The propagation-dependent spatial entanglement for the structured laser beams generated by an arbitrary incident Hermite-Gaussian (HG) mode passing through an astigmatic mode converter (AMC) is theoretically explored. The structured output beams are analytically decomposed into the expansion of HG modes for any given rotation angle of the AMC. Based on the Schmidt decomposition, the propagation-dependent spatial entanglements of the structured output modes are quantified with the von Neumann entropy. To manifest the propagation-dependent entropy, the probability distribution of the expanded HG modes in the structured output beam is quantitatively analyzed.
The propagation-dependent spatial entanglement for the structured laser beams generated by an arbitrary incident Hermite-Gaussian (HG) mode passing through an astigmatic mode converter (AMC) is theoretically explored. The structured output beams are analytically decomposed into the expansion of HG modes for any given rotation angle of the AMC. Based on the Schmidt decomposition, the propagation-dependent spatial entanglements of the structured output modes are quantified with the von Neumann entropy. To manifest the propagation-dependent entropy, the probability distribution of the expanded HG modes in the structured output beam is quantitatively analyzed.
Quantum entanglement by a beam splitter (BS) is shown to be analogous to laser mode transformation by an astigmatic mode converter (AMC). Schmidt decomposition is used to characterize the entanglement by an AMC for generating orbital angular momentum and by a BS for creating quantum photon interference. The probability distributions of Schmidt decomposition are calculated to manifest the sameness and difference between AMC and BS in generating entanglement. Finally, the theoretical patterns of mode transformations by an AMC are confirmed with experimental results to validate the present analysis.
The Schmidt decomposition is exploited to study the spatial entanglement of laser transverse modes analogous to quantum Lissajous states. Based on the inverse Fourier transform, the stationary Lissajous state can be analytically derived as a coherent superposition of degenerate Hermite–Gaussian eigenmodes. With the derived stationary state, the Schmidt modes and the participation number N can be employed to evaluate the spatial localization and the quantum entanglement. The larger the participation number, the more localized is the stationary coherent state on the Lissajous figure. Moreover, the larger the participation number, the higher is the spatial entanglement.
While information is ubiquitously generated, shared, and analyzed in a modern-day life, there is still some controversy around the ways to assess the amount and quality of information inside a noisy optical channel. A number of theoretical approaches based on, e.g., conditional Shannon entropy and Fisher information have been developed, along with some experimental validations. Some of these approaches are limited to a certain alphabet, while others tend to fall short when considering optical beams with a nontrivial structure, such as Hermite-Gauss, Laguerre-Gauss, and other modes with a nontrivial structure. Here, we propose a new definition of the classical Shannon information via the Wigner distribution function, while respecting the Heisenberg inequality. Following this definition, we calculate the amount of information in Gaussian, Hermite-Gaussian, and Laguerre-Gaussian laser modes in juxtaposition and experimentally validate it by reconstruction of the Wigner distribution function from the intensity distribution of structured laser beams. We experimentally demonstrate the technique that allows to infer field structure of the laser beams in singular optics to assess the amount of contained information. Given the generality, this approach of defining information via analyzing the beam complexity is applicable to laser modes of any topology that can be described by well-behaved functions. Classical Shannon information, defined in this way, is detached from a particular alphabet, i.e., communication scheme, and scales with the structural complexity of the system. Such a synergy between the Wigner distribution function encompassing the information in both real and reciprocal space and information being a measure of disorder can contribute into future coherent detection algorithms and remote sensing.
High-dimensional spatial entanglement generated via spontaneous parametric down-conversion (SPDC) provides a powerful resource for quantum information processing, yet its mode structure becomes increasingly complex when the pump occupies a higher-order Laguerre--Gaussian (LG) mode. Here, we develop an algebraic framework for characterizing biphoton spatial-mode entanglement by factorizing LG-pumped SPDC into a spatial-mode beam-splitter transformation followed by spatial two-mode squeezing. By choosing the biphoton LG basis radius as the geometric mean of the pump-beam radius and the crystal-induced correlation width, we obtain a natural modal basis in which the two operations can be treated separately. When the pump-beam radius matches the correlation width, spatial squeezing vanishes, and the two circular-mode components of the pump are independently conserved across the signal and idler photons, leading to conservation of both orbital angular momentum and the total spatial mode number. We further derive an analytical expression for the Schmidt number for arbitrary LG pump modes and squeezing strengths, revealing how the radial and azimuthal pump indices govern the dimensionality of the spatial entanglement. This algebraic characterization provides a unified description of the mode structure, conservation laws, and entanglement dimensionality of higher-order-LG-pumped SPDC.
It is known that the entanglement present in the transverse spatial properties of photon pairs generated by spontaneous parametric down-conversion depends explicitly on the characteristics of the pump laser beam. Here we show that, for pump beams described by a Hermite-Gaussian mode, the entanglement depends on the beam width as well as the order of the beam. The amount of entanglement generally increases with the order of the beam and is less dependent on the beam width for higher orders. We propose a method to measure the spatial entanglement directly, independent of the dimension of the subsystems.
Due to the transfer of the angular spectrum of the pump beam to the two-photon state in spontaneous parametric downconversion, the generated twin photons are entangled in their spatial degrees of freedom. This spatial entanglement can be observed through correlation measurements in any set of modes in which one may choose to perform measurements. Choosing, e.g., a Hermite-Gaussian (HG) set of spatial modes as a basis, one can observe correlations present in their spatial degrees of freedom. In addition, these modes can be used as alphabets for quantum communication. For global quantum communication purposes, we derive an analytic expression for two-photon detection probability in terms of HG modes, taking into account the effects of the turbulent atmosphere. Our result is more general as it accounts for the propagation of both signal and idler photons through the atmosphere, as opposed to other works considering one photon's propagation in vacuum. We show that while the restrictions on both the parity and order of the downconverted HG fields no longer hold, due to the crosstalk between modes when propagating in the atmosphere, the crosstalk is not uniform: there are more robust modes that tend to keep the photons in them. These modes can be employed in order to increase the fidelity of quantum communication.