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Solid State Communications

Four-wave mixing theory for two-photon generation of excitons in cuprous oxide

Igor V. Beloussov, John B. Ketterson, Yinghui Sun

Published Dec 22, 2004• 2 citations
Publisher

Abstract

The experimental situation in which excitons in Cu 2 O are pumped directly into a state with wave vector k → = 0 → by two counter-propagating laser pulses , and subsequently probed by a third, time-delayed, pulse has been theoretically investigated. It has been shown that additional electromagnetic waves are excited in the system. A specific configuration of two pump pulses and one probe pulse has been found to generate only one additional electromagnetic wave. The dependence of the time-integrated photon number in this wave on the delay time between pump and probe pulses for some values of the detuning has been investigated.

DOI
10.1016/j.ssc.2004.11.045

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•Jul 18, 1988•0 citations
The experimental situation in which excitons in Cu 2 O are pumped directly into a state with wave vector k → = 0 → by two counter-propagating laser pulses , and subsequently probed by a third, time-delayed, pulse has been theoretically investigated. It has been shown that additional electromagnetic waves are excited in the system. A specific configuration of two pump pulses and one probe pulse has been found to generate only one additional electromagnetic wave. The dependence of the time-integrated photon number in this wave on the delay time between pump and probe pulses for some values of the detuning has been investigated.
The experimental situation in which excitons in Cu2O are pumped directly into a state with wave vector = k0 by two counter-propagating laser pulses, and subsequently probed by a third, time-delayed, pulse has been theoretically investigated. It has been shown that additional electromagnetic waves are excited in the system. A specific configuration of two pump pulses and one probe pulse has been found to generate only one additional electromagnetic wave. The dependence of the time-integrated photon number in this wave on the delay time between pump and probe pulses for some values of the detuning has been investigated.
We propose and examine theoretically a four-wave-mixing experiment in which excitons in Cu2O{\text{Cu}}_{2}\text{O}Cu2​O are pumped directly into a state with wave vector k=0\mathbf{k}=\mathbf{0}k=0 by two counter-propagating cross-polarized laser pulses and which is subsequently probed by a third time-delayed pulse. Approximate analytical solutions of the time evolution equations are found which describe the time dependence of the exciton and photon densities. Most importantly, we obtain the dependence of the resulting phase-conjugated signal versus the delay time td{t}_{d}td​ between the probe and pump pulses. It is shown that the total number of photons generated in the course of the experiment is proportional to the areas of all three incoming pulses. The dependence of the resulting time-integrated signal on the delay time td{t}_{d}td​ is the same as the exciton density dependence on the time ttt. Hence, one may directly study the time evolution of the exciton condensate. The approximate analytical results are compared with a numerical solution of the evolution equations for a wide range of incident pulse intensities. It is noted that an exciton condensate arising by an alternate route can also be probed through a phase-conjugated two-photon process.
We propose and theoretically investigate a two-photon four wave mixing experiment to probe for BEC of excitons in Cu2O thin films. A relatively simple set of equations describing the dynamics of the system is obtained for a particular configuration of the exciting beams, and numerical and approximate analytical solutions are found. The resulting phase-conjugated signal vs. the delay time between the pump and probe pulses yields a direct measure of the time evolution of the exciton condensate, and exhibit peaks when the film thickness is a multiple of a half wavelength of light.
We propose and theoretically investigate a two-photon four-wave mixing experiment to probe for Bose-Einstein condensation of excitons in Cu2O{\text{Cu}}_{2}\text{O}Cu2​O thin films. A relatively simple set of equations describing the dynamics of the system is obtained for a particular configuration of the three incident beams, and numerical and approximate analytical solutions are found. When one takes into account the boundary conditions, which result in reflections from the interfaces, the characteristics of the resulting phase-conjugated signal will exhibit Fabry-Perot-type oscillations. For film thicknesses equal to a multiple of a half wavelength in the film the resulting signal is enhanced by more than an order of magnitude relative thicknesses that are an odd multiple of a quarter wavelength. In this case the contribution from accompanying excitonic condensates with wave vectors $\ifmmode\pm\else\textpm\fi{}2\mathbf{k}$ is minimal. Therefore, the resulting phase-conjugated signal vs the delay time between the pump and probe pulses yields a direct measure of the time evolution only of the exciton condensate with wave vector k=0\mathbf{k}=\mathbf{0}k=0.
We report experimental and theoretical investigations of interacting excitons of the yellow series in cuprous oxide (Cu2_22​O) with principal quantum numbers up to by means of second harmonic generation (SHG). Using picosecond pulsed laser excitation up to 10 GW/cm2^22 peak intensity we observe a pronounced change of the spectra with increasing pump laser intensity: an energetic shift to lower absolute energies and a spectral broadening. The absolute intensities of the spectral lines scale for low powers with the square of the pump power, but saturates at higher powers. At still higher powers the SHG intensity is actually reduced. To explain these results quantitively, we developed a semi-classical theory of resonant SHG where the process of SHG is fully coherent. The excitons are assumed to be bosons interacting by a distance dependent potential giving rise to both the changes in spectral line shape and the saturation by a Rydberg\it{Rydberg}Rydberg blockade. The concomitant measurement of two-photon absorption allows to derive quantitative values for the exciton-exciton interaction. While the results agree in order of magnitude with those calculated by state-of-the art atomic-like van der Waals interaction theory, the scaling with principle quantum number is quite different. As a possible screening by an electron-hole plasma created by three-photon absorption into blue and violet band states could be ruled out, our results point toward fundamental differences between excitons and atoms.
The three-pulse stimulated photon echo has been considered as one of the most fundamental transient four wave mixing processes1, where each of the three excitation pulses induces a single elementary interaction process between a laser field and an inhomogeneously broadened two-level system., resulting in a burst of the fourth wave, i.e., echo. Now if four or more optical pulses are applied to a medium, a number of photon echoes should be expected, in general. At low excitation intensities where the perturbation theory is applicable, however, the dominant nonlinear process is still the three-pulse stimulated echo and consequently each echo signal is generated only by three of the preceding pulses. It is rather a remarkable result that the echo intensity is unaffected by the presence of all the rest of the pulses other than its three generators. With this in mind, it is theoretically and experimentally shown that by applying a single optical pulse (#1) propagating along k1 followed by N pairs of counter-propagating pulses (#2j (k2j) and #3j− (k3j = −k2j); j=1,…N) in various directions, it is possible to obtain N photon echoes along the direction -k1 opposite to the first pulse #1. We will call this degenerate photon echoes, degenerate in. the sense that the echoes are spatially overlapping and temporally simultaneous. When N=l, this scheme is reduced to the ordinary backward photon echo2,3. The doubly degenerate case (N=2) is illustrated in Fig. 1. The j-th echo will be generated by the set of pulses (#1, #2j, #3j) and appear at t = t3j + t2j − t1 where t1, i2j and t3j are the arrival times of the pulses #1, #2j and #3j, respectively.