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Journal of Physics A Mathematical and General

Orthonormalized eigenstates of the operator {(aqf(Nq))}k(k⩾1) and their generation

Xiaoming Liu

Published Jul 28, 2000• 4 citations
Publisher

Abstract

The k orthonormalized eigenstates of the k th power {( a q f ( N q ))} k ( k ⩾1) of the generalized q -boson annihilation operator a q f ( N q ) are obtained, and their properties are discussed. An alternative method to construct them is proposed, and it is shown that all of them can be expressed as a linear superposition of k q - f -coherent states that have the same amplitude but different phases. Physically, they can be generated by a linear superposition of the time-dependent q - f -coherent states at different instants.

DOI
10.1088/0305-4470/33/31/312

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A new kind of k-quantum nonlinear coherent states, i.e., the κ eigenstates of the κ-th power B^κ (κ>_ 3) of the generalized annihilation operator B=a 1/f(N) of f-oscillators, are obtained and their properties are discussed. The completeness of the κ states is investigated. An alternative method to construct them is proposed. It is shown that these states may form a complete Hilbert space, and all of them can be generated by a linear superposition of κ Roy-type nonlinear coherent states. Physically, they can be generated by a linear superposition of the time-dependent Roy-type nonlinear coherent states at different instants.
Quantum statistical properties of orthonormalized eigenstates of the operator (â f (n^\hat nn^))k
The completeness of the k orthonormalized eigenstates of the operator (â f ( )) k ( k ⩾3) is investigated. We introduce a new kind of higher-order squeezing and an antibunching. The properties of the M th-order squeezing and the antibunching effect of the k states are studied. The result shows that these states may form a complete Hilbert space, and the M th-order ( M = ( n + 1/2) k ; n = 0,1,...) squeezing effects exist in all of the k states when k is even. There is an antibunching effect in all of the states. An alternative method for constructing the k states is proposed, and the result shows that all of them can be generated by linear superposition of the time-dependent nonlinear coherent states at different instants.
A method for construction of the k orthonormalized quantum eigenstates of the higher powers ak{\mathit{a}}^{\mathit{k}}ak of the quantum-annihilation operator a is proposed, and their physical meaning is explored. Nonclassical properties of these quantum eigenstates are studied and the main results are as follows: (1) All of the eigenstates can be expressed as a linear superposition of k quantum-coherent states; (2) all of them are generalized quantum-coherent states; (3) for k\ensuremath{\ge}3, they have the Nth-order quantum squeezing for N=mk+k/2 (m=0,1,2, . . .) and even k, but no Nth-order quantum squeezing for other values of N and k; (4) for k\ensuremath{\ge}3, all of them exhibit the lowest- and higher-order quantum antibunching.
Non-classical properties for the eigen-states of k-th power of q-boson annihilation operators were calculated by their representations in Fock spaces.The non-classical properties of these eigen-states were further discussed by numerically discussing their quasiprobability distributions in phase spaces.It is shown that the certain non-classical quantum states could be generated from quasiclassical coherent states by q-deformations;these eigen-states are typically non-classical quantum states,as their relevant Wigner functions reveal negatives for certain parameters.