Characterization of the pairwise correlations in different quantum networks consisting of four-wave mixers and beamsplitters
1. IntroductionQuantum correlations shared between the multiple quantum correlated beams are the fundamental ingredients of the most continuous-variable (CV) quantum technologies.[1,2] The pairwise correlations (PCs) involved in the multiple quantum correlated beams are an important research subject. For example, Refs. [3], [4] and Ref. [5] gave the classification of three-qubit correlation and four-qubit correlation respectively, which both involve the consideration of PCs. Therefore, the quantum correlation shared by any pair of the multiple quantum correlated beams is worth studying. In recent years, four-wave mixers (FWMs) based on a hot rubidium vapor[6–21] have been proved to be very successful for many practical implementations, for example, tunable delay of Einstein–Podolsky–Rosen entangled states,[9] low-noise amplification of entangled states,[10] quantum mutual information,[13] and SU(1,1) interferometer[22,23] have been experimentally realized. In 2014, our group has experimentally realized the quantum correlated triple beams by cascading two FWM processes (FWM + FWM network) and strong relative intensity squeezing has been observed between the generated triple beams.[24] Reference [25] presents a scheme to realize the versatile quantum networks by cascading several FWM processes in warm rubidium vapors, and Refs. [26] and [27] discuss the tripartite entanglement and correlation in the cascaded FWM processes, but the PCs with quantum correlation are not compared between the FWM + FWM network and the FWM+beamsplitter (BS) network, let alone the quantum correlation recovery and enhancement of the PCs. However, in this paper, it can be found that the PCs with quantum correlation can be enhanced compared with the one of the single FWM process in the FWM+BS network. Therefore, we study the PCs[28–30] in different quantum networks, such as FWM + FWM, FWM+BS, FWM + FWM + FWM, and FWM + BS + BS networks. In addition, the repulsion effect phenomena between the PCs with quantum correlation are present in the FWM + FWM and FWM + FWM + FWM networks instead of the FWM+BS and FWM+BS+BS networks. This gives the prediction that signal (idler)–frequency mode cannot be quantum correlated with the other two idler (signal)–frequency modes simultaneously.
This article is organized as follows. In Section 2, we describe the quantum correlation performance of the FWM (BS) network using the degree of relative intensity squeezing (DS) criterion. In Section 3, we use the DS criterion to analyze the PCs existing in the FWM + FWM and FWM+BS networks. In Section 4, we analyze the PCs existing in the FWM + FWM + FWM and FWM+BS+BS networks. In Section 5, we give a brief summary of this paper.
2. FWM (BS) networkIn this section, we give a simple description of the FWM (BS) network. Firstly, we will focus on the FWM network. FWM is a nonlinear process which converts two pump photons into a probe photon and an idler photon. In the cell
of Fig. 1(a), an intense pump beam and a much weaker signal beam (
) are crossed in the center of the rubidium vapor with a slight angle. Then the signal beam is amplified as 
and a new beam called idler beam is generated as 
on the other side of the pump beam at the same time. The signal beam and idler beam have different frequencies. The input–output relation of the FWM process shown in Fig. 1(a) is given by
where
is the power gain in the FWM process and
-
,
is the coherent input signal and
is the vacuum input,
is the amplified signal beam and
is the generated idler beam. See Ref. [
6] for details. The relative intensity fluctuations
can be given by
where
denotes the variance of
. The standard quantum limit (SQL) for the twin beams can be given bys
The PC with quantum correlation between the twin beams
and
can be quantified using
DS criterion, which is the ratio of the variance of the relative intensity for the twin beams to the variance at the SQL, namely
where the superscript and subscript for the
represent the network (we have three networks throughout the whole discussion, i.e., FWM (BS) network: 1 (
), FWM + FWM (BS) network: 2 (
), FWM + FWM (BS)+FWM (BS) network, 3 (
)), the
ith (
i = 1, 2, 3) beam, and the
jth (
j = 2, 3, 4) beam in the network. The PC with quantum correlation of the twin beams from the FWM process can be demonstrated when
and can be easily obtained in the experiment since
is always larger than 1. However, the values of
DS of the individual beams
and
are given by
This corresponds to a linear increase in the noise on both the signal and idler beams as gain is increased. Thus the twin beams
and
in the FWM process are both in thermal states. Equations (
4) and (
5) demonstrate that the PC with quantum correlation for the twin beams is present at the expense of increasing the fluctuations in the individual beams when
1.
Secondly, for the BS network in Fig. 1(b), the input–output relationship can be given by
where
is the transmission of the BS.
is the coherent input signal and
is the vacuum input. The values of
DS of the individual beams
and
are given by
The
DS between the beams
and
can be given by
As we can see from Eq. (
8), the
DS between the beams
and
is independent of the transmission of the BS, thus equation (
8) represents the correlation level between the two beams in both the coherent states or one beam in the coherent state, the other in the vacuum state. Equations (
4) and (
8) demonstrate that the PC with quantum correlation cannot be obtained by BS with a coherent state and a vacuum state inputs due to its passive nature and the FWM process with
=1 is equivalent to the BS with
.
3. FWM + FWM (BS) networkIn this section, we analyze the network in which one of the twin beams is amplified or attenuated by one FWM or BS, respectively. Here the FWM and BS can be considered as the gain beamsplitter[21] or lossy beamsplitter, respectively. Firstly, we will focus on the FWM + FWM network. We take the signal beam from the first FWM process (cell
) as the seed for the second FWM process (cell
) in Fig. 2(a).[24]
, 
, and 
are the triple newly-generated beams in the output stage of the FWM + FWM network. In our previous work,[24] we have shown the generation of strong quantum correlation between the triple bright beams but not the PCs with quantum correlation for any pair of the triple beams. Here we will study all the possible PCs of the triple beams 
, 
, and 
in Fig. 2(a). The input–output relationship of the FWM + FWM network in Fig. 2(a) can be written as
where
and
are the power gains for the cell
and cell
, respectively. It should be noted that the
DS of the triple beams (
,
, and
) is given by
Compared to Eq. (
4), equation (
11) means that the FWM + FWM network can enhance the quantum correlation of the system. The quantum correlation shared by the triple beams is present if
, i.e.,
or
.
PC between 
and 
can be quantified by
Equation (
12) will be reduced to 2
1 and 1/(2
) when we set
=1 and
=1 respectively, corresponding to the cases of Eq. (
5) and Eq. (
4) respectively. The physical meaning is as follows. When we set
=1, the PC between the thermal states
and
translates into the one between the vacuum state
and the thermal state
, i.e.,
. When we set
, the PC between the thermal states
and
translates into the one between the twin beams from the first FWM process, i.e.,
. The region in which
, i.e., there exists quantum correlation between the beams
and
, is shown in purple denoted as (1, 2) in Fig.
3. The value of
on the boundary (see the boundary given by
in Fig.
3) of that region reaches its maximal value of 1.33 when
, and it decreases when
and will eventually reach the value of 1. The study of
presented above is actually the question of how to preserve or even enhance the quantum correlation between the beams
and
compared with the one between the twin beams from the first FWM process under the introduction of a gain beamsplitter which brings the deterioration effect to the quantum correlation by the quantum amplification of one of the beams (
). The results shown in Fig.
3 actually show the value of
on the boundary below which the quantum correlation can always be preserved as the value of
increases. More interestingly, in this network, the PC with quantum correlation between the beams
and
cannot be quantum recovered to the one between the twin beams from the first FWM process, i.e.,
for any value of
,
due to the quantum amplification from the gain beamsplitter. The condition of
is given by
as shown in the inset of Fig.
3. With the increasing of
, the value of
always increases and eventually saturates at the value of 1, but
is required for the generation of the triple quantum correlated beams. That is to say, quantum amplification of the signal beam of the twin beams can destroy the quantum correlation between the twin beams inevitably.
PC between 
and 
can be quantified by
Equation (
13) is equal to 1 when
. Except that,
is always larger than 1, i.e., there is no quantum correlation between the beams
and
. The region in which
(
) is shown as the red line denoted as (1, 3) in Fig.
3. In this sense, there is no quantum correlation between the beams
and
for any value of
and
since
is always more than or equal to 1.
PC between 
and 
can be quantified by
The region in which
(
) is shown in cyan denoted as (2, 3) in Fig.
3. Therefore, the PC with quantum correlation between the beams
and
will be present for any
. This is not difficult to figure out if one looks at the functional form of Eq. (
14). In the region
, the existence of quantum correlation between the beams
and
eliminates the possibility of the one between the beams
and
. In other words, beam
cannot be quantum correlated with beams
and
in the same gain region. In this sense, we could call this phenomena repulsion effect of quantum correlation between the PCs in the FWM + FWM network.
Secondly, for the FWM+BS network in Fig. 2(b), the DS of the triple beams is given by
Compared to Eq. (
11), this network cannot enhance the quantum correlation existing in the triple beams
,
, and
in Fig.
2(b). We will also analyze all the possible PCs.
PC between 
and 
can be quantified by
Equation (
16) will be reduced to 1/(2
), 2
, and 1 when we set
=1,
=0, and
=1 respectively, corresponding to the cases of Eq. (
4), Eq. (
5), and Eq. (
8) respectively. These phenomena can be understood as follows. When we set
=1, the PC between the thermal states
and
translates into the one between the twin beams from the first FWM process, i.e.,
. When we set
, the PC between the thermal states
and
translates into the one between the thermal state
and the vacuum state
, i.e.,
. When we set
, the PC between the thermal states
and
translates into the one between the vacuum state
and the coherent state
, i.e.,
. This theoretical result is consistent with Eq. (
5) in Ref. [
31]. The region in which
, i.e., there exists quantum correlation between the beams
and
, is shown in green denoted as (1, 2) in Fig.
4(a). Different from the boundary of the
, with the increasing of
, the value of
on the boundary always increases and eventually saturates at the value of 1 (see the boundary given by
in Fig.
4(a)). The study of
presented above is actually the question of how to preserve or even enhance the quantum correlation between the beams
and
compared with the one between the twin beams under the introduction of a lossy beamsplitter which brings the deterioration effect to the quantum correlation by the attenuation of one of the beams (
). The results shown in Fig.
4(a) actually show the value of
on the boundary above which the quantum correlation can always be preserved as the value of
increases. More interestingly, in this case, the PC with quantum correlation between the beams
and
can be quantum recovered and enhanced compared with the one between the twin beams from the first FWM process, i.e.,
, when
. The results shown in Fig.
4(a) actually shows the boundary (the red line in Fig.
4(a)) for the values of
above which the quantum correlation enhancement can always be realized as the value of
increases. That is to say, certain attenuation of the signal beam of the twin beams can recover and enhance the quantum correlation between the twin beams from the first FWM process.
PC between 
and 
can be quantified by
The region in which
, i.e., there exists quantum correlation between the beams
and
, is shown in magenta denoted as (1, 3) in Fig.
4(b). With the increasing of
, the value of
on the boundary always decreases and eventually reaches the value of 0 (see the boundary given by
in Fig.
4(b)). As shown in Fig.
4, the sum of the boundary value of the
, i.e.,
, and the boundary value of
, i.e.,
, is always equal to 1. That is to say, the PCs with quantum correlation between
and
is limited by the lossy beamsplitter. In addition, the crossing region of
and
is present in Figs.
4(a) and
4(b), which means that the repulsion effect is absent for the FWM+BS network.
PC between 
and 
can be quantified by
which is equal to 1 when
=1 or
=1/2. As we all know, in order to generate the SQL, the shot noise limited beam, such as coherent state beam, is always preferred as shown in Eq. (
8).
means that the balanced differential measurement of a thermal state can also obtain the SQL. Except that,
is always larger than 1, i.e., there is no quantum correlation between the beams
and
for any value of
and
, since
is always more than or equal to 1.
As Section 3 mentioned, the FWM (gain beamsplitter) will inevitably destroy the quantum correlation between the twin beams 
and 
from the first FWM process, while BS (lossy beamsplitter) can preserve or even enhance the quantum correlation between the twin beams 
and 
. The repulsion effect between the PCs with quantum correlation is predicted in the gain beamsplitter case (FWM + FWM network) instead of the lossy beamsplitter case (FWM+BS network).
4. FWM + FWM (BS)+FWM (BS) networkIn this section, we analyze the network in which both of the twin beams are amplified or attenuated by two FWMs or BSs respectively. Here the FWM and BS can also be considered as the gain beamsplitter and lossy beamsplitter, respectively. Firstly, we will focus on the FWM + FWM + FWM network. We take the signal beam from the first FWM process (cell
) as the seed for the second FWM (cell
) and the idler beam as the seed for the third FWM (cell
) in Fig. 5(a). 
, 
, 
, and 
are the quadruple newly-generated beams in the output stage of the FWM + FWM + FWM network. The input–output relationship of the FWM + FWM + FWM network in Fig. 5(a) can be written as
where
is the power gain of cell
and
is the power gain of the cell
and cell
. It should be noted the
DS of the quadruple beams (
,
,
, and
) is given by
Compared to Eq. (
11), this network has also enhanced the quantum correlation of the system. The quantum correlation shared by the quadruple beams is present if
, i.e.,
or
. Next let us analyze all the possible PCs in this network.
PC between 
and 
can be quantified by
Equation (
21) will be reduced to 2
and 1/(2
) when we set
and
respectively, similar to the case of Eq. (
12). The region in which
is shown in purple denoted as (1, 2) in Fig.
6. With the increasing of
, the value of
on the boundary always increases and eventually saturates at the value of 2 (see the boundary given by
in Fig.
6). The study of
presented above is actually the question of how to preserve or even enhance the quantum correlation between the beams
and
under the introduction of two gain beamsplitters which bring the deterioration effect to the quantum correlation by the quantum amplification of both of the beams (
and
). The results shown in Fig.
6 actually show the boundary for the values of
below which the quantum correlation can always be preserved as the value of
increases. More interestingly, in this case, the PC with quantum correlation between the beams
and
can be quantum recovered to the one between the twin beams from the first FWM process only when
(the inset of Fig.
6), i.e.,
. But it means that the generation of the quadruple quantum correlated beams is impossible due to the absence of the cell
and cell
. That is to say, the quantum correlation recovery and enhancement are impossible for the FWM + FWM + FWM network.
PC between 
and 
(
and 
) can be quantified by
Equation (
23) is always larger than 1 for any value of
and
. In this sense, there is no quantum correlation between the beams
and
(
and
), since
(
) is always larger than 1 for any value of
and
.
PC between 
and 
(
and 
) can be quantified by
Equation (
22) is similar to the case of Eq. (
14). The region in which
(
) is shown in cyan denoted as ((1, 4), (2, 3)) in Fig.
6. Therefore, beams
(
) and
(
) are quantum correlated within the cyan region (
) in Fig.
6.
PC between 
and 
can be quantified by
The PC with quantum correlation between the beams
and
will be absent because
is always more than or equal to 1. This can be easily understood if one looks at the functional form of Eq. (
24). As discussed above, for this FWM + FWM + FWM network, there are three possible PCs with quantum correlation, namely
,
, and
. In addition, the existence of quantum correlation between the beams
and
eliminates the possibility of the one between the beams
(
) and
(
). In other words, beam
(
) cannot be quantum correlated with the beams
(
) and
(
) in the same gain region. These effects in this FWM + FWM + FWM network are similar to the above mentioned repulsion effect of quantum correlation between the PCs in the FWM + FWM network.
Secondly, for the FWM+BS+BS network in Fig. 5(b), the DS of the quadruple beams is given by
In this case, the quantum correlations for the quadruple beams have remained unchanged compared with the Eq. (
4). We will analyze all the possible PCs.
PC between 
and 
can be quantified by
Equation (
26) will be reduced to 1, 1/(2
), and 1 when we set
,
, and
respectively, corresponding to the cases of Eq. (
8), Eq. (
4), and Eq. (
8) respectively. We can understand these physical phenomena as follows. When we set
=1, the PC between the thermal states
and
translates into the one between the vacuum state
and the coherent state
, i.e.,
. When we set
, the PC between the thermal states
and
translates into the one between the twin beams from the first FWM process, i.e.,
. When we set
, the PC between the thermal states
and
translates into the one between the vacuum state
and the vacuum state
, i.e.,
. Its value is always less than or equal to 1 for any value of
and
. The region in which
, i.e., there exists quantum correlation between the beams
and
, is shown in Fig.
7(a). Figure
7(a) means the larger
and larger
are preferred for the PC with quantum correlation between the beams
and
. The study of
presented above is actually the question of how to preserve or even enhance the quantum correlation between the beams
and
under the introduction of two lossy beamsplitters which bring the deterioration effect to the quantum correlation by the same amount of attenuation of both of the beams (
and
). Figure
7(a) shows that the same amount of attenuation can preserve the PC with quantum correlation between the beams
and
. More interestingly, in this case, the PC with quantum correlation between the beams
and
can be quantum recovered to the one between the twin beams from the first FWM process only when
, i.e.,
. That is to say, when
, both attenuation of the twin beams can destroy the quantum correlation between the twin beams from the first FWM process inevitably.
PC between 
and 
can be quantified by
The region in which
, i.e., there exists quantum correlation between the beams
and
, is shown in magenta denoted as (1, 3) in Fig.
8(a). Figure
8(a) means that the PC with quantum correlation between the beams
and
is present for the value of
between the two boundaries
and
. With the increasing of
, the values of
on the boundaries eventually reach the value of 1/2.
PC between 
and 
(
and 
) can be quantified by
which is similar to the case of Eq. (
18). Therefore, the beams
(
) and
(
) are not quantum correlated to each other in this network.
PC between 
and 
can be quantified by
The region in which
, i.e., there exists quantum correlation between the beams
and
, is shown in green denoted as (2, 4) in Fig.
8(b). Figure
8(b) means that the PC with quantum correlation between the beams
and
is present for the value of
between the two boundaries
and
. With the increasing of
, the values of
on the boundaries eventually reach the value of 1/2. As shown in Fig.
8, the sum of the boundary value of
, i.e.,
(
), and the boundary value of
, i.e.,
(
), is always equal to 1 due to the limitation of the lossy beamsplitter.
PC between 
and 
can be quantified by
which is always less than or equal to 1 for any value of
and
. The region in which
, i.e., there exists quantum correlation between the beams
and
, is shown in Fig.
7(b). Figure
7(b) means that the larger
and smaller
are preferred for the PC with quantum correlation between
and
.
As Section 4 mentioned, both the FWM + FWM + FWM and FWM+BS+BS networks can preserve the quantum correlation between 
and 
under the condition of 
and 
, respectively. The repulsion effect between the PCs with quantum correlation is also predicted in the FWM + FWM + FWM network instead of the FWM+BS+BS network.
5. ConclusionIn summary, we theoretically characterize the performances of the PCs in different quantum networks consisting of FWMs and BSs. The quantum correlation recovery and enhancement are present in the FWM+BS network and the repulsion effect phenomena between the PCs with quantum correlation are predicted in the FWM + FWM and FWM + FWM + FWM networks. Our results presented here pave the way for the manipulation of the quantum correlation in quantum networks.
