Abstract
Frequency conversion (FC) and type-II parametric down-conversion (PDC) processes serve as basic building blocks for the implementation of quantum optical experiments: type-II PDC enables the efficient creation of quantum states such as photon-number states and Einstein–Podolsky–Rosen (EPR)-states. FC gives rise to technologies enabling efficient atom–photon coupling, ultrafast pulse gates and enhanced detection schemes. However, despite their widespread deployment, their theoretical treatment remains challenging. Especially the multi-photon components in the high-gain regime as well as the explicit time-dependence of the involved Hamiltonians hamper an efficient theoretical description of these nonlinear optical processes. In this paper, we investigate these effects and put forward two models that enable a full description of FC and type-II PDC in the high-gain regime. We present a rigorous numerical model relying on the solution of coupled integro-differential equations that covers the complete dynamics of the process. As an alternative, we develop a simplified model that, at the expense of neglecting time-ordering effects, enables an analytical solution. While the simplified model approximates the correct solution with high fidelity in a broad parameter range, sufficient for many experimental situations, such as FC with low efficiency, entangled photon-pair generation and the heralding of single photons from type-II PDC, our investigations reveal that the rigorous model predicts a decreased performance for FC processes in quantum pulse gate applications and an enhanced EPR-state generation rate during type-II PDC, when EPR squeezing values above 12 dB are considered.
Highlights
The crucial issue in these derivations is firstly the fact that multi-photon effects have to be considered during the interaction, and secondly the problem that the involved electric field operators and Hamiltonians do not commute in time. We address these issues and build two theoretical models for Frequency conversion (FC) and type-II parametric down-conversion (PDC): a rigorous numerical model extending the theoretical framework of Kolobov [30], and a simplified analytical approach
Mediated by the nonlinearity of the crystal and a strong pump beam two input fields a(in) and c(in) are interconverted into two output fields a(out) and c(out). This FC process is more commonly known as sum frequency generation (SFG), when the input beam in combination with the pump beam generates an output field at a higher frequency !out = !in + !p (figure 1(a)), or difference frequency generation (DFG), when a field with frequency !out = !in !p is created (figure 1(b))
In order to mathematically describe type-II PDC in the high-gain regime, we extend our theoretical framework for FC processes to type-II PDC, covering both degenerate and non-degenerate configurations
Summary
We first define the electric field operators of an optical wave inside a nonlinear medium as [32]. ⌧ !0) and treat the dispersion term in front of the integral n(k0) in (1) as a constant, using the value at the central wave vector k0 This approximation is justified since, in the remainder of this paper, we only consider electric fields not too broad in frequency, compared to their central frequency, and take into account the rather flat dispersion in nonlinear crystals. We restrict ourselves to electric fields in one dimension This means we assume a fully collinear propagation of the interacting fields along one axis in a single spatial mode, as e.g. given inside a waveguiding structure, since a three-dimensional treatment does not offer further physical insight into the properties of the process and complicates the calculations. In the remainder of this paper, we merge all of them into a coupling value depicting the overall efficiency of the FC process rendering the presented calculations independent of individual notations
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