Effects of losses in the hybrid atom-light interferometer

Enhanced Raman scattering can be obtained by injecting a seeded light field which is correlated with the initially prepared collective atomic excitation. This Raman amplification process can be used to realize atom-light hybrid interferometer. We numerically calculate the phase sensitivities and the signal-to-noise ratios of this interferometer with the method of homodyne detection and intensity detection, and give their differences between this two methods. In the presence of loss of light field and atomic decoherence the measure precision will be reduced which can be explained by the break of the intermode decorrelation conditions of output modes


I. INTRODUCTION
Quantum parameter estimation 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][2][3][4][5][6][7][8][9][10][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 enter the unused port and are amplified in the interferometer by the coherent light, Caves [3] suggested to replace the vacuum fluctuations with the squeezed-vacuum light to reach a sub-shotnoise 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 N number of photons or other bosons.Due to overcoming the SQL and reaching the Heisenberg limit (HL) δϕ = 1/N , it will lead to potential applications in high resolution measurements.Therefore, many theoretical proposals and experimental techniques are developed to improve the sensitivity [15][16][17].When the probe states made of correlated states, such as the NOON states of the form (|N a |0 b + e iφN |0 a |N b )/ √ 2, the HL in the phase-shift measurements can reach [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][21][22][23][24][25][26][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] in- * Electronic address: chyuan@phy.ecnu.edu.cn† Electronic address: lqchen@phy.ecnu.edu.cntroduced a new 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, SU(1,1) atomic interferometer also has been experimentally realized with Bose-Einstein Condensates [37][38][39][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][42][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][45][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 replacing the beam splitting elements in a traditional MZI [28].Different from all-optical or all-atomic interferometers, the atom-light hybrid interferometers depend on both atomic and optical phases so that we can probe the atomic phases with optical interferometric techniques.The atomic phase can be adjusted by magnetic field or Stark shifts.The atom-light hybrid interferometer is composed of two Raman amplification processes.The first nonlinear process generates the correlated optical and atomic waves in the two arms and they are decorrelated by the second nonlinear process.
In this paper, we calculate the phase sensitivities and the SNRs using the homodyne detection and the intensity detection.The differences between the phase sensitivities and the SNRs are compared.The loss of light field and atomic decoherence will degrade the measure precision.The effects of the light field loss and atomic decoherence on measure precision can be explained from the break of intermode decorrelation conditions.
Our article is organized as follows.In Sec.II, we give the model of the hybrid atom-light interferometer, and in Sec.III we numerically calculate the phase sensitivity and the SNR, and analyze and compare the conditions to obtain the optimal phase sensitivity and the maximal SNR.In Sec.IV, the LCCs of the amplitude quadrature and number operator are derived from the lightatom coupling equations in the presence of light field loss and atomic decoherence.The LCCs as a function of the transmission rate and the collisional rate are calculated and analyzed.The loss of light field and atomic decoherence will degrade the measure precision, which is explained from the intermode decorrelation conditions break.Finally, in Sec.V we conclude with a summary of our results.

II. THE MODEL OF ATOM-LIGHT HYBRID INTERFEROMETER
In this section, we review the different processes of the atom-light interferometer [49,50] as shown in Fig. 1(a)-(c), where two Raman systems replaced the BSs in the traditional MZI.Considering a three-level Lambdashaped atom system as shown in Fig. 1(d), the Raman scattering process is described by the following pair of coupled equations [51]: where η is the coupling constant, and A P is the amplitude of the pump field.The solution of above equation is where u(t) = cosh(g), v(t) = e iθ sinh(g), g = |ηA P | t, e iθ = (A P /A * P ) 1/2 , and t is the time duration of pump field E P .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).After the first Raman process of the interferometer, the Stokes field â1 and the atomic excitation b1 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 â1 will be subject to the photon loss and evolute to â′ 1 .A fictitious BS is introduced to mimic the loss of photons into the environment, then the light field â′ 1 is given by where T and R are the transmission and reflectance coefficients with T + R = 1, and V is in vacuum.The collective excitation b1 will also undergo the collisional dephasing described by the factor e −Γτ , then b′ where F = τ 0 e −Γ(τ −t ′ ) f (t ′ )dt ′ , and f (t) is the quantum statistical Langevin operator describing the collisioninduced fluctuation, and obeys f guarantees the consistency of the operator properties of b′ 1 .
Using Eqs. ( 2)-( 4), the generated Stokes field â2 and collective atomic excitation b2 can be worked out: where 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.Phase can be estimated but cannot be measured because there is not a Hermitian operator corresponding to a quantum phase [52].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]: where Ô is the measurable operator and (∆ Ô) 2 = Ô2 − Ô 2 .The precision of the phase shift measurement is not the only parameter of concern.We also need consider the SNR [11,53,54], which is given by In current optical measurement of phase sensitivity, the homodyne detection [54][55][56] and the intensity detection [32,35] are often used.That is, the observables are the amplitude quadrature operator xa2 = (â 2 + â † 2 )/2 and the number operator na2 = â † 2 â2 .For the balanced situation that is g 1 = g 2 = g, and θ 2 − θ 1 = π.Firstly, we do not consider the effect of loss on the generated Stokes field â2 and atomic collective excitation b2 .That is, R = 0 and Γτ = 0, it reduced to the ideal lossless case and we have A.

Homodyne detection
For a coherent light |α (α = |α| e iθα , N α = |α| 2 ) as the phase-sensing field, using the amplitude quadrature operator xa2 as the detected variable the phase sensitivity and the SNR are given by with where the subscript HD denotes the homodyne detection.The phase sensitivity ∆φ HD and the SNR HD depend on φ and θ α , when g and α take a certain values.From Eqs. (10) and (11), both the ∆φ HD and the SNR HD need that the term (∆x a2 ) 2 is minimal, which can be realized at φ = 0 and (∆x a2 ) 2 = 1/4 [57].When φ = 0 and θ α = π/2, we obtain the optimal phase sensitivity and the worst SNR:  But when φ = 0 and θ α = 0 or π, the maximal SNR HD is given by and the sensitivity ∆φ HD is divergent.The phase sensitivity ∆φ HD and the SNR HD 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 θ α = π/2 the sensitivity is high (i.e.∆φ small) and can beat the SQL but the SNR HD is low.At the optimal point φ = 0 and θ α = 0 the SNR HD is high, but the sensitivity is low.Ideally, of course, we would like high sensitivity ∆φ HD and high SNR HD at the same optimal point.

B. Intensity detection
If we use na2 (= â † 2 â2 ) as the detection variable, for a coherent light |α (α = |α| e iθα , N α = |α| 2 ) as the phasesensing field, the phase sensitivity and the SNR are given by where the subscript ID denotes the intensity detection, and Different from the homodyne detection, the phase sensitivity ∆φ ID and the SNR ID only depend on φ for given g and N α .Under the condition of g = 2 and N α = √ 10, the phase sensitivity ∆φ ID and the SNR ID as a function of phase shift φ are shown in Figs.3(a) and 3(b), respectively.The best phase sensitivity ∆φ ID and the maximal SNR ID (= √ N α ) are obtained at φ ≈ 0.062 and φ = 0, respectively.In Fig. 3(a) at φ ≈ 0 the slope |∂ na2 /∂φ| is very small, as well in Fig. 3(b) at φ = 0 the intensity of the signal na2 is low, but the noise is also low.It demonstrated that the noise (∆n a2 ) 2 plays a dominant role.The best phase sensitivity from the intensity detection is lower than it from the homodyne detection, i.e., ∆φ ID > ∆φ HD .The relation of maximal SNR from two detection methods is SNR HD = 2SNR ID .

C. Losses case
If the presence of loss of light field and atomic decoherence, the precision of the sensitivity and the SNR will be reduced [50,56].According to the linear error propagation, the mean-square error in parameter φ is given by ∆φ The slopes of the output amplitude quadrature operator Xa2 and the number operator na2 = â † 2 â2 are respectively given by The uncertainties of the output amplitude quadrature operator Xa2 and the number operator na2 are given by where The subscript b denotes the balanced condition when considering the losses case.The phase sensitivities ∆φ as a function of the phasesensing probe number n ph is shown in Fig. 4. The thick solid line is the SQL.The thin solid line and dotted line are sensitivities ∆φ HD from homodyne detection with and without losses cases, respectively.As well the dashed and dash-dotted lines are sensitivities ∆φ ID from intensity detection with and without losses cases, respectively.From Fig. 4, it is easy to see that the best phase sensitivities ∆φ ID are larger than ∆φ HD under the same condition.In the presence of loss and collisional dephasing (T = 0.8, Γτ = 0.1), the phase sensitivities ∆φ HD and ∆φ ID can beat the SQL under the balanced situation, which is very important for phase sensitivity measurement.
Next section, we explain the reason that the effects of the light field loss and atomic decoherence on measure precision can be explained from the break of intermode decorrelation conditions.

IV. 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 Fig. 1(a)-(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 linear correlation coefficient (LCC), which is defined as [58] where cov( The respective quadrature operators of the light and atomic excitation are xa = (â + â † )/2, ŷa = (â − â † )/2i, xb = ( b + b † )/2, and ŷb = ( 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 â, and a vacuum state in mode b, the LCC of quadratures are given by and the LCC of number operators na1 [= â † (t 1 )â(t 1 )] and nb1 [= b † (t 1 ) b(t 1 )] is given by From Eqs. ( 25)-( 27), the quadrature correlation LCCs J x1 (x a1 , xb1 ) and J y1 (ŷ a1 , ŷb1 ) are independent on the input coherent state which is different from the number correlation LCC J n1 (n a1 , nb1 ).Under θ 1 = π/2, the LCCs J x1 and J y1 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 + xb )/ √ 2, Ŷ = (ŷ a + ŷb )/ √ 2 and [ X, Ŷ ] = i/2 [58].From Eq. ( 27) the number correlation LCC J n1 is always positive so long as g = 0.If α = 0, that is vacuum state input, then J n1 (n a1 , nb1 ) = 1, this maximal value shows the strong intermode correlation and such states in optical fields are often 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 J x2 (x a2 , xb2 ) using the generated Stokes field â2 and atomic collective excitation b2 can be worked out where The LCC of number operators J n2 (n a2 , nb2 ) can also be worked out where Firstly, we do not consider the effect of loss on the generated Stokes field â2 and atomic collective excitation b2 .Under this ideal and balanced conditions, the LCCs of quadratures and number operators are respectively given by + sinh 2 g cos(θ 1 + φ) − cosh(2g) cos(θ 1 + 2φ)], (34) and where 1).When the phase shift φ is 0, V is also equal to 0, then the LCC J x2 (x a2 , xb2 ) and J n2 (n a2 , nb2 ) are 0.Under this condition, the RP2 will "undo" what the RP1 did.When the phase shift φ is π, the LCC J x2 (x a2 , xb2 ) and J n2 (n a2 , nb2 ) are respectively given by  The LCCs J x2 (x a2 , xb2 ) and J n2 (n a2 , nb2 ) as a function of the phase shift φ is shown in Fig. 5. Due to the LCC J x2 (x a2 , xb2 ) is dependent on θ 1 , the intermode correlation coefficients J x2 (x a2 , xb2 ) ranges between −1 and 0 when θ 1 = 0.The LCC J n2 (n a2 , nb2 ) is positive, and J n2 (n a2 , nb2 ) ranges between 0 and 1.This decorrelation point (φ = 0) J x2 (x a2 , xb2 ) = 0 is very important for atom-light hybrid interferometer using the homodyne detection [50].At this point (φ = 0) the noise of output field [ (∆x 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 ∆φ HD and the maximal SNR HD are obtained at this point with different θ α .The LCC J x2 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 J x2 is reduced and tend to −1.Due to large loss (T small) or large decoherence (Γτ large) one arm inside the interferomter (the optical field â′ 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.
This decorrelation point (φ = 0) J n2 (n a2 , nb2 ) = 0 is also very important 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 SNR ID .However, the slope |∂ na2 /∂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 ∆φ ID is obtained.With the decrease of the transmission rate T or the increase of Γτ , the LCC J n2 are reduced at first, then revive quickly, and finally increase to 1 as shown in Fig. 7.The behaviors of the two detection methods are different, but both of their correlations eventually tend to strong correlation due to the losses.Therefore, the serious break of decorrelation condition will degrade the sensitivity in the phase precision measurement.

V. CONCLUSIONS
We gave out the phase sensitivities and the SNRs of the atom-light hybrid interferometer with the method of homodyne detection and intensity detection.Using the homodyne detection, for given input intensity N α and coupling intensity g the optimal sensitivity ∆φ HD and the maximal SNR HD is not only dependent on the phase shift φ but also dependent on the phase θ α of the input coherent state.We obtain that the sensitivity is low (i.e.∆φ HD large) when the SNR HD is high and vice versa because the optimal point changes with θ α .Using the intensity detection, the optimal sensitivity ∆φ ID and the maximal SNR ID is only dependent on the phase shift φ for given input intensity N α and coupling intensity g.Under the balanced condition, the maximal SNR ID is obtained when the phase is 0 and the optimal phase sensitivity ∆φ ID is obtained when the phase is nearby 0. The loss of light field and atomic decoherence will degrade the sensitivity and the SNR of phase measurement, which can be explained from the break of decorrelation conditions.

FIG. 1 :
FIG. 1: (Color online) (a) The intermode correlation between the Stokes field â1 and the atomic excitation b1 is generated by spontaneous Raman process.â0 is the initial input light field.b0 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 â1 will be subject to the photon loss and evolute to â′ 1 and the collective excitation b1 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 â′ 1 and its correlated atomic excitation b′ 1 are used as initial seeding for another enhanced Raman process.(d)-(f) The corresponding energy-level diagrams of different processes are shown.

t 1 and t 2
are the durations of the pump field E P 1 and E P 2 , respectively.