Multimode higher-order antibunching and squeezing in trio coherent states

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 = 3 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.

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 computa-tion [11]. Although actual utilizations for everyday needs are still remote, nonclassical effects promise considerable poten-tial 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 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 ² ) ∑n=0∞ξn(n+p)!(n+q)!n! size 12{ Sum cSub { size 8{n=0} } cSup { size 8{ infinity } } { { {ξ rSup { size 8{n} } } over { sqrt { ( n+p ) ! ( n+q ) !n!} } } } } {}

× |n + q〉_a |n + p〉_b |n〉_c (4)

where N (p, q, r ² ) is the normalization coefficient given by

N (p, q, r ² ) = N (q, p, r ² ) = ∑n=0∞r2n(n+p)!(n+q)!n! size 12{ left ( Sum cSub { size 8{n=0} } cSup { size 8{ infinity } } { { {r rSup { size 8{2n} } } over { ( n+p ) ! ( n+q ) !n!} } } right )} {}−12 size 12{ {} rSup { size 8{ { { - 1} over {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 furtherinvestigate antibunching and squeezing of the trio coherentstate with respect to multimode and higher-order issues. Section 2 is reserved for antibunching while squeezing is dealt with in section 3. In each of the two sections, higher-order effects are studied for single-mode, two-mode and three-mode cases separately. Section 4 summarizes the main results of the paper.

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