Optical-Terahertz Biphotons

The quantum correlated pairs of photons, usually named as biphotons, are the well known quantum optical objects, comprehensively studied and exploited in modern quantum information schemes [1]. Biphotons are easily generated as pure quantum states in the process of parametric down-conversion (PDC) of optical laser radiation in a quadratic nonlinear medium, and serve further as a basis for preparation of different types of entangled photons and squeezed vacuum states. During PDC each pump photon of frequency p  decays into a pair of photons: a signal

The quantum correlated pairs of photons, usually named as biphotons, are the well known quantum optical objects, comprehensively studied and exploited in modern quantum information schemes [1]. Biphotons are easily generated as pure quantum states in the process of parametric down-conversion (PDC) of optical laser radiation in a quadratic nonlinear medium, and serve further as a basis for preparation of different types of entangled photons and squeezed vacuum states. During PDC each pump photon of , so that the biphotons can be generated in a wide spectral range. Up to now PDC processes are studied mostly in the frequency-degenerate cases, when /2 , or slightly nondegenerate ones, when the difference between signal and idler frequencies is less than the spectral band of optical transmission of the nonlinear medium. We propose to consider properties of biphotons with extremely different frequencies, when the signal photon frequency is just close to p  , while the idler photon frequency hits the terahertz range. This regime can be referred as strongly frequency non-degenerate (SFND) PDC.
The SFND regime is interesting with new possibilities that appear in the generation of correlated photons from different spectral ranges, generation of localized long-wave excitations in matter that are bound to optical delocalized states, construction of quantum single-photon sources of terahertz frequencies, quantum calibration of the terahertz wave sources and detectors, and other applications. However, direct transfer of the results and methods, obtained up to now for the PDC in the optical range, to the mixed opticalterahertz case is a quite challenging task. The main problems are caused by the presence of classical thermal field fluctuations at THz frequencies, inherent absorption of the non-linear medium at THz frequencies, and high angular diversity of the generated THz idler photons. Results of the theoretical treatment of correlation functions and frequency-angular distributions of signal and idler photons emitted via spontaneous (low-gain) PDC are presented here accounting the peculiar properties of the SFND regime.
Calculations were made using the nonlinear Kirchhoff law, which expresses the second-order moments of the output PDC-generated fields in terms of the second-order moments of the input fields via elements of the scattering matrix of a nonlinear medi-um. According to its definition, the scattering matrix Û consists of coefficients that connect linearly firstorder moments of the both signal and idler input and output fields. In particular, ' ' Here and below ()  [2]. Applying this approach, one does not need to solve exact Heisenberg equations for field operators with the introduction of special noise operators responsible for the absorbing reservoir. It is enough to consider equations for spatially varying average values of the field operators. These equations have the same structure as the classical wave equations for slowly varying field amplitudes in an absorptive crystal. They were considered accounting the multi-mode character of the parametric interaction caused by a Gaussian transverse-limited pump beam. Explicit expressions for the multimode scattering matrix elements were obtained for the case of spontaneous PDC. By substituting the scattering matrix elements into the nonlinear Kirchhoff law the second-order moments of the output fields were obtained. The results show that occupations of output modes contributions. The first-type contribution (which is true spontaneous) appears due to inherent radiation of the pumped nonlinear medium, in each signal mode, and in each idler mode. The first terms in Eqs.
are detected at the output in the signal channel. The method of quantum calibration of spectral brightness of an external terahertz radiation is based on comparison between the spontaneous and induced signals [3].
In case of the pure SPDC registration scheme there are no special sources of the external input radiation. However, the equilibrium radiation of the medium environment should be taken into account. In most cases the temperatures of the medium and of the environment are the same, and a noticeable contribution of external thermal radiation persists only at idler frequencies. As a result, the detected output signal and idler photon numbers are Following the nonlinear Kirchhoff law, the second-order moment, which describes correlation of idler and signal fields, is calculated in this case as In experimental quantum optics the intensity correlation function '' , , ', ' : : In the case of properly cooled and transparent medium, low gain and low number of involved modes this ratio is sufficiently more than its classical value 1. However, when absorption, number of modes and the medium temperature are increased, the quantum excess above 1 can become vanishingly small. Final Eq. (9) shows the influence of each of these parameters and can be used as a guide in constructing PDC schemes for generating optical-terahertz biphotons.