Appendix

A. Methods

Light sources

For coherent states, we use a picosecond-pulsed semiconductor laser at 1550 nm with a variable repetition rate, varying the mean photon number by attenuation. For multi-thermal states and heralded single photons, we pump a periodically-poled potassium titanyl phosphate waveguide designed for type II parametric down-conversion. The signal and idler modes (around 1540 nm) are split on a polarization beam splitter (PBS), and the idler is detected to herald the signal single photons, or undetected for (nearly) thermal states in the signal mode.

Loop architecture

For quantum characterisation, these states are sent to the active loop detector, which is made up of a free-space and fibre loop, with optional deterministic incoupling accomplished with an electro-optic Pockels cell [36]. The input light pulses are vertically polarized, then switched to horizontal with the Pockels cell to remain in the loop, passing through a fibre delay line (loop length 480 m, corresponding to a delay of 2.4 µs). The outcoupling half-wave plate (HWP) sets the fraction of the light outcoupled on each round trip, by rotating some of the light to vertical polarization. Note that the outcoupling optics is not the same as the incoupling optic, resulting in slightly different loss for the first bin, compared to subsequent bins. The outcoupled light is sent to a commercial superconducting nanowire single photon detector (SNSPD) from Quantum Opus. The Klyshko efficiency of the photon (when using down-conversion) coupled directly through the loop (half round-trip) is (20 ± 2)%, and the round trip loop transmission is (82 ± 2)% Depending on whether the Pockels cell was activated, we implement either the active or passive loop detector structure.

For the high dynamic range measurements, we implemented a passive loop architecture constructed exclusively from optical fibres (loop length 30 m, corresponding to a delay of 156 ns between bins) and a low-loss fibre beam splitter. All components were polarization-maintaining, such that the (polarization dependent) detector efficiency is constant for each bin, and the beam-splitter ratio (which is also typically polarization-dependent) is constant for each round trip. The round trip loop transmission in this case was 86%, which includes additional (artificially-induced) loss to produce a faster decay rate and therefore allows for a higher repetition rate. In these experiments, we used a commercial superconducting nanowire single photon detector (SNSPD) from Photon Spot.

Detector and data acquisition

In both the active and passive experiments, the detector output was recorded on a time-tagger, allowing for all events to be recorded. Gating windows of 4 ns were applied in post processing to reduce the effect of dark counts. The gating windows are separated by the corresponding loop delay, corresponding to the time for the pulse to propagate the length of the fibre loop, and well above the dead time of the detector.

B. Analytic expressions for bin click probabilities

Based on Eq. (1) and expressions for the photon number distributions ρ_in for Fock states ${\rho }_{\text{Fock}}={\delta }_{\overline{n},n}$, coherent states ${\rho }_{\text{coh}}=\exp \left[-{\overline{n}}^2\right]\frac{{\overline{n}}^n}{n!}$ and thermal states ${\rho }_{\text{therm}}=\frac{{\overline{n}}^n}{{(1+\overline{n})}^{n+1}}$, and whether the loop is actively or passively switched, the following analytic expressions for the bin click probability p_j can be derived:

Active:

Passive:

In all cases we assume a mean photon number $\overline{n}$ and a constant dark-count probability per bin (noise floor) ν_j = ν for all time bins.

C. Calibration procedure

The high dynamic range of the detector can be used to bridge the gap between bright light and quantum-based optical power standards. To calibrate the detector efficiency to a known bright light standard, we suggest the following procedure, using the setup shown in Fig. 6. Given a coherent state with a mean photon number determined with a power meter ${\overline{n}}_{\text{PM}}=\text{power}/(\text{photon}\text{energy}\times \text{repetition}\text{rate})$, one can define a system detection efficiency η_SDE as

 

Fig. 6 Schematic of calibration setup. ${\overline{n}}_{\text{in}}$ indicates the mean photon number per pulse incident on the loop. ${\overline{n}}_{\text{out}}$ is the mean photon number after the loop, summed over all bins. ${\overline{n}}_{\text{out}}$ can be switched between an on-off detector or a power meter. Reliable switching is important for the calibration procedure as systematic errors can be produced during this process.

Download Full Size | PDF

To calculate ${\overline{n}}_{\text{measured}}$, we begin with the mean photon number per bin ${\overline{n}}_{\mathrm{j}}$. For the active case this is given by

With these equations we can also calculate the total outgoing mean photon number

Which means for the active case

Experimentally we can now measure the click probability per bin p_j with the click detector under test (cf. Fig. 7(a)). Using Eqs. (8) and (11) (and substituting $\overline{n}={\overline{n}}_{\text{in}}$) for coherent states as well as Eqs. (17) and (18) each bin probability p_j gives an estimate for the mean photon number per pulse ${\overline{n}}_{\text{out}}{|}_j$ (see Fig. 8). For the active case we can write

 

Fig. 7 Histogram showing the bin probability in linear (top) and logarithmic (bottom) scale for a calibration measurement (a) and a highly attenuated input (b), using a passive loop. Red curves: Fit based on Eq. (11). The fit for the attenuated coherent state gives an estimate for R = 0.91370 ± 5 · 10⁻⁵ and η = 0.8615 ± 3 · 10⁻⁴.

Download Full Size | PDF

 

Fig. 8 Inferred mean photon number per pulse ${\overline{n}}_{\mathrm{j}}$ after the loop from measured click probabilities p_j. Red line: mean value of ${\overline{n}}_{\text{measured}}=208011$. The first bins j ≤ 25 are excluded because dead times from back reflections reduced the count rate (see next section for further details).

Download Full Size | PDF

To arrive at a final estimate of the mean photon number ${\overline{n}}_{\text{measured}}$, we use the arithmetic mean weighted by the experimental error in the individual estimates of ${\overline{n}}_{\text{out}}{|}_j$, as determined above. This provides our unbiased estimator of ${\overline{n}}_{\text{out}}$, with the weights given by the inverse of the variances ${\sigma }_{{\overline{n}}_{\text{out}}{|}_j}^2$ (see next section for details on how these errors are calculated) of each measurement of ${\overline{n}}_{\text{out}}{|}_j$, i.e.

Fig. 8 shows the measured values for ${\overline{n}}_{\text{out}}{|}_j$ and ${\overline{n}}_{\text{measured}}$. The limits on the summation over j are chosen such that the errors in the individual measurements of ${\overline{n}}_{\text{out}}{|}_j$ are not underestimated (i.e. that their uncertainties are reliable). From Fig. 9, this is from bin| 25 onwards. The use of the weighted arithmetic mean penalises values with high error bars; therefore including high bin numbers, where the uncertainty is high, does not limit our precision.

 

Fig. 9 Relative uncertainty contributions $\sqrt{\frac{{\sigma }_i^2}{{{\overline{n}}^{\prime }}^2}{\left(\frac{\mathrm{d}{\overline{n}}^{\prime }}{\mathrm{d}i}\right)}^2}$ from each error source, arising from uncertainty in the measured bin probabilities p_j (red line), the loop loss η (green line), the loop reflectivity R (blue line) and the noise ν (gold line). Note that these quantities are evaluated using the a posteriori weighted mean ${\overline{n}}^{\prime }$ i.e. the mean photon number averaged over all j. The blue dots are the total relative error contributions, and red crosses the relative error in p_j, evaluated at the individually determined photon number ${\overline{n}}_{\text{out}}{|}_j$ (in contrast to the solid lines). The deviation of these measures at low bin number j ≲ 25 arises from undercounting caused by back-reflection-induced inefficiency, resulting in an underestimate of the error associated with p_j. This is discussed further in the following section.

Download Full Size | PDF

With this summation (j ≥ 25), the mean photon number per pulse is ${\overline{n}}_{\text{measured}}=208011\pm 460$. The value of ${\overline{n}}_{\text{measured}}$ can now be compared to the value from the power meter n_PM = 251000 ± 12500 to calculate the System Detection Efficiency (SDE), i.e. Eq. (13). The results in an η_SDE = 82.8 ± 4%. The dominant error comes from the power meter reading, not ${\overline{n}}_{\text{measured}}$, and we neglect errors in laser drift and fibre coupling efficiency. The value of η_SDE includes the fibre into the cryostat and the detector efficiency. It can be seen from Eq. (19) that the value of ${\overline{n}}_{\text{out}}$ for the active case only depends of the product ηR. This product is well known as the slope of the histogram in the decay region (logarithmic plot) is 1 − ηR. The passive case, however, is different. Here the individual values of η and R must be known to infer n_out. Therefore we need an additional measurement step for the passive case in order to determine at least one of R and η. We suggest to add an unknown attenuator before or after the loop such that the first bin is non-saturated. An example measurement can be seen in Fig. 7(b). By fitting this histogram with Eq. (11) values for R and η can be determined. The full information about these two values is contained in the jump from the first bin to the second as well as in following linear decay region. Therefore the attenuator must be strong enough that the first bin in non-saturated (otherwise the uncertainties in the click probability are too high, c.f. next section) and low enough to see the linear decay region before the dark count level. The advantage of this method is that the beam splitter reflectivity R and the loop efficiency η can be determined without changing the loop.

It should be noted that losses before the loop and after the loop (up to the point where either a power meter or a click detector is placed) do not matter for the calibration procedure as this is a rescaling of the power that is already considered by the power meter reading.

This method can be easily altered to measure the power of bright pulses if the SDE is known, for example to bridge a quantum standard to bright light.

D. Estimation of uncertainty

The relative uncertainty in ${\overline{n}}_{\text{measured}}$ arises from the uncertainty in the component quantities from which ${\overline{n}}_{\text{out}}$ is calculated. For each bin j, we obtain an independent estimate of ${\overline{n}}_{\text{out}}$, i.e. ${\overline{n}}_{\text{out}}{|}_j$. This is given by Eq. (20) The relative uncertainty in each of these estimates of ${\overline{n}}_{\text{out}}{|}_j$ is given by Gaussian error propagation:

The absolute errors are all experimentally determined: σ_R, σ_η are determined by the fitting procedure, explained in the previous section. σ_ν is determined from Poissonian error from the dark-count probability measurements. Only ${\sigma }_{p_j}$, determined by Poisson error in the click statistics, depends on the bin number; the other absolute errors are constant for each bin. For the data shown above, the absolute and relative uncertainties are given in table 1: The contribution of each source of error to the relative error in the photon count rate to be determined is shown graphically in Fig. 9. The blue dots correspond to evaluating Eq. (23) as a function of the bin number j, to show the overall uncertainty in determining ${\overline{n}}_{\text{out}}{|}_j$ at each bin. The solid lines correspond to evaluating the individual contributions $\sqrt{\frac{{\sigma }_i^2}{{\overline{n}}_{\text{out}}^2}{\left(\frac{\mathrm{d}{\overline{n}}_{\text{out}}}{\mathrm{d}i}\right)}^2}$ for i ∈{p_j, R, η, ν}. Note that we use the weighted arithmetic mean as our estimate of the mean photon number ${\overline{n}}_{\text{out}}$, rather than the individually determined estimates for ${\overline{n}}_{\text{out}}{|}_j$, since these estimates under-count at low bin numbers, for reasons described in the section on limits and assumptions, below. As a comparison, we show with red crosses $\sqrt{{\frac{{\sigma }_{p_j}^2}{{\overline{n}}_{\text{out}}^2{|}_j}{\left(\frac{\mathrm{d}{\overline{n}}_{\text{out}}}{\mathrm{d}i}\right)}^2|}_j}$, i.e. the relative uncertainty contribution due to p_j, evaluated at the individual bin number j, and not the weighted average. The result of the undercounting shows a clear underestimate of the relative uncertainty at j ≲ 25.

Table 1. Absolute and relative errors of the measured values

View Table

The resulting uncertainty of the weighted arithmetic mean is thus given by the inverse root of the sum of the weights, i.e.

Given the weights calculated above, and summing from bin 25, we obtain a relative error of ${\sigma }_{\overline{n}}^2/{{\overline{n}}^{\prime }}^2=0.22\%$ in our determination of the mean photon number after the loop.

The uncertainty in determining an unknown System Detection Efficiency (SDE) stems from uncertainties in its constituent quantities. We compare our method to that used in [19], in which the expression for the SDE is SDE = PCR /[P_C a₂, a₃R_SW/(1 − ρ)/E_λ] given by the photon count rate PCR, the power on the power meter P_c, the attenuations a₂, a₃, the switch ratio R_SW, the reflectivity of the fibre connected to the switch ρ, and the photon energy E_λ. The principle uncertainties arise in the quantities PCR, P_C, a_2,3 and R_SW. The relative uncertainties in P_C and R_SW remains present in both experiments. To evaluate each approach, we compare the relative error contributions arising from calibrated attenuation and determination of the photon count rate. In [19] (using the values in Table SI 2 and Eq. (1)), this contribution is the root quadrature sum of σ_PCR/PCR = 0.14% and ${\sigma }_{\alpha }/\overline{n}=0.20\%$. Since two attenuators were used, this value contributes twice to the overall relative uncertainty. Thus the relative uncertainty arising from these two sources is $\sqrt{{({\sigma }_{\text{PCR}}/PCR)}^2+2{({\sigma }_{\alpha }/\alpha )}^2}=0.32\%$. Our method does not use calibrated attenuators, thereby removing this source of measurement error, to comparable error therefore arises in ${\sigma }_{\overline{n}}/\overline{n}=0.22\%$, as indicated above. Our method therefore provides a significant reduction in the error when compared with using calibrated attenuators.

The precision with which the inference of input mean photon number detected via the loop can be achieved depends on both the fitting error for determining R and η, as well as the statistical error from the number of data points used. Both of these depend on the number of occupied bins above the noise floor of the device. For noise ν ≪ 1, we can approximate the number of occupied bins above the noise floor as

The upper limit to the dynamic range n max is determined by the damage threshold of the detector used, i.e. the pulse energy that causes the detector to become unresponsive to subsequent pulses. For SNSPDs the first failure mode is latching at high count rates, where the device creates a self-sustaining hotspot and cannot reset without manual reduction of the bias current. We have found a large region of pulse energy (1 to 10⁷ photons per pulse) where the latching point depends only on the repetition rate and not on the pulse energy (see Fig. 11). Thus SNSPDs are ideal candidates for the loop detector, as they can handle many photons in the first bins and still be single-photon sensitive with low noise in the later bins. We have tested up to $\overline{n}=4.8\times {10}^7$ photons per 100 ps pulse at 100 kHz repetition rate with a WSi SNSPD, and found no latching or degradation of performance.

 

Fig. 10 (a) Raw time-tags for a bright pulse, zoomed in on the first 24 bins, showing clear excess counts outside of the expected loop bins (orange). This arises from spurious back-reflections present in the setup. (b) Click probability as a function of bin number. In early bins 2 ≤ j ≲ 20, back reflections cause a reduction in the expected counts since they induce dead time, which can be seen by a deviation from the expected click probabilities.

Download Full Size | PDF

 

Fig. 11 Single bin click probability as a function of incident number of photons per pulse and pulse repetition rate. Once latching occurs, no counts can be measured (upper right corner).

Download Full Size | PDF

E. Limitations and assumptions

The procedure we present relies on a few important assumptions to be applied correctly. Most generally, the detector efficiency to be calibrated must remain constant for all bins. Effects which change this assumption should be avoided, or where possible, appropriately accounted for. We neglect random drift of the efficiency or laser power, since the timescale over which we take data is relatively short (ca. 150 s). More critical is bin-to-bin variations in efficiency. This could be caused by a different polarization for each bin, and an associated polarization-dependent efficiency. We obviate this error source by using polarization-maintaining fibre components to build our loop. A further source of error arises from missed counts due to back-reflection-induced dead time losses. In early bins, there are a great many photons circulating in the setup. While most photons are in the loop, a small fraction may be reflected back towards the detector, out of synch with the actual photons to be measured. Whilst these will be detected outside the acceptance window, their detection will cause some dead time, during which time the detector is “blind” to further photons. This effect is visible in the raw time-tag data, shown in Fig. 10(a). This plot is effectively an optical-time-domain-reflectometery (OTDR) spectrum, evaluated over many orders of magnitude. It shows counts appearing outside the expected bins (shown in orange).

Thus, true events may be missed. This is effect is clearly seen in Fig. 10(b), where the second bin in particular is significantly lower than expected. Reducing back reflections is therefore crucial to the reliable operation of this system.

F. Trade-off between operating speed and dynamic range

In the case of SNSPDs, the upper limit on the dynamic range is determined by the number of occupied bins per unit time that the detector can handle before it latches. Crucially, over a certain range of repetition rates, it does not depend on the power delivered the detector. We demonstrate this by varying the repetition rate and mean photon number per pulse incident on the detector. The resulting count rate is shown in Fig. 11.

One might expect that the product of photons per pulse $\overline{n}$ and repetition rate r (i.e. $\overline{n}\times r=P$, the power delivered to the detector) would be constant, with a corresponding constant latching threshold. This is true up to a mean photon number of one photon per pulse, where the latching threshold is ∼ 10⁶ photons per second or 0.1 pW at a wavelength of 1550 nm. However, above this, the mean photon number per pulse plays almost no part in determining the latching threshold. This allows us to put much more incident power on the detector before it latches, provide the repetition rate is slightly lower. The highest we achieved is 5 × 10⁷ photons per pulse at a repetition rate of 100 kHz, yielding an optical power of 0.6 µW. This measurement was limited by available laser pulse energy; higher power may well be achievable. The consequences of this effect may also be interesting in terms of the detector dynamics and heat dissipation models, which clearly behave nonlinearly with incident optical power.