Quality estimation of non-demolition measurement with lossy atom-light hybrid interferometers

D.-H. Fan; S.-Y. Chen; S.-Y. Chen; Z.-F. Yu; Keye Zhang; Keye Zhang; L. Q. Chen; L. Q. Chen

doi:10.1364/OE.385580

1. Introduction

Quantum non-demolition measurement, also known as QND, is a measurement method to avoid the back-action of quantum measurement. The core idea is to transfer the back-action brought by the measurement to the quadrature of quantum field that we are not concerned about. This measurement method was first proposed by Braginsky [1] and Caves [2] theoretically to improve the accuracy of gravitational wave measurements [3]. In 1985, Imoto [4] proposed an interferometer scheme to link signal light and probe light using a nonlinear Kerr effect, enabling QND mechanisms on the measurement of photon number. After that, there are a lot of works about photon number QND measurement by different devices emerging for higher precision and stronger robustness [5–19]. Recently, a new type interferometer, atom-light hybrid interferometer has attracted great attentions [20,21]. The atom-light hybrid interferometer is a SU(1,1) interferometer that combines the advantages of all-optical SU(1,1) interferometers and atomic interferometers [22]. Compared with the traditional SU(2) interferometers, its sensitivity is significantly improved, and applying to QND measurements also yielded better results than traditional SU(2) interferometer solutions [23]. However, we know that in real experiment, there exist various unavoidable losses, which will bring significant effect on the QND measurements even break the quantum state to be measured. There have been some works to consider the influence of losses on the sensitivity of the atom-light interferometers [24–26], but they didn’t involve the QND measurement. In this paper, we consider a QND scheme based on an atom-light hybrid interferometer with various practical losses and derive the theoretical formula for the conditional variance $V_{S|M}$ and the transfer coefficient $T$, which are most commonly-used criteria to judge the quality of a QND scheme [27]. By the parameters obtained in the previous actual experiments we estimate and compare the influence of the losses on the QND photon number measurement with the cold atomic system and with the vapor atomic system. We find that the effect of loss on QND presents some interesting rules. Our results should be helpful for experimental realization of QND measurements via atom-light hybrid interferometer.

In the second part of this paper, we theoretically derive the formula of the QND evaluation criteria of the atom-light hybrid interferometer under the lossy conditions. With the criteria and practically experimental parameters we evaluate the performance of the cold atomic system and the vapor atomic system in the ideal lossless case and lossy case in the third part of the paper. Specially, the influences of the losses of different origins are carefully compared. The final part is the conclusion and the outlook of this work.

2 Criterion for lossy QND measurement

According to [28], we know that judging a QND device requires two independent criteria, one is based on the transfer coefficient $T_{m}$ and $T_{s}$[29], whose expressions are

(1)$$\begin{aligned}T_{s}&=\frac{SNR_{s}^{out}}{SNR_{s}^{in}}, \\ T_{m}&=\frac{SNR_{m}^{out}}{SNR_{s}^{in}}. \end{aligned}$$

$SNR_{s}^{in}$ and $SNR_{s}^{out}$ represent the signal-to-noise ratios(SNR) of the signal light before inputting into and after outputting from the measurement system, respectively. $SNR_{m}^{out}$ is the SNR of the meter light after outputting from the measurement system. The signal light is the light we need to measure. The output meter light is the light outputting from the measurement device, which contains the information of the signal light. The meter light is called as probe light in our scheme. $T_{s}$ and $T_{m}$ describe the transfer of the SNR from the input signal light to the output signal field and to the output meter field, respectively. $T_{s}=1$ represents no loss of signal light after entering the measuring device and $T_{m}=1$ represents the measuring device is noise-free. A perfect QND measurement can be achieved at $T_{s}+T_{m}=2$. A QND measurement should satisfy $1<T_{s}+T_{m}<2$. The other criterion is given by the conditional variance $V_{S|M}$, whose expression is

(2)$$V_{S|M}=V_{\hat{X}_{a}^{out}}(1-C_{\hat{X}_{a}^{out}\hat{Y}_{p}^{out}}^{2}),$$

where

(3)$$C_{\hat{X}_{a}^{out}\hat{Y}_{p}^{out}}^{2}=\frac{|<\hat{X}_{a}^{out}\hat{Y}_{p}^{out}>{-}<\hat{X}_{a}^{out}>{<}\hat{Y}_{p}^{out}>|^{2}}{V_{\hat{X}_{a}^{out}}V_{\hat{Y}_{p}^{out}}},$$

with

\begin{aligned}&\hat{X}_{a}^{out}=\hat{b}^{out}_{s}+\hat{b}^{out\dagger}_{s}, \\ &\hat{Y}_{p}^{out}=\frac{\hat{a}^{out}_{s}-\hat{a}^{out\dagger}_{s}}{i}, \\ &V_{\hat{X}_{a}^{out}}= \left\langle\hat{X}_{a}^{out2}\right\rangle-{\left\langle\hat{X}_{a}^{out}\right\rangle}^{2}, \\ &V_{\hat{Y}_{p}^{out}}= \left\langle\hat{Y}_{p}^{out2}\right\rangle-{\left\langle\hat{Y}_{p}^{out}\right\rangle}^{2}. \end{aligned}

In the above equation, $\hat {X}_{a}^{out}$ is the amplitude quadrature operator of the output signal light, $V_{\hat {X}_{a}^{out}}$ is its variance, and $\hat {Y}_{p}^{out}$ is the phase quadrature operator of the output probe light. $C_{\hat {X}_{a}^{out}\hat {Y}_{p}^{out}}$ represents the correlation degree between the output signal and the probe fields. For a perfect QND measurement $V_{S|M}=0$, which represents the amplitude quadrature operator of the output signal and the phase quadrature operator of the output probe light are completely correlated. The two criteria show how to achieve QND measurement conditions, that is, having enough transfer of SNR from the input signal light to the output light, and enough correlation between the output signal light and the output probe light. However, in actual experiments, there are many losses that cause the QND system to fall short of the optimal state. Below, we derive the specific expressions of the criteria for the atom-light hybrid interferometer with various losses.

The experimental diagram of the QND photon number measurement is shown in the Fig. 1. RA1 and RA2 represent the first and second stimulated Raman process, respectively. $\hat {a}_{s}^{in}$ is the input probe light of the interferometer, and $\hat {s}_{a}^{in}$ is the initial atomic spin wave which is in a vacuum state. $\hat {a}_{s}^{1}$ and $\hat {s}_{a}^{1}$ represent the probe light and the atomic spin wave after the first stimulated Raman process. $\hat {b}_{in}$ is the input signal light that we want to know its photon number. $\varphi$ is the phase difference between the arms of the interferometer, including the phase difference of the interferometer and the phase difference caused by the AC Stark effect. After the first stimulated Raman process, the signal light $\hat {b}$ acts on the atomic spin wave $\hat {a}_{s}^{1}$ and changes the phase of the atoms, and the phase information $\Delta \varphi _{AC}$ which is proportional to the photon number is included in the Stokes light $\hat {a}_{s}^{out}$ and atomic spin wave $\hat {s}_{a}^{out}$ after the second stimulated Raman process. The number of signal photons is known by detecting the output probe light. We assume the signal light field is in a coherent state $|\beta \rangle$ and the probe field is in $|\alpha \rangle$. The input probe field is also in a coherent state and the input atomic spin wave field is in a vacuum state. $L$ represents the loss of the optical field arm mainly caused by the propagation loss [30] and $L_{0}$ the external loss caused by propagation loss, detection loss or other losses caused by the environment. So we adopt two beam splitter models to describe them and $L, L_0=0$ for lossless case and $1$ for $100\%$ loss. For the atomic arm, we use $e^{-\Gamma \tau }$ to describe the loss due to the decoherence of the atomic ground sate with decay rate $\Gamma$ and the decoherence time $\tau$. Specially, the decoherence is caused by the collision between atoms in the vapor atomic system but by the the fluctuations in ambient temperature in the cold atoms.

Fig. 1. Schematic diagram of the QND photon number detection based atom-light hybrid interferometer. $\hat {a}_{s}^{in}$ represents the input Stokes light of the hybrid interferometer, $\hat {s}_{a}^{in}$ the initial atomic spin wave which is in a vacuum state, and $\hat {b}^{in}$ the signal light we want to know its photon number. $\varphi$ is the phase difference between the arms of the interferometer, including the original phase difference and the one caused by the AC Stark effect. $L$ and $L_{0}$ represent the internal loss of one arm of the light field and the external loss, respectively. $e^{-\Gamma \tau }$ represents the loss due to the atomic decoherence.

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Now let’s see the first criterion, the transfer coefficient $T_{S(M)}$. In this scheme, the signal light is a far detuned from the atomic level and is not absorbed. So $T_{S}$ is $1$ and $T_{s}+T_{m}$ is always greater than $1$. We only need to analyze the magnitude of $T_{m}$. But before performing the calculation, we take a comparison between $C_{\hat {X}_{a}^{out}\hat {Y}_{p}^{out}}$ and $T_{m}$ of our scheme. Under the mechanism of QND, we can treat $Y_{p}^{out}$ as the signal we want to measure, that is, the photon number, with a series of gains represented by the coefficient $\mathcal {G}$, resulting a formal transformation

(4)$$\hat{Y}_{p}^{out}=\mathcal{G}\hat{b}^{{\dagger}}_{in}\hat{b}_{in}.$$

For a far-detuned coherent signal light there is no difference between the input intensity and the output intensity, so $\hat {b}^{+}_{in}\hat {b}_{in}=\hat {b}^{\dagger }_{out}\hat {b}_{out}$. And the signal-to-noise ratios $SNR_{m}^{out}= \left \langle \hat {Y}_{p}^{out}\right \rangle /\sqrt {V_{\hat {Y}_{p}^{out}}}$ and $SNR_{s}^{in}= \left \langle \hat {X}_{a}^{in}\right \rangle /\sqrt {V_{\hat {X}_{a}^{in}}}$. Substituting them into Eq. (1) and Eq. (3) we have

(5)$$C_{\hat{X}_{a}^{out}\hat{Y}_{p}^{out}}=\frac{\mathcal{G}|\beta|}{\sqrt{V_{\hat{X}_{a}^{out}}V_{\hat{Y}_{p}^{out}}}},$$

(6)$$T_{m}=\frac{SNR_{m}^{out}}{SNR_{s}^{in}}=\frac{1}{2}\mathcal{G}|\beta|\frac{\sqrt{V_{\hat{X}_{a}^{in}}}}{\sqrt{ V_{\hat{Y}_{p}^{out}}}},$$

Considering $V_{\hat {X}_{a}^{in}}= V_{\hat {X}_{a}^{out}}=1$, we obtain the relationship

(7)$$C_{\hat{X}_{a}^{out}\hat{Y}_{p}^{out}}=2T_{m},$$

which means the two criteria of QND coincide with each other in our scheme. This is due to the coherent and far-detuned precondition of the input signal field. So in the following we focus on the conditional variance criterion $V_{S|M}$ only, which requests that the correlation degree $C_{\hat {X}_{a}^{out}\hat {Y}_{p}^{out}}=1$ for an ideal QND by our scheme. However, in this case $T_{m}=0.5$ and then the maximum value of $T_s+T_m$ is $1.5$, which means the perfect QND measurement is unavailable in our scheme even in the ideal case. This is attributed to the nonlinear amplification effect in the beam splitting steps of our atom-light hybrid interferometer.

The exact correlation degree with losses can be estimated by the input-output relationships of the hybrid interferometer, which are

(8)$$\hat{a}_{s}^{out}=\sqrt{1-L_{0}}[G^{\prime}(\varphi)\hat{a}_{s}^{in}+g^{\prime}(\varphi)\hat{s}_{a}^{in+}+g_{2}\hat{F}^{{\dagger}}+G_{2}\sqrt{L_{2}}\hat{a}_{s0}^{in}]+\sqrt{L_{0}}\hat{a}_{s0}^{out},$$

It should be noted that the operator with an subscript of 0 represents the vacuum state, where $G^{'}(\varphi )$ and $g^{'}(\varphi )$ are the phase sensitive gains of the interferometer with the expressions,

\begin{aligned} G^{'}(\varphi)=g_{1}g_{2}e^{i\varphi}e^{-\Gamma\tau}+G_{1}G_{2}\sqrt{1-L},\\ g^{'}(\varphi)=G_{1}g_{2}e^{i\varphi}\sqrt{1-L}+g_{1}G_{2}\sqrt{1-L}.\end{aligned}

Here the phase difference $\varphi =\varphi _{0}+\Delta \varphi _{AC}$, where $\varphi _{0}$ represents the original phase difference between the two arms assuming that the interferometer is not affected by the AC Stark effect and $\Delta \varphi _{AC}$ the phase difference caused by the AC Stark effect. The AC Stark phase shift $\Delta \varphi _{AC}=k\hat {b}^{\dagger }\hat {b}$ with $k$ the AC Stark effect coefficient and $\hat {b}^\dagger \hat {b}$ the signal photon number operator. For simplicity we assume $\varphi _{0}=\pi$ in the following calculations, and $\Delta \varphi _{AC}<<{\pi }$. The operator $\hat {F}=\int _{0}^{\tau }e^{-\Gamma (\tau -t')}\hat f(t')\,dt'$, where $\hat f(t)$ is the quantum Langevin operator describing the collision-induced fluctuation and obeys the correlations $\left \langle \hat {f}(t)\hat {f}^{\dagger }(t')\right \rangle =2\Gamma \delta (t-t')$ and $\left \langle \hat {f}^{\dagger }(t)\hat {f}(t')\right \rangle =0$, so $\hat {F}$ satisfies $\left \langle \hat {F}(t)\hat {F}^{\dagger }(t')\right \rangle =1-e^{-2\Gamma \tau }$. This relationship ensures that the $\hat {a}$ operator still satisfies the bosonic commutation relationship after the atomic loss $e^{-\Gamma \tau }$ is introduced. $G_1$ and $G_2$ as well as $g_1$ and $g_2$ are the gain of beam splitting and combining. We assume $G_{1}=G_{2}=G$ and $g_{1}=g_{2}=g$ below and the stimulated Raman process gives $G^2-g^2=1$.

According to the definition of the correlation degree in Eq. (3), we first derive the variance,

(9)$$\begin{aligned}V_{\hat{Y}_{p}^{out}}&=\left\langle\hat{Y}_{p}^{out2}\right\rangle-{\left\langle\hat{Y}_{p}^{out}\right\rangle}^{2} \\ &=[e^{{-}2\Gamma\tau}(4g^{4}k^{2}|\beta|^{2}|\alpha|^{2}+1+g^{4}k^{2}|\beta|^{4}+g^{4}k^{2}|\beta|^{2}) \\ &+g^{4}(\sqrt{1-L}-e^{-\Gamma\tau})^{2}+2\sqrt{1-L}g^{2}(\sqrt{1-L} \\ & -e^{-\Gamma\tau})+(1-L)+G^{2}L](1-L_{0})+G^{2}g^{2}[ \\ &(\sqrt{1-L}-e^{-\Gamma\tau})^{2}+e^{{-}2\Gamma\tau}k^{2}|\beta|^{4}+e^{{-}2\Gamma\tau}k^{2}|\beta|^{2}] \\ &(1-L_{0})+g^{2}(1-e^{{-}2\Gamma\tau})+L_{0}, \end{aligned}$$

where we set the phase difference between the probe light and the local light to be zero in Fig. 1. Because $\hat {X}_{a}^{out}=\hat {b}_{out}+\hat {b}^{\dagger }_{out}$, it’s easy to obtain

(10)$$\left\langle\hat{X}_{a}^{out}\hat{Y}_{p}^{out}\right\rangle=\sqrt{1-L}_{0}e^{-\Gamma\tau}({-}4g^{2}k|\beta|^{3}|\alpha|-2g^{2}k|\beta||\alpha|),$$

(11)$$\left\langle\hat{X}_{a}^{out}\right\rangle\left\langle\hat{Y}_{p}^{out}\right\rangle={-}4\sqrt{1-L}_{0}e^{-\Gamma\tau}g^{2}k|\beta|^{3}|\alpha|.$$

So the correlation degree is

(12)$$\begin{aligned}C_{\hat{X}_{a}^{out}\hat{Y}_{p}^{out}}^{2}&=\frac{|\left\langle\hat{X}_{a}^{out}\hat{Y}_{p}^{out}\right\rangle-\left\langle\hat{X}_{a}^{out}\right\rangle\left\langle\hat{Y}_{p}^{out}\right\rangle|^{2}}{V_{\hat{X}_{a}^{out}}V_{\hat{Y}_{p}^{out}}} \\ & =\frac{4(1-L_{0})e^{{-}2\Gamma\tau}g^{4}k^{2}|\beta|^{2}|\alpha|^{2}}{V_{\hat{Y}_{p}^{out}}V_{\hat{X}_{a}^{out}}}. \end{aligned}$$

with

(13)$$\begin{aligned}V_{\hat{Y}_{p}^{out}}V_{\hat{X}_{a}^{out}}&=[e^{{-}2\Gamma\tau}(4g^{4}k^{2}|\beta|^{2}|\alpha|^{2}+1+g^{4}k^{2}|\beta|^{4}+ \\ & g^{4}k^{2}|\beta|^{2})+g^{4}(\sqrt{1-L}-e^{-\Gamma\tau})^{2}+2\sqrt{1-L}g^{2} \\ & (\sqrt{1-L}-e^{-\Gamma\tau})+(1-L)+G^{2}L](1-L_{0})+ \\ & G^{2}g^{2}[(\sqrt{1-L}-e^{-\Gamma\tau})^{2}+e^{{-}2\Gamma\tau}k^{2}|\beta|^{4}+ \\ &e^{{-}2\Gamma\tau}k^{2}|\beta|^{2}](1-L_{0})+g^{2}(1-e^{{-}2\Gamma\tau})+L_{0}. \end{aligned}$$

3. Loss impact on QND criteria and analysis

In the present section we will use the actual parameters of two kinds of atom-light hybrid interferometers and the exact expression of the QND criterion derived above to estimate and compare the influences of various losses. The one is based on rubidium atomic vapor with a cell temperature $343 K$ and the other is on the the cold rubidium atomic system with a cell temperature $100\mu K$[31]. For the atomic vapor, we set the input probe light of the interferometer at the photon frequency $0.5GHz$ blue-detuned from the atomic transition $|5^{2}S_{1/2},F=1\rangle \rightarrow |5^{2}P_{1/2},F=2\rangle$. Its power is $50\mu W$ with $0.9mm$ spot size. The atom number in interaction region is about $10^{10}-10^{11}$. The pulse width is $300ns$, and the photon number $|\alpha |^{2}=1.17\times 10^{8}$, which is much smaller than the atom number. The signal light is $2GHz$ blue-detuned from the transition $|5^{2}S_{1/2},F=1\rangle \rightarrow |5^{2}P_{1/2},F=2\rangle$ [32]. The AC Stark coupling coefficient is inversely proportional to the detuning, so $k=3.19\times 10^{-10}$ [33]. In the case of cold rubidium atoms, due to the weak Doppler broadening the detuning of the signal light can be decreased to $120MHz$. Moreover, the correlation between the signal light and the cold atoms is enhanced by the optical cavity [34,35], so the AC Stark coupling coefficient $k=5.4\times 10^{-4}$ which is much larger than the case of atomic vapor, and the photon number $|\alpha |^{2}=1.755\times 10^{3}$, which is much smaller than the atom number $~10^{6}-10^{7}$ for cold atom system.

3.1 Lossless condition

Before analyzing the impact of the losses, in Fig. 2 we display the QND quality of the scheme in the ideal lossless case as a benchmark. Here, we substitute the different parameters of the previous two atom systems and changed the intensity of the signal light to observe the changes of the condition variance $V_{S|M}$. Because the fluctuation of the signal light $V_{\hat {X}_{a}^{out}}$ is $1$, we only need to show the correlation degree $C_{X_{a}^{out}Y_{p}^{out}}$ whose value reaches $1$ for a perfect QND. For simplicity, we use $C$ instead of $C_{X_{a}^{out}Y_{p}^{out}}$ below. In additional, the present QND system is based on an SU(1,1) atom-light hybrid interferometer whose splitting and combining processes are implemented by the Raman amplifier (RA). So the gain coefficient $g$ of the RA is an important parameter for the QND. Considering the value range of $g$ in the actual experiments, we then choose three different values of $g$ in the following comparison.

Fig. 2. (a) The correlation coefficient $C$ of the vapor atomic system for a varying photon number of the signal light (SPN) at three different gain coefficients $g$. When $g=1,2$, and $3$ the maximum value of $C$ is $0.2033$, $0.4074$, and $0.5623$, respectively. (b) The case for the cold atoms. The maximum values are $0.7727$, $0.9136$, and $0.9598$. All losses are assumed to be zero, and the other parameters are given in the main text.

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From Figs. 2(a) and 2(b) we can see that, on the whole, the value of $C$ for the cold atomic system is larger than for the vapor atomic system and for the high $g$ case is larger than for the low $g$ case. That means a better QND prefers a lager RA gain coefficient and a lower atom temperature. It is due to QND parameter depends on phase sensitivity of interferometer and the AC Stark coefficient $k$, which can be improved by the intensity of phase sensitive arm with larger $g$ and a lower atom temperature respectively. But it is still necessary to pay attention to the range of the signal photon number to be measured to ensure the QND device under its best working status. That is because as the increase of the signal photon number, the phase shift caused by Stark effect becomes larger but the noise of the signal light also increase. For the vapor atomic system, the QND device works for a larger photon number measurement and our calculation suggests that the number is preferably between $1.2\times 10^{8}$ and $5.6\times 10^{8}$ to keep $C$ above 0.5. For the cold atomic system, the device works for a low photon number case and the better range is between $26$ and $750$, where $C$ can be kept above 0.9.

3.2 External loss

In present section we add an important loss in actual experiments, the external losses to the analysis, which includes detection loss, loss of light through various optical components after the interferometer, and so on. In order to display the influence of the external loss, we have done a three-dimensional multi-layer plot of $C$ in Fig. 3 with a varying loss rate $L_0$.

Fig. 3. The influence of the external loss rate $L_0$ and the signal photon number (SPN) on the correlation coefficient $C$ of (a) the vapor atomic system and (b) the cold atomic system under the three different values of the gain coefficient $g$. The atomic decoherence rate $\Gamma$ and the internal loss rate $L$ are assumed to be zero, and the other parameters are same as in the Fig. 2.

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From the Figs. 3(a) and 3(b) we can find that for both atomic systems the larger the gain coefficient $g$, the smaller the influence of external loss on $C$. For a more detailed comparison, we choose $L_0=0.2$ which represents a usual $20\%$ loss rate in experiments, and $g=3$ and SPN$=2.4\times 10^{8}$ for the vapor atomic system to reach its optimized QND region. $C$ in this case is $0.5044$ dropped by $10\%$ than the lossless case. For the cold atomic system we choose SPN$=140$ and obtain $C=0.9551$, dropped by only $0.5\%$. So in this QND scheme, the effects of external loss is negligible in the cold atomic system but is an important loss for the vapor atomic system.

3.3 Internal loss

The internal loss refers to the propagation loss of the optical arm for the atom-light hybrid interferometer. We change the internal loss rate $L$ and show the correlation coefficient $C$ in Fig. 4. Obviously the larger the loss the smaller the correlation. This relation becomes more distinct with increasing $g$. But different from the case of external loss, $C$ doesn’t vanish even at $L=1$. That is because without the first Raman process the second one still builds correlation between the signal light and the probe light. However, there is no interference and the QND measurement is no longer feasible in this case.

Fig. 4. The influence of the internal loss rate $L$ and the signal photon number (SPN) on the correlation coefficient $C$ of (a) the vapor atomic system and (b) the cold atomic system under the different gain coefficients $g$. The atomic decoherence rate $\Gamma$ and the external loss rate $L_0$ are assumed to be zero, and the other parameters are same as in Fig. 2.

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To give a more precise comparison to the lossless case, we consider that fiber coupling loss, and other propagation losses. The sum of them is $0.20$. For this loss rate, the optimized $C$ of the vapor atomic system is $0.3961$ that means a $30\%$ decrease of the lossless case. For the cold atomic system, $C$ is $0.9072$ and the decrease is $5.5\%$. Compared with the external loss, the impact of the internal loss on the QND device is more obvious. However, we can see from the Fig. 4(a) that the surface of $C$ has a steep rise when $L$ decreases. So if there is a laser with a more pure spatial mode or a method that can improve the efficiency of single-mode coupling, then the improvement of the effect of the QND scheme of the vapor atomic system is very huge. We find the internal loss has better effect on the vapor atomic system and the cold atomic system than the external loss in the 20% loss. This is because the external loss just brings vacuum noise into the output field and then the noise is detected by homodyne detection, but the internal loss brings vacuum noise into the interference arms and then the vacuum noise will be amplified by the second Raman process and be detected by the homodyne detection.

3.4 Atomic decoherence loss

The loss of the atomic arm is mainly due to the decoherence loss caused by the collision between atoms for the vapor atomic system and to the fluctuations of ambient temperature for the cold atomic system.

As shown in Figs. 5(a) and 5(b), same as in the case of internal loss the larger the gain coefficient $g$, the steeper the curve of $C$. It has the same cause as the internal loss. In our previous experiments, for the vapor atomic system the walls of the atomic cell were plated with paraffin and the coherence time of the atoms was greatly extended, the atomic coherence time is $1ms$ and the decoherence time is $400ns$ . So $\Gamma \tau =0.0004$ and $e^{-\Gamma \tau }=0.9996$ that is the atomic decoherence loss rate is $0.04\%$. But if you use uncoated atomic pools, the loss can be as high as 40%. When the gain $g=3$, the optimized $C = 0.5613$, that is a $0.2\%$ decrease to no loss. It should be noted that the atomic loss rate could be more close to $0$ because the current most advanced coating technology allows the atomic coherence time to be $1$ minute [36]. For the cold atomic system, the atomic coherence time is $3.2ms$[37] and the decoherence time is $400ns$. In this case, $C$ = 0.9597, corresponding to a $0.01\%$ decrease to the ideal case. You can see that the atomic decoherence loss has almost no effect on the cold atoms. Here, we substitute 20% loss for comparison to observe the effect of different losses on QND under the same loss value. For the varpor atomic system, $C$ = 0.2059, corresponding to a $74\%$ decrease to the ideal case. For the cold atomic system, $C$ = 0.7232, corresponding to a $25\%$ decrease to the ideal case. Compared with the other two losses, atomic loss have the greatest impact on the QND effect, because the phase shift is coupled into atom spin wave.

Fig. 5. The influence of the atomic loss rate $e^{-\Gamma \tau }$ and the signal photon number (SPN) on the correlation coefficient $C$ of (a) the vapor atomic system and (b) the cold atomic system under the different gain coefficients $g$. The external loss rate $L_0$ and the internal loss rate $L$ are assumed to be zero, and the other parameters are same as in the Fig. 2.

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4 Conclusion and discussion

We derived the QND evaluation criteria of an atom-light hybrid SU(1,1) interferometer in lossy condition and compare the performance of the vapor and cold atomic systems with actual experimental parameters. In general, the hybrid interferometer with cold atoms works better in the QND measurement of photon number. That is understandable because the AC Stark coefficient $k$ of the cold atomic system is much larger than that of the vapor atomic system, which means a larger number-dependent phase shift and loss tolerance. Besides, the QND quality does not monotonously increase with the increase of signal photon number. The optimized working range of the signal photon number are estimated by our detailed calculation. Its value of the cold atom case is much smaller than that of the vapor atom case. Further, when input probe intensity $|\alpha |^{2}$ is constant, we find the larger the gain coefficient $g$, the greater the QND quality. Besides the gain and signal photon number, the QND quality also depends on the probe photon number. The $C$ value increases with the probe photon number, but does not monotonously increase. When the photon number is comparable with the atom number, the atoms will be insufficient for the Raman amplification process, and then the $C$ value will decrease. So in our paper, we choose the probe photon number of $|\alpha |^{2}=1.17\times 10^{8}$ and $|\alpha |^{2}=1.755\times 10^{3}$ in vapor and cold atomic systems respectively. They are much smaller than the atom numbers to assure optimized $C$ values.

Here, we achieve QND measurment via atom-light interferometer, whose phase sensitivity can break standard quantum limit. For other quantum interferometer, such as the all-optical SU(1,1) interferometer, also can achieve the same sensitivity as the atom-light hybrid interferometer, but its two interference arms are both light fields. In QND detection, its signal light and probe light can only be coupled through a nonlinear medium, which will bring many unnecessary losses. But the atoms in this paper can be coated so that the loss is very low. Therefore, atom-light hybrid interferometer is better than all-optical interferometer in QND detection. We know the application of quantum field in real life is developing rapidly, and the corresponding exploration of quantum information will also be paid more and more attention in recent years. Our work has some guiding significance for the practical application of the atom-light hybrid interferometer QND system. By choosing the appropriate operating range and taking some measures to reduce the loss, I believe that the QND system of atom-light hybrid interferometer will be widely applied in the future.

Funding

National Natural Science Foundation of China (11874152, 11574086, 11974116, 91536114); Natural Science Foundation of Shanghai (17ZR1442800); Fundamental Research Funds for the Central Universities.

Disclosures

The authors declare no conflicts of interest.

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Quality estimation of non-demolition measurement with lossy atom-light hybrid interferometers

Abstract

1. Introduction

2 Criterion for lossy QND measurement

3. Loss impact on QND criteria and analysis

3.1 Lossless condition

3.2 External loss

3.3 Internal loss

3.4 Atomic decoherence loss

4 Conclusion and discussion

Funding

Disclosures

References

Cited By

Figures (5)

Equations (15)

Optics Express