Multi-Mode Correlation in a Concurrent Parametric Amplifier
Multi-Mode Correlation in a Concurrent Parametric Amplifier
Wang, Hailong;Shi, Yunpeng
2022-06-23 00:00:00
hv photonics Article Multi-Mode Correlation in a Concurrent Parametric Amplifier Hailong Wang * and Yunpeng Shi College of Optical and Electronic Technology, China Jiliang University, Hangzhou 310018, China; p21040854081@cjlu.edu.cn * Correspondence: hlwang@cjlu.edu.cn Abstract: A concurrent parametric amplifier consisting of two pump beams is used to investigate the possibility of generating multi-mode correlation and entanglement. The existence of three-mode entanglement is demonstrated by analyzing the violation degree of three-mode entanglement criteria, including the sufficient criterion, i.e., two-condition and optimal single-condition criterion, and necessary and sufficient criterion, i.e., positivity under partial transposition (PPT) criterion. Besides, two-mode entanglement generated from any pair is also studied by using the Duan criterion and PPT criterion. We find that three-mode entanglement and two-mode entanglement of the two pairs are present in the whole parameter region. Our results pave the way for the realization and application of multi-mode correlation and entanglement based on the concurrent parametric amplifiers. Keywords: multi-mode correlation; concurrent; parametric amplifier 1. Introduction Parametric amplifier, for example, four-wave mixing (FWM), acts as a nonlinear interaction process that permits the transfer of energy and momentum between multiple optical modes, specifically, two pump modes can be converted into a signal and an idler modes via various nonlinear media [1–3]. For the atomic medium, a high-power pump beam intersects a low-power seed beam in a hot rubidium cell, causing them to interact and generate an idler beam, then the amplified seed (signal) and the newly-generated Citation: Wang, H.; Shi, Y. idler beams can be demonstrated to be quantum correlated and entangled [4–10]. Based Multi-Mode Correlation in a on this simple model of only one pump mode and only one seed mode, other variations Concurrent Parametric Amplifier. have been presented. For example, when the FWM process is seeded by the coherent Photonics 2022, 9, 443. https:// signal and idler beams and only pumped by one pump beam, a scheme of a two-mode doi.org/10.3390/photonics9070443 phase-sensitive amplifier has been constructed, and its classical and quantum properties Received: 24 May 2022 have been theoretically analyzed [11] and experimentally measured [12]. Similarly, when Accepted: 20 June 2022 the FWM process is seeded by two coherent signal beams at the same angle but in opposite Published: 23 June 2022 directions on either side of the pump beam, this dual-seed scheme has been allowed to Publisher’s Note: MDPI stays neutral achieve intensity-difference squeezing at ultra-low frequency [13]. with regard to jurisdictional claims in On the other hand, with the FWM process pumped by two coherent pump beams published maps and institutional affil- with non-degenerate frequency and only seeded by one coherent signal beam, a noiseless iations. optical amplifier has been experimentally realized, and its noise figure is always superior than that obtained with a phase-insensitive amplifier with the same gain [14]. Motivated by the above scientific advances, we propose a concurrent parametric amplifier scheme in which FWM process is pumped by the two coherent pump beams Copyright: © 2022 by the authors. with the degenerate frequency and only seeded by one coherent signal beam. As shown Licensee MDPI, Basel, Switzerland. in Figure 1, two pump beams P and P are focused and crossed in the center of a hot 1 2 This article is an open access article rubidium vapor cell. A coherent seed beam, red-shifted from pump beam as shown in distributed under the terms and the block in Figure 1, is seeded into the vapor cell, and it symmetrically crosses with the conditions of the Creative Commons two pump beams on one plane with the proper crossing angles [15] to eliminate any other Attribution (CC BY) license (https:// cascaded FWM processes. Under this experimental condition, each pump beam will interact creativecommons.org/licenses/by/ with the seed beam individually by means of FWM process. The seed beam is amplified 4.0/). Photonics 2022, 9, 443. https://doi.org/10.3390/photonics9070443 https://www.mdpi.com/journal/photonics Photonics 2022, 9, 443 2 of 10 (S) and two idler beams (I and I ) are simultaneously generated, therefore the interaction 1 2 mechanism of the triple output beams S, I and I constitute a concurrent parametric 1 2 amplification process [16]. In this work, the quantum properties of two-mode and three- mode entanglement existed in the triple output beams generated from the concurrent parametric amplifier will be discussed in detail. Firstly, the Hamiltonian describing this concurrent parametric amplifier can be written as below † † † † H = ih ¯ [# (a ˆ a ˆ a ˆ a ˆ ) + # (a ˆ a ˆ a ˆ a ˆ )], (1) 1 s 1 2 s 2 s 1 s 2 with # (i = 1 and 2) representing the interaction strength and a ˆ (i = s, 1, and 2) the bosonic i i annihilation operators. By applying the Heisenberg equation of motion to Equation (1), the solution for the annihilation operators is found to be † † a ˆ (t) = Aa ˆ (0) + Ba ˆ (0) + Ca ˆ (0), s s 1 2 a ˆ (t) = Ba ˆ (0) + Da ˆ (0) + Ea ˆ (0), (2) 1 s 1 2 a ˆ (t) = Ca ˆ (0) + Ea (0) + Fa ˆ (0), 2 1 2 with A = G, a G 1 B = , 1+a G 1 C = , 1+a 1+a G D = , (3) 1+a a( G 1) E = , 1+a a + G F = , 1+a 2 2 where G = # + # , cosh(Gt) = G, and # /# = a. Before investigating the two-mode 1 2 1 2 and three-mode entanglement existing in the system, the optical quadrature definitions should be given firstly due to the requirement of the following criteria. Concerning the three modes characterized by bosonic annihilation operators a ˆ involved in the present system, where i = s, 1, and 2, quadrature operators can be defined as follows: † † ˆ ˆ ˆ ˆ X = a + a , Y = i(a a ), (4) i i i i i i such that [X , Y ] = 2i, and X and Y are, respectively, position and momentum quadra- i i i i tures. Following the definitions of Equation (4), Equation (2) can be recast in the form of quadrature operators: X (t) = AX (0) + BX (0) + CX (0), s s 1 2 X (t) = BX (0) + DX (0) + EX (0), (5) 1 s 1 2 X (t) = CX (0) + EX (0) + FX (0), 2 1 2 and Y (t) = AY (0) BY (0) CY (0), s s 1 2 Y (t) = BY (0) + DY (0) + EY (0), (6) 1 s 1 2 Y (t) = CY (0) + EY (0) + FY (0). 2 1 2 Based on the above relations about quadrature operators, the variances and covariances of the position and momentum quadratures can be obtained to analyze the violation degree of 2 2 different entanglement criteria. V(X ) = hX i hX i represents the position quadrature i i i Photonics 2022, 9, 443 3 of 10 variance. For the covariance, it can be defined as V = (hX X i +hX X i)/2 hX ihX i. It ij i j j i i j should be noted that the covariance V will reduce to the usual variance V(X ) under the ij i condition of i = j. In reality, the variances of the three modes can be expressed as D E 2 2 2 2 X (t) = Y (t) = A + B + C = 2G 1, s s D E 2 1+a (2G 1) 2 2 2 2 X (t) = Y (t) = B + D + E = , (7) 1 2 1+a D E a +(2G 1) 2 2 2 2 2 X (t) = Y (t) = C + E + F = , 2 2 1+a and here, we used the fact that the mean values of quadrature operators are all equal to 0 and hX (0)X (0)i = hY (0)Y (0)i = d (i, j = s, 1, and 2). Similarly, the covariances can be i j i j ij expressed by hX (t)X (t)i = hY (t)Y (t)i = AB + BD + CE s 1 s 1 2a G(G 1) = , 1+a hX (t)X (t)i = hY (t)Y (t)i = AC + BE + CF (8) s s 2 2 2 G(G 1) = , 1+a X (t)X (t) = Y (t)Y (t) = BC + DE + EF h i h i 1 2 1 2 2a(G 1) = , 1+a and the above expressions can be used to discuss the entanglement properties of both two-mode and three-mode cases; this is due to the fact that the triple beams in the con- current parametric amplifier are Gaussian states, which can be fully quantified by their corresponding covariance matrix (CM). I1 I2 5P1/2 Cell 5S1/2 F=3 P2 P1 Seed F=2 Figure 1. The concurrent parametric amplifier. P and P : two pump beams; S: signal beam; I and 1 2 1 I : two idler beams. The energy level is shown in the block. 2. Two-Mode Entanglement 2.1. Duan Criterion On the one hand, two-mode entanglement in this concurrent parametric amplifier is analyzed using a sufficient criterion, i.e., Duan criterion [17], which is based on the total variance of a pair of Einstein–Podolsky–Rosen-type operators, X X and Y + Y . For i j i j physical entangled continuous variable states, this variance will rapidly reduce to zero by increasing the correlation degree. Thus, if any inequality in Equation (9) is violated, there will exist two-mode entanglement between any pair. Based on the quadrature definitions in Equation (4), the inequalities can be expressed asnumerical order. Photonics 2022, 9, 443 4 of 10 D = V(X X ) + V(Y + Y ) 4, s1 s 1 s 1 D = V(X X ) + V(Y + Y ) 4, (9) s s s2 2 2 D = V(X X ) + V(Y + Y ) 4. 12 1 2 1 2 We calculate the dependence of D , D , and D on the gain G and interaction s1 s2 12 strength ratio a as 4a G(G 1) 1 + a (2G 1) D = 2[2G 1 + ], s1 1 + a 1 + a 2 G(G 1) G 1 D = 4[G p + ], s2 1 + a 1 + a D = 4G. (10) The violation of the first, second, and third inequalities in Equation (9) can be used to claim the existence of two-mode entanglement between a ˆ and a ˆ , a ˆ and a ˆ , and a ˆ and s s 1 2 1 a ˆ , respectively. As depicted in Figure 2a, as G and a get larger, the value of D becomes 2 s1 smaller and smaller. This is because, under this condition, the concurrent parametric amplifier will reduce to a simple single pump amplifier only pumped by P , and two-mode entanglement between the modes a ˆ and a ˆ will dominate. Contrary to the dependence s 1 of D on G and a, the two-mode entanglement between a ˆ and a ˆ depicted in Figure 2b s 2 s1 can be improved by means of a larger value of G and a smaller value of a, meaning that, in this situation, the concurrent parametric amplifier will reduce to the simple single-pump amplifier only pumped by P , and two-mode entanglement between the modes a ˆ and 2 s a ˆ will dominate. This also explains the opposite behaviors between Figure 2a,b. More interestingly, as depicted in Figure 2c, two-mode entanglement between a ˆ and a ˆ is absent. 1 2 The reason for this phenomenon is that the two modes a ˆ and a ˆ are both generated from 1 2 the seed mode a and compete with each other. (a) (b) ( c) 2 5 Figure 2. The dependence of D (a), D (b), and D (c) on the gain G and interaction strength s1 s2 12 ratio a. 2.2. Positivity under Partial Transposition Criterion On the other hand, the PPT criterion as a necessary and sufficient criterion can be used to quantify two-mode entanglement in the concurrent parametric amplifier. Generally, two-mode entanglement shared by a ˆ and a ˆ can be fully quantified by the following s 1 CM [18,19]: s1 2 3 ˆ ˆ ˆ X (t) 0 X (t)X (t) 0 s 1 6 ˆ ˆ ˆ 7 0 Y (t) 0 Y (t)Y (t) s 1 6 7 CM = . (11) s1 4 ˆ ˆ ˆ 5 X (t)X (t) 0 X (t) 0 s 1 ˆ ˆ ˆ 0 Y (t)Y (t) 0 Y (t) s 1 Only when both of the symplectic eigenvalues of the partially transposed (PT) CM are s1 no less than 1, this indicates the absence of two-mode entanglement between them [18–21]. Photonics 2022, 9, 443 5 of 10 In this way, the smaller symplectic eigenvalue E can be used to quantify two-mode s1 entanglement between a ˆ and a ˆ , i.e., if E is smaller than 1, two-mode entanglement will 1 s1 exist between them. Substituting Equations (7) and (8) into Equation (11), we can obtain the detailed result for E as below: s1 2 2 4 2 2 2 4 1 + a + (1 + a )[a + 2(G + 2Ga ) 2G(1 + 3a + 4a )] E = , (12) s1 2 2 2 2 2 3/2 2(1 + a )[G + (2G 1)a ] (G 1)[G(1 + 2a ) 1] (1 + a ) and the dependence of E is depicted in Figure 3a. Due to the value of E being smaller s1 s1 than 1, thus a ˆ is quantum entangled with a ˆ in the whole parametric region (G > 1 and ˆ ˆ a > 0). Similarly, the smaller symplectic eigenvalue E of a and a can be calculated as s2 s 2 2 2 4 2 2 2 2 4 1 1 + a + (1 + a )[a + 2G (2 + a ) 2G(4 + 3a + a )] E = , (13) s2 2 2 2 2 4 2 3/2 2(1 + a )[G(2 + a ) 1] (G 1)[G(2 + a ) a ] (1 + a ) and its value is smaller than 1 in the whole parametric region as depicted in Figure 3b, meaning that a ˆ is also quantum entangled with a ˆ . s 2 Besides, the smaller symplectic eigenvalue E of a ˆ and a ˆ can be given by 12 1 2 2 2 2 2 2 2 2 2 4 1 G a + Ga + G 4a + 8Ga 2G a + G a E = , (14) 1 + a and its value is larger than 1 in the whole parametric region, as depicted in Figure 3c, indicating that a ˆ is not quantum entangled with a ˆ . This is determined by the following 1 2 fact: a ˆ and a ˆ are both generated from a ˆ and in a competitive relationship. 1 2 (a) (b) ( c) 0.5 0.4 1.5 0.6 0.3 Figure 3. The dependence of E (a), E (b), and E (c) on the gain G and interaction strength s1 s2 12 ratio a. 3. Three-Mode Entanglement 3.1. Two-Condition Criterion In the following, we analyze three-mode entanglement by using different criteria, i.e., two-condition, optimal single-condition, and PPT. First of all, a set of inequalities based on the two-condition criterion [22] is given by V = V(X X ) + V(Y + Y + O Y ) 4, s1 s 1 s 1 2 2 V = V(X X ) + V(Y + Y + O Y ) 4, (15) 2 2 s s 12 1 1 where O (i = s, 2), as arbitrary real numbers can be used to minimize the values in Equation (15). If the two inequalities in Equation (15) are both violated, then it can be deemed as a sufficient criterion to claim the presence of genuine three-mode entanglement. Following this idea, via direct differentiation of Equation (15) with regard to O , the optimal opt results of O (O ) can be given by i Photonics 2022, 9, 443 6 of 10 (hY Y i +hY Y i) s 1 s 2 O = hY i 2 G(G 1)(1 + a) = p , (2G 1) 1 + a (hY Y i +hY Y i) 2 1 2 O = 2[a Ga + G(G 1)(1 + a )] = , (16) 2G 1 + a and substituting Equation (16) into Equation (15), the detailed expression of Equation (15) can be expressed as 2 2 4(G + a )[G + 2Ga a(a + 2 G(G 1)(1 + a ))] V = , s1 2 2 (1 + a )(2G 1 + a ) 4G[G(a 1) + 2a] V = . (17) (1 + a )(2G 1) The contour plot of Equation (17) is shown in Figure 4. The dependence of V on s1 G and a is shown in Figure 4a. The region of V < 4 is enlarged compared to the one of s1 D < 4 in Figure 2a when we consider the phase quadrature of a ˆ (Y ). The variance of 2 2 s1 Y + Y + Y becomes smaller than the one of Y + Y , which claims that a ˆ has a correlation s 1 2 s 1 2 with a ˆ + a ˆ . The dependence of V on G and a is shown in Figure 4b, and the region of 1 12 V < 4 is also enlarged compared to the one of D < 4 in Figure 2c due to the same reason. 12 12 (a) (b) (c) Figure 4. The contour plot of V (a) and V (b); (c) the overlapped region between V < 4 and s1 12 s1 V < 4. The overlapped light blue region between V < 4 and V < 4 in Figure 4c means s1 12 that genuine three-mode entanglement is present in this system. 3.2. Optimal Single-Condition Criterion Secondly, the single-condition criterion using the combined quadrature variances [22] can be used to test and verify the presence of genuine three-mode entanglement. If its value in Equation (18) is no more than 2, genuine three-mode entanglement can be verified. 1 1 V = V[X p (X + X )] + V[Y + p (Y + Y )] s12 s 1 2 s 1 2 2 2 2a 2(1 + a) 2G(G 1) = 2[3G 1 + (G 1) p ]. (18) 1 + a 1 + a The dependence of V is depicted in Figure 5, and its value in most of the region is s12 no more than 2, which clearly shows the presence of genuine three-mode entanglement. Inspired by the above results, by introducing different factors [22] instead of 1/ 2, the noise unbalance between a ˆ and a ˆ in Equation (18) can be effectively canceled; this criterion 1 2 Photonics 2022, 9, 443 7 of 10 opt can be called the optimal single-condition criterion. In this way, V can be expressed s12 as follows: opt V = V[X F X F X ] + V[Y + F Y + F Y ], (19) s s 1 1 2 2 1 1 2 2 s12 opt opt where the optimal expressions (F and F ) of F and F can be calculated as 1 2 1 2 2a G(G 1) opt F = p , (2G 1) 1 + a 2 G(G 1) opt F = , (20) (2G 1) 1 + a opt respectively. Substituting Equation (20) into Equation (19), V can be simply written as s12 opt V = . (21) s12 2G 1 opt The contour plot of V is depicted in Figure 6a, and it is not dependent on interaction s12 strength ratio a, meaning that the generation of the two modes a ˆ and a ˆ is the mode X ; 1 2 s thus, F X + F X is a combined idler mode with respect to the signal mode X . 1 1 2 2 Figure 5. The dependence of V in Equation (18). s12 (a) (b) 1 0.6 opt opt Figure 6. (a) The dependence of V in Equation (21); (b) the dependence of V (trace A); B (trace s12 s12 B); (B ) = (B ) (trace C). 2 min 3 min To verify the presence of genuine three-mode entanglement, all the boundaries should be calculated according to [22]. If the value of Equation (21) is smaller than the smallest boundary, there will exist genuine three-mode entanglement. In this sense, the boundaries opt 2 2 2 2 2 2 of V are B = 2(1 + F + F ), B = 2( F + 1 F ), and B = 2( F + 1 F ) 1 2 3 s12 1 2 1 2 2 1 with the detailed expressions of Photonics 2022, 9, 443 8 of 10 B = 4 , (2G 1) 2 + 2[1 + 8G(G 1)]a B = , (22) 2 2 (2G 1) (1 + a ) 2[1 + 8G(G 1) + a ] B = , 2 2 (2G 1) (1 + a ) respectively. To find the smallest value between B , B , and B , we should obtain their 1 2 3 extreme points when the value of a is fixed. Thus, the smallest values of B and B can 2 3 be obtained: 2 + 2[1 + 8G(G 1)]a 2 (B ) = lim = , 2 min 2 2 2 a!0 (2G 1) (1 + a ) (2G 1) 2[1 + 8G(G 1) + a ] 2 (B ) = lim = . (23) 3 min 2 2 2 a!¥ (2G 1) (1 + a ) (2G 1) As Equation (23) shows, the value of B in the limit of a = 0 is equivalent to the one of B in the limit of a = ¥, and this is because under both two conditions, three-mode entanglement will reduce to two-mode entanglement only pumped by one pump beam; thus, (B ) and (B ) cannot be viewed as the boundary of three-mode entanglement. 2 min 3 min The other effective boundary B (trace B) is depicted in Figure 6b, and its value is always opt larger than the one of V (trace A); thus, genuine three-mode entanglement is present in s12 the whole gain range. Thus, the optimal single-condition criterion can be used to quantify genuine three-mode entanglement more efficiently than the single-condition criterion. 3.3. PPT Criterion Finally, the PPT criterion can also be used to quantify genuine three-mode entan- glement [18,19,23]. For three-mode entanglement, the three possible 1 2 partitions ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ (a (a , a ), a (a , a ), and a (a , a )) have to be tested. When the smallest symplectic s 1 2 1 s 2 2 s 1 eigenvalue for each of the three PT CMs is smaller than 1, all the partitions are inseparable, and genuine three-mode entanglement will exist. When the PT operation is applied to the mode a ˆ , the entanglement between a ˆ and s s the rest of the modes (a ˆ and a ˆ ) can be quantified by the smallest symplectic eigenvalue 1 2 T : s 12 2 4 6 2 3 2 2 3 2 3 1 + 3a + 3a + a 8G(1 + a ) + 8G (1 + a ) 4(2G 1)(1 + a ) G(G 1) T = . (24) s 12 2 3/2 (1 + a ) As depicted in Figure 7a, the value of T in the whole region is smaller than 1, s 12 meaning that the mode a ˆ is quantum entangled with the rest of the modes (a ˆ and a ˆ ). It s 2 should be emphasized that the value of T is independent of the interaction strength s 12 ratio a, and this is because the combination of the two idler modes a ˆ and a ˆ can be viewed 1 2 as a combined idler mode. Similarly, when the PT operation is applied to the modes a ˆ and a ˆ , the smallest symplectic eigenvalues are T and T , and their detailed results can 2 1 s2 2 s1 be written as 2 2 4 6 1 + (8G 5)a + (8G 5)a + [1 + 8G(G 1)]a 2 3 2 4[(2G 1)a 1](a + a ) (G 1)(1 + Ga ) T = , (25) 1 s2 2 3/2 (1 + a ) Photonics 2022, 9, 443 9 of 10 and 2 4 6 2 2 4 1 5a 5a + a + 8G (1 + a ) + 8G(a 1) 2 2 4(a + 1)(2G 1 + a ) (G 1)(G + a ) T = , (26) 2 s1 2 3/2 (1 + a ) respectively. The contour plots of T and T are depicted in Figure 7b,c, respectively. 1 s2 2 s1 It can be clearly seen that the values of T and T are both smaller than 1 in the whole 1 s2 2 s1 parametric region, showing that the three partitions are all inseparable and the existence of genuine three-mode entanglement in the whole parameter region. (a) ( c) (b) 0.15 0.3 0.3 Figure 7. The dependence of (a) T ; (b) T ; and (c) T . s 12 1 s2 2 s1 4. Conclusions In conclusion, we theoretically predicted that the concurrent parametric amplifier as a simple system can be used to generate two-mode and three-mode entanglement. The Duan criterion and PPT criterion can be used to quantify two-mode entanglement, which is present in the two pairs. The two-condition and optimal single-condition criterion were both analyzed to search for the entanglement region. In the case of the optimal single-condition criterion, genuine three-mode entanglement is present in the whole parameter region. More importantly, the PPT criterion was also used to claim the existence of genuine three-mode entanglement in the whole parameter region. Our concurrent parametric amplifier for generating multi-mode correlation and entanglement is integrated and phase-insensitive. Author Contributions: H.W. contributed to the development of the conceptualization, the discussions of the results, and the comments on the manuscript. Y.S. contributed to the validation, investigation, and data curation on the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Zhejiang Provincial Natural Science Foundation (LY22A040007), the National Natural Science Foundation of China (11804323), and the Fundamental Research Funds for the Provincial Universities of Zhejiang (2021YW29). Conflicts of Interest: The authors declare no conflict of interest. References 1. Carman, R.L.; Chiao, R.Y.; Kelley, P.L. 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