## Abstract

We give the general expressions of intensity-difference squeezing (IDS) generated from two types of optical parametric amplifiers [i.e. phase-sensitive amplifier (PSA) and phase-insensitive amplifier (PIA)] based on the four-wave mixing process, which clearly shows the IDS transition between the ultra-low average input photon number regime and the ultra-high average input photon number regime. We find that both the IDS of the PSA and the IDS of the PIA get enhanced with the decrease of the average input photon number especially in the ultra-low average input photon number regime. This result is substantially different from the result in the ultra-high average input photon number regime where the IDS does not vary with the average input photon number. Moreover, under the same intensity gain, we find that the optimal IDS of the PSA is better than the IDS of the PIA in the ultra-low average input photon number regime. Our theoretical work predicts the presence of strong quantum correlation in the ultra-low average input photon number regime, which may have potential applications for probing photon-sensitive biological samples.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Quantum squeezing is an important quantum resource due to its advantage in quantum noise reduction [1] and thus has been proved to be useful to improve the performance of measurement systems. For example, it has been used to enhance measurement precision of microcantilever displacement [2] and rotation angle [3,4]. It has also been exploited to increase the sensitivities of the Laser Interferometer Gravitational-Wave Observatory (LIGO) [5–7], optical magnetometer [8], cavity optomechanical magnetometer [9] and various interferometers [10–14].

A number of different techniques for the generation of quantum squeezing have been widely studied [15–20]. Among them, a phase-insensitive amplifier (PIA) based on a four-wave mixing (FWM) process in a $\sideset {^{85}}{}{\mathop {\mathrm {Rb}}}$ vapor cell with a double-$\Lambda$ energy level configuration has been shown to be a promising technique to generate quantum squeezed light due to its strong nonlinearity, multispatial mode nature and natural spatial separation of the output beams [20–35]. In this system, a strong pump beam and a weak coherent beam (i.e. probe beam) are crossed at the center of the $\sideset {^{85}}{}{\mathop {\mathrm {Rb}}}$ vapor cell. Then, a pair of intensity-difference squeezed beams (i.e. the amplified probe beam and a simultaneously generated conjugate beam) is produced, which has been used in several recent advances, such as quantum imaging [23,36–39], the tunable delay of Einstein-Podolsky-Rosen entanglement [40], the implementation of an SU(1,1) nonlinear interferometer [41–45] and the generation of the orbital-angular-momentum multiplexed continuous-variable entanglement [46,47]. Recently, our group has theoretically proposed [48] and experimentally realized [49] a phase-sensitive amplifier (PSA) in which two weak coherent beams (i.e. the probe and conjugate beams) are simultaneously and symmetrically crossed with a strong pump beam at the center of the $\sideset {^{85}}{}{\mathop {\mathrm {Rb}}}$ vapor cell. The intensity-difference squeezing (IDS) generated from this system is better than the IDS generated from the PIA under the same experimental situation.

However, the IDS generated from the PIA and the PSA mentioned above is limited to the bright-beam approximation regime [20–28,48,49], which means both the PIA and the PSA are working in the ultra-high average input photon number regime. In other words, the IDS analysis of the PIA and the PSA based on the FWM process in the ultra-low average input photon number regime is still missing. In this paper, different from the previous works considering the case without seeding [50,51], we theoretically give the general expressions of IDS generated from the PSA and the PIA based on the FWM process, which clearly shows the IDS transition between the ultra-low average input photon number regime and the ultra-high average input photon number regime. We pay more attention to the characterization of IDS in the ultra-low average input photon number regime and report several significant results. We find that both the IDS of the PSA and the IDS of the PIA get enhanced with the increase of the intensity gain. The maximal IDS of the PSA can be obtained when its phase is equal to zero. These results are similar to the bright-beam input case. More importantly, we find that both the IDS of the PSA and the IDS of the PIA get enhanced with the decrease of the average input photon number. This result is substantially different from the case of ultra-high average input photon number with the bright-beam approximation. In addition, under the same intensity gain, we find that the optimal IDS of the PSA is always better than the IDS of the PIA in the ultra-low average input photon number regime. Our theoretical results here predict the presence of strong quantum correlation in the ultra-low average input photon number regime, which may have potential applications for probing photon-sensitive biological samples [52–55].

## 2. Quantum squeezing of the PIA and the PSA

We start with the derivation of the IDS of the PSA. The basic configuration of the PSA is shown in Fig. 1(a). Two weak coherent beams [i.e. the probe ($\hat {a}$) and conjugate ($\hat {b}$) beams] are simultaneously and symmetrically crossed with a strong pump beam ($\hat {c}$) at the center of the $\sideset {^{85}}{}{\mathop {\mathrm {Rb}}}$ vapor cell. As shown in Fig. 1(b), the probe and the conjugate beams are red- and blueshifted by 3.04 GHz from the pump beam, respectively. Two pump photons are annihilated in this process, inducing the emission of one probe photon and one conjugate photon. Labeling the interaction strength of the PSA by the parameter $\kappa$, the interaction Hamiltonian can be expressed as [56,57]

where $\theta =2{\phi }_{c}$, ${\phi }_{c}$ is the phase of the pump field. The value of $\kappa$ is not only dependent on the pump power, but also dependent on one-photon detuning ($\Delta$) and two-photon detuning ($\delta$). From Eq. (1), the output probe and conjugate fields of the PSA can be written as*G*= ${\cosh }^2 (\kappa \tau )$ is the intensity gain of the PSA and $\tau$ is the interaction length.

Defining the photon number operator of the input probe and conjugate fields as $\hat {N}_{a,in}=\hat {a}^{\dagger }(0)\hat {a}(0)$ and $\hat {N}_{b,in}=\hat {b}^{\dagger }(0)\hat {b}(0)$ respectively, the average photon number of the two output fields of the PSA are given by [48]

Therefore, the degree of IDS for the PSA can be described as

When seeding a vacuum field to one of the input ports of the PSA, the PIA configuration can be obtained as shown in Fig. 2. Note that we seed a vacuum field to the conjugate port which means $\gamma =0$. From Eq. (8), the degree of IDS for the PIA can be expressed as

As we can see from Eq. (8) and Eq. (9), the terms $2(G-1)/\left \langle \hat {N}_{a,in}\right \rangle (1+\gamma)$ and $2(G-1)/\left \langle \hat {N}_{a,in}\right \rangle$ will be negligible in the ultra-high average input photon number regime with the bright-beam approximation since $\left \langle \hat {N}_{a,in}\right \rangle >>1$. Therefore, both the IDS of the PSA and the IDS of the PIA will be independent on the average input photon number. These models well explain the previous works [20–28,48,49]. However, as the decrease of the average input photon number, especially in the ultra-low average input photon number regime, these two terms can not be neglected anymore. Therefore, both the IDS of the PSA and the IDS of the PIA are dependent on the intensity gain *G* and the average input photon number of probe field $\left \langle \hat {N}_{a,in}\right \rangle$. Meanwhile, the IDS of the PSA also depends on the phase $\phi$ and the input conjugate-probe power ratio $\gamma$. Therefore, it is valuable to study how these factors influence the performance of the PSA and the PIA. Firstly, in order to analyze the effect of gain on the IDS of these two systems, we set the phase of the PSA at 0 and plot the intensity-difference noise powers of the PSA and the PIA as a function of the gain when the average input photon number of the probe field $\left \langle \hat {N}_{a,in}\right \rangle =1$ as shown in Fig. 3. It should be noted that for convenience, we consider that two input ports of the PSA have the same average input photon number (i.e. $\gamma =1$). From Fig. 3, we can directly see that for a given $\left \langle \hat {N}_{a,in}\right \rangle$, both the IDS of the PSA and the IDS of the PIA can be enhanced with the increase of the gain *G*.

Secondly, we illustrate how the IDS of the PSA and the IDS of the PIA depend on the phase $\phi$. In order to study these effects, we plot the intensity-difference noise powers of the PSA and the PIA as a function of the phase $\phi$ by setting the gain *G* at 10 and the average input photon number of the probe field $\left \langle \hat {N}_{a,in}\right \rangle$ at 1 as shown in Fig. 4. Here we also fix the ratio $\gamma$ of the PSA at 1. It can be seen intuitively that the IDS of the PSA can be manipulated by the phase and achieves its maximal degree of squeezing when $\phi =0$, while the IDS of the PIA does not depend on the phase. These results are similar to our previous studies in the ultra-high average input photon number regime [48,49].

Thirdly, in order to explore how the average input photon number of the probe field $\left \langle \hat {N}_{a,in}\right \rangle$ and the input conjugate-probe power ratio $\gamma$ together influence the IDS of the PSA and the IDS of the PIA, we fix the gain *G* at 10, the phase $\phi$ at 0, and plot the intensity-difference noise power of the PSA as a function of $\left \langle \hat {N}_{a,in}\right \rangle$ and $\gamma$ as shown in Fig. 5(a). The red solid line corresponding to the case of $\gamma =0$ is the dependence relation of the intensity-difference noise power of the PIA on $\left \langle \hat {N}_{a,in}\right \rangle$. The Fig. 5(b) is the expanded plot where $0<\gamma <0.5$ and $0.01<\left \langle \hat {N}_{a,in}\right \rangle <1$. First of all, it can be seen that both the IDS of the PSA and the IDS of the PIA get enhanced with the decrease of the average input photon number of probe field $\left \langle \hat {N}_{a,in}\right \rangle$. Then it can also be found that for a given $\left \langle \hat {N}_{a,in}\right \rangle$, the IDS of the PSA gets better first and then gets worse with the decrease of $\gamma$. In other words, there exists an optimal $\gamma$ which can optimize the IDS of the PSA. It can be expressed as

*G*and phase $\phi$. The dependence relation of the corresponding optimal intensity-difference noise power of the PSA on $\left \langle \hat {N}_{a,in}\right \rangle$ and $\gamma$ is indicated as the black dashed line in Fig. 5.

In order to better compare the IDS of the PIA and the IDS of the PSA in the ultra-low average input photon number regime, we set the phase of the PSA at 0 and plot the intensity-difference noise power of the PIA, the optimal intensity-difference noise power of the PSA and the optimal $\gamma$ of the PSA as a function of the average input photon number of the probe field $\left \langle \hat {N}_{a,in}\right \rangle$ under the same gain *G*=10 as shown in Fig. 6(a). The Figs. 6(b), (c) and (d) are the expanded plots where $0.001<\left \langle \hat {N}_{a,in}\right \rangle <1$, $0.001<\left \langle \hat {N}_{a,in}\right \rangle <0.1$ and $0.001<\left \langle \hat {N}_{a,in}\right \rangle <0.01$, respectively. As we can see that the optimal IDS of the PSA is always better than the IDS of the PIA. Moreover, as the decrease of the average input photon number, the optimal IDS of the PSA becomes almost the same as the IDS of the PIA. This means that the IDS enhancement of the PSA decreases with the decrease of the average input photon number. In addition, we can see that the optimal $\gamma$ decreases with the decrease of the average input photon number and tends to 0 when the average input photon number is sufficiently low. Since there are unavoidable loss in any real experiment, we take two types of losses into consideration, namely, the external loss that occurs after mixing (imperfect optical transmission and detection efficiency) and the internal loss (the atomic absorption in $\sideset {^{85}}{}{\mathop {\mathrm {Rb}}}$ vapor cell). In general, the external loss can be modeled by a beamsplitter with an empty port whose output state is a combination of the input and vacuum modes [58]. For simplicity, we also consider the internal loss as a beamsplitter with an empty port, contributing vacuum fluctuations to the transmitted beams. Denoting the vacuum modes introduced by loss on the probe and conjugate by the annihilation operators ${\hat {\nu }}_{i}$ ($i=1, 2, 3, 4$) respectively [59], the standard beamsplitter input-output relations give

*L*. Then, we can easily calculate the degrees of IDS of the PIA and the PSA followed by optical loss. The results are shown in Fig. 7. As shown in Fig. 7, we plot the intensity-difference noise powers of the PIA and the optimal PSA with phase $\phi =0$ as a function of the average input photon number of the probe field $\left \langle \hat {N}_{a,in}\right \rangle$ under the same gain $\textit {G}=10$ followed by optical loss. It can be found that the degrees of IDS of two types of optical parametric amplifiers decrease with the increase of loss. Moreover, as the increase of the average input photon number, the variation of IDS decreases. This is because the higher the IDS is, the more sensitive it is to the loss. In addition, the IDS of the optimal PSA is still better than the IDS of the PIA, which is consistent with the case of without loss.

## 3. Conclusion

In conclusion, we have given the general expressions of the IDS generated from the PSA and the PIA based on the FWM process, which clearly shows the IDS transition between the ultra-low average input photon number regime and the ultra-high average input photon number regime. We pay more attention to the characterization of IDS in the ultra-low average input photon number regime and report several significant results. We show that both the IDS of the PSA and the IDS of the PIA can be enhanced with the increase of the intensity gain. We also show that the maximal IDS of the PSA can be obtained when its phase is equal to zero. These results are similar to the bright-beam input case of our previous studies in the ultra-high average input photon number regime [48,49]. More importantly, we show that both the IDS of the PSA and the IDS of the PIA get enhanced with the decrease of the average input photon number. In addition, we show that the optimal IDS of the PSA is always better than the IDS of the PIA with the same intensity gain. On the one hand, these theoretical findings are valuable for better and fully understanding the quantum squeezing generated from the FWM process. On the other hand, they predict the presence of strong quantum correlation in the ultra-low average input photon number regime and hence may have potential applications for probing photon-sensitive biological samples [52–55].

## Funding

National Natural Science Foundation of China (11874155, 91436211, 11374104); Basic Research Project of Science and Technology Commission of Shanghai Municipality (20JC1416100); Natural Science Foundation of Shanghai (17ZR1442900); Minhang Leading Talents (201971); ECNU Academic Innovation Promotion Program for Excellent Doctoral Students (YBNLTS2020-046); Program of Scientific and Technological Innovation of Shanghai (17JC1400401); the National Basic Research Program of China (2016YFA0302103); the 111 project (B12024).

## Disclosures

The authors declare no conflicts of interest.

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