Quantum noise and vacuum fluctuations in balanced homodyne detections through ideal multi-mode detectors

The balanced homodyne detection as a readout scheme of gravitational-wave detectors is carefully examined from the quantum field theoretical point of view. The readout scheme in gravitational-wave detectors specifies the directly measured quantum operator in the detection. This specification is necessary when we apply the recently developed quantum measurement theory to gravitational-wave detections. We examine the two models of measurement. One is the model in which the directly measured quantum operator at the photodetector is Glauber's photon number operator, and the other is the model in which the power operator of the optical field is directly measured. These two are regarded as ideal models of photodetectors. We first show these two models yield the same expectation value of the measurement. Since it is consensus in the gravitational-wave community that vacuum fluctuations contribute to the noises in the detectors, we also clarify the contributions of vacuum fluctuations to the quantum noise spectral density without using the two-photon formulation which is used in the gravitational-wave community. We found that the conventional noise spectral density in the two-photon formulation includes vacuum fluctuations from the main interferometer but does not include those from the local oscillator. Although the contribution of vacuum fluctuations from the local oscillator theoretically yields the difference between the above two models in the noise spectral densities, this difference is negligible in realistic situations.


Introduction
Since gravitational waves are directly observed by the Laser Interferometer Gravitationalwave Observatory [1] in 2015, the gravitational-wave astronomy and the multi-messenger astronomy including gravitational-wave are developing [2]. To support these developments as more precise science, improvements of the detector sensitivity are necessary. For these improvements, it is important to continue the research and development of the science of gravitational-wave detectors together with the source sciences of gravitational waves. Current gravitational-wave detectors already reached to the fundamental noise that arises from quantum fluctuations of lasers in the detector system and the reduction of these quantum noises is an important topic in the science of gravitational-wave detectors. Therefore, to improve the sensitivity of gravitational-wave detectors, a rigorous quantum-theoretical description is required. operator in gravitational-wave detectors, because we take the time-sequence data detected at the photodetector and discuss its Fourier spectrum. This implies that the photodetection in gravitational-wave detectors is essentially a multi-mode detection.
In this paper, we re-examine the measurement process of the balanced homodyne detections for multi-mode detections. As a directly measured operator in multi-mode detections, Glauber's photon number operator N a (t) ∝Ê (−) a (t)Ê (+) a (t) (1.5) is often used in many literatures. HereÊ are the positive-and negative-frequency parts of the electric fieldÊ a (t) of the detected optical field, respectively, which are introduced in Sec. 2.1 in this paper. On the other hand, in the gravitational-wave community, it is often said that the probability of the excitation of the photocurrent is proportional to the power operatorP (1.6) of the measured optical field. It is also true that there are many literatures in which the photo-detection is treated as a classical stochastic process, in which the detection probability is proportional to the expectation value of the power operator [12][13][14]. Historically speaking, in theoretical quantum measurement of optical fields [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], there was a controversy on which variable is the directly measured by photodetectors for multifrequency optical fields. Some insisted that the direct observable of photodetectors is the above photon number operator (1.5) in the multi-mode optical fields, some insisted that the direct observable of photodetectors is the power of the optical field 1 . Although the author could not find a complete conclusion of this controversy in literature, both of the operators (1.5) and (1.6) are used in recent many literature.
Keeping in our mind this situation, in this paper, we examine the two models of directly measured operators as two ideal cases. One is the model in which the directly measured operator at photodetectors is Glauber's photon number operator (1.5), and the other is the model in which the directly measured operator is the power operator (1.6). As the results of these examinations, we reached to the conclusion that the expectation value of the operator b θ given by Eq. (1.1) is measured by the balanced homodyne detection contrary to the results in Refs. [9,10]. 1 For example, discussions in Refs. [15-17, 23, 24] are as follows: From the microscopic point of view, the interaction between photon and charged particles is proportional topÂ, wherep is the momentum of the charged particles in the photodetector andÂ is the vector potential of the laser. This interaction is also given aspÂ =p(Â (+) +Â (−) ), where A (+) (A (−) ) is the positive-(negative-) frequency part of the vector potential. In Refs. [15][16][17]23], the termpÂ (−) was ignored at their starting point. In Ref. [23], with this ignorance, it was insisted that the macroscopic photo-current is proportional to the expectation value Â (−)Â(+) under the "quasistationary field condition" even in the wideband detection. This result supports Glauber's photon number operator (1.5) as the directly measured operator in photo-detections. However, in Ref. [24], it was claimed that the termpÂ (−) gives a finite contribution to the macroscopic photo-current in the wideband photo-detections.
In any case, the connection of the microscopic process and macroscopic measured photo-current will be necessary to reach the conclusion. However, the story is not so simple but the problem becomes quite complicated if we take into account the randomness of detected photo-currents [17,28]. 3/40 In this paper, we also evaluate noise spectral density of these models. In many literatures of the gravitational-wave detection, it is written that the single sideband noise spectral densityS (s) A (ω) for an arbitrary operatorÂ(ω) with the vanishing expectation value in the "stationary" system is given by The noise spectral density (1.7) is introduced by Kimble et al. in Ref. [5] in the context of the two-photon formulation [7,8]. However, in this paper, we do not use the two-photon formulation, though some final formulae are written in terms of the two-photon formulation as much as possible. We also examine the original meaning of Kimble's noise spectral density (1.7) and derive the deviations from this noise formula (1.7) together with the noise contributions due to the imperfection of the interferometer configuration. Although Collett et al. [29] also discussed the quantum noises in the heterodyne and homodyne detections, they used the normal-ordered generating function to evaluate the noises in detections. Therefore, it is not clear whether their arguments properly includes vacuum fluctuations or not. The contributions from the vacuum fluctuations are carefully discussed in this paper. The organization of this paper as follows. In Sec. 2, we summarize the basic notation which we use in this paper and introduce Glauber's photon number operators and the power operator of the detected optical field in the multi-mode detections. In Sec. 3, we examine the expectation values measured by the homodyne detections in the two models with the above two different directly measured operators. In Sec. 4, we show the general arguments of the noise spectral density and introduce the definition of the noise spectral density in this paper. In Sec. 5, we evaluate the quantum noise in the homodyne detections. The final section (Sec. 6) devoted to our summary and discussions.

Preliminary
In this section, we describe the notation of the electric fields, quantum states, and quantum operators, which are used within this paper. In Sec. 2.1, we describe the basic notations of quantum operators for the electric field of the laser in gravitational-wave detectors. In Sec. 2.2, we introduce two quantum operators which are the candidates of the directly measured operators at the multi-mode photodetectors. One is Glauber's photon number operator and the other is the power operator as mentioned in Sec. 1.

Basic Notation
In this paper, we denote the electric field of the lasers bŷ is the positive frequency part ofÊ a (t − z): The A in the factor of Eq. (2.1) is the sectional area of the laser beam. The operatorsâ(ω) andâ † (ω) in Eq. (2.1) are annihilation and creation operators of the photon, respectively, and these satisfy the commutation relations (1.4). We explicitly denote the quadratureâ(ω) of the electric field in the subscript of the electric field (2.1) itself. We may write Eq. (2.1) so thatÊ where Θ(ω) is the Heaviside step function From the commutation relations (1.4), we can derive the commutation relation between the positive-and negative-frequency partÊ (±) a (t) of the electric field as The subscription "a" of the function ∆ a (t − t ′ ) indicates that this is the vacuum fluctuations originated from the electric fieldÊ a with the quadratureâ(ω). We note that the function ∆ a (t − t ′ ) has the ultra-violet divergence. This divergence is clearly seen from the ω-integration in the definition (2.6) of the function ∆ a (t − t ′ ) and is famous one which comes from the infinite sum of the vacuum fluctuations. However, in the actual measurement of the time sequence of the variables, the time in a measurement is discrete with a finite time bin. Namely, the time in a measurement has the minimal time interval. This time interval is adjusted in experiments so that the time scale of the interest is sufficiently resolved and it gives the maximum frequency ω max which becomes natural ultra-violet cut off of the frequency in the obtained data. Incidentally, in the actual measurement of the time sequence of the variables, the whole measurement time is also finite and this finiteness of the whole measurement time gives the minimum frequency ω min which corresponds to a natural infra-red cut off in frequency. Therefore, we may regard that the integration range over ω in the definition of the function ∆ a (t − t ′ ) in Eq. (2.6) as [ω min , ω max ] instead of [0, +∞]. For this reason, throughout this paper, we do not regard the divergence in the definition of the function ∆ a (t − t ′ ) as a serious one.
Finally, we introduce the vacuum state |0 a and the coherent state |α a of the electric field whose complex amplitude is α(ω). As usual, we introduce the vacuum state |0 a through the operation of annihilation operatorâ(ω) aŝ a(ω)|0 a = 0. (2.8) We also introduce the coherent state |α a associate with the annihilation operatorâ(ω) as an eigen state of the operatorâ(ω) aŝ a(ω)|α a = α(ω)|α a . (2.9) Here, we note that the dimension of the complex amplitude α(ω) is Hz −1/2 . Incidentally, we also note this coherent state is theoretically produced by the operation of the displacement 5 (2.10) Here, we note that the subscriptions "a" in the states |0 a and |α a indicates that these states are associated with the electric field operatorÊ a (t) with the quadratureâ(ω).
In the time domain operatorsÊ (±) a (t), the definition (2.8) of the vacuum state is given bŷ On the other hand, the definition (2.9) of the coherent state is also given bŷ Ac (2.13) These properties (2.11)-(2.13) are often used in arguments of the homodyne detections below.

Multi-mode number and power operators
It is often said that the operator (1.5) is regarded as the "photon number" in the multimode photon system from Glauber's pioneer papers [15,16]. One of reasons for regarding the operator (1.5) as the photon number is the fact that the superposition of the electric field operator is possible within the field equation, i.e., the Maxwell equations, while the superposition of the operatorn(ω) defined by Eq. (1.3) is not possible within the field equations. Therefore, it is said that the natural extension of the photon number to the multi-mode case is the operator given by Eq. (1.5). Furthermore, Glauber [15] discussed this operator through the state transition. The matrix element for the transition from the initial state |i to a final state |f in which one photon has been absorbed is given by f |Ê (+) (r, t)|i . The probability per unit time that a photon be absorbed by an ideal detector is at point "r" at time "t" is proportional to 14) The vacuum condition (2.11) implies that the rate at which photons are detected in the empty, or vacuum, state vanishes [15].
Although the above Glauber's argument seems to be plausible, the contributions of the vacuum fluctuations are not clear. On the other hand, it is consensus of the gravitational-wave community that the vacuum fluctuations are contribute to the noise in gravitational-wave detectors. For this reason, in this paper, we examine two models of the photodetection as two ideal cases. One is the model in which the directly measured operator at the photodetectors 6/40 is the multi-mode photon number defined bŷ The other is the model in which the photocurrents is proportional to the power operatorP a of optical field, which is defined bŷ (2. 16) Here, the coefficients κ n and κ p is a phenomenological constant whose dimension is [time]. These coefficients include so called "quantum efficiency". However, these are not important within our discussion of this paper.

Homodyne Detections by Multi-mode Detectors
In this section, we show the quantum field theoretical description of the homodyne detection motivated by the quantum measurement theory. In quantum measurement theories, we have to specify the directly measured operator to describe the measurement process. As mentioned above, we consider the two measurements. In Sec. 3.1, we discuss the measurement process in which the directly measured quantum operator is Glauber's photon number (2.15). In Sec. 3.2, we discuss the measurement process in which the directly measured quantum operator is the power operator (2.16) of the optical field. In this section, we concentrate only on the output expectation value of the homodyne detection.

Homodyne detections by photon-number counting detectors
Now, we describe the homodyne detections in the case where the directly measured quantum operator is Glauber's photon number (2.15). We also describe the details of the homodyne detections, here. We start our arguments from the simple homodyne detection in Sec. 3.1.1. This simple homodyne detection is extended to the balanced homodyne detection, which described in Sec. 3.1.2.
3.1.1. Simple Homodyne Detection. Here, we review the simple homodyne detection depicted in Fig. 1. In this paper, we want to evaluate the signal in the electric fieldÊ b (t). The electric field from the local oscillator is the coherent state (2.12) with the complex amplitude γ(ω). The output signal fieldÊ b (t) and the additional optical field from the local oscillator is mixed through the beam splitter with the transmissivity η. In the ideal case, this transmissivity is η = 1/2. However, in this paper, we dare to denote this transmissivity of the beam splitter for the homodyne detection by η as a simple model of the imperfection of the interferometer. As introduced in Sec. 2.1, the output electric fieldÊ b from the main interferometer is given byÊ and the electric field from the local oscillator is given bŷ Furthermore, the electric field to be detected through the photodetector is given bŷ At the beam splitter, the signal electric fieldÊ b (t) and the electric fieldÊ li (t) are mixed and the beam splitter output the electric fieldÊ co (t) to the one of the ports. These fields are related through the relation Next, we assign the states for the independent electric field described in Fig. 1. As mentioned above, the electric fieldÊ li from the local oscillator is in the coherent state |γ li (2.12) with the complex amplitude γ(ω) in the frequency domain, or equivalently, in the time domain as Eq. (2.13). In addition to this state, the electric fieldsÊ di (t) and E ci (t) are in their vacua, respectively. The junction condition for the electric fields at the beam splitter is given byÊ Due to the relation (3.6), the state associated with the quadratureâ is described by the vacuum states for the quadraturesd i andĉ i . Usually, the state associated with the quadratureb depends on the state of the input fieldÊ a into the main interferometer and the other optical fields which inject to the main interferometer [5]. Furthermore, in this paper, we consider the situation where the output electric fieldÊ b includes the information of classical forces as in Eq. (1.2) and this information are measured through the expectation value of the operatorÊ b . 8/40 To evaluate the expectation value of the signal fieldÊ b , we have to specify the state of the total system. Here, we assume that this state of the total system is given by where the state |ψ main is the state for the electric fields associated with the main interferometer, which is independent of the state |γ li , |0 ci , and |0 di . As noted above, the signal fieldÊ b may depends on the input fieldÊ a , and this input fieldÊ a is related to the quadra-turesÊ ci andÊ di through Eq. (3.6). Therefore, strictly speaking, the expectation value of the signal fieldÊ b means that but we denote simply In this section, we specify the directly measured operator of the photodetector is the multimode photon number operator defined by Eq. (2.15) associated with the optical field which enters the photodetector. In the case of the simple homodyne detection depicted in Fig. 1, the direct observable which measured at the photodetector (P.D. in Fig. 1 The expectation value of the P.D. output field N co (t) under the state (3.7) is given by The second term in Eq. (3.13) is the linear combination of the electric fieldÊ b from the main interferometer. Here, we consider the case of the monochromatic local oscillator case, in which the complex amplitude γ(ω) of the coherent state from the local oscillator is given (3.14) In this case, the Fourier transformation of this second term in Eq. (3.13) is given by Here, we denote the complex function γ as and consider the situation where ω 0 ≫ ω > 0. In this case, Eq. (3.15) is given by We note that, at this moment, ω 0 is just the central frequency of the coherent state from the local oscillator and have nothing to do with the central frequency of the signal fieldÊ b (t) from the main interferometer. Therefore, Eq. (3.17) is still valid even in the case "heterodyne detection" in which the central frequency of the coherent state from the local oscillator may not coincide with the central frequency of the signal fieldÊ b (t). Now, we choose ω 0 so that this frequency coincides with the central frequency of the signal fieldÊ b (t). This choice is the "homodyne detection". In this choice, we may identify the quadraturesb(ω 0 + ω) andb(ω 0 − ω) with the upper-and lower-sideband quadraturesb + (ω) andb − (ω) in the two-photon formulation [7,8], respectively. Therefore, we may introduce the amplitude quadratureb 1 (ω) and the phase quadratureb 2 (ω) bŷ In terms of these quadraturesb 1,2 (ω) Eq. (3.17) is given by where the operatorb θ (ω) is defined by Eq. (1.1) through the definitions (3.18) of the amplitude, and the phase quadraturesb 1 (ω) andb 2 (ω). Thus, the middle term in Eq. (3.13) yields the expectation value of the operatorb θ (ω) in the two-photon formulation [5]. 10/40 On the other hand, the Fourier transformation of the final term in Eq. (3.13) in the monochromatic local oscillator case (3.14), we obtain (3.20) Then, the Fourier transformation of the expectation value (3.13) is given by Since the final term in Eq. (3.21) is classically predictable, we can eliminate this term from the data, or we may ignore the data at the frequency ω = 0. In any case, the final term is not important, though this term is the first dominant term in Eq. (3.21) in the case where |γ| is sufficiently large. In the same case, i.e., the case where |γ| is sufficiently large, the first term in Eq. (3.21) is negligible. Then, we can measure the middle term in Eq. (3.21). Thus, we may say that the expectation value b θ (ω) can be measured through the simple homodyne detection, though the first term in Eq. (3.21) becomes a noise in the expectation value.  As depicted in Fig. 2, in the balanced homodyne detection, we detect the optical field from another port of the beam splitter (BS) in addition to the interferometer setup of the simple homodyne detection depicted in Fig. 1. We denote the photodetectors D1 and D2 as in Fig. 2. 11/40 At D2, we detect the operator At the BS, the electric fields from the main interferometer and the local oscillator are mixed asÊ . (3.23) and the detected operatorN do (t) is given bŷ The expectation value of this operator under the state (3.7) is given by From the expectation values (3.13) and (3.25), we can eliminate the term N b (t) which was a noise term in the expectation value (3.13) of the simple homodyne detection. This elimination is accomplished by the combination From this expectation value, we define the signal operatorŝ N (t) aŝ , we have already shown that the Fourier transformation of the expectation value (3.28) is proportional to the expectation value b θ (ω) . Actually, in the case where the local oscillator is monochromatic with the central frequency ω 0 and the situation ω 0 ≫ ω > 0, we conclude that From the view point of quantum measurement theories, we may regard that the operator s N (t) is the signal operator to be measured in the measurement process of the balanced homodyne detection through the measurements of Glauber's photon numberN co (t) and N do (t). 12/40

Balanced Homodyne Detections by power counting detectors
In this section, we re-examine the arguments on the balanced homodyne detection in Sec. 3.1 under the premise that the direct observable is the power operator (2.16). At the beam splitter, the signal fieldÊ b (t) and the fieldÊ li (t) from the local oscillator is mixed through the conditions (3.4) and (3.23). Through these conditions and the definition (2.16) of the operatorP a (t), the power operatorsP cl (t) at the D1 andP do (t) at D2 are given byP Here, we note that the expectation value ofP li (t) is given by Here, ∆ li (0) is the vacuum fluctuations from the local oscillator. Inspecting the arguments in Sec. 3.1.2, we define the signal operatorŝ P (t) bŷ Now, we evaluate the expectation value of the signal operatorŝ P (t). Here, we assume the commutation relation This assumption is justified in the Appendix A. Under the assumption (3.35), the expectation value of the signal operatorŝ P (t) is given by When the monochromatic local oscillator with the central frequency ω 0 and the frequency ω of interest satisfy the condition ω 0 ≫ ω > 0, the Fourier transformation of the expectation value of the operatorŝ P (t) is given by Here, we note that the factor 1/2 in the definition (3.33) is chosen so that the expectation value (3.38) coincides with Eq. (3.29). From the view point of quantum measurement theories, we may regard that the operatorŝ P (t) is the signal operator to be measured in the measurement process of the balanced homodyne detection through the measurements of the power operatorsP co (t) andP do (t). 13/40

Noise Spectral Densities
In many literature of the gravitational-wave detection, it is written that the single sideband noise spectral densityS (s) A (ω) for an arbitrary operatorÂ(ω) in the frequency domain with the condition Â (ω) = 0 in the "stationary" system is given by Eq. (1.7) in the context of the two-photon formulation. In the two-photon formulation [7,8], we consider the sideband fluctuations in the frequency ω 0 ± ω with the central frequency ω 0 of the optical field. The "single sideband" means that the noise spectral density is evaluated only in the frequency range ω > 0 of the positive sideband ω 0 + ω or the range ω > 0 of the negative sideband ω 0 − ω. This is due to implicit assumption of the symmetry of the data around the central frequency ω 0 . The frequencies ω and ω ′ in Eq. (1.7) is the sideband frequencies in the two-photon formulation.
The noise spectral density in which we take into account of the both sideband ω 0 ± ω with ω ≷ 0 is called the double sideband noise spectral density. The double sideband noise spectral densityS A (ω) for an arbitrary operatorÂ(ω) in the frequency domain with the condition Â (ω) = 0 is also given by Furthermore, we consider the (double sideband) "correlation spectral density" of the observablesÂ(ω) andB(ω) with the condition Â (ω) = B (ω) = 0 is given by Here, we examine the meaning of the noise spectral density (4.1) and noise correlation spectral density (4.2). In this paper, we do not explicitly introduce the central frequency ω 0 nor distinguish the upper-or the lower-sideband which are basis of the two-photon formulation. We consider the noise-spectral density in the frequency range ω ∈ [0, +∞]. For this reason, we first consider the double-sideband noise spectral density (4.2) in which we take into account both of the upper-and lower-sideband frequency range. We also consider the single-sideband noise spectral density (1.7), if necessary.
To examine the meaning of the spectral densities (4.2), we consider the time-domain expression of this formulae for the correlation spectral density through the Fourier transformation of the above formulae. Performing the double inverse Fourier transformations associated with ω and ω ′ in noise correlation spectral density (4.2) and introduce the time-domain variableŝ 3) Eq. (4.2) yieldsC whereC Equivalently, we may represent Eq. (4.5) and We see that the left-hand side of Eq. (4.7) depends only on τ , while the right-hand side may depend both on t and τ . This dependence implies that the "stationarity" of the system. Here, the "stationarity" of the correlation means that the correlation function does not depend on the absolute value of t but depends only on the time-difference τ = t ′ − t. If we take into account of the non-stationary cases, the correlation function may depends on t as This is general form of the correlation function. From the general correlation function (4.8), we can obtain the correlation function for the stationary noise by the time-average as We use this expression C (av)AB (τ ) defined by (4.9) of the correlation function for the stationary noise, instead ofC AB (τ ) given by Eq. (4.7). WhenÂ(t) =B(t), the auto-correlation function C (av)A (τ ) for "stationary noise" is given by If the operatorÂ(t) is self-adjoint, i.e.,Â † (t) = A(t), the auto-correlation function is given by Here, we note that [Â(t),Â(t + τ )] = 0, in general.
In the above argument, we only consider the operatorÂ(t) whose expectation value vanishes Â (t) = 0. When the operator A(t) has a non-trivial expectation value Â (t) := in|Â(t)|in under the state |in , we consider the noise operatorÂ n (t) for the operatorÂ(t), which is defined byÂ (t) =:Â n (t) + Â (t) . (4.12) Furthermore, we can also evaluate the noise correlation function by where C (av,cl)A (τ ) is the classical correlation function defined by (4.14) Thus, the quantum noise correlation function for the operator A(t) is given by Eq. (4.13), where C (av)A (τ ) and C (av,cl)A (τ ) are defined by Eq. (4.13) and (4.14), respectively. The noise 15/40 spectral density S An (ω) is given by the Fourier transformation of C (av)An (τ ) as This is the generalization of the noise spectral density of Eq. (4.1). In this paper, we evaluate the quantum noise through the noise spectral density (4.15) instead of Eq. (4.1).

Estimation of Quantum Noise
Following the discussion on the noise spectral density, we evaluate the quantum noise in homodyne detections. As discussed above, the noise spectral density (4.15) is not based on the conventional two-photon formulation and the evaluated noise spectral density here is beyond the two-photon formulation. However, we express our results in terms of the two-photon formulation as much as possible.
As the first case, we evaluate the noise spectral density in the case that the directly measured quantum operator is Glauber's photon number (2.15) in Sec. 5.1. In this case, the measured signal operator in the balanced homodyne detection is given by Eq. (3.27). Furthermore, we carefully examine the contributions of the vacuum fluctuations to the noise spectral density in this case.
As the second case, we evaluate the noise spectral density in the case where the directly measured quantum operator is the power operator (2.16) in Sec. 5.2. In this case, the measured signal operator in the balanced homodyne detection is defined by Eq. (3.33). Although we do not carry out the careful examination of the contribution of vacuum fluctuations to the noise spectral density in this case, it will be trivial from the considerations in Sec. 5.1.

Quantum Noise in Balanced Homodyne Detections by Photon-Number Detectors
Here, we evaluate the noise spectral density for the measurement operatorŝ N (t) defined by Eq. (3.27). In terms of the electric fields, the operatorŝ N (t) is given bŷ To carry out the evaluation of the noise spectral density, it is convenient to introduce the states |ŝ N (t) and ŝ N (t)|, which are defined by We also use the states |ŝ N (t + τ ) and ŝ N (t + τ )|, which are given by the replacement t → t + τ in Eqs. (5.2) and (5.3), respectively. We evaluate the noise spectral density S sNn (ω) of the noise operator By the subtraction of the classical part and usinĝ 17/40 we obtain Then, the normal-ordered noise correlation function C (normal) (av)sNn (τ ) is given by The noise spectral density S Here, we introduce the the Fourier transformed expression of the field operatorÊ (ω) = ω 0 |γ| 2 (I 1 (ω) + I 2 (ω) + I 3 (ω) + I 4 (ω)) , (5.14) where we defined I i (ω) by In Appendix C, we showed the explicit form (C10) of I 1 (ω) of the Michelson interferometer as an example. Equation (C10) is a finite result and this result implies that the expectation value b n (ω + ω 0 )b n (ω 2 ) includes the δ-function 2πδ(ω 2 − (ω 0 − ω)). Due to this δ-function, I 1 (ω) has a finite value even after the averaging process of the integration by t. The expectation value b n (ω + ω 0 )b n (ω 2 ) may have more δ-functions whose support is different from the point ω 2 = ω 0 − ω. However, even in this case, such δ function does not contribute to the result I 1 (ω) due to the property of the function (5.19) as explained in Appendix B. This situation is also true in the case of I 2 (ω), I 3 (ω), and I 4 (ω) given by Eqs. (5.16)-(5.18), respectively. These finite results of I i (ω) (i = 1, 2, 3, 4) also imply that if we force to omit the integration by ω 2 and to specify ω 2 so that the exponent in the average function vanishes in Eqs. (5.15)-(5.18), respectively, the resulting expression of I i (ω) includes 2πδ(0). This corresponds to the imposition of the stationarity to the noise spectral density. Thus, we conclude that we can obtain the correct results if we regard the expression of I 1 (ω), for 19/40 example, as (5.20) In the case where ω 0 ≫ ω > 0, we conclude Similarly, we obtain Thus, from Eqs. (5.21)-(5.24), the normal-ordered noise spectral density in the situation where ω 0 ≫ ω > 0 is given by Note that ω 0 is the central frequency of the optical field from the local oscillator. This frequency ω 0 may not coincide with the central frequency of the signal fieldÊ b (t). In this sense, the above noise spectral density includes the "heterodyne detection." Here, we regard that the central frequency ω 0 of the optical field from the local oscillator coincides with the central frequency from the main interferometer. This is the "homodyne detection." In this case, we can use the sideband pictureb ± (ω) :=b(ω 0 ± ω) and the above noise spectral density is given by Through the amplitude and phase quadraturesb 1n ,b 2n , andb θn := cos θb 1n + sin θb 2n is given by Here, we note that b θn (ω),b θn (ω ′ ) = 0. Since we only consider the positive frequency ω, the first term in the right-hand side of Eq. (5.27) is identical to the Kimble single-sideband noise spectral density S From the definition (5.1) of the signal operatorŝ N (t) and its expectation value (3.36), we defined the noise operatorŝ N n (t) (5.4) and considered states |ŝ N (t) and ŝ N (t)| as Eqs. (5.2) and (5.3), respectively. From Eqs. (5.2), (5.3), and the replacement t → t + τ , we also derived |ŝ N (t + τ ) and ŝ N (t + τ )|. From these states, here, we evaluate the inner products ŝ N n (t)|ŝ N n (t + τ ) and ŝ N n (t + τ )|ŝ N n (t) under the premises Of course, this premise is not consistent within the quantum field theory of electromagnetic fields. However, we dare to use these premise (5.29) and (5.30) to clarify from which field E b orÊ li the vacuum fluctuations contribute to the noise spectral density. Furthermore, we denote the inner products of the states ŝ N n (t)|, |ŝ N n (t + τ ) , ŝ N n (t + τ )|, and |ŝ N n (t) under the premise (5.29) and (5.30) by The straightforward calculations yields 1 2 ŝ N n (t)ŝ N n (t + τ ) +ŝ N n (t + τ )ŝ N n (t) (normal+sig.vac.) = 1 2 ( :ŝ N n (t)ŝ N n (t + τ ) +ŝ N n (t + τ )ŝ N n (t) : ) In the monochromatic local oscillator case, Here, we take into account of the vacuum fluctuations of the fieldÊ li (t) from the local oscillator in addition to the previous results. We evaluate the inner products ŝ N n (t)|ŝ N n (t + τ ) and ŝ N n (t + τ )|ŝ N n (t) under the premise This is the complete consideration which is taking into account of the vacuum fluctuations of all optical fields. We denote these inner products through this evaluation as ŝ N n (t)|ŝ N n (t + τ ) (normal+sig.vac.+loc.vac.) and ŝ N n (t + τ )|ŝ N n (t) (normal+sig.vac.+loc.vac.) . From the states given by Eqs. (5.2), (5.3), and the definition (5.4) of the noise operatorŝ N n (t), we can include the vacuum fluctuations from the local oscillator field as 1 2 ŝ N n (t)ŝ N n (t + τ ) +ŝ N n (t + τ )ŝ N n (t) = 1 2 ŝ N n (t)|ŝ N n (t + τ ) (normal+sig.vac.+loc.vac.) + ŝ N n (t + τ )|ŝ N n (t) (normal+sig.vac.+loc.vac.) = 1 2 ŝ N n (t)ŝ N n (t + τ ) +ŝ N n (t + τ )ŝ N n (t) (normal+sig.vac.) We evaluate J i (τ ) (i = 1, 2, 3, 4, 5) and their Fourier transformation J i (ω), separately. First, we evaluate J 1 (ω). Substituting Eqs. (5.13) and the definition (2.6) of the vacuum fluctuations ∆ li (τ ), we obtain Here, we use the introduce the noise-spectral density S bn (ω) by This definition of S bn (ω) has the same form of the Kimble single-sideband noise spectral density (1.7). However, we have to emphasize that the noise-spectral density S bn (ω) have nothing to do with the two-photon formulation nor upper-and lower-sideband with the central frequency ω 0 , while the Kimble single-sideband noise spectral density (1.7) defined within the sideband picture of the optical fluctuations in the two-photon formulation. The 23/40 frequencies ω 1 and ω 2 in Eq. (5.48) is not sideband frequencies, but the frequency ω in the definition (2.2) of the mode function of the electric field. Through the noise spectral density S bn (ω) defined by Eq. (5.48) and ω > 0, we obtain Second, we evaluate the Fourier transformation J 2 (ω) of J 2 (τ ) defined by Eq. (5.43). Here, we define the expectation value of the output quadratureb(ω) as b (ω) =: α(ω) + β2πδ(ω − ω 0 ), (5.50) where ω 0 is the central frequency of the signal fieldÊ b (t) and we assume that α(ω) and β are finite. From this expectation value (5.50), we can evaluate the expectation value of the electric fieldÊ are evaluated from the expectation value (5.50) and the vacuum fluctuations ∆ li (τ ) from the local oscillator is also given by Eq. (5.38) and (2.6) as in the case of J 2 (ω). Furthermore, we consider the monochromatic local oscillator case (5.11). From these, we obtain the Fourier transformation J 3 (ω) for the case ω > 0 as Similarly, we can evaluate the Fourier transformation J 4 (ω) of J 4 (τ ) defined by Eq. (5.45) for the case ω > 0 as Finally, we evaluate the Fourier transformation J 5 (ω) of J 5 (τ ) defined by Eq. (5.46). The vacuum fluctuations ∆ li (τ ) from the local oscillator is also given by Eq. (5.38) and (2.6) as in the case of J 2 (ω). We also consider the monochromatic local oscillator case (5.11). Then, we obtain the Fourier transformation J 5 (ω) for the case ω > 0 as Then the total noise spectral density is given from Eqs. ∼ ω 2 0 |γ| 2S (s) bθ (ω) Here, we consider the ideal case where the beam splitter is ideal, i.e., η = 1/2. Furthermore, we consider the situation of the signal fieldÊ b (t) is in the complete dark port β = 0 in which the leakage of the classical carrier field from the main interferometer completely is shut out. In this case the derived total spectral density of the quantum noise for the measurement of the operatorŝ N is

Quantum Noise in Balanced Homodyne Detections by Power Counting Detectors
Next, we evaluate the noise spectral density for the measurement of operatorŝ P (t) given by Eq. (3.34) in Sec. 3.2. To carry out this evaluation, as in the case of Glauber's photon number case in Sec. 5.1, it is convenient to introduce the states |ŝ P (t) and ŝ P (t)| which 25/40 are defined by We also use the states |ŝ P (t + τ ) and ŝ P (t + τ )|, which are given by the replacement t → t + τ in Eqs. (5.58) and (5.59). We evaluate the noise-spectral density S sP n (ω) for the measurement of the power-counting operatorŝ P (t) and we define the noise operatorŝ P n :=ŝ P − ŝ P . In this section, we take into account of all contribution of the vacuum fluctuations from the signal fieldÊ b and from the local oscillatorÊ li . Of course, it is possible to evaluate these contribution of the vacuum fluctuations, separately, as in the case of the number counting detector in Sec. 5.1. In this case, we have to evaluate the expectation value :ŝ P (t) : of the normal ordered operator : s P (t) : instead of the operatorŝ P (t) itself, because the subtraction of the vacuum fluctuations from the local oscillator is included in the definition (3.34) of the power-counting operator s P (t). If we consistently treat these contribution of the vacuum fluctuations, we obtain the corresponding results to the case of the number counting operatorŝ N in Secs. 5.1.1, 5.1.2, and 5.1.3, respectively. However, in this paper, we evaluate the noise-spectral density S sP n (ω) for the measurement of the operatorŝ P (t) taking into account of all contributions of vacuum fluctuations. 26/40 From the states defined in Eqs. (5.58) and (5.59), we obtain C sP n (t, τ ) = 1 2 ŝ P (t)ŝ P (t + τ ) +ŝ P (t + τ )ŝ P (t) − ŝ P (t) ŝ P (t + τ ) (5.60) All terms which include ∆ li (±τ ) are contributions from the vacuum fluctuations of the local oscillator. The averaged noise correlation function C (av)sP n (τ ) and the noise spectral density S sP n (ω) are given by dτ e +iωτ C (av)sP n (τ ) =: Here, we evaluate the noise spectral density S sP n (ω) defined by Eq. (5.63) only in the case of the monochromatic local oscillator case. In this case, K i (i = 1, 2, 3, 4, 5) should also be evaluated in this monochromatic case and these are defined as × e +iθ e −iω0(t+τ ) + e −iθ e +iω0(t+τ ) , (5.64) 27/40 First, we evaluate K 1 (ω) defined by Eq. (5.64). We useÊ bn (t) =Ê  oscillator coincides with the central frequency from the main interferometer, which is the "homodyne detection", we can use the sideband pictureb ± (ω) :=b(ω 0 ± ω) and the above 2πδ(ω − ω ′ )K 1 (ω) is given by Furthermore, using the definitions (3.18) of the amplitude-and phase-quadrature and their noise operators, the definition (1.1) of theb θ (ω) and its noise operatorb θn , we obtain Since we consider the situation ω, ω ′ > 0, the right-hand side of Eq. (5.70) is proportional to the single-sideband noise-spectral densityS (s) bθ (ω) introduced in Ref. [5]. Then, we obtain Second, we evaluate K 2 (ω) defined by Eq. (5.65). Here, we use the separationÊ b (t) = E bn (t) + Ê b (t) . Then, K 2 (ω) is separated into two parts as and We evaluate K 2−1 (ω) and K 2−2 (ω), separately. Here, we evaluate K 2−1 (ω). To evaluate this, we use the Fourier decomposition (5.13) of E bn (t). All terms in K 2−1 (ω) includes three integration by 2π due to 29/40 the Fourier transformation ofÊ bn (t + τ ),Ê bn (t), and vacuum fluctuations ∆ li , respectively. We also note that K 2−1 (ω) includes the term which have the factor Since the integrations range over ω 1 and ω 2 is [0, ∞), the term which includes the factor (5.76) vanishes. This is due to the fact that the support of the factor (5.76) is only on the single point ω 1 = −ω 2 . However, this point ω 1 = −ω 2 is out of range of the integration over ω 1 and ω 2 . Therefore, all terms which have the factor (5.76) vanish.
On the other hand, the terms including the factor may give finite contributions. From these discrimination of the terms appear in K 2−1 (ω), we obtain where we used the formula Through Eqs. (5.80) and (5.81), K 2−1 (ω) in Eq. (5.78) is given by Third, we evaluate K 3 (ω) defined by Eq. (5.66). Here, we use the separationÊ b (t) = E bn (t) + Ê b (t) , and the Fourier transformation (5.13), the explicit expression (2.6) of the vacuum fluctuation ∆ li (τ ), and the properties of the averaged function function (5.19) which are summarized in Appendix B. Furthermore, we use the situation ω 0 ≫ ω > 0 and the expression of the expectation value b (ω) given by Eq. (5.50). Then, K 3 (ω) is given by Finally, we evaluate K 4 (ω) and K 5 (ω) defined by Eqs. (5.45) and (5.46), respectively. In this evaluation, we use the explicit expression of the vacuum fluctuation ∆ li (τ ) and the situation ω 0 ≫ ω > 0. Then, we obtain the following results: In summary, from Eq. (5.63), and Eqs. (5.72), (5.84), (5.85), (5.86), and (5.87), we have obtained the noise spectral density S sP n (ω) in the situation ω 0 ≫ ω > 0 as In the case of the ideal beam splitter η = 1/2 and the complete dark port β = 0 of the main interferometer, the noise spectral density (5.88) yields S sP n (ω) = ω 2 0 |γ| 2S (s) bθ (ω) This noise spectral density is slightly different from the noise spectral density (5.57) for the number-counting detector.

Local oscillator from the main interferometer
Here, we consider the introduction of the optical field as the local oscillator from the main interferometer as depicted in Fig. 3. We consider the optical field junction at the beam splitter BS0 with the transmissivity ζ. In addition to the above notation,Ê ds (t) is the incident field directly from the light source,Ê d ′ (t) is the output field from the beam splitter BS0 to the main interferometer,Ê l ′ i (t) is the output field to the local oscillator, andÊ fi (t) 31/40 is the incident field to BS0 as depicted in Fig. 3. We assume that the state of the optical fieldÊ fi(t) is in the vacuum state |0 fi , i.e.,f i |0 fi = 0. The optical field junction conditions at BS0 yield These conditions are equivalent to Eqs. (A3) and (A4). The propagation of the optical field E d ′ (t) to the main interferometer yieldsÊ d (t) =Ê d ′ (t − x/c) and the propagation of the optical fieldÊ l ′ i (t) to BS of the balanced homodyne detection yieldsÊ li (t) =Ê(t − (x + y)/c). These are equivalent to the conditions (A6) and (A7), respectively. S1 D1 D2 BS BS0 Fig. 3 Introduction of the optical field of the local oscillator from the main interferometer.
Here, we consider the situation where the optical field from the laser source S1 in Fig. 3 is a coherent state with the complex amplitude γ s (t), i.e., Then, we obtain Ac ζγ s (t)|Ψ . Then, we obtain On the other hand, the commutation relation is unchanged. Therefore, in the expression of the noise-spectral densities, we should just replace the eigenvalue of the coherent state from the local oscillator as Eqs. (5.99) with unchanged vacuum fluctuations.

Summary and Discussion
In summary, we re-examined the estimation of quantum noise in the balanced homodyne detection. We consider the both cases in which the direct observable is Glauber's photonnumber operator (2.15) and the power operator (2.16) of the optical field, respectively. In our estimation, we did not use the two-photon formulation which is widely used in the gravitational-wave community. We concentrate on the stationary noise of the system through the time-average procedure. We also carefully treat vacuum fluctuations in our noise estimation. Furthermore, we introduce the imperfection of the beamsplitter of the balanced homodyne detection and the leakage of the classical carrier field from the main interferometer as the noise sources.
In spite of the introduction of the imperfections as the noise sources, the balanced homodyne detection of the both models of Glauber's photon-number operator and the power operator yields the expectation value of the operatorb θ (ω) as Eqs. (3.29) and (3.38). In this sense, the balanced homodyne detections enable us to measure the operatorb θ (ω) as their expectation values.
In the noise estimation, we have derived the deviations from the Kimble's noise spectral density (1.7) which are beyond the two-photon formulation in both models of Glauber's photon number operator and the power operator. As expected, the imperfection of the beamsplitter and the leakage of the classical carrier field, which are introduced as the imperfections of the interferometer configuration, contribute to the noise spectral density. These imperfections appear due to the vacuum fluctuations from the local oscillator as shown in Eqs. (5.56) and (5.88). As the result of the coupling with the vacuum fluctuations from the local oscillator, the leakage of the classical carrier field and its coupling with the imperfection of the beamsplitter leads the white noise in both cases of Glauber's photon number operator and the power operator. Even if the leakage of the classical carrier field from the 33/40 main interferometer is absent, the white noise appears due to the coupling with the vacuum fluctuation from the local oscillator the imperfection of the beamsplitter of the homodyne detection. In addition to these white noises, the coupling between the vacuum fluctuations and the imperfection of the beamsplitter leads to the frequency dependent noise in the power-counting detector model as shown in Eq. (5.88), while such terms do not appear in the model of Glauber's photon-number counting mode. Thus, the difference between the photodetector models appears the frequency dependence of the noise spectral densities, in principle.
Even in the ideal model where there is no leakage of the classical carrier field from the main interferometer (β = 0) and the beam splitter of the homodyne detection is ideal (η = 1/2), the noise spectral densities (5.57) and (5.89) have the terms due to the coupling between the vacuum fluctuations from the local oscillator and the low frequency fluctuations from the main interferometer. These ideal noise spectral densities (5.57) and (5.89) are proportional to the Kimble noise spectral density when the amplitude |γ| of the coherent state from the local oscillator is sufficiently large. In this sense, our result supports the noise spectral densities in the conventional two-photon formulation. On the other hand, when the amplitude of the coherent state from the local oscillator is small, these noise spectral densities (5.57) and (5.89) yield the deviations from Kimble's noise spectral density.
We evaluate the order of magnitude of these deviations. As noted in Sec. 2, the integration range [0, +∞] is replaced as the minimum and the maximum of the measurement time scales [ω min , ω max ] in the second term in Eq. (5.57) and the second term in Eq. (5.89). If the contribution of the noise spectral density S bn (ω 1 ) is the only vacuum fluctuations of the operatorb n , S bn (ω 1 ) becomes unity. In this case, the deviation from the Kimble's noise spectral density in Eq. (5.57) for the photon-number counting case vanishes. On the other hand, only the remaining term of the deviation from the Kimble's noise spectral density in Eq. (5.89) for the power-counting case is the last term, which yields 1 2 ω 0 dω1 2π ω 1 (ω − ω 1 ) = 1 6 ω 3 . When the output-frequency 0 < ω < 10 5 Hz, we compare the first term in Eq. (5.89) with the above integration, the condition that the deviation is dominate the Kimble's noise spectral density is given by where we used γ = I 0 / ω 0 and I 0 is the power of the laser from the local oscillator. The inequality (6.1) indicates that the deviation from Kimble's noise spectral density is extremely small in realistic situations. We may regard that the noise spectral densities (5.57) and (5.89) are regarded identical and coincide with the Kimble's noise spectral density, when the contribution to the noise spectral density S bn is the only vacuum fluctuations of the operator b n .
In the case whereb n is modulated in the frequency range [ω min , ω max ], the noise spectral density S bn (ω) may not be unity. Here, we choose S bn (ω) ∼ O (10). In this case, the dominant term in the deviations from Kimble's noise spectral density is the second terms in Eqs. (5.57) and (5.89), respectively. Comparing with the term of Kimble's noise spectral density in these equations, the conditions that the deviations from Kimble's noise spectral density dominates 34/40 the Kimble noise spectral density is given by The inequality (6.2) also indicates that the deviation from Kimble's noise spectral density is extremely small in realistic situations. Again, we may regard that the noise spectral densities (5.57) and (5.89) are regarded identical and coincide with Kimble's noise spectral density, even when the noise spectral density S bn is modulated in the frequency range [ω min , ω max ]. Even in the difference between the last term in Eqs. (5.56) and (5.88), we can evaluate the order estimate of this difference and conclude that this term is quite small compared with the Kimble noise spectral term.
Thus, we conclude that the noise spectral densities of the two ideal detector models of Glauber's photon number counting and the power counting are physically same and we cannot distinguish these models within the analyses of this paper. Our derived noise-spectral densities are based on the ideal premise that the direct observable of the photodetector is Glauber's photon number operator (2.15) or the power operator (2.16) of the optical field. Therefore, our derived noise spectral densities will characterize these ideal photodetectors model. Since these ideal models are not fully supported from theoretical point of view as mentioned in Sec. 1, it will be better to keep in our mind the possibility that the deviations from the Kimble noise-spectral density might also appear due to the deviations of the physical properties of actual photodetectors from our ideal photodetector models.
On the other hand, in the case where the directly measured operator of the photo-detector is the number operator (1.3) of each frequency modes as in Refs. [9,10], we cannot measure the operatorb θ (ω) by the balanced homodyne detection. The arguments in Refs. [9,10] together with those in this paper indicate that the choice of the directly measured operator at the photodetector affects the result not only of the noise properties but also of the expectation value of the output signal itself, in general. Therefore, we conclude that the specification of the directly measured operator is crucial in the development and applications of the mathematical quantum measurement theory.