Análisis Factorial

we expect, the upper bound is a true conservative bound that gets tighter as the photon- number cut-o↵is raised, and the lower bound approaches a true bound only in the high

m limit. Together these bounds define a quantum-Gaussian region of Ps:Pc combinations possible when measuring pure Gaussian states (blue hatched region). However, by mixing Gaussian states it is possible to decreasePs:Pcfurther. In fact, there is no lower bound on

Ps:Pc over the set of mixed Gaussian or coherent states. For example, although a bright coherent state mixed with vacuum has vanishing success probability Ps, it may have any desired coincidence probability Pc depending on the mixing ratio.

It is not possible to increase Ps:Pc by mixing Gaussian states, so the upper bound is a true bound over the set of mixed Gaussian states. The Gaussian state boundaries are shown alongside the classical state boundaries in Fig. 8.5(a). Any measurement of

Ps above this boundary is a QNG witness. Importantly, the upper bound is the relevant QNG witness for single photon sources, which operate in the region Ps Pc. For very attenuated fields (the operational region of our photon-source) the lowest-order QNG witness in Eqn. 8.21 may be reduced to

PcPs3/3, (8.23)

or, taking the limit of higher-order approximations,

PcPs3/2. (8.24)

The operational region of our source, where this approximation holds, is shown in Fig. 8.5(b). This reduced boundary is the efficient QNG witness we will use to characterize our trapped- ion photon-source in the next section.

8.3

QNG photons from a trapped-ion source

Because our trapped-ion photon-source is low noise, it beats the QNG threshold despite substantial attenuation. The performance of our red-triggered source (see Sec. 7.6) in a HBT experiment is compared to the classical and Gaussian boundaries in Fig. 8.5(b). The source is prepared in a mixture of 5D_3/2 states so that 493 nm photons may be emit- ted by triggering with a 652 nm pulse. The magnetic field is oriented perpendicular to the detection axis, such that confocal lenses map the two 6P_{1/2 !} 6S_1/2 ⇡ transitions to uniform Hˆ polarized fields. The probabilities of success and coincidence events Ps,c are determined in a 200 ns detection window from the beginning of the photon trigger. Data is shown for weak and bright trigger pulses, where trigger intensity increases efficiency at the cost of coherence. Each measurement is between 20 and 180 minutes of operation at 200 kHz repetition rate depending on the time required to collect a statistically significant number of coincidence events. Pc andPs are conservative estimators of the true probabil- ities [281], given the measured probabilities and an e↵ective beam splitter transmittance

T = 0.495(5) that includes all detection imbalances.

Red circles show the source measured in the reflected configuration, in which a dis- tant mirror recombines the two collimated fields to a single polarization-filtered mode as described in Sec. 7.1. Even without interference enhancement, the trapped-ion source beats the QNG threshold by 6 standard deviations, Fig. 8.5(b)(red circles). Performance is limited solely by the overall single-photon collection efficiency and detector dark counts. A perfect, but attenuated, single-photon source measured with the same APDs and a

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Coincidenc e prob., Pc ⇥ 10 7 Success prob., Ps Coincidenc e prob., Pc (a) (b) 0 1 2 3 4 5 Success prob., Ps⇥102 Classical Gaussian QNG A 0.0 0.2 0.4 0.6 0.8 1.0

Figure 8.5: Coincidence probabilityPc versus success probabilityPs(a) over the complete prob- ability space and (b) in the operational region of our source (arrow A). Classical light coincidence rates cannot be reduced below the NC threshold (red diagonal) and Gaussian light fields are similarly restricted to coincidence rates above the QNG threshold (blue hatched). Light fields with coincidence rates beyond this threshold (green horizontal) are unambiguously QNG. Circles correspond to measurements of our red-triggered single-photon source in the reflected (red) and symmetric (black) configuration with a 200 ns detection window. The black cross is a measurement under optimal conditions with a 500 ns detection window. Error bars indicate 95% confidence in- tervals. The performance of an ideal single-photon source measured with our APDs is shown for 200 ns (dotted line) and 500 ns (dashed line fragment) detection window.

200 ns detection window will perform according to the dotted line in Fig. 8.5(b), which is consistent with the measured data. This result confirms the single-photon purity of our trapped-ion photon-source. However, we must also be confident that we can maintain number-purity as the collection efficiency is improved.

The typical approach to improving collection efficiency from low-efficiency single pho- ton emitters employs combinations of high numerical aperture optical elements [111, 116, 253, 282–284]. This corresponds to simultaneous emission of a light field into several spatial modes. Such a multi-mode source is useful so long as the purity of emitted single- photons is not compromised. Simultaneous enhancement of spurious background light collection, spatial restrictions on excitation beams, excitation beam scattering into the photon collection modes and other processes could make enhancing collection efficiency a source of additional photon-number noise. To estimate this e↵ect, we measure the same QNG witness for a light state emitted coherently by a single-photon source into two spatial modes.

We apply this measure to our trapped-ion photon-source configured symmetrically according to Fig. 8.6. The fluorescence is detected in two spatial directions by fibre-coupled APDs positioned behind confocal HALOs. In the reflected configuration, the collected photons were combined into the same spatial mode prior to detection. In the symmetric setup they are emitted into two, in principle independent, directions. As demonstrated in [254, 255], ion fluorescence emitted in two opposite directions remains phase coherent and can be efficiently transferred into a single spatial mode, for example by recombination on

§8.4 Comparing QNG photon sources 127