Psychophysics experiments protocol

Three subjects participated in the experiments. All were male, in their twenties and had optimal vision, in Subject A’s case corrected by contact lenses. The experiments were approved by the ethics committee of the University of Vienna and participants participated entirely voluntarily and were fully informed of the aims and methods of the experiments.

All experiments were conducted inside a light-proof chamber (∼1 × 1 × 2 m) located in a dark room. The subjects wore headphones through which they could hear sounds that heralded the light and control pulses, respectively, as well as feedback on their response. Foam-filled panels provided good acoustic isolation such that no visual or auditory cues were available that could be potentially used by the subjects to infer the time interval in which the single photon stimulus was presented. After a period of ∼35–40 min, a dark-adapted subject fixated with his right eye on a barely visible red light, presented normal to the cornea, with the subject’s head kept in position by a bite bar and a headrest. The fixation light was a 660 nm LED (full-width at half-maximum=5 nm) operated at ∼6 μW output power, which was further attenuated with OD 6.5 filters before impinging normal to the cornea. The stimuli were presented in a Maxwellian view (that is, light focused onto the pupil) ∼23° temporal to the fixation light and the main optical axis of the eye. The chosen position ensured that the light stimuli were presented to the location on the retina with the highest density of rod cells, the main mediators of scotopic vision.

The subjects triggered trials by themselves and were instructed to proceed at whatever pace they felt comfortable. On the triggering of a trial, the subject was presented with two intervals each of 1 ms duration and separated by ∼800 ms, with the initiation of each interval being heralded by a synchronous acoustic signal of ∼10 ms duration. One trial on average took ∼2.5 s, but subjects could pause for a longer time if they wished to do so. One of the two intervals was pseudorandomly chosen to trigger a possible single photon from SPDC around the peak rod sensitivity (∼500 nm), while the other interval was a ‘blank’. After the second interval the subject indicated which of the two intervals they thought contained the stimulus photons (interval response) and thereafter provided a confidence rating (R1–R3; Fig. 1a). Subsequently, subjects received acoustic feedback as to whether their interval choice matched the randomly chosen interval to contain the stimulus. This feedback helped subjects to remain alert and motivated throughout the entire session. The prerequisite is, however, to ensure that subjects did not use any cues that allowed them to infer the time interval into which the photon was emitted. To do so we analysed the performance of subjects in single-photon SPDC trials where no photons were detected by the EMCCD and could use these as catch trials. In the majority of these trials (∼95%) no photon pair was generated and thus no photons were actually sent to the subjects’ eye. We found that in these catch trials the probability of correct response was not different from the 0.5 baseline for both, the combined and the high-confidence responses (0.505±0.003, P=0.08 and 0.507±0.01, P=0.3, respectively).

Before collecting data, subjects were extensively trained using a classical light source with photon number between 1 and 15 photons at the cornea (Supplementary Fig. 4a). The improved performance with experience is clearly and quantitatively visible (Supplementary Fig. 4b,c). Subjects typically required 6–8 sessions, performing one session a day, to reach their optimal performance level (Supplementary Fig. 4b,c). Each session took∼2 h, when including dark adaption.

During data acquisition, each subject went through up to 20 sessions. Still this high amount of sessions was not enough to obtain statistically significant performance for individual subjects, and therefore we pooled the data together to increase significance (Fig. 2a–d). As subject’s sensitivity and criteria used to assign the confidence ratings might vary in psychophysics trials28, we aimed to minimize or normalize possible factors causing variability to achieve maximum sensitivity and similarity across subjects by using extensive training of the subjects and using our 2AFC paradigm.

Since subjects showed significantly higher performance for the high-confidence rating R3 events (Fig. 2a), we analysed the statistical distribution of these events in more detail. The time between correct high-confidence responses was well described by an exponential distribution (R2=0.98) suggesting these correct events occur randomly in time, as expected (Supplementary Fig. 5a). In addition, the distribution of the times when correct high-confidence responses occurred during a session (Supplementary Fig. 5b) was not significantly different from a uniform distribution (P=0.07, Kolmogorov–Smirnov test) demonstrating that the performance of the subjects is approximately constant and that subjects performed equally well throughout the whole duration of the session.

Finally, we verified that performance in single-photon trials was not correlated with the feedback received in the previous trial, suggesting that the observed effect was not due to increased attention on the part of the subject (Supplementary Fig. 6).

SPDC set-up and components

In SPDC high-energy photons decay spontaneously in a non-linear crystal without inversion symmetry under energy and momentum conservation into two lower energetic photons called the signal and the idler. Typically only ∼10−9–10−12 of the incident photons lead to the generation of a pair, but due to conservation of energy and momentum, the photons of a pair are always generated together and in a correlated manner.

A diagram of the experimental set-up for the SPDC source experiments is shown in Fig. 1a. To prevent background light from impacting upon the experiment, the whole-optical SPDC set-up was constructed within a light-proof container. The single photons are produced via SPDC inside a 1-mm-thick beta-barium borate crystal cut for non-degenerate type-I phase-matching at the desired wavelengths (∼504 and 561 nm; 47.6o). The crystal was pumped with 10 ns, 266 nm laser pulses from a diode-pumped, passively Q-switched laser, capable of triggered operation of up to 10 kHz. We chose this non-degenerate configuration such that the wavelength of idler and signal mode coincided with the maximum quantum efficiency of our custom-coated Andor iXon Ultra EMCCD camera and the peak of the human rod response, respectively. The frequency modes were selected by filters (Semrock LL02-561-12.5; λ=561.4±1.1 nm and Thorlabs FELH0500+Semrock TBP01-501/15, respectively) and translatable irises. A flip mirror in the idler path allowed for the switching between the EMCCD and SPAD detectors. Both the EMCCD and the SPAD were used to measure g(2)(0) values (Supplementary Fig. 1), while only the EMCCD was used in single-photon experiments with actual subjects. Both detectors were never used simultaneously.

The signal mode was coupled into a single-mode fibre with ∼40% efficiency after spectral filtering. The angle of the tunable TBP01-501/15 filter relative to the optical path allowed for the fine-tuning of the bandwidth to ∼500–508 nm, with absorption variation across this range being negligible.

The alignment of the SPDC source was performed by fibre-coupling the frequency-filtered emission into single-mode fibres that were connected to SPADs. The correlated signal and idler modes were found by translating the fibre-coupler and optimizing the coincidences between detected photons using an electronic circuit. In the single-photon SPDC experiments, the stimulus block was generated by a single trigger pulse to the ultraviolet laser and ‘successful’ events were post-selected based on the EMCCD camera registering one and only one photon. Because of the non-unity mean photon pair per pulse rate (0.048 – corresponding to a ultraviolet laser power of ∼150 μW) and additional EMCCD noise (clock-induced charge rate of ∼0.04), an experimental session typically consisting of one thousand trials (=trigger pulses) yielded ∼80 single-photon events passing the post-selection. The optimal EMCCD operating settings were determined by a combination of the EMCCD camera’s theory of operation, numerical simulations and experimental testing. The settings of the EMCCD camera as utilized in the experiment are given below as well as in Supplementary Table 2.

Characterization and performance of the EMCCD camera

In this work, the expected photon number per EMCCD pixel in the single-photon detection regime is <0.1. Therefore, the application of a fixed count threshold, above which one photon is said to have been detected, typically yields best results29. We empirically determined the distribution of the EMCCD counts to be normal with an average of 200 and a s.d. of 3.5 counts by operating the camera in the absence of stimulus light and with its external shutter closed. Therefore, for our particular camera, we set the threshold as 220 EMCCD counts, which is the ∼6σ value above the readout noise (200 counts). Furthermore, the readout region of interest was a three-by-three pixel area and events were classified as single-photon events if this threshold was exceeded on one and one pixel only.

To minimize intrinsic noise events, the EMCCD camera (Andor iXon 897 Ultra) must operate in a continuous long series kinetic mode – this behaviour was not specified by the camera manufacturer but was noticed experimentally to produce the best results. On reception of a trigger pulse, the experimental control software waited for a signal that the camera has begun its next acquisition. As the output of the laser has a delay ranging between ∼35 and 50 μs following the reception of this trigger pulse, the exposure time of the camera was set to 100 μs to ensure synchronization between the camera and the laser. However, exposure time is a non-critical variable and can be increased without noticeable degradation of performance, as the dominant noise mechanism in this regime is clock-induced-charge, a mechanism entirely independent of the length of the exposure.

The most effective settings for the EMCCD camera were determined by a combination of the EMCCD camera’s theory of operation, data collection and numerical simulations. The settings of the EMCCD camera when run in single-photon counting mode as utilized in the experiment are given in Supplementary Table 2.

The heralding efficiency, that is, the probability for a single photon impinging on the cornea provided its partner is detected by the EMCCD, of the SPDC source in combination with the EMCCD detection was estimated at ∼20%. This was determined based on the heralding efficiency measured directly with SPADs (coincidence to trigger single photon rate), but taking into account their detection efficiency (∼40%) as well as the noise (clock-induced charge rate ∼0.04) of the EMCCD. This value of the heralding efficiency was also consistent with numerical simulations.

Elimination of two and higher number photon states

One of the key advantages of using the single photon SPDC source in combination with a multi-pixel detection array such as an EMCCD in our study was that it allowed for the identification of cases in which two or higher number photon states were generated, which could then be excluded from further analysis through post-selection. This was possible because of the high quantum efficiency of the EMCCD that we employed for detecting the idler photon and also because, unlike when conventional SPADs are used, the spreading of the spatial mode of the idler beam onto a three-by-three pixel region of the EMCCD allowed for the detection of trials in which two or multi-photon pairs were generated. At the SPDC pump power used in the experiment (laser power=150 μW) which yielded a mean photon pair rate per pulse of 0.048, the rate of two and multi-pair events was already highly reduced (∼0.11%). Moreover, using EMCCD in the above configuration allowed us to detect ∼80% of such cases where two or a higher number of photons arrived at the EMCCD and such events were discarded during the post-selection. This overall ability of the EMCCD to identify two and multi-photon events, together with the efficiency of the signal arm, the transmission efficiency of the ocular medium and the quantum efficiency of the photo isomerization left us with only ∼0.02 % of post-selected trials in which two or more photon states were generated by the crystal, misidentified by the EMCCD camera as a single-photon event and elicited a two- or multiple-photon signal on the retina. This fraction of trials that lead to multiple-photon events was obtained from a stringent analysis of the g(2)(0) function (see above) and taking into account measured values for photon loss and detector inefficiencies in the signal and idler arms. This means that from our 2,420 trials out of ∼30,000 that passed the post selection, on average <1 multiple-photon event would have occurred at the level of the retina, which we feel can be safely ignored. Thus, unlike in previous studies we can exclude the possibility that the subjects’ responses were due to two- or multiple-photon events.

Heralding efficiency of the SPDC source

A commonly used metric to quantify the quality of the single-photon source is the heralding efficiency. For a source based on SPDC this denotes the probability that a photon was present in the signal arm conditioned on the detection of the idler photon. For our set-up the heralding efficiency was ∼20% (see above).

This imperfect heralding efficiency affected neither the ability of subjects to detect single photons nor the validity of our observation on single-photon-induced modulation of the sensitivity of the visual system. This is because even a low heralding efficiency would only lead to a higher number of blank trials but would not lead to more than a single photon being present at the subjects’ cornea. Furthermore, our main results are in good agreement with the heralding efficiency (∼20%) and the overall quantum efficiency of the eye estimated previously (∼30%, see Supplementary Table 1). Based on these values, the maximum theoretical probability of correct response was estimated to be ∼0.53 (all post-selected trials, see Fig. 2a, Supplementary Fig. 2). We also note that this calculation does not hold in case of dividing the post-selected trials into different ratings, for example, in case of the high-confidence rating R3 events, this data effectively already presents a further post-selection or subset of all experimental trials, in which with the high probability visual system indeed received and detected a single-photon event.

Numerical simulations for required number of trials

The SPDC single-photon light source and the optical set-up were simulated using the Monte Carlo method, including a simulation of the output produced by the EMCCD. In the simulation, SPDC photon pairs were pseudorandomly generated from a Poisson distribution determined by the mean number of pulse pairs. Experimentally measured transmission values for both arms (signal and idler) were used to obtain probabilities of the photon numbers incident on the eye and the EMCCD.

The idler photons of a pair that survived transmission through the set-up were pseudorandomly distributed onto EMCCD pixels based on the measured beam profile. A simulated number of EMCCD counts was generated for each pixel in the three-by-three region of interest, accounting for all the stages between detection of the photon and the electronic readout, including the stochastic gain mechanism. A Monte Carlo simulation was chosen to fully account for the effects of possible serial register clock-induced-charge. First, clock-induced-charges were accounted for by the pseudorandom addition of an electron to the photoelectron counts in the pixel. The combined photoelectrons and clock-induced-charge electrons are then carried through the EMCCD register with a probabilistic gain applied to each electron individually and at each register stage. The gain (G) is given by,

where p is the probability of an electron producing a secondary electron as it is shifted through the register and r is the number of stages in the register, which in our case was 512.

Additionally, a probabilistic serial register clock-induced-charge is also accounted for at every step. This probability was determined by matching the simulated data to the experimental count distribution. Finally, the total number of electrons is then converted into a readout number of counts, based on the experimentally observed distribution, to give an EMCCD count number for each pixel.

For each trial, the number of pixels with an above-threshold number of counts was then compared with the number of photons remaining in the signal arm. Repeating this multiple times allowed for the generation of the expected photon number distribution impinging on the eye for different detected photon numbers at the EMCCD.

The number of experimental trials required was estimated by using the probability of correct response (∼0.516) based on an ideal detector limited by physiological estimates of noise and efficiency and the simulated fraction of events passing the post-selection for the different set-ups. The necessary number (n) of single-photon trials (that is, post-selected trials) required to bring the expected probability of correct response to above chance at the 95% confidence level (z=1.96) was then calculated by,

where p is the expected probability of correct response and μ is the value at random chance (that is, μ=0.5). The total number of trials that would need to be performed to reach this many post-selected events as a function of source and power could be calculated by simply dividing by the expected fraction of events passing the post-selection criteria.

Similarly, the number of required experimental trials when using an SPAD as an idler (trigger) detector was calculated, with the SPAD’s binary detection modelled by a binomial trial based on their known efficiency.

Furthermore, the expected g(2)(0) values were also simulated analogously (solid lines in Supplementary Fig. 1). We simulated propagation and simultaneous detection of signal photons in both arms behind a 50/50 beamsplitter, together with coincident detection of a photon in the idler arm.

g(2)(0) correlation function measurement

Attenuated continuous wave (CW) laser light and the Fock states of SPDC have different photon number distributions, which can be measured and distinguished with the second-order quantum correlation function g(2)(0) (ref. 30), which relates the mean photon number to its s.d.:

where Δn is the s.d. on the photon number n, and <…> represents the average. It is evident from equation (3) that in the case of Poisson distributed coherent states, in which the variance is equal to the mean (that is, ), g(2)(0)=1 and that for a one photon Fock state with zero variance, g(2)(0)=0. The second-order quantum correlation function thus represents a good and universally accepted test for the quantum nature of a light source, especially as it can be measured using a simple Hanbury Brown and Twiss interferometer31. To do this, we fibre-coupled the SPDC signal emission into a single-mode fibre-beamsplitter (Thorlabs) whose outputs were directly connected to two additional SPADs. The idler arm fibre was also detected with an SPAD. In this configuration, the g(2)(0) measurement reduces to:

where N i is the number of counts in the idler arm, N s1 and N s2 are the counts at the two detectors in the signal arm coincident with an idler count, and N c3 is the number of triple-coincidences between the idler arm and both arms of the signal. Our coincidence window was set to the laser pulse duration (10 ns), as all intra-pulse pairs are indistinguishable in the final experimental configuration.

We measured the g(2)(0) value of our SPDC source with both an SPAD and the EMCCD camera in the idler (trigger) arm. The results of both measurements are depicted in Supplementary Fig. 1, and show the expected dependence on the mean photon pair per pump pulse, which in turn is a function of pump laser intensity. The experimental results are in line with numerical simulations taking into account the parameters and theory of operation of both the SPAD and EMCCD detectors29. These numerical simulations also allow for the comparison of the ratio of multiple-to-single-photon events passing the post-selection criteria for the EMCCD and SPAD configuration respectively (Fig. 1b).

Most importantly though, using an EMCCD results in a high increase in triggered and usable single-photon events, due to the improved quantum efficiency (95% versus ∼40%) of the EMCCD compared with the SPAD, the lack of additional fibre-coupling losses (∼60%) and the EMCCDs partial ability to identify and reject cases in which more than one photon had been emitted in the idler arm (multi-pair emission), due to its pixel-array structure.

Poisson light source set-up

We used a classical Poisson light set-up for our control experiments at high photon numbers and for training purposes. A schematic showing a simplified set-up of the CW experiments is shown in Supplementary Fig. 4a. The Poisson light source used in this work was a multi-line continuous wave argon ion laser (LASOS). A filter (Semrock TBP01-501/15) in combination with an acousto-optical modulator (AOM; AA Opto-Electronic) were used to select wavelengths around the peak wavelength of human rod cell’s absorption curves between 495 and 505 nm, with the dominant line of the laser (488 nm) being filtered-out by this method. Stimulus pulses were sent to the subject’s eye by activating the AOM to deflect light into a single-mode fibre. The deflection period was 1 ms and the voltage applied to the AOM modulated the power of the deflected and hence coupled light. The output of this single-mode fibre was collimated and focused on the pupil of an eye using a lens with a focal length of 750 mm. Neutral density filters further decreased the power of the light to the desired low-photon regime. A 50/50 beamsplitter was placed in between the focusing lens and the subject’s eye, to allow the experimental photon levels to be monitored by coupling the non-experimental arm of the split light into a multi-mode fibre, whose output was attached to an SPAD (Perkin-Elmer SPCM-AQR-14-FC). The output of the SPAD was gated, and only counts simultaneous with the light being presented (1 ms). However, as the dark count rate of these particular SPADs is ∼102 s−1, there was a mean dark count signal of ∼0.1 per trigger period. No background-subtraction was performed.

The experiments were performed at seven different mean photon numbers from 20 photons with increasing steps of 20 up to a mean of 140 photons. Individually characterized neutral density filters and a low intensity power metre were used to set these photon number levels. The AOM was calibrated before each experimental session to set the precise number of photons at the cornea. Photon statistics were collected using the aforementioned SPAD for different voltages applied to the AOM to obtain a calibration curve. We ensured that for each voltage the distribution followed Poissonian statistics. Using the obtained calibration curve and known efficiency of the SPAD, the AOM voltage was set accordingly to achieve the desired photon numbers during the experiment. This was also confirmed by simultaneously recording the photon numbers in the non-experimental arm of the 50/50 beamsplitter with the SPAD during all experimental trials. Unwanted additional background light was tested for by observing both the background count rate of the SPAD in the experimental set-up and when the detection region of the SPAD was completed blocked from light. No significant difference in background level was observed.

Several sessions a week were conducted for each subject. Each session consisted of 165 semi-randomly scrambled blocks each containing five trials of the same light intensity. The ordering of the presentation of each light intensity was pseudorandomly shuffled by the computer control programme at the beginning of each experimental session without the subject or the experimenter having any knowledge of this order. Each light level was then presented in five consecutive trials, before the next light level was presented. The subject was informed that the level of the presented light was to change by a slightly prolonged beep through headphones. The different light levels were looped through in this manner until the required number of trials at each light level for that session had been performed, generally 75 per photon number per session.

Statistics

Unless stated otherwise, P values are stated as the Fisher exact test.

Mathematical modelling

Detailed description of the mathematical model based on the Signal Detection Theory that we used to fit our data is presented in the Supplementary Note 3. Parameters of the model obtained by fitting the data are shown in Supplementary Table 3.

Data availability

The data supporting the findings of this study is available from the corresponding author on request.