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Fortschritte der Physik

Star States

József Janszky, Z. Kis, Петер Адам

Published Nov 1, 1998• 1 citation
Publisher

Abstract

It is shown that the classical and quantum mechanical description of first and second order nonlinear optical processes (including squeezing) are equivalent to the effect that in both cases an initial phase space distribution is transformed in the same manner. Three-photon states – star states – are generated in third order nonlinear processes, where the classical and quantum descriptions give essentially different results. It is pointed out that the efficiency of the star state in a three photon process is superior to that of a coherent state with two order of magnitude.

DOI
10.1002/(sici)1521-3978(199811)46:6/8<829::aid-prop829>3.0.co;2-m

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It is shown that the classical and quantum mechanical description of first and second order nonlinear optical processes (including squeezing) are equivalent to the effect that in both cases an initial phase space distribution is transformed in the same manner. Three-photon states – star states – are generated in third order nonlinear processes, where the classical and quantum descriptions give essentially different results. It is pointed out that the efficiency of the star state in a three photon process is superior to that of a coherent state with two order of magnitude.
It is shown that the classical and quantum mechanical description of first and second order nonlinear optical processes (including squeezing) are equivalent in the sense that in both cases an initial phase space distribution transforms in the same manner. We investigate in detail the degenerate three wave mixing, which is a third order process, and point out that the classical and quantum descriptions give essentially different results. This process yields three-pointed star states. It is pointed out that the efficiency of the star state in a three photon process is superior to that of a coherent state with two orders of magnitude.
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 introduce nonlinear canonical transformations that yield effective Hamiltonians of multiphoton down conversion processes, and we define the associated non-Gaussian multiphoton squeezed states as the coherent states of the multiphoton Hamiltonians. We study in detail the four-photon processes and the associated non-Gaussian four-photon squeezed states. The realization of squeezing, the behavior of the field statistics, and the structure of the phase space distributions show that these states realize a natural four-photon generalization of the two-photon squeezed states.
We introduce nonlinear canonical transformations that yield effective, solvable Hamiltonians of multiphoton down-conversion processes, and we define the associated non-Gaussian multiphoton squeezed states as the coherent states of the multiphoton Hamiltonians. We study in detail the four-photon processes and the associated non-Gaussian four-photon squeezed states. The realization of squeezing, the behaviour of the field statistics, and the structure of the phase space distributions show that these states realize a natural four-photon generalization of the two-photon squeezed states.
This thesis presents a study of photon generation and conversion processes in nonlinear optics. Our results extend beyond the first order perturbative regime, in which only pairs of photons are generated in spontaneous parametric down-conversion or four wave mixing. They also allow us to identify the limiting factors for achieving unit efficiency frequency conversion, and to correctly account for the propagation of bright nonclassical photon states generated in the nonlinear material. This is done using the Magnus expansion, which allows us to incorporate corrections due to time-ordering effects in the time evolution of the state, while also respecting the quantum statistics associated with squeezed states (in photon generation) and single photons or coherent states (in photon conversion). We show more generally that this expansion should be the preferred strategy when dealing with any type of time-dependent Hamiltonian that is a quadratic form in the creation and annihilation operators of the fields involved. Using the Magnus expansion, simple figures of merit to estimate the relevance of these time-ordering corrections are obtained. These quantities depend on the group velocities of the modes involved in the nonlinear process and provide a very simple physical picture of the interactions between the photons at different times. These time-ordering effects are shown to be important only when the photons and pump beam overlap within the nonlinear material for a significant amount of time. We also discuss the possibility of preparing Fock states with many photons using parametric down-conversion sources and realistic photon-number-resolving detectors. After discussing photon generation processes, we study photon conversion processes in which a nonlinear material and a classical strong pump field are used modify the frequency of a second incoming low-intensity field. We show that time-ordering effects limit the efficiency of this conversion process, re-examine strategies to get around this problem, and show that it should be possible to have a clear observation of modified upconversion probabilities due to these effects, and thus have an experimental demonstration of time-ordering effects in nonlinear optics.
In this paper the amplitude-squared squeezing in the multiphoton nonlinear optical processes is studied. It is examined that amplitude-squared squeezing can be created in the first order approximation by k-photon unitary operators Sk(z) (K 2) acting on the coherent state |α(|α|≠0) and one can not be created by Sk(z) (K 2, K≠4) acting on the vacuum state. It is found that amplitude-squared squeezing can be created by 4-photon unitary operator S4(z) acting on the vacuum state and this shows that the amplitude-squared squeezing is a new nonclassioal effect of light field which is independent of the second order squeezing, antibunohing and sub-Poisson statistics.