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Home Search Collections Journals About Contact us My IOPscience Even and odd trio coherent states: antibunching and violation of Cauchy–Schwarz inequalities This content has been downloaded from IOPscience Please scroll down to see the full text 2002 J Opt B: Quantum Semiclass Opt 289 (http://iopscience.iop.org/1464-4266/4/5/310) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 137.132.123.69 This content was downloaded on 04/10/2015 at 02:59 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) 289–294 PII: S1464-4266(02)36820-4 Even and odd trio coherent states: antibunching and violation of Cauchy–Schwarz inequalities Nguyen Ba An1,2 and Truong Minh Duc3 Institute of Physics, PO Box 429 Bo Ho, Hanoi 10000, Vietnam Faculty of Technology, Hanoi National University, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam Physics Department, Hue Pedagogic University, 32 Le Loi, Hue, Vietnam Received May 2002, in final form July 2002 Published August 2002 Online at stacks.iop.org/JOptB/4/289 Abstract Even and odd trio coherent states are introduced by superposing two trio coherent states Their antibunching and Cauchy–Schwarz inequalities are studied in detail It is shown that, while odd trio coherent states are always antibunched and violate the Cauchy–Schwarz inequalities, even ones may or may not so, depending on the parameters Keywords: Even (odd) trio coherent states, antibunching, Cauchy–Schwarz inequalities Introduction Even and odd trio coherent states Recently, in addition to various kinds of known nonclassical states (see, e.g., [1]), there has been introduced a new one called a trio coherent state (TCS) [2, 3] The TCS, a generalization of the pair coherent state [4], has been shown to manifest intriguing nonclassical properties [2], including multimode and higher-order effects [3] A scheme to generate the TCS has also been suggested in [2] Following the way the pair coherent state goes to the even/odd pair coherent states [5], we shall consider in this paper the so-called even/odd trio coherent states (ETCS/OTCS) which are simple superpositions of two macroscopically distinguishable TCSs The study of ETCS/OTCS, in parallel with different kinds of superposition states such as even/odd coherent states [6], even/odd nonlinear coherent states [7], even/odd pair coherent states [5], etc, proves highly meaningful because such states serve as a prototype of Schrăodinger cat states [8] which exhibit quantum interferences due to the superposition principle, a cornerstone principle distinguishing the quantum from the classical world This paper is organized as follows In the next section (section 2) we define the ETCS/OTCS in several equivalent ways Then, we investigate in detail the possibility of possessing antibunching (section 3) and violating Cauchy–Schwarz inequalities (section 4) in such states In the final section (section 5) some discussions and conclusions are given Other interesting issues, such as oscillatory number distribution, squeezing, etc, are being reported in a separate paper Let us deal with two quantum states, |ξ, p, q, |ξ, p, q, , which are defined as follows: 1464-4266/02/050289+06$30.00 © 2002 IOP Publishing Ltd and |ξ, p, q, = N0 ( p, q, |ξ |2 ) (|ξ, p, q + | − ξ, p, q ), (1) 2N ( p, q, |ξ |2 ) |ξ, p, q, = N1 ( p, q, |ξ |2 ) (|ξ, p, q − | − ξ, p, q ) (2) 2N ( p, q, |ξ |2 ) where ξ = r exp(iϕ): N j (l, m, z) = ∞ z 2n+ j (2n + j + l)!(2n + j + m)!(2n + j )! n=0 j = 0, 1, N (l, m, z) = −1/2 , (3) ∞ n=0 2n z (n + l)!(n + m)!n! −1/2 (4) and |ξ, p, q is the TCS defined as the joint eigenstate of the three operators abc, Qˆ = nˆ a − nˆ c and Pˆ = nˆ b − nˆ c with nˆ a(b,c) = a + a(b+ b, c+ c) and a(b, c) the bosonic annihilation operator of mode a(b, c), i.e., Printed in the UK abc|ξ, p, q = ξ |ξ, p, q , (5) ˆ P|ξ, p, q = p|ξ, p, q , (6) ˆ Q|ξ, p, q = q|ξ, p, q (7) 289 Nguyen Ba An and Truong Minh Duc with p, q the ‘charges’ [9] which, without loss of generality, can be treated as non-negative integers From equations (1) and (2), it follows directly that |−ξ, p, q, = +|ξ, p, q, where the ξ integration is understood as (remember that ξ = r exp(iϕ)) (8) |−ξ, p, q, = −|ξ, p, q, (9) Hence, the state |ξ, p, q, and |ξ, p, q, are respectively referred to as the ETCS and OTCS Spanned in Fock space via three-mode number states |l a |m b |n c , these states |ξ, p, q, j , for j = or 1, have the form |ξ, p, q, j = N j ( p, q, |ξ |2 ) ∞ × n=0 c (17) and f ( p, q, r ) implicitly determined by ∞ 2π f ( p, q, r )r 2n+1 dr = n!(n + p)!(n + q)! Note that, in terms of the same function W ( p, q, r ) given by (17), the resolution of unity for the TCS is ∞ d2 ξ W ( p, q, r )|ξ, p, q q, p, ξ | = I p,q=0 (10) where N j ( p, q, |ξ | ) given in (3) is the right coefficient to normalize |ξ, p, q, j to unity Acting on both sides of ˆ and making use of (5), either (1) or (2) by (abc)2 ( Pˆ or Q) ((6) or (7)) yield immediately for both j = and dϕ f (r, ϕ) W ( p, q, r ) = f ( p, q, r )N −2 ( p, q, r ) ξ 2n+ j √ (2n + j + q)!(2n + j + p)!(2n + j )! × |2n + j + q a |2n + j + p b |2n + j 2π r dr with and ∞ d2 ξ f (ξ ) = (abc)2 |ξ, p, q, j = ξ |ξ, p, q, j , (11) ˆ P|ξ, p, q, j = p|ξ, p, q, j , (12) ˆ Q|ξ, p, q, j = q|ξ, p, q, j (13) Equations (11)–(13) represent another way to define the ETCS/OTCS Alternatively, the ETCS/OTCS can also be constructed as those obtained by applying the generator G( p, q, ξ, a + , b+ , c+ ) G( p, q, ξ, a + , b+ , c+ ) = N j ( p, q, |ξ |2 ) × N j−2 ( p, q, ξ a + b+ c+ )(a + )q (b+ ) p , Antibunching A mode x in a quantum state | Ax ≡ nˆ (l) c j × (14) Of course, all the definitions (1), (2) and (11)–(13), as well as (14), are equivalent with each other From the scalar product nˆ (l) b it follows that j × (15) and × N j−2 ( p, q, ξ ∗ ξ ) = δξ ξ ∞ j =0 p,q=0 d2 ξ N ( p, q, |ξ |2 ) N j2 ( p, q, |ξ |2 ) × W ( p, q, |ξ |)|ξ, p, q, j 290 j, q, p, ξ | = I (18) = |ξ |2l N j2 ( p, q, |ξ |2 ) N j−2 ( p + l, q + l, |ξ |2 ) if l = 0, 2, 4, −2 N1− j(p if l = 1, 3, 5, + l, q + l, |ξ | ) = N j2 ( p, q, |ξ |2 )  −2  N ( p − l, q, |ξ |2 )   j    if p − l = 0, 1, 2,      |ξ |2(l− p) N −2 (l − p, q + l − p, |ξ |2 ) j  if p − l = 0, −2, −4,     2(l− p) −2   N1− j (l − p, q + l − p, |ξ |2 ) |ξ |     if p − l = −1, −3, −5, (20) for mode b and (16) These imply the orthogonality with respect to j and j = j , equation (15), but non-orthogonality with respect to ξ and ξ = ξ , equation (16) The resolution of unity for the state |ξ, p, q, j is For p = an almost ideal antibunching is achievable at very small |ξ | (see figure 2(b)), but for p > the curves shift up and an ideal antibunching can never be reached (see figure 2(c)) (c) Figure The antibunching parameter Ab for mode b in the ETCS as a function of |ξ | for q = (broken), q = (chain) and q = (full) and (a) p = 0, (b) p = and (c) p = (a) (b) (c) Figure Same as in figure but in the OTCS As for the OTCS, mode b (as mode c) is always antibunched in it However, we must distinguish two kinds of behaviours For p = the function Ab emerges from zero (figure 3(a)) But for p it starts from a nonzero value (figures 3(b) and (c)) and the curves are all shifted up for increasing p (compare figure 3(c) with (b)) Violation of Cauchy–Schwarz inequalities If a quantum state | is of multimode nature there may exist correlations between modes For x = y the function [10] Cx y = |x + y + yx| |x + x| |y + y| (22) 291 Nguyen Ba An and Truong Minh Duc characterizes the degree of intermode correlation Note that, when x ≡ y, (22) reduces to (18), i.e Cx x ≡ A x Classically, the Cauchy–Schwarz inequalities Cx2y Cx x C yy (a) or, equivalently, |nˆ (2) x | |nˆ (2) y | |nˆ x nˆ y | (23) always hold We shall test in this section the inequalities (23) in our ETCS and OTCS, i.e when | ≡ | p, q, ξ, j For convenience, we shall examine the scaled Cauchy–Schwarz inequalities determined by Fx y ≡ nˆ (2) x j nˆ (2) y nˆ x nˆ y j j (24) (c) For that purpose we need to calculate the averages appearing in (24) Those in the numerator were already known from (19)– (21) and those in the denominator are calculated to be, generally for any j = 0, 1, nˆ a nˆ b = N j2 ( p, q, |ξ |2 )  −2 N j ( p − 1, q − 1, |ξ |2 )       if p 1, q    −2     N1− j (1, q, |ξ |2 )           if p = 0, q    ×  −2   N1− j ( p, 1, |ξ |2 )    |ξ |     if p 1, q =         −2    N1− j (0, 1, |ξ |2 )         if p = q = 0, j nˆ b nˆ c j = |ξ | −2 N j2 ( p, q, |ξ |2 )N1− j ( p, q Figure Fbc as a function of |ξ | in the ETCS for (a) q = and p = 0, 1, (broken, chain and full curves, respectively), (b) p = and q = 0, 1, (broken, chain and full curves, respectively) and (c) p = and q = 0, 1, (broken, chain and full curves, respectively) (a) (25) (b) + 1, |ξ | ) (b) (26) and nˆ a nˆ c j −2 = r N j2 ( p, q, |ξ |2 )N1− j ( p + 1, q, |ξ | ) (27) Thanks to the symmetry between modes a and b as mentioned in the previous section it is only necessary to deal with two pairs of modes: {b, c} (or {a, c}) and {a, b} For the pair {b, c} we discover that in the ETCS and with a fixed value of q the Cauchy–Schwarz inequality is always violated if p = 0, and ‘partially’ violated if p (a partial violation means that Fbc > at small |ξ | and then becomes less than unity for large |ξ |, as is evident from figure 4(a)) The dependence on q with a fixed p also differs for p = 0, and p For p = 0, the inequality is more violated for a greater value of q (figure 4(b)) For p 2, however, the violation is weaker (stronger) for a larger q in the small-|ξ | (large-|ξ |) region, as can be seen from figure 4(c) In the OTCS the Cauchy–Schwarz inequality is found to be always violated for the pair {b, c} For a fixed q, the violation is almost independent of p (figure 5(a)) but, for a fixed p, a greater value of q favours the violation more (figure 5(b)) As for the pair {a, b} the violation status of the Cauchy– Schwarz inequality can be summarized in tables and for the ETCS and OTCS, respectively In the tables F means a full 292 Figure Fbc as a function of |ξ | in the OTCS for (a) q = and p = 0, 1, (broken, chain and full curves, respectively) and (b) p = and q = 0, 1, (broken, chain and full curves, respectively) violation starting from zero (i.e Fab < over the whole range of |ξ | and Fab (|ξ | = 0) = 0), F˜ means a full violation starting from a non-zero value (i.e Fab < over the whole range of |ξ | and Fab (|ξ | = 0) > 0) and P means a partial violation (i.e Fab > at small |ξ | and then becoming less than unity at large |ξ |) For illustration we display in figure the situations corresponding to the last rows in table (for the ETCS) and table (for the OTCS) Discussion and conclusion From the above analysis, the OTCS is ‘fully’ nonclassical in the sense that it possesses antibunching and violates the Cauchy–Schwarz inequalities over the whole range of |ξ | for Even and odd trio coherent states: antibunching and violation of Cauchy–Schwarz inequalities (a) (b) Figure Fab as a function of |ξ | for p = 0, 1, (broken, chain and full curves, respectively) and (a) q = in the ETCS; (b) q = in the OTCS Table Cauchy–Schwarz inequality violation status for the pair {a, b} in the ETCS, depending on the charges p, q F means a full violation starting from zero (i.e Fab < in the whole range of |ξ | and Fab (|ξ | = 0) = 0), F˜ also means a full violation but starting from a non-zero value (i.e Fab < in the whole range of |ξ | and Fab (|ξ | = 0) > 0) and P means a partial violation (i.e Fab > at small |ξ | and then becoming less than unity at large |ξ |) q =0 q =1 q p=0 p=1 p F˜ F˜ P F˜ F F P F F˜ Table Same as in table but in the OTCS q=0 q p=0 p F F F F˜ any values of p, q The ETCS, on the other hand, is ‘partially’ nonclassical in the sense that violation of antibunching and Cauchy–Schwarz inequalities may or may not occur in it Of special attention is the role of charges Besides the possibility of improving the level of a nonclassical effect by altering the charges, a proper choice of charges (which can be made in preparing the initial Fock state in the process of generating the quantum state via a physical mechanism) may convert the ETCS from a partially into a fully nonclassical state (see, e.g figures 2(b) and (c) in comparison to figure 2(a); the broken and chain curves in comparison to the full curve in figure 4(a) and the full and chain curves in comparison to the broken curve in figure 6(a)) The three-mode nature of the TCS, ETCS and OTCS may be useful in the vibrational motion of the centre of mass of an ion trapped by a 3D potential In this case it is more convenient to change the notation in section as a → ax , b → a y and c → az with ax , a y and az being the annihilation operators of the quantum of the quantized vibration of a trapped ion along the x, y and z axis directions, respectively The ion trapping dynamics [11] has opened up new prospects in practical applications such as quantum computations [12] because nonclassical states realized by trapped ions are highly stable thanks to their extremely weak interaction with the environment In connection with the topic under consideration we recall that vibrational cat states can be produced in 1D by two lasers (see de Matos Filho and Vogel [6]) and in 2D by five lasers (see Gou et al [5]), whereas generation of a squeezed pair-coherent state of the motion of an ion needs a 3D trap and seven lasers [13] We shall present the technical details of the generation of vibrational ETCS and OTCS elsewhere Here we just outline the main ideas An ion trapped in a 3D isotropic harmonic potential should be driven by fourteen travelling-wave lasers, thirteen of which are chosen such that the first (2nd–13th) one drives the ion along the direction connecting the coordinate origin {x, y, z} = {0, 0, 0} to a point {x, y, z} = {1, 1, 1} ({1, −1, 1}, {1, 1, −1}, {1, −1, −1}, {1, 1, 0}, {1, −1, 0}, {1, 0, 1}, {1, 0, −1}, {0, 1, 1}, {0, 1, −1}, {1, 0, 0}, {0, 1, 0} and {0, 0, 1}) All these thirteen lasers are detuned to the sixth lower vibrational sideband The last fourteenth laser must be resonant with the ionic transition and may be arbitrarily directed Then the laser intensities and frequencies can be controlled appropriately so that, in the Lamb–Dicke limit, the effective ion–laser interaction Hamiltonian has the form Hint = ζ [ξ − (ax a y az )2 ]σ+ + h.c., with σ+ the ion raising operator and ζ , ξ depending on the lasers’ characteristics and the Lamb–Dicke parameter Thanks to the spontaneous emission the system will reach a stationary regime in which the ion and the lasers are decoupled, i.e the system density operator is of the form ρs = |0 | s s | 0|, where |0 is the ion ground state while | s describes the ion vibration and satisfies the equation (ax a y az )2 | s = ξ | s As the evolution governed by Hint conserves the charges, the ˆ s = | s is also the eigenstate of the charge operators, i.e P| ˆ s = q| s , implying | s ≡ |ξ, p, q, j The p| s , Q| desired vibrational ETCS (OTCS) |ξ, p, q, (|ξ, p, q, ) is generated if the ion vibration is initially prepared in a Fock state | = |2n + q x |2n + p y |2n z (| = |2n + + q x |2n + + p y |2n + z ) with n being some integer In conclusion, we have defined even and odd trio coherent states in several equivalent ways and studied in detail their antibunching and violation of Cauchy–Schwarz inequalities We have also shown that such states can be realized in the motion of an ion confined in a 3D trap We would expect that, as novel nonclassical states, the ETCS and OTCS could find their real implementation in the future, particularly in connection with the emergence of quantum information processing [14] Finally, the three-mode nature of the states might have something to with the GHZ state [15] Acknowledgment This work was supported in part by the National Basic Scientific Project KT-411101 References [1] Dodonov V V 2002 J Opt B: Quantum Semiclass Opt R1 [2] Nguyen B A and Truong M D 2002 J Opt B: Quantum Semiclass Opt 80 [3] Nguyen B A 2002 J Opt B: Quantum Semiclass Opt 222 [4] Agarwal G S 1986 Phys Rev Lett 57 827 Agarwal G S 1988 J Opt Soc Am B 1940 293 Nguyen Ba An and Truong Minh Duc [5] Gerry C C and Grobe R 1995 Phys Rev A 51 1698 Gou S C, Steinbach J and Knight P L 1996 Phys Rev A 54 4315 Liu X M 2001 Phys Lett A 279 123 [6] Dodonov V V et al 1974 Physica 72 597 Gerry C C 1993 J Mod Opt 40 1053 de Matos Filho R L and Vogel W 1996 Phys Rev Lett 76 608 [7] Mancini S 1997 Phys Lett A 233 291 Sivakumar S 1998 Phys Lett 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Cauchy–Schwarz inequalities (a) (a) (b) (b) Figure The antibunching parameter Ac for mode c as a function of |ξ | (a)... range of |ξ | for Even and odd trio coherent states: antibunching and violation of Cauchy–Schwarz inequalities (a) (b) Figure Fab as a function of |ξ | for p = 0, 1, (broken, chain and full curves,... coherent states [6], even/ odd nonlinear coherent states [7], even/ odd pair coherent states [5], etc, proves highly meaningful because such states serve as a prototype of Schrăodinger cat states’ [8]