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

Star states in multi-photon processes

Z. Kis, József Janszky, Петер Адам

Published Aug 19, 1998• 0 citations
Publisher

Abstract

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.

DOI
10.1117/12.320979

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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 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.
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.
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.
An investigation is carried out of higher-order squeezing in a class of three- and four-wave mixing processes that can be modeled by a master equation. In (2N)th-order squeezing, as defined by Hong and Mandel, the Nth-power of the variance of a field quadrature falls below the value it would have for a coherent state. Higher-order squeezing is of interest because (i) it occurs in many processes in which ordinary (or second-order) squeezing occurs, and (ii) it allows a much larger fractional noise reduction to be achieved than ordinary squeezing. The master equation we use models a degenerate three- or four-wave process and takes into account the losses in the signal mode. We convert the master equation into a Fokker-Planck equation for the Wigner distribution function and solve it for a wide range of initial signal states (including thermal, coherent, Fock states, and mixtures of these). Results are presented for the enhancement of fourth- and sixth-order squeezing over second-order squeezing. Differences in the character of higher-order squeezing for different input states are pointed out. The role of losses in limiting the amount of achievable squeezing in all orders is especially investigated.
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.
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 present a quantum nonlinear model of two-wave mixing in a lossless photorefractive medium. A set of equations describing the quantum nonlinear coupling for the field operators is obtained. It is found that, to the second power term, the commutation relationship is maintained. The expectation values for the photon number concur with those of the classical electromagnetic theory when the initial intensities of the two beams are strong. We also calculate the quantum fluctuations of the two beams initially in the coherent state. With an appropriate choice of phase, quadrature squeezing or number state squeezing can be produced.