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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Some properties of the states engendered by the action of photon creation and annihilation operator on a squeezed vacuum state

Hong-Yi Fan, Zhongxi Zhang

Published Oct 28, 1992• 0 citations
Publisher

Abstract

We study some properties of the states a+mS(r)0> and amS(r)0> (S(r) equals e-$ (a(superscript 2-a+2)) which are engendered by the excitations and de-excitations on a squeezed vacuum state, respectively. Using the mathematical induction method, we easily show that such states are normalized as two different forms of Legendre polynomials of squeezing parameter r. The Wigner functions of these states are also derived and analyzed.

DOI
10.1117/12.130407

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We study some properties of the states a+mS(r)0> and amS(r)0> (S(r) equals e-$ (a(superscript 2-a+2)) which are engendered by the excitations and de-excitations on a squeezed vacuum state, respectively. Using the mathematical induction method, we easily show that such states are normalized as two different forms of Legendre polynomials of squeezing parameter r. The Wigner functions of these states are also derived and analyzed.
Inverses of the harmonic oscillator creation and annihilation operators by their actions on the number states are introduced. Three of the two photon annihilation operators, viz., a(sup +/-1)a, aa(sup +/-1), and a(sup 2), have normalizable right eigenstates with nonvanishing eigenvalues. The eigenvalue equation of these operators are discussed and their normalized eigenstates are obtained. The Fock state representation in each case separates into two sets of states, one involving only the even number states while the other involving only the odd number states. It is shown that the even set of eigenstates of the operator a(sup +/-1)a is the customary squeezed vacuum S(sigma) O greater than.
Photon-added squeezed vacuum states were fabricated by act inverse operator of Boson annihilation operator ~(-1) and Boson creation operator ~+ on the squeezed vacuum states first.Then the squeezing properties of the photon-added squeezed vacuum states are discussed.The results show that only the photon-added squeezed state which was acted on ~+ can appear squeezing effects,but was acted on ~(-1) did not.The squeezing range have relation with the squeezing parameter η.So the squeezing properties of the photon-added squeezed vacuum states have closed relation with ~(-1) and ~+.
We introduce the states D(z)a\+\{+\+m\}S(r)|0〉 and examine their properties . Such states are checked to be complete vectors via the IWOP (Integration W ithin an Ordered Product of operators) technique. We find larger－scale macrosco pic oscillations of photon distribution after the squeezed vacuum states being e xcited and displaced. On contrary to the bunching photons of the squeezed vacuum states as the displacing parameter |z| and the excitation number m are both small photons coherence can exhibit antibunching effect. The quasi－proba bility distribution functions (Husimi Q( 醆+*) function and Wigner W(q p) function) are derived and their dependences on the three parameters (z m r) are discussed. The Wigner function obviously displays their nonclassical fe ature.
We introduce states a † m | ξ ⟩ and a m | ξ ⟩, where| ξ ⟩ is a two-mode squeezed vacuum state. We examine the properties of these states by analytic derivation. Because of the entanglement between the modes of the field, photon adding (or substracting) for one of the modes results in interesting outcomes. For a two-mode squeezed vacuum state, the effect of annihilation of m a-photons is just the same as creation of m b-photons. Annihilation of photons of mode a (or mode b ) actually increases the mean number of photons of mode a and mode b . We also calculate Mandel Q parameter, cross-correlation function and squeezing of the field.
We introduce the inverse of the harmonic-oscillator creation and annihilation operators by their actions on the number states. We then show that three of the two-photon annihilation operators, a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$a^, a^a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$, and a^ 2^{2}2, possess eigenstates that are analogous to the often-used coherent and squeezed states. A family of the eigenstates of the operator a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$a^ is the customary squeezed vacuum S(\ensuremath{\sigma})\ensuremath{\Vert}0〉 while another family of the eigenstates of a^a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$ is the state S(\ensuremath{\sigma})\ensuremath{\Vert}n=1〉. We find some interesting properties related to squeezing, bunching, and antibunching exhibited by these states. An alternative method of summing some series by using these states is also considered. It is hoped that these states will find applications in quantum optics and quantum mechanics in general.