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Physical Review A

Single-mode approximation of parametric down-conversion

Yongmin Li, Hideharu Mikami, Haibo Wang, Takayoshi Kobayashi

Published Dec 2, 2005• 4 citations
Publisher

Abstract

In this paper, we study the theory of spontaneous parametric down-conversion pumped by a short coherent pulse. It is shown that when single spatial mode filters and narrow band optical filters are used to filter the output state of parametric down-conversion, the postselected state of parametric down-conversion can be approximately described by a simple single-mode theory.

DOI
10.1103/physreva.72.063801

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In this paper, we study the theory of spontaneous parametric down-conversion pumped by a short coherent pulse. It is shown that when single spatial mode filters and narrow band optical filters are used to filter the output state of parametric down-conversion, the postselected state of parametric down-conversion can be approximately described by a simple single-mode theory.
The process of parametric down-conversion pumped by a short coherent pulse is analysed. The coherence properties of the down-converted field are investigated. It is found that the spectrum of the down-conversion is mostly determined by the nonlinear medium and is independent of the width of the short pumping pulse and that it is impossible to realize a transform-limited two-photon state. Fourth-order interference between a weak coherent pulse and the down-converted pulse is considered. It is found that high visibility near 100% is achievable by means of narrow optical filtering. It is shown that by gating the detection on one of the down-converted photons, a transform-limited single-photon pulse is achievable in the other conjugate field, but its pulse width is shorter than that of the down-converted pulse and its peak position is uncertain within the pulse width of the down-converted light. However, with narrow optical filtering on the gating detection process, a transform-limited single-photon state can be obtained in the gated field with its peak position synchronized to the pump pulse. The possibility for the realization of a multi-photon interference experiment is explored for the quantum state measurement of the single-photon state.
Parametric down-conversion (PDC) forms one of the basic building blocks for quantum optical experiments. However, the intrinsic multimode spectral-temporal structure of pulsed PDC often poses a severe hindrance for the direct implementation of the heralding of pure single photon states or, for example, continuous-variable entanglement distillation experiments. To get rid of multimode effects narrowband frequency filtering is frequently applied to achieve a single-mode behaviour. To date, the theoretical models of filtered PDC, however, only account for filtering in the low gain regime, where only a few photon-pairs are emitted at any single point in time. In this paper we extend these theoretical descriptions and present a rigorous theoretical model for filtered type-II PDC. Our theoretical investigations reveal that filtering a multimode type-II PDC state, followed by a careful optimization of the measurement basis, effectively achieves a single-mode behaviour and further enables the generation of single-mode EPR-entanglement. Considering the ever increasing brightness of PDC sources our developed theoretical framework, enables the accurate description of filtered PDC in quantum optical experiments.
Single-photon sources are necessary components for many prospective quantum technologies. One candidate for a single-photon source is spontaneous parametric down-conversion combined with a heralding photon detection. The heralded light pulse from such a source is typically treated as single mode; this treatment, however, is incomplete. We develop a full multimode description based on the exact Bogoliubov treatment of the down-conversion process. We then provide a perturbative and effective treatment, which illustrates the most important physical mechanisms and permits analytical estimates of the success probability and purity of single-photon states under practical heralding conditions, both without relying on the precise detection time of the heralding photon and when accepting photons only in a narrow window around the time of the detection. This permits us to characterize the emitted light under three different assumptions for the pump pulse. For spontaneous parametric down-conversion with a very short pump pulse, we find the single-mode description to be accurate, while for longer pump pulses and continuous pumping, a multimode description is necessary. Our findings can be used to guide the design of quantum information protocols based on heralded single-photon sources, as their performance may depend on the multimode nature of the sources.
Using the approach of coupled wave equations, we consider spontaneous parametric down conversion (SPDC) in the narrow-band regime and its relationship to classical nonlinear processes such as sum-frequency generation. We find simple expressions in terms of mode-overlap integrals for the absolute pair-production rate into single-spatial modes and simple relationships between the efficiencies of the classical and quantum processes. The results, obtained with Green function techniques, are not specific to any geometry or nonlinear crystal. The theory is applied to both degenerate and nondegenerate SPDC. We also find a time-domain expression for the correlation function between filtered signal and idler fields.
A theoretical investigation of the performance of single-mode coupled spontaneous parametric downconversion sources is proposed, which only requires very few assumptions of practical interest: quasi-degenerate collinear generation and narrow bandwidth obtained through spectral filtering. Other assumptions, such as pump-beam spatial and temporal envelopes, target single-mode profile and size, and nonlinear susceptibility distribution, are only taken into account in the final step of the computation, thus making the theory general and flexible. Figures of merit for performance include absolute coupled brightness and conditional coupling efficiency. Their optimization is investigated using functions that only depend on dimensionless parameters, so that the results provide the best experimental configuration for a whole range of design choices (e.g., crystal length, pump power, etc.). A particular application of the theory is validated by an experimental optimization obtained under compatible assumptions. A comparison with other works and proposals for numerically implementing the theory under less stringent assumptions is also provided.
Parametric down-conversion (PDC) forms one of the basic building blocks for quantum optical experiments. However, the intrinsic multimode spectral-temporal structure of pulsed PDC often poses a severe hindrance for the direct implementation of the heralding of pure single-photon states or, for example, continuous-variable entanglement distillation experiments. To get rid of multimode effects narrowband frequency filtering is frequently applied to achieve a single-mode behavior. A rigorous theoretical description to accurately describe the effects of filtering on PDC, however, is still missing. To date, the theoretical models of filtered PDC are rooted in the discrete-variable domain and only account for filtering in the low-gain regime, where only a few photon pairs are emitted at any single point in time. In this paper we extend these theoretical descriptions and put forward a simple model, which is able to accurately describe the effects of filtering on PDC in the continuous-variable domain. This developed straightforward theoretical framework enables us to accurately quantify the tradeoff between suppression of higher-order modes, reduced purity, and lowered Einstein--Podolsky--Rosen entanglement, when narrowband filters are applied to multimode type-II PDC.