aa r X i v : . [ qu a n t - ph ] M a y Quantum Feedback Amplification
Naoki Yamamoto
Department of Applied Physics and Physico-Informatics,Keio University, Hiyoshi 3-14-1, Kohoku, Yokohama 223-8522, Japan (Dated: October 8, 2018)Quantum amplification is essential for various quantum technologies such as communication andweak-signal detection. However, its practical use is still limited due to inevitable device fragilitythat brings about distortion in the output signal or state. This paper presents a general theory thatsolves this critical issue. The key idea is simple and easy to implement: just a passive feedback of theamplifier’s auxiliary mode, which is usually thrown away. In fact, this scheme makes the controlledamplifier significantly robust, and furthermore it realizes the minimum-noise amplification evenunder realistic imperfections. Hence, the presented theory enables the quantum amplification to beimplemented at a practical level. Also, a nondegenerate parametric amplifier subjected to a specialdetuning is proposed to show that, additionally, it has a broadband nature.
PACS numbers: 42.65.Yj, 02.30.Yy, 03.65.Yz, 42.50.Lc
I. INTRODUCTION
The amplifier is clearly one of the most important com-ponents incorporated in almost all current technologicaldevices. The basic function of an autonomous ampli-fier is simply to transform an input signal u to y = Gu with gain G >
1. However, such an amplifier is frag-ile in the sense that the device parameters change easily,and eventually distortion occurs in the output y . Thiswas indeed a most serious issue which had prevented anypractical use of amplifiers in, e.g., telecommunication.Fortunately, this issue was finally resolved back in 1927by Black [1, 2]; there are a huge number of textbooksand articles reviewing this revolutionary work, and herewe refer to Refs. [3, 4]. The key idea is the use of feed-back shown in Fig. 1; that is, an autonomous amplifiercalled the “plant” is combined with a “controller” in sucha way that a portion of the plant’s output is fed back tothe plant through the controller. Then the output of thewhole controlled system is given by y = G (fb) u, G (fb) = G GK , (1)where K is the gain of the controller. Now, if the planthas a large gain G ≫
1, it immediately follows that G (fb) ≈ /K . Hence, the whole system works as an am-plifier, simply provided that the controller is a passivedevice (i.e., an attenuator) with K <
1. Importantly, apassive device such as a resistor is very robust, and itsparameters contained in K almost do not change. This isthe mechanism of robust amplification realized by feed-back control. Note, of course, that this feedback archi-tecture is the core of an operational amplifier (op-amp).Surely there is no doubt about the importance of quan-tum amplifiers. A pertinent quantum counterpart to theclassical amplifier is the phase-preserving linear amplifier [5, 6] (in what follows, we simply call it the “amplifier”).In fact, this system has a crucial role in diverse quantumtechnologies such as communication, weak-signal detec-tion, and state processing [7, 8, 9, 10, 11, 12, 13, 14]. ControllerPlant
Gu K y+ -
FIG. 1: Classical feedback-amplification scheme: G is thegain of an autonomous amplifier, and K is the gain of a passivecontroller. In particular, recent substantial progress in both theoryand experiments [15, 16, 17, 18, 19, 20, 21, 22, 23, 24]has further advanced this field. An important fact is that,however, an amplifier must be an active system poweredby external energy sources, implying that its parametersare fragile and can change easily. Because of this pa-rameter fluctuation, the amplified output signal or statesuffers from distortion [25, 26, 27]. As a consequence,the practical applicability of the quantum amplificationis still severely limited. That is, we are now facing thesame problem we had 90 years ago.To make the discussion clear, let us here describethe general quantum-amplification process. Ideally theamplifier transforms a bosonic input mode b to ˜ b = g b + g b † , where b is an auxiliary mode, and the coeffi-cients satisfy | g | −| g | = 1 from [˜ b , ˜ b † ] = 1. Hence, theoutput ˜ b is an amplified mode of b with gain | g | > g and g are frequency dependent as shown later. How-ever, note again that the system parameters, especiallythe coupling strength of the pumped crystal, cannot bekept exactly constant, and eventually the amplified out-put mode ˜ b has to be distorted.Now the motivation is clear; we need a quantum ver-sion of the feedback-amplification method described inthe first paragraph. The contribution of this paper is, infact, to develop a general theory for quantum feedbackamplification that resolves the fragility issue of quan-tum amplifiers. The key idea is simple and easy to im-plement, i.e., feedback of the auxiliary output mode ˜ b through a passive controller to the auxiliary input mode b . Indeed, it is proven that the whole controlled systempossesses a strong robustness property against parame-ter fluctuations, which thus enables quantum amplifiersto be implemented at a practical level. This type of con-trol scheme is, in general, called the coherent feedback [28, 29, 30, 31, 32, 33], meaning that an output field isfed back to an input field through another quantum sys-tem without involving any measurement process; hence,an excess classical noise is not introduced in the feedbackloop. Now note that the auxiliary output ˜ b has some in-formation about ˜ b due to their entanglement, though ˜ b is usually thrown away in the scenario of quantum am-plification. Thus, we have an interpretation that the pre-sented scheme utilizes the signal-recycling technique [34]for reducing the sensitivity, unlike the conventional use ofit for enhancing the sensitivity of the gravitational-wavedetector.In addition to the above-described main contribution,some important results are obtained. First, we see thatthe controlled system reaches the fundamental quantumnoise limit [6] even if some imperfections are present inthe feedback loop. This means that precise fabrication ofthe feedback control is not necessary, which thus againemphasizes the feasibility of the presented scheme. Next,this paper proposes a type of NDPA subjected to a spe-cial detuning that circumvents the usual gain-bandwidthtrade-off in the amplification process. A drawback of thismodified amplifier is that, as will be shown, it is very sen-sitive to the parameter fluctuation. The presented the-ory has a distinct advantage in such a situation; that is,this issue can now be resolved by constructing a feedbackloop. Therefore, as a concrete application of the theory,this paper proposes a robust, near-minimum-noise, andbroadband amplifier.Finally, note that there are a variety of quantum ampli-fiers considered in the literature such as an optical back-action evasion amplifier [35]; however, the schematic pre-sented in this paper is essentially different from all thosemodifications in the following sense. While those mod-ified amplifiers have their own purposes for improvingthe performance or achieving the goal in some specificsubjects (e.g. back-action evasion), the feedback schemeis a device-independent and purpose-independent funda-mental architecture that must be incorporated in all am-plifiers. In fact, in the classical regime, the “operation”part of an op-amp has its own purpose (e.g., differenti-ation and integration), but any op-amp does not workwithout feedback. II. MODEL OF PHASE-PRESERVING LINEARQUANTUM AMPLIFIER
Let us begin with a specific model: the NDPA. Thisis an optical cavity system with two internal modes a and a . They are orthogonally polarized and obey thefollowing Hamiltonian: H = ω a † a + ω a † a + iλ ( a † a † e − iω t − a a e iω t ) , with λ ∈ R the coupling strength between the modes, ω i the resonant frequencies of a i , and 2 ω the pumpfrequency. Also, in the above expression the rotating-wave approximation is taken under the assumption 2 ω ≈ ω + ω . The system couples with a signal input b and an auxiliary (idler) input b with strength κ . Then,in the rotating frame at frequency ω , the dynamics ofthe NDPA is given by the following Langevin equations[10, 36, 37]: da dt = (cid:16) − κ − i ∆ (cid:17) a + λa † − √ κb , (2) da † dt = (cid:16) − κ i ∆ (cid:17) a † + λa − √ κb † , (3)where ∆ = ω − ω and ∆ = ω − ω are detuning. Also,the output equations (boundary conditions) are given by˜ b = √ κa + b , ˜ b † = √ κa † + b † . (4)Now the Laplace transformation of an observable x t inthe Heisenberg picture is defined by x ( s ) := Z ∞ e − st x t dt, where Re( s ) >
0. Then the Laplace transforms of b ,etc., are connected by the following linear equations:˜ b ( s ) = g ( s ) b ( s ) + g ( s ) b † ( s ) ,g ( s ) = ( s − κ + i ∆ )( s + κ − i ∆ ) − λ D ( s ) , g ( s ) = − κλD ( s ) ,D ( s ) = (cid:16) s + κ i ∆ (cid:17)(cid:16) s + κ − i ∆ (cid:17) − λ . The stability analysis can be conducted in the Laplacedomain; that is, for the amplifier to be stable, all rootsof the characteristic equations of the transfer functions(i.e., poles ) must lie in the left-hand complex plane. Inthe above case, particularly when ∆ = ∆ = 0, thecharacteristic equation is D ( s ) = s + κs + κ / − λ = 0;hence, κ / − λ > s = iω with ω the frequency; that is, weconsider the linear transformation at the steady state,˜ b ( iω ) = g ( iω ) b ( iω ) + g ( iω ) b † ( iω ). Note that g and g satisfy | g ( iω ) | − | g ( iω ) | = 1 for all ω . In partic-ular, when ∆ = ∆ = 0, the amplification gain at the ControllerPlant b b b b b b b b d d dGK d FIG. 2: Coherent feedback configuration for the autonomousamplifier G (plant). d , d , d , and d are unwanted noisyinput fields. resonant frequency ω = 0 is given by | g (0) | = κ + 4 λ | κ − λ | , and it takes a large number nearly at the threshold λ ≈ κ/ −
0. Thus, ˜ b is in fact an amplified mode of b withgain | g | .The above example can be generalized; any phase-preserving linear quantum amplifier is modeled as anopen dynamical system with two inputs and two out-puts. Let us represent the input-output relation in theLaplace domain as follows: (cid:20) ˜ b ( s )˜ b † ( s ) (cid:21) = G ( s ) (cid:20) b ( s ) b † ( s ) (cid:21) ,G ( s ) = (cid:20) G ( s ) G ( s ) G ( s ) G ( s ) (cid:21) , (5)where b ( s ) is the Laplace transformation of b , etc. Thetransfer function matrix G ( s ) at s = iω (i.e. the scatter-ing matrix) satisfies | G ( iω ) | − | G ( iω ) | = | G ( iω ) | − | G ( iω ) | = 1 ,G ( iω ) G ∗ ( iω ) − G ( iω ) G ∗ ( iω ) = 0 ∀ ω. (6)Thus, | G ( iω ) | represents the amplification gain. III. THE QUANTUM FEEDBACKAMPLIFICATIONA. Feedback configuration
Our control scheme is based on coherent feedback; thatis, the controller is also given by a quantum system andis connected to the plant through the input and outputfields. Note that, if a measurement process is involvedin the feedback loop, it inevitably introduces additionalnoise. Now we take a passive system (e.g. a beam splitterand an optical cavity) as the controller, with two inputs b , b and two outputs ˜ b , ˜ b ; note that a single-input and single-output passive system has a gain equal to 1and thus does not work as an attenuator. We representthe input-output relation of this system in the Laplacedomain as follows: (cid:20) ˜ b † ( s )˜ b † ( s ) (cid:21) = K ( s ) (cid:20) b † ( s ) b † ( s ) (cid:21) ,K ( s ) = (cid:20) K ( s ) K ( s ) K ( s ) K ( s ) (cid:21) . (7)Here, the creation operator representation is taken tomake the notation simple. Because of the passivity prop-erty, the transfer function matrix K ( s ) is unitary in theFourier domain; i.e., K ( iω ) † K ( iω ) = I holds for all ω .We now consider connecting the controller to the plant.But unlike the classical case, where both the plant andthe controller can be a single input-output system andarbitrary split or addition of signal is allowed, designinga feedback scheme in the quantum case is not trivial. Forexample, we could divide ˜ b into two paths by a beamsplitter and use one of them for feedback purpose, butin this case the resultant whole controlled system is nota minimum-noise amplifier. Instead, this paper proposesthe following feedback connection as shown in Fig. 2:˜ b = b , b = ˜ b , (8)which is, of course, equivalent to ˜ b † = b † and b † = ˜ b † .Note that in Fig. 2 practical unwanted noises d , . . . , d are illustrated, but these modes are ignored for the mo-ment. From Eqs. (5), (7) and (8), the whole controlledsystem, with inputs b , b † and outputs ˜ b , ˜ b † , has the fol-lowing input-output relation in the Laplace domain: (cid:20) ˜ b ( s )˜ b † ( s ) (cid:21) = " G (fb)11 ( s ) G (fb)12 ( s ) G (fb)21 ( s ) G (fb)22 ( s ) b ( s ) b † ( s ) (cid:21) , where G (fb)11 = [ G − K ( G G − G G )] / (1 − K G ) ,G (fb)12 = ( G K ) / (1 − K G ) ,G (fb)21 = ( G K ) / (1 − K G ) ,G (fb)22 = [ K + G ( K K − K K )] / (1 − K G ) . The matrix entries satisfy the condition correspondingto Eq. (6), i.e., | G (fb)11 ( iω ) | − | G (fb)12 ( iω ) | = 1 ∀ ω , etc.Finally, as remarked in Sec. II, for the whole controlledsystem to be stable, the controller should be carefullydesigned so that all poles of G (fb) ij ( s ) must lie in the left-hand complex plane, as demonstrated in Sec. V. B. Robust amplification via feedback
We now focus on the output of the controlled systemin the Fourier domain, i.e.,˜ b ( iω ) = G (fb)11 ( iω ) b ( iω ) + G (fb)12 ( iω ) b † ( iω ) , and the amplification gain | G (fb)11 ( iω ) | especially when theoriginal gain | G ( iω ) | is large. Note that G (fb)11 lookssomewhat different from the classical counterpart (1);hence, it is not immediate to see if | G (fb)11 ( iω ) | can be ap-proximated by a function of only the controller. Nonethe-less, the analogous result to the classical case indeedholds as shown below.For the proof we use Eq. (6) (below, we omit the vari-able iω ). First, from | G || G | = | G || G | togetherwith the other two equations, we have | G | = | G | and | G | = | G | . Also G G − G G = G /G ∗ holds.Here, in the limit | G | → ∞ , it follows that | G G − G G || G | = | G || G | = 1 | G | → . This implies that ( G G − G G ) / | G | converges tozero in this limit. As a consequence, we have | G (fb)11 | = (cid:12)(cid:12)(cid:12) G / | G | − K ( G G − G G ) / | G | / | G | − K G / | G | (cid:12)(cid:12)(cid:12) → (cid:12)(cid:12)(cid:12) G / | G |− K G / | G | (cid:12)(cid:12)(cid:12) = 1 | K | . Hence, in the frequency range where the plant has a largegain | G ( iω ) | ≫
1, the whole controlled system amplifiesthe input b ( iω ) with gain | G (fb)11 ( iω ) | ≈ / | K ( iω ) | > G changes while main-taining a large value, the whole controlled system carriesout robust amplification with stable gain 1 / | K | . C. Feedback gain synthesis
Here we conduct a quantitative analysis on the robust-ness property, which provides a guideline for synthesizingthe feedback gain K . To see the idea clearly, let us againconsider the classical case (1). Let ∆ G be the fluctua-tion that occurs in the plant G ; then the fluctuation thatoccurs in the whole controlled system G (fb) is calculatedas∆ G (fb) = G + ∆ G G + ∆ G ) K − G GK ≈ ∆ G (1 + GK ) , which as a result leads to∆ G (fb) G (fb) = 11 + GK · ∆ GG . (9)Hence, the gain sensitivity to the unwanted fluctuationcan be reduced by the factor 1 / | GK | by feedback.Equation (9) suggests to us not to design G and K sep-arately; rather, what determines the performance of thecontrolled amplifier is the loop gain GK . Actually, whilethe controlled amplification gain, G (fb) ≈ /K , can be made bigger by taking a smaller value of K , we shouldnot design a too small K such that GK ≈
0; in this case,Eq. (9) yields ∆ G (fb) /G (fb) = ∆ G/G , and thus there isno improvement in the sensitivity. The so-called
Bodeplot developed by Bode (e.g., see Ref. [40]) is a powerfulgraphical method for synthesizing K as well as G , andit is now the standard tool for general feedback circuitdesign.Now let us try to establish the quantum version of theabove discussion. Note in advance that a straightforwardcalculation, like Eq. (9), cannot be carried out in thequantum case, but nonetheless a similar useful equationfor determining the controller parameter K is shown.First, due to | G | = | G | and G G − G G = G /G ∗ , we find G (fb)11 = | G | − K G G ∗ (1 − K G ) = G G ∗ · G ∗ − K − K G , which thus leads to | G (fb)11 | = | G ∗ − K || − K G | . The fluctuation of the controlled gain is given by∆ | G (fb)11 | = | ( G + ∆ G ) ∗ − K || − K ( G + ∆ G ) | − | G ∗ − K || − K G | . Then, from the general relation | x + ǫ | ≈ | x | + ( xǫ ∗ + x ∗ ǫ ) / | x | with x, ǫ ∈ C and | ǫ | ≪
1, the normalized fluc-tuation of the amplification gain of the controlled systemcan be explicitly calculated as∆ | G (fb)11 || G (fb)11 | = 1 − | K | | G ∗ − K | · Re (cid:16) G ∗ − K − K G ∆ G (cid:17) . (10)Next, noting that | G | = | G | ≫ | K | ≪| G | , we find from Eq. (10) that (cid:12)(cid:12)(cid:12) ∆ | G (fb)11 || G (fb)11 | (cid:12)(cid:12)(cid:12) ≈ | G | · (cid:12)(cid:12)(cid:12) Re (cid:16) G ∗ − K G ∆ G (cid:17)(cid:12)(cid:12)(cid:12) . Thus, from the general relation | Re( xy ) | ≤ | x || y | with x, y ∈ C , it follows that (cid:12)(cid:12)(cid:12) ∆ | G (fb)11 || G (fb)11 | (cid:12)(cid:12)(cid:12) ≤ | − K G | · | ∆ G || G | . (11)Equation (11) has a similar form to Eq. (9) and indeedprovides us a guideline for feedback design. That is, asin the classical case, the balance of K and G deter-mines the ability of the controlled system to suppress thefluctuation (note that | ∆ G | ≥ ∆ | G | = ∆ | G | ). Inparticular, we now deduce a similar conclusion as in theclassical case; if we choose a too small K in addition to | G | ≫ K G is almost zero,then substituting the relation∆ | G | = ∆ | G | = Re( G ∗ ∆ G ) / | G | into Eq. (10) we obtain ∆ | G (fb)11 | / | G (fb)11 | = ∆ | G | / | G | .That is, in this case the fluctuation is not at all sup-pressed via feedback. To design an appropriate controllergain K , the Bode plot of the loop gain K ( iω ) G ( iω )is useful. IV. QUANTUM NOISE LIMIT
Let us define the noise magnitude of b by h| ∆ b | i := 12 h ∆ b ∆ b † + ∆ b † ∆ b i , ∆ b = b − h b i . Then, through the ideal amplification process ˜ b = g b + g b † , the noise magnitude must be also amplified as h| ∆˜ b | i = | g | h| ∆ b | i + | g | /
2, where b is assumedto be in the vacuum. This implies the degradation of thesignal-to-noise ratio: ^ (S / N) = |h ˜ b i| h| ∆˜ b | i = |h b i| h| ∆ b | i + A < |h b i| h| ∆ b | i = (S / N) . Hence, the added noise A := | g | | g | = | g | − | g | quantifies the fidelity of the amplification process [5, 6].In particular, in the large amplification limit | g | → ∞ we find A → /
2, which is called the quantum noise limit .Up to now, the ideal setup is assumed, and the con-trolled system is driven by only the signal b and the aux-iliary input b , implying that it actually reaches the quan-tum noise limit in the large amplification limit. Hence,here we consider the following general case where someexcess noise exists, as illustrated in Fig. 2; the plant issubjected to an unwanted noise d that enters into thesystem in the form ˜ b = G b + G b † + G d ; thecontroller is also affected by a noise d ; furthermore, thefeedback transmission lines are lossy, which is modeledby inserting fictitious beam splitters with additional in-puts d and d . Note that d , . . . , d are all annihilationmodes. Then the output of the whole controlled systemhas the form˜ b = G (fb)11 b + G (fb)12 b † + G (fb)13 d + G (fb)14 d † + G (fb)15 d † + G (fb)16 d † . Then, if the excess noises are all vacuum, the added noisein the feedback-controlled amplification process, denotedby A (fb) , satisfieslim | G (fb)11 |→∞ A (fb) = lim | G |→∞ A (o) = 12 + | G | | G | , (12)where A (o) is the added noise of the plant. The proof ofEq. (12), including the detailed forms of G (fb) k , is given inAppendix A. This is a very useful result for the followingreasons. First, in the large amplification limit the two G a i n [ d B ] −2 −1 Frequency [ rad/s] κ −2 −1 λ = 5 κ λ = 3 κ λ = κ (b)(a) β = 0.0 (No Feedback) β = 0.05 β = 0.1 FIG. 3: (a) Gain profile of the specially detuned NDPAwithout feedback. (b) Gain profile of the feedback-controlledsystem with parameters λ = 5 κ and various β . added noises A (fb) and A (o) are equal; as a consequencethe second term in the right-hand side of Eq. (12) is afunction of only the plant and cannot be further alteredby feedback control. Hence, we have the following no-go theorem: If the original amplifier does not reach thequantum noise limit (i.e., lim | G |→∞ | G | / | G | > V. APPLICATION TO OPTICAL BROADBANDAMPLIFICATION
In any practical situation, it is important to carefullyengineer an amplifier so that it has a proper frequencybandwidth in which nearly constant amplification gainis realized. On the other hand, it is known in boththe classical and quantum cases that, particularly for anamplifier with a single pole, the effective bandwidth be-comes smaller if the amplification gain is taken to bebigger. That is, there is a gain-bandwidth constraint .However, this constraint is not necessarily applied to amore complex amplifier with multiple poles. In fact, re-cently in Ref. [21], the authors propose a hybrid amplifiercomposed of two cavity modes and an additional opto-mechanical mode that circumvents the gain-bandwidthconstraint. In this section, we study another system thatis also free from this constraint, that yet does not need anadditional degree of freedom. Then, the effectiveness offeedback is discussed, demonstrating its ability to makethe system robust.
ControllerPlant b b b b b b b b FIG. 4: The NDPA (plant) and its coherent feedback; bysimply feeding the auxiliary output ˜ b back to the auxiliaryinput b through just a beam splitter, the robustness propertyis drastically improved, as discussed in the main text. A. NDPA with special detuning
The plant system is the NDPA with dynamics (2), (3)and output (4). Here we consider the ideal case wherethe unwanted noises d , . . . , d shown in Fig. 2 are notpresent. Without any invention, this system is subjectto a gain-bandwidth constraint [10], but now let us takethe specific detuning as ∆ = ∆ = λ . The transferfunction matrix of this system is then given by G ( s ) = 1( s + κ/ (cid:20) s − κ / iκλ − κλ − κλ s − κ / − iκλ (cid:21) . The maximum gain is | G (0) | = p λ /κ , whichbecomes larger by increasing λ . Remarkably, this ampli-fication can be carried out without sacrificing the band-width. Figure 3 (a) shows the three cases correspondingto λ = κ, κ, κ , all of which have the same effectivebandwidth ∼ κ/
10. A clear advantage of this system isin its implementability; that is, it is composed of onlyoptical devices, and there is no need to prepare an auxil-iary system such as an opto-mechanical oscillator. Also,note that the system is always stable (the pole of thetransfer function matrix is − κ/ λ , in contrast to the standard NDPA whichimposes | λ | < κ/ B. The feedback effect
Next, let us consider the feedback control of this am-plifier, again in the ideal setup. Here, as shown in Fig. 4,a beam splitter with transmissivity α and reflectivity β is taken as a controller. This device has no internal dy-namics, and its transfer function matrix is constant: K ( s ) = (cid:20) α ββ − α (cid:21) , α, β ∈ R . Thus, K = β represents the attenuation level. Theamplification gain of the whole controlled system is then G (fb)11 ( s ) = (1 − β ) s + βκs − (1 + β ) κ / iκλ (1 − β ) s + κs + (1 + β ) κ / iβκλ . G a i n [ d B ] −2 −1 Frequency [ rad/s] κ FIG. 5: The upper blue lines represent the gain profileof the specially detuned NDPA without feedback, | G ( iω ) | ,while the lower red lines correspond to the controlled case | G (fb)11 ( iω ) | with β = 0 .
1. In both cases, λ = 5 κ . This expression shows that, as expected from the generaltheory discussed in Sec. III B, in the limit | G ( iω ) | → ∞ (i.e., λ → ∞ ) the gain becomes | G (fb)11 ( iω ) | → /β in acertain frequency bandwidth. To determine the attenua-tion level β , we need the stability condition; it is shownin Appendix B that, for all the poles of G (fb)11 ( s ) to lie inthe left-hand complex plane, the parameters must satisfy | λ | < κ s ββ (1 − β ) . (13)This yields | β | < κ/ | λ | when β ≪
1; hence, let us herechoose λ = 5 κ , leading to | β | < .
1. Figure 3 (b) showsthe gain | G (fb)11 ( iω ) | for the two cases β = 0 . β = 0 . β = 0). Wethen observe that the gain of the controlled amplifier be-comes smaller than that without feedback; in exchangefor this reduced gain, the controlled amplifier obtains agreat robustness property against the parameter fluctu-ation as demonstrated later. Note that a larger value of β (thus smaller amplification gain) induces a wider fre-quency bandwidth; hence, the controlled NDPA has thegain-bandwidth constraint. But the point here is ratherthat the gain and bandwidth can be easily tuned by justchanging the reflectivity of the beam splitter. That is, aneasily adjustable amplification can be realized, and thisis also a clear advantage of feedback. C. Robustness property
As repeatedly emphasized in this paper, the mainstrength of feedback is that the controlled system pos-sesses a robustness property. To see this, let us consideran imperfect case as follows. First, the device parame-ters are fragile; the coupling strength λ fluctuates in sucha way that λ = (1 + 0 . ǫ ) λ , where λ is the nominalvalue; similarly, the detunings ∆ and ∆ can be slightlydeviated from λ , which is modeled by ∆ = (1+0 . ǫ ) λ and ∆ = (1+0 . ǫ ) λ . Here ( ǫ , ǫ , ǫ ) are independentrandom variables subjected to the uniform distributionin [ − , A γ α κ / (a) (b)= 0.10 γ κ = 0.01 γ κ (fb)(o) A FIG. 6: The added noise of the controlled amplifier withattenuation level β = 0 . β = 0) versus (a) γ/κ with fixed α = α = 0 .
5, and (b) α = α with fixed γ/κ . In both figures, the red solid linesrepresent A (fb) , while the blue dotted lines are A (o) , at ω = 0. the signal mode a is subjected to optical loss, which ismodeled by adding the extra term − γa / − √ γd tothe right-hand side of Eq. (2), with γ the magnitude ofthe loss and d the unwanted vacuum noise. The feed-back transmission lines are also lossy, which is modeledby Eq. (A6).The blue lines in Fig. 5 are 50 sample values of theautonomous gain | G ( iω ) | in the case λ = 5 κ and α = α = 0 .
99. That is, in fact, due to the parameterfluctuation described above, the amplifier becomes fragileand the amplification gain significantly varies. Nonethe-less, this fluctuation can be suppressed by feedback; thered lines in Fig. 5 are 50 sample values of the controlledgain | G (fb)11 ( iω ) | with attenuation level β = 0 .
1, whosefluctuation is indeed much smaller than that of | G ( iω ) | [53]. (Note that, because the fluctuation of | G (fb)11 ( iω ) | is very small, the set of sample values looks like a thickline.) That is, the controlled system is certainly robustagainst the realistic fluctuation of the device parameters. D. Added noise
Finally, let us investigate how much the excess noiseis added to the output of the controlled or noncontrolledspecially detuned NDPA. Again we set λ = 5 κ , andthe feedback control is conducted with attenuation level β = 0 .
1. Also, the same imperfections considered in theprevious subsection are assumed; that is, the system suf-fers from the signal loss (represented with γ ) and theprobabilistic fluctuation of the parameters ( λ, ∆ , ∆ );furthermore, the feedback control is implemented withthe lossy transmission lines (represented with α and α ).With this setup Fig. 6 is obtained, where the red solidlines are sample values of the added noise at the centerfrequency ω = 0 for the controlled system, A (fb) given byEq. (A11), while the blue dotted lines represent those ofthe non-controlled system, A (o) given by Eq. (A3). (Inthe figure it appears that six thick lines are plotted, buteach is the set of 50 sample values.) Figure 6 (a) shows A (fb) and A (o) versus the signal loss rate γ/κ , where for the controlled system we fix α = α = 0 . α = α ,with fixed signal loss γ/κ .The first crucial point is that, in both figures (a) and(b), A (fb) and A (o) are close to each other. This is thefact that can be expected from Eq. (12), which states that A (fb) and A (o) coincide in the large amplification limit. Itis also notable that, for all sample values, A (fb) is smallerthan A (o) [54] ; in other words, the feedback controller re-duces the added noise, although in the large amplificationlimit this effect becomes negligible as proven in Eq. (12).Another important feature is that, as seen in Fig. 6 (a),the signal loss γ is the dominant factor increasing theadded noise, and the feedback loss 1 − α (= 1 − α ) doesnot have a large impact on it, as seen in Fig. 6 (b). Asconsequence, when γ is small, the controlled amplifiercan perform amplification nearly at the quantum noiselimit 1 /
2, with almost no dependence on the feedbackloss; this fact is also consistent with Eq. (12).In summary, the specially detuned NDPA with feed-back control functions as a robust, near-minimum-noise(if γ ≪ κ ), and broadband amplifier. VI. CONCLUSION
The presented feedback control theory resolves the crit-ical fragility issue in phase-preserving linear quantumamplifiers. The theory is general and thus applicableto many different physical setups, such as optics, opto-mechanics, superconducting circuits, and their hybridiza-tion. Moreover, the feedback scheme is simple and easyto implement, as demonstrated in Sec. V. Note also thatthe case of phase-conjugating amplification [38, 39] canbe discussed in a similar way; see Appendix C.In a practical setting, the controller synthesis prob-lem becomes complicated, implying the need to develop amore sophisticated quantum feedback amplification the-ory, which indeed was established in the classical case[3, 40, 41]. The combination of those classical approacheswith the quantum control theory [42, 43, 44] should ad-vance this research direction. Another interesting futurework is to study genuine quantum-mechanical settings,e.g., probabilistic amplification [26, 45, 46, 47]. Finally,note that feedback control is used in order to reach thequantum noise limit, in a different amplification scheme(the so-called op-amp mode) [10, 48]; connection to theseworks is also to be investigated.This work was supported in part by JSPS Grant-in-AidNo. 15K06151.
Appendix A: Proof of Eq. (12)
First, we again describe the setup of the imperfect sys-tem depicted in Fig. 2, in a more detailed way. The plantsystem is subjected to an unwanted noise d , such thatthe input-output relationship is given by (cid:20) ˜ b ( s )˜ b † ( s ) (cid:21) = (cid:20) G ( s ) G ( s ) G ( s ) G ( s ) G ( s ) G ( s ) (cid:21) b ( s ) b † ( s ) d ( s ) . (A1)The transfer function matrix in this case satisfies | G | − | G | + | G | = | G | − | G | − | G | = 1 ,G G ∗ − G G ∗ + G G ∗ = 0 , (A2)for all s = iω . Thus, the added noise of the plant systemis given by A (o) = | G | + | G | | G | = | G | + 2 | G | − | G | = 12 − | G | + | G | | G | . (A3)This leads to lim | G |→∞ A (o) = 12 + | G | | G | , (A4)where the second term is assumed to exist. Note that ina more general setup some creation noise modes (e.g. d † )can be contained in Eq. (A1), but this modification doesnot change the conclusion. The controller also containsan unwanted noise field d in the following form: (cid:20) ˜ b † ( s )˜ b † ( s ) (cid:21) = (cid:20) K ( s ) K ( s ) K ( s ) K ( s ) K ( s ) K ( s ) (cid:21) b † ( s ) b † ( s ) d † ( s ) . (A5)The point here is that only the creation modes appearin Eq. (A5), unlike Eq. (A1) that involves both creationand annihilation modes; this is indeed due to the pas-sivity property of the controller. Finally, the transmis-sion lines (optical fields) for feedback are assumed to belossy. This setting is modeled by inserting two fictitiousbeam splitters; the beam splitter in the output field ˜ b has transmissivity α and reflectivity δ , and also thebeam splitter in the output field ˜ b has transmissivity α and reflectivity δ (we assume α i , δ i ∈ R without loss ofgenerality). Then, the coherent feedback connection isrepresented by the following relations: b † = α ˜ b † + δ d † , b † = α ˜ b † + δ d † . (A6)Combining Eqs. (A1), (A5), and (A6), we end up with˜ b = G (fb)11 b + G (fb)12 b † + G (fb)13 d + G (fb)14 d † + G (fb)15 d † + G (fb)16 d † , where G (fb)11 = [ G − α α K ( G G − G G )] / G ,G (fb)13 = [ G − α α K ( G G − G G )] / G ,G (fb)12 = α G K / G , G (fb)14 = α G K / G ,G (fb)15 = α δ G K / G , G (fb)16 = δ G / G , and G = 1 − α α K G . Note that the transfer func-tions satisfy at s = iω | G (fb)11 | − | G (fb)12 | + | G (fb)13 | − | G (fb)14 | − | G (fb)15 | − | G (fb)16 | = 1 . (A7)Here we derive some preliminary results that are usedlater. First, Eq. (A2) leads to | G G ∗ − G G ∗ | = | G | | G | ; together with the other two equations, wethen have (cid:12)(cid:12)(cid:12) G G − G G G (cid:12)(cid:12)(cid:12) = 1 + (cid:12)(cid:12)(cid:12) G G (cid:12)(cid:12)(cid:12) − | G | + | G | + 1 | G | . (A8)Similarly, (cid:12)(cid:12)(cid:12) G G − G G G (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) G G (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) G G (cid:12)(cid:12)(cid:12) − . (A9)Furthermore, we can prove that | G /G | → | G | → ∞ as follows. If | G /G | →
0, this leads to G /G → G /G →
0, and fur-thermore, | G /G | → | G | → G → G G ∗ − G G ∗ →
0, which leads to a con-tradiction.Now we are concerned with the amplification gain | G (fb)11 | in the limit | G | → ∞ . It is given by | G (fb)11 | = (cid:12)(cid:12)(cid:12) − α α K ( G G − G G ) /G /G − α α K ( G /G ) (cid:12)(cid:12)(cid:12) ≈ (cid:12)(cid:12)(cid:12) − α α K ( G /G ) + ( G G − G G ) /G G /G (cid:12)(cid:12)(cid:12) . The second term is upper bounded, because fromEq. (A8) (cid:12)(cid:12)(cid:12) G G − G G G (cid:12)(cid:12)(cid:12).(cid:12)(cid:12)(cid:12) G G (cid:12)(cid:12)(cid:12) ≤ r .(cid:12)(cid:12)(cid:12) G G (cid:12)(cid:12)(cid:12) (A10)and also | G /G | does not converge to zero. Therefore,we need K ( G /G ) → K →
0) to havethe condition | G (fb)11 | → ∞ .Finally, the added noise in the controlled system iscomputed as follows: A (fb) = | G (fb)12 | + | G (fb)13 | + | G (fb)14 | + | G (fb)15 | + | G (fb)16 | | G (fb)11 | = | G (fb)11 | + 2 | G (fb)13 | − | G (fb)11 | = 12 − | G (fb)11 | + | G (fb)13 | | G (fb)11 | , (A11)where Eq. (A7) is used; also note that all the noise fieldsare now vacuum. The third term is given by | G (fb)13 | | G (fb)11 | = (cid:12)(cid:12)(cid:12) G − α α K ( G G − G G ) G − α α K ( G G − G G ) (cid:12)(cid:12)(cid:12) . (A12)Then from Eq. (A8) we find (cid:12)(cid:12)(cid:12) K · G G − G G G (cid:12)(cid:12)(cid:12) ≤ | K | + (cid:12)(cid:12)(cid:12) K · G G (cid:12)(cid:12)(cid:12) → , in the limit K ( G /G ) → K →
0. Also | K ( G G − G G ) /G | → | G (fb)11 | → ∞ is given by lim | G (fb)11 |→∞ A (fb) = 12 + | G | | G | . Hence, together with Eq. (A4), we obtain Eq. (12).The point of this result is that, due to the strong con-straint on the noise input fields, which is represented byEq. (A7), the added noise does not explicitly contain theterms that stem from the creation input modes d † , d † ,and d † . This is because of the passivity property of thecontroller (A5) and the feedback transmission lines (A6)that are composed of only the creation modes. Appendix B: Proof of Eq. (13)
The stability of the controlled amplifier is guaranteedif and only if all the poles of G (fb) ( s ) lie in the left-handcomplex plane. In our case, those are given by the solu-tions of the following characteristic equation:(1 − β ) s + κs + (1 + β ) κ / iβκλ = 0 . In the standard case where the coefficients of the char-acteristic equation are all real, the Routh-Hurwitz crite-rion can be used for the stability test, but now the aboveone contains an imaginary coefficient. Hence here we set s = x + iy , x, y ∈ R , transforming the above equation to h x + κ − β ) i − y = κ β − β ) , y = − βκλ − β ) x + κ . The poles are given by the intersections of these curvesin the complex plane; Fig. 7 shows the case for βλ > βλ < κ s β − β . Considering the other case (i.e. βλ <
Nyquist method [41] is a very useful graphical tool that can dealwith such cases, although an exact stability condition isnot available. Another way is a time-domain approachbased on the so-called small-gain theorem [49, 50], thatproduces a sufficient condition for a feedback-controlledsystem to be stable; the quantum version of this method[51] will be useful to test the stability of the controlledfeedback amplifier. −βλ 1+β1−βκ− 2
Re(s)Im(s)
FIG. 7: The poles represented by the cross points betweentwo curves.
Appendix C: Phase-conjugating case
The Hermitian conjugate of the second element ofEq. (3) is given by ˜ b = G ∗ b † + G ∗ b . That is, theoutput ˜ b is the amplified signal of the conjugated input b † , with gain | G | ; this is called the phase-conjugatingamplification. The feedback control in this case is almostthe same as for the phase-preserving amplification. Weconsider the ideal feedback configuration shown in Fig. 2(i.e., the noise fields d , . . . , d are ignored) and now fo-cus on the auxiliary output ˜ b = ( G (fb)21 ) ∗ b † + ( G (fb)22 ) ∗ b .Then the amplification gain is evaluated, in the large am-plification limit | G | → ∞ , as | G (fb)21 | = (cid:12)(cid:12)(cid:12) K / | G | − K G / | G | (cid:12)(cid:12)(cid:12) → (cid:12)(cid:12)(cid:12) K − K e iθ (cid:12)(cid:12)(cid:12) = | K || K | = s | K | − . In the first line of the above equation, we have used | G /G | = 1 + 1 / | G | →
1; also the last equalitycomes from the unitarity of K , i.e., | K | + | K | = 1.Therefore, when the original amplification gain is large( | G | ≫ p / | K | − >
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