Quantum squeezing and entanglement from a two-mode phase-sensitive amplifier via four-wave mixing in rubidium vapor

To show the amplification and de-amplification of the probe field, we need to derive the relation between the average photon number of the input and the output fields. From equation (2), the mean value of number operator ${{\hat{N}}_{a,{\rm out}}}$ of the output probe beam is given by:

$$\begin{eqnarray}&&\langle {{\hat{N}}_{a,{\rm out}}}\rangle =G\langle \widehat{{{N}_{a}}}\rangle +(G-1)\langle \widehat{{{N}_{b}}}\rangle +G-1+2\sqrt{\langle \widehat{{{N}_{a}}}\rangle }\sqrt{\langle \widehat{{{N}_{b}}}\rangle }\sqrt{G(G-1)}{\rm cos} \phi \end{eqnarray} \tag{ 4 }$$

where $\langle \widehat{{{N}_{a}}}\rangle$ and $\langle \widehat{{{N}_{b}}}\rangle$ represent the average input photon number of the probe and conjugate fields respectively. ϕ is equal to $2{{\phi }_{c}}-{{\phi }_{a}}-{{\phi }_{b}}$, where ${{\phi }_{a}}$ and ${{\phi }_{b}}$ are the phase of the probe and conjugate fields respectively. Considering the probe beam is bright, $\langle \widehat{{{N}_{a}}}\rangle \gg 1$. Therefore, the third term $G-1$ in equation (4), which is due to the spontaneous emission during the FWM process, is negligible. In particular, when the conjugate port is seeded with vacuum, which means $\langle \widehat{{{N}_{b}}}\rangle =0$, we have $\langle {{\hat{N}}_{a,{\rm out}}}\rangle =G\langle \widehat{{{N}_{a}}}\rangle$. This is consistent with the result of the PIA in [27]. It is obvious from equation (4) that the intensity of output probe field depends not only on the phase ϕ but also the ratio β (defined as $\langle {{\hat{N}}_{a}}\rangle /\langle {{\hat{N}}_{b}}\rangle$)between the average photon number of the input probe and conjugate beams. For simplicity, we focus on the situation when $\beta =1$ at present. This will reduce equation (4) as follows:

$$\begin{eqnarray}&&\langle {{\hat{N}}_{a,{\rm out}}}\rangle =(2G-1)\langle \widehat{{{N}_{a}}}\rangle +2\sqrt{G(G-1)}\langle \widehat{{{N}_{a}}}\rangle {\rm cos} \phi \end{eqnarray} \tag{ 5 }$$

To study the amplification of the probe beam, it is convenient to define the effective gain ${{G}_{{\rm eff},{\rm probe}}}$ for probe light as $\langle {{\hat{N}}_{a,{\rm out}}}\rangle /\langle {{\hat{N}}_{a,{\rm in}}}\rangle$. It turns out to be:

$$\begin{eqnarray}&&{{G}_{{\rm eff},{\rm probe}}}=2G-1+2\sqrt{G(G-1)}{\rm cos} \phi \end{eqnarray} \tag{ 6 }$$

As the phase ϕ varied, the effective gain ${{G}_{{\rm eff},{\rm probe}}}$ varies between ${{G}_{{\rm eff},{\rm probe}}}\gt 1$ (amplification) and ${{G}_{{\rm eff},{\rm probe}}}\lt 1$ (de-amplification) as shown in figure 2. When ${{G}_{{\rm eff},{\rm probe}}}\gt 1$, two pump photons convert to one probe photon and one conjugate photon, otherwise one probe photon and one conjugate photon convert to two pump photons. The conversion efficiency depends on the phase of input beams. From the inset plot of figure 2, it is clear that with higher gain, smaller effective gain is obtained at $\phi =\pi$.

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It is therefore worth studying the influence of ϕ and β on the interference visibility for the two output ports. We define the visibility V as $({{N}_{{\rm max} }}-{{N}_{{\rm min} }})/({{N}_{{\rm max} }}+{{N}_{{\rm min} }})$. The visibility for the probe and conjugate output ports are given by:

$$\begin{eqnarray}&&{{V}_{{\rm probe}}}=\frac{2\sqrt{\beta }\sqrt{G(G-1)}}{\beta G+G-1}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{{V}_{{\rm conjugate}}}=\frac{2\sqrt{\beta }\sqrt{G(G-1)}}{\beta G+G-\beta }\end{eqnarray} \tag{ 8 }$$

The result, as a function of G under different β, is presented in figure 3. The visibility saturates as the gain increases and when $\beta =1$, we obtain the maximal saturated visibility (close to 1) for both beams simultaneously. Thus it is good to work in this condition ($\langle \widehat{{{N}_{a}}}\rangle =\langle \widehat{{{N}_{b}}}\rangle$) for achieving high interference visibility. Additionally, when β is closer to the unit, the saturated visibility increases. From these curves we also can clearly see the symmetrical role of the two input beams.

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The interference phenomena can be explained as two individual phase insensitive FWM processes seeded only by the probe or conjugate beam interfering with each other. Each FWM process produces its own output of the probe and conjugate beams respectively. The two output probes of the same frequency overlap spatially under our current symmetric geometry and so are the two output conjugate beams. Thus these two output probe (conjugate) beams will interfere with each other, leading to the amplification or de-amplification of the beams. From this point of view, our two-mode PSA induces interference signals for each output port.

From the above discussion, we find that the probe beam could be amplified or de-amplified determined by the phase ϕ of the input beams. However, the probe beam always has increased noise after an amplification or de-amplification process. We will show the result by studying the noise level of probe beam. Usually, it is quantified by the ratio of the variance of the output after the PSA to the variance at the standard quantum limit (SQL), namely:

$$\begin{eqnarray}&&\frac{Var{{({{{\hat{N}}}_{a}})}_{{\rm PSA}}}}{Var{{({{{\hat{N}}}_{a}})}_{{\rm SQL}}}}=2G-1\end{eqnarray} \tag{ 9 }$$

Here one can easily calculate the term $Var{{({{\hat{N}}_{a}})}_{{\rm PSA}}}$ from equation (2), and $Var{{({{\hat{N}}_{a}})}_{{\rm SQL}}}$ is just the mean value of the output photon number, i.e. $\langle {{\hat{N}}_{a,{\rm out}}}\rangle$ . The result corresponds to the linear increase of the noise on the probe beam as the gain increases. It is not dependent on the phase of the input beams whether or not the system is amplifying or de-amplifying the input signal. The second term on the right side of equation (2) represents the added noise induced by the amplifier [49]. This kind of additive noise is independent of the input probe mode. Therefore this kind of amplifier always introduces excess noise to the input signal. The same result can be obtained for the conjugate beam due to the symmetrical role of probe and conjugate beams in this system.

In conclusion, this system behaves as an amplifier whose gain is determined by the phase ϕ of the input beams. From the point of interference, it can be seen as an interferometer whose visibility depends on the intensity ratio β of the input beams.

The degree of squeezing of the intensity difference between the probe and conjugate beams with respect to the SQL is given by:

$$\begin{eqnarray}&&D{{S}_{{\rm diff}}}=\;\frac{Var{{({{{\hat{N}}}_{a}}-{{{\hat{N}}}_{b}})}_{{\rm PSA}}}}{Var{{({{{\hat{N}}}_{a}}-{{{\hat{N}}}_{b}})}_{{\rm SQL}}}}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&=\;\frac{1}{(2G-1)+4\sqrt{G(G-1)}\frac{\sqrt{\langle {{{\hat{N}}}_{a}}\rangle \langle {{{\hat{N}}}_{b}}\rangle }}{\langle {{{\hat{N}}}_{a}}\rangle +\langle {{{\hat{N}}}_{b}}\rangle }{\rm cos} \phi }\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&=\;\frac{1}{(2G-1)+4\sqrt{G(G-1)}\frac{\sqrt{\beta }}{1+\beta }{\rm cos} \phi }\end{eqnarray} \tag{ 12 }$$

When we seed a vacuum field to the conjugate port, i.e. $\langle \widehat{{{N}_{b}}}\rangle =0$ there is a quantum noise reduction of $1/(2G-1)$ in intensity difference, which agrees with the result in [27]. The degree of squeezing depends not only on the gain G but also the ratio β and the phase ϕ as shown in figures 4 and 5. In figure 4, we set the phase of input beams ϕ as zero to study the influence of the intensity ratio β on the degree of intensity difference squeezing. As shown in figure 4, when the ratio β is above 1, the squeezing level decreases as β increases. It slowly approaches the squeezing level of the PIA configuration when β tends to infinity (not shown in the figure 4), which corresponds to the case of seeding a vacuum field to the conjugate port. As β decreases to zero, which means seeding a vacuum field to the probe port, the squeezing level also returns to the PIA configuration. We also find the maximal squeezing is achieved when $\beta =1$, which corresponds to the maximal visibility for both the probe and conjugate beams as discussed in figure 3. Most importantly, nearly 3 dB enhancement (e.g. for G = 5, 2.98 dB is possible) on the squeezing level of intensity difference between the probe and conjugate beams can be achieved in this system compared to the PIA configuration with the same gain. As shown in figure 5, the squeezing level varies with the phase, here we set $\beta =1$. When $\phi \in [0,\pi -{\rm arccos} \sqrt{\frac{G-1}{G}})$, this system produces relative intensity squeezing between the two output fields. In particular, when $\phi \in [0,\frac{\pi }{2})$, the squeezing level is even lower than the PIA configuration with the same gain. When $\phi \in (\pi -{\rm arccos} \sqrt{\frac{G-1}{G}},\pi +{\rm arccos} \sqrt{\frac{G-1}{G}})$, we get anti-squeezing on the intensity difference between the two output fields. We could either enhance the squeezing level compared to the PIA configuration or get an anti-squeezing on intensity difference via varying the phase ϕ of the input beams. The maximal squeezing level on intensity difference is obtained when $\phi =0$ and $\beta =1$, which corresponds to the situation that two pump photons convert to one probe photon and one conjugate photon. The numbers of the generated photons on both beams are the same in principle, thus the noise of intensity difference is reduced in this case.

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The nearly 3 dB enhancement can be interpreted by the interference between the two individual phase insensitive FWM processes seeded only by the probe or conjugate beam. Here we assume both FWM processes are operating at the same gain of G. Considering the FWM process seeded only by the probe beam, the intensity of the output probe beam is $G\langle {{\hat{N}}_{a}}\rangle$ and the newly generated conjugate beam has an intensity of $(G-1)\langle {{\hat{N}}_{a}}\rangle$. The same results can be obtained for the FWM process seeded only by the conjugate beam. The intensity of the amplified conjugate beam is $G\langle {{\hat{N}}_{b}}\rangle$, and $(G-1)\langle {{\hat{N}}_{b}}\rangle$ for the newly generated probe beam. For simplicity, we assume $\langle {{\hat{N}}_{a}}\rangle =\langle {{\hat{N}}_{b}}\rangle ={{N}_{0}}$. Therefore, the maximal intensity of the probe beam of the PSA seeded by both the probe and conjugate beams is $(2G-1+2\sqrt{G(G-1)})\langle {{\hat{N}}_{0}}\rangle$. When $G\gg 1$, this maximal intensity is approximately given by $2(2G-1)\langle {{\hat{N}}_{0}}\rangle$. Since the number difference operator ${{\hat{N}}_{a}}-{{\hat{N}}_{b}}$ is invariant during the FWM process, the maximal squeezing degree of the intensity difference will occur when two FWM processes exhibit constructive interference. Thus

$$\begin{eqnarray}&&DS_{{\rm diff}}^{{\rm max} }=\frac{Var{{({{{\hat{N}}}_{a}}-{{{\hat{N}}}_{b}})}_{{\rm PSA}}}}{Var{{({{{\hat{N}}}_{a}}-{{{\hat{N}}}_{b}})}_{{\rm SQL}}}}\approx \frac{2{{N}_{0}}}{4(2G-1){{N}_{0}}}=\frac{1}{2(2G-1)}\end{eqnarray} \tag{ 13 }$$

The maximal squeezing degree of the intensity difference in decibels is approximately $3+10{{{\rm log} }_{10}}(1/(2G-1))$. As mentioned above, the second term corresponds to the squeezing degree of the phase insensitive amplifier. In other words, the nearly 3 dB enhancement comes from the constructive interference of the two individual FWM processes.

The noise of the intensity sum between the probe and conjugate beams is always amplified and well above its corresponding SQL during the FWM process in the PIA configuration. Due to the interference phenomena on both the probe and conjugate beams, we find that the noise of the intensity sum of the two output fields could be well below the SQL within a very narrow phase and intensity ratio range in the PSA configuration.

From equations (2) and (3), the degree of squeezing of the intensity sum with respect to the SQL is given by:

$$\begin{eqnarray}&&D{{S}_{{\rm sum}}}=\frac{Var{{({{{\hat{N}}}_{a}}+{{{\hat{N}}}_{b}})}_{{\rm PSA}}}}{Var{{({{{\hat{N}}}_{a}}+{{{\hat{N}}}_{b}})}_{{\rm SQL}}}}\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&=\;2(2G-1)-\alpha \end{eqnarray} \tag{ 15 }$$

where $\alpha =1/[(2G-1)+4\sqrt{G(G-1)}\frac{\sqrt{\beta }}{1+\beta }{\rm cos} \phi ]$. In the PIA configuration, the $D{{S}_{{\rm sum}}}$ is approximately $2(2G-1)-1/(2G-1)$ independent of ϕ. Due to the term α in equation (15), when ϕ nears π, it is possible to obtain a squeezing level below the SQL. Additionally, the noise property depends on the ratio β. In figure 6, we set the phase of input beams ϕ as π and study how the intensity ratio β affects the degree of intensity sum squeezing. In figure 7, we set $\beta =1$ to study the influence of the phase on the degree of intensity sum squeezing. From these two figures it is clear that the squeezing level below the SQL can only be observed within a very narrow range of both β and ϕ. The size of the range depends on G. The higher the gain, the narrower the range. Moreover, a larger squeezing level can be achieved with higher gain G, e.g. for a given G = 5, more than 10 dB of squeezing is possible, shown as the red curves in figures 6 and 7. The maximal squeezing level is achieved when $\beta =1$ and $\phi =\pi$, which corresponds to the situation in which two input photons convert to two pump photons.

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In conclusion, in order to observe the squeezing for the intensity difference between probe and conjugate beams, the input signals should be amplified. Moreover nearly 3 dB of enhancement is theoretically predicted compared to the PIA configuration with the same gain. In contrast, the input signal should be de-amplified in order to observe the squeezing for the intensity sum. Additionally, this type of intensity sum squeezing can never be realized in the PIA configuration.

The 'amplitude' and 'phase' quadrature operators of the field are defined by:

$$\begin{eqnarray}&&\hat{X}=\frac{1}{\sqrt{2}}({{\hat{a}}^{\dagger }}+\hat{a}),\hat{Y}=\frac{{\rm i}}{\sqrt{2}}({{\hat{a}}^{\dagger }}-\hat{a})\end{eqnarray} \tag{ 16 }$$

Based on the above discussion, it is better to have two input signals of equal power to show high non-classical properties of this system. Thus for the following calculations, we assume $\langle \widehat{{{N}_{a}}}\rangle =\langle \widehat{{{N}_{b}}}\rangle$.

From equation (16) the sum/difference combinations of the amplitude and phase quadratures of probe and conjugate are:

$$\begin{eqnarray}&&{{\hat{X}}_{a}}+{{\hat{X}}_{b}}=\frac{1}{\sqrt{2}}({{\hat{a}}^{\dagger }}+\hat{a}+{{\hat{b}}^{\dagger }}+\hat{b})\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{{\hat{X}}_{a}}-{{\hat{X}}_{b}}=\frac{1}{\sqrt{2}}({{\hat{a}}^{\dagger }}+\hat{a}-{{\hat{b}}^{\dagger }}-\hat{b})\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{{\hat{Y}}_{a}}+{{\hat{Y}}_{b}}=\frac{i}{\sqrt{2}}({{\hat{a}}^{\dagger }}-\hat{a}+{{\hat{b}}^{\dagger }}-\hat{b})\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{{\hat{Y}}_{a}}-{{\hat{Y}}_{b}}=\frac{i}{\sqrt{2}}({{\hat{a}}^{\dagger }}-\hat{a}-{{\hat{b}}^{\dagger }}+\hat{b})\end{eqnarray} \tag{ 20 }$$

The variances for each combination are:

$$\begin{eqnarray}&&Var({{\hat{X}}_{a}}+{{\hat{X}}_{b}})=2G-1+2\sqrt{G(G-1)}{\rm cos} \theta \end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&Var({{\hat{X}}_{a}}-{{\hat{X}}_{b}})=2G-1-2\sqrt{G(G-1)}{\rm cos} \theta \end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&Var({{\hat{Y}}_{a}}+{{\hat{Y}}_{b}})=2G-1-2\sqrt{G(G-1)}{\rm cos} \theta \end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&Var({{\hat{Y}}_{a}}-{{\hat{Y}}_{b}})=2G-1+2\sqrt{G(G-1)}{\rm cos} \theta \end{eqnarray} \tag{ 24 }$$

The amplitude quadratures sum and phase quadratures sum (amplitude quadratures difference and phase quadratures difference) between the probe and conjugate beams forms one pair of Heisenberg uncertainty conjugate variables, since $[{{\hat{X}}_{a}}+{{\hat{X}}_{b}},{{\hat{Y}}_{a}}+{{\hat{Y}}_{b}}]=2{\rm i}$ ($[{{\hat{X}}_{a}}-{{\hat{X}}_{b}},{{\hat{Y}}_{a}}-{{\hat{Y}}_{b}}]=2{\rm i}$). But the amplitude quadratures sum and phase quadratures difference (amplitude quadratures difference and phase quadratures sum) are commutative since $[{{\hat{X}}_{a}}+{{\hat{X}}_{b}},{{\hat{Y}}_{a}}-{{\hat{Y}}_{b}}]=0$ ($[{{\hat{X}}_{a}}-{{\hat{X}}_{b}},{{\hat{Y}}_{a}}+{{\hat{Y}}_{b}}]=0$). In order to show entanglement, the noise reduction in the two generalized quadratures (${{\hat{X}}_{a}}-{{\hat{X}}_{b}},{{\hat{Y}}_{a}}+{{\hat{Y}}_{b}}$ or ${{\hat{X}}_{a}}+{{\hat{X}}_{b}},{{\hat{Y}}_{a}}-{{\hat{Y}}_{b}}$) must fulfill the in-separability criterion: $E=Var({{\hat{X}}_{a}}-{{\hat{X}}_{b}})+Var({{\hat{Y}}_{a}}+{{\hat{Y}}_{b}})\lt 2$ or $E=Var({{\hat{X}}_{a}}+{{\hat{X}}_{b}})+Var({{\hat{Y}}_{a}}-{{\hat{Y}}_{b}})\lt 2$ [50, 51].The value of E for our system is changing with the phase of the pump field, as shown in figure 8. The solid lines represent the amplitude quadratures difference and phase quadratures sum entanglement (denoted as type I entanglement) and the dash lines represent the amplitude quadratures sum and phase quadratures difference entanglement (denoted as type II entanglement). The strongest entanglement for type I is obtained when $\theta =2\pi$ which corresponds to the maximal amplification for the input beams. On the other hand, the strongest entanglement for type II entanglement is achieved at $\theta =\pi$ which corresponds to the maximal de-amplification for the input beams. These two types of quadrature entanglement can be achieved by changing the phase θ. Both types of quadrature entanglement can only be observed within a phase range which is determined by the gain G. With higher gain, the value of E gets smaller meaning stronger entanglement, but the phase range for observing the entanglement becomes narrower. E.g. with a given G = 5, the smallest value of E is equal to 0.11. When $\theta \in (2\pi -{\rm arccos} \frac{2}{\sqrt{5}},2\pi +{\rm arccos} \frac{2}{\sqrt{5}})$, it shows type I entanglement. When $\theta \in (\pi -{\rm arccos} \frac{2}{\sqrt{5}},\pi +{\rm arccos} \frac{2}{\sqrt{5}})$, it shows type II entanglement. The size of phase ranges for observing these two types of quadrature entanglement are the same.

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