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Transverse Correlation in Entangled Photons and Light-Matter Interaction

Jianming Wen

Published Jan 1, 2007• 0 citations
Publisher

Abstract

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.

DOI
10.13016/m2gd51

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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.
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.
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.
Spontaneous parametric down-conversion (SPDC) is one of the most important sources of entangled photons in quantum optics and underpins a wide range of applications in quantum communication, quantum imaging, quantum sensing, and photonic quantum information processing. In this nonlinear optical process, a pump photon is converted into a pair of lower-energy photons that exhibit strong correlations in multiple degrees of freedom. While the generation of photon pairs is governed by energy and momentum conservation, the resulting biphoton state possesses a rich spatial structure whose properties are determined by the interplay among the pump field, the nonlinear medium, and subsequent propagation. This thesis investigates the spatial properties of entangled photon pairs generated through SPDC, focusing on the evolution of their correlations in the near field and on the behavior of orbital-angular-momentum (OAM) entangled states under tight focusing conditions. A common theme throughout this work is the examination of physical effects that emerge when the full biphoton wavefunction is retained rather than reduced through the simplifying approximations often employed in theoretical treatments. The principal contribution of this thesis concerns the near-field propagation of SPDC photon pairs. The spatial correlations of SPDC-generated photons are commonly analyzed within the thin-crystal approximation, where the biphoton wavefunction is assumed to factorize into independent functions of the sum and difference coordinates. We demonstrate that birefringence-induced transverse walk-off breaks this factorization and introduces an intrinsic coupling between these coordinates. More importantly, this coupling survives free-space propagation and remains observable even for crystals that nominally satisfy the thin-crystal condition. As a result, the joint spatial correlations develop a characteristic tapering near the crystal image plane that cannot be captured by conventional factorized models. Numerical simulations and experimental measurements are found to be in excellent agreement, revealing a previously overlooked feature of SPDC and providing a more complete description of photon-pair generation and propagation in birefringent nonlinear media. The second part of the thesis explores the propagation of OAM-entangled photon pairs beyond the paraxial regime. While OAM correlations are typically studied under weak-focusing conditions, many emerging quantum technologies require structured quantum states to be manipulated in highly confined optical systems. We show that tight focusing modifies the OAM correlation spectrum, leading to a broadening of the accessible mode distribution. This broadening exhibits distinct transverse and longitudinal characteristics and arises from the emergence of nonparaxial field components absent in conventional scalar descriptions. These results provide new insight into the behavior of structured quantum light in strongly focused optical fields and establish a framework for understanding OAM entanglement beyond the paraxial approximation. Together, the results presented in this thesis reveal previously unexplored aspects of the structure and propagation of entangled photon pairs. By moving beyond widely used approximations, this work uncovers physical effects that influence the generation, evolution, and measurement of spatially entangled states, with implications for quantum imaging, high-dimensional quantum communication, and the broader development of structured-light-based quantum technologies.
High-dimensional entanglement in the form of transverse spatial correlation between a pair of photons generated via spontaneous parametric downconversion is not only a valuable resource in many academic and real-life applications but also provides access to several intriguing quantum phenomena. One such non-intuitive phenomenon is phase entanglement, in which the biphoton state is correlated in the complex phase of its wavefunction. This state, which emerges during the propagation of the biphoton wavefunction, exhibits neither position nor momentum correlation, yet retains full entanglement. In this work, we experimentally explore this state in two distinct ways. The first is by tracking the vanishing spatial photon number correlation over propagation distances lying in [0,∞)\left[0,\infty\right)[0,∞), folded into a finite range using single-lens imaging. These observations show excellent agreement with our theoretical predictions based on the Double Gaussian (DG) approximation of the biphoton state. The second approach involves performing a two-photon interference experiment using a double slit and this state, which reveals the correlated phase front. We show, both theoretically and experimentally, that the observed two-photon interference structure is markedly different from that produced by position-correlated photons, as confirmed by computing the joint probability distribution of photons (JPD) and related metrics. Such interference using phase-entangled light has not been attempted before and opens avenues for advanced experiments and applications in the field of spatial entanglement.
Efficient Generation of Entangled Photon-Pairs The first experiments with correlated photons have been performed in the context of EPR-Bell experiments on the realistic and local properties of quantum mechanics. The source used there produced pairs of polarization entangled photons from a 2-photon decay of Calcium atoms. The technical requirements of these experiments were high (vacuum systems, stronge dye-lasers, etc.) whereas the efficiency of the source was quite low. An important step forward was the introduction of spontaneous parametric down conversion (SPDC), which has become the most common source in quantum optics for generating correlated or entangled photon pairs. In this process photons of an intense pump laser convert to photon pairs in an optical nonlinear crystal. Conservation of energy and momentum leads to strong correlations between the generated photons. With this kind of two-photon source it was possible to realize or improve many experiments on the foundations of quantum mechanics addressing the EPR-Paradoxon and in the new field of quantum information. But again, more advanced experiments and applications suffer from the limited ef- ficiency of the fluorescence process. Many photon pairs are lost by spatial and spectral filtering, which is necessary to achieve polarization entanglement and long coherence times. Different techniques have been implemented to increase the number of photon pairs using two-crystal arrangements, focusing techniques or periodically poled crystals. Most of these methods have the disadvantage that no entangled photons have been observed. It is the aim of this work to increase the yield and to improve the mode definition of entangled photon pairs generated by resonant enhancement of the pump mode and the fluorescence modes. As a first step a linear cavity for the pump mode was realized. Since the conversion probability is proportional to the pump power it was possible to increase the photon pair count rate by factor of 7 over the previous source. Besides the possibility of further improvement on already established pair correlation experiments, such an enhancement allows to build a compact source for photon pairs, in which an expensive argon-ion laser is replaced by a cheap diode laser. Among other applications such sources are of strong interest for quantum cryptography. 3 In many quantum information experiments optical fibers are use to carry the photons over long distance. Therefore, light from the parametric down-conversion source has to be efficiently coupled into fibers. In the second part we report on a new method to optimize collection efficiency by matching the angular distribution of the parametric fluorescence to the spatial mode of an optical fiber. By using this technique, we detected 366500 polarization-entangled photon pairs per second in the near-infrared region in single-mode optical fibers for 465 mW pump power (at 351.1 nm) with a 2 mm BBOcrystal. The entanglement of the photon pairs was verified by measuring polarization correlations of more than 96% in a HV-basis and in a ±45◦-basis. To our knowledge, such enormous count rates of highly entangled photon pairs have not been reached yet with any other technique. In the third part of this thesis we investigated the process of parametric downconversion in a cavity which is resonant to certain longitudinal down-conversion modes only. The idea of placing the parametric down-conversion source inside a cavity is not new. Such a device is usually referred to as a single or double resonant optical parametric oscillator (OPO) and is mainly used to generate squeezed quantum states. In that kind of application the system is operating close to but still under the threshold of oscillation. In our application the situation is quite different. The system is operating far below threshold so that mainly spontaneous emission occurs. In that mode correlations between single photons can still be observed. But bouncing the light back and forth inside the cavity increases the interaction length and hence enhances the signal levels of the down-conversion fields. Further, by resonating two certain modes only, the bandwidth is reduced by orders of magnitude and the coherence time is found to be inverse proportional to the bandwidth. A similar experiment has already been realized with a type-I parametric down-converter in the resonator. We have tried to realize a compact double resonant OPO far below threshold with a type-II parametric down-converter in a high-finesse cavity to realize a bright source of entangled photon pairs with extremely narrow bandwidth.
This dissertation contains new theoretical work on how to utilize the entangled photons produced in a parametric down conversion to improve resolution and sensitivity in quantum interferometry, magnetometry and lithography. Chapter 1 introduces some basics of quantum imaging and sensing systems and gives an outline of the thesis. Chapter 2 deals with the use of entangled photons in Sagnac interferometry. We introduce the Sagnac ring interferometer and analyze the Sagnac phase shift with classical and quantum inputs of light. We compare the results obtained from entangled photon pairs input with classical and single-photon inputs. We show how the photon-photon correlation measurements increase the resolution and the sensitivity in the phase shift. In chapter three, we present an analysis of how parametric down-converted photons could be utilized in getting better spectroscopic information about a medium which has magnetic field induced anisotropy. We demonstrate how the improvement in magneto-optical rotation (MOR) of light could be realized by employing two different schemes with collinear and noncollinear down-conversion geometries compared to the use of coherent light. We calculate the resolution that can be achieved in the MOR's both by the use of coherent light and down-converted light. We discuss the possibility that the Heisenberg limit could be reached in magnetometry by the use of down-converted light. In chapter 4, we propose a new idea of using coherent beam stimulated parametric processes along with spontaneous ones to produce resolution improvement in two-photon interferometry. The stimulated processes enhance the count rate by several orders of magnitude while at the same time maintaining high visibility at large gains of the parametric process. We further find that the phases of coherent fields can also be used as tuning knobs to control the visibility of the pattern. We use coherent beams at the signal and idler frequencies as the seed light for the down-conversion process. In chapter 5, we analyze the minimum phase uncertainty of the phase measurement in a Mach-Zehnder interferometer using stimulated parametric down-conversion. We calculate the phase sensitivities for different measurement schemes. We conclude the thesis in chapter 6 and give an outlook for future work.