The development of efficient protocols for pure and mixed states preparation is challenging task. Most of the theory of quantum information applications has been developed for fully coherent or completely incoherent light. However, in many situations of interest partially coherent light has been proved to be a more robust model of radiation. In this paper the underpinning theory of two photon amplitude functions for down-converted fields with partially coherent pump beams is investigated. By using the generalised Siegert relations and the coherent mode representation of the cross spectral density matrix the two photon amplitude is fully characterised for partially coherent beams. A number of correlation properties from modern coherence theory are demonstrated to be preserved under parametric down-conversion. Based on the generalised Siegert relations and the Cauchy-Schwarz inequality a measure of entanglement for the two photon amplitude is proposed. Upper bounds for this measure are found in terms of the \emph{golden ratio} for maximally entangled states. Two inequalities are derived for the two photon amplitude for which there exist a transition zone from super-Poisson statistics to sub-Poisson statistics for down-converted partially coherent fields.
The development of efficient protocols for pure and mixed states preparation is challenging task. Most of the theory of quantum information applications has been developed for fully coherent or completely incoherent light. However, in many situations of interest partially coherent light has been proved to be a more robust model of radiation. In this paper the underpinning theory of two photon amplitude functions for down-converted fields with partially coherent pump beams is investigated. By using the generalised Siegert relations and the coherent mode representation of the cross spectral density matrix the two photon amplitude is fully characterised for partially coherent beams. A number of correlation properties from modern coherence theory are demonstrated to be preserved under parametric down-conversion. Based on the generalised Siegert relations and the Cauchy-Schwarz inequality a measure of entanglement for the two photon amplitude is proposed. Upper bounds for this measure are found in terms of the \emph{golden ratio} for maximally entangled states. Two inequalities are derived for the two photon amplitude for which there exist a transition zone from super-Poisson statistics to sub-Poisson statistics for down-converted partially coherent fields.
We show that in parametric down-conversion the coherence properties of a temporally partially coherent pump field get entirely transferred to the down-converted entangled two-photon field. Under the assumption that the frequency bandwidth of the down-converted signal-idler photons is much larger than that of the pump, we derive the temporal coherence functions for the down-converted field, for both infinitely fast and time-averaged detection schemes. We show that in each scheme the coherence function factorizes into two separate coherence functions with one of them carrying the entire statistical information of the pump field. In situations in which the pump is a Gaussian Schell-model field, we derive explicit expressions for the coherence functions. Finally, we show that the concurrence of time-energy-entangled two-qubit states is bounded by the degree of temporal coherence of the pump field. This study can have important implications for understanding how correlations of the pump field manifest as two-particle entanglement as well as for harnessing energy-time entanglement for long-distance quantum communication protocols.
We study the spatial coherence properties of the entangled two-photon field produced by parametric down-conversion (PDC) when the pump field is, spatially, a partially coherent beam. By explicitly treating the case of a pump beam of the Gaussian Schell-model type, we show that in PDC the spatial coherence properties of the pump field get entirely transferred to the spatial coherence properties of the down-converted two-photon field. As one important consequence of this study, we find that, for two-qubit states based on the position correlations of the two-photon field, the maximum achievable entanglement, as quantified by concurrence, is bounded by the degree of spatial coherence of the pump field. These results could be important by providing a means of controlling the entanglement of down-converted photons by tailoring the degree of coherence of the pump field.
Partially coherent optical beams present phenomena that may not appear in perfectly coherent beams. In this regard, spontaneous parametric downconversion is an intriguing physical process, since the optical properties of the pump laser beam are transferred to the quantum properties of the down-converted photon pairs. Here, the authors study the transfer of twist phase, a novel property of the so-called twisted Gaussian Schell model beams, from a pump beam to down-converted photons. The authors show that a non-zero twist phase can lead to highly entangled yet highly mixed states. The correlations that appear in near/far-field intensity measurements, in the two-photon covariance matrix as well as in orbital angular momentum modes, are explored. The authors derive expressions for the purity of the two-photon state and relate it to the transverse spatial coherence of the pump beam. Their main results utilize a Wigner function formalism that can be applied to a number of scenarios involving partially coherent downconversion sources and contribute to the recent interest in spatially correlated photon pairs produced in mixed states, with possible applications in quantum communication and imaging.
Entangled photons produced by spontaneous parametric down-conversion have been of paramount importance for our current understanding of quantum mechanics and advances in quantum information. In this process, the quantum correlations of the down-converted photons are governed by the optical properties of the pump beam illuminating the nonlinear crystal. Extensively, the pump beam has been modeled by either coherent beams or by the well-known Gaussian-Schell model, which leads to the natural conclusion that a high degree of optical coherence is required for the generation of highly entangled states. Here, we show that when a novel class of partially coherent Gaussian pump beams is considered, a distinct type of quantum state can be generated for which the amount of entanglement increases inversely with the degree of coherence of the pump beam. This leads to highly incoherent yet highly entangled multiphoton states, which should have interesting consequences for photonic quantum information science.
We study partially coherent fields that have a coherent-mode representation in the orbital-angular-momentum-mode basis. For such fields, we introduce the concepts of the angular coherence function and the coherence angle. Such fields are naturally produced by the process of parametric down-conversion---a second-order nonlinear optical process in which a pump photon breaks up into two entangled photons, known as the signal and idler photons. We show that the angular coherence functions of the signal and idler fields are directly related to the angular Schmidt (spiral) spectrum of the down-converted two-photon field and thus that the angular Schmidt spectrum can be measured directly by measuring the angular coherence function of either the signal or the idler field, without requiring coincidence detection.
Entangled-photon pairs (biphotons) generated by spontaneous optical parametric down-conversion exhibit a number of properties that are analogous to those of ordinary photons generated by incoherent sources. The spatial pump-field distribution and the two-particle wave function in the biphoton case play the respective roles of the source intensity distribution and the second-order coherence function in the incoherent case. The van Cittert--Zernike theorem, which is applicable for incoherent optical sources emitting independent photons, has a counterpart for biphotons. Likewise, the partial-coherence theory of image formation has an analogous counterpart for biphoton beams transporting spatial information. However, an underlying duality, rather than analogy, accompanies the mathematical similarity between incoherent and biphoton emissions if the comparison is made between the photon count rate in the incoherent case and the biphoton count rate in the entangled-photon case. The smaller the size of an incoherent source, the more separable is the coherence function and the more coherent is the field, and therefore the higher the visibility of ordinary interference fringes. In contrast, the narrower the size of a biphoton pump source, the more separable is the wave function and the less entangled is the field, and therefore the lower the visibility of biphoton interference fringes. This duality is similar to the complementarity between single and two-photon interference exhibited for biphotons.