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Physical Review A

Orthonormalized eigenstates of the quantum operator ak and their nonclassical properties

Xiao-Ming Liu, Shun-Jin Wang, ZHANG WEN-ZHONG

Published Jun 1, 1995• 6 citations
Publisher

Abstract

A method for construction of the k orthonormalized quantum eigenstates of the higher powers ${\mathit{a}}^{\mathit{k}}$ 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.

DOI
10.1103/physreva.51.4929

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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.
In this paper, we study first the connection of the k orthonormalized eigenstates of ak{\mathit{a}}^{\mathit{k}}ak with coherent states, then, according to the higher-order squeezing defined by Zhang et al. [Phys. Lett. A 150, 27 (1990)], study the properties of the higher-order squeezing of k orthonormalized eigenstates. Our results show that the k orthonormalized eigenstates can be represented as a linear superposition of k coherent states, and that all of them are minimum uncertainty states of the operators Z1{\mathit{Z}}_{1}Z1​(N) and Z2{\mathit{Z}}_{2}Z2​(N) (N=mk, m=1,2,3, . . .) for even and odd k, and all of them have the Nth-order squeezing [N=(m+1/2)k, m=0,1,2, . . .] for even k.
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.
We have studied the orthonormalized eigenstates of the higher powers ak{\mathit{a}}^{\mathit{k}}ak (k\ensuremath{\ge}3) of the annihilation operator a and their properties. Our conclusions are as follows: (1) The orthonormalized eigenstates of ak{\mathit{a}}^{\mathit{k}}ak construct a complete representation with respect to \ensuremath{\alpha}. (2) None of the orthonormalized eigenstates of ak{\mathit{a}}^{\mathit{k}}ak shows the squeezing effect in the usual sense. (3) All of the eigenstates of ak{\mathit{a}}^{\mathit{k}}ak show the antibunching effect.
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.