Multi-directional higher-order amplitude squeezing

Fan-even K -quantum nonlinear coherent states are introduced and higher-order amplitude squeezing is investigated in such states. It is shown that for a given K the lowest order in which an amplitude component can be squeezed is 2K and the squeezing appears simultaneously in K directions separated successively in phase by π/K . © 2001 Published by Elsevier Science B.V.

During a few last years, there has been growing interest in the nonlinear coherent state (NCS) [1-5] defined as the eigenstate |χ ; f 〉 of the non-boson non-Hermitian operator af ( ˆn),

af ( ˆn)|χ ; f 〉 = χ |χ ; f 〉, (1)

with ˆn = a+a, a the bosonic annihilation operator, χ a complex eigenvalue and f an arbitrary (assumed to be real) nonlinear operator-valued function of ˆn. Re-cently, the so-called K -quantum nonlinear coherent state (KNCS) has been introduced [6-8] as a gener-alization of the NCS to K (K an arbitrary positive in-teger) eigenstates |ξ ; K, j, f 〉 of the non-boson non-Hermitian operator aK f ( ˆn),

aK f ( ˆn)|ξ ; K, j, f 〉 = ξK |ξ ; K, j, f 〉, (2)

with j = 0, 1, . . . , K − 1 and ξ a complex number.

Note that the notations in [6-8] are not the same. In[6] (af(ˆn))^k is used instead of a^K f(ˆn).In [7] it is aN +1f ( ˆn) with N counted from zero and the state is named NCS of order N + 1. The even (odd) NCS [9-11] corresponds to K = 2, j = 0 (j = 1). The lin-ear case (see, e.g., [12-18]) is recovered when f ≡ 1. Other types of multi-quantum states are also devel-oped (see, e.g., [19,20]). In [7,8] the KNCS has been shown physically realizable in the quantized vibration of the center-of-mass motion of a harmonically trapped ion which is further properly driven by two laser beams one of which is resonant and the other is detuned to the Kth lower sideband. The KNCS dis-plays a multi-peaked structure in the number distribu-tion depending on both the character of the nonlinear function f and the power K. The normalization dif-ficulty connected with the wildly oscillating behavior at large n and Lamb-Dicke parameter has been dealt with in [7] while nonclassical effects have been stud-ied in [8]. In this Letter we further explore the KNCS under another angle of view. Namely, a property of the KNCS will be used to construct the so-called fan-even K -quantum nonlinear coherent states (FEKNCS’s) in which higher-order amplitude squeezing will be inves-tigated. Unlike in usual states, in FEKNCS’s multi-directional squeezing is possible.

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