Theory of high-efficiency sum-frequency generation for single-photon waveform conversion

John M. Donohue, Michael D. Mazurek, and Kevin J. Resch

Phys. Rev. A 91, 033809 – Published 9 March 2015

Abstract

The optimal properties for single photons may vary drastically between different quantum technologies. Along with central frequency conversion, control over photonic temporal waveforms will be paramount to the effective coupling of different quantum systems and efficient distribution of quantum information. Through the application of pulse shaping and the nonlinear optical process of sum-frequency generation, we examine a framework for manipulation of single-photon waveforms. We use a nonperturbative treatment to determine the parameter regime in which both high-efficiency and high-fidelity conversion may be achieved for Gaussian waveforms and study the effect such conversion techniques have on energy-time entanglement. Additionally, we prove that aberrations due to time ordering are negligible when the phase matching is nonrestrictive over the input bandwidths. Our calculations show that ideal quantum optical waveform conversion and quantum time lensing may be fully realized using these techniques.

Received 17 December 2014

DOI:https://doi.org/10.1103/PhysRevA.91.033809

Authors & Affiliations

John M. Donohue^*, Michael D. Mazurek, and Kevin J. Resch

Institute for Quantum Computing and Department of Physics & Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

^*jdonohue@uwaterloo.ca

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Issue

Vol. 91, Iss. 3 — March 2015

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Figure 1
Sum-frequency generation for optical waveform manipulation. (a) A pair of photons may be created through, for example, spontaneous parametric down-conversion, with a signal photon in mode 1 and its herald in mode $h$ . The signal photon is then mixed with a strong escort pulse in mode 2 to produce an up-converted signal photon in mode 3. (b) Temporal waveforms may be customized by applying dispersion (represented by the chirp parameter $A_{i})$ to the input photon and escort pulse before SFG. A third chirp applied to the up-converted signal allows for complete temporal magnification. Bandwidth compression can be achieved in this scenario as shown, with equal-and-opposite chirps applied to the input photon and escort pulse and no third chirp required. Bold lines represent strong pulses and thin lines represent single photons.
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Figure 2
Single-photon up-conversion efficiency and fidelity. (a) The probability of successful up-conversion $〈 {\hat{n}}_{3} 〉$ is shown as a function of the scaled coupling constant $p$ for various pulse width ratios $q$ , with the dimensionless time delay $T$ held constant. As $p$ increases in the regime where the escort is much broader in time than the input photon (low $q)$ , the efficiency of up-conversion follows a sine pattern, reaching unit efficiency at $p = π$ . In the regime where the escort is much narrower in time than the photon, high up-conversion efficiency is not achievable. (b) Here we show the probability as a function of $p$ for various dimensionless time delays $T$ , with the pulse width ratio $q$ held at one. As the time delay is increased, the pulses cease to overlap well and the maximum efficiency is seen to drop. Notably, the peak efficiency is no longer well defined past $| T | \approx 1.5$ , as the first local maximum in efficiency is no longer the global maximum. (c) The maximum possible efficiency is numerically calculated as a function of $q$ and $T$ , with the optimal $p$ estimated as the first peak of a fourth-order expansion of Eq. (14). In the low- $q$ regime, the efficiency is also robust against time delays $T$ . (d) By numerically calculating the fidelity of the temporal waveform at the estimated optimal efficiency with that expected from first-order perturbation theory via Eq. (19), we see that the first-order approximation is an excellent description in the low- $q$ regime. Time delays disturb the symmetry of the system and further reduce the fidelity.
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Figure 3
Effectiveness and efficiency of bandwidth compression. (a) The success probability $〈 {\hat{n}}_{3} 〉$ of bandwidth compression is shown as a function of the absolute coupling constant $γ$ , with $τ = 0, σ_{1} = σ_{2} = σ, A_{1} = - A_{2} = A$ , and $A_{3} = 0$ . As the chirp applied is increased, the compression achieved is stronger at the expense of peak power in the escort pulse; however, while more power is required to achieve optimal efficiency, the potential peak efficiency is constant. (b) In the regime where the pulse length ratio $q$ is low, the spectral width of the up-converted signal $σ_{3}$ is seen to be identical to the width $σ_{3}^{(1)}$ expected from a first-order approximation regardless of the input bandwidths. However, as $q$ grows, the ratio of the two widths is seen to depend on the pulse length ratio before chirp $q_{0}$ . The lined region is algebraically inaccessible for real-valued chirp parameters $A_{1} = - A_{2}$ , with the $q = q_{0}$ line corresponding to zero applied chirp and the $q = \frac{1}{q_{0}}$ line corresponding to the large-chirp limit.
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Figure 4
Rényi 2-entropy after up-conversion at peak efficiency. The Renyi 2-entropy, a measure of entanglement, is shown for the up-converted subsystem as a function of the initial Renyi 2-entropy for various values of $q$ . The bipartite energy-time entanglement is unaltered after postselecting on successful up-conversion if the pulse length ratio $q$ is small, i.e., in the time-lensing limit. However, if the input photon is of temporal length comparable to or longer than the strong escort pulse (high $q)$ , the postselection results in an effective loss of entanglement in addition to imperfect efficiency.
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