Increase of the sensitivity of interferometry as demonstrated in Ref. [1] is based on photon subtraction obtained by diverting with a beamsplitter a small part (10%) of the thermal light at one output port of the interferometer (see Fig. 1), and heralding: only events where one photon is measured by APD1 are retained.

 

Fig. 1. Heralding scheme: measurements on APD2 are retained if and only if a single photon is detected on APD1 (reproduced from Ref. [1]).

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This heralding scheme, named photon subtraction in Ref. [1], is claimed to double the mean number of photons measured on APD2. For a photon thermal state after the action of an annihilation operator, this is a straightforward result: it is proved in Ref. [2] that for every state the average number of photons ${\overline{n}}_{-}$ after the action of the annihilation operator is given by

On the one hand, this result is not surprising at low flux. Indeed, the second-order coherence function, whose value is 2 for a thermal beam, can be interpreted as a conditional probability. Hence, detecting a photon is a rare event, which increases the probability of detecting in the same temporal mode more photons than the mean, because thermal photons are bunched, even at very low fluxes [3]. On the other hand, the doubling in Eq. (3) does not depend on the average flux $\overline{n}$, which is troubling. Indeed, at high flux the probability of detection of at least one photon by APD1 tends to unity and doubling the conditional average means doubling the average. The fact that, most often, more than one photon is detected by APD1 does not improve the plausibility of Eq. (3): it is proved in Ref. [1] that the conditional average at APD2 is even higher than $2\overline{n}$ for multiple detections at APD1. Hence, it is necessary to avoid using a model based on an annihilation operator for the experimental scheme in Ref. [1] and to use rather the well-known physics of the beamsplitter. We will see, as foreseen above, that both models are equivalent at very low flux but differ for higher flux.

Let $R=r{r}^{*}$ be the reflection coefficient of the beamsplitter, corresponding to the part of the flux diverted to APD1, and $T=t{t}^{*}=1-R$ its transmission coefficient. The wave function of the thermal beam in the mode before the beamsplitter, named in, can be written as

The wave function after the beamsplitter is obtained [4] by applying the same input creation operators on the output modes $o1$ and $o2$, corresponding, respectively, to APD1 and APD2:

This function looks quite complicated. However, we are interested only in its elements corresponding to exactly one photon in $o1$. This reduced wave function reads

Equation (6) can be translated in a more intuitive way by using conditional probabilities:

In Eq. (7) and in all the following, $P$ means “probability of” and the vertical bar means “knowing.” $P(1_{o1})$, corresponding to $\Vert A^2\Vert$ in Eq. (6), is given by

Equation (7) means that the quantum formalism is rigorously equivalent to a semi-classical model where the conditional probabilities at the output are given by the Bayes theorem:

The first expression under Eq. (10) means that there are as many possibilities of obtaining one photon in 1 as the number $n_{\text{in}}$ of input photons, each one occurring with a probability of $R{T}^{n_{\text{in}}-1}$. Because $R+T=1$, $n_{\text{in}}R{T}^{n_{\text{in}}-1}$ is clearly smaller than 1 whatever $n_{\text{in}}$.

It is now straightforward to calculate all the quantities of interest by using well-known formulae on series. We find

Note that this result can be established more directly by observing that the number of photons in mode $o1$ can be viewed as resulting from a random deletion from a Bose–Einstein statistics, and, hence, obeys also a Bose–Einstein statistics with a reduced mean $\overline{n}R$ [5]. The conditional mean reads

For $\overline{n}R\ll 1$, we retrieve the limit $2\overline{n}$. The one-photon subtraction model appears correct only in this limit. With the values in Ref. [1], $R=0.1$ and $\overline{n}=1.14$, we find $(\overline{n_{o2}}|1_{o1})=1.84$, in good agreement with the experimental value of 1.81 given in the supplementary material in Ref. [1], and quite far from the value $2\overline{n}=2.28$. We conclude that the discrepancy between this value and the experimental one is not due to loss, as argued in the supplementary material in Ref. [1], but rather to the use of a model based on the annihilation operator for a not sufficiently low value of $\overline{n}R$.

A key point is the rate of heralding success. It is argued in Ref. [1] that the heralding scheme will have a practical interest if one is able to reduce the subtraction loss to at least 50%. What is actually relevant is the mean

The coefficient multiplying $\overline{n}$ on the right-hand side of inequality (13) has a maximum value of 0.30, obtained for $\overline{n}R=0.5$. Hence, the number of “useful photons” cannot be increased beyond $\overline{n}$. This property has a simple physical interpretation: a passive linear beamsplitter cannot increase the total number of photons, and, if we do not consider the detector noise, the best strategy is to use all the photons without heralding. This issue has been extensively discussed in the context of weak value amplification [6]: selecting temporal modes with the highest signal-to-noise ratio has an interest only if the detector noise is so high that the signal does not emerge beyond this detector noise for the other modes.

One may wonder about the possibility of using all the temporal modes where at least a click is registered on APD1. Indeed a double click, for example, induces a higher conditional mean than a simple click [1]. Rather than a lengthy calculation of the conditional mean for each number of clicks, we present in the following a simple argument, which shows that the result is worse than a non-conditional detection, at least in absence of detector noise. We consider in the following the use at the output of the total number of photons $n_o$, detected by either APD1 or APD2. This number is equal to $n_{\text{in}}$ and we can write

All the terms in Eq. (15) are easy to calculate using series like in Eqs. (8) and (12), with the results

As expected, the result tends to $2\overline{n}+1$ if $\overline{n}R\ll 1$. For higher $\overline{n}R$, the result is smaller than $2\overline{n}+1$, though greater than the conditional mean for one click, $2\frac{T\overline{n}}{1+\overline{n}R}+1$. Moreover, the conditional mean multiplied by the success rate remains smaller than $\overline{n}$, by construction of Eq. (14).

All these results are valid whatever the quantum efficiency of the photodiodes. Indeed, a non-unity quantum efficiency keeps the statistics of $n_o$ Bose–Einstein, with a reduced mean [5]. Then, all the above computations concern the repartition of these detected photons between the photodiodes. Because the detection in a reference experiment without a beamsplitter is also performed with the same non-unit quantum efficiency, it can be considered that the total mean photon number $\overline{n}$ refers to a number of detected photons. With this convention, the quantum efficiency has no role in the computations.

We have validated all the above results by generating a million of random numbers obeying a Bose–Einstein distribution. Each of these numbers represents a number of photons at the input and each photon is randomly either reflected with a probability $R$ or transmitted, using random numbers obeying a uniform distribution. Last, they are randomly detected or not depending on the quantum efficiency. Results are in perfect agreement with the analytical results above.

As quoted in Ref. [1], other schemes were proposed in order to subtract exactly a photon from a field. As explained in Ref. [7], such a scheme does not correspond to applying an annihilation operator, since the probability of success is ideally unity, whatever the nonzero number of photons in the field. Hence, there is no multiplication of the elements of the density operator by the number of photons, unlike for the annihilation operator. This multiplication comes fundamentally from a probability of success proportional to the number of photons. It seems difficult to imagine other schemes that are compatible with this feature than heralding at very low flux. Hence, the above conclusions seem general, and the realization with a beamsplitter seems generic.

To conclude, the increase of the conditional mean obtained by heralding is by principle not sufficient to compensate for the low heralding success rate. Hence, in the absence of a strong detector noise, using all photons is preferable.