Quantum metrology of solid-state single-photon sources using photon-number-resolving detectors

The non-classical light source used in our experiments is represented by a deterministically fabricated quantum dot (QD) based SPS. This type of SPS was previously exploited to demonstrate the efficient generation of indistinguishable single photons [12], twin-photon states [13], and polarization entangled photon pairs [14]. The device is based on a single pre-selected InGaAs QD embedded precisely at the center of a monolithic microlens made out of GaAs. Together with a distributed Bragg reflector (AlGaAs/GaAs) beneath the QD, this photonic nanostructure enhances the photon extraction efficiency of the emitter to typical values of 20%–30% into a numerical aperture (NA) of 0.4. For the deterministic fabrication of the QD-microlens aligned to a pre-selected target emitter, we employed a technique called 3D in situ electron-beam lithography. For a detailed review on this type of quantum light source, we refer the reader to [15].

The experimental setups used for optical quantum metrology are illustrated in figure 1(a). A confocal micro-photoluminescence (μPL) setup is employed, with the sample mounted onto the cold finger of a Helium-flow cryostat at a temperature of 8 K. The sample is excited using a pulsed wavelength-tunable titanium-sapphire (Ti:Sa) laser operated in ps-mode (repetition rate: 80 MHz) at a wavelength of 909 nm, corresponding to p-shell excitation of the QD. PL from the QD microlens is collected via a microscope objective (NA: 0.4) and spectrally analysed using a monochromator with attached charge-coupled device camera (overall spectral resolution: 0.017 nm/25 μeV). For photon statistics experiments, the emission of the charge-neutral QD exciton X⁰ is spectrally filtered using the exit slit of the monochromator (100 μeV bandwidth) and then coupled either to a fiber-based HBT- or HOM-type setup. In case of the HBT experiment the QD is excited using laser pulses with a period of 12.5 ns (see figure 1(a), inset). The resulting emission of the X⁰-state leaving the monochromators' exit slit is coupled to a 50:50 beamsplitter based on optical multimode fibers. Both output ports of the beamsplitter are then coupled either to the PNR detection system (see section 2.3) or standard click detectors (SPCMs). In case of the HOM experiments, the QD is excited using a sequence of double-pulses with a pulse-separation of 4 ns. The resulting emission of the X⁰-state is coupled to a fiber-based polarization-maintaining asymmetric Mach–Zehnder interferometer attached to the output port of the monochromator. The polarization of the photons in one interferometer arm can be switched, either being co- or cross-polarized with respect to the other arm. The interference of consecutively emitted photons at the second beam-splitter is ensured by a 4 ns delay-line in one interferometer arm. Details on the HOM-setup can be found in [16]. At the two output ports of the HOM-setup photons are again detected using PNR or click detectors, respectively. To obtain reference measurements for the second-order photon autocorrelation as well as the photon indistinguishability (see section 3.1), the HBT and HOM setup are used in combination with the click detectors based on silicon avalanche photo diodes with an overall timing resolution of 350 ps in combination with time-correlated single-photon counting electronics.

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PNR experiments are enabled by employing a two-channel detection system based on fiber-coupled tungsten TESs operated in a cryogenic environment. Each of the TESs acts as a highly sensitive calorimeter, which is able to detect smallest amounts of energy dissipated during photon absorption. The energy dissipation results in a rise of temperature, which ultimately leads to a detectable change in electrical current. The latter is measured via an inductively coupled two-stage direct-current superconducting quantum interference device (SQUID). The TES/SQUID detector units are mounted onto the cold stage of an adiabatic demagnetization refrigerator stabilized at a temperature of 100 mK. This compact custom-made state-of-the-art detection system enables detections efficiencies of >87% in the spectral region of interest (850–950 nm) and discriminates photon numbers of up to 25. A detailed characterization of the PNR detection system is reported in [17]. For the PNR experiments, it was additionally necessary to reduce the excitation repetition rate to 1 MHz by pulse picking the excitation laser (80 MHz repetition rate), to accommodate for the thermal recovery time (∼1 μs) of the TES chips. To achieve the required suppression between picked pulses, we applied a two-stage pulse-picking scheme as follows. First, an acousto-optical modulator (AOM) is synchronized to the pulsed Ti:Sa laser to diffract every 80th pulse into 1st order. The AOM shows an overall extinction ratio of >1/2000, yet the subsequent pulse of the picked one is only suppressed by a factor of ∼50, due to the finite decay time constant of the AOM. For this reason, a fiber-coupled electro-optical modulator is added as second pulse-picking stage to prevent re-excitation of the quantum emitter. Carefully matching the time windows of the two modulators results in an overall extinction ratio of >1/10000 and a suppression of the subsequent pulse of >200, which is sufficient for the experiments in this work. The resulting laser pulses are then coupled to a fiber-based asymmetric Mach–Zehnder interferometer forming the final pulse sequence, launching a double-pulse (4 ns pulse-separation) every 1 μs to the sample.

To classify the experiments exploiting the photon-number resolution of our TES-based detector system presented in section 3.2, we performed measurements with click detectors for reference. Figure 2(a) depicts the corresponding measurement data of the second-order photon-autocorrelation ${{g}^{(2)}}_{{\rm{HBT}}}(\tau ).$ Almost negligible coincidences at τ = 0 proof pronounced single-photon emission. A quantitative analysis of the experimental data reveals ${{g}^{\left(2\right)}}_{{\rm{HBT}},{\rm{click}}}\left(0\right)$ = 0.035 ± 0.003. Next, we address the photon-indistinguishability in HOM-type experiments using click detectors. Figure 2(b) displays the obtained coincidence histograms of two-photon detection events at the output ports of the HOM setup. In case of co-polarized photons (solid blue curve), quantum-mechanical two-photon interference manifests in a strongly reduced number of coincidences at τ = 0, compared to the measurement in cross-polarized configuration (dashed gray curve). Quantitatively evaluating the experimental data for co-polarized measurement configuration according to [16], we extract a two-photon interference visibility of V_HOM,click = (90 ± 5)%.

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For the quantum metrology with photon-number resolution, we use the PNR detector system described in section 2.3 with the experimental configuration displayed in figure 1(a). This experimental configuration allows us to measure the number of photons at the two output ports of the HOM setup, providing direct access to the result of the two-photon interference experiment. Due to the limited temporal resolution (∼1 μs) of the PNR detectors, and in contrast to the corresponding experiment with click detectors, the different pathway combinations of two photons resulting from one excitation cycle cannot be distinguished. Therefore we evaluate the correlated photon-number-distribution of either measuring one or two photons at each channel of the PNR detector. From this measurement we deduce the conditional probability P(TES₁, TES₂) to detect n photons in TES detectors 1 and 2 in co-polarized and cross-polarized configuration. Out of four possible pathway combinations at the first beam splitter, only one allows the two photons to arrive simultaneously and interfere at the second beam splitter. This is comparable to the central peak of the cluster at zero-delay in figure 2(b). For cross-polarized configuration the probabilities to measure one photon in each output port P(1/1) or to measure two photons at either output port P(2/0, 0/2) are equally distributed, as the photons are distinguishable, leaving the Mach–Zehnder interferometer statistically uncorrelated. For co-polarized configuration, however, the probability P(2/0, 0/2) is enhanced while the P(1/1) is reduced. The resulting contrast $\tfrac{P\left(2/0,0/2\right)}{P(1/1)}$ = $\tfrac{{\eta }_{1}^{2}+{\eta }_{2}^{2}}{2{\eta }_{1}{\eta }_{2}}$ × $\tfrac{(3/4+u)}{(5/4-u)}$ reveals the degree of photon-indistinguishability ${V}_{{\rm{HOM}}}=4u-1,$ where a finite detection efficiency mismatch η₁/η₂ between the two detection channels is taken into account. Note, that in cases of reasonably high detection efficiencies (>30%), it is sufficient to consider the easily accessible efficiency mismatch η₁/η₂, although the visibility depends on the absolute values of η₁ and η₂.

In our PNR experiment, we record the time traces of the response of both TES/SQUID detector units, which are coupled to the HOM experiment via single-mode fibers. Examples of time traces recorded in the co-polarized (HH) configuration are displayed in figure 3(a). The conditional probabilities described above are evaluated, by post-selecting pairs of traces at TES 1 and TES 2 from the raw data corresponding to events (1/1), (2/0) and (0/2). For this purpose we analyse the pulse area of measured time traces. It is worth mentioning, that other methods for evaluating the time traces are possible, such as the principal component analysis or simply the pulse height [17]. In this case, the pulse-area analysis in combination with moderate constraints to the pulse shape (i.e. slope of rising/falling edge) enabled the optimum performance in terms of energy resolution and efficient filtering of erroneous events due to 'pile-ups' of consecutive photons or 'cosmics'. Figure 3(b) depicts count histograms of the pulse area extracted from time traces of both TES units in HH configuration. The occurrence of zero ('0'), one ('1') and two-photon ('2') detection events is clearly discriminated, demonstrating the high energy resolution of our TESs. Note, that the overlap between the peaks of the photon number states is only visible in logarithmic scaling, which was chosen in order to show the two-photon events more clearly. The corresponding conditional propabilities are dipicted in figure 3(c) for HH- and HV-configuration, respectively, where error bars correspond to one standard deviation. For HH setting, the probability P(2/0, 0/2) to detect a photon pair at one of the two HOM output ports is significantly increased, while the probability P(1/1) to detect single photons at both TESs is reduced with respect to the probabilistic case (indicated as dashed line). This observation confirms the presence of quantum mechanical two-photon interference of indistinguishable photons at the HOM beamsplitter. For HV setting, i.e. distinguishable photons at the HOM beamsplitter, the probabilities P(1/1) and P(2/0, 0/2) are the same within the measurement error. Taking into account the observed detection efficiency mismatch of the two detection channels η₁/η₂ = 0.68 and using the formula discussed above, we determine a photon-indistinguishability of V_HOM,PNR = (90 ± 7)%. In addition, we also perform and analyse PNR measurements in HBT-configuration to access the suppression of two-photon emission events of our SPS. Using the approximation ${g}_{{\rm{HBT}}}^{\left(2\right)}\left(0\right)$ ≈ 2P₂/${{P}_{1}}^{2}$ [22], where P_n is the probability to detect n photons, we gain the antibunching value ${g}_{{\rm{HBT}},{\rm{PNR}}}^{\left(2\right)}\left(0\right)$ = 0.08 ± 0.02 of our PNR experiment.

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