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Physical Review Letters

Sub-Poissonian Photon Statistics in a Three-Wave Interaction Starting in the Out-of-Phase Regime

A. Bandilla, G. Drobný, Igor Jex

Published Nov 27, 1995• 13 citations
Publisher

Abstract

We present for the first time numerical and analytical calculations for the nonlinear interaction of three quantized waves all sizably excited from the beginning and having different phase relations. With a Kerr-state ansatz for the signal we get strongly sub-Poissonian photon statistics and conclude on similar effects by initially entangled states.

DOI
10.1103/physrevlett.75.4019

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We present for the first time numerical and analytical calculations for the nonlinear interaction of three quantized waves all sizably excited from the beginning and having different phase relations. With a Kerr-state ansatz for the signal we get strongly sub-Poissonian photon statistics and conclude on similar effects by initially entangled states.
We consider the classical and quantum-mechanical processes of three-wave interactions in different phase regimes and present numerical calculations for the quantum case, where all three modes are sizably excited from the beginning. These excitations are coherent so that various important phase regimes can be adjusted. In addition, one mode can also be prepared in a squeezed or Kerr state. The classical solutions are well known and are briefly summarized, but certain phase regimes are classically unexplored and we show here that they give interesting and surprising results. In the out-of-phase regime (where the photon numbers do not change in the first order of time) we get, with an initial Kerr state, strongly sub-Poissonian photon statistics in the signal after a short interaction time. This effect is limited by the classically described phase shifts that are present even in the parametric approximation. This nonclassical phenomenon (due to the Kerr state) helps us to understand similar nonclassical effects generated by entangled states of the pump and signal during sum-frequency generation. \textcopyright{} 1996 The American Physical Society.
We study the statistical properties of squeezed Kerr states generated by squeezing, with a two-photon parametric process, the output of a Kerr medium whose input is coherent light. The squeezing has the effect of allowing the photon statistics to depend on the Kerr nonlinearity. Without the squeezing the statistics are Poissonian, but after squeezing they may be either sub-Poissonian or super-Poissonian. We also study the Q-function quasiprobability distribution for these states and related distributions associated with the quantum-mechanical phase. Finally, we show that quadrature squeezing can be enhanced and that squeezing and sub-Poissonian statistics can coexist.
In the present paper we study higher-order sub-Poissonian photon statistics in interaction of a single mode radiation initially in the coherent state Ŧα〉 with an assembly of two excited two-level atoms using the Hamiltonian, H=ω(a+a+Sz)+g(aS++a+S.), in the natural system of units. Here, α=|α|eiθα, a is the annihilation operator for radiation, Sz and S± are the collective Dicke operators, g is the coupling constant, ω is the energy of the photons and the energy difference between the two atomic levels. We solve it exactly and conclude that higher-order sub-Poissonian photon statistics can be obtained for an arbitrary order by choosing suitably the square root of mean photon number |α| of the coherent radiation and coupling time gt. Variations of higher-order sub-Poissonian photon statistics with parameters |α| and gt have also been discussed.
The squeezing and sub-Poissonian eects of light in third harmonic generation are investigated based on the fully quantum mechanical approach up to the rst order Hamiltonian interaction in gt, where g is the coupling constant between the modes per second and t is the interaction time between the waves during the process in a nonlinear medium.The coupled Heisenberg equations of motion involving real and imaginary parts of the quadrature operators are established.The occurrence of amplitude squeezing eects in both the quadratures of the radiation eld in the fundamental mode is investigated and found to be dependent on the selective phase values of the eld amplitude.The photon statistics of the pump mode in this process have also been investigated and found to be sub-Poissonian in nature.It is shown that for particular phase values the amplitude squeezing and sub-Poissonian photon statistics of light occur simultaneously.It is observed that there is no possibility to produce squeezed light in the harmonic mode up to rst-order interaction in gt.Further, it is found that the normal squeezing in the harmonic mode directly depends upon the amplitude-cubed squeezing of the initial pump eld to the case of second-order interaction in gt.This gives a method of converting higher-order (amplitude-cubed) squeezing of the fundamental mode into normal squeezing of the harmonic mode and vice versa.
The statistical fluctuations of light can, under certain circumstances, exhibit a number of interesting nonclassical features. These include photon antibunching, sub-Poissonian statistics, and squeezing, all of which exist only in a quantum field. The three effects are distinct, and do not necessarily go together, although the first two are sometimes mistakenly treated as interchangeable. Both photon antibunching and sub-Poissonian statistics have been observed in experiments on resonance fluorescence from single atoms, which is described. Squeezing has not yet been seen experimentally, although many nonlinear processes are expected to give rise to it. The squeezed state can be recognized in heterodyne or homodyne experiments, where the squeezing becomes manifest as sub-Poissonian photon fluctuations.