_2s^2+α_1s+α_0/s^4+β_3s^3+β_2s^2+β_1s+β_0,


where


    α_3=    -γ/2,  α_2=g_24^2+g_34^2-λ^2,  α_1=-(g_24^2+g_34^2)γ/2, 
    
    α_0=   -λ^2g_34^2,  β_3=γ/2,  β_2=g_24^2+g_34^2-λ^2, 
    
    β_1=    (g_24^2+g_34^2)γ/2,  β_0=-λ^2g_34^2.


Now we show that Z(iΩ) approximates the target phase-cancellation filter 
e^2iΩ L_arm/c in the high-gain limit γ→2λ+0. 
First, by setting κ_1=2c/L_arm and taking this limit, the coefficients 
in Z(s) become 


    α_3 =    -β_3=-λ,  α_2 = β_2=2c(c+L_armλ)/L_armL_4-λ^2,
    
    α_1=    -β_1=-2cλ(c+L_armλ)/L_armL_4,  α_0 = β_0=-2c^2λ^2/L_armL_4.


Then, in the s-domain with |s|≪γ, or equivalently |s|≪λ, we have 


    Z(s) ≈α_1s+α_0/β_1s+β_0
             =- s+cλ/c+L_armλ/ s-cλ/c+L_armλ.


Now we set an additional assumption κ_1≪γ, which leads to 
c≪ L_armλ and as a result 


    Z(s)≈- s+c/L_arm/ s-c/L_arm.


This is exactly the same as the transfer function G_11^(fb)(s) in 
Eq. (<ref>) with κ_1=2c/L_arm. 
Hence Z(iΩ)≈ e^2iΩ L_arm/c=e^-iϕ_arm(Ω) 
holds, meaning that the system depicted in Fig. <ref>(a) may 
approximate the target phase-cancellation filter. 
This can be actually seen in Fig. <ref>(c) showing the phase 
plot of Z(iΩ) given in Eq. (<ref>), 
where L_4=0.5 m and the other parameters are the same as those used in 
Fig. <ref>(b). 
This shows that the exact model incorporating the loop cavity a_4 certainly 
has the desired phase cancelling effect. 








 §.§ The entire system and stabilizing control







In the previous subsection we have seen that the constructed filter certainly has a desired 
phase-cancellation property, from which we expect that this active filter broadly enhances 
the sensitivity of the gravitational-wave detector in the high-frequency regime. 
Here we model the entire system composed of the interferometer and the phase-cancellation 
filter depicted in Fig. <ref>. 
This entire system must be stabilized, because the phase cancellation filter is itself an unstable 
system; 
here we employ measurement-based feedback for this purpose. 

Note that this measurement feedback is not an additional requirement over the existing proposals; 
the stabilization is necessary as well in the optomechanics-based implementation 
<cit.>. 





Let us begin with the dynamics of the entire system without stabilization. 
Here we assume that ω_p=ω_k=ω_0 (k=1, …, 4). 
Then in the rotating frame at frequency ω_0, the Hamiltonian of the entire system 
is given by 


    H_tot=   P^2/2M+Δ_d d^† d
                    -ħ G_arm(d+d^†)X-F_GWX
    
        + ħ g_NI(d^† a_1+da_1^†)
               +i ħλ(a_1^† a_2^†-a_1a_2)
    
        + ħ g_24(a_2^† a_4+a_2a_4^†)
                + ħ g_34(a_3^† a_4+a_3a_4^†),


where again (X, P) are the differential (position, momentum) operators of ETMs and 
d is the sideband mode of the interferometer field. 
We assume that only a_1 couples with d, with strength 
g_NI=√(cγ/(2L_arm)). 
The signal leaks to outside through the SRM where the vacuum input d_in 
must enter. 
Then the dynamical equation of the entire system is given by


    Ẋ= 1/MP,  Ṗ= ħ G_arm(d+d^†)+F_GW,  
    
       ḋ=  -iΔ_d d-γ_IFO/2d+iG_armX
                            -ig_NIa_1-√(γ_IFO)d_in,
    
       ȧ_1= -γ_1loss/2a_1-ig_NId
                             +λ a_2^†-√(γ_1loss)b_1loss,
    
       ȧ_2 = λ a_1^†-ig_24a_4,  ȧ_3 = -κ_3loss/2a_3-ig_34a_4
                             -√(κ_3loss)b_3loss,
    
       ȧ_4 = -κ_4loss/2a_4
                              -ig_24a_2-ig_34a_3-√(κ_4loss)b_4loss,
    
               d_out = √(γ_IFO)d + d_in,


where b_kloss (k=1, 3, 4) are the noise field representing the optical losses 
of the internal modes a_k with magnitude κ_kloss. 
We use the quadrature representation 
q_k=(a_k+a_k^†)/√(2), p_k=(a_k-a_k^†)/(√(2)i) (k=d, 1, 2, 3, 4), 
Q_d^in,out=(d_in,out+d_in,out^†)/√(2), 
P_d^in,out=(d_in,out-d_in,out^†)/(√(2)i), 
Q_nloss=(b_nloss+b_nloss^†)/√(2), 
P_nloss=(b_nloss-b_nloss^†)/(√(2)i) (n=1, 3, 4). 
Also we define the dimensionless operators X_M=X√(MΩ_M/ħ) and 
P_M=P/√(ħ MΩ_M), with Ω_M the eigenfrequency of the ETM; 
they satisfy [X_M,P_M]=i. 
Then the above dynamical equations are summarized to 


    ẋ=Ax+B_ww,   y=Cx+Dw,


where x=[  X_M  P_M  q_d  p_d  q_1  p_1  q_2  p_2  q_3  p_3  q_4  p_4  ]^T, 
w=[  F_GW  Q_d^in  P_d^in  Q_1loss  P_1loss  
Q_3loss  P_3loss  Q_4loss  P_4loss  ]^T, and
y=[  Q_d^out  P_d^out  ]^T. 
The matrices A∈ℝ^12×12, B_w∈ℝ^12×9, 
C∈ℝ^2×12, D∈ℝ^2×9 are shown in 
Appendix <ref>. 
Note that A has eigenvalues with positive real part, meaning that the uncontrolled 
entire system is unstable.


To stabilize the system, we apply the measurement-based quantum feedback control, 
particularly the quantum linear quadratic Gaussian (LQG) feedback control 
<cit.>, which has the same form as the classical version 
<cit.>. 
This control is generally conducted by feeding a measurement output back to control the 
system. 
In our case we measure P_d^out by the photodetector 
(note that measuring both Q_d^out and P_d^out is prohibited by quantum mechanics); 
the measurement result is used to construct the estimate x̂, which is fed back to 
control the ETMs directly by implementing a piezo-actuator <cit.>. 
This control is modeled by adding the classical input u=-F_ux̂ to the dynamics 
of the oscillator, where F_u∈ℝ^1×12 is the feedback gain to be designed. 
In the LQG setting, the (quantum) Kalman filter is used to obtain the least squared estimate 
x̂. 
The entire controlled system are then given by


    ẋ = Ax+B_ww+B_uu, 
         y_m =  P_d^out=C_mx+D_mw,
    
       ẋ̂̇ = Ax̂+B_u u+K_u(y_m-C_mx̂),    u=-F_ux̂,


where K_u∈ℝ^12 is the Kalman gain shown later. 
B_u=[ 0, 1, 0, ⋯, 0]^T∈ℝ^12 (only the second element is 
non-zero) represents that the actuator directly drives P_M of the oscillator. 
C_m∈ℝ^1×12 and D_m∈ℝ^1×9 are second 
row vectors of C and D, respectively. Here we define e=x̂-x. 
Then the above dynamical equation is rewritten as 


    [
    	[ ẋ; ė ]]=A_tot[
    	[ x; e ]]+B_totw,   
    	y_m=C_tot[
    	[ x; e ]]+D_totw,


where


    A_tot=   [
    	[ A-B_uF_u  -B_uF_u;        0 A-K_uC_m ]],   
    	C_tot=[
    	[ C_m   0 ]],
    
     
    	B_tot=   [
    	[        B_w; K_uD_m-B_w ]],   
           D_tot=D_m.


The entire system becomes stable when A_tot has no eigenvalue with 
positive real part. 
Since the eigenvalues of A_tot are the same as those of A-B_uF_u and 
A-K_uC_m, we can stabilize the system by determining appropriate F_u and K_u. 
The necessary and sufficient condition for such F_u and K_u to exist is that 
the system is controllable and observable; that is, the following controllability 
matrix 𝒞_u and observability matrix 𝒪_y_m are of full-rank: 


    𝒞_u
             =[[     B_u    AB_u       ⋯ A^11B_u ]], 
    
       𝒪_y_m
             =[[         C_m^T     A^T C_m^T             ⋯ (A^T)^11C_m^T ]]^T.


In the LQG setup, F_u and K_u are determined from the policy to minimize the 
following cost function 𝒥 and the estimation error ϵ: 


    𝒥 =    lim_t→∞1/t⟨∫_0^t (x^T(τ)Qx(τ)+Ru^2(τ))dτ⟩, 
    
    ϵ =    ⟨ (x-x̂)^T(x-x̂)⟩,


where Q∈ℝ^12×12 and R∈ℝ are the weighing matrices. 
From the separation principle of the LQG control, these two optimization problems 
can be solved separately. 
If the optimal solutions of F_u and K_u are uniquely determined, then they stabilize 
the entire system and are given by 


    F_u=R^-1B_u^TP_F,   
           K_u=(P_KC_m^T+B_wVD_m^T)(D_mD_m^T)^-1,


where P_F∈ℝ^12×12 and P_K∈ℝ^12×12 are the 
solutions of the following algebraic Riccati equations:


    P_FA+A^TP_F-P_FB_uR^-1B_u^TP_F+Q=0, 
    
       P_KA^T+AP_K+B_wVB_w^T-(P_KC_m^T+B_wVD_m^T)
    
           ×(D_mVD_m^T)^-1(P_KC_m^T+B_wVD_m^T)^T=0.


V is the covariance matrix of the vector w. 
Note that F_GW, the first element of w, is assumed to be a Gaussian noise with 
known variance, but in reality it is an unknown signal whose noise part is not necessarily Gaussian; 
as we will describe later, this assumption can be weakened so that only a stabilizing controller exists. 








 §.§ Quantum noise of the stabilized system







The quantum noise observed at the detector is calculated as follows. 
First we have 


    y_m(s)=   [C_tot(sI-A_tot)^-1B_tot
                    +D_tot]w(s)
    
    
              =     Ψ_Q_d(s)Q_d^in(s)+Ψ_P_d(s)P_d^in(s) + Ψ_h(s)h(s) 
    
        + ∑_k=1,3,4( 
                 Ψ_Q_k(s)Q_kloss(s) + Ψ_P_k(s)P_kloss(s) ),


where the functions Ψ_⋆ are the transfer functions from the 
corresponding noise fields and the gravitational-wave strain signal h(t) 
to the output y_m. 
As shown in Eq. (<ref>), the quantum noise operator is defined as 
F_N(iΩ)=y_m(iΩ)/Ψ_h(iΩ)-h(iΩ). 
Then from Eq. (<ref>), we obtain the noise spectral density:


    S(Ω)=   1/2|Ψ_h|^2(|Ψ_Q_d|^2+|Ψ_P_d|^2+|Ψ_Q_1|^2+|Ψ_P_1|^2
    
              + |Ψ_Q_3|^2+|Ψ_P_3|^2+|Ψ_Q_4|^2
                   +|Ψ_P_4|^2 ).





The value of parameters chosen in this study are shown in Table <ref>. 
For the interferometer part, unlike the setup in Fig. <ref>, a non-zero value of Δ_d 
is taken, which is necessary for the system to be controllable and observable; 
actually 𝒞_u and 𝒪_y_m in 
Eq. (<ref>) are both of full-rank in these parameter choice. 
This non-zero value of Δ_d and the value of γ_IFO are calculated from the 
scaling law <cit.> of the gravitational-wave detector containing a SRM; 
consequently the coupling constant γ_IFO is effectively changed to 1062 Hz from 
2π× 200 Hz. 
For the phase-cancellation filter part, we emphasize that κ_1 = 2c/L_arm is the 
condition to cancel the phase and it draws the connection between the interferometer and the 
phase-cancellation filter. 
For the LQG controller part, the value of 1/2 in V= diag{10^-22, 1/2, ⋯, 1/2} 
(all 1/2 except the (1,1) element) denotes the vacuum fluctuation. 
On the other hand, the (1,1) element of V denotes the variance of F_GW, which 
is unknown as mentioned at the end of the previous subsection. 
Hence we used Fig. 1 in Ref. <cit.> to have an estimate value 10^-22, meaning 
that the Kalman filter does not produce the optimal estimate x̂. 
However, note that we do not need a very accurate estimate of this value but require the LQG 
control only to stabilize the entire control system. 
In fact with the above parameter choice, this purpose is fulfilled, and we end up with 
Fig. <ref>; 
this shows that the proposed phase-cancellation filter can enhance the bandwidth in 
the high-frequency regime without sacrificing the peak-sensitivity. 



















We conclude this section with discussion on the possible advantages and 
disadvantages of the proposed filter. 
Figures <ref>(a) and (b) show the quantum noise 
of the entire controlled system with several optical loss magnitudes in 
(a) NDPA and (b) the loop cavity. 
Recall that the loss magnitude of the cavity modes are represented as  
γ_1loss=cT_1loss/L_1 and 
κ_4loss=cT_4loss/L_4, 
where (T_1loss, T_4loss) and (L_1, L_4) are the 
optical loss ratios and the round trip cavity lengths of the corresponding 
cavity modes, respectively. 
In the figures, the cavity lengths are fixed to L_1=1.5 m and L_4=0.5 m, 
and we change optical loss ratios to plot the noise spectral densities with 
several loss magnitudes γ_1loss or κ_4loss. 
The other parameters are the same as those used in 
Fig. <ref>. 
Importantly, the figures show that the sensitivity is not largely affected 
by the optical losses both in NDPA (γ_1loss) and the loop 
cavity (κ_4loss). 
In particular, the loss in the loop cavity has almost no effect on the 
sensitivity, as expected from the fact that the feedback amplification scheme 
is in general robust against the imperfection in the feedback loop 
<cit.>. 
As for the loss in NDPA, there is certainly some impact on the sensitivity 
in the high frequency regime, but this can be reduced by making the length 
of NDPA longer. 

On the other hand, the parameter κ_3loss, i.e., the optical 
loss magnitude in the control cavity with mode a_3, has a large impact on 
the sensitivity, as indicated in Fig. <ref>(c); 
note that the parameters other than κ_3loss are the same 
as those used in Fig. <ref>. 
Figure <ref>(c) tells why κ_3loss 
is chosen to be much smaller than γ_1loss and κ_4loss in Fig. <ref>. 
To achieve such a small loss, the optical path length of the control cavity 
should be long; 
from κ_3loss=cT_3loss/L_3 with T_3loss 
the loss ratio and L_3 the round trip length of the control cavity, if 
κ_3loss=100 Hz is required, we need, e.g., 
T_3loss=0.01% and L_3 = 300 m. 
That is, although the proposed phase-cancellation filter based on the feedback amplification 
method can be constructed in all-optics way in contrast to the opto-mechanical proposal 
<cit.>, a very careful fabrication for the control cavity is required. 
In fact, to experimentally implement the proposed all-optics phase-cancellation filter requires 
a number of phase locks around the devices <cit.>; also note that the large power 
level of the laser injected to the interferometer is required. 












§ CONCLUSION




In this paper, we have shown that a variety of quantum functionalities are generated 
under the concept of feedback amplification. 
We hope that, combined with the several established quantum information methods 
such as entanglement generation <cit.> and analogue information processing 
<cit.>, those basic functionalities may be effectively applied to enhance 
the performance of existing quantum technological devices and moreover to create an
useful quantum mechanical machine. 




This work was supported in part by JST PRESTO Grant No. JPMJPR166A. 
N.Y. acknowledges helpful discussions with H. Yonezawa, E. Huntington, M. Woolley, 
I. Petersen, M. James, and V. Ugrinovskii. 












§ PROOF OF EQ. (<REF>)



The proof is composed of the following four steps. 




  Step 1: 
Derive the relation between (b_1,b_2,b_3) and (b̃_1,b̃_2,b̃_3) by using 
Eqs. (<ref>), (<ref>), and 
(<ref>). 
More precisely, we find the transfer function H(s) that satisfies the following relation:
 

    [ [     b̃_1(s); b̃_2^†(s^∗);     b̃_3(s);             ]]
               = H(s)
                [ [ b_1^†(s^∗);     b_2(s);     b_3(s);            ]],  
    
             H(s)=   [[ H_11(s) H_12(s) H_13(s); H_21(s) H_22(s) H_23(s); H_31(s) H_32(s) H_33(s) ]].



  Step 2: Prove that, in the high gain limit regime, 

    H(s)
        →[ [                0               -1                0;       -1/K_22(s)                0 -K_21(s)/K_22(s); -K_12(s)/K_22(s)                0   [K(s)]/K_22(s);                  ]].



  Step 3: 
Derive the relation between (b̃_2, b̃_3,b̃_4) and (b_1, b_3, b_4) using 
Eqs. (<ref>), (<ref>), and (<ref>). 
That is, we aim to have the expression of G^(fb)(s) in terms of 
{H_ij(s)} and {K_ij^∗(s)}. 


  Step 4: 
Substitute {H_ij(s)} in Eq. (<ref>) to { G^(fb)_ij(s)} obtained 
in Step 3, which leads to Eq. (<ref>). 


First, in Step 1, from Eqs. (<ref>), (<ref>), 
and (<ref>), the entries in H(s) are found to be 


    H_11=   G_12-G_21K_22[G]/1-G_21G_21K_22,  
        H_12=
            G_11G_22K_22/1-G_21G_21K_22,
    
        H_13=   G_11K_21/1-G_21G_21K_22,  
        H_21=
            G_11G_22/1-G_21G_21K_22,
    
        H_22=
               G_12+G_21K_22[G]/1-G_21G_21K_22,  
        H_23=
            G_11G_21K_21/1-G_21G_21K_22,
    
        H_31=   K_12G_21G_22/1-G_21G_21K_22,  
        H_32=
            K_12G_22/1-G_21G_21K_22,
    
        H_33=   K_11-G_21G_21[K]/1-G_21G_21K_22,


where we have omitted the Laplace index s for simplicity. 


The proof of Step 2 is similar to that for deriving Eq. (<ref>) in Sec. <ref>. 
That is, we take the “quantum ideal op-amp assumption" as follows;


    [G(s)]/G_22(s)→ 0,   G_12(s)/G_22(s)→ 1,   G_21(s)/G_22(s)→ 1,
       [G(s)]/G_22(s)→ 0,   G_12(s)/G_22(s)→ 1,   G_21(s)/G_22(s)→ 1,


and G_11(s)=G_22(s) and G_11(s)=G_22(s) in the domain 
s∈ℂ such that |G_11(s)|→∞ 
(⟺ |G_22(s)|→∞) and |G_11(s)|→∞ 
(⟺ |G_22(s)|→∞). 
Then in this high gain limit, the transfer functions are calculated as follows; 


    H_11=   G_12-G_21K_22[G]/1-G_21G_21K_22
    
        =   (G_12/G_22)/G_22-(G_21/G_22)K_22([G]/G_22)/1/(G_22G_22)-(G_21/G_22)(G_21/G_22)K_22
    →   0,
    
    H_12
        =   G_11G_22K_22/1-G_21G_21K_22
    
        =   K_22/1/(G_22G_22)-(G_21/G_22)(G_21/G_22)K_22→ 1,
    
    H_13
        =   G_11K_21/1-G_21G_21K_22
    
        =   K_21/G_22/1/(G_22G_22)-(G_21/G_22)(G_21/G_22)K_22→0,
    
    H_21
        =   G_11G_22/1-G_21G_21K_22
    
        =   1/1/(G_22G_22)-(G_21/G_22)(G_21/G_22)K_22→ -1/K_22,
    
    H_22
        =   G_12+G_21K_22[G]/1-G_21G_21K_22
    
        =   (G_12/G_22)/G_22+(G_21/G_22)K_22([G]/G_22)/1/(G_22G_22)-(G_21/G_22)(G_21/G_22)K_22
    →   0,
    
    H_23
        =   G_11G_21K_21/1-G_21G_21K_22
    
        =   (G_21/G_22)K_21/1/(G_22G_22)-(G_21/G_22)(G_21/G_22)K_22→ -K_21/K_22,
    
    H_31
        =   G_21G_22K_12/1-G_21G_21K_22
    
         =   (G_21/G_22)K_12/1/(G_22G_22)-(G_21/G_22)(G_21/G_22)K_22→-K_12/K_22,
    
    H_32
        =   G_22K_12/1-G_21G_21K_22
    
         =   K_12/G_22/1/(G_22G_22)-(G_21/G_22)(G_21/G_22)K_22→0,
    
    H_33
        =   K_11-G_21G_21[K]/1-G_21G_21K_22
    
         =   K_11/(G_22G_22)-(G_21/G_22)(G_21/G_22)[K]/1/(G_22G_22)-(G_21/G_22)(G_21/G_22)K_22
    →   [K]/K_22.


Now we have proved Eq. (<ref>). 


Step 3 can be completed by combining Eq. (<ref>), (<ref>), 
and (<ref>). 
The resulting expressions are found to be 


    G_11^(fb)
        =    
                H_21
                + (
                        H_11H_22 - H_12 H_21
                  )
                K_22^∗/
                    1 -H_12 K_22^∗,
        
    
    G_12^(fb)
        =    
                H_23 + ( H_13H_22- H_12 H_23 )  K_22^∗/
                1 - H_12 K_22^∗,
         
    
    G_13^(fb)
        =    
                H_22K_12^∗/
                1 - H_12 K_22^∗,
         
    
    G_21^(fb)
        =    
                H_31
                +
                (
                    H_11H_32
                    -
                    H_12 H_31
                )
                K_22^∗/
                1 - H_12 K_22^∗,
         
    
    G_22^(fb)
        =    
                H_33
                +
                (
                    H_13H_32
                    -
                    H_12 H_33
                )
                K_22^∗/
                1 - H_12 K_22^∗,
         
    
    G_23^(fb)
        =    H_32K_12^∗/1 - H_12 K_22^∗,
         
    
    G_31^(fb)
        =    H_11 K_21^∗/1 - H_12 K_22^∗,
         
    
    G_32^(fb)
        =    H_13K_21^∗/1 - H_12 K_22^∗,
         
    
    G_33^(fb)
        =    
                K_11^∗ + H_12(
                    K_12^∗ K_21^∗ - K_11^∗ K_22^∗)
            /
                1 - H_12 K_22^∗.


Step 4 is done by simply applying Eq. (<ref>) to the above equations 
from Eq. (<ref>) to Eq. (<ref>), which leads to 
Eq. (<ref>). 








§ THE MATRIX ENTRIES OF A, B_W, C, D






    A=[ 
    [          0        Ω_M          0          0          0          0          0          0          0          0          0          0;          0          0    √(2)G_M          0          0          0          0          0          0          0          0          0;          0          0   -γ_IFO/2          Δ          0       g_NI          0          0          0          0          0          0;    √(2)G_M          0         -Δ   -γ_IFO/2      -g_NI          0          0          0          0          0          0          0;          0          0          0       g_NI -γ_1loss/2          0          λ          0          0          0          0          0;          0          0      -g_NI          0          0 -γ_1loss/2          0         -λ          0          0          0          0;          0          0          0          0          λ          0          0          0          0          0          0       g_24;          0          0          0          0          0         -λ          0          0          0          0      -g_24          0;          0          0          0          0          0          0          0          0 -κ_3loss/2          0          0       g_34;          0          0          0          0          0          0          0          0          0 -κ_3loss/2      -g_34          0;          0          0          0          0          0          0          0       g_24          0       g_34 -κ_4loss/2          0;          0          0          0          0          0          0      -g_24          0      -g_34          0          0 -κ_4loss/2 ]],




    B_w=[
    [           0           0           0           0           0           0           0           0           0; 1/√(ħ MΩ_M)           0           0           0           0           0           0           0           0;           0   -√(γ_IFO)           0           0           0           0           0           0           0;           0           0   -√(γ_IFO)           0           0           0           0           0           0;           0           0           0 -√(γ_1loss)           0           0           0           0           0;           0           0           0           0 -√(γ_1loss)           0           0           0           0;           0           0           0           0           0           0           0           0           0;           0           0           0           0           0           0           0           0           0;           0           0           0           0           0 -√(κ_3loss)           0           0           0;           0           0           0           0           0           0 -√(κ_3loss)           0           0;           0           0           0           0           0           0           0 -√(κ_4loss)           0;           0           0           0           0           0           0           0           0 -√(κ_4loss) ]],


    C=[
    [        0        0 √(γ_IFO)        0        0        0        0        0        0        0        0        0;        0        0        0 √(γ_IFO)        0        0        0        0        0        0        0        0 ]],  
    D=[
    [ 0 1 0 0 0 0 0 0 0; 0 0 1 0 0 0 0 0 0 ]],

where G_M=G_arm√(ħ/(MΩ_M)).










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