Abstract

Photon-number squeezing and correlations enable measurement of absorption with an accuracy exceeding that of the shot-noise limit. However, sub-shot noise imaging and sensing based on these methods require high detection efficiency, which can be a serious obstacle if measurements are carried out in “difficult” spectral ranges. We show that this problem can be overcome through the phase-sensitive amplification before detection. Here we propose an experimental scheme of sub-shot-noise imaging with tolerance to detection losses.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Quantum metrology, one of the rapidly developing quantum technologies, uses quantum resources for overcoming the limits set by classical measurement methods [1,2]. Phase sensitivity is one example [3], measurement of absorption [4] is another one. Together with the spatial resolution, these two types of measurement form the basis for quantum imaging [5]. With spectral resolution, quantum-enhanced measurements of both types can be used for spectroscopy.

In this work we will focus on the measurement of absorption, especially for the case of weakly absorbing objects. In classical optics, in order to measure a weak absorption 𝒜 ≪ 1, the object under test is placed into one of the output beams of a 50% beamsplitter, see Fig. 1(a). The second beam is used as a reference, for suppressing the effect of the amplitude fluctuations present in the incident beam. The absorption of the object is found by subtracting the output signals of the detectors placed into both channels.

On the fundamental level, the sensitivity of absorption measurements is limited by the quantum fluctuations of the light intensity. In the simplest case of a coherent quantum state at the input, the sensitivity scales as Δ𝒜~1/N, see Eq. (11), where N is the number of photons used [6,7]. This characteristic dependence, which originates from the Poissonian distribution of the photon number in the coherent state, is known as the shot noise limit (SNL). Although it can be improved by simply increasing the number of photons, this may not be an option in environmental or biological measurements where the use of high intensity should be avoided.

 figure: Fig. 1

Fig. 1 Schemes of quantum imaging and sensing. (a) Classical differential scheme. The object with absorption 𝒜 is placed into one of the outputs of a 50% beamsplitter, and the absorption is retrieved through intensity subtraction measurement. (b) Conventional scheme of sub-shot-noise quantum imaging. The object is placed into one of the twin beams, emerging from a NOPA with the parametric gain r. Two photodetectors with equal quantum efficiencies η measure the numbers of photons in signal and idler beams, and the difference is calculated. (c) An alternative scheme, where only one beam is squeezed by a DOPA before probing the object, the other (much stronger) one being coherent. In schemes (b) and (c), additional DOPAs with gain R (shown by dashed lines) can be placed into both arms (b) or into the signal arm (c) to overcome the detrimental effect of the detection loss.

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More advanced quantum states of light could provide better sensitivity for weak illumination. The ideal case evidently corresponds to the Fock quantum states |N〉. In this case, Δ𝒜 is considerably reduced as it gets multiplied by a factor 𝒜, see Eq. (14), [8]. The practical implementation of this idea uses twin beams produced by a non-degenerate parametric amplifier (NOPA) and relies on their high degree of photon-number correlation [9–11]. In this scheme, see Fig. 1(b), the absorbing object is placed into one of the beams (the signal one), with the second one serving as a reference. The photon-number measurement in the reference beam projects the signal one into the quantum state with a well-defined N (in the ideal case — into the Fock state |N〉), enabling thus the sub-shot-noise sensitivity [4,6,12–14]. It is worth noting that in this case, the sub-shot-noise sensitivity is achieved not by improving the scaling with the number of photons used for the measurement, but due to the reduction of uncertainty by a factor of 𝒜.

A similar scheme was proposed in Ref. [15] for improving the signal-to-noise ratio (SNR) in imaging. In this scheme, the object to be imaged is placed into one of multimode twin beams. The image is retrieved by subtracting the outputs of spatially resolving detectors placed into both beams. The SNR is then improved compared to the case of classical differential imaging due to the noise suppression below the SNL. This technique of sub-shot-noise imaging was demonstrated experimentally in works [16–18].

Closely related to abovementioned technique is the recently proposed “quantum illumination” protocol [19, 20], where the introduced entanglement between the signal and ancilla qubits could help to detect a presence of the target object, which was demonstrated in optical [21], and microwave domains [22].

Instead of twin beams, one can use a single sub-Poissonian squeezed beam, produced in a degenerate optical parametric amplifier (DOPA), as a probe [23]. A coherent beam fed from the same laser source can be used in this case as a reference, see Fig. 1(c). Although not suitable for imaging in the case of a single-mode beam, this scheme is convenient for spectroscopy. It was indeed used to enhance the sensitivity of spectroscopic measurements, with squeezed light from a parametric amplifier [24] and from an amplitude-squeezed semiconductor laser [25]. This method can be very useful for sensing through absorption measurement, for instance, for monitoring the concentration of various gases [26].

For both these versions of sub-shot-noise quantum sensing, the increase in SNR strongly depends on the detection efficiency of the optical setup. In the experiments [4, 14, 16, 17] performed in the visible range, the total detection efficiency of the setup exceeded 90%. However, in some cases, the absorption should be measured in other spectral ranges, where detection is inefficient. Moreover, as we show further, in order to reveal the full potential of quantum sensing, the detectors inefficiency should not exceed the measured absorption. In the case of weak absorption measurements, 𝒜 ≲ 10−2, this requirement can not be fulfilled with the existing photodetectors.

Here we show that sub-shot-noise quantum imaging can work in the presence of any loss provided that the beams carrying the information about the absorption are amplified before the detection using phase-sensitive parametric amplifiers (DOPAs), shown by dashed lines in Figs. 1(b) and 1(c). This approach is similar to the strategy of overcoming loss in high-precision interferometry, first proposed in Ref. [27], elaborated theoretically in Refs. [28–31], and demonstrated experimentally in Refs. [32–35]. It was also proposed as a method of “quantum phase magnification” [36], and for “enhanced quantum tomography” [37,38].

In all these cases, the quadrature containing the signal is amplified and this way protected against loss. In the twin beam case which we explore here, the amplification is also phase-sensitive, but it does not matter which phase is amplified as long as it is the same for both twin beams. It is important that an ideal phase-sensitive amplifier does not introduce additional noise, and for this reason, it can be used for the noiseless amplification of images [39].

The present work considers the case where the image to be amplified is a sub-shot-noise one. In order to emphasize the effect, in our numerical estimates we will target very weak absorption 𝒜 ≈ 10−5, typical, in particular, for gas absorption measurements.

The paper is organized as follows. Section 2 considers the fundamental bounds for the absorption measurement. In Section 3 we discuss sub-shot-noise quantum imaging using twin beams and the tolerance to inefficient detection provided by phase-sensitive amplification, see Fig. 1(b). Section 4 describes an alternative method, using a single squeezed coherent beam, also followed by phase-sensitive amplification for overcoming the detection loss, see Fig. 1(c). In Section 5 we outline possible experiments, and Section 6 is the conclusion.

2. Cramer-Rao bounds

Statistics of the number of photons n after an absorbing object are described by a conditional probability distribution W(n/𝒜) parameterized by the power absorption factor 𝒜. The ultimate accuracy of estimating this parameter of the probability distribution is given by the Cramer-Rao bound [40],

Δ𝒜CR={n=01W(n/𝒜)[W(n/𝒜)𝒜]2}1/2.
This equation describes the ideal case of a precise measurement of n (that is, photon counting with 100% efficiency) as well as an ideal (lossless) preparation of the incident quantum state of light. However, it is easy to modify it to take into account the non-ideal detection efficiency and the non-ideal preparation. We model them by imaginary gray filters with the power transmissivities ηd and ηp, located, respectively, after and before the object. In this case, the object transmissivity
𝒯=1𝒜
in Eq. (1) has to be replaced by the corresponding combined one,
1𝒜η=η𝒯,
where
η=ηpηd
is the total quantum efficiency. This gives
Δ𝒜CR={n=01W(n/𝒜η)[W(n/𝒜η)𝒜]2}1/2.
Combining Eqs. (5) and (3), we obtain
Δ𝒜CR=1η{n=01W(n/𝒜η)[W(n/𝒜η)𝒜η]2}1/2.
Here we calculate this bound for two most important particular cases.

We start with a coherent state, which describes the laser radiation in the idealized case where the excess (technical) noise is absent. This “classical” light is characterized by the Poissonian photon-number distribution with some mean value N0. After passing through the absorbing object and the imaginary filter, which models the detector inefficiency, the photon-number distribution still remains Poissonian, but with a reduced mean photon number (1 − 𝒜η)N0:

W(n/𝒜)=e(1𝒜η)N0[(1𝒜η)N0]nn!.
Substituting this distribution into Eq. (6), we obtain the Cramer-Rao bound for coherent light:
Δ𝒜CRcoh=𝒯ηdN,
where
N=ηpN0
is the mean photon number at the object. Equation (8) takes into account the object transmissivity and the detector inefficiency in the known expression for the shot-noise limit [6,7].

From a practical viewpoint, most important is the case of a highly transparent object and good preparation and detection efficiencies,

𝒜1,1ηp,d1.
In this case, the sensitivity is virtually not affected by the object absorption and by the preparation inefficiency,
Δ𝒜CRcoh=1N.

Consider now a photon-number (Fock) state |N0〉. Taking into account that in the absorption measurements the information is encoded into the probe light intensity, this state should provide the best possible sensitivity. Indeed, for this state, the probability distribution for the number of detected photons is the binomial one,

W(n/𝒜)=N0!n!(N0n)!𝒜ηN0n(1𝒜η)n,
and the corresponding Cramer-Rao bound is
Δ𝒜CRFock=[𝒜+(1η)𝒯]𝒯ηdN.
In the particular case of (10),
Δ𝒜CRFock=𝒜+2N,
where
2=p2+d2
is the total inefficiency of the scheme and
p,d2=1ηp,dηp,d.
It is interesting that opposite to the phase measurements (see e.g. [31] and the references therein), Eqs. (13) and (14) still scale with the number of quanta as 1/N [8], because the absorption, opposite to the phase shift, is a stochastic process which introduces an additional uncertainty, scaling as N, into the number of photons. However, the additional factor in the numerator could provide a significant sensitivity gain in the case of the weak (𝒜 ≪ 1) absorption measurements with efficient preparation and detection (1 − η ≪ 1).

Given the same photon number N of the beam hitting the object and the same values of ηp, ηd, the quantum advantage can be described by the ratio Q of the uncertainties for the shot-noise measurement Δ𝒜CRcoh and the one using some nonclassical state of light,

Q=Δ𝒜CRcohΔ𝒜.
In the Fock state case (Δ𝒜 = Δ𝒜CRFock), it follows from Eqs. (11) and (14) that this quantum advantage is
Q=1𝒜+2.
Clearly, the asymptotic value of this gain 1/𝒜can be reached only if 2𝒜. In the opposite case, the gain is limited by the scheme inefficiency 2.

3. Twin-beams absorption measurement and its enhancement through phase-sensitive amplification

Consider now the “twin beams” scheme shown in Fig. 1(b). Let the twin beams be produced by a NOPA with the parametric gain r. The object, with the absorption 𝒜, is placed into beam 1. Then both beams pass through DOPAs, which stretch some arbitrary but synchronized quadratures of these beams by eR. After that, the beams are detected by two photodetectors.

There are various strategies of processing the output data. Here we consider two linear ones used, respectively, in [16] and [4].

The first and simpler procedure is based on the estimator

𝒜˜=nd1nd2G,
where nd1 and nd2 are the photon numbers registered by the first and the second photodetectors and
G=nd1𝒜
is the transfer function. The measurement error for this procedure is calculated in Appendix B, see Eq. (55). This general equation is quite cumbersome. In order to reveal its structure, we consider here the case (10) of a small absorption and good efficiency of the setup, assuming in addition for simplicity that the photon number is large,
N1.
Also for simplicity, we present here the equations for only two most interesting cases: either the output DOPAs are not used at all (R = 0), or they provide strong amplification,
e2R1.

In the case of no amplification, R = 0, Eq. (55) simplifies to

(Δ𝒜)2=𝒜2+𝒜+22N.
This limit differs from the Cramer-Rao bound (14) in two aspects. The first one is the additional term 𝒜2, which stems from the asymmetry introduced into the twin beams by the absorption and, as we will show below, can be removed by using an estimator more advanced than (19). Second, the terms imposed by the preparation and detector inefficiency are twice as large due to the two beams used in this protocol.

In the case (22) of strong amplification,

(Δ𝒜)2=2[𝒜2+𝒜+2(p2+d2e2R)N]+1N2.
One can see that in this case the detection inefficiency is suppressed by the factor e−2R. The origin for this is evident, namely amplification of the signal (that is, amplitude) quadrature of the light before the detection, which suppresses the influence of the noise introduced by the detectors’ inefficiency. At the same time, the additional photon-number fluctuations originating from the imbalance due to object are enhanced by the DOPAs, thus adding: (i) yet another factor 2, and (ii) the additional term 1/N2.

In Ref. [4], a more sophisticated but still linear estimator was proposed:

𝒜˜=nd1knd2G,
where the factor k should be optimized to provide the minimum of Δ𝒜. The corresponding minimized Δ𝒜 is also calculated in Appendix B, see Eq. (59). The resulting asymptotic equations for the absorption uncertainty and the quantum advantage for the same “no amplification” and “strong amplification” cases as above are, respectively,
(Δ𝒜)2=𝒜+22N,Q=1𝒜+22,
and
(Δ𝒜)2=2[𝒜+2(p2+d2e2R)N]+1N2,Q=12[𝒜+2(p2+d2e2R)]+1/N.
Equations (25) differ from the previous ones (23) by the absence of the additional term 𝒜2. The price for this is that this procedure requires a priori information on 𝒜. In this article we assume that the value of 𝒜 could always be estimated using for example the conventional approach of classical differential scheme, and then a series of consecutive measurements could be performed in order for 𝒜 to converge towards the true value.

The quantum advantage Q achieved using the estimator (24) in the case of R = 0 is shown in Fig. 2 as a function of the detection inefficiency d2for various values of the preparation inefficiency [note that the plots in Fig. 2, as well as in Fig. 3 below are drawn using the exact Eqs. (59) and (69), respectively]. Among them, p2=103can be considered as a realistic one. It corresponds to absorption inside the nonlinear crystal on the order of 10−4mm−1 [41] and reflection at each surface of the crystal of about 10−4 [42]. Clearly, Q is above unity only for sufficiently high detection efficiency. Similarly to the Fock state case (see Sec. 2), the quantum advantage reaches its maximal value 1/𝒜only if d,p2𝒜. In the opposite case, the quantum advantage is limited by the scheme inefficiency:

Q12.

However, as one can see from Eqs. (23b) and (25b), by increasing the amplification R the effect of the inefficient detection can be made arbitrarily small. In the practical case of 𝒜2, the corresponding sensitivity gain can more than compensate for the above-mentioned additional factor 2 before 𝒜 and the additional term 1/N2 in the equation for Δ𝒜. This is demonstrated by Fig. 3, where the quantum advantage achieved with the estimator (24) is plotted as a function of the parametric gain R for several values of the detector’s efficiency, from the very optimistic to the very low ones. One can see that for any quantum efficiency the asymptotic value of the quantum advantage,

Q=12(𝒜+2p2)+1/N,
can be reached, provided there is sufficiently strong parametric amplification.

 figure: Fig. 2

Fig. 2 The quantum advantage Q of the ‘twin beams’ scheme, see Eqs. (17) and (59), for the case of no parametric amplification (R = 0) as a function of detection efficiency ηd for p2=0(black, maximum efficiency), 10−5 (red), 10−4 (green), 10−3 (blue), and 10−2 (magenta). The object absorption is 𝒜 = 10−5 and the mean photon number is N = 107.

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It is interesting to note that in the case of a low quantum efficiency, this value can be even exceeded, see the blue and magenta curves in Fig. 3. This is because the baseline value Δ𝒜CRcoh is also affected by the detectors inefficiency, and this effect becomes significant if d21, see Eq. (8).

 figure: Fig. 3

Fig. 3 The quantum advantage Q of the ‘twin beams’ scheme in the presence of additional parametric amplification, see Eqs. (17) and (59), as a function of the gain R for different values of the detection efficiency: η = 0.99 (red), 0.9 (green), 0.5 (blue), and 0.1 (magenta). The mean photon number is N = 107 and the absorption is 𝒜 = 10−5.

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4. Sub-shot-noise sensing with squeezed light

An alternative way to overcome the shot noise limit is to use another quantum state with reduced photon-number fluctuations, namely the squeezed coherent state. Consider the scheme shown in Fig. 1(c). Here a coherent beam is split on a beamsplitter, so that part (top) is amplitude squeezed in a degenerate parametric amplifier (DOPA) with the parametric gain r and used as a probe beam with the photon number N, and another one (bottom) is used as a reference. Then the probe beam passes through another DOPA, which has the parametric gain R.

Note that while the optical power in the probe beam could be limited by the probed object fragility, there is no such limitation for the reference beam. Therefore, it is reasonable to have the reference beam much stronger than the signal one, in order to suppress the reference beam shot noise. In this case, an asymmetric beamsplitter has to be used, and the reference photodetector output has to be proportionally scaled down. We assume this case here, taking into account the quantum noise of the signal beam only.

The measurement uncertainty for this scheme is calculated in Appendix C, see Eq. (69). Similarly to the previous section, we consider several characteristic asymptotic cases.

We start with the baseline case of R = 0. If the input squeezing is absent as well, then Eq. (69) reduces to the shot-noise limit (8). Suppose that the input optical field is strongly squeezed,

e2r1,
and assume also conditions (10, 21). In this case it follows from Eq. (69) that
(Δ𝒜)2=e2|r|+𝒜+2N+e4|r|8N2
and
Q=1e2|r|+𝒜+2+e4|r|8N.
One can see that the optimal value of the input squeezing exists which provides the minimum of the initial uncertainty of the number of quanta and therefore the minimum of Δ𝒜. This optimal squeezing corresponds to
e2|r|=(4N)1/3,
which gives
(Δ𝒜)2=𝒜+2N+325/3N4/3,
Q=1𝒜+2+325/3N1/3.
Note that the first two terms in Eq. (31a) coincide with the Cramer-Rao bound for a Fock state (14). The third term originates from the photon-number fluctuations in the incident light, which can not be made arbitrary small within the constraints of Gaussian (displacement and squeezing) transformations. Note that for the values of 𝒜 and N that we use in this paper, it is this term that dominates if the losses are small, p,d2103. At the same time, in the more realistic case of stronger losses, the quantum advantage for the squeezed coherent state case can be 2higher than in the twin-beam case, compare Eqs. (25a) and (31). The reason for this is evident: the former scheme requires only one noisy channel while the latter, two of them.

The quantum advantage for the squeezed coherent input state in the case of R = 0 is shown in Fig. 4 as a function of the detection inefficiency factor d2for various values of preparation inefficiency, and can be compared with the corresponding plots for the twin-beam scheme shown in Fig. 2.

 figure: Fig. 4

Fig. 4 The quantum advantage Q of the “squeezed coherent” scheme, see Eqs. (17) and (69), for the case of no parametric amplification (R = 0) as a function of detection efficiency ηd for p2=0(black), 10−3 (blue), and 10−2 (magenta). The object absorption is 𝒜 = 10−5 and the mean photon number is N = 107.

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In the “strong amplification” case of (22), assuming also the asymptotic conditions (10, 21, 28), Eq. (69) can be reduced to

(Δ𝒜)2=e2|r|+𝒜+p2+d2e2RNe2|r|4+e4|r|8R8(Ne2|r|4)2.
This equation shows once again that the detector inefficiency can be overcome by the amplification.

Dependence of Δ𝒜 on the input squeeze factor r is non-trivial. However, in all reasonable practical cases, one can assume that

e2|r|4N,
which gives
(Δ𝒜)2=e2|r|+𝒜+p2+d2e2RN+e4|r|8R8N2.
The minimum of this equation in |r| is provided by
e2|r|=(4N)1/3e8R/3,
which gives the following simple equations:
(Δ𝒜)2=𝒜+p2+d2e2RN+3e8R/325/3N4/3,
Q=1𝒜+p2+d2e2R+3e8R/325/3N1/3.
Then it follows from Eqs. (33) and (35) that in this case
e4R4N.
Comparison of Eq. (36) with Eq. (31) shows that not only the detection inefficiency term, but also the one which stems from the Gaussianity of the input quantum state [the last one in (36a)] is suppressed by the amplification.

In comparison with the twin-beam scheme of Sec. 3, the effect of the object absorption on the quantum advantage is reduced by a factor of 2, and the effect of the preparation and detection losses, by a factor of 2, compare Eqs. (25b) and (36b). On the other hand, the last term in Eq. (36b) is always larger than its counterpart in Eq. (25b) due to the condition (37). However, for the realistic parameters that we use in this paper, this term is smaller than the previous ones, making the “squeezed coherent” scheme more sensitive.

Fig. 5 shows the quantum advantage Q and the corresponding numerically optimized squeeze factor r for the ‘squeezed coherent’ scheme as a function of amplification R for the same values of the detection efficiency and the other parameters as in Fig. 3.

 figure: Fig. 5

Fig. 5 The quantum advantage Q (top) and the corresponding numerically optimized squeeze factor r (bottom) for the “squeezed coherent” scheme, see Eqs. (17) and (69), as a function of the gain R for different values of the detection efficiency: η = 0.99 (red), 0.9 (green), 0.5 (blue), and 0.1 (magenta). The straight black line in the bottom plot shows the asymptotic (35). The mean photon number is N = 107 and the absorption is 𝒜 = 10−5.

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5. Experimental implementation

From the above analysis, it follows that the scheme with a squeezed coherent beam provides a better sensitivity in the absorption measurement than the twin-beams one. This scheme can be relatively easily implemented using a cavity-based DOPA, where the state-of-the-art squeezing reaches 15 dB [43]. The same device can serve for the amplification of the beam after the object to overcome the detection losses. This single-mode scheme will be suitable for sub-shot-noise spectroscopy or sensing without spatial resolution.

More challenging is to overcome detection loss in sub-shot-noise imaging, where spatially multimode beams are involved. A sub-Poissonian multimode beam can be produced from multimode twin beams through heralding [44] but the resulting amount of photon-number squeezing is not high because of the inefficiency of the detector used in this method. Much more promising is to utilize the twin-beam squeezing available in spatially multimode traveling-wave NOPAs [16,17,44] where the noise reduction down to 7.8 dB has been reported [44].

The principal scheme of a possible experiment is shown in Fig. 6. Multimode twin beams are produced by a phase-insensitive traveling-wave NOPA. A certain technical difficulty arises from the necessity to provide phase-sensitive amplification with the same phase in each beam. To avoid matching the mode structure of phase sensitive amplifiers in both beams, one can take advantage of the transformation between phase sensitive and phase-insensitive amplification through mode conversion [39].

 figure: Fig. 6

Fig. 6 Scheme of a possible experiment. NOPA1 generates beams in modes â1 and â2, and a weakly absorbing object is placed into mode â1. After the beam in mode â2 acquires a π/2 phase shift, both beams are overlapped on a 50% beamsplitter to form modes a^±=(a^1±ia^2)/2 at the output. These two modes are amplified by NOPA2, which is equivalent to the phase-sensitive amplification of modes â1 and â2. In the end, photon numbers in modes â1 and â2 should be measured, which requires overlapping the modes on another 50% beamsplitter.

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The operation of the NOPA producing the twin beams is described by the Hamiltonian

H^=iγa^1a^2+h.c.,
where γ quantifies the interaction strength.

Phase sensitive amplification with the same phase and the same parametric gain in modes â1, â2 after the absorbing object is described by the Hamiltonian

H^=iΓ[(a^1)2+(a^2)2]+h.c.,
with the interaction strength characterized by Γ. This Hamiltonian can be represented as one of a phase-insensitive amplifier,
H^=iΓa^+a^+h.c.,
for the modes
a^±=a^1±ia^22.
The modes â1,2 can be transformed into â± by using a 50% beamsplitter, with the π/2 phase shift introduced into â2. After the second NOPA, a second beamsplitter should be used to convert the “ ±” modes back into the “1, 2” ones.

Although Fig. 6 looks very complicated, the same idea can be implemented much more simply using polarization optics, see Fig. 7. The vertically and horizontally polarized twin beams are produced through type-II parametric down-conversion, and the object is absorbing just one polarization. As phase sensitive amplifiers, one can use type-I parametric amplifiers placed into both beams. The first one is then acting just on the vertically polarized beam, mode v, and the second one, just on the horizontally polarized beam, mode h (panel a). In the end, both beams are detected separately, for instance, by different sensitive areas of the same camera after a birefringent beamsplitter. To retrieve the image, the intensity distributions obtained by the camera should be subtracted pixel-by-pixel.

 figure: Fig. 7

Fig. 7 Polarization implementation of the experiment on sub-shot-noise imaging.

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However, it is experimentally challenging to provide amplification with the same phase for both polarizations. A solution is to use, as in Fig. 6, instead of phase-sensitive amplification of vertical and horizontal polarizations (type-I OPA), a phase-insensitive two-mode amplification (type-II OPA), shown in panel b. This NOPA should amplify modes (h±iv)/2, i.e., right- and left-circularly polarized modes. Because a type-II OPA can only amplify linearly polarized beams, it has to be preceded by a quarter wave plate placed into both beams. Note that the type-II OPA should have the polarization direction tilted by 45°.

6. Conclusion

We have considered two methods of sub-shot-noise measurement of weak absorption, one using twin beams and the other one, squeezed coherent light. Both are “substitutes” for the ideal case where an absorptive object is probed by a Fock state, which has zero uncertainty in the photon number. The “squeezed coherent” scheme is easier to implement, with a single-mode coherent beam and a single-mode squeezer. However, while it is suitable for sensing or absorption measurement, for instance, in spectroscopy, it cannot be used for imaging, where the object should be illuminated by a spatially multimode beam. In this case, the ‘twin-beams’ scheme is more convenient: a traveling-wave NOPA is always strongly multimode, unless special measures are taken. Both methods give advantage over the classical differential method, the “squeezed coherent” one providing a factor of 2better performance than the other one and reaching the Cramer-Rao bound for Fock states in the ideal case of infinite squeezing and no loss.

Meanwhile, both schemes turn out to be useless whenever the detection efficiency is low. This will be the case if imaging, sensing, or absorption spectroscopy is carried out in the mid-infrared or even terahertz spectral range. As we show, a way around it is to apply phase sensitive amplification before detection. We show that at any value of the detection efficiency, by sufficiently amplifying the beams after the absorbing object one can reach the same quantum advantage as in the case of high efficiency.

The experimental implementation of this technique is most simple in sensing or spectroscopy, where no spatial resolution is required. Single-mode cavity-based OPAs can be used in this case, both for generating a squeezed coherent beam and for amplifying it after the object under study. For imaging, we consider a polarization setup with a traveling-wave type-II OPA producing orthogonally polarized twin beams. Another type-II OPA preceded by a quarter-wave plate can then be used for the amplification before detection.

A. Detection inefficiency

We model the detectors’ inefficiency by imaginary beamsplitters with the power transmissivity ηd:

d^1,2=ηdc^1,2+1ηdu^1,2=ηd(c^1,2+du^1,2),
where ĉ1,2 are the fields at the input of the detectors, d̂1,2 are the effective input fields of the corresponding imaginary lossless detectors, and û1,2 are vacuum fields.

It is easy to show that the statistics of the effective input fields depend on the statistics of the real ones as

n^d1,2=ηdn^c1,2,
(δn^d1,2)2=ηd2[(δn^c1,2)2+d2n^c1,2],
δn^d1δn^d2=ηd2δn^c1δn^c2,
where
n^c1,2=c^1,2c^1,2,
n^d1,2=d^1,2d^1,2,
δn^c1,2=n^c1,2n^c1,2,
δn^d1,2=n^d1,2n^d1,2.

B. Twin beams

The annihilation operators for two vacuum input fields, see Fig. 8, are ẑ1,2 . The non-ideal NOPA transforms them into

a^1,2=ηp(z^1,2coshr+z^2,1sinhr)+1ηpv^1p,2p,
where v̂1p, v̂2p are two independent vacuum fields.

 figure: Fig. 8

Fig. 8 Measurement of absorption using twin-beams scheme with amplification before detection. The object under study is placed into the first beam, and both beams are simultaneously amplified before the direct detection. The state preparation loss is modeled by means of two identical beamsplitters with power transmissivity ηp, and the detection loss, by a similar beamsplitter ηd. The pump of the non-linear crystals is not depicted.

Download Full Size | PPT Slide | PDF

The object partly absorbs the first beam, leaving the second one unchanged:

b^1=𝒯a^1+𝒜v^obj=𝒯p(z^1coshr+z^2sinhr)+𝒜pv^1,
b^2=a^2=ηp(z^2coshr+z^1sinhr)+1ηpv^2,
where
𝒯=1𝒜,
𝒯p=ηp𝒯,𝒜p=1ηp𝒯,
v^1=(1ηp)𝒯v^1p+𝒜v^obj𝒜p,
v^2=v^2p,
and v̂obj is the vacuum noise introduced by the object. Note that v̂1,2 again are two independent vacuum fields.

Finally, the DOPAs give

c^1=b^1coshR+b^1sinhR=𝒯p(C^1+S^1)+𝒜p(v^1coshR+v^1sinhR),
c^2=b^2coshR+b^2sinhR=ηp(C^2+S^2)+1ηp(v^2coshR+v^2sinhR),
where
C^1,2=z^1,2coshrcoshR+z^2,1sinhrsinhR,
S^1,2=z^1,2coshrsinhR+z^2,1sinhrcoshR.
It follows from these equations that the mean values and the second moments of the photon numbers in the output beams are
n^c1=(𝒯N+12)cosh2R12,
n^c2=(N+12)cosh2R12,
(δn^c1)2=(𝒯N+12)2cosh4R14,
(δn^c2)2=(N+12)2cosh4R14,
δn^c1δn^c2=𝒯N(N+ηp)cosh4R,
where
N=ηpsinh2r
is the mean photon number at the object.

In the case of estimator (19), the absorption can be measured with the uncertainty [see Eqs. (43)]

(Δ𝒜)2=(δn^d1δn^d2)2G2=1(G/ηd)2[(δn^c1)2+(δn^c2)22δn^c1δn^c2+2(n^c1+n^c2)].
where
G=n^d1𝒜=ηdNcosh2R
is the transfer function.

In the case of estimator (24), the uncertainty is

(Δ𝒜)2=(δn^d1kδn^d2)2G2=(δn^d1)22kδn^d1δn^d2+k2(δn^d2)2G2.
where k is the factor which has to be optimized. It is easy to see that the minimum of (57) occurs at
k=δn^d1δn^d2(δn^d2)2,
and is [see Eqs. (43)]
(Δ𝒜)2=1G2[(δn^d1)2δn^d1δn^d22(δn^d2)2]=1(G/ηd)2[(δn^c1)2+2nc1δn^c1δn^c22(δn^c2)2+2nc1].
In the case of R = 0 and (10, 21), Eqs. (55) and (59) simplify, respectively, to Eqs. (23a) and (25a).

In the case of eR ≫ 1 and (21)), Eqs. (55) and (59) give, respectively,

(Δ𝒜)2=2{𝒜2+1N[𝒜+2(1ηp)𝒯+2(1+𝒯)e2R]}+1N2,
(Δ𝒜)2=2𝒯N[𝒜+2(1ηp)𝒯+2(1+𝒯)e2R]+1N2[1+𝒯22+2𝒯(1𝒯ηp2)].
If, in addition, assumptions (10) are fulfilled, then these equations simplify, respectively, to Eqs. (23b) and (25b).

 figure: Fig. 9

Fig. 9 Measurement of absorption using a squeezed coherent state and amplification before detection. Since the reference beam is considered to have a very large amplitude in order to suppress the shot-noise in the corresponding channel, one can take into account only the noise of the signal channel. Here α is rescaled, see Eq. (63).

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C. Squeezed coherent state

With an account for the preparation imperfection, the annihilation operator for the field incident at the object, see. Fig. 9, is

a^=α+ηp(z^coshr+z^sinhr)+1ηpv^p,
where ẑ, v̂p are vacuum fields and
α=Nηpsinh2r
(we assume that α is real). The object modifies this field as follows:
b^=𝒯a^+𝒜v^obj=𝒯α+𝒯p(z^coshr+z^sinhr)+𝒜pv^,
where
v^=(1ηp)𝒯v^p+𝒜v^obj𝒜p
and v̂obj is the vacuum noise introduced by the object. Note that v̂ again is a vacuum noise.

Finally, the DOPA gives

c^=b^coshR+b^sinhR=𝒯αeR+𝒯p[z^cosh(r+R)+z^sinh(r+R)]+𝒜p(v^coshR+v^sinhR).
It follows from (66) that the mean value and the variance of the photon number in the output beam are
n^c=𝒯α2e2R+𝒯psinh2(r+R)+𝒜psinh2R,
(δn^c)2=𝒯𝒯pα2e2r+4R+𝒯p22sinh22(r+R)+𝒯𝒜pα2e4R+𝒯p𝒜psinh2(r+2R)+𝒜p22sinh22R.

With an account for Eqs. (43), the absorption measurement error is

(Δ𝒜)2=(δn^d)2G2=(δn^c)2+2n^c(G/ηd)2,
where
G=n^d𝒜=ηd[α2e2Rηpsinh2(r+R)+ηpsinh2R]
is the transfer function.

Funding

Deutsche Forschungsgemeinschaft-RFBR project CH1591/2-1 - 16-52-12031 NNIOa.

References

1. V. B. Braginsky and F. Ya. Khalili, “Quantum nondemolition measurements: the route from toys to tools,” Rev. Mod. Phys. 68, 1–11 (1996). [CrossRef]  

2. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306, 1330–1336 (2004). [CrossRef]   [PubMed]  

3. R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kolodynski, Progress in Optics (Elsevier, 2015), Chap. 4.

4. P.-A. Moreau, J. Sabines-Chesterking, R. Whittaker, S. K. Joshi, P. M. Birchall, A. McMillan, J. G. Rarity, and J. C. F. Matthews, “Demonstrating an absolute quantum advantage in direct absorption measurement,” Sci. Rep. 7, 6256 (2017). [CrossRef]   [PubMed]  

5. M. I. Kolobov, Quantum Imaging (Springer, 2007). [CrossRef]  

6. E. Jakeman and J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmission measurements,” Opt. Comm. 59(3), 219–223 (1986). [CrossRef]  

7. A. Monras and M. G. A. Paris, “Optimal quantum estimation of loss in bosonic channels,” Phys. Rev. Lett. 98, 160401 (2007). [CrossRef]   [PubMed]  

8. G. Adesso, F. Dell’Anno, S. De Siena, F. Illuminati, and L. A. M. Souza, “Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states,” Phys. Rev. A 79, 040305 (2009). [CrossRef]  

9. O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93243601 (2004). [CrossRef]  

10. G. Brida, L. Caspani, A. Gatti, M. Genovese, A. Meda, and I. Ruo-Berchera, “Measurement of sub-shot-noise spatial correlations without background subtraction,” Phys. Rev. Lett. 102, 213602 (2009). [CrossRef]   [PubMed]  

11. I. N. Agafonov, M. V. Chekhova, and G. Leuchs, “Two-color bright squeezed vacuum,” Phys. Rev. A 82, 011801 (2010). [CrossRef]  

12. P. R. Tapster, S. F. Seward, and J. G. Rarity, “Sub-shot-noise measurement of modulated absorption using parametric down-conversion,” Phys. Rev. A 44, 3266–3269 (1991). [CrossRef]   [PubMed]  

13. P. H. Souto Ribeiro, C. Schwob, A. Maître, and C. Fabre, “Sub-shot-noise high-sensitivity spectroscopy with optical parametric oscillator twin beams,” Opt. Lett. 22(24), 1893–1895 (1997). [CrossRef]  

14. R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Absorption spectroscopy at the ultimate quantum limit from single-photon states,” New J. Phys. 19(2), 023013 (2017). [CrossRef]  

15. E. Brambilla, L. Caspani, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “High-sensitivity imaging with multi-mode twin beams,” Phys. Rev. A 77, 053807 (2008). [CrossRef]  

16. G. Brida, M. Genovese, and I. Ruo-Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227 (2010). [CrossRef]  

17. N. Samantaray, I. Ruo-Berchera, A. Meda, and M. Genovese, “Realization of the first sub-shot-noise wide field microscope,” Light: Sci. Appl. 6, e17005 (2017). [CrossRef]  

18. E. Losero, I. Ruo-Berchera, A. Meda, A. Avella, and M. Genovese, “Unbiased estimation of an optical loss at the ultimate quantum limit with twin-beams,” Sci. Rep. 8(1), 7431 (2018). [CrossRef]   [PubMed]  

19. S. Lloyd, “Enhanced sensitivity of photodetection via quantum illumination,” Science 321(5895), 1463–1465 (2008). [CrossRef]   [PubMed]  

20. S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, “Quantum illumination with Gaussian states,” Phys. Rev. Lett. 101, 253601 (2008). [CrossRef]   [PubMed]  

21. E. D. Lopaeva, I. Ruo-Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, “Experimental realization of quantum illumination,” Phys. Rev. Lett. 110, 153603 (2013). [CrossRef]   [PubMed]  

22. S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114, 080503 (2015). [CrossRef]   [PubMed]  

23. M. Xiao, L.-A. Wu, and H. J. Kimble, “Detection of amplitude modulation with squeezed light for sensitivity beyond the shot-noise limit,” Opt. Lett. 13(6), 476–478 (1988). [CrossRef]   [PubMed]  

24. E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992). [CrossRef]   [PubMed]  

25. S. Kasapi, S. Lathi, and Y. Yamamoto, “Sub-shot-noise frequency-modulation spectroscopy by use of amplitude-squeezed light from semiconductor lasers,” J. Opt. Soc. Am. B 17(2), 275–279 (2000). [CrossRef]  

26. J. Hodgkinson and R. P. Tatam, “Optical gas sensing: a review,” Meas. Sci. Technol. 24(1), 012004 (2013). [CrossRef]  

27. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981). [CrossRef]  

28. B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986). [CrossRef]  

29. A. M. Marino, N. V. C. Trejo, and P. D. Lett, “Effect of losses on the performance of an su(1,1) interferometer,” Phys. Rev. A 86, 023844 (2012). [CrossRef]  

30. C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016). [CrossRef]  

31. M. Manceau, F. Ya. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys. 19(1), 013014 (2017). [CrossRef]  

32. J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011). [CrossRef]  

33. J. Kong, J. Jing, H. Wang, F. Hudelist, C. Liu, and W. Zhang, “Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers,” Appl. Phys. Lett. 102, 011130 (2013). [CrossRef]  

34. F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014). [CrossRef]   [PubMed]  

35. M. Manceau, G. Leuchs, F. Ya. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an SU(1,1) interferometer,” Phys. Rev. Lett. 119, 223604 (2017). [CrossRef]   [PubMed]  

36. O. Hosten, R. Krishnakumar, N. J. Engelsen, and M. A. Kasevich, “Quantum phase magnification,” Science 352(6293), 1552–1555 (2017). [CrossRef]  

37. U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasiprobability distributions,” Phys. Rev. A 48, 4598–4604 (1993). [CrossRef]   [PubMed]  

38. E. Knyazev, K.Yu. Spasibko, F. Ya. Khalili, and M. Chekhova, “Quantum tomography enhanced through parametric amplification,” New J. Phys. 20(1), 013005 (2018). [CrossRef]  

39. S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999). [CrossRef]  

40. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 2010).

41. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, 1999). [CrossRef]  

42. Newlight Photonics Inc., “Thin BBO crystal for ultrafast lasers,” http://newlightphotonics.com/v1/ultrathin-bbo-crystals.html.

43. H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117, 110801 (2016). [CrossRef]  

44. T. S. Iskhakov, V. C. Usenko, U. L. Andersen, R. Filip, M. V. Chekhova, and G. Leuchs, “Heralded source of bright multi-mode mesoscopic sub-poissonian light,” Opt. Lett. 41(10), 2149–2152 (2016). [CrossRef]   [PubMed]  

References

  • View by:

  1. V. B. Braginsky and F. Ya. Khalili, “Quantum nondemolition measurements: the route from toys to tools,” Rev. Mod. Phys. 68, 1–11 (1996).
    [Crossref]
  2. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306, 1330–1336 (2004).
    [Crossref] [PubMed]
  3. R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kolodynski, Progress in Optics (Elsevier, 2015), Chap. 4.
  4. P.-A. Moreau, J. Sabines-Chesterking, R. Whittaker, S. K. Joshi, P. M. Birchall, A. McMillan, J. G. Rarity, and J. C. F. Matthews, “Demonstrating an absolute quantum advantage in direct absorption measurement,” Sci. Rep. 7, 6256 (2017).
    [Crossref] [PubMed]
  5. M. I. Kolobov, Quantum Imaging (Springer, 2007).
    [Crossref]
  6. E. Jakeman and J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmission measurements,” Opt. Comm. 59(3), 219–223 (1986).
    [Crossref]
  7. A. Monras and M. G. A. Paris, “Optimal quantum estimation of loss in bosonic channels,” Phys. Rev. Lett. 98, 160401 (2007).
    [Crossref] [PubMed]
  8. G. Adesso, F. Dell’Anno, S. De Siena, F. Illuminati, and L. A. M. Souza, “Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states,” Phys. Rev. A 79, 040305 (2009).
    [Crossref]
  9. O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93243601 (2004).
    [Crossref]
  10. G. Brida, L. Caspani, A. Gatti, M. Genovese, A. Meda, and I. Ruo-Berchera, “Measurement of sub-shot-noise spatial correlations without background subtraction,” Phys. Rev. Lett. 102, 213602 (2009).
    [Crossref] [PubMed]
  11. I. N. Agafonov, M. V. Chekhova, and G. Leuchs, “Two-color bright squeezed vacuum,” Phys. Rev. A 82, 011801 (2010).
    [Crossref]
  12. P. R. Tapster, S. F. Seward, and J. G. Rarity, “Sub-shot-noise measurement of modulated absorption using parametric down-conversion,” Phys. Rev. A 44, 3266–3269 (1991).
    [Crossref] [PubMed]
  13. P. H. Souto Ribeiro, C. Schwob, A. Maître, and C. Fabre, “Sub-shot-noise high-sensitivity spectroscopy with optical parametric oscillator twin beams,” Opt. Lett. 22(24), 1893–1895 (1997).
    [Crossref]
  14. R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Absorption spectroscopy at the ultimate quantum limit from single-photon states,” New J. Phys. 19(2), 023013 (2017).
    [Crossref]
  15. E. Brambilla, L. Caspani, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “High-sensitivity imaging with multi-mode twin beams,” Phys. Rev. A 77, 053807 (2008).
    [Crossref]
  16. G. Brida, M. Genovese, and I. Ruo-Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227 (2010).
    [Crossref]
  17. N. Samantaray, I. Ruo-Berchera, A. Meda, and M. Genovese, “Realization of the first sub-shot-noise wide field microscope,” Light: Sci. Appl. 6, e17005 (2017).
    [Crossref]
  18. E. Losero, I. Ruo-Berchera, A. Meda, A. Avella, and M. Genovese, “Unbiased estimation of an optical loss at the ultimate quantum limit with twin-beams,” Sci. Rep. 8(1), 7431 (2018).
    [Crossref] [PubMed]
  19. S. Lloyd, “Enhanced sensitivity of photodetection via quantum illumination,” Science 321(5895), 1463–1465 (2008).
    [Crossref] [PubMed]
  20. S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, “Quantum illumination with Gaussian states,” Phys. Rev. Lett. 101, 253601 (2008).
    [Crossref] [PubMed]
  21. E. D. Lopaeva, I. Ruo-Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, “Experimental realization of quantum illumination,” Phys. Rev. Lett. 110, 153603 (2013).
    [Crossref] [PubMed]
  22. S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114, 080503 (2015).
    [Crossref] [PubMed]
  23. M. Xiao, L.-A. Wu, and H. J. Kimble, “Detection of amplitude modulation with squeezed light for sensitivity beyond the shot-noise limit,” Opt. Lett. 13(6), 476–478 (1988).
    [Crossref] [PubMed]
  24. E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
    [Crossref] [PubMed]
  25. S. Kasapi, S. Lathi, and Y. Yamamoto, “Sub-shot-noise frequency-modulation spectroscopy by use of amplitude-squeezed light from semiconductor lasers,” J. Opt. Soc. Am. B 17(2), 275–279 (2000).
    [Crossref]
  26. J. Hodgkinson and R. P. Tatam, “Optical gas sensing: a review,” Meas. Sci. Technol. 24(1), 012004 (2013).
    [Crossref]
  27. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [Crossref]
  28. B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
    [Crossref]
  29. A. M. Marino, N. V. C. Trejo, and P. D. Lett, “Effect of losses on the performance of an su(1,1) interferometer,” Phys. Rev. A 86, 023844 (2012).
    [Crossref]
  30. C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
    [Crossref]
  31. M. Manceau, F. Ya. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys. 19(1), 013014 (2017).
    [Crossref]
  32. J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
    [Crossref]
  33. J. Kong, J. Jing, H. Wang, F. Hudelist, C. Liu, and W. Zhang, “Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers,” Appl. Phys. Lett. 102, 011130 (2013).
    [Crossref]
  34. F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
    [Crossref] [PubMed]
  35. M. Manceau, G. Leuchs, F. Ya. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an SU(1,1) interferometer,” Phys. Rev. Lett. 119, 223604 (2017).
    [Crossref] [PubMed]
  36. O. Hosten, R. Krishnakumar, N. J. Engelsen, and M. A. Kasevich, “Quantum phase magnification,” Science 352(6293), 1552–1555 (2017).
    [Crossref]
  37. U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasiprobability distributions,” Phys. Rev. A 48, 4598–4604 (1993).
    [Crossref] [PubMed]
  38. E. Knyazev, K.Yu. Spasibko, F. Ya. Khalili, and M. Chekhova, “Quantum tomography enhanced through parametric amplification,” New J. Phys. 20(1), 013005 (2018).
    [Crossref]
  39. S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999).
    [Crossref]
  40. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 2010).
  41. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, 1999).
    [Crossref]
  42. Newlight Photonics Inc., “Thin BBO crystal for ultrafast lasers,” http://newlightphotonics.com/v1/ultrathin-bbo-crystals.html .
  43. H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117, 110801 (2016).
    [Crossref]
  44. T. S. Iskhakov, V. C. Usenko, U. L. Andersen, R. Filip, M. V. Chekhova, and G. Leuchs, “Heralded source of bright multi-mode mesoscopic sub-poissonian light,” Opt. Lett. 41(10), 2149–2152 (2016).
    [Crossref] [PubMed]

2018 (2)

E. Losero, I. Ruo-Berchera, A. Meda, A. Avella, and M. Genovese, “Unbiased estimation of an optical loss at the ultimate quantum limit with twin-beams,” Sci. Rep. 8(1), 7431 (2018).
[Crossref] [PubMed]

E. Knyazev, K.Yu. Spasibko, F. Ya. Khalili, and M. Chekhova, “Quantum tomography enhanced through parametric amplification,” New J. Phys. 20(1), 013005 (2018).
[Crossref]

2017 (6)

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Absorption spectroscopy at the ultimate quantum limit from single-photon states,” New J. Phys. 19(2), 023013 (2017).
[Crossref]

P.-A. Moreau, J. Sabines-Chesterking, R. Whittaker, S. K. Joshi, P. M. Birchall, A. McMillan, J. G. Rarity, and J. C. F. Matthews, “Demonstrating an absolute quantum advantage in direct absorption measurement,” Sci. Rep. 7, 6256 (2017).
[Crossref] [PubMed]

N. Samantaray, I. Ruo-Berchera, A. Meda, and M. Genovese, “Realization of the first sub-shot-noise wide field microscope,” Light: Sci. Appl. 6, e17005 (2017).
[Crossref]

M. Manceau, F. Ya. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys. 19(1), 013014 (2017).
[Crossref]

M. Manceau, G. Leuchs, F. Ya. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an SU(1,1) interferometer,” Phys. Rev. Lett. 119, 223604 (2017).
[Crossref] [PubMed]

O. Hosten, R. Krishnakumar, N. J. Engelsen, and M. A. Kasevich, “Quantum phase magnification,” Science 352(6293), 1552–1555 (2017).
[Crossref]

2016 (3)

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

T. S. Iskhakov, V. C. Usenko, U. L. Andersen, R. Filip, M. V. Chekhova, and G. Leuchs, “Heralded source of bright multi-mode mesoscopic sub-poissonian light,” Opt. Lett. 41(10), 2149–2152 (2016).
[Crossref] [PubMed]

2015 (1)

S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114, 080503 (2015).
[Crossref] [PubMed]

2014 (1)

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

2013 (3)

E. D. Lopaeva, I. Ruo-Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, “Experimental realization of quantum illumination,” Phys. Rev. Lett. 110, 153603 (2013).
[Crossref] [PubMed]

J. Hodgkinson and R. P. Tatam, “Optical gas sensing: a review,” Meas. Sci. Technol. 24(1), 012004 (2013).
[Crossref]

J. Kong, J. Jing, H. Wang, F. Hudelist, C. Liu, and W. Zhang, “Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers,” Appl. Phys. Lett. 102, 011130 (2013).
[Crossref]

2012 (1)

A. M. Marino, N. V. C. Trejo, and P. D. Lett, “Effect of losses on the performance of an su(1,1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

2011 (1)

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

2010 (2)

I. N. Agafonov, M. V. Chekhova, and G. Leuchs, “Two-color bright squeezed vacuum,” Phys. Rev. A 82, 011801 (2010).
[Crossref]

G. Brida, M. Genovese, and I. Ruo-Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227 (2010).
[Crossref]

2009 (2)

G. Adesso, F. Dell’Anno, S. De Siena, F. Illuminati, and L. A. M. Souza, “Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states,” Phys. Rev. A 79, 040305 (2009).
[Crossref]

G. Brida, L. Caspani, A. Gatti, M. Genovese, A. Meda, and I. Ruo-Berchera, “Measurement of sub-shot-noise spatial correlations without background subtraction,” Phys. Rev. Lett. 102, 213602 (2009).
[Crossref] [PubMed]

2008 (3)

S. Lloyd, “Enhanced sensitivity of photodetection via quantum illumination,” Science 321(5895), 1463–1465 (2008).
[Crossref] [PubMed]

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, “Quantum illumination with Gaussian states,” Phys. Rev. Lett. 101, 253601 (2008).
[Crossref] [PubMed]

E. Brambilla, L. Caspani, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “High-sensitivity imaging with multi-mode twin beams,” Phys. Rev. A 77, 053807 (2008).
[Crossref]

2007 (1)

A. Monras and M. G. A. Paris, “Optimal quantum estimation of loss in bosonic channels,” Phys. Rev. Lett. 98, 160401 (2007).
[Crossref] [PubMed]

2004 (2)

O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93243601 (2004).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref] [PubMed]

2000 (1)

S. Kasapi, S. Lathi, and Y. Yamamoto, “Sub-shot-noise frequency-modulation spectroscopy by use of amplitude-squeezed light from semiconductor lasers,” J. Opt. Soc. Am. B 17(2), 275–279 (2000).
[Crossref]

1999 (1)

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999).
[Crossref]

1997 (1)

P. H. Souto Ribeiro, C. Schwob, A. Maître, and C. Fabre, “Sub-shot-noise high-sensitivity spectroscopy with optical parametric oscillator twin beams,” Opt. Lett. 22(24), 1893–1895 (1997).
[Crossref]

1996 (1)

V. B. Braginsky and F. Ya. Khalili, “Quantum nondemolition measurements: the route from toys to tools,” Rev. Mod. Phys. 68, 1–11 (1996).
[Crossref]

1993 (1)

U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasiprobability distributions,” Phys. Rev. A 48, 4598–4604 (1993).
[Crossref] [PubMed]

1992 (1)

E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
[Crossref] [PubMed]

1991 (1)

P. R. Tapster, S. F. Seward, and J. G. Rarity, “Sub-shot-noise measurement of modulated absorption using parametric down-conversion,” Phys. Rev. A 44, 3266–3269 (1991).
[Crossref] [PubMed]

1988 (1)

M. Xiao, L.-A. Wu, and H. J. Kimble, “Detection of amplitude modulation with squeezed light for sensitivity beyond the shot-noise limit,” Opt. Lett. 13(6), 476–478 (1988).
[Crossref] [PubMed]

1986 (2)

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

E. Jakeman and J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmission measurements,” Opt. Comm. 59(3), 219–223 (1986).
[Crossref]

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[Crossref]

Adesso, G.

G. Adesso, F. Dell’Anno, S. De Siena, F. Illuminati, and L. A. M. Souza, “Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states,” Phys. Rev. A 79, 040305 (2009).
[Crossref]

Agafonov, I. N.

I. N. Agafonov, M. V. Chekhova, and G. Leuchs, “Two-color bright squeezed vacuum,” Phys. Rev. A 82, 011801 (2010).
[Crossref]

Andersen, U. L.

T. S. Iskhakov, V. C. Usenko, U. L. Andersen, R. Filip, M. V. Chekhova, and G. Leuchs, “Heralded source of bright multi-mode mesoscopic sub-poissonian light,” Opt. Lett. 41(10), 2149–2152 (2016).
[Crossref] [PubMed]

Avella, A.

E. Losero, I. Ruo-Berchera, A. Meda, A. Avella, and M. Genovese, “Unbiased estimation of an optical loss at the ultimate quantum limit with twin-beams,” Sci. Rep. 8(1), 7431 (2018).
[Crossref] [PubMed]

Bache, M.

O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93243601 (2004).
[Crossref]

Barzanjeh, S.

S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114, 080503 (2015).
[Crossref] [PubMed]

Berry, M.

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Absorption spectroscopy at the ultimate quantum limit from single-photon states,” New J. Phys. 19(2), 023013 (2017).
[Crossref]

Birchall, P. M.

P.-A. Moreau, J. Sabines-Chesterking, R. Whittaker, S. K. Joshi, P. M. Birchall, A. McMillan, J. G. Rarity, and J. C. F. Matthews, “Demonstrating an absolute quantum advantage in direct absorption measurement,” Sci. Rep. 7, 6256 (2017).
[Crossref] [PubMed]

Braginsky, V. B.

V. B. Braginsky and F. Ya. Khalili, “Quantum nondemolition measurements: the route from toys to tools,” Rev. Mod. Phys. 68, 1–11 (1996).
[Crossref]

Brambilla, E.

E. Brambilla, L. Caspani, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “High-sensitivity imaging with multi-mode twin beams,” Phys. Rev. A 77, 053807 (2008).
[Crossref]

O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93243601 (2004).
[Crossref]

Brida, G.

E. D. Lopaeva, I. Ruo-Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, “Experimental realization of quantum illumination,” Phys. Rev. Lett. 110, 153603 (2013).
[Crossref] [PubMed]

G. Brida, M. Genovese, and I. Ruo-Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227 (2010).
[Crossref]

G. Brida, L. Caspani, A. Gatti, M. Genovese, A. Meda, and I. Ruo-Berchera, “Measurement of sub-shot-noise spatial correlations without background subtraction,” Phys. Rev. Lett. 102, 213602 (2009).
[Crossref] [PubMed]

Cable, H.

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Absorption spectroscopy at the ultimate quantum limit from single-photon states,” New J. Phys. 19(2), 023013 (2017).
[Crossref]

Carri, J.

E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
[Crossref] [PubMed]

Caspani, L.

G. Brida, L. Caspani, A. Gatti, M. Genovese, A. Meda, and I. Ruo-Berchera, “Measurement of sub-shot-noise spatial correlations without background subtraction,” Phys. Rev. Lett. 102, 213602 (2009).
[Crossref] [PubMed]

E. Brambilla, L. Caspani, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “High-sensitivity imaging with multi-mode twin beams,” Phys. Rev. A 77, 053807 (2008).
[Crossref]

Caves, C. M.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[Crossref]

Chekhova, M.

E. Knyazev, K.Yu. Spasibko, F. Ya. Khalili, and M. Chekhova, “Quantum tomography enhanced through parametric amplification,” New J. Phys. 20(1), 013005 (2018).
[Crossref]

M. Manceau, F. Ya. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys. 19(1), 013014 (2017).
[Crossref]

M. Manceau, G. Leuchs, F. Ya. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an SU(1,1) interferometer,” Phys. Rev. Lett. 119, 223604 (2017).
[Crossref] [PubMed]

Chekhova, M. V.

T. S. Iskhakov, V. C. Usenko, U. L. Andersen, R. Filip, M. V. Chekhova, and G. Leuchs, “Heralded source of bright multi-mode mesoscopic sub-poissonian light,” Opt. Lett. 41(10), 2149–2152 (2016).
[Crossref] [PubMed]

I. N. Agafonov, M. V. Chekhova, and G. Leuchs, “Two-color bright squeezed vacuum,” Phys. Rev. A 82, 011801 (2010).
[Crossref]

Choi, S.-K.

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999).
[Crossref]

Danzmann, K.

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

De Siena, S.

G. Adesso, F. Dell’Anno, S. De Siena, F. Illuminati, and L. A. M. Souza, “Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states,” Phys. Rev. A 79, 040305 (2009).
[Crossref]

Degiovanni, I. P.

E. D. Lopaeva, I. Ruo-Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, “Experimental realization of quantum illumination,” Phys. Rev. Lett. 110, 153603 (2013).
[Crossref] [PubMed]

Dell’Anno, F.

G. Adesso, F. Dell’Anno, S. De Siena, F. Illuminati, and L. A. M. Souza, “Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states,” Phys. Rev. A 79, 040305 (2009).
[Crossref]

Demkowicz-Dobrzanski, R.

R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kolodynski, Progress in Optics (Elsevier, 2015), Chap. 4.

Di Trapani, P.

O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93243601 (2004).
[Crossref]

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, 1999).
[Crossref]

Engelsen, N. J.

O. Hosten, R. Krishnakumar, N. J. Engelsen, and M. A. Kasevich, “Quantum phase magnification,” Science 352(6293), 1552–1555 (2017).
[Crossref]

Erkmen, B. I.

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, “Quantum illumination with Gaussian states,” Phys. Rev. Lett. 101, 253601 (2008).
[Crossref] [PubMed]

Erven, C.

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Absorption spectroscopy at the ultimate quantum limit from single-photon states,” New J. Phys. 19(2), 023013 (2017).
[Crossref]

Fabre, C.

P. H. Souto Ribeiro, C. Schwob, A. Maître, and C. Fabre, “Sub-shot-noise high-sensitivity spectroscopy with optical parametric oscillator twin beams,” Opt. Lett. 22(24), 1893–1895 (1997).
[Crossref]

Filip, R.

T. S. Iskhakov, V. C. Usenko, U. L. Andersen, R. Filip, M. V. Chekhova, and G. Leuchs, “Heralded source of bright multi-mode mesoscopic sub-poissonian light,” Opt. Lett. 41(10), 2149–2152 (2016).
[Crossref] [PubMed]

Gatti, A.

G. Brida, L. Caspani, A. Gatti, M. Genovese, A. Meda, and I. Ruo-Berchera, “Measurement of sub-shot-noise spatial correlations without background subtraction,” Phys. Rev. Lett. 102, 213602 (2009).
[Crossref] [PubMed]

E. Brambilla, L. Caspani, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “High-sensitivity imaging with multi-mode twin beams,” Phys. Rev. A 77, 053807 (2008).
[Crossref]

O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93243601 (2004).
[Crossref]

Genovese, M.

E. Losero, I. Ruo-Berchera, A. Meda, A. Avella, and M. Genovese, “Unbiased estimation of an optical loss at the ultimate quantum limit with twin-beams,” Sci. Rep. 8(1), 7431 (2018).
[Crossref] [PubMed]

N. Samantaray, I. Ruo-Berchera, A. Meda, and M. Genovese, “Realization of the first sub-shot-noise wide field microscope,” Light: Sci. Appl. 6, e17005 (2017).
[Crossref]

E. D. Lopaeva, I. Ruo-Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, “Experimental realization of quantum illumination,” Phys. Rev. Lett. 110, 153603 (2013).
[Crossref] [PubMed]

G. Brida, M. Genovese, and I. Ruo-Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227 (2010).
[Crossref]

G. Brida, L. Caspani, A. Gatti, M. Genovese, A. Meda, and I. Ruo-Berchera, “Measurement of sub-shot-noise spatial correlations without background subtraction,” Phys. Rev. Lett. 102, 213602 (2009).
[Crossref] [PubMed]

Giovannetti, V.

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, “Quantum illumination with Gaussian states,” Phys. Rev. Lett. 101, 253601 (2008).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref] [PubMed]

Guha, S.

S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114, 080503 (2015).
[Crossref] [PubMed]

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, “Quantum illumination with Gaussian states,” Phys. Rev. Lett. 101, 253601 (2008).
[Crossref] [PubMed]

Gurzadyan, G. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, 1999).
[Crossref]

Hodgkinson, J.

J. Hodgkinson and R. P. Tatam, “Optical gas sensing: a review,” Meas. Sci. Technol. 24(1), 012004 (2013).
[Crossref]

Hosten, O.

O. Hosten, R. Krishnakumar, N. J. Engelsen, and M. A. Kasevich, “Quantum phase magnification,” Science 352(6293), 1552–1555 (2017).
[Crossref]

Hudelist, F.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

J. Kong, J. Jing, H. Wang, F. Hudelist, C. Liu, and W. Zhang, “Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers,” Appl. Phys. Lett. 102, 011130 (2013).
[Crossref]

Illuminati, F.

G. Adesso, F. Dell’Anno, S. De Siena, F. Illuminati, and L. A. M. Souza, “Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states,” Phys. Rev. A 79, 040305 (2009).
[Crossref]

Iskhakov, T. S.

T. S. Iskhakov, V. C. Usenko, U. L. Andersen, R. Filip, M. V. Chekhova, and G. Leuchs, “Heralded source of bright multi-mode mesoscopic sub-poissonian light,” Opt. Lett. 41(10), 2149–2152 (2016).
[Crossref] [PubMed]

Jakeman, E.

E. Jakeman and J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmission measurements,” Opt. Comm. 59(3), 219–223 (1986).
[Crossref]

Jarzyna, M.

R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kolodynski, Progress in Optics (Elsevier, 2015), Chap. 4.

Jedrkiewicz, O.

E. Brambilla, L. Caspani, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “High-sensitivity imaging with multi-mode twin beams,” Phys. Rev. A 77, 053807 (2008).
[Crossref]

O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93243601 (2004).
[Crossref]

Jiang, Y.-K.

O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93243601 (2004).
[Crossref]

Jing, J.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

J. Kong, J. Jing, H. Wang, F. Hudelist, C. Liu, and W. Zhang, “Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers,” Appl. Phys. Lett. 102, 011130 (2013).
[Crossref]

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

Joshi, S. K.

P.-A. Moreau, J. Sabines-Chesterking, R. Whittaker, S. K. Joshi, P. M. Birchall, A. McMillan, J. G. Rarity, and J. C. F. Matthews, “Demonstrating an absolute quantum advantage in direct absorption measurement,” Sci. Rep. 7, 6256 (2017).
[Crossref] [PubMed]

Kasapi, S.

S. Kasapi, S. Lathi, and Y. Yamamoto, “Sub-shot-noise frequency-modulation spectroscopy by use of amplitude-squeezed light from semiconductor lasers,” J. Opt. Soc. Am. B 17(2), 275–279 (2000).
[Crossref]

Kasevich, M. A.

O. Hosten, R. Krishnakumar, N. J. Engelsen, and M. A. Kasevich, “Quantum phase magnification,” Science 352(6293), 1552–1555 (2017).
[Crossref]

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 2010).

Khalili, F. Ya.

E. Knyazev, K.Yu. Spasibko, F. Ya. Khalili, and M. Chekhova, “Quantum tomography enhanced through parametric amplification,” New J. Phys. 20(1), 013005 (2018).
[Crossref]

M. Manceau, G. Leuchs, F. Ya. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an SU(1,1) interferometer,” Phys. Rev. Lett. 119, 223604 (2017).
[Crossref] [PubMed]

M. Manceau, F. Ya. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys. 19(1), 013014 (2017).
[Crossref]

V. B. Braginsky and F. Ya. Khalili, “Quantum nondemolition measurements: the route from toys to tools,” Rev. Mod. Phys. 68, 1–11 (1996).
[Crossref]

Kimble, H. J.

E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
[Crossref] [PubMed]

M. Xiao, L.-A. Wu, and H. J. Kimble, “Detection of amplitude modulation with squeezed light for sensitivity beyond the shot-noise limit,” Opt. Lett. 13(6), 476–478 (1988).
[Crossref] [PubMed]

Klauder, J. R.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

Knyazev, E.

E. Knyazev, K.Yu. Spasibko, F. Ya. Khalili, and M. Chekhova, “Quantum tomography enhanced through parametric amplification,” New J. Phys. 20(1), 013005 (2018).
[Crossref]

Kolobov, M. I.

M. I. Kolobov, Quantum Imaging (Springer, 2007).
[Crossref]

Kolodynski, J.

R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kolodynski, Progress in Optics (Elsevier, 2015), Chap. 4.

Kong, J.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

J. Kong, J. Jing, H. Wang, F. Hudelist, C. Liu, and W. Zhang, “Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers,” Appl. Phys. Lett. 102, 011130 (2013).
[Crossref]

Krishnakumar, R.

O. Hosten, R. Krishnakumar, N. J. Engelsen, and M. A. Kasevich, “Quantum phase magnification,” Science 352(6293), 1552–1555 (2017).
[Crossref]

Kumar, P.

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999).
[Crossref]

Lathi, S.

S. Kasapi, S. Lathi, and Y. Yamamoto, “Sub-shot-noise frequency-modulation spectroscopy by use of amplitude-squeezed light from semiconductor lasers,” J. Opt. Soc. Am. B 17(2), 275–279 (2000).
[Crossref]

Leonhardt, U.

U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasiprobability distributions,” Phys. Rev. A 48, 4598–4604 (1993).
[Crossref] [PubMed]

Lett, P. D.

A. M. Marino, N. V. C. Trejo, and P. D. Lett, “Effect of losses on the performance of an su(1,1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

Leuchs, G.

M. Manceau, G. Leuchs, F. Ya. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an SU(1,1) interferometer,” Phys. Rev. Lett. 119, 223604 (2017).
[Crossref] [PubMed]

T. S. Iskhakov, V. C. Usenko, U. L. Andersen, R. Filip, M. V. Chekhova, and G. Leuchs, “Heralded source of bright multi-mode mesoscopic sub-poissonian light,” Opt. Lett. 41(10), 2149–2152 (2016).
[Crossref] [PubMed]

I. N. Agafonov, M. V. Chekhova, and G. Leuchs, “Two-color bright squeezed vacuum,” Phys. Rev. A 82, 011801 (2010).
[Crossref]

Liu, C.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

J. Kong, J. Jing, H. Wang, F. Hudelist, C. Liu, and W. Zhang, “Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers,” Appl. Phys. Lett. 102, 011130 (2013).
[Crossref]

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

Lloyd, S.

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, “Quantum illumination with Gaussian states,” Phys. Rev. Lett. 101, 253601 (2008).
[Crossref] [PubMed]

S. Lloyd, “Enhanced sensitivity of photodetection via quantum illumination,” Science 321(5895), 1463–1465 (2008).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref] [PubMed]

Lopaeva, E. D.

E. D. Lopaeva, I. Ruo-Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, “Experimental realization of quantum illumination,” Phys. Rev. Lett. 110, 153603 (2013).
[Crossref] [PubMed]

Losero, E.

E. Losero, I. Ruo-Berchera, A. Meda, A. Avella, and M. Genovese, “Unbiased estimation of an optical loss at the ultimate quantum limit with twin-beams,” Sci. Rep. 8(1), 7431 (2018).
[Crossref] [PubMed]

Lugiato, L. A.

E. Brambilla, L. Caspani, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “High-sensitivity imaging with multi-mode twin beams,” Phys. Rev. A 77, 053807 (2008).
[Crossref]

O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93243601 (2004).
[Crossref]

Maccone, L.

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, “Quantum illumination with Gaussian states,” Phys. Rev. Lett. 101, 253601 (2008).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref] [PubMed]

Maître, A.

P. H. Souto Ribeiro, C. Schwob, A. Maître, and C. Fabre, “Sub-shot-noise high-sensitivity spectroscopy with optical parametric oscillator twin beams,” Opt. Lett. 22(24), 1893–1895 (1997).
[Crossref]

Manceau, M.

M. Manceau, F. Ya. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys. 19(1), 013014 (2017).
[Crossref]

M. Manceau, G. Leuchs, F. Ya. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an SU(1,1) interferometer,” Phys. Rev. Lett. 119, 223604 (2017).
[Crossref] [PubMed]

Marino, A. M.

A. M. Marino, N. V. C. Trejo, and P. D. Lett, “Effect of losses on the performance of an su(1,1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

Matthews, J. C. F.

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Absorption spectroscopy at the ultimate quantum limit from single-photon states,” New J. Phys. 19(2), 023013 (2017).
[Crossref]

P.-A. Moreau, J. Sabines-Chesterking, R. Whittaker, S. K. Joshi, P. M. Birchall, A. McMillan, J. G. Rarity, and J. C. F. Matthews, “Demonstrating an absolute quantum advantage in direct absorption measurement,” Sci. Rep. 7, 6256 (2017).
[Crossref] [PubMed]

McCall, S. L.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

McMillan, A.

P.-A. Moreau, J. Sabines-Chesterking, R. Whittaker, S. K. Joshi, P. M. Birchall, A. McMillan, J. G. Rarity, and J. C. F. Matthews, “Demonstrating an absolute quantum advantage in direct absorption measurement,” Sci. Rep. 7, 6256 (2017).
[Crossref] [PubMed]

Meda, A.

E. Losero, I. Ruo-Berchera, A. Meda, A. Avella, and M. Genovese, “Unbiased estimation of an optical loss at the ultimate quantum limit with twin-beams,” Sci. Rep. 8(1), 7431 (2018).
[Crossref] [PubMed]

N. Samantaray, I. Ruo-Berchera, A. Meda, and M. Genovese, “Realization of the first sub-shot-noise wide field microscope,” Light: Sci. Appl. 6, e17005 (2017).
[Crossref]

G. Brida, L. Caspani, A. Gatti, M. Genovese, A. Meda, and I. Ruo-Berchera, “Measurement of sub-shot-noise spatial correlations without background subtraction,” Phys. Rev. Lett. 102, 213602 (2009).
[Crossref] [PubMed]

Mehmet, M.

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

Monras, A.

A. Monras and M. G. A. Paris, “Optimal quantum estimation of loss in bosonic channels,” Phys. Rev. Lett. 98, 160401 (2007).
[Crossref] [PubMed]

Moreau, P.-A.

P.-A. Moreau, J. Sabines-Chesterking, R. Whittaker, S. K. Joshi, P. M. Birchall, A. McMillan, J. G. Rarity, and J. C. F. Matthews, “Demonstrating an absolute quantum advantage in direct absorption measurement,” Sci. Rep. 7, 6256 (2017).
[Crossref] [PubMed]

Neville, A.

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Absorption spectroscopy at the ultimate quantum limit from single-photon states,” New J. Phys. 19(2), 023013 (2017).
[Crossref]

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, 1999).
[Crossref]

O’Brien, J. L.

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Absorption spectroscopy at the ultimate quantum limit from single-photon states,” New J. Phys. 19(2), 023013 (2017).
[Crossref]

Olivares, S.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

E. D. Lopaeva, I. Ruo-Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, “Experimental realization of quantum illumination,” Phys. Rev. Lett. 110, 153603 (2013).
[Crossref] [PubMed]

Ou, Z. Y.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

Paris, M. G. A.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

A. Monras and M. G. A. Paris, “Optimal quantum estimation of loss in bosonic channels,” Phys. Rev. Lett. 98, 160401 (2007).
[Crossref] [PubMed]

Paul, H.

U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasiprobability distributions,” Phys. Rev. A 48, 4598–4604 (1993).
[Crossref] [PubMed]

Pirandola, S.

S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114, 080503 (2015).
[Crossref] [PubMed]

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, “Quantum illumination with Gaussian states,” Phys. Rev. Lett. 101, 253601 (2008).
[Crossref] [PubMed]

Polzik, E. S.

E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
[Crossref] [PubMed]

Rarity, J. G.

P.-A. Moreau, J. Sabines-Chesterking, R. Whittaker, S. K. Joshi, P. M. Birchall, A. McMillan, J. G. Rarity, and J. C. F. Matthews, “Demonstrating an absolute quantum advantage in direct absorption measurement,” Sci. Rep. 7, 6256 (2017).
[Crossref] [PubMed]

P. R. Tapster, S. F. Seward, and J. G. Rarity, “Sub-shot-noise measurement of modulated absorption using parametric down-conversion,” Phys. Rev. A 44, 3266–3269 (1991).
[Crossref] [PubMed]

E. Jakeman and J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmission measurements,” Opt. Comm. 59(3), 219–223 (1986).
[Crossref]

Ruo-Berchera, I.

E. Losero, I. Ruo-Berchera, A. Meda, A. Avella, and M. Genovese, “Unbiased estimation of an optical loss at the ultimate quantum limit with twin-beams,” Sci. Rep. 8(1), 7431 (2018).
[Crossref] [PubMed]

N. Samantaray, I. Ruo-Berchera, A. Meda, and M. Genovese, “Realization of the first sub-shot-noise wide field microscope,” Light: Sci. Appl. 6, e17005 (2017).
[Crossref]

E. D. Lopaeva, I. Ruo-Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, “Experimental realization of quantum illumination,” Phys. Rev. Lett. 110, 153603 (2013).
[Crossref] [PubMed]

G. Brida, M. Genovese, and I. Ruo-Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227 (2010).
[Crossref]

G. Brida, L. Caspani, A. Gatti, M. Genovese, A. Meda, and I. Ruo-Berchera, “Measurement of sub-shot-noise spatial correlations without background subtraction,” Phys. Rev. Lett. 102, 213602 (2009).
[Crossref] [PubMed]

Sabines-Chesterking, J.

P.-A. Moreau, J. Sabines-Chesterking, R. Whittaker, S. K. Joshi, P. M. Birchall, A. McMillan, J. G. Rarity, and J. C. F. Matthews, “Demonstrating an absolute quantum advantage in direct absorption measurement,” Sci. Rep. 7, 6256 (2017).
[Crossref] [PubMed]

Samantaray, N.

N. Samantaray, I. Ruo-Berchera, A. Meda, and M. Genovese, “Realization of the first sub-shot-noise wide field microscope,” Light: Sci. Appl. 6, e17005 (2017).
[Crossref]

Schnabel, R.

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

Schwob, C.

P. H. Souto Ribeiro, C. Schwob, A. Maître, and C. Fabre, “Sub-shot-noise high-sensitivity spectroscopy with optical parametric oscillator twin beams,” Opt. Lett. 22(24), 1893–1895 (1997).
[Crossref]

Seward, S. F.

P. R. Tapster, S. F. Seward, and J. G. Rarity, “Sub-shot-noise measurement of modulated absorption using parametric down-conversion,” Phys. Rev. A 44, 3266–3269 (1991).
[Crossref] [PubMed]

Shapiro, J. H.

S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114, 080503 (2015).
[Crossref] [PubMed]

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, “Quantum illumination with Gaussian states,” Phys. Rev. Lett. 101, 253601 (2008).
[Crossref] [PubMed]

Souto Ribeiro, P. H.

P. H. Souto Ribeiro, C. Schwob, A. Maître, and C. Fabre, “Sub-shot-noise high-sensitivity spectroscopy with optical parametric oscillator twin beams,” Opt. Lett. 22(24), 1893–1895 (1997).
[Crossref]

Souza, L. A. M.

G. Adesso, F. Dell’Anno, S. De Siena, F. Illuminati, and L. A. M. Souza, “Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states,” Phys. Rev. A 79, 040305 (2009).
[Crossref]

Sparaciari, C.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

Spasibko, K.Yu.

E. Knyazev, K.Yu. Spasibko, F. Ya. Khalili, and M. Chekhova, “Quantum tomography enhanced through parametric amplification,” New J. Phys. 20(1), 013005 (2018).
[Crossref]

Tan, S.-H.

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, “Quantum illumination with Gaussian states,” Phys. Rev. Lett. 101, 253601 (2008).
[Crossref] [PubMed]

Tapster, P. R.

P. R. Tapster, S. F. Seward, and J. G. Rarity, “Sub-shot-noise measurement of modulated absorption using parametric down-conversion,” Phys. Rev. A 44, 3266–3269 (1991).
[Crossref] [PubMed]

Tatam, R. P.

J. Hodgkinson and R. P. Tatam, “Optical gas sensing: a review,” Meas. Sci. Technol. 24(1), 012004 (2013).
[Crossref]

Trejo, N. V. C.

A. M. Marino, N. V. C. Trejo, and P. D. Lett, “Effect of losses on the performance of an su(1,1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

Usenko, V. C.

T. S. Iskhakov, V. C. Usenko, U. L. Andersen, R. Filip, M. V. Chekhova, and G. Leuchs, “Heralded source of bright multi-mode mesoscopic sub-poissonian light,” Opt. Lett. 41(10), 2149–2152 (2016).
[Crossref] [PubMed]

Vahlbruch, H.

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

Vasilyev, M.

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999).
[Crossref]

Vitali, D.

S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114, 080503 (2015).
[Crossref] [PubMed]

Wang, H.

J. Kong, J. Jing, H. Wang, F. Hudelist, C. Liu, and W. Zhang, “Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers,” Appl. Phys. Lett. 102, 011130 (2013).
[Crossref]

Weedbrook, C.

S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114, 080503 (2015).
[Crossref] [PubMed]

Whittaker, R.

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Absorption spectroscopy at the ultimate quantum limit from single-photon states,” New J. Phys. 19(2), 023013 (2017).
[Crossref]

P.-A. Moreau, J. Sabines-Chesterking, R. Whittaker, S. K. Joshi, P. M. Birchall, A. McMillan, J. G. Rarity, and J. C. F. Matthews, “Demonstrating an absolute quantum advantage in direct absorption measurement,” Sci. Rep. 7, 6256 (2017).
[Crossref] [PubMed]

Wu, L.-A.

M. Xiao, L.-A. Wu, and H. J. Kimble, “Detection of amplitude modulation with squeezed light for sensitivity beyond the shot-noise limit,” Opt. Lett. 13(6), 476–478 (1988).
[Crossref] [PubMed]

Xiao, M.

M. Xiao, L.-A. Wu, and H. J. Kimble, “Detection of amplitude modulation with squeezed light for sensitivity beyond the shot-noise limit,” Opt. Lett. 13(6), 476–478 (1988).
[Crossref] [PubMed]

Yamamoto, Y.

S. Kasapi, S. Lathi, and Y. Yamamoto, “Sub-shot-noise frequency-modulation spectroscopy by use of amplitude-squeezed light from semiconductor lasers,” J. Opt. Soc. Am. B 17(2), 275–279 (2000).
[Crossref]

Yurke, B.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

Zhang, W.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

J. Kong, J. Jing, H. Wang, F. Hudelist, C. Liu, and W. Zhang, “Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers,” Appl. Phys. Lett. 102, 011130 (2013).
[Crossref]

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

Zhou, Z.

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

Appl. Phys. Lett. (2)

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

J. Kong, J. Jing, H. Wang, F. Hudelist, C. Liu, and W. Zhang, “Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers,” Appl. Phys. Lett. 102, 011130 (2013).
[Crossref]

J. Opt. Soc. Am. B (1)

S. Kasapi, S. Lathi, and Y. Yamamoto, “Sub-shot-noise frequency-modulation spectroscopy by use of amplitude-squeezed light from semiconductor lasers,” J. Opt. Soc. Am. B 17(2), 275–279 (2000).
[Crossref]

Light: Sci. Appl. (1)

N. Samantaray, I. Ruo-Berchera, A. Meda, and M. Genovese, “Realization of the first sub-shot-noise wide field microscope,” Light: Sci. Appl. 6, e17005 (2017).
[Crossref]

Meas. Sci. Technol. (1)

J. Hodgkinson and R. P. Tatam, “Optical gas sensing: a review,” Meas. Sci. Technol. 24(1), 012004 (2013).
[Crossref]

Nat. Commun. (1)

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

G. Brida, M. Genovese, and I. Ruo-Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227 (2010).
[Crossref]

New J. Phys. (3)

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Absorption spectroscopy at the ultimate quantum limit from single-photon states,” New J. Phys. 19(2), 023013 (2017).
[Crossref]

E. Knyazev, K.Yu. Spasibko, F. Ya. Khalili, and M. Chekhova, “Quantum tomography enhanced through parametric amplification,” New J. Phys. 20(1), 013005 (2018).
[Crossref]

M. Manceau, F. Ya. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys. 19(1), 013014 (2017).
[Crossref]

Opt. Comm. (1)

E. Jakeman and J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmission measurements,” Opt. Comm. 59(3), 219–223 (1986).
[Crossref]

Opt. Lett. (3)

P. H. Souto Ribeiro, C. Schwob, A. Maître, and C. Fabre, “Sub-shot-noise high-sensitivity spectroscopy with optical parametric oscillator twin beams,” Opt. Lett. 22(24), 1893–1895 (1997).
[Crossref]

M. Xiao, L.-A. Wu, and H. J. Kimble, “Detection of amplitude modulation with squeezed light for sensitivity beyond the shot-noise limit,” Opt. Lett. 13(6), 476–478 (1988).
[Crossref] [PubMed]

T. S. Iskhakov, V. C. Usenko, U. L. Andersen, R. Filip, M. V. Chekhova, and G. Leuchs, “Heralded source of bright multi-mode mesoscopic sub-poissonian light,” Opt. Lett. 41(10), 2149–2152 (2016).
[Crossref] [PubMed]

Phys. Rev. A (8)

U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasiprobability distributions,” Phys. Rev. A 48, 4598–4604 (1993).
[Crossref] [PubMed]

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

A. M. Marino, N. V. C. Trejo, and P. D. Lett, “Effect of losses on the performance of an su(1,1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

E. Brambilla, L. Caspani, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “High-sensitivity imaging with multi-mode twin beams,” Phys. Rev. A 77, 053807 (2008).
[Crossref]

I. N. Agafonov, M. V. Chekhova, and G. Leuchs, “Two-color bright squeezed vacuum,” Phys. Rev. A 82, 011801 (2010).
[Crossref]

P. R. Tapster, S. F. Seward, and J. G. Rarity, “Sub-shot-noise measurement of modulated absorption using parametric down-conversion,” Phys. Rev. A 44, 3266–3269 (1991).
[Crossref] [PubMed]

G. Adesso, F. Dell’Anno, S. De Siena, F. Illuminati, and L. A. M. Souza, “Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states,” Phys. Rev. A 79, 040305 (2009).
[Crossref]

Phys. Rev. D (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[Crossref]

Phys. Rev. Lett. (10)

E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
[Crossref] [PubMed]

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, “Quantum illumination with Gaussian states,” Phys. Rev. Lett. 101, 253601 (2008).
[Crossref] [PubMed]

E. D. Lopaeva, I. Ruo-Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, “Experimental realization of quantum illumination,” Phys. Rev. Lett. 110, 153603 (2013).
[Crossref] [PubMed]

S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114, 080503 (2015).
[Crossref] [PubMed]

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 Schemes of quantum imaging and sensing. (a) Classical differential scheme. The object with absorption 𝒜 is placed into one of the outputs of a 50% beamsplitter, and the absorption is retrieved through intensity subtraction measurement. (b) Conventional scheme of sub-shot-noise quantum imaging. The object is placed into one of the twin beams, emerging from a NOPA with the parametric gain r. Two photodetectors with equal quantum efficiencies η measure the numbers of photons in signal and idler beams, and the difference is calculated. (c) An alternative scheme, where only one beam is squeezed by a DOPA before probing the object, the other (much stronger) one being coherent. In schemes (b) and (c), additional DOPAs with gain R (shown by dashed lines) can be placed into both arms (b) or into the signal arm (c) to overcome the detrimental effect of the detection loss.
Fig. 2
Fig. 2 The quantum advantage Q of the ‘twin beams’ scheme, see Eqs. (17) and (59), for the case of no parametric amplification (R = 0) as a function of detection efficiency ηd for p 2 = 0(black, maximum efficiency), 10−5 (red), 10−4 (green), 10−3 (blue), and 10−2 (magenta). The object absorption is 𝒜 = 10−5 and the mean photon number is N = 107.
Fig. 3
Fig. 3 The quantum advantage Q of the ‘twin beams’ scheme in the presence of additional parametric amplification, see Eqs. (17) and (59), as a function of the gain R for different values of the detection efficiency: η = 0.99 (red), 0.9 (green), 0.5 (blue), and 0.1 (magenta). The mean photon number is N = 107 and the absorption is 𝒜 = 10−5.
Fig. 4
Fig. 4 The quantum advantage Q of the “squeezed coherent” scheme, see Eqs. (17) and (69), for the case of no parametric amplification (R = 0) as a function of detection efficiency ηd for p 2 = 0(black), 10−3 (blue), and 10−2 (magenta). The object absorption is 𝒜 = 10−5 and the mean photon number is N = 107.
Fig. 5
Fig. 5 The quantum advantage Q (top) and the corresponding numerically optimized squeeze factor r (bottom) for the “squeezed coherent” scheme, see Eqs. (17) and (69), as a function of the gain R for different values of the detection efficiency: η = 0.99 (red), 0.9 (green), 0.5 (blue), and 0.1 (magenta). The straight black line in the bottom plot shows the asymptotic (35). The mean photon number is N = 107 and the absorption is 𝒜 = 10−5.
Fig. 6
Fig. 6 Scheme of a possible experiment. NOPA1 generates beams in modes â1 and â2, and a weakly absorbing object is placed into mode â1. After the beam in mode â2 acquires a π/2 phase shift, both beams are overlapped on a 50% beamsplitter to form modes a ^ ± = ( a ^ 1 ± i a ^ 2 ) / 2 at the output. These two modes are amplified by NOPA2, which is equivalent to the phase-sensitive amplification of modes â1 and â2. In the end, photon numbers in modes â1 and â2 should be measured, which requires overlapping the modes on another 50% beamsplitter.
Fig. 7
Fig. 7 Polarization implementation of the experiment on sub-shot-noise imaging.
Fig. 8
Fig. 8 Measurement of absorption using twin-beams scheme with amplification before detection. The object under study is placed into the first beam, and both beams are simultaneously amplified before the direct detection. The state preparation loss is modeled by means of two identical beamsplitters with power transmissivity ηp, and the detection loss, by a similar beamsplitter ηd. The pump of the non-linear crystals is not depicted.
Fig. 9
Fig. 9 Measurement of absorption using a squeezed coherent state and amplification before detection. Since the reference beam is considered to have a very large amplitude in order to suppress the shot-noise in the corresponding channel, one can take into account only the noise of the signal channel. Here α is rescaled, see Eq. (63).

Equations (87)

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Δ 𝒜 CR = { n = 0 1 W ( n / 𝒜 ) [ W ( n / 𝒜 ) 𝒜 ] 2 } 1 / 2 .
𝒯 = 1 𝒜
1 𝒜 η = η 𝒯 ,
η = η p η d
Δ 𝒜 CR = { n = 0 1 W ( n / 𝒜 η ) [ W ( n / 𝒜 η ) 𝒜 ] 2 } 1 / 2 .
Δ 𝒜 CR = 1 η { n = 0 1 W ( n / 𝒜 η ) [ W ( n / 𝒜 η ) 𝒜 η ] 2 } 1 / 2 .
W ( n / 𝒜 ) = e ( 1 𝒜 η ) N 0 [ ( 1 𝒜 η ) N 0 ] n n ! .
Δ 𝒜 CRcoh = 𝒯 η d N ,
N = η p N 0
𝒜 1 , 1 η p , d 1 .
Δ 𝒜 CRcoh = 1 N .
W ( n / 𝒜 ) = N 0 ! n ! ( N 0 n ) ! 𝒜 η N 0 n ( 1 𝒜 η ) n ,
Δ 𝒜 CRFock = [ 𝒜 + ( 1 η ) 𝒯 ] 𝒯 η d N .
Δ 𝒜 CRFock = 𝒜 + 2 N ,
2 = p 2 + d 2
p , d 2 = 1 η p , d η p , d .
Q = Δ 𝒜 CRcoh Δ 𝒜 .
Q = 1 𝒜 + 2 .
𝒜 ˜ = n d 1 n d 2 G ,
G = n d 1 𝒜
N 1 .
e 2 R 1 .
( Δ 𝒜 ) 2 = 𝒜 2 + 𝒜 + 2 2 N .
( Δ 𝒜 ) 2 = 2 [ 𝒜 2 + 𝒜 + 2 ( p 2 + d 2 e 2 R ) N ] + 1 N 2 .
𝒜 ˜ = n d 1 k n d 2 G ,
( Δ 𝒜 ) 2 = 𝒜 + 2 2 N , Q = 1 𝒜 + 2 2 ,
( Δ 𝒜 ) 2 = 2 [ 𝒜 + 2 ( p 2 + d 2 e 2 R ) N ] + 1 N 2 , Q = 1 2 [ 𝒜 + 2 ( p 2 + d 2 e 2 R ) ] + 1 / N .
Q 1 2 .
Q = 1 2 ( 𝒜 + 2 p 2 ) + 1 / N ,
e 2 r 1 ,
( Δ 𝒜 ) 2 = e 2 | r | + 𝒜 + 2 N + e 4 | r | 8 N 2
Q = 1 e 2 | r | + 𝒜 + 2 + e 4 | r | 8 N .
e 2 | r | = ( 4 N ) 1 / 3 ,
( Δ 𝒜 ) 2 = 𝒜 + 2 N + 3 2 5 / 3 N 4 / 3 ,
Q = 1 𝒜 + 2 + 3 2 5 / 3 N 1 / 3 .
( Δ 𝒜 ) 2 = e 2 | r | + 𝒜 + p 2 + d 2 e 2 R N e 2 | r | 4 + e 4 | r | 8 R 8 ( N e 2 | r | 4 ) 2 .
e 2 | r | 4 N ,
( Δ 𝒜 ) 2 = e 2 | r | + 𝒜 + p 2 + d 2 e 2 R N + e 4 | r | 8 R 8 N 2 .
e 2 | r | = ( 4 N ) 1 / 3 e 8 R / 3 ,
( Δ 𝒜 ) 2 = 𝒜 + p 2 + d 2 e 2 R N + 3 e 8 R / 3 2 5 / 3 N 4 / 3 ,
Q = 1 𝒜 + p 2 + d 2 e 2 R + 3 e 8 R / 3 2 5 / 3 N 1 / 3 .
e 4 R 4 N .
H ^ = i γ a ^ 1 a ^ 2 + h . c . ,
H ^ = i Γ [ ( a ^ 1 ) 2 + ( a ^ 2 ) 2 ] + h . c . ,
H ^ = i Γ a ^ + a ^ + h . c . ,
a ^ ± = a ^ 1 ± i a ^ 2 2 .
d ^ 1 , 2 = η d c ^ 1 , 2 + 1 η d u ^ 1 , 2 = η d ( c ^ 1 , 2 + d u ^ 1 , 2 ) ,
n ^ d 1 , 2 = η d n ^ c 1 , 2 ,
( δ n ^ d 1 , 2 ) 2 = η d 2 [ ( δ n ^ c 1 , 2 ) 2 + d 2 n ^ c 1 , 2 ] ,
δ n ^ d 1 δ n ^ d 2 = η d 2 δ n ^ c 1 δ n ^ c 2 ,
n ^ c 1 , 2 = c ^ 1 , 2 c ^ 1 , 2 ,
n ^ d 1 , 2 = d ^ 1 , 2 d ^ 1 , 2 ,
δ n ^ c 1 , 2 = n ^ c 1 , 2 n ^ c 1 , 2 ,
δ n ^ d 1 , 2 = n ^ d 1 , 2 n ^ d 1 , 2 .
a ^ 1 , 2 = η p ( z ^ 1 , 2 cosh r + z ^ 2 , 1 sinh r ) + 1 η p v ^ 1 p , 2 p ,
b ^ 1 = 𝒯 a ^ 1 + 𝒜 v ^ obj = 𝒯 p ( z ^ 1 cosh r + z ^ 2 sinh r ) + 𝒜 p v ^ 1 ,
b ^ 2 = a ^ 2 = η p ( z ^ 2 cosh r + z ^ 1 sinh r ) + 1 η p v ^ 2 ,
𝒯 = 1 𝒜 ,
𝒯 p = η p 𝒯 , 𝒜 p = 1 η p 𝒯 ,
v ^ 1 = ( 1 η p ) 𝒯 v ^ 1 p + 𝒜 v ^ obj 𝒜 p ,
v ^ 2 = v ^ 2 p ,
c ^ 1 = b ^ 1 cosh R + b ^ 1 sinh R = 𝒯 p ( C ^ 1 + S ^ 1 ) + 𝒜 p ( v ^ 1 cosh R + v ^ 1 sinh R ) ,
c ^ 2 = b ^ 2 cosh R + b ^ 2 sinh R = η p ( C ^ 2 + S ^ 2 ) + 1 η p ( v ^ 2 cosh R + v ^ 2 sinh R ) ,
C ^ 1 , 2 = z ^ 1 , 2 cosh r cosh R + z ^ 2 , 1 sinh r sinh R ,
S ^ 1 , 2 = z ^ 1 , 2 cosh r sinh R + z ^ 2 , 1 sinh r cosh R .
n ^ c 1 = ( 𝒯 N + 1 2 ) cosh 2 R 1 2 ,
n ^ c 2 = ( N + 1 2 ) cosh 2 R 1 2 ,
( δ n ^ c 1 ) 2 = ( 𝒯 N + 1 2 ) 2 cosh 4 R 1 4 ,
( δ n ^ c 2 ) 2 = ( N + 1 2 ) 2 cosh 4 R 1 4 ,
δ n ^ c 1 δ n ^ c 2 = 𝒯 N ( N + η p ) cosh 4 R ,
N = η p sinh 2 r
( Δ 𝒜 ) 2 = ( δ n ^ d 1 δ n ^ d 2 ) 2 G 2 = 1 ( G / η d ) 2 [ ( δ n ^ c 1 ) 2 + ( δ n ^ c 2 ) 2 2 δ n ^ c 1 δ n ^ c 2 + 2 ( n ^ c 1 + n ^ c 2 ) ] .
G = n ^ d 1 𝒜 = η d N cosh 2 R
( Δ 𝒜 ) 2 = ( δ n ^ d 1 k δ n ^ d 2 ) 2 G 2 = ( δ n ^ d 1 ) 2 2 k δ n ^ d 1 δ n ^ d 2 + k 2 ( δ n ^ d 2 ) 2 G 2 .
k = δ n ^ d 1 δ n ^ d 2 ( δ n ^ d 2 ) 2 ,
( Δ 𝒜 ) 2 = 1 G 2 [ ( δ n ^ d 1 ) 2 δ n ^ d 1 δ n ^ d 2 2 ( δ n ^ d 2 ) 2 ] = 1 ( G / η d ) 2 [ ( δ n ^ c 1 ) 2 + 2 n c 1 δ n ^ c 1 δ n ^ c 2 2 ( δ n ^ c 2 ) 2 + 2 n c 1 ] .
( Δ 𝒜 ) 2 = 2 { 𝒜 2 + 1 N [ 𝒜 + 2 ( 1 η p ) 𝒯 + 2 ( 1 + 𝒯 ) e 2 R ] } + 1 N 2 ,
( Δ 𝒜 ) 2 = 2 𝒯 N [ 𝒜 + 2 ( 1 η p ) 𝒯 + 2 ( 1 + 𝒯 ) e 2 R ] + 1 N 2 [ 1 + 𝒯 2 2 + 2 𝒯 ( 1 𝒯 η p 2 ) ] .
a ^ = α + η p ( z ^ cosh r + z ^ sinh r ) + 1 η p v ^ p ,
α = N η p sinh 2 r
b ^ = 𝒯 a ^ + 𝒜 v ^ obj = 𝒯 α + 𝒯 p ( z ^ cosh r + z ^ sinh r ) + 𝒜 p v ^ ,
v ^ = ( 1 η p ) 𝒯 v ^ p + 𝒜 v ^ obj 𝒜 p
c ^ = b ^ cosh R + b ^ sinh R = 𝒯 α e R + 𝒯 p [ z ^ cosh ( r + R ) + z ^ sinh ( r + R ) ] + 𝒜 p ( v ^ cosh R + v ^ sinh R ) .
n ^ c = 𝒯 α 2 e 2 R + 𝒯 p sinh 2 ( r + R ) + 𝒜 p sinh 2 R ,
( δ n ^ c ) 2 = 𝒯 𝒯 p α 2 e 2 r + 4 R + 𝒯 p 2 2 sinh 2 2 ( r + R ) + 𝒯 𝒜 p α 2 e 4 R + 𝒯 p 𝒜 p sinh 2 ( r + 2 R ) + 𝒜 p 2 2 sinh 2 2 R .
( Δ 𝒜 ) 2 = ( δ n ^ d ) 2 G 2 = ( δ n ^ c ) 2 + 2 n ^ c ( G / η d ) 2 ,
G = n ^ d 𝒜 = η d [ α 2 e 2 R η p sinh 2 ( r + R ) + η p sinh 2 R ]

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