Multi-photon Absorption Rates for N00N States William Plick, Christoph F. Wildfeuer, Jonathan P. Dowling: Hearne Institute for Theoretical Physics, LSU Multi-Photon Absorption Two photon absorption was first predicted by Maria Goppert-Mayer [1] in her doctoral Thesis in It was first observed by Werner Kaiser in 1961, shortly after the development of the laser. Rates for two photon absorption were calculated via the perturbative approach by B. R. Mollow [2] who found that the two photon absorption rate goes as: In the limit where the width (inverse decay time) of the final state is small compared to the bandwidth of the field (FWHM), where g is a tensor function that includes information about the line widths and other atomic factors. The term of interest in this equation is the second order correlation function G, which contains all necessary information about the field statistics. Note that in many circumstances these terms may be separated out from the integral, making field statistics the critical quantity when calculating absorption rates. G.S. Agarwal generalized this approach to the case of N-photon absorption [3]. Finding that for the field: Where λ is the absorption coefficient. Showing that the rate of change of the average number of photons in the field (in point of fact the absorption rate) depends upon the following ratio: Showing that the absorption continues to exhibit a strong dependence on the field statistics for this generalization. Another interesting aspect of two photon absorption is the relationship between the penetration depth of the light and its intensity. This is best seen by comparing Beer’s Law for single photon absorption from an Optical Parametric Amplifier (OPA) output to its two photon absorption. N00N States Resulting in much less absorption per amount of material traversed. The cause of this is the fact that the process is second order and thus dependent upon the square of the intensity. N00N States are maximally path entangled states of the form |N0>-|0N>. They have become objects of interest because it has been shown that they exhibit suppersensitvity and supperesolution, which may have application to lithography [4]. Thus it is useful to consider the absorption rates of these states. A simple approach is to compare the ratios given previously: Note that for N=2 absorption rates are as good as for coherent states. Also, absorption remains competitive with coherent states, even for somewhat higher N. Especially when compared to Fock states, which quickly drop off. However, ideal N00N states in an interferometer or cavity are not realistic for lithography. Realistic N00N state pulses have envelope functions and frequency distributions, for example the output of SPDC [5]: For this a more complex model for absorption must be constructed. Furthermore, since schemes for producing them depend on at least second order non- linearities, fluxes of N00N states are very low. So it would be helpful to optimize the absorption rates. There are several parameters which can be adjusted in the 2002 case (which is produced with a BBO crystal and a beam splitter): the length of the BBO source (which affects the frequency distribution through dispersion), whether to use type I or II down conversion and how to pulse the pump laser. The best prospect for modeling this process is to write an equation of motion for the density matrix of the combined field-atom system, which includes all of these parameters explicitly. The perturbative approach would not be able to do this in a straightforward fashion. Extension to N>2 would require details about the generation scheme. So far up to N=4 has been demonstrated. [1] Göppert-Mayer M (1931). "Über Elementarakte mit zwei Quantensprüngen". Ann Phys 9: 273–95. [2] B. R. Mollow (1968). “Two Photon Absorption and Field Correlation Functions”. Phys. Rev [3] G. S. Agarwal (1969). “Field-Correlation Effects in Multiphoton Absorption Processes”. Phys. Rev. A [4] A. N. Boto et. al. (2000). “Quantum Interferometric Optical Lithography”. PRL [5] Yoon-Ho Kim (2001). “Two-Photon Quantum Entanglement”. PhD Thesis, University of Maryland. relative