Joint spectral characterization of photon-pair sources

Abstract

The ability to determine the joint spectral properties of photon pairs produced by the processes of spontaneous parametric downconversion (SPDC) and spontaneous four-wave mixing (SFWM) is crucial for guaranteeing the usability of heralded single photons and polarization-entangled pairs for multi-photon protocols. In this paper, we compare six different techniques that yield either a characterization of the joint spectral intensity or of the closely related purity of heralded single photons. These six techniques include: (i) scanning monochromator measurements, (ii) a variant of Fourier transform spectroscopy designed to extract the desired information exploiting a resource-optimized technique, (iii) dispersive fibre spectroscopy, (iv) stimulated-emission-based measurement, (v) measurement of the second-order correlation function $g^{(2)}$ for one of the two photons, and (vi) two-source Hong–Ou–Mandel interferometry. We discuss the relative performance of these techniques for the specific cases of a SPDC source designed to be factorable and SFWM sources of varying purity, and compare the techniques’ relative advantages and disadvantages.

Keywords:

• Nonlinear optics
• quantum optics
• spontaneous parametric downconversion
• spontaneous four-wave mixing
• purity
• joint spectral intensity

Acknowledgements

We thank Thomas Gerrits for helpful discussions and a loan of equipment, and Alan Migdall and Offir Cohen for helpful discussions.

Notes

No potential conflict of interest was reported by the authors.

1 A practical note: calculating the Schmidt decomposition is functionally equivalent to singular value decomposition, which is easily performed by any capable linear algebra package.

2 Confusingly, the Schmidt number is also sometimes defined as the total number of eigenmodes without weighting by eigenvalues, and K is sometimes referred to as the ‘cooperativity parameter’.

3 It is worth noting that a filtered correlated source can actually be more desirable in some cases. For example, it is possible to take advantage of correlations to improve heralding efficiency (the chance of detecting an idler photon in a given mode conditional on the detection of a signal photon). However, this is achieved at the expense of reduced source brightness, i.e. fewer pairs overall.

4 Due to the broad bandwidth involved, one must take care to use waveplates with a flat retardance (that is, the optical path-length difference between ordinary and extraordinary polarization $L_e-L_o$) in the wavelength region of interest.

5 The system jitter is slightly more nuanced for a two-dimensional measurement; the photodiode jitter is correlated with both the signal and idler detection, as both are referenced to the photodiode signal. This shared reference leads to the signal and idler detection times appearing more correlated than they actually are. This manifests in the positive diagonal direction in Figure 13, which corresponds to the sum of the two detection times, and therefore the jitter of the reference does not cancel. This contrasts with the anti-diagonal direction which corresponds to the difference of detection times, where the photodiode jitter does cancel. However, Monte Carlo simulations of this effect indicate that it is minor, leading to a purity bias of no more than 0.01 in our measurements.

6 Richardson-Lucy is an iterative maximum-likelihood-based deconvolution algorithm developed for image processing, specifically for removing a Gaussian blur from an image. Initially, we attempted to use a naive matrix-inversion-based deconvolution, but this proved to be numerically unstable for our application due to the high condition number of the joint-spectral matrix.

7 For this to be exactly true, the complementary modes must be collected in the other arm. Otherwise, it is the (smaller) set of modes collected by both arms that determines heralded single-photon purity. This effect can cause the single-arm measurement to underestimate the coincidence post-selected purity.

8 Defined as $V\equiv 1-C_{\text{min}}$, where $C_{\text{min}}$ is the minimum count rate normalized by the baseline count rate.

9 One could instead implement spatial pre-compensation on the pump, causing the two idler modes to directly overlap.

Additional information

Funding

Funding for this work has been provided in part by NSF [grant number PHY 09-03865 ARRA], NSF [grant number 1521110], [grant number 1640968], by Office of Naval Research (ONR) [grant number N00014-13-1-0627], by US Army ARO DURIP [grant number W911NF-12-1-0562], by CONACYT, Mexico, by DGAPA (UNAM) and by AFOSR [grant number FA9550-13-1-0071].