Abstract

Collective atomic excitation can be realized by the Raman scattering. Such a photon-atom interface can form an SU(1,1)-typed atom-light hybrid interferometer, where the atomic Raman amplification processes take the place of the beam splitting elements in a traditional Mach-Zehnder interferometer. We numerically calculate the phase sensitivities and the signal-to-noise ratios (SNRs) of this interferometer with the method of homodyne detection and intensity detection, and give their differences of the optimal phase points to realize the best phase sensitivities and the maximal SNRs from these two detection methods. The difference of the effects of loss of light field and atomic decoherence on measure precision is analyzed.

© 2016 Optical Society of America

1. Introduction

Quantum enhanced metrology is the use of quantum techniques to improve measurement precision than purely classical approaches, which has been received a lot of attention in recent years [1–11]. Interferometers can provide the most precise measurements. Recently, physicists with the advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) observed the gravitational waves [12]. The Mach–Zehnder interferometer (MZI) and its variants have been used as a generic model to realize precise measurement of phase. In order to avoid the vacuum fluctuations entering the unused port, Caves [3] suggested to replace the vacuum fluctuations with the squeezed-vacuum light to reach a sub-shot-noise sensitivity. Xiao et al. [13] and Grangier et al. [14] have demonstrated the experimental results beyond the standard quantum limit (SQL) δφ=1/N with ϕ phase shift and N number of photons or other bosons. Due to surpassing the SQL and reaching the Heisenberg limit (HL) δϕ = 1/N, it will lead to potential applications in high precision measurements. Therefore, many theoretical proposals and experimental techniques are developed to improve the sensitivity [15–17]. When the probe states are made of correlated states, such as the NOON states of the form (|Na|0b+eiϕN|0a|Nb)/2, the phase-sensing measurements can reach the HL [18, 19]. But, high-N NOON states is very hard to synthesize. In the presence of realistic imperfections and noise, the ultimate precision limit in noisy quantum-enhanced metrology was also studied [20–27].

However, most of the current atomic and optical interferometers are made of linear devices such as beam splitters and phase shifters. In 1986, Yurke et al. [28] introduced a new type of interferometer where two nonlinear beam splitters take the place of two linear beam splitters (BSs) in the traditional MZI. It is also called the SU(1,1) interferometer because it is described by the SU(1,1) group, as opposed to SU(2) for BSs. The detailed quantum statistics of the two-mode SU(1,1) interferometer was studied by Leonhardt [29]. SU(1,1) phase states also have been studied theoretically in quantum measurements for phase-shift estimation [30, 31]. An improved theoretical scheme of the SU(1,1) optical interferometer was presented by Plick et al. [32] who proposed to inject a strong coherent beam to “boost” the photon number. Experimental realization of this SU(1,1) optical interferometer was reported by different groups [33, 34]. The noise performance of this interferometer was analyzed [11, 35] and under the same phase-sensing intensity condition the improvement of 4.1 dB in signal-to-noise ratio was observed [36]. By contrast, an SU(1,1) atomic interferometer also has been experimentally realized with Bose-Einstein Condensates [37–40]. Gabbrielli et al. [40] realized a nonlinear three-mode SU(1,1) atomic interferometer, where the analogy of optical down conversion, the basic ingredient of SU(1,1) interferometry, is created with ultracold atoms.

Collective atomic excitation due to its potential applications for quantum information processing has attracted a great deal of interest [41–43]. Collective atomic excitation can be realized by the Raman scattering. Initially prepared collective atomic excitation can be used to enhance the second Raman scattering [44–46]. Subsequently, we proposed another scheme to enhance the Raman scattering using the correlation-enhanced mechanism [47]. That is, by injecting a seeded light field which is correlated with the initially prepared collective atomic excitation, the Raman scattering can be enhanced greatly, which was also realized in experiment recently [48]. Such a photon-atom interface can form an SU(1,1)-typed atom-light hybrid interferometer [49], where the atomic Raman amplification processes take the place of the beam splitting elements in a traditional MZI [28]. That is to say, the atom-light hybrid interferometer is composed of two Raman amplification processes (see Fig. 1). The first nonlinear process generates the correlated optical and atomic waves in the two arms and they are decorrelated by the second nonlinear process. Different from all-optical or all-atomic interferometers, two features are evident in this scheme: (1) we can probe the atomic phases with optical interferometric techniques; (2) the main loss mechanism, one arm is from optical loss another arm is from the collisional dephasing. The atomic phase can be adjusted by magnetic field or Stark shifts. The atom-light correlations have also been used to enhance the sensitivity of atom interferometry [50–53]. This SU(1,1)-type hybrid correlated interferometer was also realized in the circuit quantum electrodynamics system [54], which provides a different method for basic measurement using the hybrid interferometers.

 figure: Fig. 1

Fig. 1 (a) The intermode correlation between the Stokes field a^1 and the atomic excitation b^1 is generated by spontaneous Raman process. a^0 is the initial input light field. b^0 is in vacuum or an initial atomic collective excitation which can be prepared by another Raman process or electromagnetically induced transparency process. (b) During the delay time τ, the Stokes field a^1 will be subject to the photon loss and evolute to a^1 and the collective excitation b^1 will undergo the collisional dephasing to b^1. A fictitious beam splitter (BS) is introduced to mimic the loss of photons into the environment. V^ is the vacuum. (c) After the delay time τ, the light field a^1 and its correlated atomic excitation b^1 are used as initial seed for another enhanced Raman process. (d)–(f) The corresponding energy-level diagrams of different processes are shown.

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In this paper, we calculate the phase sensitivities and the SNRs using the homodyne detection and the intensity detection. The differences of the optimal phase points to realize the best phase sensitivities and the maximal SNRs from two different detection methods are given. The loss of light field and atomic decoherence degrade the measure precision, and the differences between the two loss processes are compared. Although the two detection methods have different behaviors of the linear correlation coefficients (LCCs), both of their correlations eventually tend to be strong with increasing of losses, which can be used to explain the measure precision reduction.

2. The Model of atom-light hybrid interferometers

In this section, we review the different processes of the atom-light interferometer [49, 55] as shown in Figs. 1(a)–1(c), where two Raman systems replaced the BSs in the traditional MZI. Considering a three-level Lambda-shaped atom system as shown in Fig. 1(d), the Raman scattering process is described by the following pair of coupled equations [56]:

a^(t)t=ηAPb^(t),b^(t)t=ηAPa^(t),
where η is the coupling constant, and AP is the amplitude of the pump field. For alkali-metal atoms, the excited hyperfine levels are separated by few hundred megahertz. For the Raman process in Ref. [49], the detuning is about 1 – 3 GHz and is larger than the energy level splitting of excited state, then the excited states can be adiabatically eliminated. The solution of above equation is
a^(t)=u(t)a^(0)+v(t)b^(0),b^(t)=u(t)b^(0)+v(t)a^(0),
where u(t) = cosh(g), v(t) = e sinh(g), g = |ηAP| t, eiθ=(AP/AP*)1/2, and t is the time duration of pump field EP. We use different subscripts to differentiate the two processes, where 1 denotes the first Raman process (RP1) and 2 denotes the second Raman process (RP2). t1 and t2 are the durations of the pump field EP1 and EP2, respectively.

After the first Raman process of the interferometer, the Stokes field a^1 and the atomic excitation b^1 are generated as shown in Fig. 1(a). Then after a small delay time τ, the second Raman process of the interferometer takes place which is used as beams combination as shown in Fig. 1(c). During the small delay time τ shown in Fig. 1(b), the Stokes field a^1 will be subject to the photon loss and evolute to a^1. A fictitious BS is introduced to mimic the loss of photons into the environment, then the light field a^1 is given by

a^1=Ta^1(t1)eiϕ+RV^,
where T and R are the transmission and reflectance coefficients with T + R = 1, and V^ is in vacuum. The collective excitation b^1 will also undergo the collisional dephasing described by the factor eΓτ, then b^1 is
b^1=b^1(t1)eΓτ+F^,
where F^=0τeΓ(τt)f^(t)dt, and f^(t) is the quantum statistical Langevin operator describing the collision-induced fluctuation, and obeys f^(t)f^(t)=2Γδ(tt) and f^(t)f^(t)=0. Then F^F^=1e2Γτ guarantees the consistency of the operator properties of b^1. Here Γ describes the decoherence in the ground states which is much slower than the decay of the excited states. In the Raman process of hot atoms, the decoherence is mainly from the collisional dephasing. If using laser cooling technique for atoms, the collisional dephasing will be reduced.

Using Eqs. (2)(4), the generated Stokes field a^2 and collective atomic excitation b^2 can be worked out:

a^2(t2)=U1a^1(0)+V1b^1(0)+Ru2V^+v2F^,
b^2(t2)=eiϕ[U2b^1(0)+V2a^1(0)]+Rv2V^+u2F^,
where
U1=Tu1u2eiϕ+eΓτv1*v2,V1=Tv1u2eiϕ+eΓτu1*v2U2=eΓτu1u2eiϕ+Tv1*v2,V2=eΓτv1u2eiϕ+Tu1*v2.

Next, we use the above results to calculate the phase sensitivity and the SNR, and analyze and compare the conditions to obtain optimal phase sensitivity and the maximal SNR.

3. Phase sensitivity and SNR

Phase can be estimated but cannot be measured because there is not a Hermitian operator corresponding to a quantum phase [57]. In phase precision measurement, the estimation of a phase shift can be done by choosing an observable, and the the relationship between the observable and the phase is known. The mean-square error in parameter ϕ is then given by the error propagation formula [18]:

Δϕ=(ΔO^)21/2|O^/ϕ|,
where O^ is the measurable operator and (ΔO^)2=O^2O^2. The precision of the phase shift measurement is not the only parameter of concern. We also need consider the SNR [11,54,58], which is given by
SNR=O^(ΔO^)21/2.

In current optical measurement of phase sensitivity, the homodyne detection [54, 59, 60] and the intensity detection [32, 35] are often used. That is, the observables are the amplitude quadrature operator x^a2=(a^2+a^2)/2 and the number operator n^a2=a^2a^2. For the balanced situation that is g1 = g2 = g, and θ2θ1 = π. Firstly, we do not consider the effect of loss on the generated Stokes field a^2 and atomic collective excitation b^2. That is, R = 0 and Γτ = 0, it reduced to the ideal lossless case and we have U1=U2=U=[cosh2geiϕsinh2g], V1=V2=V=12sinh2g[eiϕ1]eiθ1, where |U|2|V|2=1.

3.1. Homodyne detection

For a coherent light |α⟩ (α = |α| eiθα, Nα = α|2) as the phase-sensing field, using the amplitude quadrature operator x^a2 as the detected variable the phase sensitivity and the SNR are given by

ΔϕHD=(Δx^a2)21/2Nαcosh2g|sin(ϕ+θα)|,
SNRHD=Nα[cosh2gcos(ϕ+θα)sinh2gcos(θα)](Δx^a2)21/2,
with
(Δx^a2)2=14[cosh2(2g)sinh2(2g)cosϕ],
where the subscript HD denotes the homodyne detection. The phase sensitivity ΔϕHD and the SNRHD depend on ϕ and θα, when g and α take a certain values. From Eqs. (10) and (11), both the ΔϕHD and the SNRHD need that the term (Δx^a2)2 is minimal, which can be realized at ϕ = 0 and (Δx^a2)2=1/4 [61].

When ϕ = 0 and θα = π/2, we obtain the optimal phase sensitivity and the worst SNR:

ΔϕHD=1Na12cosh2g,
SNRHD=0.

But when ϕ = 0 and θα = 0 or π, the maximal SNRHD is given by

SNRHD=2Nα,
and the sensitivity ΔϕHD is divergent. The phase sensitivity ΔϕHD and the SNRHD of above two different cases are shown in Figs. 2(a) and 2(b), respectively. We find that at the optimal point ϕ = 0 and θα = π/2 the sensitivity is high (i.e. small Δϕ) and can beat the SQL but the SNRHD is low. At the optimal point ϕ = 0 and θα = 0 the SNRHD is high, but the sensitivity is low. Ideally, of course, we would like high sensitivity ΔϕHD and high SNRHD at the same optimal point.

 figure: Fig. 2

Fig. 2 The phase sensitivity ΔϕHD and the SNRHD versus the phase shift ϕ using the method of homodyne detection with (a) θα = π/2; (b) θα = 0. Parameters: g = 2, |α| = 10.

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3.2. Intensity detection

If we use n^a2(=a^2a^2) as the detection variable, for a coherent light |α(α=|α|eiθα,Nα=|α|2) as the phase-sensing field, the phase sensitivity and the SNR are given by

ΔϕID=(Δn^a2)21/22(Nα+1)sinh2(2g)|sinϕ|,
SNRID=1(Δn^a2)21/2[Nα|cosh2gsinh2geiϕ|2+12sinh2(2g)(1cosϕ)],
where the subscript ID denotes the intensity detection, and
(Δn^a2)2=Nα|cosh2gsinh2geiϕ|4+12(1+Nα)×sinh2(2g)|cosh2gsinh2geiϕ|2(1cosϕ).

Different from the homodyne detection, the phase sensitivity ΔϕID and the SNRID only depend on ϕ for given g and Nα. The phase sensitivity ΔϕID and the SNRID as a function of phase shift ϕ are shown in Figs. 3(a) and 3(b), respectively. The maximal SNRID(=Nα) is obtained at ϕ = 0, but the best phase sensitivity ΔϕID is not at ϕ = 0 because the slope n^a2/ϕ is 0 (sin ϕ = 0). Using ∂ΔϕID/∂ϕ = 0 we obtain

(1+cos2ϕ)2sinh2(2g)[(2Nα+1)cosh2(2g)+Nα]cosϕ[4Nα(2cosh2(2g)+sinh4(2g))+(1+Nα)sinh24g]=0.

 figure: Fig. 3

Fig. 3 (a) Δna2, |⟨∂⟨na2⟩/∂ϕ⟩|, and the phase sensitivity ΔϕID; (b) ⟨na2⟩, Δna2 and the SNRID versus the phase shift ϕ using the method of intensity detection. Parameters: g = 2, |α| = 10.

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Under the condition of g = 2 and Nα = 100, the best phase sensitivity ΔϕID is obtained at ϕ ≃ 0.062, which is also demonstrated in Fig. 3(a). At ϕ ≃ 0.062 the slope |n^a2/ϕ| is very small, as well in Fig. 3(b) at ϕ = 0 the intensity of the signal n^a2 low. But these two phases are the best measurement points for phase sensitivity and SNR, respectively. Because the noise is also low, the noise (Δn^a2)2 plays a dominant role. The homodyne detection has a better optimal phase sensitivity than the intensity detection as shown in Fig. 4(a). The relation of maximal SNR between two detection methods is SNRHD = 2SNRID.

 figure: Fig. 4

Fig. 4 (a) The optimal phase sensitivities Δϕ and (b) the maximal SNR versus the phase-sensing probe number nph. The optimal phase sensitivities ΔϕHD and ΔϕID are obtained at ϕ = 0 and ϕ = 0.062, respectively. The maximal SNRs are obtained at ϕ = 0 and θα = 0. Parameter: g = 2.

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3.3. Losses case

In the presence of loss of light field and atomic decoherence, the precision of the sensitivity and the SNR will be reduced [55, 60]. The intensities and slopes of the output amplitude quadrature operator x^a2 and the number operator n^a2=a^2a^2 are respectively given by

x^a2=NαeΓτsinh2gcosθα+TNαcosh2gcos(ϕ+θα),
n^a2=(Tcosh4g+e2Γτsinh4g)Nα+sinh2g(1e2Γτ)+14sinh2(2g)(T+e2Γτ)12TeΓτ(Nα+1)sinh2(2g)cosϕ,
and
|x^a2ϕ|=TNcosh2g|sin(ϕ+θα)|,
|n^aϕ|=12TeΓτ(Nα+1)sinh2(2g)|sin(ϕ)|.

We note that only the slope |x^a2/ϕ| does not depend on the collisional dephasing rate Γ. Therefore, the adverse effect on phase sensitivity from homodyne detection is smaller than that from intensity detection under the same uncertainties of measurement operators.

The uncertainties of the output amplitude quadrature operator x^a2 and the number operator n^a2 are given by

(Δx^a2)2=14[sinh2(2g)(T2TeΓτcosϕ)+2e2Γτsinh4g+cosh(2g)],
(Δn^a2)2=|Ub|4Nα+|UbVb|2(1+Nα)+Rcosh2g(|Ub|2Nα+|Vb|2)+sinh2g[|Ub|2(1+Nα)+Rcosh2g](1e2Γτ),
where
|Ub|2=(Tcosh2g+eΓτsinh2g)22TeΓτsinh2gcosh2g(1+cosϕ),
|Vb|2=12sinh2(2g)(T+e2Γτ2TeΓτcosϕ),
where the subscript b denotes the balanced condition with consideration of losses case.

Here, the phase-sensing field is not the input field as in the traditional MZI, but the amplified field inside the interferometer. Although the phase shift is generated on the light field, the light field and the atomic collective excitation are quantized. The phase-sensing probe number includes not only the photon number a^1(t1)a^1(t1) but also the atomic collective excitation number b^1(t1)b^1(t1), which is given by

nph=Nαcosh(2g)+2sinh2g.

Now, we compare the optimal sensitivities with SQL ( 1/nph). The phase sensitivities Δϕ as a function of the phase-sensing probe number nph is shown in Fig. 4(a). The thick solid line is the SQL. The thin solid line and dotted line are sensitivities ΔϕHD from homodyne detection with and without losses cases, respectively. As well the dashed and dash-dotted lines are sensitivities ΔϕID from intensity detection with and without losses cases, respectively. From Fig. 4(a), it shows that the best phase sensitivities ΔϕID are larger than ΔϕHD under the same condition. In the presence of the loss and collisional dephasing (T = 0.8, Γτ = 0.1), the phase sensitivities ΔϕHD and ΔϕID can beat the SQL under the balanced situation, which is very important for phase sensitivity measurement. The maximal SNRs as a function of the phase-sensing probe number nph is shown in Fig. 4(b). It shows that the value of SNR are reduced largely in the presence of the loss of light field and atomic decoherence.

Next section, we will present that the effects of the light field loss and atomic decoherence on measure precision can be explained from the break of intermode decorrelation conditions.

4. The correlations of atom-light hybrid interferometer

In this section, we use the above results to calculate the intermode correlations of the different Raman amplification processes of the atom-light interferometer as shown in Figs. 1(a)–1(c) [49]. We also study the effects of the loss of light field and the dephasing of atomic excitation on the correlation. The intermode correlation of light and atomic collective excitation can be described by the LCC, which is defined as [62]

J(A^,B^)=cov(A^,B^)(ΔA^)21/2(ΔB^)21/2,
where cov(A^,B^)=(A^B^+B^A^)/2A^B^ is the covariance of two-mode field and (ΔA^)2=A^2A^2, (ΔB^)2=B^2B^2.

The respective quadrature operators of the light and atomic excitation are x^a=(a^+a^)/2, y^a=(a^a^)/2i, x^b=(b^+b^)/2, and y^b=(b^b^)/2i. After the first Raman scattering process, the intermode correlations between the light field mode and the atomic mode are generated. We start by injecting a coherent state |α⟩ in mode a^, and a vacuum state in mode b^, the LCC of quadratures are given by

Jx1(x^a1,x^b1)=cosθ1tanh(2g),
Jy1(y^a1,y^b1)=cosθ1tanh(2g),
and the LCC of number operators n^a1[=a^(t1)a^(t1)] and n^b1[=b^(t1)b^(t1)] is given by
Jn1(n^a1,n^b1)=(1+2|a|2)[4coth2(2g)(|a|2+|a|4)+1]1/2.

From Eqs. (30)(32), the quadrature correlation LCCs Jx1(x^a1,x^b1) and Jy1(y^a1,y^b1) are independent on the input coherent state which is different from the number correlation LCC Jn1(n^a1,n^b1). Under θ1π/2, the LCCs Jx1 and Jy1 are opposite and not zero, which shows the correlation exists. Due to their opposite intermode correlations, the squeezing of quantum fluctuations is in a superposition of the two-modes, i.e., X^=(x^a+x^b)/2, Y^=(y^a+y^b)/2 and [X^,Y^]=i/2 [62].

From Eq. (32) the number correlation LCC Jn1 is always positive so long as g ≠ 0. If α = 0, namely vacuum state input, then Jn1(n^a1,n^b1)=1, this maximal value shows the strong intermode correlation and such states in optical fields are usually called “twin beams”. For this vacuum state input case, the state of atomic mode and light mode is similar to the two-mode squeezed vacuum state.

After the second Raman process of the interferometer, the LCC of quadratures Jx2(x^a2,x^b2) using the generated Stokes field a^2 and atomic collective excitation b^2 can be worked out

Jx2(x^a2,x^b2)=cov(x^a2,x^b2)(Δx^a2)21/2(Δx^b2)21/2,
where
cov(x^a2,x^b2)=14Re[eiϕ(V1U2+U1V2)+u2v2(R+1e2Γτ)],(Δx^a2)2=14[|U1|2+|V1|2+R|u2|2+|v2|2(1e2Γτ)],(Δx^b2)2=14[|U2|2+|V2|2+R|u2|2+|v2|2(1e2Γτ)].

The LCC of number operators Jn2(n^a2,n^b2) can also be worked out

Jn2(n^a2,n^b2)=cov(n^a2,n^b2)(Δn^a2)21/2(Δn^b2)21/2,
where
cov(n^a2,n^b2)=|U1V1|2|α|2+(1+|α|2)Re[U1*U2V1V2*]+R[Re[eiϕU2V1u2*v2*]+|α|2Re[eiϕU1V2u2*v2*]]+[R|u2v2|2+(1+|α|2)Re[eiϕU1*V2*u2v2]](1e2Γτ),
(Δn^a2)2=|U1|4|α|2+|U1V1|2(1+|α|2)+R|u2|2(|V1|2+|U1|2|α|2)+|v2|2[|U1|2(|α|2+1)+R|u2|2](1e2Γτ),
(Δn^b2)2=|V2|4|α|2+|U2V2|2(1+|α|2)+R|v2|2(|U2|2+|V2|2|α|2)+|u2|2[|V2|2(1+|α|2)+R|v2|2](1e2Γτ).

First, we do not consider the effect of loss on the generated Stokes field a^2 and atomic collective excitation b^2. Under this ideal and balanced condition, the LCCs of quadratures and number operators are respectively given by

Jx2(x^a2,x^b2)=2Re[VUeiϕ]|U|2+|V|2=sinh(2g)cosh2(2g)sinh2(2g)cosϕ[cosh2gcos(θ1+3ϕ)+sinh2gcos(θ1+ϕ)cosh(2g)cos(θ1+2ϕ)]
and
Jn2(n^a2,n^b2)=|UV(1+2|α|2)|(UV)1/2=(1+2|α|2)×[4[1+sinh2(2g)(1cosϕ)]2(|α|2+|α|4)[1+sinh2(2g)(1cosϕ)]21+1]1/2,
where U¯=|U|2|α|2+|V|2(|α|2+1), V¯=|V|2|α|2+|U|2(|α|2+1). When the phase shift ϕ is 0, V reduces to 0, then the LCCs Jx2(x^a2,x^b2) and Jn2(n^a2,n^b2) simplify to 0. Under this condition, the RP2 will “undo” what the RP1 did. When the phase shift ϕ is π, the LCCs Jx2(x^a2,x^b2) and Jn2(n^a2,n^b2) are respectively given by
Jx2(x^a2,x^b2)=tanh(4g)cos(θ1),
Jn2(n^a2,n^b2)=1+2|α|24coth2(2g1)(|α|2+|α|4)+1=Jn1(n^a1,n^b1).

The LCCs Jx2(x^a2,x^b2) and Jn2(n^a2,n^b2) as a function of the phase shift ϕ are shown in Fig. 5. Due to the LCC Jx2(x^a2,x^b2) is dependent on θ1, the intermode correlation coefficients Jx2(x^a2,x^b2) ranges between −1 and 0 when θ1 = 0. The corresponding LCC Jn2(n^a2,n^b2) is positive, and ranges between 0 and 1.

 figure: Fig. 5

Fig. 5 The linear correlation coefficients (a) Jx2; (b) Jn2 as a function of the phase shift ϕ for lossless case. Parameters: θ1 = 0, g = 2, |α| = 10.

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This decorrelation point (ϕ = 0, Jx2(x^a2,x^b2)=0) is very important for atom-light hybrid interferometer using the homodyne detection [55]. At this point (ϕ = 0) the noise of output field [(Δx^a2)2=1/4] is the same as that of input field and it is the lowest in our scheme as shown in Fig. 2. The optimal phase sensitivity ΔϕHD and the maximal SNRHD are obtained at this point with different θα. The LCC Jx2 as a function of the transmission rate T and the collisional dephasing rates Γτ are shown in Fig. 6. With the decrease of the transmission rate T or the increase of Γτ, the LCCs Jx2 tends to −1. Due to large loss (T small) or large decoherence (Γτ large) one arm inside the interferomter (the optical field a^1 or the atomic excitation b^1) is vanished, the decorrelation condition does not exist. Therefore, the serious break of decorrelation condition will degrade the sensitivity in the phase precision measurement.

 figure: Fig. 6

Fig. 6 The linear correlation coefficients Jx2 as a function of (a) the transmission rate T; (b) the collisional rate Γτ. Parameters: g = 2, |α| = 10, θα = π/2 and ϕ = 0.

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This decorrelation point (ϕ = 0, Jn2(n^a2,n^b2)=0) plays a significant role on precision phase estimation for the intensity detection. The low noise is dominant in realizing the optimal sensitivity and the maximal SNR. At this point (ϕ = 0), we can obtain the maximal SNRID. However, the slope |n^a2/ϕ| is equal to 0 at this decorrelation point as shown in Fig. 3(a). In Fig. 3(b) at nearby the decorrelation point, the noise is amplifed a little and the optimal phase sensitivity ΔϕID is obtained. With the decrease of the transmission rate T or the increase of Γτ, the LCCs Jn2 decrease at first, then revive quickly, and finally increase to 1 as shown in Fig. 7. Although the two detection methods have different behaviors of LCCs, both of their correlations eventually tend to be strong with increasing of loss. Therefore, the serious break of decorrelation condition will degrade the sensitivity in the phase precision measurement.

 figure: Fig. 7

Fig. 7 The linear correlation coefficients Jn2 as a function of (a) the transmission rate T; (b) the collisional dephasing rate Γτ, where g = 2, |α| = 10, and θ2θ1 = π, and ϕ = 0.062.

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5. Conclusions

We present the phase sensitivities and the SNRs of the atom-light hybrid interferometer with the method of homodyne detection and intensity detection, respectively. Using the homodyne detection, for given input intensity Nα and coupling strength g the optimal sensitivity ΔϕHD and the maximal SNRHD is not only dependent on the phase shift ϕ but also dependent on the phase θα of the input coherent state. At the optimal point ϕ = 0, we obtain that the sensitivity is low (i.e. large ΔϕHD) when the SNRHD is high and vice versa, due to different θα.

Using the intensity detection, the optimal sensitivity ΔϕID and the maximal SNRID is only dependent on the phase shift ϕ for given input intensity Nα and coupling strength g. Under the balanced condition, the maximal SNRID is obtained when the phase is 0 and the optimal phase sensitivity ΔϕID is obtained when the phase is nearby 0. The homodyne detection has a better optimal phase sensitivity than the intensity detection, and the relation of maximal SNR between two detection methods is SNRHD = 2SNRID.

Although the two detection methods have different behaviors of LCCs, both of their correlations eventually tend to be strong with increasing of losses. The loss of light field and atomic decoherence degrade the sensitivities and the SNRs of phase measurement, which can be explained from the break of decorrelation conditions of the optimal points.

Acknowledgments

This work is supported by the National Key Research Program of China under Grant number 2016YFA0302000, the National Natural Science Foundation of China (grant numbers 11474095, 11274118, 11234003, 91536114 and 11129402) and Supported by Innovation Program of Shanghai Municipal Education Commission (grant number 13ZZ036), and the Fundamental Research Funds for the Central Universities.

References and links

1. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).

2. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Springer Science and Bussiness Media, 2011). [CrossRef]  

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

4. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994). [CrossRef]   [PubMed]  

5. S. L. Braunstein, C. M. Caves, and G. J. Milburn, “Generalized uncertainty relations: theory, examples, and Lorentz invariance,” Ann. Phys. 247, 135 (1996). [CrossRef]  

6. H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325 (2002). [CrossRef]  

7. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006). [CrossRef]   [PubMed]  

8. M. Zwierz, C. A. Pérez-Delgado, and P. Kok, “General optimality of the Heisenberg limit for quantum metrology,” Phys. Rev. Lett. 105, 180402 (2010). [CrossRef]  

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

10. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222 (2011). [CrossRef]  

11. Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012). [CrossRef]  

12. B. P. Abbott et al., (LIGO Scientific Collaboration and Virgo Collaboration), “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016). [CrossRef]  

13. M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the short noise limit,” Phys. Rev. Lett. 59, 278 (1987). [CrossRef]   [PubMed]  

14. P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153 (1987). [CrossRef]   [PubMed]  

15. G. Tóth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A 47, 424006 (2014). [CrossRef]  

16. L. Pezzè and A. Smerzi, in Proceedings of the international school of physics “Enrico Fermi”, Course CLXXXVIII Atom Interferometry edited by G. Tino and M. Kasevich, eds. (Società Italiana di Fisica and IOS, 2014).

17. R. D. Dobrzanski, M. Jarzyna, and J. Kolodyński, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345 (2015). [CrossRef]  

18. J. P. Dowling, “Quantum optical metrology–the lowdown on high-N00N states,” Contemp. Phys. 49, 125 (2008). [CrossRef]  

19. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000). [CrossRef]   [PubMed]  

20. R. D. Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009). [CrossRef]  

21. B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406 (2011). [CrossRef]  

22. R. Demkowicz-Dobrzanski, J. Kolodyński, and M. Gutǎ, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012). [CrossRef]   [PubMed]  

23. D. W. Berry, Michael J. W. Hall, and Howard M. Wiseman, “Stochastic Heisenberg limit: optimal estimation of a fluctuating phase,” Phys. Rev. Lett. 111, 113601 (2013). [CrossRef]   [PubMed]  

24. R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013). [CrossRef]   [PubMed]  

25. W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, “Improved quantum metrology using quantum error correction,” Phys. Rev. Lett. 112, 080801 (2014). [CrossRef]  

26. E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, “Quantum error correction for metrology,” Phys. Rev. Lett. 112, 150802 (2014). [CrossRef]   [PubMed]  

27. S. Alipour, M. Mehboudi, and A. T. Rezakhani, “Quantum metrology in open systems: dissipative Cramér-Rao bound,” Phys. Rev. Lett. 112, 120405 (2014). [CrossRef]  

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

29. U. Leonhardt, “Quantum statistics of a two-mode SU(1,1) interferometer,” Phys. Rev. A 49, 1231 (1994). [CrossRef]   [PubMed]  

30. A. Vourdas, “SU(2) and SU(1,1) phase states,” Phys. Rev. A 41, 1653 (1990). [CrossRef]   [PubMed]  

31. B. C. Sanders, G. J. Milburn, and Z. Zhang, “Optimal quantum measurements for phase-shift estimation in optical interferometry,” J. Mod. Opt. 44, 1309 (1997).

32. W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted sub-shot noise quantum interferometry,” New J. Phys. 12, 083014 (2010). [CrossRef]  

33. 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]  

34. T. S. Horrom, B. E. Anderson, P. Gupta, and P. D. Lett, “SU(1,1) interferometry via four-wave mixing in Rb,” in 45th Winter Colloquium on the Physics of Quantum Electronics (PQE, 2015).

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

36. 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]  

37. C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010). [CrossRef]   [PubMed]  

38. D. Linnemann, “Realization of an SU(1,1) interferometer with spinor Bose-Einstein condensates,” Master thesis, University of Heidelberg, 2013.

39. J. Peise, B. Lücke, L. Pezzè, F. Deuretzbacher, W. Ertmer, J. Arlt, A. Smerzi, L. Santos, and C. Klempt, “Interaction-free measurements by quantum Zeno stabilization of ultracold atoms,” Nat. Commun. 6, 6811 (2015). [CrossRef]   [PubMed]  

40. M. Gabbrielli, L. Pezzè, and A. Smerzi, “Spin-mixing interferometry with Bose-Einstein condensates,” Phys. Rev. Lett. 115, 163002 (2015). [CrossRef]   [PubMed]  

41. L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413 (2001). [CrossRef]   [PubMed]  

42. K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041 (2010). [CrossRef]  

43. L. Li, Y. O. Dudin, and A. Kuzmich, “Entanglement between light and an optical atomic excitation,” Nature 498, 466 (2013). [CrossRef]   [PubMed]  

44. L. Q. Chen, G. W. Zhang, C.-H. Yuan, J. Jing, Z. Y. Ou, and W. Zhang, “Enhanced Raman scattering by spatially distributed atomic coherence,” Appl. Phys. Lett. 95, 041115 (2009). [CrossRef]  

45. L. Q. Chen, G. W. Zhang, C.-L. Bian, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “Observation of the Rabi oscillation of light driven by an atomic spin wave,” Phys. Rev. Lett. 105, 133603 (2010). [CrossRef]  

46. C.-H. Yuan, L. Q. Chen, J. Jing, Z. Y. Ou, and W. Zhang, “Coherently enhanced Raman scattering in atomic vapor,” Phys. Rev. A 82, 013817 (2010). [CrossRef]  

47. C.-H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Correlation-enhanced phase-sensitive Raman scattering in atomic vapors,” Phys. Rev. A 87, 053835 (2013). [CrossRef]  

48. B. Chen, C. Qiu, L. Q. Chen, K. Zhang, J. Guo, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “Phase sensitive Raman process with correlated seeds,” Appl. Phys. Lett. 106, 111103 (2015). [CrossRef]  

49. B. Chen, C. Qiu, S. Chen, J. Guo, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015). [CrossRef]   [PubMed]  

50. J. Jacobson, G. Björk, and Y. Yamamoto, “Quantum limit for the atom-light interferometer,” Appl. Phys. B 60, 187 (1995). [CrossRef]  

51. S. A. Haine, “Information-recycling beam splitters for quantum enhanced atom interferometry,” Phys. Rev. Lett. 110, 053002 (2013). [CrossRef]   [PubMed]  

52. S. S. Szigeti, B. Tonekaboni, W. Y. S. Lau, S. N. Hood, and S. A. Haine, “Squeezed-light-enhanced atom interferometry below the standard quantum limit,” Phys. Rev. A 90, 063630 (2014). [CrossRef]  

53. S. A. Haine and W. Y. S. Lau, “Generation of atom-light entanglement in an optical cavity for quantum enhanced atom interferometry,” Phys. Rev. A 93, 023607 (2016). [CrossRef]  

54. Sh. Barzanjeh, D. P. DiVincenzo, and B. M. Terhal, “Dispersive qubit measurement by interferometry with parametric amplifiers,” Phys. Rev. B 90, 134515 (2014). [CrossRef]  

55. H. Ma, D. Li, C.-H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “SU(1,1)-type light-atom-correlated interferometer,” Phys. Rev. A 92, 023847 (2015). [CrossRef]  

56. M. G. Raymer, “Quantum state entanglement and readout of collective atomic-ensemble modes and optical wave packets by stimulated Raman scattering,” J. Mod. Opt. 51, 1739 (2004). [CrossRef]  

57. R. Lynch, “The quantum phase problem: a critical review,” Phys. Rep. 256, 367 (1995). [CrossRef]  

58. T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, “Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons,” Phys. Rev. A 57, 4004 (1998). [CrossRef]  

59. O. Steuernagel and S. Scheel, “Approaching the Heisenberg limit with two-mode squeezed states,” J. Opt. B: Quantum Semiclass. Opt. 6, S66 (2004). [CrossRef]  

60. D. Li, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “The phase sensitivity of an SU(1,1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16, 073020 (2014). [CrossRef]  

61. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

62. C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).

References

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  1. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).
  2. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Springer Science and Bussiness Media, 2011).
    [Crossref]
  3. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981).
    [Crossref]
  4. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
    [Crossref] [PubMed]
  5. S. L. Braunstein, C. M. Caves, and G. J. Milburn, “Generalized uncertainty relations: theory, examples, and Lorentz invariance,” Ann. Phys. 247, 135 (1996).
    [Crossref]
  6. H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325 (2002).
    [Crossref]
  7. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
    [Crossref] [PubMed]
  8. M. Zwierz, C. A. Pérez-Delgado, and P. Kok, “General optimality of the Heisenberg limit for quantum metrology,” Phys. Rev. Lett. 105, 180402 (2010).
    [Crossref]
  9. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330 (2004).
    [Crossref] [PubMed]
  10. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222 (2011).
    [Crossref]
  11. Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
    [Crossref]
  12. B. P. Abbott and et al., (LIGO Scientific Collaboration and Virgo Collaboration), “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
    [Crossref]
  13. M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the short noise limit,” Phys. Rev. Lett. 59, 278 (1987).
    [Crossref] [PubMed]
  14. P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153 (1987).
    [Crossref] [PubMed]
  15. G. Tóth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A 47, 424006 (2014).
    [Crossref]
  16. L. Pezzè and A. Smerzi, in Proceedings of the international school of physics “Enrico Fermi”, Course CLXXXVIII Atom Interferometry edited by G. Tino and M. Kasevich, eds. (Società Italiana di Fisica and IOS, 2014).
  17. R. D. Dobrzanski, M. Jarzyna, and J. Kolodyński, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345 (2015).
    [Crossref]
  18. J. P. Dowling, “Quantum optical metrology–the lowdown on high-N00N states,” Contemp. Phys. 49, 125 (2008).
    [Crossref]
  19. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
    [Crossref] [PubMed]
  20. R. D. Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
    [Crossref]
  21. B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406 (2011).
    [Crossref]
  22. R. Demkowicz-Dobrzanski, J. Kolodyński, and M. Gutǎ, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
    [Crossref] [PubMed]
  23. D. W. Berry, Michael J. W. Hall, and Howard M. Wiseman, “Stochastic Heisenberg limit: optimal estimation of a fluctuating phase,” Phys. Rev. Lett. 111, 113601 (2013).
    [Crossref] [PubMed]
  24. R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013).
    [Crossref] [PubMed]
  25. W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, “Improved quantum metrology using quantum error correction,” Phys. Rev. Lett. 112, 080801 (2014).
    [Crossref]
  26. E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, “Quantum error correction for metrology,” Phys. Rev. Lett. 112, 150802 (2014).
    [Crossref] [PubMed]
  27. S. Alipour, M. Mehboudi, and A. T. Rezakhani, “Quantum metrology in open systems: dissipative Cramér-Rao bound,” Phys. Rev. Lett. 112, 120405 (2014).
    [Crossref]
  28. B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1 1) interferometers,” Phys. Rev. A 33, 4033 (1986).
    [Crossref]
  29. U. Leonhardt, “Quantum statistics of a two-mode SU(1,1) interferometer,” Phys. Rev. A 49, 1231 (1994).
    [Crossref] [PubMed]
  30. A. Vourdas, “SU(2) and SU(1,1) phase states,” Phys. Rev. A 41, 1653 (1990).
    [Crossref] [PubMed]
  31. B. C. Sanders, G. J. Milburn, and Z. Zhang, “Optimal quantum measurements for phase-shift estimation in optical interferometry,” J. Mod. Opt. 44, 1309 (1997).
  32. W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted sub-shot noise quantum interferometry,” New J. Phys. 12, 083014 (2010).
    [Crossref]
  33. 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]
  34. T. S. Horrom, B. E. Anderson, P. Gupta, and P. D. Lett, “SU(1,1) interferometry via four-wave mixing in Rb,” in 45th Winter Colloquium on the Physics of Quantum Electronics (PQE, 2015).
  35. A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU(1,1) interferometer,” Phys. Rev. A 86, 023844 (2012).
    [Crossref]
  36. 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]
  37. C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010).
    [Crossref] [PubMed]
  38. D. Linnemann, “Realization of an SU(1,1) interferometer with spinor Bose-Einstein condensates,” Master thesis, University of Heidelberg, 2013.
  39. J. Peise, B. Lücke, L. Pezzè, F. Deuretzbacher, W. Ertmer, J. Arlt, A. Smerzi, L. Santos, and C. Klempt, “Interaction-free measurements by quantum Zeno stabilization of ultracold atoms,” Nat. Commun. 6, 6811 (2015).
    [Crossref] [PubMed]
  40. M. Gabbrielli, L. Pezzè, and A. Smerzi, “Spin-mixing interferometry with Bose-Einstein condensates,” Phys. Rev. Lett. 115, 163002 (2015).
    [Crossref] [PubMed]
  41. L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413 (2001).
    [Crossref] [PubMed]
  42. K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041 (2010).
    [Crossref]
  43. L. Li, Y. O. Dudin, and A. Kuzmich, “Entanglement between light and an optical atomic excitation,” Nature 498, 466 (2013).
    [Crossref] [PubMed]
  44. L. Q. Chen, G. W. Zhang, C.-H. Yuan, J. Jing, Z. Y. Ou, and W. Zhang, “Enhanced Raman scattering by spatially distributed atomic coherence,” Appl. Phys. Lett. 95, 041115 (2009).
    [Crossref]
  45. L. Q. Chen, G. W. Zhang, C.-L. Bian, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “Observation of the Rabi oscillation of light driven by an atomic spin wave,” Phys. Rev. Lett. 105, 133603 (2010).
    [Crossref]
  46. C.-H. Yuan, L. Q. Chen, J. Jing, Z. Y. Ou, and W. Zhang, “Coherently enhanced Raman scattering in atomic vapor,” Phys. Rev. A 82, 013817 (2010).
    [Crossref]
  47. C.-H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Correlation-enhanced phase-sensitive Raman scattering in atomic vapors,” Phys. Rev. A 87, 053835 (2013).
    [Crossref]
  48. B. Chen, C. Qiu, L. Q. Chen, K. Zhang, J. Guo, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “Phase sensitive Raman process with correlated seeds,” Appl. Phys. Lett. 106, 111103 (2015).
    [Crossref]
  49. B. Chen, C. Qiu, S. Chen, J. Guo, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
    [Crossref] [PubMed]
  50. J. Jacobson, G. Björk, and Y. Yamamoto, “Quantum limit for the atom-light interferometer,” Appl. Phys. B 60, 187 (1995).
    [Crossref]
  51. S. A. Haine, “Information-recycling beam splitters for quantum enhanced atom interferometry,” Phys. Rev. Lett. 110, 053002 (2013).
    [Crossref] [PubMed]
  52. S. S. Szigeti, B. Tonekaboni, W. Y. S. Lau, S. N. Hood, and S. A. Haine, “Squeezed-light-enhanced atom interferometry below the standard quantum limit,” Phys. Rev. A 90, 063630 (2014).
    [Crossref]
  53. S. A. Haine and W. Y. S. Lau, “Generation of atom-light entanglement in an optical cavity for quantum enhanced atom interferometry,” Phys. Rev. A 93, 023607 (2016).
    [Crossref]
  54. Sh. Barzanjeh, D. P. DiVincenzo, and B. M. Terhal, “Dispersive qubit measurement by interferometry with parametric amplifiers,” Phys. Rev. B 90, 134515 (2014).
    [Crossref]
  55. H. Ma, D. Li, C.-H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “SU(1,1)-type light-atom-correlated interferometer,” Phys. Rev. A 92, 023847 (2015).
    [Crossref]
  56. M. G. Raymer, “Quantum state entanglement and readout of collective atomic-ensemble modes and optical wave packets by stimulated Raman scattering,” J. Mod. Opt. 51, 1739 (2004).
    [Crossref]
  57. R. Lynch, “The quantum phase problem: a critical review,” Phys. Rep. 256, 367 (1995).
    [Crossref]
  58. T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, “Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons,” Phys. Rev. A 57, 4004 (1998).
    [Crossref]
  59. O. Steuernagel and S. Scheel, “Approaching the Heisenberg limit with two-mode squeezed states,” J. Opt. B: Quantum Semiclass. Opt. 6, S66 (2004).
    [Crossref]
  60. D. Li, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “The phase sensitivity of an SU(1,1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16, 073020 (2014).
    [Crossref]
  61. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
  62. C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).

2016 (2)

B. P. Abbott and et al., (LIGO Scientific Collaboration and Virgo Collaboration), “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref]

S. A. Haine and W. Y. S. Lau, “Generation of atom-light entanglement in an optical cavity for quantum enhanced atom interferometry,” Phys. Rev. A 93, 023607 (2016).
[Crossref]

2015 (6)

H. Ma, D. Li, C.-H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “SU(1,1)-type light-atom-correlated interferometer,” Phys. Rev. A 92, 023847 (2015).
[Crossref]

B. Chen, C. Qiu, L. Q. Chen, K. Zhang, J. Guo, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “Phase sensitive Raman process with correlated seeds,” Appl. Phys. Lett. 106, 111103 (2015).
[Crossref]

B. Chen, C. Qiu, S. Chen, J. Guo, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

R. D. Dobrzanski, M. Jarzyna, and J. Kolodyński, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345 (2015).
[Crossref]

J. Peise, B. Lücke, L. Pezzè, F. Deuretzbacher, W. Ertmer, J. Arlt, A. Smerzi, L. Santos, and C. Klempt, “Interaction-free measurements by quantum Zeno stabilization of ultracold atoms,” Nat. Commun. 6, 6811 (2015).
[Crossref] [PubMed]

M. Gabbrielli, L. Pezzè, and A. Smerzi, “Spin-mixing interferometry with Bose-Einstein condensates,” Phys. Rev. Lett. 115, 163002 (2015).
[Crossref] [PubMed]

2014 (8)

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]

W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, “Improved quantum metrology using quantum error correction,” Phys. Rev. Lett. 112, 080801 (2014).
[Crossref]

E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, “Quantum error correction for metrology,” Phys. Rev. Lett. 112, 150802 (2014).
[Crossref] [PubMed]

S. Alipour, M. Mehboudi, and A. T. Rezakhani, “Quantum metrology in open systems: dissipative Cramér-Rao bound,” Phys. Rev. Lett. 112, 120405 (2014).
[Crossref]

G. Tóth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A 47, 424006 (2014).
[Crossref]

S. S. Szigeti, B. Tonekaboni, W. Y. S. Lau, S. N. Hood, and S. A. Haine, “Squeezed-light-enhanced atom interferometry below the standard quantum limit,” Phys. Rev. A 90, 063630 (2014).
[Crossref]

Sh. Barzanjeh, D. P. DiVincenzo, and B. M. Terhal, “Dispersive qubit measurement by interferometry with parametric amplifiers,” Phys. Rev. B 90, 134515 (2014).
[Crossref]

D. Li, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “The phase sensitivity of an SU(1,1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16, 073020 (2014).
[Crossref]

2013 (5)

C.-H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Correlation-enhanced phase-sensitive Raman scattering in atomic vapors,” Phys. Rev. A 87, 053835 (2013).
[Crossref]

S. A. Haine, “Information-recycling beam splitters for quantum enhanced atom interferometry,” Phys. Rev. Lett. 110, 053002 (2013).
[Crossref] [PubMed]

L. Li, Y. O. Dudin, and A. Kuzmich, “Entanglement between light and an optical atomic excitation,” Nature 498, 466 (2013).
[Crossref] [PubMed]

D. W. Berry, Michael J. W. Hall, and Howard M. Wiseman, “Stochastic Heisenberg limit: optimal estimation of a fluctuating phase,” Phys. Rev. Lett. 111, 113601 (2013).
[Crossref] [PubMed]

R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013).
[Crossref] [PubMed]

2012 (3)

R. Demkowicz-Dobrzanski, J. Kolodyński, and M. Gutǎ, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref] [PubMed]

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
[Crossref]

A. M. Marino, N. V. Corzo 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 (3)

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406 (2011).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222 (2011).
[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]

2010 (6)

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted sub-shot noise quantum interferometry,” New J. Phys. 12, 083014 (2010).
[Crossref]

C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010).
[Crossref] [PubMed]

M. Zwierz, C. A. Pérez-Delgado, and P. Kok, “General optimality of the Heisenberg limit for quantum metrology,” Phys. Rev. Lett. 105, 180402 (2010).
[Crossref]

K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041 (2010).
[Crossref]

L. Q. Chen, G. W. Zhang, C.-L. Bian, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “Observation of the Rabi oscillation of light driven by an atomic spin wave,” Phys. Rev. Lett. 105, 133603 (2010).
[Crossref]

C.-H. Yuan, L. Q. Chen, J. Jing, Z. Y. Ou, and W. Zhang, “Coherently enhanced Raman scattering in atomic vapor,” Phys. Rev. A 82, 013817 (2010).
[Crossref]

2009 (2)

L. Q. Chen, G. W. Zhang, C.-H. Yuan, J. Jing, Z. Y. Ou, and W. Zhang, “Enhanced Raman scattering by spatially distributed atomic coherence,” Appl. Phys. Lett. 95, 041115 (2009).
[Crossref]

R. D. Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[Crossref]

2008 (1)

J. P. Dowling, “Quantum optical metrology–the lowdown on high-N00N states,” Contemp. Phys. 49, 125 (2008).
[Crossref]

2006 (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref] [PubMed]

2004 (3)

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

M. G. Raymer, “Quantum state entanglement and readout of collective atomic-ensemble modes and optical wave packets by stimulated Raman scattering,” J. Mod. Opt. 51, 1739 (2004).
[Crossref]

O. Steuernagel and S. Scheel, “Approaching the Heisenberg limit with two-mode squeezed states,” J. Opt. B: Quantum Semiclass. Opt. 6, S66 (2004).
[Crossref]

2002 (1)

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325 (2002).
[Crossref]

2001 (1)

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413 (2001).
[Crossref] [PubMed]

2000 (1)

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

1998 (1)

T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, “Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons,” Phys. Rev. A 57, 4004 (1998).
[Crossref]

1997 (1)

B. C. Sanders, G. J. Milburn, and Z. Zhang, “Optimal quantum measurements for phase-shift estimation in optical interferometry,” J. Mod. Opt. 44, 1309 (1997).

1996 (1)

S. L. Braunstein, C. M. Caves, and G. J. Milburn, “Generalized uncertainty relations: theory, examples, and Lorentz invariance,” Ann. Phys. 247, 135 (1996).
[Crossref]

1995 (2)

R. Lynch, “The quantum phase problem: a critical review,” Phys. Rep. 256, 367 (1995).
[Crossref]

J. Jacobson, G. Björk, and Y. Yamamoto, “Quantum limit for the atom-light interferometer,” Appl. Phys. B 60, 187 (1995).
[Crossref]

1994 (2)

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
[Crossref] [PubMed]

U. Leonhardt, “Quantum statistics of a two-mode SU(1,1) interferometer,” Phys. Rev. A 49, 1231 (1994).
[Crossref] [PubMed]

1990 (1)

A. Vourdas, “SU(2) and SU(1,1) phase states,” Phys. Rev. A 41, 1653 (1990).
[Crossref] [PubMed]

1987 (2)

M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the short noise limit,” Phys. Rev. Lett. 59, 278 (1987).
[Crossref] [PubMed]

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153 (1987).
[Crossref] [PubMed]

1986 (1)

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

1981 (1)

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

Abbott, B. P.

B. P. Abbott and et al., (LIGO Scientific Collaboration and Virgo Collaboration), “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref]

Abrams, D. S.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

Acín, A.

R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013).
[Crossref] [PubMed]

Agarwal, G. S.

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted sub-shot noise quantum interferometry,” New J. Phys. 12, 083014 (2010).
[Crossref]

Alipour, S.

S. Alipour, M. Mehboudi, and A. T. Rezakhani, “Quantum metrology in open systems: dissipative Cramér-Rao bound,” Phys. Rev. Lett. 112, 120405 (2014).
[Crossref]

Anderson, B. E.

T. S. Horrom, B. E. Anderson, P. Gupta, and P. D. Lett, “SU(1,1) interferometry via four-wave mixing in Rb,” in 45th Winter Colloquium on the Physics of Quantum Electronics (PQE, 2015).

Apellaniz, I.

G. Tóth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A 47, 424006 (2014).
[Crossref]

Arlt, J.

J. Peise, B. Lücke, L. Pezzè, F. Deuretzbacher, W. Ertmer, J. Arlt, A. Smerzi, L. Santos, and C. Klempt, “Interaction-free measurements by quantum Zeno stabilization of ultracold atoms,” Nat. Commun. 6, 6811 (2015).
[Crossref] [PubMed]

Banaszek, K.

R. D. Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[Crossref]

Barzanjeh, Sh.

Sh. Barzanjeh, D. P. DiVincenzo, and B. M. Terhal, “Dispersive qubit measurement by interferometry with parametric amplifiers,” Phys. Rev. B 90, 134515 (2014).
[Crossref]

Berry, D. W.

D. W. Berry, Michael J. W. Hall, and Howard M. Wiseman, “Stochastic Heisenberg limit: optimal estimation of a fluctuating phase,” Phys. Rev. Lett. 111, 113601 (2013).
[Crossref] [PubMed]

Bian, C.-L.

L. Q. Chen, G. W. Zhang, C.-L. Bian, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “Observation of the Rabi oscillation of light driven by an atomic spin wave,” Phys. Rev. Lett. 105, 133603 (2010).
[Crossref]

Björk, G.

J. Jacobson, G. Björk, and Y. Yamamoto, “Quantum limit for the atom-light interferometer,” Appl. Phys. B 60, 187 (1995).
[Crossref]

Boto, A. N.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

Brask, J. B.

R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013).
[Crossref] [PubMed]

Braunstein, S. L.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

S. L. Braunstein, C. M. Caves, and G. J. Milburn, “Generalized uncertainty relations: theory, examples, and Lorentz invariance,” Ann. Phys. 247, 135 (1996).
[Crossref]

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
[Crossref] [PubMed]

Caves, C. M.

S. L. Braunstein, C. M. Caves, and G. J. Milburn, “Generalized uncertainty relations: theory, examples, and Lorentz invariance,” Ann. Phys. 247, 135 (1996).
[Crossref]

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
[Crossref] [PubMed]

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

Chaves, R.

R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013).
[Crossref] [PubMed]

Chen, B.

B. Chen, C. Qiu, S. Chen, J. Guo, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

B. Chen, C. Qiu, L. Q. Chen, K. Zhang, J. Guo, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “Phase sensitive Raman process with correlated seeds,” Appl. Phys. Lett. 106, 111103 (2015).
[Crossref]

Chen, L. Q.

B. Chen, C. Qiu, L. Q. Chen, K. Zhang, J. Guo, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “Phase sensitive Raman process with correlated seeds,” Appl. Phys. Lett. 106, 111103 (2015).
[Crossref]

B. Chen, C. Qiu, S. Chen, J. Guo, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

H. Ma, D. Li, C.-H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “SU(1,1)-type light-atom-correlated interferometer,” Phys. Rev. A 92, 023847 (2015).
[Crossref]

C.-H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Correlation-enhanced phase-sensitive Raman scattering in atomic vapors,” Phys. Rev. A 87, 053835 (2013).
[Crossref]

C.-H. Yuan, L. Q. Chen, J. Jing, Z. Y. Ou, and W. Zhang, “Coherently enhanced Raman scattering in atomic vapor,” Phys. Rev. A 82, 013817 (2010).
[Crossref]

L. Q. Chen, G. W. Zhang, C.-L. Bian, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “Observation of the Rabi oscillation of light driven by an atomic spin wave,” Phys. Rev. Lett. 105, 133603 (2010).
[Crossref]

L. Q. Chen, G. W. Zhang, C.-H. Yuan, J. Jing, Z. Y. Ou, and W. Zhang, “Enhanced Raman scattering by spatially distributed atomic coherence,” Appl. Phys. Lett. 95, 041115 (2009).
[Crossref]

Chen, S.

B. Chen, C. Qiu, S. Chen, J. Guo, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

Cirac, J. I.

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413 (2001).
[Crossref] [PubMed]

Corzo Trejo, N. V.

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

Davidovich, L.

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406 (2011).
[Crossref]

de Matos Filho, R. L.

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406 (2011).
[Crossref]

Demkowicz-Dobrzanski, R.

R. Demkowicz-Dobrzanski, J. Kolodyński, and M. Gutǎ, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref] [PubMed]

Deuretzbacher, F.

J. Peise, B. Lücke, L. Pezzè, F. Deuretzbacher, W. Ertmer, J. Arlt, A. Smerzi, L. Santos, and C. Klempt, “Interaction-free measurements by quantum Zeno stabilization of ultracold atoms,” Nat. Commun. 6, 6811 (2015).
[Crossref] [PubMed]

DiVincenzo, D. P.

Sh. Barzanjeh, D. P. DiVincenzo, and B. M. Terhal, “Dispersive qubit measurement by interferometry with parametric amplifiers,” Phys. Rev. B 90, 134515 (2014).
[Crossref]

Dobrzanski, R. D.

R. D. Dobrzanski, M. Jarzyna, and J. Kolodyński, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345 (2015).
[Crossref]

R. D. Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[Crossref]

Dorner, U.

R. D. Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[Crossref]

Dowling, J. P.

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted sub-shot noise quantum interferometry,” New J. Phys. 12, 083014 (2010).
[Crossref]

J. P. Dowling, “Quantum optical metrology–the lowdown on high-N00N states,” Contemp. Phys. 49, 125 (2008).
[Crossref]

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325 (2002).
[Crossref]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

Duan, L.-M.

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413 (2001).
[Crossref] [PubMed]

Dudin, Y. O.

L. Li, Y. O. Dudin, and A. Kuzmich, “Entanglement between light and an optical atomic excitation,” Nature 498, 466 (2013).
[Crossref] [PubMed]

Dür, W.

W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, “Improved quantum metrology using quantum error correction,” Phys. Rev. Lett. 112, 080801 (2014).
[Crossref]

Ertmer, W.

J. Peise, B. Lücke, L. Pezzè, F. Deuretzbacher, W. Ertmer, J. Arlt, A. Smerzi, L. Santos, and C. Klempt, “Interaction-free measurements by quantum Zeno stabilization of ultracold atoms,” Nat. Commun. 6, 6811 (2015).
[Crossref] [PubMed]

Escher, B. M.

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406 (2011).
[Crossref]

Estève, J.

C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010).
[Crossref] [PubMed]

Fröwis, F.

W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, “Improved quantum metrology using quantum error correction,” Phys. Rev. Lett. 112, 080801 (2014).
[Crossref]

Gabbrielli, M.

M. Gabbrielli, L. Pezzè, and A. Smerzi, “Spin-mixing interferometry with Bose-Einstein condensates,” Phys. Rev. Lett. 115, 163002 (2015).
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Gerry, C. C.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222 (2011).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref] [PubMed]

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

Grangier, P.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153 (1987).
[Crossref] [PubMed]

Gross, C.

C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010).
[Crossref] [PubMed]

Guo, J.

B. Chen, C. Qiu, S. Chen, J. Guo, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

B. Chen, C. Qiu, L. Q. Chen, K. Zhang, J. Guo, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “Phase sensitive Raman process with correlated seeds,” Appl. Phys. Lett. 106, 111103 (2015).
[Crossref]

Gupta, P.

T. S. Horrom, B. E. Anderson, P. Gupta, and P. D. Lett, “SU(1,1) interferometry via four-wave mixing in Rb,” in 45th Winter Colloquium on the Physics of Quantum Electronics (PQE, 2015).

Guta, M.

R. Demkowicz-Dobrzanski, J. Kolodyński, and M. Gutǎ, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref] [PubMed]

Haine, S. A.

S. A. Haine and W. Y. S. Lau, “Generation of atom-light entanglement in an optical cavity for quantum enhanced atom interferometry,” Phys. Rev. A 93, 023607 (2016).
[Crossref]

S. S. Szigeti, B. Tonekaboni, W. Y. S. Lau, S. N. Hood, and S. A. Haine, “Squeezed-light-enhanced atom interferometry below the standard quantum limit,” Phys. Rev. A 90, 063630 (2014).
[Crossref]

S. A. Haine, “Information-recycling beam splitters for quantum enhanced atom interferometry,” Phys. Rev. Lett. 110, 053002 (2013).
[Crossref] [PubMed]

Hall, J. L.

T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, “Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons,” Phys. Rev. A 57, 4004 (1998).
[Crossref]

Hall, Michael J. W.

D. W. Berry, Michael J. W. Hall, and Howard M. Wiseman, “Stochastic Heisenberg limit: optimal estimation of a fluctuating phase,” Phys. Rev. Lett. 111, 113601 (2013).
[Crossref] [PubMed]

Hammerer, K.

K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041 (2010).
[Crossref]

Helstrom, C. W.

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).

Holevo, A. S.

A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Springer Science and Bussiness Media, 2011).
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Holland, M. J.

T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, “Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons,” Phys. Rev. A 57, 4004 (1998).
[Crossref]

Hood, S. N.

S. S. Szigeti, B. Tonekaboni, W. Y. S. Lau, S. N. Hood, and S. A. Haine, “Squeezed-light-enhanced atom interferometry below the standard quantum limit,” Phys. Rev. A 90, 063630 (2014).
[Crossref]

Horrom, T. S.

T. S. Horrom, B. E. Anderson, P. Gupta, and P. D. Lett, “SU(1,1) interferometry via four-wave mixing in Rb,” in 45th Winter Colloquium on the Physics of Quantum Electronics (PQE, 2015).

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]

Jacobson, J.

J. Jacobson, G. Björk, and Y. Yamamoto, “Quantum limit for the atom-light interferometer,” Appl. Phys. B 60, 187 (1995).
[Crossref]

Jarzyna, M.

R. D. Dobrzanski, M. Jarzyna, and J. Kolodyński, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345 (2015).
[Crossref]

Jing, J.

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Figures (7)

Fig. 1
Fig. 1 (a) The intermode correlation between the Stokes field a ^ 1 and the atomic excitation b ^ 1 is generated by spontaneous Raman process. a ^ 0 is the initial input light field. b ^ 0 is in vacuum or an initial atomic collective excitation which can be prepared by another Raman process or electromagnetically induced transparency process. (b) During the delay time τ, the Stokes field a ^ 1 will be subject to the photon loss and evolute to a ^ 1 and the collective excitation b ^ 1 will undergo the collisional dephasing to b ^ 1 . A fictitious beam splitter (BS) is introduced to mimic the loss of photons into the environment. V ^ is the vacuum. (c) After the delay time τ, the light field a ^ 1 and its correlated atomic excitation b ^ 1 are used as initial seed for another enhanced Raman process. (d)–(f) The corresponding energy-level diagrams of different processes are shown.
Fig. 2
Fig. 2 The phase sensitivity ΔϕHD and the SNRHD versus the phase shift ϕ using the method of homodyne detection with (a) θα = π/2; (b) θα = 0. Parameters: g = 2, |α| = 10.
Fig. 3
Fig. 3 (a) Δna2, |⟨∂⟨na2⟩/∂ϕ⟩|, and the phase sensitivity ΔϕID; (b) ⟨na2⟩, Δna2 and the SNRID versus the phase shift ϕ using the method of intensity detection. Parameters: g = 2, |α| = 10.
Fig. 4
Fig. 4 (a) The optimal phase sensitivities Δϕ and (b) the maximal SNR versus the phase-sensing probe number nph. The optimal phase sensitivities ΔϕHD and ΔϕID are obtained at ϕ = 0 and ϕ = 0.062, respectively. The maximal SNRs are obtained at ϕ = 0 and θα = 0. Parameter: g = 2.
Fig. 5
Fig. 5 The linear correlation coefficients (a) Jx2; (b) Jn2 as a function of the phase shift ϕ for lossless case. Parameters: θ1 = 0, g = 2, |α| = 10.
Fig. 6
Fig. 6 The linear correlation coefficients Jx2 as a function of (a) the transmission rate T; (b) the collisional rate Γτ. Parameters: g = 2, |α| = 10, θα = π/2 and ϕ = 0.
Fig. 7
Fig. 7 The linear correlation coefficients Jn2 as a function of (a) the transmission rate T; (b) the collisional dephasing rate Γτ, where g = 2, |α| = 10, and θ2θ1 = π, and ϕ = 0.062.

Equations (42)

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a ^ ( t ) t = η A P b ^ ( t ) , b ^ ( t ) t = η A P a ^ ( t ) ,
a ^ ( t ) = u ( t ) a ^ ( 0 ) + v ( t ) b ^ ( 0 ) , b ^ ( t ) = u ( t ) b ^ ( 0 ) + v ( t ) a ^ ( 0 ) ,
a ^ 1 = T a ^ 1 ( t 1 ) e i ϕ + R V ^ ,
b ^ 1 = b ^ 1 ( t 1 ) e Γ τ + F ^ ,
a ^ 2 ( t 2 ) = U 1 a ^ 1 ( 0 ) + V 1 b ^ 1 ( 0 ) + R u 2 V ^ + v 2 F ^ ,
b ^ 2 ( t 2 ) = e i ϕ [ U 2 b ^ 1 ( 0 ) + V 2 a ^ 1 ( 0 ) ] + R v 2 V ^ + u 2 F ^ ,
U 1 = T u 1 u 2 e i ϕ + e Γ τ v 1 * v 2 , V 1 = T v 1 u 2 e i ϕ + e Γ τ u 1 * v 2 U 2 = e Γ τ u 1 u 2 e i ϕ + T v 1 * v 2 , V 2 = e Γ τ v 1 u 2 e i ϕ + T u 1 * v 2 .
Δ ϕ = ( Δ O ^ ) 2 1 / 2 | O ^ / ϕ | ,
SNR = O ^ ( Δ O ^ ) 2 1 / 2 .
Δ ϕ HD = ( Δ x ^ a 2 ) 2 1 / 2 N α cosh 2 g | sin ( ϕ + θ α ) | ,
SNR HD = N α [ cosh 2 g cos ( ϕ + θ α ) sinh 2 g cos ( θ α ) ] ( Δ x ^ a 2 ) 2 1 / 2 ,
( Δ x ^ a 2 ) 2 = 1 4 [ cosh 2 ( 2 g ) sinh 2 ( 2 g ) cos ϕ ] ,
Δ ϕ HD = 1 N a 1 2 cosh 2 g ,
SNR HD = 0 .
SNR HD = 2 N α ,
Δ ϕ ID = ( Δ n ^ a 2 ) 2 1 / 2 2 ( N α + 1 ) sinh 2 ( 2 g ) | sin ϕ | ,
SNR ID = 1 ( Δ n ^ a 2 ) 2 1 / 2 [ N α | cosh 2 g sinh 2 g e i ϕ | 2 + 1 2 sinh 2 ( 2 g ) ( 1 cos ϕ ) ] ,
( Δ n ^ a 2 ) 2 = N α | cosh 2 g sinh 2 g e i ϕ | 4 + 1 2 ( 1 + N α ) × sinh 2 ( 2 g ) | cosh 2 g sinh 2 g e i ϕ | 2 ( 1 cos ϕ ) .
( 1 + cos 2 ϕ ) 2 sinh 2 ( 2 g ) [ ( 2 N α + 1 ) cosh 2 ( 2 g ) + N α ] cos ϕ [ 4 N α ( 2 cosh 2 ( 2 g ) + sinh 4 ( 2 g ) ) + ( 1 + N α ) sinh 2 4 g ] = 0 .
x ^ a 2 = N α e Γ τ sinh 2 g cos θ α + T N α cosh 2 g cos ( ϕ + θ α ) ,
n ^ a 2 = ( T cosh 4 g + e 2 Γ τ sinh 4 g ) N α + sinh 2 g ( 1 e 2 Γ τ ) + 1 4 sinh 2 ( 2 g ) ( T + e 2 Γ τ ) 1 2 T e Γ τ ( N α + 1 ) sinh 2 ( 2 g ) cos ϕ ,
| x ^ a 2 ϕ | = T N cosh 2 g | sin ( ϕ + θ α ) | ,
| n ^ a ϕ | = 1 2 T e Γ τ ( N α + 1 ) sinh 2 ( 2 g ) | sin ( ϕ ) | .
( Δ x ^ a 2 ) 2 = 1 4 [ sinh 2 ( 2 g ) ( T 2 T e Γ τ cos ϕ ) + 2 e 2 Γ τ sinh 4 g + cosh ( 2 g ) ] ,
( Δ n ^ a 2 ) 2 = | U b | 4 N α + | U b V b | 2 ( 1 + N α ) + R cosh 2 g ( | U b | 2 N α + | V b | 2 ) + sinh 2 g [ | U b | 2 ( 1 + N α ) + R cosh 2 g ] ( 1 e 2 Γ τ ) ,
| U b | 2 = ( T cosh 2 g + e Γ τ sinh 2 g ) 2 2 T e Γ τ sinh 2 g cosh 2 g ( 1 + cos ϕ ) ,
| V b | 2 = 1 2 sinh 2 ( 2 g ) ( T + e 2 Γ τ 2 T e Γ τ cos ϕ ) ,
n ph = N α cosh ( 2 g ) + 2 sinh 2 g .
J ( A ^ , B ^ ) = c o v ( A ^ , B ^ ) ( Δ A ^ ) 2 1 / 2 ( Δ B ^ ) 2 1 / 2 ,
J x 1 ( x ^ a 1 , x ^ b 1 ) = cos θ 1 tanh ( 2 g ) ,
J y 1 ( y ^ a 1 , y ^ b 1 ) = cos θ 1 tanh ( 2 g ) ,
J n 1 ( n ^ a 1 , n ^ b 1 ) = ( 1 + 2 | a | 2 ) [ 4 coth 2 ( 2 g ) ( | a | 2 + | a | 4 ) + 1 ] 1 / 2 .
J x 2 ( x ^ a 2 , x ^ b 2 ) = c o v ( x ^ a 2 , x ^ b 2 ) ( Δ x ^ a 2 ) 2 1 / 2 ( Δ x ^ b 2 ) 2 1 / 2 ,
c o v ( x ^ a 2 , x ^ b 2 ) = 1 4 Re [ e i ϕ ( V 1 U 2 + U 1 V 2 ) + u 2 v 2 ( R + 1 e 2 Γ τ ) ] , ( Δ x ^ a 2 ) 2 = 1 4 [ | U 1 | 2 + | V 1 | 2 + R | u 2 | 2 + | v 2 | 2 ( 1 e 2 Γ τ ) ] , ( Δ x ^ b 2 ) 2 = 1 4 [ | U 2 | 2 + | V 2 | 2 + R | u 2 | 2 + | v 2 | 2 ( 1 e 2 Γ τ ) ] .
J n 2 ( n ^ a 2 , n ^ b 2 ) = c o v ( n ^ a 2 , n ^ b 2 ) ( Δ n ^ a 2 ) 2 1 / 2 ( Δ n ^ b 2 ) 2 1 / 2 ,
c o v ( n ^ a 2 , n ^ b 2 ) = | U 1 V 1 | 2 | α | 2 + ( 1 + | α | 2 ) Re [ U 1 * U 2 V 1 V 2 * ] + R [ Re [ e i ϕ U 2 V 1 u 2 * v 2 * ] + | α | 2 Re [ e i ϕ U 1 V 2 u 2 * v 2 * ] ] + [ R | u 2 v 2 | 2 + ( 1 + | α | 2 ) Re [ e i ϕ U 1 * V 2 * u 2 v 2 ] ] ( 1 e 2 Γ τ ) ,
( Δ n ^ a 2 ) 2 = | U 1 | 4 | α | 2 + | U 1 V 1 | 2 ( 1 + | α | 2 ) + R | u 2 | 2 ( | V 1 | 2 + | U 1 | 2 | α | 2 ) + | v 2 | 2 [ | U 1 | 2 ( | α | 2 + 1 ) + R | u 2 | 2 ] ( 1 e 2 Γ τ ) ,
( Δ n ^ b 2 ) 2 = | V 2 | 4 | α | 2 + | U 2 V 2 | 2 ( 1 + | α | 2 ) + R | v 2 | 2 ( | U 2 | 2 + | V 2 | 2 | α | 2 ) + | u 2 | 2 [ | V 2 | 2 ( 1 + | α | 2 ) + R | v 2 | 2 ] ( 1 e 2 Γ τ ) .
J x 2 ( x ^ a 2 , x ^ b 2 ) = 2 Re [ V U e i ϕ ] | U | 2 + | V | 2 = sinh ( 2 g ) cosh 2 ( 2 g ) sinh 2 ( 2 g ) cos ϕ [ cosh 2 g cos ( θ 1 + 3 ϕ ) + sinh 2 g cos ( θ 1 + ϕ ) cosh ( 2 g ) cos ( θ 1 + 2 ϕ ) ]
J n 2 ( n ^ a 2 , n ^ b 2 ) = | U V ( 1 + 2 | α | 2 ) | ( U V ) 1 / 2 = ( 1 + 2 | α | 2 ) × [ 4 [ 1 + sinh 2 ( 2 g ) ( 1 cos ϕ ) ] 2 ( | α | 2 + | α | 4 ) [ 1 + sinh 2 ( 2 g ) ( 1 cos ϕ ) ] 2 1 + 1 ] 1 / 2 ,
J x 2 ( x ^ a 2 , x ^ b 2 ) = tanh ( 4 g ) cos ( θ 1 ) ,
J n 2 ( n ^ a 2 , n ^ b 2 ) = 1 + 2 | α | 2 4 coth 2 ( 2 g 1 ) ( | α | 2 + | α | 4 ) + 1 = J n 1 ( n ^ a 1 , n ^ b 1 ) .

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