We continue the study, from a semiclassical viewpoint, of Calvo and Picon's operators, as manipulating photon states in quantum communication. In a previous paper, we defined a one-dimensional model operator and studied it analytically before moving on to Calvo and Picon's full two-dimensional operators. In the present paper, we show how the one-dimensional operator may also be useful as an experimental model, since it allows manipulations of two-dimensional Laguerre-Gaussian modes; the intensity distributions (in physical space) of the Laguerre-Gaussian modes then approximately flow along the elliptic curves studied earlier. Since the Wigner transform is fundamental in the study of Laguerre-Gaussian modes, we give a slightly expanded and improved treatment of the semiclassical Wigner transform, which was only briefly mentioned in the previous paper.
We continue the study, from a semiclassical viewpoint, of Calvo and Picon's operators, as manipulating photon states in quantum communication. In a previous paper, we defined a one-dimensional model operator and studied it analytically before moving on to Calvo and Picon's full two-dimensional operators. In the present paper, we show how the one-dimensional operator may also be useful as an experimental model, since it allows manipulations of two-dimensional Laguerre-Gaussian modes; the intensity distributions (in physical space) of the Laguerre-Gaussian modes then approximately flow along the elliptic curves studied earlier. Since the Wigner transform is fundamental in the study of Laguerre-Gaussian modes, we give a slightly expanded and improved treatment of the semiclassical Wigner transform, which was only briefly mentioned in the previous paper.
Calvo and Picón defined a class of operators, for use in quantum communication, that allows arbitrary manipulations of the three lowest two-dimensional Hermite–Gaussian modes HT={|0,0⟩,|1,0⟩,|0,1⟩}. Our paper continues the study of those operators, and our results fall into two categories. For one, we show that the generators of the operators have infinite deficiency indices, and we explicitly describe all self-adjoint realizations. And second, we investigate semiclassical approximations of the propagators. The basic method is to start from a semiclassical Fourier integral operator ansatz and then construct approximate solutions of the corresponding evolution equations. In doing so, we give a complete description of the Hamilton flow, which in most cases is given by elliptic functions. We find that the semiclassical approximation behaves well when acting on sufficiently localized initial conditions, for example, finite sums of semiclassical Hermite–Gaussian modes, since near the origin the Hamilton trajectories trace out the bounded components of elliptic curves.
Abstract. Gabriel F. Calvo and Antonio Picón defined a class of operators, for use in quantum communication, that allows arbitrary manipulations of the three lowest two-dimensional Hermite-Gaussian modes HT = {|0, 0〉, |1,0〉, |0, 1〉}. Our paper continues the study of those operators, and our results fall into two categories. For one, we show that the generators of the operators have infinite deficiency indices, and we explicitly describe all selfadjoint extensions. And secondly we investigate semiclassical approximations of the propagators. The basic method is to start from a semiclassical Fourier Integral Operator ansatz and then construct approximate solutions of the corresponding evolution equations. In doing so, we give a complete description of the Hamilton flow, which in most cases is given by elliptic functions. We find that the semiclassical approximation behaves well when acting on sufficiently localized initial conditions, for example, finite sums of semiclassical Hermite-Gaussian modes, since near the origin the Hamilton trajectories are periodic, tracing out the bounded components of elliptic curves. 1.
Recent developments in laser physics have called renewed attention to Laguerre–Gaussian (LG) beams of paraxial light. In this paper we consider the corresponding LG modes for the two-dimensional harmonic oscillator, which appear in the transversal plane at the laser beam's waist. We see how they arise as Wigner transforms of Hermite–Gaussian modes, and we proceed to find a closed form for their own Wigner transforms, providing an alternative to the methods of Simon and Agarwal. Our main observation is that the Wigner transform intertwines the creation and annihilation operators for the two classes of modes.
Quantum systems entanglement in continuous variables has been receiving increasing attention from the scientific community. The transverse spatial shape of photons (i.e. transverse position and momentum degree of freedom) is such a continuous variable systems, which due to its ease of control is prone to be used for testing some of the new physical predictions that continuous variables quantum systems can offer, as the improvement of the security of quantum key distribution cryptography and the efficiency enhancement of quantum communication protocols. Following this line, the mean goal of this Thesis has been the characterization and measurement of the spatial shape of photons. Another outstanding problem in Quantum Optics Information Technology (QOIT) is the coherent and reversible storage of photons states in matter, i.e. the necessity of a quantum memory. Theoretical and experimental results about this topic have been showed.
Below we summarize the more important contributions of this Thesis:
Demonstration about the non-validity of the OAM transformation rule FORr highly focused light beams: We have shown from a theoretical and experimental point of view that the OAM transformation rule is not valid for highly focused light beams, since Dove prisms inherently introduce astigmatism, and therefore further OAM changes. We have analyzed how light beams with a well defined value of the OAM per photon, after traversing the Dove prism, are transformed into a superposition of states with well defined OAM. The violation of the rule, turns out to be more important for highly focused light beams. We have provided a quantitative study of the properties of the Dove prism, and we have verified experimentally the validity of the theoretical results in a series of experiments with a commercially available Dove prism.
Presentation of a new method to measure the amplitude and phase of Laguerre-Gaussian like beams: A method to measure the amplitude an phase of Laguerre-Gaussian like beams have been presented, from a theoretical and experimental point of view. This method is based on a small variation of the spiral phase interferometry technique, which allows to avoid some technical problems that can be found in the reconstruction of finite sized beams and beams with phase singularities. A few examples of the use of our technique for the characterization of complex beams have been presented.
Description of the transverse spatial shape of bipartite entangled photons generated in Spontaneous Parametric Down Conversion (SPDC) using the Wigner function formulism: We have show some theoretical and experimental results regarding the spatial Wigner function of entangled paired photons generated in SPDC. Concretely, we have studied the remote preparation of one of the photons of the pair, for the photon being in a Gaussian beam and in a superposition of two modes. Also, we have analyzed and measured the case of the Wigner function of only one of the two photons of the pair, which shows that one photon of the pair is in a statistical mixture of modes. Finally, an experiment to completely measure the spatial state of the two photons was proposed, and also analytical expressions for the results one would expect in this case were given.
Investigation about the storage of spatial light polarization grating into the Zeeman ground states of cold cesium atoms: We have investigated theoretically and experimentally the dynamics of light grating stored in electromagnetically induced transparent (EIT) medium associated with a degenerate two-level system. The dependence of the stored light grating with the intensities of the incident and reading beams has been investigated. Bragg diffraction into the stored grating has been employed to probe the dynamics under different conditions showing reasonable results in agreement with the theoretical model developed. Moreover, we also have observed the coherent evolution of the stored grating in the presence of an applied magnetic field, which allows collapses and revivals of the stored grating, which is associated with the Larmor precession of the induced grating around the applied magnetic field.
We think that the results of this Thesis are an important step in order to explore the quantum capabilities of the transverse spatial shape of photons and can encourage going further in this line of research. Moreover, the investigation on the dynamics of light grating stored in an EIT medium is of considerable importance for a better understanding of the coherent memory for multidimensional state spaces. Thus, a further and more detailed investigation of the topics presented here should be of special interest for the scientific community.
We extend shapelets for the analysis of astronomical images to be available in a phase space, introducing \textit{Wigner function shapelets}(WFSs). Whereas conventional shapelets expand images separately in configuration or Fourier space using Hermite-Gaussian or Laguerre-Gaussian modes, WFSs represent images directly in the four-dimensional phase space with symplectic group Sp(4,R), which is quantized by a phase-space cell $2π\lambdabar$ that determines a resolution limit of a telescope. WFSs consist of a bilinear form of the cross-Wigner function of the Laguerre-Gaussian modes as an orthogonal and complete basis for the Wigner function of an image, carrying out SU(2) irreducible representations of the phase space with the Hopf tori. We introduce a scalar function Wkℓ(Q0,Q2) from the U(1)×U(1) - covariant tori to a two-dimensional space of constants of motion (Q0,Q2) -- the harmonic energy and axial angular momentum -- thereby yielding a natural phase-space ``band structure'', given a pair of winding number (k,ℓ)∈Z2. % The WFSs leverage key properties of the Wigner function for image analysis: (i) it encodes full information of an image in a symmetry-preserving way; (ii) its trasport equation naturally involves with a Liouville equation at $\lambdabar \rightarrow 0$; (iii) it admits positive/negative oscillatory patterns on (Q0,Q2) plane that can be sensitive spatial coherent structure of galaxy morphology and cosmological imprints; and (iv) systematics and noise can be manipulated as a quantum channel operation. This paper aims to bring together all the formulae related to the Wigner function in the context of astrophysics and cosmology, formally organizing them in terminologies of both astronomy and of quantum information theory.