Home Search Collections Journals About Contact us My IOPscience Multimode higher-order antibunching and squeezing in trio coherent states This content has been downloaded from IOPscience Please scroll down to see the full text 2002 J Opt B: Quantum Semiclass Opt 222 (http://iopscience.iop.org/1464-4266/4/3/310) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 129.93.16.3 This content was downloaded on 07/09/2015 at 09:59 Please note that terms and conditions apply INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS J Opt B: Quantum Semiclass Opt (2002) 222–227 PII: S1464-4266(02)32962-8 Multimode higher-order antibunching and squeezing in trio coherent states∗ Nguyen Ba An Institute of Physics, PO Box 429 Bo Ho, Hanoi 10000, Vietnam and Faculty of Technology, Hanoi National University, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam Received 23 January 2002, in final form 22 March 2002 Published 21 May 2002 Online at stacks.iop.org/JOptB/4/222 Abstract We study the multimode higher-order nonclassical effects of novel trio coherent states We show that such states exhibit antibunching to all orders in the single-mode case However, the two-mode higher-order antibunching may or may not exist depending on the parameters We also show that in such states squeezing is fully absent in both single-mode and two-mode situations As for the three-mode case, the so-called sum-squeezing is impossible but another kind of squeezing may arise for the orders K that are a multiple of three The degree of the lowest allowable K = order squeezing can reach a remarkable amount of 18% Of interest is the following property: when the order grows, the degree of antibunching increases but that of squeezing decreases Keywords: Antibunching, squeezing, multimode, higher-order, trio coherent state Introduction Current progresses, both theoretical and experimental, in quantum information science have led to the common recognition that nonclassical features in the quantum world may be utilized in communication networks to achieve various tasks that are impossible classically, such as quantum cryptography [1– 3], quantum teleportation [4, 5], quantum computation [6–9], etc For example, squeezed light can be applied to teleport entangled quantum bits [10] and antibunched light is useful to perform quantum communication and quantum computation [11] Although actual utilizations for everyday needs are still remote, nonclassical effects promise considerable potential applications in the future Therefore, searches for and study of new nonclassical states are welcome In fact, it is not impossible that a really adequate nonclassical state is still undiscovered or that it is among the discovered ones but its useful properties remain not properly exploited In addition to numerous known-to-date kinds of nonclassical states, a novel kind has recently been introduced [12] These are trio coherent states which generalize the so-called pair coherent states [13–20] The * This work is devoted to my teacher, Professor Nguyen Van Hieu, on the occasion of his 65th birthday 1464-4266/02/030222+06$30.00 © 2002 IOP Publishing Ltd trio coherent state |ξ, p, q is defined as the right eigenstate simultaneously of the operators abc, n a − n c and n b − n c where n x = x + x, x = {a, b, c} with a, b and c being bosonic annihilation operators of three independent boson modes (Note that different, more convenient, notations are used here rather than those in [12].) That is, abc|ξ, p, q = ξ |ξ, p, q (1) (n b − n c )|ξ, p, q = p|ξ, p, q (2) (n a − n c )|ξ, p, q = q|ξ, p, q (3) where ξ = r exp(iφ) with real r, φ is the complex eigenvalue and p, q are referred to as ‘charges’ which, without loss of generality, can be regarded as fixed non-negative integers These ‘charges’ serve as constants of motion in processes in which the boson number changes only in trios (each trio consists of one boson in mode a, one boson in mode b and one boson in mode c) Among various representations [12] of the trio coherent state, the most useful one is via Fock states |n x |ξ, p, q = N ( p, q, r ) ∞ n=0 × |n + q a |n + p b |n Printed in the UK c ξn √ (n + p)!(n + q)!n! (4) 222 Multimode higher-order antibunching and squeezing in trio coherent states where N ( p, q, r ) is the normalization coefficient given by N ( p, q, r ) = N (q, p, r ) = ∞ n=0 r 2n (n + p)!(n + q)!n! −1/2 (5) The mathematical properties of the state |ξ, p, q were studied in detail in [12] in which it was also shown that the trio coherent state exhibits sub-Poissonian number distribution, a type of squeezing and violates Cauchy–Schwartz inequalities An experimental scheme towards generation of such states was also proposed in [12] In the present paper we further investigate antibunching and squeezing of the trio coherent state with respect to multimode and higher-order issues Section is reserved for antibunching while squeezing is dealt with in section In each of the two sections, higher-order effects are studied for single-mode, two-mode and three-mode cases separately Section summarizes the main results of the paper Higher-order antibunching 2.1 Single-mode antibunching Single-mode higher-order antibunching is defined by the fulfilment of the following inequality [21, 22] n (l+1) n (m−1) < n (l) n (m) x x x x (6) where denotes the quantum average, x = {a, b, c}, l, m are integers satisfying the conditions l m and l−1 n (l) x ≡ j =0 (n x − j ) The usual (i.e first-order) antibunching corresponds to m = l = for which equation (6) reduces to n (2) < nx x (7) and the well-known inequality results ( nx ) ≡ n 2x − nx < nx n (l+1) x n (l) nx x < = r |q−l|−(q−l) N ( p, q, r )N −2 n (l) a |q − l| − (q − l) (12) × p+ , |q − l|, r Similarly, for mode b,   N ( p, q, r )N −2 ( p − l, q, r )     if p l = n (l) b 2(l− p)  N ( p, q, r )N −2 (l − p, q + l − p, r ) r     if p l (13) which can also be rewritten jointly as = r | p−l|−( p−l) N ( p, q, r )N −2 n (l) b | p − l| − ( p − l) × q+ , | p − l|, r As for mode c we obtain a simple formula (14) (15) (8) (9) or, likewise, A x;l ≡ We observe that n (l) a depends in fact on p, q and q − l rather than on p, q, l separately and we can rewrite equation (11) jointly as = r 2l N ( p, q, r )N −2 ( p + l, q + l, r ) n (l) c Here we consider the case of l m = for which the lth-order antibunching exists for mode x if < n (l) nx n (l+1) x x Figure Single-mode third-order antibunching of mode a, Aa;3 , as a function of r for p = and q = 0, 1, 2, 3, 4, (upwards at r = 8) (10) The function A x;l measures the degree of single-mode lth-order antibunching; the smaller (larger) A x;l the larger (smaller) the antibunching degree In the trio coherent state |ξ, p, q represented by equation (4) we obtain for an arbitrary l   N ( p, q, r )N −2 ( p, q − l, r )     if q l n (l) = a 2(l−q) r N ( p, q, r )N −2 ( p + l − q, l − q, r )     if q l (11) Using equations (10), (12), (14) and (15) allows us to study the modal antibunching in dependence on the parameters involved For mode a we find that it is antibunched to any order l over the entire range of r, but the dependence of Aa,l on r differs for q ∈ Q ≡ [1, l] (i.e q = 1, 2, , l) and for q ∈ Q˜ ∈ / [1, l] (i.e q = 0, l + 1, l + 2, ) When q ∈ Q, Aa,l equals zero at r = and then grows with increasing ˜ Aa,l starts from a nonzero value at r However, when q ∈ Q, r = and then also grows with r These behaviours are clearly seen from figure Furthermore, this figure indicates the ˜ and q1 < q2 , i.e relation Aa,l (q1 ) < Aa,l (q2 ) if q1,2 ∈ Q ( Q) the antibunching degree decreases with increasing q for given l, p and r Concerning the p-dependence, an opposite relation holds: Aa,l ( p1 ) > Aa,l ( p2 ) if p1 < p2 , i.e the antibunching degree increases with increasing p for fixed l, q and r A somewhat surprising feature is that antibunching of a higher order turns out to be more prominent, i.e the antibunching degree increases with growing l for parameters otherwise fixed (see figure 2) Thus, speculatively, higher-order antibunching might play a role superior to usual antibunching Note that such a prominence of higher-order antibunching was discovered for the first time in [22] for two-mode coherent states 223 Nguyen Ba An (a) (b) Figure Single-mode lth-order antibunching of mode a, Aa;l , as a function of the order l at r = and p = q = (circles) and p = q = (triangles) The above-mentioned properties remain true for mode b with the roles of p and q being exchanged As for mode c, it is also antibunched to any order l over the whole range of r, but a different behaviour arises Namely, when p (q) is fixed and q ( p) grows, Ac,l decreases at small r but increases at large r Nevertheless, it is found that, similar to the situation with modes a and b, mode c is more antibunched for a higher order as well For two arbitrary modes x and y (x = y = {a, b, c}), the twomode higher-order antibunching is defined by the fulfilment of the inequality [21] (16) where l m For simplicity we limit ourselves to the case with l m = for which equation (16) reduces to (l) n (l+1) + n (l+1) < n (l) x y x n y + n y nx (17) Likewise, two modes x and y are said to be antibunched to order l if A x,y;l < where A x,y;l = n (l+1) + n (l+1) x y (l) n (l) x n y + n y nx (18) is a measure of two-mode lth-order antibunching The two-mode higher-order antibunching was dealt with for pair and generalized pair coherent states in [21] Here we study this effect for the trio coherent state for which our calculations yield for any positive integers l, m the following analytic expressions of the averaged values of interest: (m) n (l) a nb  N ( p, q, r )N −2 ( p − m, q − l, r )      if p m&q l      r 2(l−q) N ( p, q, r )N −2 ( p − m + l − q, l − q, r ) = (19)  if l − q m − p&l q       r 2(m− p) N ( p, q, r )N −2 (m − p, m − p + q − l, r )    if m p&m − p l − q 224 (m) n (l) a nc   r 2m N ( p, q, r )N −2 ( p + m, q − l + m, r )     if m l − q (20) = 2(l−q)  N ( p, q, r )N −2 ( p + l − q, l − q − m, r ) r     if l − q m and 2.2 Two-mode antibunching (m) (l) (m) n (l+1) n (m−1) + n (l+1) n (m−1) < n (l) x y y x x n y + n y nx Figure Two-mode lth-order antibunching Aa,b;l as a function of r for p = 0, q = 0, 1, 2, 3, 4, (upwards) and (a) l = and (b) l = (m) n (l) b nc   r 2m N ( p, q, r )N −2 (q + m, p − l + m, r )     if m l − p (21) = 2(l− p)  N ( p, q, r )N −2 (q + l − p, l − p − m, r ) r     if l − p m The above expressions assign delicate dependences of the twomode higher-order antibunching on the problem parameters Our treatment is confined to the case with p = and q, r , l varying As a result, we discover that no antibunching arises for q > l When q l the two-mode antibunching occurs with distinct behaviours for q < l and q = l In the former situation A x,y;l equals zero at r = and increases with r , while in the latter situation A x,y;l equals unity at r = and decreases with increasing r Figure plots, for example, Aa,b;l as a function of r for p = and various values of l and q to confirm the above conclusions concerning the role of the charge q on antibunching of order l Also, as well as for the singlemode case, we find that the two-mode antibunching degree is larger for a higher order Higher-order squeezing 3.1 Single-mode squeezing Following Hillery [23] we consider the modal K th power amplitude component operator Q x (K , ϕ) = 12 ((x + ) K eiϕ + x K e−iϕ ) (22) where x = {a, b, c}, K = 1, 2, 3, and ϕ is a phase determining the direction of x K in the complex plane Multimode higher-order antibunching and squeezing in trio coherent states The operators (22) for phases differing by π/2 obey the commutation relation i [Q x (K , ϕ), Q x (K , ϕ + π/2)] = Fx (K ) (23) with Fx (K ) given by [24] K Fx (K ) = l=1 K !K (l) (x + ) K −l x K −l (K − l)!l! (24) where K (l) ≡ l−1 j =0 (K − j ) A state is said to be squeezed in mode x to an order K if there exists an angle ϕ such that Sx (K , ϕ) ≡ ( Q x (K , ϕ))2 − Fx (K ) < ) n (K x (26) which is ϕ-independent and always positive This implies the absence of single-mode squeezing to any order in the trio coherent state 3.2 Two-mode squeezing (27) where x = y = {a, b, c}, K = 1, 2, 3, and ϕ is a phase determining the direction of (x + y) K in the complex plane These operators obey the commutation relation [Q x y (K , ϕ), Q x y (K , ϕ + π/2)] = i Fx y (K ) Fx y (K ) = (x + y) K (x + + y + ) K − (x + + y + ) K (x + y) K (29) A state is said to be two-mode squeezed to an order K if there exists an angle ϕ such that Fx y (K ) < (30) The K = case reproduces the usual two-mode squeezing introduced by Loudon and Knight [26] It is not difficult to check that in the trio coherent state Sx y (K , ϕ) = K l=0 K! (K − l)!l! ) (K −l) n (K x ny (32) where K = 1, 2, 3, and ϕ is a phase determining the direction of (abc) K in the complex plane For any ϕ, the following commutation relation holds i L(K ) [P(K , ϕ), P(K , ϕ + π/2)] = (33) where L(K ) is given by L(K ) = (abc) K (a + b+ c+ ) K − (a + b+ c+ ) K (abc) K (34) A state is said to be three-mode sum-squeezed to an order K if there exists an angle ϕ such that U (K , ϕ) ≡ ( P(K , ϕ))2 − L(K ) < (35) The lowest-order K = case reproduces the usual three-mode sum-squeezing [29] In the trio coherent state we obtain (36) ) (K ) (K ) × cos[2(K φ − ϕ)] + n (K ] a nb nc (37) At first glance, both equations (36) and (37) contain a phase dependence and sum-squeezing is expected to occur Yet, a simple trigonometric manipulation in U (K , ϕ) ≡ P (K , ϕ) − P(K , ϕ) − L(K ) (38) casts it into (28) with Fx y (K ) given by Sx y (K , ϕ) ≡ ( Q x y (K , ϕ))2 − P(K , ϕ) = 12 [(a + b+ c+ ) K eiϕ + (abc) K e−iϕ ] P(K , ϕ) = r K cos(K φ − ϕ) P (K , ϕ) = 14 [ L(K ) + 2r 2K In the two-mode situation let us consider the operator Q x y (K , ϕ) = √ ((x + + y + ) K eiϕ + (x + y) K e−iϕ ) 2 The first kind of three-mode squeezing which we consider in this sub-section is related to the so-called sum-squeezing [27] The concept of the general multimode (first-order) sumsqueezing has been introduced in [28] Higher-order sumsqueezing in the three-mode case can be defined in terms of the operators (25) The lowest-order K = case reproduces the usual squeezing introduced by Stoler [25] In the trio coherent state it is easy to verify for arbitrary K and x that Sx (K , ϕ) = 3.3 Three-mode squeezing (31) for any order K as well as any pair of modes This equation (31) indicates that there are no angles ϕ that can make Sx y (K , ϕ) negative Hence, no two-mode squeezing can appear in the trio coherent state It is worth noting, however, that two-mode squeezing exists in the pair coherent state [13, 14] (K ) ) (K ) U (K , ϕ) = 21 [ n (K − r 2K ] a nb nc (39) which abandons any phase dependences Thus, unexpectedly, the sum-squeezing defined by equation (35) is impossible to any order K To gain more insight into the physics implied ) (K ) (K ) by equation (39) let us calculate the quantity n (K a nb nc For later use, we have derived the general analytic expression (m) (s) in the trio coherent state for arbitrary positive of n (l) a nb nc integers l, m and s in the form: (m) (s) n (l) = r 2s N ( p, q, r )N −2 (s + p − m, s + q − l, r ) a nb nc (40) if l, m and s meet one of the following four conditions (i) p m, q l, (ii) p < m, q l, s m − p, (iii) p m, q < l, s l − q and (iv) p < m, q < l, s {l − q, m − p}; (m) (s) = r 2(l−q) N ( p, q, r ) n (l) a nb nc × N −2 (l − q + p − m, l − q − s, r ) (41) if l, m and s meet one of the following two conditions (i) p m, q < l, l − q s, (ii) p < m, q < l, l − q {s, m − p} and (m) (s) = r 2(m− p) N ( p, q, r ) n (l) a nb nc × N −2 (m − p + q − l, m − p − s, r ) (42) 225 Nguyen Ba An if l, m and s meet one of the following two conditions (i) p < m, q l, m− p s, (ii) p < m, q < l, m− p {s, l−q} ) (K ) (K ) = Making use of the above formulae yields n (K a nb nc 2K r for any positive integer K That is, U (K , ϕ) ≡ due to equation (39) and, hence, the trio coherent state is a state of minimum uncertainty in which the uncertainty of P(K , ϕ) is equal in all directions The uncertainty region is a circle of √ radius R = 12 L(K ) with L(K ) determined explicitly by K K K L(K ) = l=0 m=0 s=0 (m) (s) (K !)6 n (l) a nb nc (l!m!s!)2 (K −l)!(K −m)!(K −s)! ) (K ) (K ) − n (K a nb nc (43) Numerical calculations of equation (43) show that the uncertainty circle radius quickly increases with both K and r Another kind of three-mode higher-order squeezing is associated with the operators Q(K , ϕ) = √ [(a + + b+ + c+ ) K eiϕ + (a + b + c) K e−iϕ ] (44) where K = 1, 2, 3, and ϕ is a phase determining the direction of (a + b + c) K in the complex plane These operators, for an arbitrary ϕ, obey the commutation relation [Q(K , ϕ), Q(K , ϕ + π/2)] = i F(K ) (45) with F(K ) given by F(K ) = (a+b+c) K (a + +b+ +c+ ) K −(a + +b+ +c+ ) K (a+b+c) K (46) A state is said to be three-mode squeezed to an order K if there exists an angle ϕ such that S (K , ϕ) ≡ ( Q(K , ϕ))2 − 12 F(K ) < [ (a + b + c) K e−iϕ ]} (48) In the trio coherent state our calculations yield (a + b + c) K = δ K ,3m (a + b + c)2K K! ξ K /3 ((K /3)!)3 (2K )! = δ K ,3m ξ 2K /3 ((2K /3)!)3 (49) (50) where m is a positive integer, and (a + + b+ + c+ ) K (a + b + c) K K l = l=0 s=0 K! (K − l)!(l − s)!s! −l) (l−s) (s) n (K nb nc a (51) −l) (l−s) (s) The averages n (K n b n c appearing in equation (51) are a to be evaluated by virtue of equations (40), (41) or (42) From equations (48)–(51) we observe that, if K = 3m, then S (K , ϕ) = K l l=0 s=0 K! (K − l)!(l − s)!s! −l) (l−s) (s) n (K nb nc a (52) 226 S (K , ϕ) = + × 2K /3 (2K )! cos[2(K φ/3 − ϕ)] r ((2K /3)!)3 2(K !)2 cos2 (K φ/3 − ϕ) + ((K /3)!)6 K! (K − l)!(l − s)!s! l K l=0 s=0 −l) (l−s) (s) n (K nb nc a (53) The phase dependence is transparent from equation (53) and for appropriate φ and ϕ one may have negative S (K , ϕ) resulting in a three-mode squeezing The squeezing degree is assessed by the quantity S(K , ϕ) = 12S (K , ϕ) F(K ) (54) where F(K ) is explicitly determined by S (K , ϕ) = 16 { (a + + b+ + c+ ) K (a + b + c) K which is phase-independent and always positive, meaning no three-mode squeezing In other words, three-mode squeezing, in the sense of equation (47), may occur only for K being a multiple of three For such K (47) In terms of annihilation/creation operators S (K , ϕ) reads + [ (a + b + c)2K e−2iϕ ] + Figure Three-mode squeezing degree S, equation (54), as a function of r for p = q = 0, K φ/3 − ϕ = π/2 and K = and K = K l F(K ) = l=0 s=0 K! (K − l)!(l − s)!s! K −l l−s s m=0 u=0 v=0 (u) (v) ((K − l)!(l − s)!s!)2 n (m) a nb nc × (m!u!v!)2 (K − l − m)!(l − s − u)!(s − v)! −l) (l−s) (s) − n (K nb nc a (55) By the definition (54), the ideal squeezing corresponds to S = −1 An extensive graphical work based on the above analytically derived formulae has indicated that squeezing is most favourable when p = q = and K φ = 3(ϕ + π/2) Under such conditions we draw in figure the squeezing degree S as a function of r for the two lowest allowable orders K = and K = For each value of K , squeezing appears in the small-r side and disappears in the large-r side The r range in which squeezing occurs widens for increasing order K It is also clear that squeezing becomes worse for a higher order, a fact which is opposed to antibunching (see section 2) Quantitatively, for K = the maximal degree of three-mode squeezing is about 18%, whereas for K = it is just around 9% Multimode higher-order antibunching and squeezing in trio coherent states Conclusion In conclusion, we have extended the consideration of trio coherent states introduced in a previous paper [12] to the case of multimode higher-order antibunching and squeezing We have proven that single-mode antibunching exists to all orders but two-mode antibunching arises only under certain constraints imposed upon l, p and q We have elucidated in detail the dependences on the parameters involved Of particularity is the increase of antibunching degree when the order grows, a fact emphasizing the role of higher-order antibunching as compared to that of the usual antibunching We have also explicitly shown that, in the trio coherent state, squeezing to any order has nothing to with the single-mode and twomode situations In the three-mode case, sum-squeezing does not appear either However, its analysis has revealed the trio coherent state as a kind of minimum uncertainty state Only the kind of three-mode higher-order squeezing defined by equation (47) turns out to be possible but not for any orders K The allowed orders have been found to be a multiple of three In contrast to antibunching, the degree of such a three-mode squeezing decreases with increasing K This kind of squeezing is worth attention since it possesses a remarkable maximal amount of squeezing: 18% for K = and 9% for K = An extension of the trio coherent state to nonlinear trio coherent states including odd/even trio coherent states is now under way Finally, it is important to remember that, though an experimental scheme for generating the trio coherent state was suggested in [12], further work is worthwhile towards a real implementation of such a state Acknowledgments The author is indebted to Professor Nguyen Van Hieu, his teacher, for the constant attention and encouragement in the author’s entire scientific career The present paper is respectfully dedicated to Professor Nguyen Van Hieu on his 65th birthday This work was supported by the National Basic Science Project KT-04.1.2 and by the Faculty of 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Introduction Current progresses, both theoretical and