Enhancement of quantum entanglement between twin beams via a four-wave mixing process with double feedback

Two-port feedback has been theoretically studied and proved to be effective in enhancing the degree of entanglement of the output beams for a four-wave mixing process. Here, to further consider the loss effects and phase delays, we use a model that is closer to practical implementation. By reasonably tuning the feedback coefficients, we find that a higher degree of entanglement and higher power of the output beams can be achieved, and a loose requirement for phase-locking accuracy can be obtained. This scheme may have promising applications in practical quantum computation and quantum communication.


Q
uantum information science has become a popular research field in recent years, and multipartite quantum entanglement provides an important resource for quantum information processing. 1) In quantum optics, discrete variable and continuous variable entanglement generated by different techniques have been studied, [2][3][4][5][6][7][8][9] such as the Greenberger-Horne-Zeilinger (GHZ) state generated by using polarization-entangled photons and beam splitters in a discrete variable system, 10) and the Einstein-Podolsky-Rosen (EPR) entangled state generated by an optical parametric oscillator (OPO) using titanyl phosphate (KTP) in a continuous variable system, 11) and so on.Apart from the above-mentioned methods, the four-wave mixing (FWM) process in the hot Rb vapor cell developed by Lett [12][13][14] as an optical parametric amplifier (OPA) was proved to be a promising technique to generate quantum squeezed and entangled beams in the continuous variable systems, having the virtues of higher efficiency because of its strong nonlinearity and natural spatial separation of the output beams.[15][16][17][18][19] In addition, the FWM process in the hot Rb vapor cell can be multiplexed to efficiently generate multipartite quantum correlated and entangled beams.[20][21][22][23] For different cases of applications of quantum information science, such as guaranteeing the high fidelity of quantum protocols and increasing the precision of quantum measurement, a high degree of quantum correlation and entanglement of the generated beams is necessary.[24][25][26][27] For the case of the FWM process in the Rb cell, the traditional way to strengthen the degree of quantum correlation and entanglement is to enhance the FWM gain by increasing the intensity of the pump beam.[29][30][31][32] However, in the experimental study on FWM coherent feedback, 28) only the conjugate beam is fed back, while in the theoretical study 30,31) of the two-port feedback FWM process, the approach of modeling loss effect of the Rb vapor cell is simply to add beam splitters to the output ports.Taking into account the continuous loss effect during the propagation of beams in the Rb vapor cell, we present the double feedback FWM (DFFWM) scheme with feedback for both output beams using a more accurate theoretical model, demonstrating that this DFFWM process can generate beams with stronger quantum entanglement and higher power than both cases of the single feedback and without feedback, and it is possible to loosen the requirement of phase-locking accuracy.In addition, according to Ref. 33, the model in this paper could be used to deal with any double feedback FWM process because of the independence of implementation details in the amplifier.
The considered DFFWM model is illustrated in Fig. 1(a), where the FWM process is used as the OPA with a pump beam, two inputs (probe beam a 0 and conjugate beam b 0 ) and two outputs (probe beam a N and conjugate beam b N ).The Rb FWM process in the double Λ-type three-level Rb 85 system is shown in Fig. 1(b), where two pump photons convert to a probe photon and a conjugate photon.Four tunable beam splitters (BS) are used in the feedback path, where BS 1 and BS 2 with reflectivity of f 1 and f 2 act as the feedback controllers [see Fig. 1(a)], and BS 3 and BS 4 with reflectivity of f 3 and f 4 are used to simulate the propagation loss in the feedback path.The DFFWM is seeded by a coherent state probe beam a in and a vacuum state conjugate beam b , in which are introduced into the feedback controller BS 1 and BS 2 along with a N and b , N respectively.a out and b out are the outputs of BS 1 and BS , 2 which are also the outputs of the DFFWM process.a f and b f represent the intermediate operators in the feedback path from BS 1 and BS 2 to BS 3 and BS , 4 respectively.v a and v b are introduced into BS 3 and BS 4 as vacuum losses, respectively, the abandoned parts of the feedback beams a f and b f are represented by "Loss" in the left of Fig. 1(a), and the outputs a 0 and b 0 act as the inputs of the FWM process.In addition, according to Ref. 33, the beam trajectories through the medium are separated into N distinct stages, and each stage consists of an ideal squeezing process (with squeezing parameter s) and a loss process (with transmission coefficients t a and t b for probe and conjugate beams, respectively).For the propagation process of the beams through the vapor cell, the overall squeezing parameter is = S Ns, and the overall transmission coefficients of probe and conjugate beams are If f 1 or f 2 is equal to 0, the DFFWM turns into a single feedback FWM (SFFWM), when both f 1 and f 2 are equal to 0, the DFFWM is equivalent to a non-feedback FWM (NFFWM).We assume that the phase change of a f (b f ) from BS 1 (BS 2 ) to BS 3 (BS 4 ) is f 1 (f 2 ), and the phase changes of other processes in the feedback path are 0, then f 1 and f 2 can be equivalently used to represent the phase changes of the two feedback beams during the propagation in the whole feedback path, respectively.
For the FWM process in the hot Rb vapor cell, by using the method in Ref. 33, the relationship of a , b N can be obtained as 33,34) ( ) where a , i b i are correlated with the FWM gain G as well as the transmissivities T a and T ; ) are all vacuum operators, representing the injected vacuum states during the propagation of probe and conjugate beams in the hot Rb vapor cell.The number N means how many pairs of vacuum states are injected, and is taken to be infinity.This model is more accurate than that only using two beam splitters to simulate loss effect in Refs.30, 31.The relationships of the inputs and outputs of the feedback controllers can be expressed as 2 The relationships of the inputs and outputs of the loss simulation BS are given by ( ) According to the above equations, we can get the expressions of the outputs where the coefficients p , a i b i By substituting Eqs. ( 1) and ( 2) into Eq.( 3), it can be found that the output beams of the system can be most quantum entangled with f 1 and f 2 equal to π. 18,32,[35][36][37][38][39] Equation ( 4) can be used to analyze any double feedback FWM process, because the model is independent of experimental details even when loss effects are well considered.The coefficients are set according to practical experiments to be = T 0.92, The transmissivity T a is smaller than T , b because the absorption of probe beam is stronger than the conjugate beam in the hot Rb vapor cell.
In the following, we use the positive partial transpose (PPT) criterion, [40][41][42][43][44] which is a necessary and sufficient criterion for Gaussian states at specific conditions, to study the entanglement properties of twin beams generated by the DFFWM process. Acording to the PPT criterion, quantum entanglement between the output beams exists when the smallest symplectic eigenvalue V m of the partial transposed (PT) covariance matrix (CM) s (i.e.,  s) is smaller than 1.The symplectic eigenvalues of  s can be computed as the absolute value of the eigenvalues of  s W i , where . The relationship between s and  s is 1,1,1, 1 .The covariance matrix s of the twin beams can be written as where (A, B = X, Y; j, k = a,b) is defined as the covariance of the quadrature of the output beams, and can be derived from Eq. (4).To make the  PPT the stronger the entanglement of the generated twin beams.
Firstly, we study the effect of feedback coefficients on entanglement, with the phase delays f 1 and f 2 are set to be p.The existence of an optimal set of feedback coefficients for generating strongest entanglement is due to the fact that, on one hand, the introduction of feedback equivalently amplifies the reflected portion of the output beams multiple times, enhancing the quantum entanglement, on the other hand, the introduction of feedback would also bring losses from BS 3 and BS , 4 which would deteriorate the entanglement properties of the output beams.To see the differences between DFFWM, SFFWM and NFFWM more visually, we show in Fig. 2(b) the minimum E PPT versus the FWM gain for four different schemes, DFFWM, SFFWM with = f 0, and NFFWM.The overall trend is that the minimum E PPT would decrease with the increase of G.At the same G, E PPT of the SFFWM is always lower than that of the NFFWM, and E PPT of the DFFWM is further lower than that of the SFFWM, indicating that the generated twin beams of the DFFWM could achieve a higher degree of entanglement than that of the SFFWM.In addition, it can be seen that the four curves would converge gradually as G increases.The reason is that when G becomes larger, in order to minimize E , PPT the feedback coefficients have to take smaller values, which results in a decrease of the degree of entanglement enhancement.
Then, we investigate how phase delay affects entanglement of the output beams.The feedback coefficients f 1 and f 2 are set to the values that minimize E PPT when G = 2.5 in Fig. 2(a), then we plot the variation of E PPT with respect to f 1 and f 2 shown in Fig. 3(a).It is obvious that when E PPT can achieve the minimum value, meaning the degree of entanglement of the output beams is the strongest, and in general, the closer the phases are to p, the smaller E PPT is.However, it can be observed that even if f 1 and f 2 are not close to p, E PPT can still achieve small values.In order to show this more clearly and to compare DFFWM with SFFWM and NFFWM, the dependence of E PPT on the phase f for different cases is plotted in Fig. 3 ).As depicted in Fig. 3(b), it is apparent that the phase region for DFFWM with f f = 1 2 to achieve higher degree of entanglement than NFFWM is smaller than that for the two cases of SFFWM.Conversely, for the case of DFFWM with the phase region to achieve higher degree of entanglement than NFFWM is larger than that for the two cases of SFFWM.This can be attributed to the effect of the terms ( ) f f + e i 1 2 in the expression of E .
PPT In previous theoretical studies on double feedback, to simplify the discussion, the phase delay in one of the feedback paths is set to a fixed value p, or the phase delays of two feedback paths are set to equal; consequently, it was assumed that higher phase sensitivity or higher phase-locking accuracy should be satisfied for the double feedback model.However, from the curve of DFFWM with f f p in Fig. 3(b), it is possible to achieve higher degree of entanglement with looser requirement of phase-locking accuracy than that for the two cases of SFFWM.This provides more valuable information for the practical applications of DFFWM.
Finally, we study the effect of feedback on the intensities of the output beams of DFFWM and SFFWM while E PPT having the optimal values.The intensity of either of the output beams of DFFWM, SFFWM and NFFWM with optimal E PPT is proportional to the photon number denoted as N , opt which can be derived from Eq. ( 4).We define an intensity enhancement parameter P , opt which can be calculated by dividing N opt of an output beam by N opt of the output probe beam of NFFWM with the same G, representing the degree of intensity enhancement.Larger value of P opt means stronger enhancement of intensity.For the output probe beam of NFFWM itself, = P 1.
opt In Fig. 4, we plot the variation of P opt with respect to the FWM gain G for output probe beams (a) and conjugate beams (b) for the cases of DFFWM, SFFWM and NFFWM.Obviously, the output beams of DFFWM and SFFWM have stronger intensity than that of NFFWM, and P opt of DFFWM is much larger than that of SFFWM.For example, when G = 2, the intensity of the probe beam generated by DFFWM is 2.61, 4.08 and 10.20 times stronger than that of SFFWM with = f 0, 1 = f 0, 2 and NFFWM, respectively.When G is relatively small, the value of P opt increases rapidly; however, as G increases, the rate of increase in P opt would slow down.The reason is that as G increases, the feedback coefficients corresponding to the smallest E PPT would decrease, thus the degree of amplification of intensity would increase slower.
In conclusion, we have investigated the DFFWM model that is closer to practical implementation.It is shown that DFFWM can generate twin beams with a higher degree of entanglement and much higher power than that for the cases of SFFWM and NFFWM, also, the requirement for phaselocking accuracy of DFFWM can be less stringent than that of SFFWM.The results can, in principle, be extended to other optical FWM systems.This method provides the possibility for producing bright beams with a higher degree of entanglement more easily, which is quite useful for quantum communication, quantum computation and quantum-enhanced metrology.

2
respectively.The relationship of FWM gain G and squeezing parameter S is = G S cosh .2

Fig. 1 . 2 ©
Fig. 1.(a) The model of DFFWM.The FWM process in the hot Rb vapor cell acts as the optical parametric amplifier with a pump beam and the gain of G, a 0 and b 0 are the input beams of this FWM process, and a N and b N are outputs.Two tunable beam splitters BS 1 and BS 2 (with reflectivity of f 1 and f , 2 respectively) represent the feedback controller, and two beam splitters BS 3 and BS 4 (with reflectivity of f 3 and f , 4 respectively) simulate the loss effect.(b) The FWM process in the double L-type three-level Rb 85 system.

Figure 2 (
a) shows E PPT with respect to f 1 and f 2 at G = 2.5.It can be found that a point exists where E PPT reaches the minimum value −12.95 with = f (b) with feedback coefficients f 1 and f 2 minimizing E , PPT where f f = 1 (f 2 ) for the case of SFFWM with = f f f = 1 (f 2 ) for the case of DFFWM with f f = 1 2

Fig. 2 .
Fig. 2. (a) E PPT versus feedback coefficients f 1 and f 2 at G = 2.5. (b) The optimal E PPT versus the FWM gain G.

Fig. 4 .
Fig. 4. Intensity enhancement parameter P opt of the output probe (a) and conjugate (b) beams with respect to G. Feedback coefficients f 1 and f 2 take the values corresponding to the optimal E .

Fig. 3 .
Fig. 3. (a) E PPT versus phase delay f 1 and f 2 at G = 2.5. (b) E PPT versus phase f at G = 2.5.Feedback coefficients take the values to minimize E PPT when f f p = = .