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Physical Review A

Transverse effects in paired-photon generation via an electromagnetically induced transparency medium. I. Perturbation theory

Jianming Wen, Morton H. Rubin

Published Aug 15, 2006• 40 citations
Publisher

Abstract

Generation of counterpropagating paired Stokes and anti-Stokes photons at the two-photon level in an electromagnetically induced transparency system is examined in quantum theory. In particular, we focus on the transverse effects of generated entangled photon pairs. Using first-order perturbation theory, we have studied the two-photon state including the phase matching conditions. The comparison of single photon counting and two-photon coincidence counting measurement is provided, which show some interesting properties.

DOI
10.1103/physreva.74.023808

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Transverse effects in paired-photon generation via an electromagnetically induced transparency medium. II. Beyond perturbation theory

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Generation of counterpropagating paired Stokes and anti-Stokes photons in a double-$\ensuremath{\Lambda}$- (or $M$-) type electromagnetically induced transparency (EIT) system is examined in quantum theory in the three-dimensional space. We use the Maxwell-Bloch equations to study the transverse effects of generated entangled photon pairs via such an EIT medium. We reexamine correlation measurements for both amplitude and intensity correlations. In the low parametric gain region $[\sqrt{{v}_{g}∕8{\ensuremath{\gamma}}_{31}L}\ensuremath{\gg}\sqrt{{k}_{s}∕{k}_{as}}(\ensuremath{\mid}{\ensuremath{\Omega}}_{p}\ensuremath{\mid}∕\ensuremath{\mid}{\ensuremath{\Gamma}}_{p}\ensuremath{\mid})]$, when the paired Stokes and anti-Stokes are at the single-photon level, we obtain the results shown in Wen and Rubin, Phys. Rev. A 74, 023807 (2006). If $\sqrt{{k}_{s}∕{k}_{as}}(\ensuremath{\mid}{\ensuremath{\Omega}}_{p}\ensuremath{\mid}∕\ensuremath{\mid}{\ensuremath{\Gamma}}_{p}\ensuremath{\mid})$ is comparable to or greater than $\sqrt{{v}_{g}∕8{\ensuremath{\gamma}}_{31}L}$, i.e., in the high parametric gain regime, the system may not reach the single-photon level, and accidental coincidences occur which cause loss of visibility in the interference experiments.

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Physical Review Letters•May 9, 2005•393 citations
Generation of counterpropagating paired Stokes and anti-Stokes photons at the two-photon level in an electromagnetically induced transparency system is examined in quantum theory. In particular, we focus on the transverse effects of generated entangled photon pairs. Using first-order perturbation theory, we have studied the two-photon state including the phase matching conditions. The comparison of single photon counting and two-photon coincidence counting measurement is provided, which show some interesting properties.
Generation of counterpropagating paired Stokes and anti-Stokes photons in a double-$\ensuremath{\Lambda}$- (or MMM-) type electromagnetically induced transparency (EIT) system is examined in quantum theory in the three-dimensional space. We use the Maxwell-Bloch equations to study the transverse effects of generated entangled photon pairs via such an EIT medium. We reexamine correlation measurements for both amplitude and intensity correlations. In the low parametric gain region $[\sqrt{{v}_{g}∕8{\ensuremath{\gamma}}_{31}L}\ensuremath{\gg}\sqrt{{k}_{s}∕{k}_{as}}(\ensuremath{\mid}{\ensuremath{\Omega}}_{p}\ensuremath{\mid}∕\ensuremath{\mid}{\ensuremath{\Gamma}}_{p}\ensuremath{\mid})]$, when the paired Stokes and anti-Stokes are at the single-photon level, we obtain the results shown in Wen and Rubin, Phys. Rev. A 74, 023807 (2006). If $\sqrt{{k}_{s}∕{k}_{as}}(\ensuremath{\mid}{\ensuremath{\Omega}}_{p}\ensuremath{\mid}∕\ensuremath{\mid}{\ensuremath{\Gamma}}_{p}\ensuremath{\mid})$ is comparable to or greater than $\sqrt{{v}_{g}∕8{\ensuremath{\gamma}}_{31}L}$, i.e., in the high parametric gain regime, the system may not reach the single-photon level, and accidental coincidences occur which cause loss of visibility in the interference experiments.
In recent years, quantum entanglement has attracted much; attention, because its unique properties provide potential; applications, which could not be achieved using conventional; techniques, such as quantum computing, quantum imaging and; lithography. To realize these advancements, one has to obtain an; entanglement-generation source, thoroughly master its physical; properties, and fully understand the light-matter interaction.; This dissertation is an attempt to address such issues as stated; above.; Conventionally, paired photons are created from; \textit{spontaneous parametric down-conversion} (SPDC). It is; known that the transverse correlation in biphotons may improve the; visibility and resolution in quantum imaging and lithography. In; this thesis, we described an alternative biphoton source --; Raman-EIT (\textit{electromagnetically induced transparency}); generator, and emphasize on its geometrical and optical; properties. We found that to utilize the transverse effects in; paired Stokes-anti-Stokes, it is necessary to make the product of; the EIT window times the group delay much greater than unity.; To gain further insight into quantum imaging and lithography, we; studied the transverse correlation in triphoton entanglement; theoretically. We found that in the two-image process, the quality; of images is determined by the optical path-lengths, even though; the Gaussian thin lens equations are satisfied. The ghost; interference-diffraction patterns of double slits show one more; fold interference, which is essentially different from the; biphoton case. Klyshko's advanced-wave model is still applicable,; with some modifications. We also generalized the transverse; correlation to the case of multi-photon entangled states.; To implement quantum computing, one key element is quantum memory.; In this thesis, we have theoretically explored the feasibility of; such a memory by using nonclassical SPDC light in an EIT system at; the single-photon level. We found that both the quantum coherence; of SPDC and atomic coherence of EIT can survive after interacting; within a vapor cell. Due to the inherent mismatch of magnitude; between the spectral bandwidth of SPDC and the very narrow; transmission width of EIT, the coincidence counts of the; two-photon interference is reduced to one pair per second, which; is barely doable in the current experimental situation.
We describe the theory of paired photon generation in a double-$\ensuremath{\Lambda}$ atomic system. Using the Heisenberg-Langevin formalism we evaluate and analyze the spectral generation rates and intensity correlation function for the output fields. Different regimes of parametric down conversion are analyzed. We discuss the influence of the optical depth and Langevin noise fluctuations on the paired photon generation and predict that at the optical depth of 100 or more the contribution of Langevin noise fluctuations is small and therefore Stokes and anti-Stokes photons are generated mostly in pairs. Comparison between the theory and experiments shows good agreement.
The transverse correlation of triphoton entanglement generated within a single crystal is analyzed. Among many interesting features of the transverse correlation, they arise from the spectral function FFF of the triphoton state produced in the parametric processes. One consequence of transverse effects of entangled states is quantum imaging, which is theoretically studied in photon counting measurements. Klyshko's two-photon advanced-wave picture is found to be applicable to the multiphoton entanglement with some modifications. We found that in the two-photon coincidence counting measurement by using triphoton entanglement, although the Gaussian thin lens equation (GTLE) holds, the imaging shown in coincidences is obscure and has a poor quality. This is because of tracing the remaining transverse modes in the untouched beam. In the triphoton imaging experiments, two kinds of cases have been examined. For the case that only one object with one thin lens is placed in the system, we found that the GTLE holds as expected in the triphoton coincidences and the effective distance between the lens and imaging plane is the parallel combination of two distances between the lens and two detectors weighted by wavelengths, which behaves as the parallel combination of resistors in the electromagnetism theory. Only in this case, a point-point correspondence for forming an image is well-accomplished. However, when two objects or two lenses are inserted in the system, though the GTLEs are well-satisfied, in general a point-point correspondence for imaging cannot be established. Under certain conditions, two blurred images may be observed in the coincidence counts. We have also studied the ghost interference-diffraction experiments by using double slits as apertures in triphoton entanglement. It was found that when two double slits are used in two optical beams, the interference-diffraction patterns show unusual features compared with the two-photon case. This unusual behavior is a destructive interference between two amplitudes for two photons crossing two double slits.